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MATHEMATICAL

AND

PHYSICAL PAPEES.

Honlron: C. J. CLAY and sons,

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MABIA LANE.

ffilassoto: 60, WELLINGTON STREET.

5Lcip?ig: F. A. BROCKHAUS.

j^eta gorii: THE MACMILLAN COMPANY.

36omtas atiti Calcutta; MACMILLAN AND CO., Ltd.

[All Eights reserved.]

MATHEMATICAL

AND

PHYSICAL PAPEES

BY THE LATE

Sir GEORGE GABRIEL STOKES, Bart,
ScD., LL.D., B.C.L., Past Prbs. E.S.,

KT PRUSSIAN ORDER POUJi LE MERITE^ FOR. ASSOC. INSTITUTE OF FRAUCE, ETC,

MASTER OF PEMBROKE COLLEOE AND LUOABIAN PROFESSOR OF MATHEMATICS

IN THE UNIVERSITY OF CAMBRIDGE.

Reprinted from the Original Journals and Transactwna,
with brief Historioal Notes and References.

VOL. IV.

CAMBRIDGE:

AT THE UNIVERSITY PRESS.

1904

FEINTED BY J. AND 0. J!'. CLAY,
AT THE UNIVEESITY PRESS.

PEEFACE.

TX the preface to the third volume Sir George Stokes expressed
his intention, should life and health last, to complete the
republication of his Scientific Papers Anthout delay. These
conditions were not destined to be fulfilled; and the task of
completing this memorial of his genius has fallen to other hands.

This fourth volume includes the Papers that were published
between 1853 and 1876. In accordance with the plan folloAved in
the previous volumes, only memoirs and notes invoh-ing distinct
additions to scientific knowledge have been included; this re-
striction has however had little effect except in the omission of
some addresses, extracts from which will be collected together in
another place. The texts of the papers have been reproduced
as they originally appeared, with the exception that obvious
misprints have been corrected. A few historical and elucidatory
footnotes, those within square brackets, have been added by the
Editor.

Advice has been received, in the selection and treatment of
the material, from Lord Kelvin and Lord Eajleigh, who have seen
most of the proof sheets. Acknowledgment is also due to Prof
Liveing who has very kindly examined the papers dealing with
technical chemical and spectroscopic subjects, and to Mr F. G.
Hopkins, who has gone through the work on the reactions
of haemoglobin.

VI PREFACE.

The remaining papers, together with a biographical notice
which is being prepared for the Koyal Society by Lord Rayleigh,
will appear in a fifth volume. It is hoped that it will be possible
to arrange a selection from Sir George Stokes' scientific corre-
spondence, for publication either there or in a separate form:
some extracts from it, relating to the contents of the present
volume, are here printed in an Appendix.

The portrait prefixed to this volume is taken from an oil
painting made in 1874 by Mr Lowes Dickinson, which belongs to
Pembroke College.

J. LARMOR.

St John's College, Cambridge,
January, 1904.

CONTENTS.

PAGE

1853. On the Change of Eefrangibility of Light.— II 1

1852. On the Optical Properties of a recently discovered Salt of Quinine 18

1853. On the Change of Befrangibility of Light and the exhibition thereby of

the Chemical Rays 22

1853. On the Cause of the Occurrence of Abnormal Figures in Photographic

Impressions of Polarized Eings 30

1853. On the Metallic Reflexion exhibited by certain Non-Metallic Substances 38

1854. Extracts from Letter to Dr W. Haidinger: on the Direction of the

Vibrations in Polarized Light : on Shadow Patterns and the
Chromatic Aberration of the Eye : on Haidiuger's Brushes . . 50

1854. On the Theory of the Electric Telegraph. By Prof. W. Thomson.

(Extract) ... 61

1855. On the Achromatism of a Double Object-glass ..... 63

1856. Remarks on Professor Challis's paper, entitled "A Theory of the

Composition of Colours, etc." ........ 65

1856. Supplement to the "Account of Pendulum Experiments undertaken

in the Harton Colliery...". By G. B. Airy, Esq., Astronomer Royal 70

1857. On the Polarization of Diffracted Light 74.

1857. On the Discontinuity of Arbitrary Constants which appear in Divergent

Developments ........... 77

1857. On the Effect of Wind on the Intensity of Sound . . . .110

1859. On the Existence of a Second Crystallizable Fluorescent Substance

(Paviin) in the Bark of the Horse-Chestnut ..... 112

1859. On the bearing of the Phenomena of Diffraction on the Direction of

the Vibrations of Polarized Light, with Remarks on the Paper of

Professor P. Eisenlohr 117

1860. Note on Paviin .... 119

1860. On the Colouring Matters of Madder. By Dr E. Schunck. (Extract) 122

1860. Extracts relating to the Early History of Spectrum Analysis . . 127

1861. Note on Internal Radiation 137

1862. On the Intensity of the Light reflected from or transmitted through

a Pile of Plates 145

1862. Report on Double Refraction 157

1862. On the Long Spectrum of Electric Light . . . . . .208

VIU CONTENTS.

PAGE

1863. On the Change of Form assumed by Wrought Iron and other Metals

when heated and then cooled by partial Immersion in Water. By
Lieut.-Col. H. Clark, E.A., F.E.S. Note appended by Prof. Stokes . 234

1864. On the supposed Identity of Biliverdin with Chlorophyll, with remarks

on the Constitution of Chlorophyll 236

1864. On the Discrimination of Organic Bodies by their Optical Properties . 238
1864. On the Application of the Optical Properties of Bodies to the Detection

and Discrimination of Organic Substances 249

1864. On the Beduotion and Oxidation of the Colouring Matter of the Blood 264

1867. On a Property of Curves which fulfil the condition i;^ + ^ = 0- By

W. J. Macquorn Eankine. (Supplement) 276

1867. On the Internal Distribution of Matter which shall produce a given

Potential at the surface of a Gravitating Mass 277

1868. Supplement to a paper on the Discontinuity of Arbitrary Constanta

which appear in Divergent Developments ..... 283
1868. On the Communication of Vibration from a Vibrating Body to

a surrounding Gas 299

1868. Account of Observations of the Total Eclipse of the Sun. ...By J. Pope

Henuessy. Note added by Prof. Stokes 325

1869. On a certain Reaction of Quinine 327

1872. Explanation of a Dynamical Paradox 334

1872. On the Law of Extraordinary Befraction in Iceland Spar . . 336

1873. Sur Temploi du prisme dans la verification de la loi de la double

refraction 337

1871. Notice of the Eesearches of the late Eev. W. Vernon Harcourt on the
Conditions of Transparency in Glass and the Connexion between the
Chemical Constitution and Optical Properties of Different Glasses . 339

1873. On the Principles of the Chemical Correction of Object-glasses . 344

1874. On the Improvement of the Spectroscope. By Thomas Grubb, P.E.S.

Note appended by Prof. Stokes 335

1874. On the Construction of •■ perfectly Achromatic Telescope . . 356

1875. On the Optical Properties of a Titano-Silicio Glass .... 358

1876. On a Phenomenon of Metallic Eefleotion 361

1876. Preliminary Note on the Compound Nature of the Line-Spectra of

Elementary Bodies. By J. N. Lockyer, F.E.S. (Extract) . . 365

Appendix (Correspondence of Prof. G. G. Stokes and Prof. W. Thomson
on the nature and possibilities of Spectrum Analysis) . . . 367

Index 377

MATHEMATICAL AND PHYSICAL PAPERS.

On the Change of Refrangibility of Light. — No. II.

[From the PhUosophical Transactions for 1853, pp. 385 — 396.
Received June 16, read June 26, 1853.]

The chief object of the present communication is to describe
a mode of observation, which occurred to me after the publication
of my former paper, which is so convenient, and at the same time
so delicate, as to supersede for many purposes methods requiring
the use of sun-light. On account of the easiness of the new
method, the cheapness of the small quantity of apparatus required,
and above all, on account of its rendering the observer independent
of the state of the weather, it might be immediately employed
by chemists in discriminating between different substances.

I have taken the present opportunity of mentioning some
other matters connected with the subject of these researches.
The articles are numbered in continuation of those of the former
paper *.

Method of observing hy the use of Absorbing Media.

241. Conceive that we had the power of producing at will
media which should be perfectly opaque with regard to rays
belonging to any desired regions of the spectrum, from the
extreme red to the most refrangible invisible rays, and perfectly
transparent with regard to the remainder. Imagine two such
media prepared, of which the second was opaque with regard
to those rays of the visible spectrum with regard to which the
first was transparent, and vice versa. It is clear that if both
media were held in front of the eye no light would be perceived.
The same would still be the case if the first medium were removed
* [Ante, Vol. iii. p. 267.]
S. IV. 1

2 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

from the eye, and placed so as to intercept all the rays which fell
on certain objects, which were then viewed through the second,
provided the objects did nothing more than reflect, refract, scatter,
or absorb the incident rays. But if any of the objects had the
property of emitting rays of one refrangibility under the influence
of rays of another, it might happen that some of the rays so
emitted were capable of passing through the second medium, in
which case the object would appear luminous in a dark field.

242. Let us consider now how the media must be arranged
so as to bring out to the utmost the sensibility of a given substance.
To take a particular instance, suppose the substance to be glass
coloured by uranium. In this case the sensibility of the' medium
begins, with almost absolute abruptness, near the fixed line b of
Fraunhofer, and continues from thence onwards. The dispersed
light has the same, or at least almost rigorously the same, com-
position throughout, and consists exclusively of rays less refrangible
than b. Consequently, we should have to prepare a first medium
which was opaque with regard to the visible rays less refrangible
than b, and transparent with respect to the rays, whether visible
or invisible, more refrangible, and a second mediuiii complementary
to the former in the manner described in the preceding article.
If the pair of media were still strictly complementary in this
manner, but the point of the spectrum at which the transparency
of the first medium began and that of the second ended were
situated at some distance from b, the sensibility of the glass would
be exhibited as before, only the maximum effect would not be
produced, on account of the absorption of a portion either of the
active or of the dispersed rays, according as the point in question
was situated above or below b.

Now, although the commencement of the sensibility of canary
glass is unusually abrupt, it generally happens that the sensibility
of a medium, or at least the main part of it, comes on with great
rapidity, and lasts throughout the rest of the spectrum, though
frequently it is most considerable in a region extending not very
greatly beyond the point where it commenced. In those cases in
which the dispersion of different tints commenced at two or three
different places in the spectrum, I have almost always had evidence
of the independent presence of different sensitive principles, to
which the observed effects were respectively due.

ON THE CHANGE OF REFRANGIBILITY OP LIGHT. 3

Hence, if we could prepare absorbing media at pleasure, we
should get ready for general use in these observations a few pairs
of media complementary in the particular manner already described,
but having the points of the spectrum at which the transparency
of the first medium commenced and that of the second ended
different in different pairs, situated say in the yellow for one pair,
in the blue for another, in the extreme violet for a third.

243. It . is not of course possible to prepare media in this
manner at pleasure, and all we can do is to select from among
those which occur in nature. Nevertheless it is useful, as a guide
ia the selection, to consider what constitutes the ideal perfection
of absorbing media for this particular purpose. But before pro-
ceeding to mention the media which I have found convenient,
I will describe the arrangement which I have adopted for ad-
mitting the light.

A hole was cut in the window-shutter of a darkened room,
and through this the light of the clouds and external objects
entered in all directions. The diameter of the hole was four
inches, and it might perhaps have been still larger with advantage.
A small shelf, blackened on the top, which could be screwed on to
the shutter immediately underneath the hole, served to support
the objects to be examined, as well as the first absorbing medium.
This, with a few coloured glasses, forms all the apparatus which
it is absolutely necessary to employ, though for the sake of some
experiments it is well to be provided also with a small tablet
of white porcelain, and an ordinary prism, and likewise with one
or two vessels for holding fluids.

244. In the observation, the first medium is placed resting
on the shelf so as to cover the hole; the object is placed on the
shelf immediately in front of the hole ; the second medium is held
anywhere between the eyes and the object. As it is not possible
to obtain media which are strictly complementary, it will happen
that a certain quantity of light is capable of passing through both
media. This might no doubt be greatly reduced by increasing the
absorbing power of the media, but it is by no means advisable
to do so to any great extent, because it is important that the
second medium should transmit as many as possible of the rays
which are of such refrangibilities as to be stopped by the first.

1—2

4 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Accordingly, it might sometimes be doubtful whether the illumina-
tion perceived on the object were due merely to scattered light
which had passed through both media, or to really " degraded "
light *. To remove all doubt, it is generally sufficient to transfer
the second medium from before the eyes to the front of the hole.
The light merely scattered by the object will necessarily be the
same as before, if the room be free from stray light ; and even if
there be a little stray light, the illumination, so far as it is
concerned, will be increased instead of diminished ; whereas if the
illumination previously observed were due to fluorescence, and the
media were properly chosen, the object which before was luminous
will now be comparatively dark.

Sometimes, in the case' of substances which have only a low
degree of sensibility, it is better to leave the second medium in
front of the eyes, and use a third medium, which is held alternately
in the path of the rays incident on the object and between the

* This term, which was suggested to me by my friend Prof. Thomson, appears
to me highly significant. The expression degradation of light might be substituted
with advantage for true internal dispersion to designate the general phenomenon ;
but it is perhaps a little too wide in its signification, and might be taken to include
phosphorescence (if indeed in this case the refrangibility be really always lowered),
as well as the emission of non-luminous radiant heat by a body which had been
exposed to the red rays of the spectrum. As to the term internal dispersion,
though I employed it, following Sir David Brewster, I confess I never liked it. It
seems especially awkward when applied to a washed paper or dyed cloth ; it was
adopted at a time when the phenomenon was confounded with opalescence ; and, so
far as it implies theoretical notions at all, it seems rather to point to a theory now
no longer tenable : 1 allude to the theory of suspended particles. Indeed, this
theory, as it seems to me, ceased to be tenable as soon as Sir John Hersohel had
discovered the peculiar analysis Of light connected with epipolic dispersion, and
Sir David Brewster had connected the phenomenon with internal dispersion, so far
at least as the common appearance of a continuous and coloured dispersed beam
formed a connexion. The expression dependent emission is awkward, but would be
significant, because the light is emitted in the manner of self-luminous bodies, but
only in dependence upon the active rays, and so long as the body is under their
influence. In this respect the phenomenon differs notably from phosphorescence.
It is quite conceivable that a continuous transition may hereafter be traced by
experiment from the one phenomenon to the other, but no such transition has yet
been traced, nor is it by any means certain that the phenomena are not radically
distinct. On this account it would, I conceive, be highly objectionable to call true
internal dispersion phosphorescence. In my former paper I suggested the term
fluorescence, to denote the general appearance of a solution of sulphate of quinine
and similar media. I have been encouraged to give this expression a wider signi-
fication, and henceforth, instead of true internal dispersion, I intend to use the
term fluorescence, which is a single word not implying the adoption of any theory.

ON THE CHANGE OF REFBANGIBILITY OF LIGHT. 5

object and the eyes. Such a medium, though not at all necessary,
may be used also in the case of highly sensitive substances, for the
sake of varpng the experiment and rendering the result more
striking.

As it will be convenient to have names for the media fulfilling
these different offices, I will call the first medium, or that with
which the whole is covered, the principal absorbent, the second
medium the complementary absorbent, and the third medium,
when such is employed, the transfer medium. For the transfer
medium we may choose a medium of the same nature as the
complementary absorbent, but paler. This is perhaps the best
kind to employ in the methodical examination of various sub-
stances ; but if the object of the observer be merely to illustrate
the phenomena of the change of refrangibility of light, he may vary
the experiments by using other media.

245. I have hitherto spoken only of the increase of illumina-
tion due to the sensibility of the substance under examination.
But independently of illumination, the colour of the emitted
light affords in most cases a ready means of detecting fluorescence.
Thus, suppose the principal absorbent to transmit no visible rays
but deep blue and violet, and the substance examined to appear,
when viewed through the complementary absorbent, of a bright
orange colour. Since no combination of the rays transmitted
by the principal absorbent can make an orange, we may instantly
conclude that the substance is sensitive. However, I do not
consider it safe, at least for a beginner, to trust very much to
absolute colour, for few who have not been used to optical experi-
ments can be aware to what an extent the eye under certain
circumstances is liable to be deceived. The relative colour of two
objects seen at the same time under similar circumstances may
usually be judged of safely enough ; that is, of two such colours
it may be possible to say with certainty that one inclines more
to blue or to red than the other. Of course in many cases the
change of colour is so great that there can be no mistake; still
I think it a safe rule for a person employing these modes o
observation without having been previously used to optical ex
periraents, to require some other proof of a change of refrangibility
than merely a change of colour. Experience will soon show what
appearances may safely be relied on.

6 ON THE CHANGE OF EEPRANGIBILITY OF LIGHT.

246. If it be desired to view the object isolated as much
as possible, it may be placed directly on the shelf, or better
still, on black velvet. But it is generally preferable to have
for comparison a standard object which reflects freely the visible
rays, of whatever kind, incident upon it, and does not possess
any sensible degree of fluorescence. It is in this way that the
white porcelain tablet is useful ; and in observing, I generally
place the tablet on the shelf and the object on it. A white plate
would answer, but a tablet is better, on account both of its shape
and of the comparative dullness of its surface. It is true that the
tablet used exhibited a very sensible 'amount of fluorescence when
examined in a linear spectrum formed by a quartz train ; still the
effect was so small, and so much of it was due to those highly
refrangible rays which are stopped by glass, that for practical
purposes the tablet may be regarded as insensible. However,
an observer is not obliged either to assume that all tablets are
alike, or to apply to the particular tablet which he proposes to use,
methods of observation requiring the use of a,pparatus which he is
not supposed to possess. The methods of observation described
in the present paper are complete in themselves; the observer
has it in his power to test for himself the tablet he proposes
to employ ; and he is bound to do so before taking it for a standard
of comparison. It may easily be tested by means of a prism, as
will be explained presently.

247. The following are the combinations of media which
I have chiefly employed: —

First Combination. — In this combination the principal ab-
sorbent is a glass coloured deep violet by manganese with a little
cobalt, or else a glass coloured deep purple by manganese alone,
combined with a rather pale blue glass coloured by cobalt, or with
a deeper blue glass in case the day be bright. It is very easy
to tell by means of a prism, with candle-light, whether a purple
glass contains any sensible quantity of cobalt, on account of the
very peculiar mode of absorption which is characteristic of this
metal. In the examination of substances by this combination no
complementary absorbent is usually required ; but if it be wished
to employ one, a pale yellow glass, of the kind mentioned in
connexion with the next combination, may be used.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 7

Second Combination.— In this case the principal absorbent
is a solution of the ammoniaco-sulphate of copper, employed in
such thickness as to give a deep blue. In my experiments the
fluid was contained in a cell with parallel sides of glass, which was
closed at the top for greater convenience ; but a very broad flat
bottle would answer as well, because in the case of the principal
absorbent the regularity of refraction of rays across it is of
no consequence. Such a bottle, however, would have to be ordered
expressly. The complementary absorbent in this combination is
a yellow glass coloured by silver, and slightly overburnt. These
glasses, as commonly prepared, are opaque with regard to most
of the violet, but become transparent again with regard to the
invisible rays beyond ; and, in the case of a pale glass, the com-
mencing transparency in the extreme violet may even be perceived
by means of light received directly into the eye. I have got
a glass of a pretty deep orange-yellow colour, which is more
transparent than common window-glass with regard to rays of
such high refrangibility as to be situated near the end of the
region of the solar spectrum which it requires a quartz train
to show. But when too much heat is used in the preparation,
the glass acquires, on the interior of the coloured face, a delicate
blue appearance, having a good deal of the aspect of a solution
of sulphate of quinine, though it has in reality nothing to do with
fluorescence ; and in this state the glass is nearly opaque with
regard to the invisible rays of the solar spectrum beyond the
violet, though it still transmits a few among those which are
nearly the most refrangible. Of course, if the complementary
absorbent were always left in its position between the eyes and
the object, its transparency or opacity with regard to invisible
rays would be a matter of indifference ; but as it is desirable that
its transference from that position to the front of the hole should
produce as much difference as possible, it is important that it
should be opaque, or nearly so, with regard to the ultra-violet rays
transmitted by the principal absorbent. Hence one of these
slightly over-burnt glasses should be selected for the present
purpose, and such are very commonly met with.

Third Combination. — In this case everything is the same
as in the preceding combination, except that the fluid is replaced
by a glass of a pretty deep blue, coloured by cobalt.

8 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Fourth Combination. — In this the principal absorbent is
a solution of nitrate of copper, and the complementary absorbent
a light red or deep orange glass.

248. In the first combination the darkness is tolerably
complete without the use of any complementary absorbent,
since no visible rays are transmitted except violet and some
extreme red. The latter are no inconvenience, but rather help
to set off the tint of the light due to fluorescence. This is,
I think, the best combination to employ when the fluorescent
light is blue, or at least deep blue ; because in that case much
of the light is lost by absorption in the yellow glass employed
in the second combination. It has the advantage, too, of allowing
the fluorescent light to enter the eye without being modified
by absorption. Nevertheless no correct estimate can be formed
of the absolute colour of the fluorescent light without making
very great allowance for the effects of contrast, especially when
the body, instead of being isolated as far as possible, is placed
on the porcelain tablet.

The second combination is on the whole the most powerful.
The media in this case make a very fair approach towards the
ideal perfection explained in Art. 242. The darkness is so far
complete, or else may easily be made so by increasing a little
the strength of either absorbent, that if the tablet be written
on with ink, and placed on the shelf between the media, the
writing cannot be read. It forms a striking experiment, after
having treated the tablet in this manner, to introduce between
the media a piece of canary glass or a similar medium. The glass
is not only luminous itself, but it emits so much light as to
illuminate the whole tablet, so that the writing is instantly visible.
In those cases in which the fluorescent light is yellow, orange,
or red, it is shown a good deal more strongly by this combination
than by the preceding.

The third combination is applicable to the same cases as
the second. The blue glass answers extremely' well, but is not
quite so good as the blue fluid. The darkness is less complete,
on account of the red and yellow transmitted by the glass. Never-
theless this combination is sometimes useful in observing with
a prism, and at any rate it may very well be employed by a

ox THE CHAXGE OF EEFEAXGIBILITT OF LIGHT. 9

person who does not happen to have a vessel of the proper shape
for holding the fluid.

In the second and third combinations the point of the
spectrum at which the transparency of the principal absorbent
begins, and that of the complementary absorbent ends, or rather
the point which most nearly possesses this character, is situated
in the blue. Thinking that the fluorescence of those substances
which emit light of low refrangibiKty might be better brought
out if this point was situated lower down in the spectrum, I tried
the fourth combination. In this case the media have very fairly
the required complementarj- character ; the darkness is pretty
complete, and the fluorescence of scarlet cloth and similar
substances is very well exhibited. However, the eflect in these
cases is shown so well by the second combination, that, except
it be for the sake of varying the experiment, I do not think
it worth while to employ the fourth combination, more especiallv
as it has the disadvantage of leaving the observer in doubt
whether the red or orange light perceived constitutes the whole
of the fluorescent light, or only that part of it which alone has
been able to get through the complementary absorbent.

249. The mode of observation may be altered in various ways
which afford pleasing illustrations of the theory, though in the
regular examination of a set of substances it is best to proceed in
a more methodical manner. Thus, if nothing but a violet or blue
glass or a blue fluid be used as a principal absorbent, and the
substances under examination be highly sensitive, their appearance
■ftill be remarkably changed if the coloured medium be transferred
from before the hole to before the eyes. Again, if the complemen-
tar\- absorbent be made to exchange places with the principal
absorbent, the result will be similar, although the very same media
are merely interposed in different parts of the compound path
of the Hght from the clouds to the eye. If a transfer medium be
employed, and it be, as has hitherto been supposed, of the same
general nature as the complementary absorbent, it will not produce
much effect when it is interposed between the object and the eyes,
but when it is placed in the path of the rays incident on the
object, the fluorescent light will be nearly if not entireh" cut oft'.
If, however, we take for a transfer medium a glass or fltiid having
the same general character as the principal absorbent, the effect

10 ON THE CHANGE OF EEFEANGIBILITY OF LIGHT.

will be just the reverse. This is strikingly shown in the case
of a substance, which, like scarlet cloth, emits a red fluorescent
light, by taking for a transfer medium a solution of nitrate of
copper, and in the case of turmeric paper or yellow uranite, by
taking the same solution, or else a blue glass. In the case of the
two substances last mentioned, if we take for a transfer medium
a red solution of mineral chameleon, diluted so as to be merely
pink, the intensity of the light emitted will, under certain
conditions, be not much different in the two positions of the
medium, because a portion of the active rays in one position
and a portion of the degraded rays in the other will be absorbed ;
but the colour of the portion of the emitted light which reaches
the eye will be altogether different in the two positions of the
transfer medium.

Mode of observing by means of a Prism.

250. In this method no absorbing medium is required except
the principal absorbent. The white tablet being laid on the shelf,
a slit is first held in such a position as to be seen projected
against the sky, and the light thus coming directly into the eye,
after having passed through the principal absorbent, is analysed
by a prism held in the other hand. The slit is now held so that
the tablet is seen through it, and the light coming from the
tablet is analysed. It will be found that the spectrum seen
in the first instance is faithfully reproduced, being merely less
luminous, as must necessarily happen. At least, this was the case
in those tablets which I have examined; and in this way each
observer ought to test for himself the tablet he proposes to
employ. After having been thus tested, the tablet may be used
as a standard of comparison.

Suppose now that it is wished to examine a slip of turmeric
paper, or a riband, or other similar object. The object is laid on
the tablet, and the slit held immediately in front of it, in such
a manner that one part, suppose the central portion, of the slit
is seen projected on the object, and the remainder on the tablet.
The light coming through the slit is then analysed by the prism,
and the fluorescence, if any, of the object is indicated by light
appearing in those regions of the spectrum in which, in the
case of the light scattered by the tablet, there is nothing but
darkness.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

11

Occasionally in these observations a blue glass is preferable
to a solution of the ammoniaco-sulphate of copper, because the
extreme red and the greenish yellow bands transmitted by the
glass, while too faint to interfere with the fluorescent light,
are useful as points of reference.

251. The general appearance of the spectrum in this mode
of observation may be gathered from the accompanying figures,
of which the first represents turmeric paper seen under the blue
glass, and the second represents a mass of crystals of nitrate
of uranium seen under the copper solution. In fig. 1, RR', YY'

Fig. 1.

are the red and yellow bands transmitted by the glass, which
are seen equally in the light scattered by the tablet and that
scattered by the paper. BVB'V is the blue and violet light
transmitted by the glass. Of this a considerable portion,
especially in the more refrangible part, is absorbed by the
turmeric paper, which on the other hand emits a quantity of
red, yellow, and green light, not found among the incident rays.
Fig. 2 sufficiently explains itself In this case the fluorescent

Fig. 2.

light is decomposed by the prism into bright bands, of which
six may be readily made out. No blue or violet light enters
the eye fi-om the part of the slit which is seen projected on

12 ON THE CHANGE OF REFEANGIBILITY OF LIGHT.

the mass of crystals, except where a crystalline face happens
to be situated in such a position as to reflect the light of the
sky into the eye, as represented in the figure. In the case
of a substance so highly sensitive as nitrate of uranium, and
which does not, like a slip of paper, lie flat on the tablet, the
spectrum of the fluorescent light in reality extends, at least on
the side next the window, though with less intensity, to some
distance beyond the part of the slit which corresponds to the
object, because the tablet is lighted up by the rays emitted by
the object; but this is not represented in the figure.

252. The mode of using the prism just explained is that
by which the phenomenon of the change of refrangibility is most
strikingly illustrated ; but in the actual examination of substances
the chief use of the prism is to determine, in the case of substances
which are sufficiently sensitive to admit with advantage of such
a mode of observation, the composition of the fluorescent light.
For this purpose it is often better to isolate the object by placing
it on black velvet. This is especially the case with very minute
crystals, or other objects, which are best placed on black velvet,
and viewed through the prism as a whole, no slit being required.

Examples of the application of the preceding methods
of observation.

253. The peculiar properties of paper washed with tincture
of turmeric or stramonium seeds, of yellow uranite, and other
highly sensitive substances, come out in a remarkable manner
under the modes of examination described in this paper. I need
not say that such is the case with solutions of sulphate of quinine,
or horse-chestnut bark, or other clear and highly sensitive media,
seeing that in this case the appearance due to fluorescence is
obvious to common observation. If a piece of horse-chestnut bark
be put to float in a glass of water close to the hole covered by the
principal absorbent, the appearance of the descending streams
of solution of esculine is very singular and beautiful. My present
object is however rather to illustrate the power of these methods
by their application to substances which stand much lower in the
scale.

By the use of absorbing media alone, as well by a principal
absorbent and a prism, I have been able to detect without

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 13

difficulty the sensibility of white paper on a day of continuous
clouds and rain. Even cotton-wool, which stands very much
lower in the scale, is shown by the use of absorbing media with
ordinary daylight to be sensitive. In the case of such substances
as bone, ivory, white leather, the white part of a quill, which stand
much higher in the scale, the most inexperienced observer could
hardly fail instantly to detect the fluorescence. All plates of
colourless glass which I have examined, and other pieces which
were of such a shape as to admit of being looked into edgeways
to a considerable depth, were found by the second combination
to be sensitive. Crystals of sulphate of quinine, which may be
readily prepared from the disulphate of commerce, show their
fluorescence extremely well by the first combination. These
crystals are much less sensitive than their solution, and the light
which they emit is of a much deeper blue. It must in reality be
of a very deep blue colour ; for it nearly matches the fluorescent
light of fluor-spar, although when the crystals are viewed under
the violet glass the tint in both cases appears comparatively pale,
from contrast with the violet. A solution of nitrate of uranium
on the other hand has only a low degree of sensibility compared
with the crystals of that salt. If a drop of the solution be placed
on the porcelain tablet when the hole is covered with the deep
blue copper solution, it appears comparatively dark, because much
more illumination is lost by the absorption of the indigo and
violet than is gained by the fluorescence of the solution. But
when the tablet is viewed through the complementary absorbent,
the solution is seen to be more luminous than the tablet, and
to emit yellow rays, which are not found in the incident light.

The reactions of quinine mentioned in my former paper
(Arts. 205-208), may very conveniently be observed by means
of drops of the solution placed on the tablet; and in this way
it is possible to work in a perfectly satisfactory manner with
excessively minute quantities of quinine. The statement there
made, that the blue colour was destroyed by hydrochloric acid, etc.,
must be understood only with reference to the mode of observation
there supposed to be adopted, which was suSicient for the object
in view. When the solutions are examined in a pure spectrum
formed by sunlight, or even by the method described in the
present paper, it is seen that the blue colour is not absolutely
destroyed by hydrochloric acid, and is even developed to a slight

14 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

extent on the addition of hydrochloric acid to a previously alkaline
solution. Still there is a broad distinction between the two
classes of solutions, which is all that is required. I have since
extended a good deal these results, and mean to pursue the
subject further. Meanwhile I may be permitted to correct an
error in Art. 205, relative to the effect of hydrocyanic acid, which
was there stated to develope the blue colour. The experiment
was made with the acid of commerce, containing a foreign acid,
to which the effect was probably due.

Comparison of the relative advantages of different modes
of observation.

254. At first sight it might have been supposed that day-
light could never be more than a poor substitute for sunlight
in any observations relating to fluorescence. Such, however, I
consider to be by no means the case. In the first place, when
sunlight is used it is made to enter a room in a definite direction ;
whereas in using absorbing media all the rays are employed
whose directions lie within a solid angle having the object
examined for vertex, and the hole for base. If we leave out
the part of this solid angle which corresponds to trees or houses,
the part which corresponds to sky will still be so large as to make
up in a good measure for the superior brilliancy of the light of
the sun. In the second place, stray light is much more perfectly
excluded than when a beam of sunlight, containing rays of all
kinds, is admitted into a room. When indeed the use of sunlight
is combined with that of absorbing media, it is possible to detect
very minute degrees of sensibility. Still for general purposes
I consider the methods depending upon the use of absorbing
media with ordinary daylight quite comparable with, if not equal
to, those methods involving the use of sunlight which are applic-
able to opaque bodies ; I allude especially to the method of a linear
spectrum. The peculiarities in the composition of the fluorescent
light, when such exist, can be made out about equally well by
both methods.

But when the substance to be investigated is a solution, or
a clear solid of sufiicient size to be examined as such, rhethods
of observation can be put in practice with sunlight which surpass
anything that can be done merely by the use of absorbing media.
In consequence of the absence of stray light, which would other-

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 15

wise dazzle the eye, an amount of concentration of the rays can
be brought to bear on the object, which enables the observer
to detect excessively minute degrees of sensibility. Thus, when
the sun's light is condensed by a rather large lens, and made
to pass through a strong solution of the ammoniaco-sulphate of
copper, the condensed beam of violet and invisible rays serves
to detect fluorescence in almost all fluids. This, however, is no
great advantage; the method is in fact too powerful; and the
observer is left in doubt whether the effect perceived be due to the
fluid deemed to be examined, or to some impurity which it con-
tains in an amount otherwise perhaps inappreciable. The great
advantage which sunlight observations possess in the examination
of substances, which, however, is only applicable to clear media,
is that they enable the observer to make out the distribution
of activity in the incident spectrum. In some cases this con-
stitutes the chief peculiarity in the mode of fluorescence of a
particular substance ; in other cases it enables the observer to
see, as it were, independently of each other, different sensitive
substances existing together in solution. Another advantage
of sunlight, which applies equally to clear and to opaque media,
is that it enables the observer, with the assistance of a quartz
train, to make out fluorescence which does not commence till
that region of the spectrum which it requires a quartz train
to show. But such cases are too rare to render this a point
of much consequence. Of course there are observations such
as those which relate to the fixed lines of the invisible rays,
or to the determination of the absorbing action of a medium
with regard to invisible rays of each degree of refrangibility
in particular, which imperatively require sunlight : I am speaking
at present only with reference to those observations of which
the object is to investigate the mode of fluorescence of a particular
substance.

As to the method of observation in which a prism is combined
with a principal absorbent, its chief use is to determine, in the
case of the more sensitive substances, the composition of the
fluorescent light. It is not generally so convenient as the method
which involves the use of absorbing media alone for determining
which among a group of objects are sensitive, and which not,
especially when the objects are minute.

16 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

255. Although the description of the mode of observing
by means of absorbing media has run to some little length,
the reader must not suppose that the observations are at all
difficult. Of course observations of all kinds become more or less
difficult when they are pushed to the extreme limit of refinement
of which they are susceptible. But in the case of substances
which are at all highly sensitive, and this comprises almost all the
more interesting instances, the observations are extremely easy.
I have spoken of a darkened room, which is certainly the most
convenient when it can be had. But I have no doubt that an
observer who could not procure such might easily arrange for
himself a darkened box, which would answer the purpose. Indeed
the fluorescence of highly sensitive substances, though they be
opaque, may be exhibited by means of absorbing media in broad
daylight.

Platinocyanides.

256. In the Report of the Twentieth Meeting of the British
Association (Edinburgh, 1850, Transactions of the Sections, p. 5),
is a notice by Sir David Brewster of " The Optical Properties
of the Cyanurets of Platinum and Magnesia, and of Barytes and
Platinum," salts which he had received from M. Haidinger of
Vienna. The notice is chiefly devoted to the properties of the
reflected light ; but with respect to the latter of the salts, Sir
David remarks that " it possesses the property of internal dis-
persion, the dispersed light being a brilliant green, while the
transmitted light is yellow." Although the distinction between
true internal dispersion and opalescence was not at the time
understood, there could be little doubt from the nature of the
case that the internal dispersion mentioned by Sir David Brewster
was, in fact, an instance of the former of these phenomena; but
I could not try for want of a specimen of the salt. Some months
ago I received from M. Haidinger a specimen of the first of the salts
mentioned at the beginning of this paragraph, namely M. Quadrat's
cyanide of platinum and magnesium, a salt of great optical interest
on account of the remarkable metallic reflexion which it exhibits.
On examining the salt, I was greatly interested by finding that
it was highly sensitive, the fluorescent light being red. This
induced me to form some of Gmelin's platinocyanide of potassium,
and I found that the blue light which this salt exhibits in certain

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 17

aspects is, in fact, due to fluorescence, a property which the salt
possesses in an eminent degree. Having afterwards received
some of the same salt pure from Professor Gregory, I applied
it to the formation, on a small scale, of the platinocyanides of
calcium, barium, strontium, and two or three others. The three
salts last named, of which the second is that mentioned by Sir
David Brewster, are all eminently sensitive, the fluorescent light
being of different shades of green. It is only in the solid state
that the platinocyanides are sensitive; their solutions look like
mere water. The precipitates which a solution of platinocyanide
of potassium gives with salts of the heavy metals are, in most
cases that I have yet observed, insensible. With a solution of
pemitrate of mercury, however, a bright yellow precipitate is
produced which is exceedingly sensitive, so as to look brighter
than even yellow uranite. The light emitted is yellow. It forms
a very striking experiment to place side by side on- the tablet
with the second combination a drop of a solution of platinocyanide
of potassium, and another of pemitrate of mercury. The drops
look like water on the dark field of view; but when they are
mixed, a precipitate is produced which glows like a self-luminous
body with a yellow light. The precipitate with nitrate of silver
is also sensitive, though not in so high a degree. The platino-
cyanides are extremely interesting ; first, as forming a third case,
or rather class of cases, in which the property of fluorescence
is attached to substances chemically isolated in a satisfactory
manner (though I believe chemists are acquainted with a few
other organic compounds to which the property belongs), the
other two cases being salts of quinine and of peroxide of uranium ;
and secondly, as exhibiting a new and remarkable feature, which
consists in the polarization of the fluorescent light. I content
myself at present with this notice; the salts require a more
extended study.

S. IV.

On the Optical Properties of a recently discovered
Salt of Quinine.

[From the Report of the British Association, Belfast, 1852, pp. 15-16.]

This salt is described by Dr Herapath in the Philosophical
Magazine for March 1852, and is easily formed in the way there
recommended, namely, by dissolving disulphate of quinine in
warm acetic acid, adding a few drops of a solution of iodine in
alcohol, and allowing the liquid to cool, when the salt crystallizes
in thin scales reflecting (while immersed in the fluid) a green
light with a metallic lustre. When taken out of the fluid the
crystals are yellowish-green by reflected light, with a metallic
aspect. The following observations were made with small crystals
formed in this manner ; aiid an oral account of them was given at
a meeting of the Cambridge Philosophical Society, shortly after
the appearance of Dr Herapath's paper.

The crystals possess in an eminent degree the property of
polarizing light, so that Dr Herapath proposed to employ them
instead of tourmalines, for which they would form an admirable
substitute, could they be obtained in sufficient size. They appear
to belong to the prismatic system; at any rate they are sym-
metrical (so far as relates to their optical properties and to the
directions of their lateral faces) with respect to two rectangular
planes perpendicular to the scales. These planes will here be
called respectively the principal plane of the length and the prin-
cipal plaiie of the breadth, the crystals being usually longest in the
direction of the former plane.

When the crystals are viewed by light directly transmitted,
which is either polarized before incidence or analysed after trans-
mission, so as to retain only light polarized in one of the principal
planes, it is found that with respect to light polarized in the

ON THE OPTICAL PROPERTIES OF A SALT OF QUININE. 19

principal plane of the length the crystals are transparent, and
nearly colourless, at least when they are as thia as those which
are usually formed by the method above mentioned. But with
respect to light polarized in the principal plane of the breadth,
the thicker crystals are perfectly black, the thinner ones only
transmitting light, which is of a deep red colour.

When the crystals are examined by light reflected at the
smallest angle with which the observation is practicable, and the
reflected light is analysed, so as to retain, first, light polarized in
the priacipal plane of the length, and secondly, light polarized in
the other principal plane, it is found that in the first case the
crystals have a vitreous lustre, and the reflected light is colourless ;
while in the second case the light is yellowish-green, and the
crystals have a metallic lustre. When the plane of incidence is
the principal plane of the length, and the angle of incidence is
increased from 0° to 90", the part of the reflected pencil which
is polarized in the plane of incidence undergoes no remarkable
change, except perhaps that the lustre becomes somewhat
metallic. When the part which is polarized in a plane perpen-
dicular to the former is examined, it is found that the crystals
have no angle of polarization, the reflected light never vanishing,
but only changing its colour, passing from yellowish-green, which
it was at first, to a deep steel-blue, which colour it assumes at a
considerable angle of- incidence. When the light reflected in the
principal plane of the breadth is examined in a similar manner,
the pencil which is polarized in the plane of incidence undergoes
no remarkable change, continuing to have the appearance of being
reflected from a metal, while the other or colourless pencil vanishes
at a certain angle, and afterwards reappears, so that in this plane
the crystals have a polarizing angle.

If, then, for distinction's sake, we call the two pencils which
the crystals, as belonging to a doubly refracting medium, transmit
independently of each other, ordinary and extraordinary, the
former being that which is transmitted with little loss, we may
say, speaking approximately, that the medium is transparent with
respect to the ordinary ray and opaque with respect to the extra-
ordinary, while, as regards reflexion, the crystals have the proper-
ties of a transparent medium or of a metal, according as the
refracted ray is the ordinary or the extraordinary. If common

2—2

20 ON THE OPTICAL PROPERTIES OF

light merely be used, both refracted pencils are produced, and
the corresponding reflected pencils are viewed together; but by
analysing the reflected light by means of a Nicol's prism, the
reflected pencils may be viewed separately, at least when the
observations are confined to the principal planes. The crystals
are no doubt biaxal, and the pencils here called ordinary and
extraordinary are those which in the language of theory cor-
respond to different sheets of the wave surface. The reflecting
properties of the crystals may be embraced in one view by re-
garding the medium as not only doubly refracting and doubly
absorbing, but doubly metallic. The metallicity, so to speak, of
the medium of course alters continuously with the point of the
wave surface to which the pencil considered belongs, and doubtless
is not mathematically null even for the ordinary ray.

If the reflexion be really of a metallic nature, it ought to
produce a relative change in the phases of vibration of light
polarized in and perpendicularly to the plane of incidence. This
conclusion the author has verified by means of the effect produced
on the rings of calcareous spar. Since the crystals were too small
for individual examination in this experiment, the observation
was made with a mass of scales deposited on a flat black surface,
and arranged at random as regards the azimuth of their principal
planes. The direction of the change is the same as in the case
of a metal, and accordingly the reverse of that which is observed
in total internal reflexion.

In the case of the extraordinary pencil the crystals are least
opaque with respect to red light, and accordingly they are less
metallic with respect to red light than to light of higher refrangi-
bility. This is shown by the green colour of the reflected light
when the crystals are immersed in fluid, so that the reflexion
which they exhibit as a transparent medium is in a good measure
destroyed.

The author has examined the crystals for a change of refrangi-
bility, and found that they do not exhibit it. Safilower-red, which
possesses metallic optical properties, does change the refrangibility
of a portion of the incident light ; but the yellowish-green light
which this substance reflects is really due to its metallicity and
not to the change of refrangibility, for the light emitted from

A RECENTLY DISCOVERED SALT OF QUININE. 21

the latter cause is red, besides which it is totally different in
other respects from regularly reflected light.

In conclusion, the author observed that the general fact of the
reflexion of coloured polarized pencils had been discovered by
Sir David Brewster in the case of chrysammate of potash*, and
in a subsequent communication he had noticed, in the case of
other crystals, the difference of effect depending upon the azimuth
of the plane of incidence f. Accordingly, the object of the present
communication was merely to point out the intimate connexion
which exists (at least in the case of the salt of quinine) between
the coloured reflexion, the double absorption, and the metallic
properties of the medium.

Note added during printing. — When the above communication
was made to the Association, the author was not aware of M. Hai-
dinger's papers on the subject of the coloured reflexion exhibited
by certain crystals. The general phenomenon of the reflexion of
oppositely polarized coloured pencils had in fact been discovered
independently by M. Haidinger and by Sir David Brewster, in the
instances, respectively, of the cyanide of platinum and magnesium,
and of the chrysammate of potash. A brief notice of the optical
properties of the former crystal will be found in Poggendorff's
Annalen, Bd. Lxviii. (1846), S. 302, and further communications
from M. Haidinger on the subject are contained in several of
the subsequent volumes of that periodical. The relation of the
coloured reflexion to the azimuth of the plane of incidence was
noticed by M. Haidinger from the first.

* Eeport of the Meeting of the British Association at Southampton, 1846,
Part n. p. 7.

t Ibid. Edinburgh, 1847, p. 5.

On the Change of Refbangibility op Light and the

EXHIBITION thereby OF THE ChEMICAL RaYS.

[From the Proceedings of the Royal Institution of Great Britain, I, pp. 259-
264; Friday evening lecture, Feh. 18, 1853. Also Pogg. Ann. lxxxix,
1853, pp. 627-8.]

Before proceeding to the more immediate subject of the
Lecture, it was necessary to refer to certain discoveries of Sir
John Herschel and Sir David Brewster, more especially as it was
the discovery by the former of these philosophers of the epipolic
dispersion of light, and of the peculiar analysis of light which
accompanies the phenomenon, that led to the researches respecting
the change of refrangibility.

When a weak acid solution of quinine is prepared, by dissolving,
suppose, one part of the commercial disulphate in 200 parts of
water acidulated with sulphuric acid, a fluid is obtained which
appears colourless and transparent when viewed by transmitted
light, but which exhibits nevertheless in certain aspects a peculiar
sky-blue colour. This colour of course had frequently been noticed;
but it is to Sir John Herschel that we owe the first analysis of
the phenomenon*. He found that the blue light emanates in
all directions from a very thin stratum of fluid adjacent to the
surface (whether it be the free surface or the surface of contact
of the fluid with the containing glass vessel), by which the inci-
dent rays enter the fluid. His experiments clearly show that
what here takes place is not a mere subdivision of light into
a portion which is dispersed and a portion which passes on, but
an actual analysis. For after the rays have once passed through
the stratum from which the blue dispersed light comes, they are
deprived of the power of producing the same effect ; that is, they
do not exhibit any blue stratum when they are incident a second
time on a solution of quinine. To express the modification which
* Philosophical Transactions for 1845, p. 143.

ON THE CHANGE OF EEFRANGIBILITT OF UGHT ETC. 23

the transmitted light had undergone, the further nature of which
did not at the time appear, Sir John Herschel made use of the
term " epipolized."

Sir David Brewster had several years before discovered a
remarkable phenomenon in an alcoholic solution of the green
colouring matter of leaves, or, as it is called by chemists, chloro-
phyll. This fluid when of moderate strength and viewed across
a moderate thickness is of a fine emerald green colour; but Sir
David Brewster found that when a bright pencil of rays, formed
by condensing the sun's light by a lens, was admitted into the
fluid, the path of the rays was marked by a bright beam of a
blood red colour*. This singular phenomenon he has designated
internal dispersion. He supposed it to be due to suspended
particles which reflected a red light, and conceived that it might
be imitated by a fluid holding in suspension an excessively fine
coloured precipitate. A similar phenomenon was observed by
him iu a great many other solutions, and in some soKds : and in
a paper read before the Royal Society of Edinburgh in 1846 he
has entered fully into the subject". In consequence of Sir John
Herschel's papers, which had just appeared, he was led to examine
a solution of sulphate of quinine: and he concluded firom his
observations that the " epipolic " dispersion of light exhibited by
this fluid was only a particular iostance of internal dispersion,
distinguished by the extraoi-dinary rapidity ^vith which the rays
capable of dispersion were dispersed.

The Lecturer stated, that, having had his attention called
some time ago to Sir John Herschel's papers, he had no sooner
repeated some of the experiments than he felt an extreme interest
in the phenomenon. The reality of the epipoKc analysis of Hght
was at once evident from the experiments ; and he felt confident
that certain theoretical views respecting the nature of %ht had
onlv to be followed fearlessly into their legitimate consequences,
in order to explain the real nature of epipolized hght.

The exhibition of a richly coloured beam of light in a perfectly
clear fluid, when the observation is conducted in the manner of
Sir David Brewster, seemed to point to the dispersions exhibited

* Edinburgh Phil. Trans. VoL xh. p. 512.

t Vol. XVI. Part n., and Phil Mag. June, 1S48.

24 ON THE CHANGE OF EEFEANGIBILITY OF LIGHT

by the solutions of quinine and chlorophyll as one and the same
phenomenon. The latter fluid, as has been already stated, dis-
perses light of a blood red colour. When the transmitted light
is subjected to prismatic analysis, there is found a remarkably
intense band of absorption in the red, besides certain other ab-
sorption bands, of less intensity, in other parts of the spectrum.
Nothing at first seemed more likely than that, in consequence of
some action of the ultimate molecules of the medium, the inci-
dent rays belonging to the absorption band in the red, withdrawn,
as they certainly were, from the incident beam, were given out
in all directions, instead of being absorbed in the manner usual
in coloured media. It might be supposed that the incident vibra-
tions of the luminiferous ether generated synchronous vibrations
in the ultimate molecules, and were thereby exhausted, and that
the molecules in turn became centres of disturbance to the ether.
The general analogy between the phenomena exhibited by the
solutions of chlorophyll and of quinine would lead to the ex-
pectation of absorption bands in the light transmitted by the
latter. If these bands were but narrow, the light belonging to
them might not be missed in the transmitted beam, unless it
were specially looked for; and the beam might be thus "epi-
polized," without, to ordinary inspection, being changed in its
properties in any other respect. But on subjecting the light to
prismatic analysis, first with the naked eye, and then with a
magnifying power, no absorption bands were perceived.

A little further reflection showed that even the supposition
of the existence of these bands would not alone account for the
phenomenon. For the rays producing the dispersed light (if we
confine our attention to the thin stratum in which the main part
of the dispersion takes place) are exhausted by the time the
incident light has traversed a stratum the fiftieth of an inch
thick, or thereabouts, whereas the dispersed rays traverse the
fluid with perfect freedom. This indicates a difference of nature
between the blue-producing rays and the blue rays produced.
Now, as the Lecturer stated, he felt very great confidence in
the principle that the nature of light is completely defined by
specifying its refrangibility and its state as to polarization. The
difference of nature, then, indicated by the phenomenon, must
be referred to a difference in one or other of these two respects.

AND THE EXHIBITION THEREBY OF THE CHEMICAL BAYS. 25

At first he took for granted that there could be no change of
refrangibility. The refrangibility of light had hitherto been
regarded as an attribute absolutely invariable*- To suppose that
it had changed would, on the undulatory theory, be equivalent
to supposing that periodic vibrations of one period could give
rise to periodic vibrations of a different period, a supposition
presenting no small mechanical difficulty. But the hypotheses
which he was obliged to form on adopting the other alternative,
namely, that the difference of nature had to do with the state
of polarization, were so artificial as to constitute a theory which
appeared utterly extravagant. He was thus led to contemplate
the possibility of a change of refrangibility. No sooner had he
dwelt in his mind on this supposition, than the mystery respecting
the nature of epipolized light vanished; all the parts of the phe-
nomenon fell naturally into their places. So simple did the whole
explanation become, when once the fundamental hypothesis was
admitted, that he could not help feeling strongly impressed that
it would turn out to be true. Its truth or fallacy was a question
easily to be. decided by experiment; the experiments were per-
formed, and resulted in its complete establishment.

The Lecturer then described what may be regarded as the
fundamental experiment. A beam of sunlight was reflected hori-
zontally through a vertical slit into a darkened room, and a pure
spectrum was formed in the usual manner, namely, by transmitting
the light through a prism at the distance of several feet fi-om the
slit, and then through a lens close to the prism. In the actual
experiment, two or three prisms were used, to produce a greater
angular separation of the colours. Instead of a screen, there was
placed at the focus of the lens a vessel containing a solution of
sulphate of quinine. It was found that the red, orange, etc., in
fact, nearly the whole of the visible rays, passed through the fluid
as if it had been mere water. But on arriving about the middle
of the violet, the path of the rays within the fluid was marked

* It is true that the phenomenon of phosphorescence is in a certain sense an
exception ; but the effect is in this case a work of time, which seems at once to
remove it from all the ordinary phenomena of light, which, as far as sense can
judge, take place instantaneously. It is true that there now appears a close
analogy in many respects between true internal dispersion and phosphorescence.
But while the nature of epipolized light remained yet unexplained, there was
nothing in the former phenomenon to point to the latter.

26 ON THE CHANGE OF REFEANGIBILITY OF LIGHT

by a sky-blue light, which emanated in all directions from the
fluid, as if the medium had been self-luminous. This blue light
continued throughout the region of the violet, and far beyond, in
the region of the invisible rays. The posterior surface of the
luminous portion of the fluid marked the distance to which the
incident rays were able to penetrate into the medium before
they were exhausted. This distance, which at first exceeded
the diameter of the vessel, decreased with great rapidity, so that
in the greater part of the invisible region it amounted to only a
very small fraction of an inch. The fixed lines of the extreme
violet, and of the more refrangible invisible rays, were exhibited
by dark planes interrupting the dispersed light. When a small
portion of the incident spectrum was isolated, by stopping the
rest by a screen, and the corresponding beam of blue dispersed
light was refracted sideways by a prism held to the eye, it was
found to consist of light having various degrees of refrangibility,
with colour corresponding, the more refrangible rays being more
abundant than the less refrangible. The nature of epipolized
light is now evident; it is nothing but light from which the highly
refrangible invisible rays have been withdrawn by transmitting
it through a solution of quinine, and does not differ from light
from which those rays have been withdrawn by any other means.

The fundamental experiment, excepting that part of it which
relates to the analysis of the dispersed light, was then exhibited
by means of the powerful voltaic battery belonging to the Insti-
tution, which was applied to the combustion of metals. The rays
emanating from the voltaic arc were applied to form a pure
spectrum, which was received on a slab of glass coloured by
peroxide of uranium, a medium which possesses properties similar
to those of a solution of sulphate of quinine in a still more
eminent degree.

The difference of nature of the illumination produced by
a change of refrangibility, or " true internal dispersion," from
that due to the mere scattering of light, may be shown in a
very instructive form by placing paper washed with sulphate of
quinine, or a screen of similar properties, so as to receive a long
narrow horizontal spectrum, and refracting this upwards by a
prism held to the eye. Were the luminous band formed on the
paper due merely to the scattering of the incident rays, it ought

AND THE EXHIBITION THEBEBY OF THE CHEMICAL BAYS. 27

of course to be thrown obliquely upwards ; whereas it is actually
decomposed by the prism into two bands, one ascending obliquely,
and consisting of the usual colours of the spectrum in their
natural order, the other running horizontally, and extending far
beyond the more refrangible end of the former. Whatever be
the screen, the horizontal band is always situated below the
oblique, since there appears to be no exception to the law, that
when the refrangibility of light is changed in this manner it is
always lowered.

The general appearance of some highly "sensitive" media in the
invisible rays was then exhibited by means of the flame of sulphur
burning in oxygen, a source of these rays which Dr Faraday,
to whose valuable assistance the Lecturer was much indebted,
had in some preliminary trials found very efficacious. The chief
media used were articles made of glass coloured by uranium, and
solutions of quinine, of horse-chestnut bark, and of the seeds of
the datura stramonium. A tall cylindrical jar filled with water
showed nothing remarkable ; but when a solution of horse-chestnut
bark was poured in, the descending fluid was strongly luminous.
The experiment was varied by means of white paper on which
words had been written with a pretty strong solution of sulphate
of quinine, an alcoholic solution of the seeds of the datura stra-
monium, and a purified aqueous solution of horse-chestnut bark.
By gas-light the letters were invisible ; but by the sulphur light,
especially when it had been transmitted through a blue glass,
which transmits a much larger proportion of the invisible than
of the visible rays, the letters appeared luminous, on a compara-
tively dark ground. A glass vessel containing a thin sheet of a
very weak solution of chromate of potash allowed the letters to
be seen as well, or very nearly as well as before, when it was
interposed between the eye and the paper; but when it was
interposed between the flame and the paper the letters wholly
disappeared, — the medium being opaque with respect to the
rays which caused the letters to be luminous, but transparent
with respect to the rays which they emitted.

It was then remarked what facilities are thus afforded for
the study of the invisible rays. When a pure spectrum is once
formed, it is as easy to determine the mode of absorption of an
absorbing medium with respect to the invisible, as with respect

28 ON THE CHANGE OF EEFRANGIBILITY OF LIGHT

to the visible rays. It is sufficient to interpose the medium in
the path of the incident rays, and to notice the effect. Again,
the effect of various flames and other sources of light on solutions
of quinine, and on similar media, indicates the richness or poverty
of those sources with respect to the highly refrangible invisible
rays. Thus, the flames of alcohol, of hydrogen, etc., of which the
illuminating power is so feeble, were found to be very rich in
invisible rays. This was still more the case with a small electric
spark, while the spark from a Leyden jar was found to abound in
rays of excessively high refrangibility. These highly refrangible
rays were stopped by glass, but passed freely through quartz.
These results, and others leading to the same conclusion, had
induced the Lecturer to order a complete train of quartz. A
considerable portion of this was finished before the end of last
August, and was applied to the examination of the solar spectrum.
A spectrum was then obtained extending beyond the visible
spectrum, that is, beyond the extreme violet, to a distance at
least double that of the formerly known chemical spectrum. This
new region was filled with fixed lines like the regions previously
known.

But a spectrum far surpassing this was obtained with the
powerful electrical apparatus belonging to the Institution. The
voltaic arc from metallic points furnished a spectrum no less than
six or eight times as long as the visible spectrum. This was in
fact the spectrum which had already been exhibited in connexion
with the fundamental experiment. The prisms and lens which
the Lecturer had been employing in forming the spectrum were
actually made of quartz. The spectrum thus obtained was filled
from end to end with bright bands. When a piece of glass was')
interposed in the path of the incident rays, the length of ire
spectrum was reduced to a small fraction of what it had \-'
all the more refi-acted part being cut away. A strong dischas,
of a Leyden jar had been found to give a spectrum at least as
long as the former, but not, like it, consisting of nothing but
isolated bright bands.

The Lecturer then explained the grounds on which he con-
cluded that the end of the solar spectrum on the more refrangible
side had actually been reached, no obstacle existing to the
exhibition of rays still more refrangible if such were present.

AND THE EXHIBITION THEEEBY OF THE CHEMICAL BAYS. 29

He stated also that during the winter, even when the sun shone
clearly, it was not possible to see so far as before. As spring
advanced he found the light continually improving, but still he
was not able to see so far as he had seen at the end of August.
It was plain that the earth's atmosphere was by no means trans-
parent with respect to the most refrangible of the rays belonging
to the solar spectrum.

In conclusion, there was exhibited the effect of the iuvisible
rays coming firom a succession of sparks from the prime conductor
of a large electrifying machine, in illuminating a slab of glass
coloured by uranium.

On the Cause of the Occurrence of Abnormal Figures
IN Photographic Impressions of Polarized Rings.

[From the Philosophical Magazine, vi, Aug. 1853, pp. 107 — 113.
Also Pogg. Ann. xc, 1853, pp. 488—497.]

The object of the following paper is to consider the theory
of some remarkable results obtained by Mr Crookes in applying
photography to the study of certain phenomena of polarization.
An account of these results, taken from the Journal of the
Photographic Society, is published in the last number of the
Philosophical Magazine*.

In the ordinary applications of photography, certain objects
and parts of objects are to be represented which differ from one
another in colour, or in brightness, or in both, according to the
nature of the substances, and the way in which the lights and
shadows fall. In the photograph the objects are represented as
simply light or dark. Inasmuch as the photographic power,
in relation to a given sensitive substance, of a heterogeneous
pencil of rays is not proportional to its illuminating power, the
darkness of the objects in a negative photograph is not propor-
tional to, nor even always in the same order of sequence as, their
brightness as they appear to the eye. Still, the outlines of the
objects and of their parts are faithfully preserved. For although
it is conceivable that two adjacent parts of an object, which the
eye instantly distinguishes by their colour, should reflect rays
of almost exactly equal photographic power in relation to the
particular sensitive substance employed, so as to be absolutely
undistinguishable on the photograph, or on the other hand, that
two parts of an object between which the eye can see absolutely
no distinction should yet come out distinct on the photograph,
the conditions which would have to be satisfied in order that

* Vol. VI. p. 73.

PHOTOGRAPHIC IMPRESSIONS OF POLARIZED RINGS. 31

the forms of a set of objects, suppose coloured patterns, or
a painting of the rings of crystals, should be changed in this
way by the substitution of one set of outlines for another are
so very peculiar, that the chances may be regarded as infinity
to one that no such changes of form will be produced to any
material extent.

But when photography is applied to the representation of
phenomena of interference, such as the rings of crystals seen by
means of polarized light, the case is in some respects materially
different. To take a particular instance, let us suppose the
rings of calcareous spar to be viewed by white light, the planes
of polarization of the polarizer and analyser being crossed, so as
to give the black cross ; and consider the alternations which take
place in going outwards from the centre of the field, suppose in
a direction inclined at angles of 45° to the arms of the cross.
At first there are evident alternations of intensity; but very
soon the eye, which under such circumstances is but a bad judge
of differences of intensity even when the lights to be compared
have the same colour, can no longer perceive the differences of
illumination, but judges entirely by the difference of tint. The
same takes place with nitre, sugar, and other colourless biaxal
crystals. Except in the immediate neighbourhood of the optic
axis or axes, the rings, which owe their existence and their forms
in the first instance to the laws of double refraction and of the
interference of polarized light, are in other respects created and
their forms determined by the condition of maximum contrast
of tint.

Now consider what takes place when an image of such a
system is thrown on a sensitive plate, prepared suppose by
means of bromide of silver. The rays of any one refrangibility
would together form a regular system of rings, which, if these
rays were alone present, and if the refrangibility were comprised
within the limits between which the substance is acted on, would
impress on the plate a system of rings exactly like those seen by
means of the same homogeneous rays, provided they belong to
the visible spectrum. The same would take place for rays of each
refrangibility in particular, and the several elementary systems
of rings thus formed are actually superposed when heterogeneous
light is used. When the photograph is finished, it exhibits

32 ON THE CAUSE OF THE OCCURRENCE OF ABNORMAL FIGURES

certain alternations of light and shade corresponding to alterna-
tions in the total photographic intensity of the rays which had
acted on the plate, without any distinction being preserved be-
tween the action of rays of one refrangibility and that of rays of
another; whereas, when the rings are viewed directly, the eye
catches the differences of tint without noticing the difference of
intensity, except in the neighbourhood of the optic axis or axes.
Of course I am now speaking only of the alternations perceived
in following a line drawn across the rings, not of the dark
brushes, or of the variation of intensity perceived in passing
along a given ring. Hence, when heterogeneous light is used,
the circumstances which determine the rings are so different in
the two cases that it is no wonder that the character of the rings
seen on a photograph should differ in some respects from that of
the rings seen directly.

But not only is a difference of character indicated as likely to
take place ; a more detailed consideration of the actual mode of
superposition will serve to explain some of the leading features
of the abnormal rings as observed by Mr Crookes. Let us take
for example calcareous spar, and suppose the transmitted rays
to be all of the same refrangibility. In this case the intensity
along a given radius vector, drawn from the centre of the cross,
varies as the square of the sine of half the retardation of phase
of the ordinary relatively to the extraordinary pencil (see Airy's
Tract). If i be the angle of incidence, the retardation varies
nearly as sitf i ; and if sin^ i = r, we may take, as representing
the variations of intensity /,

/ = sitf(mr2) = |(l-cos2mr") (1).

In this expression m is a constant depending upon the refrangi-
bility of the rays. In the case of calcareous spar the tints of
the rings follow Newton's scale, and m is very nearly propor-
tional to the reciprocal of the wave-length.

Suppose now that rays of two different degrees of refran-
gibility pass through the crystal together, and that the photo-
graphic intensities of the two kinds are equal. Suppose also
that the aggregate effect which the two systems produce together
on the plate is the sum of the effects which they are capable of

IN PHOTOGRAPHIC IMPRESSIONS OF POLARIZED RINGS. 33

producing separately. .The latter supposition, if not strictly-
true, will no doubt be approximately true if the plate be not too
long exposed. Then, if m' be what the parameter m becomes
for the second system, we may represent the variation of inten-
sity along a given radius vector by

/ = sin'' (mr^) + sin'' (mV^) = 1 — cos (m - m'r') cos (m + mV^) ... (2).

Suppose the refrangibilities of the two systems to be mode-
rately different; then'the difference between the two parameters
m, m' will be small, but not extremely small, compared with
either of them. Hence of the two factors in the expression for
, I the second will fluctuate a good deal more rapidly than the
first, and will be that which mainly determines the radii, etc. of
the rings. If the first factor were constant and equal to 1, its
value when r = 0, the expression (2) would be of precisely the
same form as (1), the parameter being the mean of the two, m, m.
However, the first factor is not constant, but decreases as r
increases, and presently vanishes, and then changes sign. Hence
the rings become less distinct than with homogeneous rays, and
presently there takes place a sort of dislocation amounting to
half an order, that is, the bright rings beyond a certaia point,
or in other words, outside the circle determined by a certain value
of r, correspond, in regard to the series formed by their radii,
with the dark rings inside this circle, and vice versd. At some
distance beyond that at which the dislocation takes place the
rings become very distinct again ; but it is useless to trace further
the variations of the expression for /, because the circumstances
supposed to exist in forming that expression are too remote from
those of actual experiment to allow the interpretation of the
formula to be carried far.

According to the numerical values of m and m', a dark ring might
be converted, by the change of sign of the factor cos (m — m') r^,
iato a bright ring, or a bright ring into a dark ring, or a ring of
either kind might be rendered broader or narrower than it would
regularly have been. The coalescence of the fourth and fifth
bright rings in Mr Crookes's photographs when bromide of
silver was used seems to be merely an effect of this nature.

But in order that a dislocation of the kind above explained
should take place, it is not essential that two kinds only of rays

8. IV. 3

34 ON THE CAUSE OF THE OCCURRENCE OF ABNORMAL FIGURES

should act, nor even that the curve of photographic intensity
should admit of two distinct maxima within the spectrum.
Suppose that rays of all refrangibilities lying within certain
limits pass through the crystal and fall on the plate. For the
sake of obtaining an expression which admits of being worked
out numerically without too much trouble, and yet results from
a hypothesis not very remote from the circumstances of actual
experiment, I will suppose the total photographic power of the
rays whose parameters lie between m £ind'm + dm to be propor-
tional to sin TO dm between the limits m = 27r and m = Stt, and to
vanish beyond those limits. Since m is very nearly proportional
to the reciprocal of the wave-length, and the ratio Sir to 27r or
3 to 2 is nearly that of the wave-lengths of the fixed lines D,
H, this assumption corresponds to the supposition that the rays
less refrangible than D are inefficient, that the action there
commences, then increases according to a certain law, attains a
maximum, decreases, and finally vanishes at H. The action
would really terminate at H if a bath of a solution of sulphate of
quinine of a certain strength were used. On this assumption,
and supposing, as before, that the rays of different refrangibilities
act independently of each other, we have

I = I sin'' (mr^) sin to dm.

J 2n

On working out this expression, and writing x for 2r^, we find
1 = 1 + cos (l™) cos (I to)

As the full discussion of this formula presents no difficulty it
may be left to the reader. The last factor in the numerator of
the fraction is that where fluctuations correspond to the rings.
Whenever x passes through an odd integer greater than 1 the
first factor changes sign, and there is a dislocation or displace-
ment of half an order, but when x passes through the value 1
the denominator changes sign along ' with both factors of the
numerator, and there is no dislocation. When x becomes con-
siderable the denominator x^—1 becomes very large, and the
fluctuations of intensity become insensible.

The following table contains the values of I calculated from
the formula (3) for 16 values of x in each of the flrst 7 orders

IN PHOTOGBAPHIC IMPRESSIONS OF POLARIZED RINGS.

35

of rings. In passing from one ring to its consecutive the angle
^trx increases by 27r, and therefore x by 0"8. The sixteenth
part of this, or 0"05, is the increment of a; in. the table. Elach
rertical column corresponds to one order. The value of x cor-
responding to any number in the table will be found by adding
together the numbers in the top and left-hand columns.

X

0^00

0^80

1-60

2-40

3-20

4-00

4^80

•00

0-000

0^142

0-481

0-830

1033

1^067

1^014

■05

0-080

0-223

0-543

0-860

1-037

1^060

1-010

•10

0-295

0^418

0-667

0-905

1032

1-044

1-005

•15

0-619

0-692

0-829

0-955

1-020

1-021

1-001

■20

1-000

1-000

1-000

1-000

1-000

1-000

1-000

•25

1-377

1-293

1-154

1033

0-977

0-979

1-001

•30

1-692

1-527

1-268

1-051

0-956

0-964

1^004

•35

1-898

1-669

1-327

1-054

0-939

0-956

1-008

•40

1-963

1-702

1-3.33

1-045

0-932

0-956

1^012

■45

1-881

1-791

1-286

1-030

0-936

0-963

1-013

•50

1-669

1-465

1-205

1-014

0^950

0-974

1-012

•55

1-356

1-243

1103

1-004

0^973

0-987

1-007

■60

1^000

1-000

l^OOO

1-000

1^000

1-000

1-000

•65

0-654

0-775

0-913

1-004

1-027

1^010

0-991

■TO

0373

0-600

0-853

1-013

1049

1015

0-983

■75

0^192

0-499

0-826

1-024

1^063

1016

0-977

A curve of iutensity might easily be constructed from this
table by taking ordinates proportional to the numbers in the
table, and abscissae proportional to the values of r, and therefore
to the square roots of the numbers 0, 1, 2, 3, 4, etc. But the
form of the curve will be understood well enough either from
the formula (3), or from an inspection of the numbers in the
table.

It will be seen that iu the first three columns the numbers lying
between the hori^iontal lines beginning -20 and -60 correspond
to bright rings, and the remainder of each colunm, together with
the beginning of the next, corresponds to a dark ring. But the
dark ring, which would regularly follow the fourth bright ring,
is converted, by the change of sign of the factor cos {^irx), into
a bright ring, forming with the former one broad bright ring
having a minimum corresponding to x = S, where however the

3—2

36 ON THE CAUSE OF THE OCCURRENCE OF ABNORMAL FIGURES

intensity falls only down to its mean value unity. A similar
displacement occurs in the seventh column, but here the whole
variation of intensity is comparatively small.

In the case of calcareous spar the character of the rings is
the same all the way round, but in the photographs of the rings
of nitre a new feature presents itself Mr Crookes's figure of
the abnormal rings of nitre is rather too small to be clear, but
with the assistance of his description there is no difficulty in
imagining what takes place. With reference to these photographs
he observes, "But here a remarkable dislocation presented itself;
each quadrant of the interior rings, instead of retaining its usual
regular figure, appeared as if broken in half, the halves being
alternately raised and depressed towards the neighbouring rings."
This effect admits of easy explanation as a result of the super-
position of systems of rings which separately are perfectly regular,
when we consider that the poles of the lemniscates of the several
elementary systems do not coincide, since in nitre the angle be-
tween the optic axes increases from the red to the blue. Now
the change of character which may be described as a displace-
ment of half an order is due to the circumstance that the smaller
rings corresponding to the more refi:angible rays are, as it were,
overtaken by the larger rings corresponding to the less refran-
gible. It is plain that the variation of position of the poles of
the lemniscates would tend to retard this effect in directions
lying outside the optic axes, and to accelerate it in directions
lying between those axes. Hence what was a bright ring in one
part of its course would become a dark ring in another part,
so that each quadrant would exhibit a dislocation of half an
order in the rings. In order to show this dislocation to the
greatest advantage, a crystal of a certain thickness should be used.
With a very thin crystal there would be no dislocation of this
nature, but only a displacement like that which takes place with
calcareous spar. With a very thick crystal the effect of the
chromatic variation of position of the optic axes would be too
much exaggerated.

It appears then that all the leading features of the abnormal
rings are perfectly explicable as a result of the superposition of
separately regular systems. But if known causes suffice for the
explanation of phenomena, we must by no means resort to agents.

lif PHOTOGRAPHIC IMPRESSIONS OF POLARIZED RINGS. 37

whose existence is purely hypothetical, such for example as in-
visible rays accompanying, but distinct from, visible rays of the
same refrangibility. Some of the minor details of the abnormal
rings may require further explanation or more precise calculation ;
but such calculations are of no particular interest unless the
phenomena offered grounds for suspecting the agency of hitherto
unrecognized causes.

The difference between the photographs taken with iodide
and bromide of silver is easily explained, when we consider the
manner in which those substances are respectively affected by
the rays of the spectrum. With iodide of silver there is such a
concentration of photographic power extending from about the
fixed line G of Fraunhofer to a little beyond H, that even when
white light is employed we may approximately consider that we
are dealing with homogeneous rays. On this account, and not
because the rays of high refrangibility are capable of producing
a more extended system of rings than those of low refrangibility,
the rings visible on the photograph are much more numerous
than those seen directly by the eye with the same white light.
Moreover, the rings do not exhibit the same abnormal character
as with bromide of silver, in relation to which substance the
photographic power of the rays is more diffused over the spec-
trum.

It is not possible to place the eye and a sensitive plate pre-
pared with bromide of silver under the same circumstances with
regard to the formation of abnormal rings. It would be easy,
theoretically at least, to place the eye and the plate in the same
circumstances as regards rings, by using homogeneous light;
but then, I feel no doubt, the rings visible on the plate would
be as regular as those seen by the eye. On the other hand, if
differences of colour exist in the figure viewed by the eye, they
inevitably arrest the attention, and it is impossible to get rid of
them without at the same time rendering the light so nearly
homogeneous that on that account nothing abnormal would be
shown. Hence Mr Crookes's abnormal lings afford a very curious
example of the creation, so to speak, by photography of forms
which do not exist in the object as viewed by the eye.

On the Metallic Reflexion exhibited by Certain
Non-Metallic Substances.

[From the Philosophical Magazine, vi, Dec. 1853, pp. 393—40.3. Also Pogg.
Ann., xci, 1854, pp. 300—14; Ann. de Chimie, xlvi, 1856, pp. 504—8.]

In the October Number of the Philosophical Magazine is
a translation of a paper by M. Haidinger of Vienna, containing
an account of his observations relating to the optical properties
of Herapathite. In this paper he refers to a communication
which I made to the British Association at the meeting at
Belfast*; and indeed one great object of his examination of this
salt was to see whether a law which he had discovered, and
already extensively verified, relating to the connexion between the
reflected and transmitted tints of bodies which have the property
of reflecting a different tint from that which they transmit,
would be verified in this case. The report of my communication
published in the Abb^ Moigno's Cosmosf had led him to suppose
that my observations were at variance with his law.

My attention was first directed to this subject while engaged
in some observations on safflower-red (carthamine), which I was
led to examine with reference to its fluorescence. In following
out the connexion which I had observed to exist between the
absorbing power of a medium and its fluorescence, I was induced
to notice particularly the composition of the light transmitted by
the powder; and I found that the medium, while it acted power-
fully on all the more refrangible rays of the visible spectrum,
absorbed green light with remarkable energy. I need not now
describe the mode of absorption more particularly. During these
experiments I was struck with the metallic yellowish-green re-
flexion which this substance exhibits. It occurred to me that the
almost metallic opacity of the medium with respect to green light

[* p. 18, ante.]
t Vol. I. p. 574.

ON METALLIC REFLEXION BY NON-METALS. 39

was connected with the reflexion of a greenish light with a metallic
aspect. I found, in fact, that the medium agreed with a metal in
causing a retardation in the phase of vibration of light polarized
perpendicularly to the plane of incidence relatively to light
polarized in that plane. The observation was made by reflecting
light polarized at an azimuth of about 45° from the surface of the
medium to be examined, the angle of incidence being considerable,
about equal to the angle of maximum polarization, and viewing
the reflected light through a Nicol's prism capped by a plate of
calcareous spar cut perpendicularly to the axis. Now by using
different absorbing media in succession, it was found that with red
light, for which safflower-red is comparatively transparent, the
reflected light was sensibly plane-polarized, while for green and
blue light the ellipticity was very considerable. ■

In the case of a transparent medium, light would be polarized
by reflexion, or at least very nearly so, at a proper angle of
incidence. Hence if the light reflected by such a medium as
safflower-red were decomposed into two pencils, one, which for
distinction's sake may be called the ordinary, polarized in the
plane of incidence, and the other, or extraordinary, polarized
perpendicularly to the plane of incidence, the extraordinary pencil
would vanish at the polarizing angle, except in so far as the laws
of the reflexion deviate from those belonging to a transparent
substance. Hence the light remaining in the extraordinary
pencil might be expected to be more distinctly related to the light
absorbed with such energy. Accordingly, it was found that
under these circumstances the extraordinary pencil (in the case
of safflower-red) was of a very rich green colour, whereas without
analysis the light was of a yellowish-green colour. Similar obser-
vations were extended to specular iron.

These phenomena recalled to my mind a communication which
Sir David Brewster made at the meeting of the British Associa-
tion at Southampton in 1846*; and on referring to his paper,
I found that the appearance of differently coloured ordinary and
extraordinary pencils in the light reflected by safflower-red was
the same phenomenon as he has there described with reference to
chrysammate of potash.

* Notice of a new property of light exhibited in the action of chrysammate of
potash upon common and polarized hght. (Transactions of the Sections, p. 7.)

40 ON THE METALLIC REFLEXION

The observations above-mentioned were made towards the end
of 1851. Accordingly, when Dr Herapath's first paper on the
new salt of quinine appeared*, I was prepared to connect the
metallic green reflected light with an intense absorbing action
with respect to green rays. Having prepared some crystals
according to his directions, I was readily able to trace the pro-
gress of the absorption in the case of light polarized in a plane
perpendicular to what is usually the longest dimension of the
crystalline plates, and to observe how the light passed from red
to black as the thickness increased. Even the thickest of these
crystals was so thin as to show hardly any colour by light
polarized in the plane of the length. The result of crossing two
such plates is of course obvious to any one who is conversant
with physical optics. The intense absorption was readily con-
nected with the metallic reflexion. An oral account of these
observations was given at a meeting of the Cambridge Philo-
sophical Society on March 15, 1852; but it was not till the obser-
vations were a second time described, with some slight additions,
at the meeting of the British Association at Belfast, that any
account of them was published. A notice of this communication
appeared in the Athenceum of September 25, 1852, and from this
the report in Cosmos seems to be taken, though the latter is
not free from errors. In the report in the Athenceum, the colour
of the more rapidly absorbed pencil is briefly described in these
words : " But with respect to light polarized in the principal
plane of the breadth, the thicker crystals are perfectly black,
the thinner ones only transmitting light, which is of a deep red
colour." The comparative transparency of the crystals with
regard to red light is afterwards expressly connected with the
green colour of the light reflected as if by a metal. But in the
report in Gosvios, the passage just quoted is replaced by " tandis
que pour le cas de la lumiere polaris^e dans le plan principal de
la largeur ils sont opaques et noirs, quelque minces qu'ils soient
d'ailleurs." This error led M. Haidinger to suppose that my
observations were opposed to his law ; whereas the fact is, that,
without knowing at the time what he had done, I had been led
independently to a similar conclusion.

In mentioning my own observations on safflower-red, Hera-
pathite, etc., nothing is further from my wish than to neglect
* Fhil. Mag. for March 1852 (Vol. iii. p. 161).

the pricritT of :"r.c<s« to vhom prioriry bekr.c^. M. HA:i:"c;r
had Skvcr.-j. yo.irs before dij-Aweied the pheiKuaenota ::' the
refle3don of ctirterviitly coikmied ,np^>i:;.y p>';:«-:rc\i -vnci,?. xrhioh
^•.r I>»rki BTv\vj:rr s"r.:rr'-v :\r:er\vard5, s-ji indeiviideiiTlv. ,v.5-
coTeied in rhe .m« of :r.ry5.v-..: ...:e of r-, --,<h. M. H.v.i:r.c;r had
fivua The firs: "riserred a .,.:<?: imp.nc-.r:: char^oTcr of the rhe'z:-
meiK® m T~e .w;^; :: iiiivny "vst^lj, namely, rhe .riiv.Tari.xn of
the y«liunzati«.ui of :>.f r^rccted '.:i:hr. which Sir r\-,\-;d BietrsTer
d.v* not <x- 1" to have norioed ir. the :.>se :: . rr.fkv.v. ..ate >jf -.::.-#>.
and which r^rispt? .v.-^ r;:: Tiery erkl^at in that Si,:. I., a ,:-.:rr
paper M. Hv.r: c^r had aiir.iin.vd the Ar.r.e-aciirary rrlr::.!
of the rcr.cVTcV. and :r,-:.5-... ::ted ::r.:5* Theie i? n, thine new in
emp-:y;v.c :'r.e r.-.-.p? of cakaiecij siwr r.5 a :v.;.\-:l5 ;:' iit:e-:::v.c
e.'.".y:ic po'=.r:--\rior: : ;\rd the prvn^fTty of pivxv.t-'ir.c elliptic
p lariiati." -.r. red;::,,-;: ri;\t:^--'>y-..ri.-<:-d iicr.t had vr; -:v.<iv
been observc-d in sv.histvW-.v^ even of v^ r-^rable rtcttti", I ;--"j:
r,:t swaie, however, that the ohr^.^'-n.^tio Taan.>ti;t"_j of th^ chancr
>.:' pha*: had be^n e^jvritnetitaliy .^^n-.tf .ted w::h thr chiomatic
vsr.r^ts.v.j- of an iiLtc'.ije .VriS-rhir..j a.'ti:" oaa the twrt of the
medium. I hsvo hitherto mentiono.i hv.t one ."ftance of tV.i?
cwmeiioii. Imt I shsli yr;«~t y haveoeea^i.m to allnde to anther.

.U-* i-ri.-te*x-**»«a ILu'i:;!^ isitiss«r S.Vpia-,'' rr'.-cx Ae Ja=:i&rT y^nVr -:" dM
lV«««uSnp^ trf tiM Vi:>jeaaat>eil sr^S ?i;.-*;.s: Cliss of the Aoaoj— t of <>.if:i.-^s s:
■^-itniii Sen- ti>? v;-ir l>fi. tlU be feoai i ■-;>^ ,-;' M. Hii.linp;r'f yr;T:,--j pir^rf OB
;.::? ju-.-A-:. ~-:> paXKH- .vrrii-s ft ia«tiK!i£:i<.-. s-jfonn: of «J»s rr.rifr^r-i, »;:J;
li^a«»of u- *:rriA- »zii ?u':.?rir.-t ivl.-c:;. «f dK scbjiirA-s -p W> tint i=i^
examiBc<J kvtlK isji.r. MB.-cnr-.; is -uziitr to ihinr. F.r a .-jpT of :Ji;s is
■w^il i* fft • ;ri_ .Oij-r^ <>f i.i> p»j«rf. I »i^ ir^iriis-i lo ms '< - ji»e~s of iiir i^iiisr.

" 0* ike Pi««^J3«a« izpd Xaws ef SZ;r li; Baitnattioa. .^j fiiiHTei ia i2i-i smwa
of Mesus -jwa Li^i;." rcizr>ta cc! the iKifiScatiaB i<(oesR«<i v3 die rr-cj of

(fif.'. r-««». r.? 1n.^,K r, 231. I" a ««^mamfgtii.-n 10 dK ?r::::<i A:^.-c;i;:-n il
tl-~ ~-ef:i-c it <-^ tVt— t-jtxB in l>i\ Mr ])s^ rir^ericrs »*>- aaB-'cu: »;<£:>>:
~S;.T:f-uv-< gc ^*i.'*! fe iad 4e<ees«d rlip'd.- rofauxsunB fcv aaeaa- .^i: tii'r ra«^ ~i
^il.-ir^cc< >r.ir Is l:i:; -isf. V:>«-«»er, h; ^-^rr^Mcj ii~ vr.'peiTj-. c-: »iiji dte

I ak> -.* i>Me =i==i.-ii ibe "r-r:^ i'crres cf :-;iir.:ii:rv wk-h rave '.ctz i~re..-T<rd
i;: vcilirtLieii H,— ■: i«Seeieii ~.=i :r-ii syajness s^:*rir.-s in pjaessIiT i.,; iflx-aie
r;;-s«:-!ir.'ii* af M. JEuu&, pirij^ >.K3;:.>i ibey iirv sc ?'".4 " sf i-j W »ii;"ly >cpir-i;ji
~.~ ii= eCliriiMiT IB iisi ,-is<:" of oudtuune. e«., piriy btx-i^se ii-ev si^iii re hi-r;

42 ON THE METALLIC REFLEXION

I think it but justice to myself to point out the error in Cosmos
(from whence M. Haidinger derived his information respecting
my observations), in consequence of which I would appear to
have been guilty of a grievous oversight in the examination of
Herapathite : but I would hardly have ventured to mention my
observations on carthamine, etc., were it not that, when different
persons arrive independently at a similar conclusion, it frequently
happens that views present themselves to the mind of one which
may not have occurred to another. In the present case, in stating
in detail my own views as to the nature of the phenomenon,
I hope to be able to add something to what has been already
done by M. Haidinger and Sir David Brewster, and it seemed
not out of place to mention the observations ia which those
views originated.

It appears, then, that certain substances, many of them of
vegetable origin, have the property of reflecting (not scattering)
light which is coloured and has a metallic aspect. The sub-
stances here referred to are observed to possess an exceedingly
intense absorbing action with respect to rays of the refrangibilities
of these which constitute the light thus reflected, so that for
these rays the opacity of the substances is comparable with that
of metals. Contrary, however, to what takes place in the case of
metals, this intense absorbing action does not usually extend to
all the colours of the spectrum, but is subject to chromatic varia-
tions, in some cases very rapid. The aspect of the reflected light,
which itself alone would form but an uncertain indication, is not
the only nor the principal character which distinguishes these
substances. In the case of transparent substances, or those of
which the absorbing power is not extremely intense (for example,
coloured glasses, solutions, etc.), the reflected light vanishes, or
almost entirely vanishes, at a certain angle of incidence, when it
is analysed so as to retain only light polarized perpendicularly
to the plane of incidence*, which is not the case with metals.
In the case of the substances at present considered, the reflected
light does not vanish, but at a considerable angle of incidence

* I do not here take into account the peculiar phenomena which have been
observed by Sir David Brewster with reference to the influence of the doable
refraction of a medium (such as calcareous spar) on the polarization of the reflected
light, which, indeed, are not very conspicuous unless the medium be placed in
optical cohtact with a fluid having nearly the same refractive index.

EXHIBITED BY CERTAIN NON-METALLIC SUBSTANCES. 43

the pencil polarized perpendicularly to the plane of incidence
becomes usually of a richer colour, in consequence of the removal,
in a great measure, of that portion of the reflected light which
is independent of the metallic properties of the medium ; it com-
monly becomes, also, more strictly related to that light which
is absorbed with such great intensity. The reflexion from a
transparent medium is weakened or destroyed by .bringing the
medium into optical contact with another having nearly or exactly
the same refractive index. Accordingly, in the case of these
optically metallic substances, the colours which they reflect by
virtue of their metallicity* are brought out by putting the
medium in optical contact with glass or water. A remarkable
character of metallic reflexion consists in the circumstance, that
as the angle of incidence increases from 0°, the phase of vibra-
tion of light polarized in the plane of incidence is accelerated
relatively to that of light pojarized in the perpendicular plane.
Accordingly, the same change takes place, with the same sign,
in the case of these optically metallic substances ; but the amount
of the change is subject to most material chromatic variations,
being considerable for those colours which are absorbed with
great energy, but insensible for those colours for' which the
medium is comparatively transparent, so that the absorption
may be neglected which is produced by a stratum of the medium
having a thickness amounting to a small multiple of the length
of a wave of light. If the medium be crystallized, it may happen
that one only of the oppositely polarized pencils which it transmits
suffers, with respect to certain colours, an exceedingly intense
absorption; or, if that is the case with both pencils, that the
colours so absorbed are different. It may happen, likewise, that
the absorption varies with the direction of the ray within the
crystal. In such cases the light reflected by virtue of the metal-
licity of the medium will be subject to corresponding variations,
so that the medium is to be regarded as not only doubly refracting
and doubly absorbing, but doubly metallic.

The views which I have just explained are derived from a
combination of certain theoretical notions with some experiments.

• I use this word to signify the assemblage of optical properties by which
a metal differs from a transparent medium, or one moderately coloured, such as
a coloured glass.

44 ON THE METALLIC REFLEXION

They have need of being much more extensively verified by
experiment; but, so far as I at present know, they are in con-
formity with observation.

To illustrate the effect of bringing a transparent medium into
optical contact with an optically metallic substance, I may refer
to safBower-red. If a portion of this substance be deposited on
glass by means of water, and the water be allowed to evaporate,
a film is obtained which reflects on the upper surface a yellowish-
green light, but on the surface of contact with the glass a very
fine green inclining to blue. A green of the latter tint appears
to be more truly related to the colours absorbed with greatest
energy. Similar remarks apply to the light reflected by Hera-
pathite, according as the crystals are in air or in the mother-
liquor. If a small portion of Quadratite (platinocyanide of
magnesium) be dissolved on glass in a drop or two of water, and
the fluid be allowed to evaporate, the tints reflected by the upper
and under surfaces of the film of crystals are related to one
another much in the same way as in the case of saSlower-red.
For a fine specimen of the salt last mentioned I am indebted to
the kindness of M. Haidinger. I may mention in passing, that
the platinocyanides as a class are of extreme optical interest.
The crystals are generally at the same time doubly refracting,
doubly absorbing, doubly metallic, and doubly fluorescent. By
the last expression I mean that the fluorescence, which the crystals
generally exhibit in an eminent degree, is related to directions
fixed relatively to the crystal, and to the azimuths of the planes of
polarization of the incident and emitted rays.

M. Haidinger has expressed the relation between the surface
and substance colours of bodies by saying that they are comple-
mentary. This expression was probably not intended to be
rigorously exact ; and that it cannot be so, is shown by the follow-
ing simple consideration. The tint of the light transmitted
across a stratum of a given substance almost always, if not
always, varies more or less according to the thickness of the
stratum. Now one and the same tint, namely, that of the
reflected light, cannot be rigorously complementary to an infinite
variety of tints of different shades, namely, those of the light
transmitted across strata of different thickness. In most cases,
indeed, the variation of tint may not be so great as to prevent

EXHIBITED BY CERTAIN NON-METALLIC SUBSTANCES. 45

US from regarding the reflected and transmitted tints in a general
sense as complementary. But as media exist (for example, salts
of sesquioxide of chromium, solutions of chlorophyll) which
change their tint in a remarkable manner according to the
thickness of the stratum through which the light has to pass, it
is probable that instances may yet be observed in which M. Hai-
dinger's law would appear at first sight to be violated, although
in reality, when understood in the proper sense, it would be
found to be obeyed. As the existence of surface-colour seems
necessarily to imply a very intense absorption of those rays which
are reflected according to the laws which belong to metals, it
follows that it is in the very thinnest crystals or films of those
which it is commonly practically possible to procure, that the
transmitted tint is to be sought which is most properly comple-
mentary to the tint of the reflected light.

I will here mention another instance of the connexion between
metallic reflexion and intense absorption. I choose this instance
because a different explanation from that which I am about to
offer has been given of a certain phenomenon observed in the
substance. The instance I allude to is specular iron. As it is
already known that various metallic oxides and sulphurets possess
the optical properties of metals, there is nothing new in bringing
forward this particular mineral as a substance of that kind. It
is to the chromatic variation of the metallicity that I wish to
direct attention. If light polarized at an azimuth of about 45°
be reflected from a scale of this substance at about the polarizing
angle, and the reflected light be viewed through a plate of cal-
careous spar and a Nicol's prism, it will be found, by using
different absorbing media in succession, that the change of phase,
as indicated by the character of the rings, while it is very evident
for red light, becomes much more considerable in the highly
refrangible colours. Now specular iron is almost opaque for light
of all colours, but as it gives a red streak it appears that the
substance-colour is red; and, in fact, it is known that very
thin laminae are blood-red by transmitted light. Accordingly,
the chromatic variation of the change of phase corresponds to
that of 'the intense absorbing power.

The light reflected by specular iron is not extinguished by
analysation, whatever be the angle of incidence ; but at the

46 ON THE METALLIC BEFLEXION

angle of incidence which gives the nearest approach to complete
polarization, a quantity of blue light is observed to remain. This
has been explained by comparing specular iron to a substance of
high dispersive power, so that the polarizing angle for red light
is considerably less than for blue ; and accordingly on increasing ,
the angle of incidence, the light (which is here supposed to be
analysed so as to retain only the portion polarized perpendicularly
to the plane of incidence), while it becomes much less copious
near the polarizing angle, becomes at the same time of a decided
blue colour*. I believe, however, that the blue light is mainly
due to the chromatic variation of the metallicity, the medium,
considered optically, being much more metallic for blue light
than for red, though it may in some measure be due to the cause
previously assigned.

Specular iron is a good example of a substance forming a
connecting link between the true metals and substances like
safflower-red. It resembles metals in the circumstance that the
absorbing power, as inferred from the chromatic variation of the
metallicity, and as indicated by the tint of the streak, is not
subject to the same extensive chromatic variations as in the case
of colouring matters like safflower-red. It resembles safflower-
red in being sufficiently transparent with respect to a portion
of the spectrum to allow the connexion between the metallicity
and the substance-colour to be observed; whereas the substance-
colour of metals is not known from direct observation, except,
perhaps, in the case of gold, which in the state of gold-leaf lets
through a greenish light.

I am now able to bring , forward a striking confirmation of the
relation which seems to exist between the light reflected as if
from a metal, and that absorbed with great energy. On reading
M. Haidinger's paper, of which the title has been already quoted,
I was particularly interested by finding crystallized permanga-
nate of potash mentioned among the substances which exhibit
distinct surface- and substance-colours. I had previously noticed
the very remarkable mode of absorption of light by red solution
of mineral chameleonf. This solution, which may be regarded
as an optically pure solution of permanganate of potash, inas-

* See Dr Lloyd's Lectures on the Wave-Theory of Light, Part n. p. 18.
t Phil. Tram, for 1852, p. 558. [Ante, Vol. iii. p. 402.]

EXHIBITED BY CERTAIN NON-METALLIC SUBSTANCES. 47

much as it is associated with only a colourless salt of potash,
absorbs green light with great energy, as is indicated by the
tint, even without the use of a prism. But when the light
transmitted by a pale solution is analysed by a prism, it is found
that there are five remarkable dark bands of absorption, or
minima of transparency, which are nearly equidistant, and are
situated mainly in the green region. The first is situated on
the positive or more refrangible side of the fixed line D, at
a distance, according to a measurement recently taken, of about
four-sevenths of the interval between consecutive bands ; the last
nearly coincides with F. The first minimum is less conspicuous
than the second and third, which are the strongest of the set.
Now it occurred to me, that as the solution is so opaque for rays
having the refrangibilities of these minima of transparency, cor-
responding maxima might be expected in the light reflected from
the crystals. This expectation has since been realized by obser-
vations made on some small crystals. On analysing the reflected
light by a prism, I was readily able to observe four bright bands,
or maxima, in the spectrum. These, as might have been expected,
were more easily seen when the light was incident nearly per-
pendicularly than at a large angle of incidence. The first
band was yellow, the others green, passing on to bluish-green.
On decomposing the light reflected at a considerable angle of
incidence, in a plane parallel to the axis, into two streams,
polarized respectively in, and perpendicularly to, the plane of
incidence, and analysing them by a prism, the bands were hardly
or not at all perceptible in the spectrum of the former pencil,
while that of the latter consisted of nothing but the bright bands.
The tint alone of the first bright band already indicated that
the band was more refrangible than the light lying on the nega-
tive side of the fiirst dark band seen in the spectrum of the light
transmitted by the solution, and less refrangible than the light
lying between the first and second dark bands, so that its position
would correspond, or nearly so, to the first dark band. However,
the eye is greatly liable to be deceived, in experiments on absorp-
tion, by the effects of contrast, and therefore an observation of the
nature of that just mentioned cannot, I conceive, be altogether
relied on. The smallness of the crystals occasioned some difficulty ;
but a more trustworthy observation was made in the following
manner.

48 ON THE METALLIC REFLEXION

The sun's light was reflected horizontally into a darkened
room, and allowed to fall on a crystal. The reflected light was
limited by a slit, placed at the distance of two or three feet from
the crystal. This precaution was taken to ensure making the
observation on the regularly reflected light. Had no slit been
used, or else a slit placed close to the crystal, it might have been
supposed that the light observed was not regularly reflected, but
merely scattered, as it would be by a coloured powder. The
appearance of a spot of green light on a screen held at the
place of the slit showed that the light was really regularly
reflected. The slit was also traversed by the light scattered by
the support of the crystal, etc. The slit was viewed through a
prism and small telescope ; and the position of the dark bands,
or minima of brilliancy, in the reflected light could thus be
compared with the fixed lines, which were seen by means of the
scattered light in the uninterrupted spectrum corresponding to
that portion of the slit through which the light reflected from
the crystal did not pass. The minimum situated on the positive
side of the first bright band lay at something more than a band-
interval on the positive side of the fixed line D ; the minimum
beyond the fourth bright band lay at the distance of about half
a band-interval on the negative side of F. It thus appears that
the minima in the light reflected by the crystal were inter-
mediate in position between the minima seen in the light trans-
mitted through the solution, so that the maxima of the former
corresponded to the minima of the latter.

It might have been considered satisfactory to compare the
reflected light with the light transmitted, not by the solution,
but by the crystals themselves. But the crystals absorb light
with such energy as to be opaque ; and even when they are
spread out on glass, the film thus obtained is too deeply coloured
for the purpose. For to show the bands well, the solution must
be so dilute, or else seen through so small a thickness, as to be
merely pink. As M. Haidinger states that the phenomena of
the reflected light are the same for all the faces in all azimuths,
and for the polished surface of a mass of crushed crystals, it may
be presumed that the absorption is not much affected by the
crystalline arrangement, and that the composition of the light
transmitted by the solution is sensibly the same as that which

EXHIBITED BY CERTAIN NON-METALLIC SUBSTANCES. 49

would be observed across a crystalline plate, were it possible to
obtain one of sufficient thinness.

The first bright band in the reflected light does not usually
appear to be very distinctly separated from the continuous light
of lower refrangibility ; but the latter may be got rid of by
observing the light reflected about the polarizing angle, and
analysing it so as to retain only the portion polarized perpendi-
cularly to the plane of incidence. As the surface of the crystals
is liable to become spoiled, it is safest in observations on the
reflected light to make use of a crystal recently taken out of the
mother-liquor. I have only observed four bright bands in the
reflected light, whereas there are five distinct minima in the
light transmitted by the solution. However, the extreme minima
are less conspicuous than the intervening ones, besides which
the fifth occurs in a comparatively feeble region of the spectrum.
The fourth bright band in the reflected light was rather feeble,
but with finer crystals perhaps even a fifth might have been
visible. As the metallicity of the crystals is almost or perhaps
quite insensible in the parts of the spectrum corresponding to
the maxima of transparency, we may say, that, as regards the
optical properties of the reflected light, the medium changes
four or five times from a transparent substance to a metal and
back again, as the refrangibility of the light changes fi-om a little
beyond the fixed line D to a little beyond F.

S. IV.

Extracts from Letter to Dr W. Haidinger: on the
Direction of the Vibrations in Polarized Light: on
Shadow Patterns and the Chromatic Aberration op
THE Eye: on Haidinger's Brushes. {Fehruary 9, 1854.)

Die Richtung der Schwingungen des Lichtdthers im polarisirten
Lichte. Mittheilung aus einem Schreiben des Hrn. Prof.
Stokes, nebst Bemerkungen von W. Haidinger.

[Fogg. Ann. xcvi, 1855, pp. 287—292. Mitgetheilt vom Hrn. Verf.
aus d. SiUungsberichten d. Wiener Akademie {April 1854).]

EiN Abschnitt des Schreibens vom Hrn. Prof. Stokes, den ich
heute der hochverehrten mathematisch-naturwissenschafblichen
Classe vorzulegen die Ehre babe, beziebt sicb auf die Richtung
der Schwingungen des Lichtatbers in Bezug auf die Polarisations-
Ebene, und zwar enthalt er nicht nur eine Beurtheilung der
Tragweite der Bemerkungen, welche ich als Beweis fur die senk-
rechte Richtung dieser Schwingungen gegen diese Polarisations-
Ebene aus den Erscheinungen an pleochromatiscben Krystallen
darstellen zu diirfen glaubte*, sondem auch seine Ansicht iiber
den Gegenstand selbst, iibereinstimmend mit seinen eigenen
frtiberen Arbeiten

" Die Thatsachen, deren ich in Bezug auf die Polarisation
des Fluorescenz-Lichtes der Kalium-Platin-Cyanide (in einem
andern Tbeile des Schreibens) gedachte, und die Art wie die
Polarisirung der einfallenden Strablen auf dieses Licht wirkt,
stimmen, so viel ich glaube, viel besser mit der Annahme
iiberein, dass die Schwingungen im polarisirten Lichte senk-
recht auf der Polarisations-Ebene stehen, als mit der anderen
Theorie.

* Sitzungsberichte der kais. Akademie der Wissenscliaften, Math.-naturw. Classe,
1852, vm, S. 52.

DIEECTION OF VIBRATIONS IN POLARIZED LIGHT. 51

" Diess veranlasst mich, der Beweisgrunde zu erwahnen, welche
Sie anfiihrten, um zu zeigen, dass im polarisirten Lichte die
Schwingungen senkrecht auf der Polarisations-Ebene stehen.
Da ich glaube, Sie wiirden gem meine Ansicht dariiber kennen,
so will ich sie ausfiihrlich anfuhren. Zu allererst kann ich sagen,
dass ich es nicht fur moglich halte, durch irgend eine Combination
von anerkannten Ergebnissen die Frage zu entscheiden. Unter
den anerkannten Ergebnissen betrachte ich solche, wie diese — •
dass die Schwingungen transversal sind — dass im linear-polari-
sirten Lichte die Schwingungen geradlinig siad, und symmetrisch
mit Beziehung der Polarisations-Ebene, und daher entweder parallel
oder senkrecht auf diese Ebene — dass im elliptisch-polarisirten
Lichte die Schwingungen elliptisch sind u. s. w. Die Ent-
scheidung muss sich immer auf eine oder die andere Art auf
dynamische oder physikalische Betrachtungen stiitzen, welche,
mogen sie an sich noch so wahrscheinlich seyn, doch nicht zu
den anerkannten Ergebnissen gezahlt werden konnen. Es ist
auch nicht schwierig zu sehen, welche die Betrachtungen dieser
Art in dem Falle Ihrer Beweisfuhrung sind. Nehmen wir den
Fall eines doppelt absorbirenden einaxigen Krystalles, wie Tur-
malin. Es sey oG Fig. 6 parallel der Axe, oA oB zwei Richt-
ungen senkrecht auf die Axe. Die eine Farbe (ich will sie
nennen) sieht man in der Richtung der Axe Co, und in alien
Richtungen in der Ebene BoA (oder senkrecht auf die Axe)
in dem oG parallel polarisirten Lichte. Die andere Farbe {E)
sieht man in alien Richtungen in der Ebene BoA, wenn das
Licht in dieser Ebene polarisirt ist, und man sieht sie gar nicht
in der Richtung Go. ' Wenn diese Farbe nun von Transversal-
Schwingungen abhangt, so sind alle solche Schwingungen, trans-
versal oder senkrecht gegen die Axe, mit einem Male ausge-
schlossen, und die einzigen Schwingungen, welche moglicherweise
zu der Farbe des extraordinaren Strahles, der in dem Krystall
entsteht, gehoren konnen, sind die parallel der Richtimg der Axe.'
Aber wenn von Schwingungen gesprochen wird, welche zu dieser
oder jener Farbe gehoren, so wird stillschweigend vorausgesetzt,
dass in der That die Farbe abhangig ist von der Richtung der
Schwingungen. Nun kann man sich allerdings ganz gut vor-
stellen (so unwahrscheinlich es auch immer seyn mag), dass
Absorption von dem gleichzeitigen Einflusse der Richtung der
Schwingungen und der Richtung der Fortpfianzung abhange,

4—2

52

BEARING OF DOUBLE ABSORPTION IN CRYSTALS

dergestalt, dass sie als gegeben betrachtet werden kann, wenn
die Richtung einer Linie gegeben ist, welche senkrecht auf den
beiden vorerwahnten Richtungen steht. Nimmt man diesen Satz
an in Bezug auf die Natur der Absorption, so ist vollkommen

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

klar, dass die experimentalen Thatsachen, welche Sie in Bezug
auf die doppelte Absorption anfiihren, zu dem Schlusse fiihren
wurde, die Schwingungen waren im polarisirten Lichte der
Polarisations-Ebene parallel. Die Wahrscheinlichkeit, welche der
Grund Ihrer Beiyeisfuhrung der Wahrheit von Fresnel's Annahme
giebt, reicht also nicht bis zur absoluten Gewissheit, sondem
entspricht nur der Unwahrscheinlichkeit, dass Absorption gleich-
zeitig von der Richtung der Schwingungen und von der Richtung
der Fortpflanzung abhangig seyn sollte, in der Art wie ich es
oben erwahnte.

" Sie sagen : Wenn die Farbe E, welche man in der Richtung
Go nicht sehen kann, irgendwie auf Transversal-Schwingungen
beruht, so sind alle solche Schwingungen transversal oder senk-
recht auf die Axe ausgeschlossen, und die einzigen Schwingungen,
welche moglicherweise dem in dem Krystalle entstehenden extra-
ordinaren Strahl angehoren konnen, sind dann der Richtung der
Axe parallel. Nun konnte aber eine besondere Farbe in einer
Richtung von transversalen Schwingungen abhangen, und nicht von
transversalen Schwingungen abhangen in einer anderen Richtung.
Sie konnte von transversalen Schwingungen in der Richtung ab-
hangen, in welchen sie abhftngig ist von der Wirkung des Mittels.

ON DIRECTION OF VIBRATIONS IN POLARIZED LIGHT. 53

auf das Licht, und das Licht besteht aus transversalen Schwing-
ungen : sie konnte nicht von transversalen Schwingungen ab-
hangen in der Richtung, wo sie nicht bloss von der Richtung
der Schwingungen bestimmt ist, sondern auch von der Richtung
der Fortpflanzung abhangt.

" Daher kann ich Ihre Folgerungen nicht als einen Beweis in
dem strengen Sinne des Wortes betrachten. Ein solcher hangt
am Ende von gewissen physikalischen Betrachtungen ab, welche
sich auf die Absorption beziehen. Meine eigenen Ansichten in
Bezug auf die Ursache der Absorption fuhren mich sehr stark
zu der Meinung, dass sie bloss von der Richtung der Schwin-
gungen und der Schwingungszeit {periodic time) und gar nicht
von der Fortpflanzungsrichtung abhangt. In meinem Sinne
haben daher Ihre Griinde sehr grosses Gemcht. Aber da diess
von meinen eigenen individuellen Ansichten abhangt, so be-
trachte ich dieselben nicht als Etwas, was nothwendiger Weise
zu allgemeiner Beistimmung zwingen muss.

" Da ich bei diesem Gegenstande bin, so erlauben Sie mir Ihre
Auftnerksamkeit auf gewisse Untersuchungen zu lenken, welche
mich in einer ganzlich verschiedenen Weise zu einer ahnlichen
Schlussfassung fiihrten. Sie sind in dem 9. Band der Cambridge
Philosophical Transactions, Part I, veroffentlicht. Eine dyna-
mische Untersuchung des Problems der Beugung, in anderen
Worten eine mathematische Untersuchung der Beugung, be-
handelt wie ein dynamisches Problem, ftihrte mich zu folgendem
Gesetz : Wenn linear polarisirtes Licht der Beugung untenvorfen
wird, so ist jeder Strahl nach der Beugung linear polarisirt, und
die Schwingungsebene des gebeugten Strahles ist parallel der
Schwingungsrichtung des einfallenden Strahles. Unter" Schwing-
ungsebene ist die Ebene verstanden, welche durch den Strahl
und durch die Schwingungsrichtung geht. Es sey AB Fig. 7
in der Ebene des Papieres der einfallende Strahl, der bei
B gebeugt wird. BE auch in der Ebene des Papieres ein ge-
beugter Strahl, der in das Auge eintritt, CD, in einer Ebene
senkrecht auf AB, sey die Schwingungsrichtung des einfallenden
Strahles. Man lege eine Ebene durch BE und CD, diess wird
die Schwingungsebene des gebeugten Strahles seyn und wenn
in dieser Ebene FG senkrecht auf BE gezogen wird, so ist FQ

54 POLARIZATION OF DIFFRACTED LIGHT.

die Schwingungsrichtung. Mit anderen Worten, die Schwing-
ungsrichtung in dera gebeugten Strahle ist so nahe als moglich
der Schwingungsrichtung in dem einfallenden Strahle parallel,
als diess nur immer unter der Bedingung geschehen kann, dass
sie senkrecht auf dem gebeugten Strahle steht. Dieses Gesetz
erscheint sehr natilrlich, selbst unabhangig von allem Calcill.
Nun folgt aber aus demselben, dass wenn die Schwingungsebene
zuerst mit der auf ABE senkrecht stehenden Ebene zusammen-
fallt, und dann allmahlich durch gleiche Winkel herumgedreht
wird, dass dann die Schwingungsebenen des gebeugten Strahles
nicht gleichformig ausgetheilt seyn werden, sondern sie werden
mehr angehauft gegen eine Ebene durch BE senkrecht auf die
Ebene ABE erscheinen. Wenn a^, a^ die Azimuthe der Schwing-
ungsebene des einfallenden und des gebeugten Strahles sind,
erhalten von Ebenen senkrecht auf ABE, und 6 das Supplement
des Winkels ABE, so haben wir tang a,; = cos ^ tangcfi. Nun
setzt uns aber der Versuch in den Stand die Richtung und das
Maass jener Anhaufung der Pularisationsebenen zu bestimmen,
und nach dem Ergebnisse werden wir uns geleitet finden, sie als
parallel oder senkrecht auf die Schwingungsebenen zu betrachten.
Wenn nun Fig. 8 die Projection der Polarisations-Ebenen des
einfallenden Strahles auf einer senkrecht auf diesem Strahl
stehenden Ebene in verschiedene Stellungen des Polarisirers (z. B.
eines Nicol'schen Prismas in einer kreisfdrmig getheilten Fassung)
darstellt, und Fig. 9 oder Fig. 10 dasselbe ftir den gebeugten
Strahl vorstellt, so wilrden die Ebenen mehr gehauft seyn wie
in Fig. 9 und 10, je nachdem die Polarisations-Ebenen parallel
oder senkrecht auf die Schwingungsebenen sind. Die Horizon-
tallinien in Fig. 8, 9, 10 stellen die Projectionen auf der Ebene
ABE dar. Bei einem Glasgitter geschieht die Beugung unter
einem so«bedeutenden Winkel, dass der theoretische Azimuth der
Polarisations-Ebene des gebeugten Strahles in manchen Fallen
bis zwanzig Grad variiren kann, je nachdem man voraussetzt,
dass die Schwingungen des polarisirten Lichtes parallel oder
senkrecht auf die Polarisations-Ebene stehen. Das Ergebniss
der Versuche war vollstandig zu Gunsten von Fresnel's Voraus-
setzung."

COLOURED SHADOW PATTERNS FORMED WITH A LENS. 55

Mittheilung aus einem Schreiben des Hrn. Prof. Stokes, iiber
das optische Schachhrettmuster ; von W. Haidinger.

[Pogg. Ann. xcvi, 1855, pp. 305—312. Mitgetheilt vom Hrn. Verf.
aus d. Sitsungsberiehten d. Wiener Akademie {April 1854).]

"Ich habe ahnliche Erscheinungen," wie das Interferenz-

Schachbrettmuster, " auf einem Schirme dargestellt, indem ich
das Sonnenlicht horizontal in ein finsteres Zimmer reflectirte,
in dem Fenster, auf dem Wege des einfallenden Lichtes ein
durchlochertes Zinkblech, wie es ftlr Fensterblenden dient, an-
brachte, mit einer grossen Linse in einiger Entfernung von dem
Bleche, und das Bild des Bleches nun auf einem Blatte Papier
auffing, welches von dem Bilde nach beiden Seiten gegen die
Linse zu und von derselben weg bewegt werden konnte. Ich
iiberzeugte mich, dass die Erscheinung nicht auf Interferenz
beruht, sondern einen viel einfacheren Charakter besitzt, und
dass die Erklarung derselben aus der geometrischen Theorie der
Schatten und Halbschatten folgt. In der That kenne ich kein
Interferenz-Phanomen, das auf einer breiten Lichtflache, wie die
des Himmels ist, beruht ; es ist immer erforderlich, die ein-
fallenden Strahlen zu begranzen, indem sie etwa durch ein Loch
oder einen Spalt gehen, oder indem man sich des Sonnenbildes
einer Linse mit kurzer Brennweite bedient. In meinen Ver-
suchen brachte ich nicht nur die Erscheinungen hervor, welche
den von Ihnen beschriebenen ahnlich sind, indem ich den vollen
Sonnenstrahl anwandte, sondern ich untersuchte auch die Inter-
ferenz- Wirkungen, indem ich das Bild der Sonne durch eine
Linse von ziemlich kurzer Brennweite beniitzte und das Zink-
blech nun in einige Entfernung vom Fenster rtickte, und ich
bin liberzeugt, dass Interferenz- Wirkungen, selbst wenn sie nicht
ganz unsichtbar seyn sollten, doch in Ihrem Phanomen so schwach
sind, dass sie vernachlassigt werden konnen.

"Man betrachte zuerst Licht von einem einzigen Grade der
Brechbarkeit (Fig. 11).

"Es seyen LL' die einfallenden Strahlen, SS' sey der durch-
locherte Schirm, W die Linse, aa' das Bild der Sonne, ss' das
Bild des Schirmes. Man betrachte nur den Lichtbundel, der
durch eine einzige Oeffiiung geht. Nach der Brechung durch
die Linse werden sich diese gerade so fortpflanzen (siehe Fig. 12)

56 COLOURED SHADOW PATTERNS FORMED WITH A LENS

als ob (7(7' eine helle Scheibe ware, von der die Strahlen aus-
gehen, von welchen uns aber nur diejenigen angehen, welche
durch das Loch aa' durchgehen (aa' ist aber das Bild der Oeifnung),
Oder welche von g<t' in solchen Richtungen ausgehen, dass sie
durch das Loch aa' hindurchgehen wiirden, wenn man sie nicht
durch den Schirm aufgefangen hatte. Es entsteht dadurch also,
was man einen negativen Schatten vol va.' und Halbschatten Aa.(7
A' a! a' nennen konnte, das heisst Raume, welche fur Beleuchtung
eben dasjenige sind, was Schatten und Halbschatten fiir Fin-
sterniss. Auf einem Schirme, mit welchem man die Strahlen
auffangt, wiirde eine Kreisflache beleuchtet seyn, am schmalsten
bei aa' (vorausgesetzt, dass tro-' grosser ist als aa'), und in dieser
Entfernung auch gleichformig hell, wahrend bei anderen Ent-
fernungen die Mitte heller seyn wird als der Rand.

"Man betrachte nun die Wirkung des Uebereinanderfallens
der hellen Kreise, welche den benachbarten Oeffnungen ent-
sprechen. Um die Frage auf das Aeusserste zu vereinfachen,
nehme ich die Oeffnungen sehr klein an, so dass man 0.0! als
Punkt betrachten kann. Ich nehme dabei die Anordnung der
Oeffnungen als die namliche an, wie in der von Ihnen gegebenen
Figur*. Stellt man den Schirm in den Focus, so erscheint
eine Reihe heller Flecke (Fig. 13). Bewegt man den Schirm
ein wenig in die Richtung gegen die Linse oder von derselben
weg, so offnen sich die lichten Flecke zu lichten Kreisflachen
(Fig. 14). Es sey d die Entfernung zwischen den Mittelpunkten
zweier benachbarter Kreise, in verticaler oder horizontaler
Richtung. Bewegt man den Schirm so weit, bis die Radien der
Kreise grosser sind als \d aber kleiner als djtj^, so werden die
Rander der Scheiben iiber einander fallen, etwa so wie in Fig. 15.
Der Schirm wird dadurch dem gi'ossten Theile nach beleuchtet,
mit Ausnahme von dunkeln quadratartigen, regelmassig geord-
neten Raumen (Fig. 16). Man bewege nun den Schirm so weit,
bis die Radien der vergrbsserten hellen Scheiben grosser sind als
d/\/2, aber noch immer kleiner als d, dann ist der Mittelpunkt
der bisher dunklen Raume durch vier iiber einander fallende
Kj-eisscheiben beleuchtet (Fig. 17), wahrend der Mittelpunkt
jedes friiher hellen Raumes immer noch nur von einer einzigen
beleuchtet wird. Die am hellsten beleuchteten Raume des Schir-

* Sitzungsberichte u. s. w., Bd. vii, S. 291.

AND THEIR ALTERATION WITH DISTANCE OF SCREEN.

57

mes sind also nun in regelmassiger Anordnung die Punkte wie a,
welche in ihrer Lage denjenigen Gegenden des Schirmes ent-
sprechen, welche, wenn dieser im Focus stand, die Mittelpunkte

I,

Fig. 11.

Fig. 12.

• •

Fig. 13.

o o o o
o o o o
o o o o

O O O -O

Fig. 14.

Fig. 15.

Fio. 16.

Fig. 17.

Fio. 18.

<

der dunkeln Zivischenrdume waren. Es ist nicht nothwendig,
die Details waiter zu verfolgen, nur das Eine bemerke ich noch,
dass wenn der Kadius nur noch urn weniges kleiner ist als d.

58 COLOURED SHADOW PATTERNS FORMED WITH A LENS

man dann nicht helle Punkte auf dunklem Felde, sondern dunkle
Punkte auf hellem Felde hat, wobei die dunkeln Punkte in
ihrer Lage den Mittelpunkten der Scheiben entsprechen, oder
jenen Punkten, welche hell waren, als sich der Schirm im Focus
befand.

" Im Allgemeinen wird das gleiche Ergebniss folgen, auch wenn
die Oeffnungen nicht ganz klein sind aber doch noch in einigem
Verhaltnisse zu den Zwischenraumen stehen. Doch diirften die
verschiedenen Phasen der Erscheinung in diesem Falle sich we-
niger auiFallend darstellen, als in jenem.

"Betrachten wir nun die Wirkung der Ueberlagerung der
verschiedenen Bestandtheile des weissen Lichtes. Anstatt des
einfachen Bildes der Sonne aa' und des durchlocherten Schirmes
ss haben wir eine unendliche Menge von Bildem (Fig. 18),
von welchen die starker gebrochenen, wie o-jcr'j, Sis'b, naher
an der Linse liegen als die weniger gebrochenen. Da der
Schirm, auf welchem das Bild aufgefangen wird, ziemlich nahe
an dem Orte der deutlichsten Erscheinung des durchlocherten
Schirmes aufgestellt ist, so werden die chromatischen Abweich-
ungen des Bildes ss' viel weniger wichtig seyn, als die von aa ,
und sie mogen daher hier der Einfachheit wegen ganzlich iiber-
gangen werden. Wird nun also der Papierschirm aus seiner
fruheren Stellung in Brennpunkte von der Linse hinweggeriickt,
so folgen sich die verschiedenen Phasen der Erscheinung schneller
fur die mehr als fiir die weniger brechbaren Farben, und diess
aus dem Grunde, well der Schirm von Sjs't weiter absteht, als
von SrS'r. Wenn dagegen der Schirm gegen die Linse zu bewegt
wird, so geschehen die Aenderungen fruher fur die weniger als
fur die mehr brechbaren Farben.

" Das gleiche Princip erklart auch die Erscheinung im Auge.
Die Homhaut, Krystall-Linse u. s. w. nehmen den Platz der
Linse ein, die Netzhaut ersetzt den Papierschirm. Die Haupt-
verschiedenheit liegt in der Art, wie der durch jede einzelne
Oeffnung kommende Strahlenbtindel begranzt ist. In dem oben
betrachteten Falle war er begranzt durch oder in Folge der
begranzten Ausdehnung der Sonnenscheibe selbst, in dem gegen-
wartigen Falle geschieht diess in Folge der begranzten Pupille des
Auges. Ferner, anstatt dass der Schirm bewegt wird, wahrend
die Bilder s^s's, s^sV eine feste Lage haben, so ist es hier der

OR RECEIVED DIRECT BY THE EYE. 59

Schirm (die Netzhaut), welcher fest steht, wahrend die Bilder ss
bewegt werden, in Folge der Bewegung des Gegenstandes, der
diese Bilder hervorbringt, eine Bewegung, welche in alien Fallen
von Brechung eine Bewegung des Bildes in derselben Richtung
zur Folge hat. Aber keiner dieser beiden Umstande hat einen
Einfluss auf einen Erklarungsgrund.

"Man halte nun ein Stilck durchlocherte Karte oder durch-
lochertes Papier gegen den Himmel, in der Entfemung der deut-
lichsten Sehweite, und nahere es dann allmahlich dem Auge.
Die wahren Bilder der Karte, welche den verschiedenen Farben
entsprechen, fallen nun hinter die Netzhaut ; da aber die mehr
gebrochenen Bilder vor den weniger gebrochenen liegen, so sind
sie weniger ausserhalb des Brennpunktes. Daher finden die
Veranderungen der Erscheinung schneller statt fur die weniger
als fur die mehr brechbaren Farben. Wenn daher die dunkeln
Zwischenraume in die dunkeln Flecken iiberzugehen beginnen,
so sind sie roth umsaumt, weil die rothen Kreisscheiben auf der
Netzhaut grosser sind als die blauen. Diese Umsaumung durch
Roth, oder vielmehr durch die mehr brechbaren Farben in ihrer
Folge, konnte vielleicht zu wenig lebhaft sejTi, um einen Eindruck
hervorzubringen. Wenn durch das Uebereinanderfallen der Kreise
die dunkeln Flecke in helle Flecke verwandelt wurden, so sind
die letzteren gelblich von dem Vorwalten der weniger brechbaren
Farben, wahrend das allgemeine Feld blaulich ist, von dem
Vorwalten der mehr brechbaren. Wird die durchlocherte Karte,
aus der friiheren Stellung in der Entfemung des deutlichsten
Sehens in eine grossere Entfemung vom Auge geriickt, so liegen
die von den verschiedenen im weissen Licht enthaltenen Farben
herrlihrenden Bilder der Karte vor der Netzhaut, zu ausserst die
mehr brechbaren, und sie sind daher entfernter vom Brenn-
punkte als die weniger brechbaren. Daher sind die Farben der
Flecken und Zwischenraume die entgegengesetzten von denen in
der friiheren Lage.

"Ich bemerke hier, dass das Auge nicht achromatisch ist.
Schon Fraunhofer hat diess in seinen Bemerkungen liber das
Spectrum gezeigt. Sehr auffallend zeigt es eine Erscheinung,
welche ich langst beobachtete, und welche ich spater von Prof
Dove in Berlin in einer Abhandlung angegeben fand, die er mir
sandte. Wenn man einen hellen, wohlbegranzten Gegenstand,

60 CHROMATIC ABERRATION OF THE EYE : HAIDINGER'S BRUSHES.

wie ein Licht oder die Soiinenscheibe, durch ein tiefblaues Glas
oder durch ejne Verbindung mehrerer solcher Glaser betrachtet,
welche keinen anderen sichtbaren Strahlen den Durchgang ge-
statten ausser den aussersten rothen und violetten, so sieht man
die rothen und die violetten Bilder der Gegenstande nicht gleich
deutlich zusammen. Wenn ich die Sonnenscheibe durch eine
Combination dieser Art betrachte, was ohne die geringste Un-
bequemlichkeit ausfilhrbar ist, wenn man nur ein hinlanglich
dunkles Glas oder eine, hinlangliche Anzahl von Glasem an-
wendet, so sehe ich eine wohl begranzte rothe Scheibe und eine
undeutliche violette Scheibe von etwa dem doppelten Uurch-
messer der ersteren. Die letztere kann durch die Anwendung
einer convexen Linse deutlich gemacht warden, aber dann wird
jene andere undeutliche. In der That kann ich entfemte Gegen-
stande deutlich vermittelst der aussersten rothen Strahlen sehen,
bin aber entschieden kurzsichtig in Bezug auf die violetten
Strahlen. Fur mittlere Strahlen, und iibereinstimmend fur ge-
wohnliches Licht, sollte ich daher etwas weniger kurzsichtig seyn,
welches auch der Fall ist."

Einige neuere Ansichten ilber die Natur der Polarisations-
huschel ; von W. Haidinger.

[Pogg. Ann. xcvi, 1855, p. 314.' Mitgetheilt vom Hrn. Verf.
aus d. Sitzungsberichten d. Wiener Akademie {Mai 1854).]

" Ich bin keinesweges durch irgend welche der Erklarungsarten
befriedigt, welche ich bisher liber die Ursache Ihrer Biischel
gesehen habe. Man kann alien, vorztiglich aber der des Hrn.
Jamin, einen Einwurf machen, der unwiderlegbar scheint. Ich
will diesen Gegenstand aber hier nicht weiter verfolgen, weil
ich daran bin demnachst einen Aufsatz dariiber an das Philo-
sophical Magazine zu schicken. Ich bin iiberzeugt, dass die
Erscheinung entweder in oder knapp an der Netzhaut ihren
Sitz hat. Ich werde eine Muthmassung in Bezug auf die Ursache
derselben aufstellen, nach welcher sie von der Art abhangen, wie
die letzten Nervenfasem die Empfindung des Lichtes aufnehmen.
Ich bin uberzeugt, dass die sogenannte Nachahmung der Er-
scheinung durch Uhrglaser oder Linsen, welche Hr. Jamin vor-
schlug, mit derselben nichts zu thun hat.''

On the Theory of the Electric Telegraph.
By Prof. W. Thomson.* (Extract.)

[From the Proceedings of the Royal Society, May 1855 : also Lord Kelvin's
Mathematical and Physical Papers, II, pp. 61-76.]

Extract of Letter from Prof Stokes to Prof. W. Thomson
{dated Nov. 1854).

"In working out for myself various forms of the solution of

the equation ~j- = -i— j under the conditions « = when i = from

x = to a; = oo; v =f(t), when x = from t=Otot=<x),l found
that the solution with a single integral only (and there must
necessarily be this one) was got out most easily thus : —

" Let V he expanded in a definite integral of the form

V = I OT (^, ->( ) sin ax dx,
Jo

which we know is possible.

d^v
" Since v does not vanish when x=Q, -^-^ is not obtained by

2

differentiating under the integral sign, but the term — avx=a must

TT

be supplied f, so that (observing that Vx=o=f(t) by one of the
equations of condition) we have

Hence

-r-T= ■{— at (t)— cc'tjt} sm ax dx.
dx^ Jo {-^ )

dv d'v f^ (d'ST 2 I .

[* This investigation was commenced in consequence of a letter received by the
author from Prof. Stokes dated Oct. 16, 1854, and consists mainly of two letters
addressed to Prof. Stokes, followed by a reply from him as above.]

t According to the method explained in a paper " On the Critical Values of the
Sums of Periodic Series," Gamb. Phil. Trans. Vol. viii, p. 533. [Mathematical
and Physical Papers, G. G. Stokes, Vol. i, p. 236.]

62 ON THE PROPAGATION OF SIGNALS IN ELECTRIC CABLES.

and the second member of the equation being the direct develop-
ment of the first, which is equal to zero, we must have

at TT "^

whence

Jo "■
the inferior limit being an arbitrary function of a. But the other
equation of condition gives

therefore

V = [-V' f [/(*') ae"''*^ sin axdadt'.

r 1 /7r\i -51

But 6-'"''cos6oic^a=- (-1 e *«,

therefore

Jo ao i2Va

e 4a

whence writing t — i!, x, for a, 6, and substituting, we have

r(i-0"^e*l<-'')/(«')<^*'-
Jo

"Your conclusion as to the American wire follows from the
differential equation itself which you have obtained. For the

equation kc-^ = -j—^ shows that two submarine wires will be

similar, provided the squares of the lengths x, measured to simi-
larly situated points, and therefore of course those of the whole
lengths I, vary as the times divided by ck; or the time of any
electrical operation is proportional to kcl^.

dv o^v
"The equation kc j- = -^ — hv gives h x l-^ for the additional

condition of similarity of leakage."

On the Achromatism of a Double Object-glass.

[From the Report of the Brituh Association, Glasgow, 1855, pp. 14-15.]

The general theory of the mode of rendering an object-glass
achromatic by combining a flint-glass with a crown-glass lens, is
well known. The achromatism is never perfect, on account of the
irrationality of dispersion. The defect thence arising cannot
possibly be obviated, except by altering the composition of the
glass. It seemed worthy of consideration whether much improve-
ment might not be effected in this direction; but the problem
which the author proposed for consideration was only the follow-
ing : — Given the kinds of glass to be employed, to find what
ought to be done so as to produce the best effect ; in other words,
to determine the ratio of the focal lengths which gives the nearest
approach to perfect achromatism. Two classes of methods may be
employed for this purpose. In the one, compensations are effected
by trial on a small scale; in the other, the refractive indices of
each kind of glass are determined for certain well-defined objects
in the spectrum, such for example as the principal fixed lines.
The former has this disadvantage, that compensations on a small
scale do not furnish so delicate a test as the performance of a
large object-glass. The observation of refractive indices, on the
other hand, admits of great precision; but it does not immediately
appear what ought to be done with the refractive indices when
they are obtained. After alluding to the method proposed by
Fraunhofer for combining the refractive indices, which, however,
as he himself remarked, did not lead to results in exact accordance
with observation, the author proposed the following as the con-
dition of nearest approach to achromatism : — that the point of the
spectrum for which the focal length of the combination is a
minimum shall be situated at the brightest part, namely, at about
one-third of the interval DE firom the fixed line B towards E.

64 ON THE ACHROMATISM OF A DOUBLE OBJECT-GLASS.

The refractive index of the flint-glass may be regarded as a
function of the refractive index of the crown-glass, and may be
expressed with sufficient accuracy by a series with three terms
only. The three arbitrary constants may be determined by the
values of three refractive indices determined for each kind of glass.
The result is as follows: — Let /Xj, /Xj, fis be the refractive indices
for the crown-glass ; /u./, yLcg', A's' the same for the flint-glass ] fJ-, jj!
the refractive indices of the two glasses for any arbitrary ray ; m
the value of jm for the point at which the focal length is to be
made a minimum; r the ratio of A^u.' to Ayu. to be employed in
the ordinary formula for achromatism. Then having calculated
numerically

'1,2 — ' t '2,3 — I

t^-i -f^l /"3 — M2

we shall have*

^ = n,2+ — - — ~ K3-n,2).

For the value of m it will be sufficient to take

On applying this formula to calculate r for the object-glass for
which Fraunhofer has given both the refractive indices of the com-
ponent glasses and the value of r, which, as observation showed,
gave the best results, and taking in succession various combina-
tions of three lines each out of the seven used by Fraunhofer, the
author found that whenever the combination was judiciously
chosen, the resulting value of r was the same, whatever might
have been the combination, and equal to 1'980, which is precisely
the value determined by Fraunhofer from observation, as giving
the best effect f.

[* In fact the value of the relative dispersion r for the ray of index fi, may be
obtained from r = dfi'jdn, where ii'=A+Bx + Cx^, in which x represents /x - yUj-l

[t The subject of this note is resumed and amplified in a lecture "On the
Principles of the Chemical Correction of Object-glasses " delivered to the Photo-
graphic Society, and published in the Photographic Journal, Feb. 15, 1873 ;
reprinted infra.]

Remakks ox Professor Challiss paper, entitled ' A Theory

OF THE COMPOSmOX OF COLOUES ETC."
[FiX)m the Philosophical Magazine, sn, 1856, pp. 421-5.]

Mt object in the present communication is not to discuss
Professor Challiss theory, but to rectity some statements as to
the experimental facts of the case, as well as one relating to the
extent of some researches of my own. I have, however, on some
points expressed opinions, respecting the justice of which it is
only one who is familiar with certain classes of optical ejrperi-
ments who can feel the confidence that I entertain.

From the paragraph commencing at the foot of page 330, it
is plain that Professor ChalUs has made some confusion between
three perfectly distinct things : Sir David Brewster's controverted
analysis of the solar spectrum by means of absorbing media*;
his discovery of the phenomenon of internal dispersion + : and my
own discovery, that a beam of rays of prismatic purity (whether
belonging to the visible or invisible portion of the spectrum is
indiflerent) may. by their action on certain media, produce light
■which may be decomposed by the prism into portions extending
over a wide range of refirangibility, and having colours answering
to their refi^angibilities *

As to the first, it was asserted by Sir David Brewster that
light of prismatic purity may have its colour changed by passing
through absorbing media. This has nothing to do with "internal"'
or ■■ epipolic" dispersion, or " fluorescence." Glass coloured blue
bv cobalt, for instance, has none of these properties, although it
is one of the media which exhibit most strikingly the phenomena
adduced by Sir David Brewster. Were such a change of colour

* Edinburgh Transactions, Vol. in, p. 133.

+ Edinb. Trans. Vol. STi, p. Ill : and PhiL Mag. Vol. xsxn, 1S4S. p. 401.

J Philosophical Transactions for l?-52, p. 463.

S. IV. 5

66 EEMAEKS ON PROFESSOR CHALLIS'S PAPER, ENTITLED

made out, it would be a point of the utmost importance to
consider in reference to any physical theory of light. But
while none deny that the appearances are as stated by Sir David
Brewster, the inference to be drawn from those appearances
remains open to discussion. Airy*, Helmholtzf, and Bernardj,
by operating in a different manner, have come to the conclusion
that the colour is not changed; and Helmholtz has attributed
the apparent change partly to the mixture of a very small
quantity of stray light, partly to the effects of contrast. Having
been much in the habit of analysing the light transmitted by
coloured solutions, and having repeatedly seen the phenomena
on which Sir David Brewster relies, I may be permitted to
express my belief that the change of colour is only apparent,
being an illusion depending upon contrast, and that this is one
of the cases in which the direct evidence of the senses must
be controlled. Were the change of colour real. Prof Challis's
statement (p. 330), that " experiment has proved that both the
colour and the angle of refraction for a given angle of incidence
depend, the substance being given, only on the value of \," would
cease to be true.

As to the second, the principal phenomenon consists in this :
that when a beam of sunlight, condensed by a lens, is admitted
into certain perfectly clear (i.e. not muddy) media, the path of
the rays is marked by light, of different colours in different cases,
which emanates in all directions. As the real nature of this
remarkable phenomenon was not at the time understood, and
the phenomenon itself was confounded with the effects of mere
suspended particles, it is needless to discuss its possible bearing
on any theory of the sensation of colour under this head.

As to the third, the new light emanating from the media
which possess the property in question is just like any other light
of the same prismatic composition. In its physical properties
it retains no traces of its parentage, and its colour depends simply
upon its new refrangibility, having nothing to do with that of
the producing rays, nor to the circumstance of their belonging

* Phil. May. Vol. xxx, p. 73.
t Poggendorff's Annalen, Vol. lxxxvi, p. 501.

X Report of the Meeting of the British Association at Liverpool in 1854,
2nd Part, p. 5.

"A THEORY OF THE COMPOSITION OF COLOUES ETC." 67

to the visible or the invisible part of the spectrum. Hence,
in speculating on the sensation of colour, this phenomenon may
be set aside as not bearing upon the question. I may re-
mark, however, that with regard to the sensation of colour, an
analogy has often struck me between the retina and a fluorescent
substance, or rather a mixture of three or more fluorescent
substances: but this is only an analogy.

It is not true, as Professor Challis seems to suppose (p. 332),
that absorption is always, or even generally, accompanied by
epipolic dispersion. Among the great variety of coloured metallic
solutions, I have hitherto found that property only in solutions
of salts of sesquioxide of uranium. I make this remark merely
by the way, to prevent misconception: I perfectly agree with
Professor Challis in believing that a ray of definite refrangibility
is uncompounded ; in fact, it was my firm belief in that doctrine
which led me to make out the phenomenon of the change of
refrangibility of light.

The superposition of two coloured glasses or ribbons by no
means gives the effect of the mixture of the two colours. Various
methods of mixing colours are enumerated by Mr Maxwell at
the end of his paper, entitled " Experiments on Colour, etc.," in
the twenty-first volume of the Edinburgh Transactions, p. 275.
The production of white by a mixture of blue and yellow is
by no means confined to prismatic blue and yellow, but takes
place just as well with the colours of coloured bodies. In making
experiments with the spectrum, in order to neutralize, when
possible, a prismatic colour of given intensity by another prismatic
colour, so as to produce white, two points must be attended to :
the place of the second colour in the spectrum must be properly
chosen, and the intensity of the light properly regulated. Hence
any speculations as to the cause of the variations of intensity in
the solar spectrum can have no bearing on the subject before us,
seeing that the relation between the intensities of the mixed
colours necessary for the production of whiteness is a matter
of experimental adjustment.

The reason why the superposition of two coloured bodies does
not give the mixture of the colours is known, and is very simple.
The composition of the light transmitted through a coloured
glass may very conveniently be represented by a curve, in the

5—2

68 EEMARKS ON PEOFESSOR CHALLIS'S PAPER, ENTITLED

manner of Sir John Herschel, in which the abscissa x denotes
refrangibility, measured, suppose, by the distance from the ex-
treme red in some standard spectrum, and the ordinate y denotes
the intensity; so that ydx is the quantity of light between the
refrangibilities x and x + dx, the intensity in the incident light
being taken equal to unity, for simplicity's sake, whatever be the
value of X, as we only care to compare intensities for the same
value of X. Let y, y' be the ordinates in the curves belonging
to two glasses, ys the ordinate belonging to the tint obtained by
superposing the glasses, ym the ordinate belonging to the mixed
tint, as procured, for instance, by a double-image prism, in which
case each of the superposed differently coloured images has half
the brightness of the original. Then y^ = yy', but 2/m = 2 (y + y') ]
and it is easy to see how different may be the curves whose
ordinates are ys, ym respectively. Thus, let the scale of abscissae
be such that the spectrum extends from x = Q to x = ir, and let
y = ^{\ — cosxy, y =\(1 -'rcoaxy. In this case yg = ^^ svo*x,
which vanishes at the extremities, and is a maximum in the
middle; whereas ^^, = 1(1 +cos^a;), which is a maximum at the
two extremities, and a minimum in the middle. In the former
case, the tint would be a sort of green, a pretty full colour; in
the latter, a sort of dilute purple. The colours of two ribbons
may very conveniently be mixed in equal proportion by placing
them side by side, and viewing them through a double-image
achromatic prism; and it will be seen how different the mixed
colour is from that seen on superposing the ribbons and holding
them up to the light.

I cannot agree with Professor Challis, that " the coloured light
of substances, though derived from sunlight, is in fact new
light," except so far as relates to that portion which arises from
fluorescence. But fluorescence is often absent altogether; and
even when it exists, the colour thence arising must in most
cases be but a small fraction of the whole colour observed when
the substance is freely exposed to white light, not viewed under
absorbing media. I think that any one who has been in the
constant habit of analysing by the prism the light transmitted
through clear coloured fluids or solids, and the light transmitted
through or reflected from dyed or other coloured substances,
must be forced to admit, that, setting aside a comparatively

" A THEORY OF THE COMPOSITION OF COLOURS ETC." 69

small number of cases in which the colour observed is referable
to other causes, the colours of natural bodies are due to absorption.
The exceptions are colours due to fluorescence, as in the case of
solutions of quinine, or to regular chromatic reflexion, as in the
case of gold, copper, platino-cyanide of magnesium, murexide, etc.,
not to mention such colours as those of the rainbow, etc., which
result from the general properties of bodies with regard to their
action on light, not from any speciality of the substance by which
the colours happen to be produced. The mode in which I conceive
absorption to operate in occasioning the colours observed in
dyed ribbons, flowers, coloured powders, etc., I have more fully
explained elsewhere*. Now absorption is best studied in clear
solids or solutions, where it is not complicated by irregular
reflexions or refractions. But when such media are studied by
the aid of a pure spectrum, there cannot be a moment's hesitation
that the colour of the transmitted light is due to the abstraction
from the incident white light of some of the component rays, as
explained by Newton. The colour results, not from the light
acted on by the medium, but precisely from the portion left
unaffected. Hence its origin is celestial (supposing the sun to
be the source of the light employed), not terrestrial. But if
the colours of natural bodies arise from absorption, the origin
of those colours must be deemed celestial too. To make the
origin of the green colour of a leaf terrestrial, but that of the
green colour of the light transmitted through an alcoholic solution
of the colouring matter celestial, notwithstanding that the two
greens agree in their very remarkable prismatic composition,
would be needlessly and most capriciously to multiply the causes
of natural phenomena. The light which gives us the sensation
of greenness when we look at a leaf is, I conceive, no more
terrestrial in its origin than the sun's light reflected from a
mirror is terrestrial, as not retaining the direction which it had
in travelling to us from the sun. It is only in the phenomenon
of fluorescence, and the closely allied phenomenon of phosphor-
escence, that the light emitted can be considered as new light
having" a terrestrial origin.

* Philosophical Transactions for 1852, p. 527. [Ante, Vol. iii, p. 356; of. also
p. 396. See also p. 153, infra.]

Supplement to the "Account of Pendulum Experiments

UNDERTAKEN IN THE HaRTON CoLLIEEY. ..". By G. B.

Airy, Esq., Astronomer Koyal.

[From the Philosophical Transactions for 1856. Received Feh. 13,
read Mar. 6, 1856.]

ADDENDUM (pp. 353-355).

On communicating with Professor Stokes, in reference to the
effect of the Earth's rotation and ellipticity in modifying the
numerical results of the Harton Experiment, I was favoured by
that gentleman with an investigation, which, with his permission,
I subjoin as a valuable addition to my own paper.

"I shall suppose the surface of the Earth to be an ellipsoid
of revolution, and will employ the notation made use of in my
paper on Clairaut's Theorem, published in the fourth volume of
the Cambridge and Dublin Mathematical Journal *. In this,

V is the potential of the Earth's mass.

r, are the polar coordinates of any point in or exterior to
the Earth's surface ; r being measured from the centre, and 9 from
the axis of rotation.

a is the equatorial radius.

6 the ellipticity.

(o the angular velocity.

m the ratio of the centrifugal force to gravity at the equator.

E the mass of the Earth.

V the angle between the normal and radius vector at any point
of the surface.

In the following investigation, small quantities of the second
order are neglected, e and m being regarded as small quantities
of the first order.

[* Ante, Vol. ii, p. 104. See also p. 153, infra.]

IXFLUEXCE OF EARTH'S ROTATION' AXD ELLIPTICITY.

If U=V+jr'sm'd,

the differential coefficients

dU

IdU

dr'

r dd

■will give the components of the force along and perpendicular to
the radius vector ; and, g being the force of gravity,

dU . IdU

"which becomes, since v and j-^ are small quantities of the first

order,

dU

Let V be measured along the vertical ; then
or, to the first order.

= cos V -f- — smv --j^,
dv dr r dd

dg _dg _ d'-U
dv dr dn

Let c be the depth of the mine; then if (C/ay be neglected,

we shall have for the vahie of the fraction ^ r-= — = (which

gravity above

I will call F), calculated on the supposition that aU the attracting

mass is internal to both stations,

g dv'

where, after differentiation, r is to be put equal to the radius
vector of the surface, namely a (1 — e cos- 0). Xow the value of V
(Article 5 of the paper referred to) is

^ E (Ee 1 , „\ aV , . 1\

which is true, independently of any particular hypothesis re-
specting the distribution of matter in the interior of the Earth;
so that

fr=--^--2a>-a-j^(^cos^^-3j+2'^sin-^

72 INFLUENCE OF EARTH'S ROTATION AND ELLIPTICITY

dU

and 9 = —

dr

E „fE6 1 , ,
r^ \a 2

') p Las' <^ - g) - oy'r sin^ 6,

, c^o dq

whence — ^ = — f-

dv dr

= ?| _ 12 f^ - 1 „w) ^; fcos^ ^ - Jl + 0,= sin^ ^.
r^ \a 2 / r^\ 3/

Putting now r = a (1 - e cos'' ^), to^ = m - , we find

Cfc

j,-|(l+2«co.'«)-^(,-2)(coe'#-g)-»,|(l-co8'«)

Whence

_1 ^^2

'l + (^_3e)cos^^ + 2e-|

3m

/Sm \ „ . uiii

and therefore

= -(l-2ecos2^ + e + m};

2c
i^=l+- {l-26C0s^^+e + m]

Now the method adopted in the ' Account of Experiments/ etc.,
Article 57, gives

whence

1 dg 2c 2c ,^
--^^ = -=-(1+6 cos^ 0),
g dr r a '

i?'=H--(l + ecos»(?).

ON THE HARTON PIT EXPERIMENT ON GRAVITY. 73

Therefore, if R be the ratio of the value oi F—1 given above,
to i'' — 1 as calculated by the method of the ' Account of Experi-
ments,'

„ l-26COs''^ + e-|-m _ „ „.

-^ = r^ ro = 1-36 cos^ e + e+m.

1 + e cos^

If I be the geocentric latitude of the place, we may in the
small term replace 6 by 90° — Z; and since

COS" 6 = sin'' 1 = ^ {1-cos 21),

we

find

R=l+m-^+ ~coa2l.

Jit Zi

Now

'^-^■^-

0-00346

1

^ ~ 300-8

= 000333

I, for Harton, = 54° 48'

3

= 1 + 0-00346

- 0-00334 =

1-00012,

That R should have been so very nearly equal to unity,
depends upon an accidental numerical relation between the
values of m, e, and I. At the equator, R — \ would have been
as great as 0-00679.

In Article 60 qf the 'Account,' i^- 1 was found =-00012032 ;
whence iJ(.^- 1)= -00012033 ; which only alters the final value
of the mean density in the ratio of 6836 to 6835, giving for
result

6-565.

At the equator, the correction to the deduced value 6-566 would
have been - -077."

On the Polaeization of Diffracted Light.

[From the Philosophical Magazine, XIII, 1857, pp. 159-61 :
also Pogg. Ann. ci, 1857, pp. 154-7.]

On considering the recent interesting experimental researches
of M. Holtzmann on this subject*, I am induced to make the
following remarksf.

In the more common phenomena of diffraction, in which the
angle of diffraction is but small, we know that the character of
the diffracting edge, and the nature of the body by which the
light is obstructed, are matters of indifference. This was made
the object of special experimental investigation by Fresnel ; and
its truth is further confirmed by the wonderful accordance which
he found between the results of the most careful measurements
and the predictions of a theory in which it is assumed that the
office of the opaque body is merely to stop a portion of the inci-
dent light. But when diffraction is produced by a fine grating,
the angle of diffraction is no longer restricted to be small; and
it becomes an open question whether the precise circumstances
of the diffraction may not have to be taken into account, and not
merely the form and dimensions of the apertures through which
the light passes. If so, the problem becomes one of extreme
complexity. In my memoir on the Dynamical Theory of Dif-
fraction, published in the ninth volume of the Cambridge Philo-
sophical Transactions, I investigated the problem on the hypo-
thesis that in diffraction at a large angle, as we know to be the
case in diffraction at a small one, the office of the opaque body is
merely to stop a portion of the incident light. I distinctly stated
this as a hypothesis, and I always regarded it as rather pre-
carious. I was guided by the following consideration. Let AB

[* Pogg. Ann. 1856 ; also Phil. Mag. xiii, 1857, pp. 12.5-9.]
[t Cf. also the author's remarks of date 1882, ante Vol. ii, p. 327, in which
Lorenz's experiments of 1860, confirming his own results, are referred to.]

ON THK POLARIZATION OF DIFFRACTED LIGHT. 75

he the section of a transparent interval by the plane of diffrac-
tion, supposing for simplicity the diffraction to take place in air
or in a homogeneous medium, and not at the confines of two
different media ; let AB = b; let /9 be the angle of diffraction,
and X the wave-length in the medium. Supposing the light to
be incident perpendicularly on the grating, the difference of
phase of the secondary waves which started from A, B, respect-
ively, will be determined by the length of path b sin /3 within the
medium. In experiment this will usually be a considerable
multiple of X. In the line AB take two points, A', B', equidistant
from A, B, respectively, and comprising between them as large a
multiple as possible of Xcosec/8. If we suppose the influence
of the opaque body insensible at the distance AA' or BB' from A
or B, the secondary waves which start from all points in the
interval A'-B' will neutralize each other by interference, so that the
whole effect will be due to the secondary waves which start from
AA' and BB'. Suppose the angle yS to belong to the brightest
part of a " spectrum of the first class " (Fraunhofer) ; then
AA' + BB' = ^X cosec /3, X referring to mean rays, so that A A' or
BB' is only equal to ^X cosec /3. If, for example, /3 = 30°, AA'
is only equal to ^X. At such very small distances it maj' well
be doubted whether the influence of the opaque body may not
have to be taken into account.

When diffraction takes place at the confines of two different
media, suppose air and glass, the problem is still further com-
plicated. We may, however, apply the theory to which reference
has been made on the two extreme suppositions, first, that the
diffraction takes place wholly in the first, secondly, that it takes
place wholly in the second medium. The results of my own ex-
periments were very fairly represented by theory, the vibrations
being supposed perpendicular to the plane of polarization, pro-
vided the diffraction be conceived to take place in the first
medium, or in other words, just before the light reaches the
grating; but they would not at all fit the hypothesis of vibra-
tions parallel to the plane of polarization. I put forth some
considerations, founded on probable reasoning, to show that the
supposition of diffraction taking place in the first medium was
in accordance with the physical circumstances of the case. So
decided was the result obtained, that it seemed to me a strong

76 OK THE POLARIZATION OF DIFFRACTED LIGHT.

argument in favour of the hypothesis that the vibrations are
perpendicular to the plane of polarization, though I still felt the
necessity of repeating the experiments under varied circum-
stances.

But since the appearance of M. Holtzmann's researches the
state of the question is changed. I have no reason to doubt the
correctness of his results, while on the other hand the result I
myself obtained was far too decided to be passed by. The con-
clusion which, in the present state of the question, seems to me
most probable is, that the polarization of light diffracted at a
large angle is, in fact, influenced by the nature of the diffracting
body. The subject demands a much more extensive experi-
mental investigation, in which the circumstances of diffraction
shall be varied as much as possible. I hope to have leisure to
undertake such an investigation : meanwhile it would be prema-
ture to offer any decided opinion. It seems to me, however,
worthy of attentive consideration, whether a glass grating may
not offer a fairer experiment for the decision of the question as
to the direction of vibration in polarized light than a smoke
grating, inasmuch as in the former we have to do with an unin-
terrupted medium, glass, the surface of which is merely rendered
irregular, whereas in the latter the problem is complicated by
the existence of two distinct media, glass and so9t, placed alter-
nately. I call the layer of soot a medium, for though no light
can pass through any sensible thickness of it, we must not con-
clude from that that it is without influence on the light which
passes excessively close to it.

I have not mentioned the effect of oblique refraction in the
experiments of M. Holtzmann, because if it were allowed for,
the character of the results obtained would remain unchanged,
the magnitude of the observed effect would only be somewhat
diminished.

Ox THE DiSCOXTIXUITY OF ArBITEAEY CONSTANTS WHICH
APPEAR IX DiVEEGEXT DEVELOPMENTS.

[From the Transactions of the Cambridge Philosophical Society, Vol. x,
pp. 106-128. Resvd May 11, 1857.]

[Abstract. Proc. Camh. Phil. Soc. Vol. I, pp. 181-2.]

In a paper '' On the Numerical Calculation of a class of Definite Integrals
and Infinite Series" printed in the ninth volume of the Cambridge Philo-
sophical Transactions, the author succeeded in putting the integral

I

cos — {vfi — mw) dw

vmder a form which admits of extremely easy numerical calculation when
m is large, whether positive or negative. The integral is obtained in the first
instance under the form of circular functions for m positive, or an exponential
for »i negative, multiplied by series according to descending powers of m.
These series, which are at first convergent, though ultimately divergent, have
arbitrary constants as coefficients, the determination of which is all that
remains to complete the process. From the nature of the series, which are
applicable only when m is large, or when it is an imaginary quantity with
a large modulus, the passage from a large positive to a large negative value
of m cannot be made through zero, but only by making m imaginary and
altering its amplitude by tt. The author succeeded in determining directly
the arbitrary constants for m positive, bvit not for m negative. It was fovmd
that if, in the analytical expression apphcable in the case of m positive, — m
were written for m, the result would become correct on throwing away the
part involving an exponential with a positive index. There was nothing
however to show <i priori that this process was legitimate, nor, if it were, at
what value of the amplitude of m a change in the analytical expression ought
to be made, although the occurrence of radicals in the descending and
ultimately divergent series, which did not occur in ascending convergent
series by which the function might always be expressed, showed that some
change analogous to the change of sign of a radical ought to be made in
passing through some values of the amplitude of the variable m. The method
which the author applied to this function is of very general application, but
is subject throughout to the same difficulty.

In the present paper the author has resiuned the subject, and has pointed
out the character by which the liability to discontinviity in the arbitrary
constants may be ascertained, which consists in this, that the terms of an
associated divergent series come to be regularly positive. It is thus found
that, notwithstanding the discontinuity, the complete integrals, by means of
divergent series, of the difiFerential equations which the functions treated of

78 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

satisfy, are expressed in such a manner as to involve only as many unknown
constants as correspond to the degree of the equation.

Divergent series are usually divided into two classes, according as the
terms are regularly positive, or alternately positive and negative. But
according to the view here taken, series of the former kind appear as
singularities of the general case of divergent series proceeding according to
powers of an imaginary variable, as indeterminate forms in passing through
which a discontinuity of analytical expression takes place, analogous to
a change of sign of a radical.

In a paper " On the Numerical Calculation of a class of Definite
Integrals and Infinite Series," printed in the ninth volume of the
Transactions of this Society, I succeeded in developing the integral

f COS n (w' — Tnw) dw in a form which admits of extremely easy

numerical calculation when m is large, whether positive or negative,
or even moderately large. The method there followed is of very
general application to a class of functions which frequently occur
in physical problems. Some other examples of its use are given
in the same paper; and I was enabled by the application of it
to solve the problem of the motion of the fluid surrounding a
pendulum of the form of a long cylinder, when the internal
friction of the fluid is taken into account*.

These functions admit of expansion, according to ascending
powers of the variables, in series which are always convergent, and
which may be regarded as defining the functions for all values of
the variable real or imaginary, though the actual numerical calcu-
lation would involve a labour increasing indefinitely with the
magnitude of the variable. They satisfy certain linear differential
equations, which indeed frequently are what present themselves in
the first instance, the series, multiplied by arbitrary constants,
being merely their integrals. In my former paper, to which the
present may be regarded as a supplement, I have employed these
equations to obtain integrals in the form of descending series
multiplied by exponentials. These integrals, when once the
arbitrary constants are determined, are exceedingly convenient for
numerical calculation when the variable is large, notwithstanding
that the series involved in them, though at first rapidly convergent,
become ultimately rapidly divergent.

* Cambridge Philosophical Transactions, Vol. ix. Part ii. [Ante, Vol. ii, p. 329.
Further papers on this subject of dates 1868, 1889, 1902, are reprinted infra.}

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 79

The determination of the arbitrary constants may be effected
in two ways, numerically or analytically. In the former, it will be
sufficient to calculate the function for one or more values of the
variable from the ascending and descending series separately, and
equate the results. This method has the advantage of being
generally applicable, but is wholly devoid of elegance. It is better,
when possible, to determine analytically the relations between the
arbitrary constants in the ascending and descending series. In the
examples to which I have applied the method, with one exception,
this was effected, so far as was necessary for the physical problem,
by means of a definite integral, which either was what presented
itself in the first instance, or was employed as one form of the
integral of the differential equation, and in either case formed a
link of connexion between the ascending and the descending series.
The exception occurs in the case of Mr Airy's integral for m nega-
tive. I succeeded in determining the arbitrary constants in the
divergent series for m positive ; but though I was able to obtain
the correct result for ni negative, I had to profess myself (p. 177,
[343]) unable to give a satisfactory demonstration of it.

But though the arbitrary constants which occur as coefficients
of the divergent series may be completely determined for real
values of the variable, or even for imaginary values with their
amplitudes lying between -restricted limits, something yet remains
to be done in order to render the expression by means of divergent
series analytically perfect. I have already remarked in the former
paper (p. 176, [342]) that inasmuch as the descending series
contain radicals which do not appear in the ascending series, we
may see, a priori, that the arbitrary constants must be discon-
tinuous. But it is not enough to know that they must be
discontinuous; we must also know where the discontinuity takes
place, and to what the constants change. Then, and not till then,
will the expressions by descending series be complete, inasmuch as
we shall be able to use them for all values of the amplitude of the
variable.

I have lately resumed this subject, and I have now succeeded
in ascertaining the character by which the liability to discontinuity
in these arbitrary constants may be ascertained. I may mention
at once that it consists in this ; that an associated divergent series
comes to have all its terms regularly positive. The expression

80 ON THE DISCONTINUITY OF AEBITRARY CONSTANTS

becomes thereby to a certain extent illusory; and thus it is that
analysis gets over the apparent paradox of famishing a discon-
tinuous expression for a continuous function. It will be found
that the expressions by divergent series will thus acquire all the
requisite generality, and that though applied without any re-
striction as to the amplitude of the variable they will contain only
as many unknown constants as correspond to the degree of the
differential equation. The determination, among other things, of
the constants in the development of Mr Airy's integral will thus
be rendered complete *.

1. Before proceeding to more difficult examples, it will be
well to consider a comparatively simple function, which has been
already much discussed. As my object in treating this function is
to facilitate the comprehension of methods applicable to functions
of much greater complexity, I shall not take the shortest course,
but that which seems best adapted to serve as an introduction to
what is to follow.

Consider the integral

M=2 e-^ sm laccdx = — - \r—^ + ^ .' (1).

Jo -I ^.oo.4.5

[* The considerations developed in this memoir form the complement of
Section m of the memoir " On the Critical Values of the Sums of Periodic Series "
{Camh. Phil. Trans. 1847 ; ante Vol. i, p. 279), in which the cause of discontinuity
in the values of series and integrals was traced to infinitely alow convergence at the
place of the sudden change. Here the converse phenomenon of discontinuous
changes in the constants multiplying the semi-convergent series, which together
represent a continuous function, is traced to the same kind of origin, in the
domain however of the complex variable, — namely (p. 103) the multiplier of a series
can change disoontinuously only in crossing places at which one of the other series
involved in the expression of the function loses its semi-convergence.

The formulae for the Bessel functions, employed here (pp. 100 — 103) as an
illustration, were again obtained eleven years later by H. Hankel, in his memoir On
the cylinder functions. Math. Annalen, Vol. i, Deo. 1868, p. 498, with the same
explanation of the discontinuities of their constants. At that time the method of
complex contour integration, developed by Eiemann in his fundamental memoir
on the hypergeometric series, a few months before the date of the present paper,
afforded the natural means of procedure. It is of great interest to trace the
approaches (cf. p. 91) toward that general method in analysis, also probably the
result of physical considerations, to which the solution of the present most intricate
problem of discontinuous representation had independently given rise. The pro-
cedure of the present paper also applies when no expression of the form of a
complex integral is available.]

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 81

The integral and the series are both convergent for all values
of a, and either of them completely defines u for all values real or
imaginary of a. We easily find from either the integral or the
series

(I'ii

Ta^^-- = ^ (2)-

This equation gives, if we observe that m = when a = 0,

« = 2.-«f%-^a = 2e-«i + j^3 + ^-^+^-^ + ...}...(3).

This integral or series like the former gives a determinate and
unique value to u for any assigned value of a real or imaginary.
Both series, however, though ultimately convergent, begin by
diverging rapidly when the modulus of a is large. For the sake of
brevity I shall hereafter speak of an imaginary quantity simply as
large or small when it is meant that its modulus is large or small.

2. In order to obtain m in a form convenient for calculation
when a is large, let us seek to express u by means of a descending
series. We see from (2) that when the real part of a^ is positive,
the most important terms of the equation are 2au and 2, and the
leading term of the development is a~\ Assuming a series with
arbitrary indices and coefficients, and determining them so as to
satisfy the equation, we readily find

11 1.3

" ~ a "^ 2a» "•" 2V '^ '"

This series can be only a particular integral of (2), since it

wants an arbitrary constant. To complete the integral we must

add the complete integral of

du „ .

-=- + 2au = 0,
da

whence we get for the complete integral of (2)

„ . 1 1 1.3 ,1.3.5 ,.-

"=^^"''^a + 2a^+2^ + ^W- + ^^^^

This expression might have been got at once from (3) by

integration by parts. It remains to determine the arbitrary

constant C.

3. The expression (1) or (3) shows that u is an odd function
of a, changing sign with a. But according to (4) u is expressed as
the sum of two functions, the first even, the second odd, unless

8. IV. 6

82 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

C = 0, in which case the even function disappears. But since, as
we shall presently see, the value of C is not zero, it must change
sign with a. Let

a = p (cos + V — 1 sin 6).

Since in the application of the series (^4) it is supposed that p is
large, we must suppose a to change sign by a variation of 6, which
must be increased or diminished (suppose increased) by vr. Hence,
if we knew what G was for a range tt of 9, suppose from ^ = « to
= « + TT, we should know at once what it was from = a + tt to
6 = 0.+ 27r, which would be sufficient for our purpose, since we may
always suppose the amplitude of a included in the range a to
a + Stt, by adding, if need be, a positive or negative multiple of 27r,
which as appears from (1) or (3) makes no difference in the value
of M.

4. When p is large the series (4) is at first rapidly convergent,
but be p ever so great it ends by diverging with increasing rapidity.
Nevertheless it may be employed in calculation provided we do
not push the series too far, but stop before the terms get large
again. To show in a general way the legitimacy of this, we may
observe that if we stop with the term

1.3.5...(2{-1)

the value of u so obtained will satisfy exactly, not (2), but the
differential equation

da 2W+2 ^ '

Let Mo be the true value of u for a large value of a^ of a, and
suppose that we pass from «„ to another large value of a keeping
the modulus of a large all the while. Since u ought to satisfy (2),
we ought to have

w = Mj + 2e"

ra

' e^da,

whereas since our approximate expression for u actually satisfies
(5) we actually have, putting Ai for the last term.

u=u„ + e-'''r{2-Ai)e^'da (6).

J a.

If a be very large, and in using the series (4) we stop about
where the moduli of the terms are smallest, the modulus of A{ will

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 83

be very small. Hence in general Ai may be neglected in com-
parison with (2), and we may use the expression (4), though we
stop after i + 1 terms of the series, as a near approximation to u.

5. But to this there is an important restriction, to understand
which more readily it will be convenient to suppose the integration
from tto to a performed, first by putting

da = (cos 6 + 'J — 1 sin 6) dp,
and integrating from po to p, 6 remaining equal to d^, and then

da = p{ — sin 9 + "J — 1 cos 6) dO,
and integrating from do to 0, p remaining unchanged. This is
allowable, since u is a finite, continuous, and determinate function
of a, and therefore the mode in which p and 6 vary when a passes
from its initial value a,, to its final value a is a matter of in-
difference. The modulus of e"^ will depend on the real part
p^ cos 26 of the index. Now should cos 29 become a maximum
within the limits of integration, we can no longer neglect Ai in the
integration. For however great may be the value previously
assigned to i, the quantity p-^^-^e^'cosse ^jj become, for values of 9
comprised within the limits of integration, infinitely great, when p
is infinitely increased, compared with the value of eP^™s2« ^^ either
Umit. And though the modulus of the quantity 26"' under the
integral sign will become far greater still, inasmuch as it does not
contain the factor p~^~^, yet as the mutual destruction of positive
and negative parts may take place qiiite differently in the two

integrals l2e"^(Za and iAg^^da, we can conclude nothing as to

their relative importance.

6. Now cos 29 will continually increase or decrease from one
limit to the other, or else will become a maximum, according as
the two limits 9^ and 9 lie in the same interval to tt or tt to 2ir,
or else lie one in one of the two intervals and the other in the
other. Hence we may employ the expression (4), with an in-
variable value of C yet to be determined, so long &s Q < < ir, and
we may employ the expression obtained by writing C for G so long
as TT < ^ < 27r, but we must not pass from one interval to the
other, retaining the same expression. Now we have seen (Art. 3)
that the constant changes sign when 9 is increased by tt, and
therefore C = —G. And since u is unchanged when 9 is increased
by any multiple of 27r, we readily see that in order to make the

6—2

84 ON THE DISCONTINUITY OF ARBITEAEY CONSTANTS

expression (4) generally applicable, it will be sufficient to change
the sign of the constant whenever 6 passes through zero or a
multiple of tt.

7. We may arrive at the same conclusion in another way,
which will be of more general or at least easier application, as not
involving the integration of the differential equation.

The modulus of the general term (Art. 4) of the series (4),
expressed by means of the function V, is

r(^ + i)

Suppose i very large. Employing the formula

r (« + 1 ) = ^l-irx ( - ) , nearly, when x is large.

observing that V (|^) = tt^, and calling the modulus yiti, we find

/^i = 2^ (i - \y- e-'+ip-^-\

which, since (i + cf = iV, nearly, becomes

IMi = 2H^e-y^'-^ (7).

We easily get, either from this expression or from the general
term,

't^ = ^. nearly (8).

Hence when p is large the ratio of consecutive moduli becomes
very nearly equal to unity for a great number of terms together,
about where the modulus is a minimum. To find approximately
the minimum modulus fj,, we must put i = p' in (7), which gives

/i = 25/3-16-"' (9).

If we knew precisely at what term it would be best to stop,
the expression for /j, would be a measure of the uncertainty to
which we were liable in using the series (4) directly, that is, with-
out any transformation. For although it is clear that we must
stop somewhere about the term with a minimum modulus, in order
that the differential equation (5) which our function really satisfies
may be as good an approximation as can be had to the true
differential equation (2), the number of terms comprised in this
about will increase with i, the order of the term of minimum
modulus. If we suppose that we are uncertain to the extent of

WHICH APPEAH IN DIVERGENT DEVELOPMENTS. 85

n terms, the stim of the moduli of these n nearly equal terms
will be

nearly. It seems as if n must increase with i, but not so fast as i.
If we suppose that it is of the form Mi or kp, the sum of the
n terms will be a quantity of the order e~''^ But even if n
increased as any power p of i, however great, still the sum of
the n terms would be a quantity of the order p^~^e~i^, which when
p was infinitely increased would become infinitely small in com-
parison with the modulus e"''^'^^ of the term multiplied by G
in (4), provided ' had any given value differing from zero or a
multiple of tt. Hence if d have any value Ijring between a and
TT — a, or else between tt + a and 27r — a, where a is a small
positive quantity which in the end may be made as small as we
please, the quantity G in (4) cannot pass from one of its values to
another without rendering the function u discontinuous, which it
is not. But when ^ = or = tt, the term Ce^'^ becomes merged in
the vagueness with which, in this case, the divergent series defines
the function. Hence we arrive in a way quite different from that
of Art. 5 at the conclusions enunciated in Art. 6.

8. Nor is this all. When the terms of a regular series are
alternately positive and negative, the series may be converted by
the formulae of finite differences into others which converge rapidly.
In the present case the terms are not simply positive and negative
alternately, except when 6 is an odd multiple of ^tt, but the same
methods will apply with the proper modification. Suppose that
we sum the series (4) directly as far as terms of the order i — 1
inclusive. Omitting the common factor e~<'^+"*^-', which may be
restored in the end, we have for the rest of the series

If we denote by Z) or 1 + A the operation of passing from fii to
fi4+i, and separate symbols of operation, this becomes

(1 + e-^«^'D + e-*«^i)^ + ...)h,

or {!-(!+ A) e-29^^}->i.

/^-eWrr

Now 1 - e-^«v/-i = 1 - cos 2^ + V - 1 sin 20 = 2 sin 0e\a /

which reduces the expression to

(2 sin 0)-'e("' ^^''^\l - (2 sin 0)-ie~V2 +'')^-i Aj-^i,

86 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

or, putting q for (2 sin 6)~'^, to

Now if p be very large, and ^f belong to the part of the series
where the moduli of consecutive terms are nearly equal, the
successive differences A/if, A^;, . . . will decrease with great rapidity.
Hence if 6 have any given value different from zero or a multiple
of TT, by taking p sufficiently great, we may transform the series
about where it ceases to converge into one which is at first rapidly
convergent, and thus a quantity which may be taken as a measure
of the remaining uncertainty will become incomparably smaller
even than /x, much more, incomparably smaller than the modulus
of e~'^. But if ^ = or = TT, the above transformation fails, since q
becomes infinite. In this case if we want to calculate u closer
than to admit of the uncertainty to which we are liable, knowing
only that we must stop somewhere about the place where the series -
begins to diverge after having been convergent, we must have
recourse to the ascending series (1) or (3), or to some perfectly
distinct method. The usual method by which '2ux is made to
depend on Ju^dx would evidently fail, in consequence of the
divergence of the integral.

9. In applying practically the transformation of the last
article to the summation of the series (4), it would not usually,
when p was very large, be necessary to go as far as the part of the
series where the moduli of consecutive terms are nearly equal. It
would be sufficient to deduct I, 21 ... from the logarithms of /ii+i,
fjbi^.,..., where I is nearly equal to the mean increment of the
logarithms at that part of the series, to associate the factor /
whose logarithm is I with the symbol D, and take the differences of
the numbers

Mi> f~'^l^i+i, J~''l^i+i, etc.
However, my object leads me to consider, not the actual summation
of the series, but the theoretical possibility of summation, and
consequent interpretation of the equation (4).

10. The mode of discontinuity of the constant C having been
now ascertained, nothing more remains except to determine that

constant, which is done at once. Writing V — la for a in (4) after
having put for u its first expression in (3), we have

2^ re-'^da = -'^^l Ce"' - - + ^,- ... ,
J a 2a'

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 87

whence, putting a == oo , we have (7 = V — Itt^ Hence we get for
the general expression for C in (4),

.(10),

G= \J - Itt*, when < ^ < tt, 1
(7 = — V — Itt* when irKd <2it\]
and therefore from (3) and (4)

Jo ~ a 2a' 2^a^^ 2W ^ '''

the sign being + or — according as 6, the amplitude of a, is com-
prised within the limits and tt, or tt and 27r.

Writing aV — 1 for a in (11), which comes to altering the
origin of 6 by ^tt, we find

Jo -a 2a' 2^0" 2'a^ ^ ''

the sign being + or — according as the amplitude of a lies within
the limits — ^-n- and ^tt, or ^ir and f tt. It is worthy of remark
that in this expression the transcendental quantity tt^ appears as
a true radical, admitting of the double sign.

Two cases of the integral / ef^da occur in actual investiga-

rt

tions, namely when 6 = \7r, when the integral leads to I e*''dt,

Jo
which occurs in the theory of probabilities, and when 6 = ^tt, when

it leads to Fresnel's integrals I cos {^tts") ds and I sin (Itts^) ds.

Jo Jo

In the latter case the expression (11) is equivalent to the develop-
ment of these integrals which has been given by M. Cauchy.

11. If in equation (11) we put a= p (cos ^ + V — 1 sin 6^),
where ^ is a small positive quantity, and after equating the real
parts of both sides of the equation make 6 vanish, we find, which-
ever sign be taken,

1 1 1.3 1.3.5 „ 2 f ,j /.ox

p+2p^ + 2y+-¥pr+ = 2e-^'j^e^'dp ...(13).

88 ON THE DISCONTINUITY OF AEBITEARY CONSTANTS

The expression which appears on the second side of this
equation may be regarded as a singular value of the sum of the
series

1 1 1.3 1.3.5 ....

a+2a»+2W+^^ + ('*^'

a series which when vanishes takes the form of the first member
of the equation. The equivalent of the series for general values
of the variable is given, not by (13), but by (11). It may be
remarked that the singular value is the mean of the general values
for two infinitely small values of 0, one positive and the other
negative.

These results, to which we are led by analysis, may be com-
pared with the known theory of periodic series. If /(*') be a finite
function of cc, the value of which changes abruptly from a to b as
cc increases through the value c, a quantity lying between and tt,
and /(«) be expanded between the limits and tt in a series of
sines of multiples of x, and if <f) (n, a;) be the sum of n terms of the
series, the value of cj) (n, x) for an infinitely large value of n and a
value of X infinitely near to c is indeterminate, like that of the
fraction

{x+yy-^x-y

{x-yj+x + y'

which takes the form § when x and y vanish, but of which the
limiting value is wholly indeterminate if x and y are independent.
We may enquire, if we please, what is the limit of the fraction
when X first vanishes and then y, or the limit when y first vanishes
and then x, for each of these has a perfectly clear and determinate
signification. In the former case we have, calling the fraction

lim.j,_„ lim.^_„ -f {x, y) = lim.y.„ ^^— | = - 1

in the latter

lim.a,=„ lim.y_„ i|r {x, y) = ^vca.x^,-^^- = 1.

00 T~ oc

So in the case of the periodic series if we denote by f a small
positive quantity

lim.f_„ lim,„_c„ 4>(n,c-^) = lim.f.„ f(c - ^) = a,

lim.j.o lim.„^,^, </> (n, c + ^) = lim.f_o/(c + ^) = 6 ;

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 89

but we know that

lim.„„„ lim.f_(| <j) (n, c ± ^) = lim.„_„ <f> (n, c) = ^(a + b).

Similarly in the case of the series (14) if we denote its sum by
')((a) = -sT (p, 6), and use the term limit in an extended sense, so as
to understand by lim.p=oo F{p) a function of p to which -F(p) may
be regarded as equal when p is large enough, and if we suppose d
to be a small positive quantity, we have from (11)

lim.9=„lim.p_„ -sr (p, 6) = lim.s^o {26-"' ["e«Va - V^ttV"'}

.0

= 26-"'' r eP'dp - V -" 1 -n-h-"' ;
Jo

fa

lim.8=o lim.p_„ ot (p, -6)= lim.e„„ 126""' e'^'^da + V - 1 ttV}

Jo

= 26-'-' l' ef^dp + V^lTr^e-"',
Jo

whereas equation (13) may be expressed by

lim.p_«lim.9„„ ra- {p, ±6) = lim.p=„o % (/>) = 2e-''' ( e^'cip.

There is however this difference between the two cases, that
in the case of the periodic series the series whose general term
is ^<f)(n, c) is convergent, and may be actually summed to any
assigned degree of accuracy, whereas the series (13), though at
first convergent, is ultimately divergent ; and though we know that
we must stop somewhere about the least term, that alone does not
enable us to find the sum, except subject to an uncertainty com-
parable with e"*""- Unless therefore it be possible to apply to the
series (13) some transformation rendering it capable of summation
to a degree of accuracy incomparably superior to this, the equation
(13) must be regarded as a mere symbolical result. We might
indeed define the sum of the ultimately divergent series (13) to
mean the sum taken to as many terms as should make the
equation (13) true, and express that condition in a manner which
would not require the quantity taken to denote the number of
terms to be integral ; but then equation (13) would become a mere
truism. However I shall not pursue this subject further, as these
singular values of divergent series appear to be merely matters oi
curiosity.

90 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

12. In order still further to illustrate the subject, before going
on to the actual application of the principles here established, let
us consider the function defined by the equation

■, 1 1-1 , 1.1.3 3 ,-_,

^=^+r-2T4*+ 27476 '^ ^^^)-

Suppose that we have to deal with such values only of the
imaginary variable x as have their moduli less than unity. For
such values the series (15) is convergent, and the equation (15)
assigns a determinate and unique value to u. Now we happen to
know that the series is the development of (1 + xf. But this
function admits of one or other of the following developments
according to descending powers of x : — ■

^ = ^.i + |^-i_^_^^-i + ___^H_ (16),

« = _,i_i^-. + Li2^-t_Ll^^-a + (17).

Let x = p (cos ^ + V — 1 sin 6), and let x^ denote that square
root of X which has ^6 for its amplitude. Although the series (16),
(17) are divergent when p < 1, they may in general, for a given
value of 6, be employed in actual numerical calculation, by
subjecting them to the transformation of Art. 8, provided p do not
differ too much from 1. The greater be the accuracy required,
6 being given, the less must p differ from 1 if we would employ
the series (16) or (17) in place of (15). It remains to be found
which of these series must be taken.

If 6 lie between (2i— l)7r + a and (2i + l)7r — a, where i is
any positive or negative integer or zero, and a a small positive
quantity which in the end may be made as small as we please,
either series (16) or (17) may by the method of Art. 8 be converted
into another, which is at first sufficiently convergent to give u
with a sufficient degree of accuracy by employing a finite number
only of terms. If m terms be summed directly, and in the formula
of Art. 8 the ?i"' difference be the last which yields significant
figures, the number of terms actually employed in some way or
other in the summation will be 'in + n + \. And in this case we
cannot pass from one to the other of the two series (16), (17)
without rendering u discontinuous. But when 6 passes through

WHICH APPEAR IX DIVERGENT DEVELOPMENTS 91

an odd multiple of tt "^ve may have to pass from one of the two
series to the other. Xow when is increased by 'Iir the series
(16) or (17) changes sign, whereas (15) remains unchanged.
Therefore in calculating u for two values of 6 differing by 2Tr we
must employ the two series (16) and (17), one in each case.
Hence we must employ one of the series from 0= — tt to 6 = ir,
the other from d = Tr to 6 = Stt, and so on : and therefore if we
knew which series to take for some one value of x everything
would be determined.

Xow when p = 1 the series (15) becomes identical with (16)
when 6 has the particular value 0. Hence (16) and not (17) gives
the true value of u when — tt < ^ < tt.

13. Let p, 6 be the polar co-ordinates of a point in a plane,
the origin, G a circle described round with radius unity, S the
point determined by x=—'l, that is, by p=l, ^=7r. To each
value of X corresponds a point in the plane ; and the restriction
laid down as to the moduli of x confines our attention t() points
within the circle, to each of which corresponds a determinate value
of u. If Pf, be any point in the plane, either within the circle or
not, and a moveable point P start from Pj, and after making any
circuit, without passing through 5^. return Xo P^ again, the functiijn
(1 + x)^ will regain its primitive value u„, or else become equal to
— ((,, according as the circuit excludes or includes the point »S,
which for the present purpose may be called a singular point.
Suppose that we wished to tabulate u, using when possible the
divergent series (16) in place of the convergent series (15). For a
given value of 0, in commencing with small values of p we should
have to begin with the series (15), and when p became large
enough we might have recourse to (16). Let OP be the smallest
value of p for which the series (16) may be employed: for which,
suppose, it will give u correctly to a certain number of decimal
places. The length OP vrill depend upon 0, and the locus of P
will be some curve, sjTnmetrical with respect to the diameter
through S. As increases the curve will gradually approach the
circle G, which it will run into at the point 5. For points IjTng
between the curve and the circle we may employ the series (16),
but we cannot, keeping within this space, make pass through the
value TT. The series (16), (17) are convergent, and their sums
vary continuously with x, when p >1; and if we employed the

92 ON THE DISCONTINUITY OF AEBITRARY CONSTANTS

same series (16) for the calculation of u for values of cc having
amplitudes tt — /3, ir + ^, corresponding to points P, P', we should
get for the value of m at P' that into which the value of w at P
passes continuously when we travel from P to P' outside the point
S, which as we have seen is minus the true value, the latter being
defined to be that into which the value of m at P passes con-
tinuously when we travel from P to P' inside the point S.

In the case of the simple function at present under considera-
tion, it would be an arbitrary restriction to confine our attention to
values of a: having moduli less than unity, nor would there be any
advantage in using the divergent series (16) rather than the
convergent series (15). But in the example first considered we
have to deal with a function which has a perfectly determinate
and unique value for all values of the variable a, and there is the
greatest possible advantage in employing the descending series for
large values of p, though it is ultimately divergent. In the case
of this function there are (to use the same geometrical illustration
as before) as it were two singular points at infinity, corresponding
respectively to ^ = and 6 = Tr.

14. The principles which are to guide us having been now
laid down, there will be no difficulty in applying them to other
cases, in which their real utility will be perceived. I will now
take Mr Airy's integral, or rather the differential equation to
which it leads, the treatment of which will exemplify the subject
still better. This equation, which is No. 11 of my paper " On the
Numerical Calculation, etc.," becomes on writing u for U, — 3x for n

^-9-«« = (18).

The complete integral of this equation in ascending series,
obtained in the usual way, is

"^ ^ 2.3 • 2.3.5.6"'' 2T3T5T6T8T9"'""
+ ^ ■>"' + 3 . 4 ■'' 3747677 + 37476777971 + • •

.(19).

These series are always convergent, and for any value of x real or
imaginary assign a determinate and unique value to w.

WHICH APPEAR IN DIVERGENT DEVELOPMENTS.

9a

The integral in a form adapted for calculation when a; is large,
obtained by the method of my former paper, is

1-

1

5 1.5.7.11

" 1.2. 144V

1 +

1 . 144a;'
1.5

1.5.7.11

+ +

1.5.7.11.13.17
1.2.3. 144'a;*
1.5.7.11.13.17

- + ..

1 . 144r' 1.2. 144V 1.2.3. 144^a;'^

.(20).

The constants C, D must however be discontinuous, since
otherwise the value of u determined by this equation would not
recur, as it ought, when the amplitude of x is increased by 27r.
We have now first to ascertain the mode of discontinuity of these
constants, secondly, to find the two linear relations which connect.
A, B with G, D.

Let the equation (20) be denoted for shortness by

u=Gx-^Mx)+Dx-^f^{x) (21);

and let f{x), when we care only to express its dependence on the-
amplitude of x, be denoted by F{6). We may notice that

F,{e + l'rr) = F,{e); F,{0 +^7r) = F,(e) (22).

15. In equation (21), let that term in which the real part of
the index of the exponential is
positive be called the superior,
and the other the inferior
term. In order to represent
to the eye the existence and
progress of the functions /i (a;),
/2(x) for different values of 6,
draw a circle with any radius,
and along a radius vector in-
clined to the prime radius at
the variable angle 6 take two
distances, measured respec-
tively outwards and inwards
from the circumference of the

circle, proportional to the real part of the index of the exponential
in the superior and inferior terms, alone being supposed to vary,.

Fig. 1.

94 ON THE DISCONTINUITY OF ARBITEARY CONSTANTS

or in other words proportional to cos f ft For greater convenience
suppose these distances moderately small compared with the
radius. Consider first the function F^{6) alone. The curve will
evidently have the form represented in the figure, cutting the
circle at intervals of 120°, and running into itself after two
complete revolutions. The equations (22) show that the curve
corresponding to Fzi^) is already traced, since F2 (0) = F-^(6 + lir).
If now we conceive the curve marked with the proper values of the
constants C, D, it will serve to represent the complete integral of
equation (18).

In marking the curve we may either assume the amplitude 6
of X to lie in the interval to Itt, and determine the values of
C, D accordingly, or else we may retain the same value of C or i)
throughout as great a range as possible of the curve, and for that
purpose permit to go beyond the above limits. The latter
course will be found the more convenient.

16. We must now ascertain in what cases it is possible for
the constant C or D to alter discontinuously as 6 alters con-
tinuously. The tests already given will enable us to decide.

The general term of either series in (20), taken without regard
to sign, is

1.5...(6^•-5)(6^•-l) .
1.2...i(14>4.x^y '

and the modulus of this term, expressed by means of the function
r, is

^(^•+^)^(^•+»)
r(i)r(f)r(i + i)(V)»'

which when i is very large becomes by the transformations employed
ip Art. 7, very nearly,

Denoting this expression by fjn, and putting for r(|-) r(|) its value
77 cosec ^TT or 27r, we have

'^'• = (''^^)"*(-4^ <23);

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 95

whence for very large values of i

W'i? <^^>-

For large values of p the moduli of several consecutive terms
are nearly equal at the part of the series where the modulus is a
minimum, and for the minimum modulus fi we have very nearly
from (24), (23)

i = 4|0*, fj, = (2m)-ie-' = (27n)-^e-^''l

If the exponential in the expression for /j, be multiplied by the
modulus of the exponential in the superior term, the result will be

g— {4=F2COS§9) p^

the sign — or + being taken according as cosf^ is positive or
negative. Hence even if the terms of the divergent series were all
positive, the superior term would be defined by means of its series
within a quantity incomparably smaller, when p is indefinitely
increased, than the inferior term, except only when +cos|^ = l,
and in this case too and this alone are the terms of the divergent
series in the superior term regularly positive. In no other case
then can the coefficient of the inferior term alter discontinuously,
and the coefficient of the other term cannot change so long as that
term remains the superior term. Keferring for convenience to the
figure (Fig. 1), we see that it is only at the points a, b, c, at the
middle of the portions of the curve which lie within the circle, that
the coefficient belonging to the curve can change.

It might appear at first sight that we could have three distinct
coefficients, corresponding respectively to the portions aAb, hBc,
cCa of the curve, which would make three distinct constants
occurring in the integral of a differential equation of the second
order only. This however is not the case ; and if we were to assign
in the first instance three distinct constants to those three portions
of the curve, they would be connected by an equation of condition.

To show this assume the coefficient belonging to the part of
the curve about B to be equal to zero. We shall thus get an
integral of our equation with only one arbitrary constant. Since
there is no superior term fi:om 6 = — ^-jr to 6 = + ^tt, the coefficient

96

ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

Fig. 2.

of the other term cannot change discontinuously at a (i.e. when 6
passes through the value zero); and by what has been already
shown the coefBcient must re-
main unchanged throughout
the portion bBc of the curve,
and therefore be equal to zero ;
and again the coefficient must
remain unchanged throughout
the portion cCaAb, and there-
fore have the same value as at
a ; but these two portions be-
tween them take in the whole
curve. The integral at present
under consideration is repre-
sented by Fig. 2, the coefficient
having the same value through-
out the portion of the curve
there drawn, and being equal
to zero for the remainder of the
course *

The second line on the
right-hand side of (20) is what
the first becomes when the
origin of 6 is altered by + |7r,
and the arbitrary constant
changed. Hence if we take
the term corresponding to the
curve represented in Fig. 3, and Fio. 3.

having a constant coefficient

throughout the portion there represented, we shall get another
particular integral with one arbitrary constant, and the sum of
these two particular integrals will be the complete integral.

In Fig. 3 the uninterrupted interior branch of the curve is
made to lie in the interval ^tt to tt. It would have done equally
well to make it lie in the interval — Jtt to — tt ; we should thus in
fact obtain the same complete integral merely somewhat differently
expressed.

* A numerical verification of the discontinuity here represented is given as an
Appendix to this paper.

WHICH APPEAR IX Dn-ERGEXT DEVELOPMEXTS. 97

The integral (^20^ mar now be conveniently expressed in the
foUovnuir form, in which the discontinuity of the constants is
exhibited :

K--^^-

In this equation the expression (,— 4- to +4—^ denotes that
the fonction written after it is to be taken wheneTer an angle in
the indefinite series

...t^--t77. ^-2- e. t>J--2T7, i> + -i:7r....

fells within the specified limits, which will be either once or twice
according to the value of i?.

17. If we put X) = in (^2oy the resiilrinir value of u wiU be
equal to ilr Airy s inregral. multiplied by an arbitrary oonsranr.

J- beincr equal to —^1 t^- When f^=0 we have the inreorr^U

belonginir to the dark side of the eaiisric. when f' = — that be-
loniriiii: to the briirhr side. We easilv see fix>m i 25 \ or bv referrincr
:o Fi^. 2. in what way ro pass fi>>m one of these inrecrals to the
other, the inteirrjls being sippv^sei.i to be expressed by me^ms of
the divergent series. If we have ;v>t the analytical expression
K-Ioncin^ to the dark side we m-.\s: add + tt, — tt in siicoession to
the aniplirude of jc, and take the sum of the results. If we have
£^>r the analytical expression beloiiirinir ro rhe brio-h' side, we miisr
alter the amplitude of j- by — . and ivjeot the superior function in
the i-es-.;I:ini: expression. It is shown in Art. 9 of my paper "" On
the Numerical Calculation, ero. that the latter process leads lo a
correct lesiUt, but I \vas unable then to give a demonsrration.
This desideraf.-.ui is now supplied.

IS. It now only remains to coimect the ocnsranrs A, B with
C, D in the two different forms ^^19^ and i 2-? > of the integnd of 1 1 S i.
This may be done by means of the complete integral of (ISi
ext^resse-i in the form of definite inTetrrals.

s. IV. 7

98

ON

THE DISCONTINUITY OF ARBITEART CONSTANTS

Let

Jo

then
whence

= 3-3"^'

d^ . cv c' c^

■-4- ;r ow —

dec' ' 3 3

(26).

In order to make the left-hand member of this equation agree
with (18), we must have c' = — 27, and therefore

c = — 3, or 3a, or 3/3,
a, /3 being the imaginary cube roots of — 1, of which a will be
supposed equal to

cos I^TT + V — 1 sin ^TT.
Whichever value of c be taken, the right-hand member of equation
(26) will be equal to — 9, and therefore will disappeaa^sOTL taking
the difference of any two functions cv corresponding to two difljbfent
values of c. This difference multiplied by an arbitrary copistant
will be an integral of (18), and accordingly we shall haveE for the
complete integral J

u = e\ e-"-' (e^'^ + ae-'""^^) d\ + F e"'^' (e^^^ + I3e-'^'^'')d\. . .(27).
Jo Jo

That this expression is in fact equivalent to (19) might be
verified by expanding the exponentials within parentheses, and
integrating term by term.

To find the relations between E, F and A,B,it will be sufficient
to expand as far as the first power of iv, and equate the results.
We thus get

A+Bx=r e-^ {(l + a)E + (l+^)F

+ 3i{l-cL')E + (l-^)F]xX}d\
which gives, since

^ e-^VX = 1 r a), j^ e~>^'XdX = i r (f),

A=iTq){{l + a)E + (l+0)F})

B= r(f){(i+/3)^+(i + «)i?'}J ^^^''-

WHICH APPEAR IN DIVEEGENT DEVELOPMENTS. 99

19. We have now to find the relations between E, F and G, D,
for which purpose we must compare the expressions (25), (27),
supposing X indefinitely large.

In order that the exponentials in (25), may be as large as
possible, we must have ^ = fir in the term multiplied by G, and
= in the term multiplied by D. We have therefore for the
leading term of u

Ce-i^^p-ie^'>^ when 6' = ^ ;
Bp-^e^"^, when ^ = 0.

Let us now seek the leading term of m from the expression (27),
taking first the case in which ^ = 0. It is evident that this must
arise from the part of the integral which involves ^"'^ or in this
case e'''^ which is

Jo

e-'-'+^o'^dX.

^ow 3/)\ — \' is a maximum for X = p*. Let \ = p* + f ; then

3p\ -\' = 2pi - Spi^' - ?^
and our integral becomes

-ft
Put ^=3~*/)~*|; then the integral becomes

8-ip-ie'i>^ r e-^-'-^f-^^d^.

Let now p become infinite ; then the last integral becomes

e-^d^ or TT*. For though the index - f " - 3-*p~^f ' becomes

positive for a sufficient large negative value of |, that value lies far
beyond the limits of integration, within which in fact the index
continually decreases with |, having at the inferior Umit the value
- 2/3*. Hence then for ^ = 0, and for very large values of p, we
have ultimately

u = 3-iTri{E + F)p-h'i-\

100 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

Next let 6 = ^. In this case ax = - p, and we get for the
leading part of u

Jo
which when p is very large becomes, as before,

S-i'jr^aEp-ie'''^-

Comparing the leading terms of lo both for ^ = |7r and for

5 / —

0, we find, observing that a = e^

D = S-i7ri(E + F)] ^ ^'

Ehminating E, F between (28) and (29) we have finally

(30).

A= ■7r-ir(^){ G+e ^^ '-D)
B = 37r-i r Q) {- C + e^^~^ B}]

20. As a last example of the principles of this paper, let us
take the differential equation

(Pu 1 du _ . ,„ ,

dx^ X dx

The complete integral of this equation in series according to
ascending powers of x involves a logarithm. If the arbitrary
constant multiplying the logarithm be equated to zero we shall
obtain an integral with only one arbitrary constant. This integral,
or rather what it becomes when v —Ix is written for x, occurs in
many physical investigations, for example the problem of annular
waves in shallow water, and that of diffraction in the case of
a circular disk. I had occasion to employ the integral with a
logarithm in determining the motion of a fluid about a long
cylindrical rod oscillating as a pendulum, the internal friction of
the fluid itself being taken into account*. In that paper the
integral of (31) both in ascending and in descending series was
employed, but the discussion of the equation was not quite com-
pleted, one of the arbitrary constants being left undetermined. A
knowledge of the value of this constant was not required for

* Camb. Phil. Trans. Vol. ix, Part ii, [1850], p. [38]. [Ante, Vol. in, p. 38.]

|-...(32)

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 101

determining the resultant force of the fluid on the pendulum,
which was the great object of the investigation, but would have
been required for determining the motion of the fluid at a great
distance from the pendulum.

21. The three forms of the integral of (31) which we shall
require are given in Arts. 28 and 29 of my paper on pendulums.
The complete integral according to ascending series is

^ = (^+^l°8-^)(l+S + 2^^ + 2^f:6^+-)

where Si = l"' + 2"' + 3-^ . . . + i-\

The series contained in this equation are convergent for all
real or imaginary values of a;, but the value of u determined by the
equation is not unique, inasmuch as log a; has an infinite number
of values. To pass from one of these to another comes to the same
thing as changing the constant A by some multiple of 2irB V — 1.
If p, 6, the modulus and amplitude of x, be supposed to be polar
co-ordinates, and the expression (32) be made to vary continuously
by giving continuous variations to p and without allowing the
former to vanish, the value of log x will increase by 27r V— 1 in
passing from any point in the positive direction once round the
origin so as to arrive at the starting point again. In order to
render everything definite we must specify the value of the
logarithm which is supposed to be taken.

The complete integral of (31) expressed by means of descending
series is

n -i^ J 1 J. 1' 1^3' P. 3^5"

'^^'^ ^ ^^ + 274^+2. 4(4*')^ + 2. 4. 6(4^)= + '

.(33).

These series are ultimately divergent, and the constants C, D
are discontinuous. It may be shown precisely as before that the
values of 6 for which the constants are discontinuous are

...-47r, — 27r, 0, 2ir, 47r...for(7,
...— Stt, — TT, IT, Stt, ... for D.

102 ON THE DISCONTINUITy OF ARBITRARY CONSTANTS

Hence the equation (33) may be written, according to the
notation employed in Art. 16, as follows :

w = (0 to 27r)Gx-ie-''(l-^^ + ...]

+ (-Tr to + -7r) Dx-h" (l + ~^+ ...] (34).

22. It remains to connect A, B with G, D. For this purpose
we shall require the third form of the integral of (31), namely

IT

u= \ {i7+i?'log(a;sin''a))}(e*<=»«" + e-*<'o«")rf(u...(35).

Jo

As to the value of log « to be taken, it will suffice for the present
to assume that whatever value is employed in (32), the same shall
be employed also in (35).

To connect A, B with E, F, it will suffice to compare (32) and
(35), expanding the exponentials, and rejecting all powers of as.
We have

IT

A+B\oga: = 2 I {E + F\og{xsin^(o)}d(o
Jo

= '7r(E + Floga;) + 27r log (^) . F;

whence A = TrE —2irlog2.F]

B = -7rF [ (^^>

To connect C, D with E, F, we must seek the ultimate value
of u when p is infinitely increased. It will be convenient to
assume in succession 6 = and 6 = Tr. We have ultimately
ii-om (34)

u = Bp-ieP when 0=0; m = -V-1 Cp'^e'' when = tt. . .(37).

It will be necessary now to specify what value of logo; we
suppose taken in (35). Let it be log p + >•/ — 10, 6 being supposed
reduced within the limits and 27r by adding or subtracting if
need be 2i-ir, where i is an integer.

The limiting value of m for = from (35) may be found as in
Art. 29 of my paper on pendulums, above referred to. In fact, the
reasoning of that Article will apply if the imaginary quantity
there denoted by m be replaced by unity. The constants

C, D, C, B', G", D",

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 103

of the former paper correspond to

A, B, G, D, E, F,
of the present. Hence we have for the ultimate value of u for

u^{^0e<'{E + {',r-^V'{\)^\og2)F] (38).

For 6 = 17, (35) becomes

u=\ \E + 'irF \/ -\+F\og{pBXD? (o)]{e-'"^^'' + ei"^'')da);

and to find the ultimate value of u we have merely to write
E + ■jtF'^ — 1 for £^ in the above, which gives ultimately for ^ = tt

TT

if=J— j e''[JSr+7ri^V-l+ {7r-ir'(^) + log2}i?]...(39).

Comparing the equations (38), (,39) with (37), we get
(7- (I? [^ V^I - ■jrF+ {7r-^r'(4) + log2} s/^^\F]
B=ifj[E+{',r-^ri^) + \og2]F]

...(40).

Eliminating E, F between (36) and (40), we get finally
G = (2-rr)-i [V^T^ + {(tt-* V (i) + log 8) s/^1 - tt) 5] |
n = (27r)-4 [A + {-TT-i r (i) + log 8} 5] j- • • •(- >

' Conclusion.
23. It has been shown in the foregoing paper.
First, That when functions expressible in convergent series
according to ascending powers of the variable are transformed so
as to be expressed by exponentials multiplied by series according
to descending powers, applicable to the calculation of the functions
for large values of the variable, and ultimately divergent, though
at first rapidly convergent, the series contain in general dis-
continuous constants, which change abruptly as the amplitude of
the imaginary variable passes through certain values.

Secondly, That the liability to discontinuity in one of the
constants is pointed out by the circumstance, that for a particular
value of the amplitude of the variable, all the terms of an associated
divergent series become regularly positive.

104 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

Thirdly, That a divergent aeries with all its terms regularly-
positive is in many cases a sort of indeterminate form, in passing
through which a discontinuity takes place.

Fourthly, That when the function may be expressed by means
of a definite integral, the constants in the ascending and cIcHoonding
series may usually be connected by one uniform process. The
comparison of the leading terms of the ascending series with the
integral presents no difficulty. The comparison of the leading
terms of the descending series with the integral may usually be
effected by assigning to the amplitude of the variable such a
value, or such values in succession, as shall render the real part of
the index of the exponential a maximum, and then seeking what
the integral becomes when the modulus of the variable increases
indefinitely. The leading term obtained from the integral will be
found within a range of integration comjmsing the maximum
value of the real part of the index of the exponential under the
integral sign, and extending between limits which may be supposed
to become indefinitely close after the modulus of the original
variable has been made indefinitely great, whereby the integral
will be reduced to one of a simpler form. Hhould a definite
integral capable of expressing the function not be discovered, the
relations between the constants in the ascending and descending
series may still be obtained numerically by calculating from the
ascending and descending series separately and equating the
results*.

Appendix.

[Added since the reading of the Paper.]

On account of the strange appearance of figures 2 and 3, the
reader may be pleased to see a numerical verification of the
discontinuity which has been shown to exist in the values of the
arbitrary constants. I subjoin therefore the numerical calculation
of the integral to which Fig. 2 relates, for two values of x, from the
ascending and descending series separately. For this integral
D = 0, and I will take C= 1, which gives (equations 30),

^ = 7r-ira); 5 = -37r-ir(^);
and log A = 0-1793878 ; log (- B) = 0-3602028.

[* Of. also Proc. Camb. Phil. Soc. vi, 1889, pp. 362-6, reprinted infra.]

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 105

The two values of x chosen for calculation have 2 for their
common modulus, and 90°, 150°, respectively, for their amplitudes,
so that the correspondiag radii in Fig. 2 are situated at 30° on each
side of the radius passing through the point of discontinuity c.
The terms of the descending series are calculated to 7 places of
decimals. As the modulus of the result has afterwards to be
multiplied by a number exceeding 40, it is needless to retain more
than 6 decimal places in the ascending series. In the multiplica-
tions required after summation, 7-figure logarithms were employed.
The results are given to 7 significant figures, that is, to 5 places of
decimals.

The following is the calculation by ascending series for the ampli-
tude 90° of X. By the first and second series are meant respectively
those which have A, B hi their coefiicients in equation (19).

First Series.

Second Series.

Order of Coefficient

Coefficient

term

Real part of V - 1

Real part of ^^

+ 1-000000

+ 2-000000

1

- 12-000000

-1-12-000000

2

-28-800000

-20^571429

3

-1-28-800000

- 16-457143

4

4-15-709091

-f- 7-595605

5

- 5-385974

+ 2-278681

6

- 1-267288

- 0-479722

7

-I- 0-217249

- 0-074762

8

+ 0-028337

-1- 0-008971

9

- 0-002906

-1- 0-000855

10

- 0-000240

- 0-000066

11

-1- 0-000016

- 0-000004

12

-1- 0-000001

/

Sum -13-330099 -H 11-628385 V-l - 2-252373-11-446641 V- 1

Sum multiplied

by 4, - 20-14750 -1-17-57548 V^ ! ^J S, + 5-16230 +26-23499 V^-

When the amplitude of x becomes 150° in place of 90°, the
amplitude of a^ is increased by 180°. Hence in the first series it
will be sufficient to change the sign of the imaginary part. To see
what the second series becomes, imagine for a moment the factor x
put outside as a coefficient. In the reduced series it would be
sufficient to change the sign of the imaginary part ; and to correct
for the change in the factor x it would be sufficient to multiply by

106

ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

COS 60° + V — 1 sin 60°. But since the amplitude of x was at first
90°, the real and imaginary parts of the series calculated correspond
respectively to the imaginary and real parts of the reduced series.
Hence it will be sufficient to change the sign of the real part in
the product of the sum of the second series by B, and multiply by
^ (1 + \/3 V — 1), which gives the result

- 25-30132 + 8-64681 V^.

Hence we have for the result obtained from the ascending
series :

for amp. x = 90°, for amp. x — 150°,

From first series

- 20-14750 + 17-57548 V^l - 20-14750 - 17-57548 V^^

From second series

+ 5-16230 + 26-23499 V"^

-25-30132+ 8-64681 V-1

Total - 14-98520 + 43-81047 V - 1 - 45-44882 - 8-92867 V - 1

On account of the particular values of amp. x chosen for
calculation, the terms in the ascending series were either wholly
real or wholly imaginary. In.the case of the descending series this
is only true of every second term, and therefore the values of the
moduli are subjoined in order to exhibit their progress. The
following is the calculation for amp. x = 90°, in which case there is
no inferior term.

Coefficient

Order

Modulus

Real part

of V^

1-0000000

+ 1-0000000

1

0-0122762

+ 0-0086806

+ 0-0086806

2

0-0011604

+0-0011604

3

0-0002099

-0-0001484

+ 0-0001484

4

0-0000563

-0-0000563

5

0-0000200

-0-0000142

- 0-0000142

6

0-0000089

-0-0000089

7

0-0000047

+0-0000033

-0-0000033

8

0-0000029

+00000029

9

0-0000021

+ 0-0000015

+0-0000015

10

0-0000017

+0-0000017

11

00000015

-0-0000010

+0-0000010

R

emainder
Sum

-0-0000007

-0-0000017

+ 1-0084677

+ 0-0099656 V-l

WHICH APPEAR IX DIVERGENT DEVELOPMENTS. 107

The modulus of the term of the order 12 is 14 in the seventh
place, and is the least of the moduli. Those of the succeeding
terms are got by multiplying the above by the factors 1-0616,
1'220S, I0II6, 20053, etc., and the successive differences of the
series of tactoi-s headed by unity are

A' = + 00616, A= = + 00976, A' = + 00340, A^ = + 00373, etc.

These differences when multiplied by 14 are so small that in
the application of the tiimsformation of Art. 8, for which in the
present case 9 = 1, the differences may be neglected, and the
series there given reduced to its firat term. It is thus that the
remainder given above was calculated.

The sum of the series is now to be reduced to the form
p (cos 0+\/ — Ismd), and thus multiplied by e"**^ and by x-i. We
have

for series log. mod. = 00036832 amp. = + 0' 33' 5S"-21

for exponential log. mod. = 17371779 amp. = + 130° 49' 0"-78
for.)-i log. mod. = 1-9247425 amp.=- 22 30'

1-6656036 + 108" 52 5S"-99

When the amplitude of .r is 150", there are both superior and
inferior terms in the expression of the function by means of
descending series. It -svill be most convenient, as has been ex-
plained, to put in succession, in the function multiplied by C in
equation (20), amp. .1:= 150" and amp. .r= — 210°, and to take the
sum of the results. The first -svill give the superior, the second the
inferior term.

For the amplitudes 90". 150" of .i-, or more generally for any
two ampKtudes equidistant from 120", the amplitudes of a^ vriW be
equidistant from ISO , so that for any rational and real function of
a^ we may pass from the result in the one case to the result in the
other bv simply changing the sig-n of \ — 1, or, which comes to the
same, changing the sign of the amplitude of the result. The series
and the exponential are both such frmctions, and for the iactor x~i
we have simply to replace the amplitude —22" 30' by — 37" 30'.
Hence we have for the superior term

log. mod. = 1-6656036 : amp. = - 168" 52' 58"-99.

108 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

When amp. x is changed from 150° to — 210°, amp. a;* is
altered by 3 x 180°, and therefore the sign of x^ is changed.
Hence the log. mod. of the exponential is less than it was by
2 X 1'737... or by more than 3. Hence 4 decimal places will be
sufficient in calculating the series, and 4-figure logarithms may be
employed in the multiplications. The terms of the series will be
obtained from those already calculated by changing first the signs
of the imaginary parts, and secondly the sign of every second term,
or, which comes to the same, by changing the signs of the real
parts in the terms of the orders 1, 3, 5 ..., and of the imaginary
parts in the terms of the orders 0, 2, 4 . . . Hence we have

Real part Coefficient of «y - 1
+ 1-0000

-0-0087 +0-0087

-0-0012

+ 0-0001 +0-0001

+ 0-9914 +0-0076 \/-l.
log. mod. = 1-9963 ; amp. = +26' -5.

Hence we have altogether for the inferior term,
log. mod. = 2-1838 ; amp. = + 183° 45'-5.

Hence reducing each imaginary result from the form
p (cos + V — 1 sin ^) to the form a + V — 1 6, we have for the final
result, obtained fi:om the descending series :

For amp. x = 90°. For amp. x = 150°.

From superior term

- 14-98520 + 43-81046 V^T; - 45-43360 - 8-92767 V^HT
From inferior term - 0-01524 - 0-00100 V^T

- 45-44884 - 8-92867 V"^^

Had the asserted discontinuity in the value of the arbitrary
constant not existed, either the inferior term would have been
present for amp. x = 90°, or it would have been absent for
amp. a; =160°, and we see that one or other of the two results
would have been wrong in the second place of decimals.

WHICH APPEAR IX DIVERGEXT DEVELOPMENTS. 109

In considering the ivlatire difficulty of the calculation by the
ascending and descending series, it must be remembered that the
blanks only occur in consequence of the special values of the
amplitude of .r chosen for calculation: for general values they
^ould have been all filled up bv fig^ires. Hence even for so low
a value of the modulus of x as 2 the descending series have a
decided advantage over the ascending.

On the Effect of Wind on the Intensity of Sound.

[From the Report of the British Association, Dublin, 1857, p. 22.]

The remarkable diminution in the intensity of sound, which
is produced when a strong wind blows in a direction from the
observer towards the source of sound, is familiar to everybody, but
has not hitherto been explained, so far as the author is aware.
At first sight we might be disposed to attribute it merely to the
increase in the radius of the sound-wave which reaches the
observer. The whole mass of air being supposed to be carried
uniformly along, the time which the sound would take to reach
the observer, and consequently the radius of the sound-wave,
would be increased by the wind in the ratio of the velocity of
sound to the sum of the velocities of sound and of the wind, and
the intensity would be diminished in the inverse duplicate ratio.
But the effect is much too great to be attributable to this cause.
It would be a strong wind, whose velocity was a twenty-fourth
part of- that of sound ; yet even in this case the intensity would
be diminished by only about a twelfth part. The first volume of
the Annates de Ghimie (1816) contains a paper by M. Delaroche,
giving the results of some experiments made on this subject.
It appeared from the experiments, — first, that at small distances,
the wind has hardly any perceptible efiect, the sound being propa-
gated almost equally well in a direction contrary to the wind and in
the direction of the wind; secondly, that the disparity between the ;
intensity of the sound propagated in these two directions becomes
proportionately greater and greater as the distance increases;
thirdly, that sound is propagated rather better in a direction perpen-
dicular to the wind than even in the direction of the wind. The
explanation offered by the author of the present communication is
as follows*. If we imagine the whole mass of air in the neighbour-
hood of the source of disturbance divided into horizontal strata,
these strata do not all move with the same velocity. The lower

[* Cf. Osborne Eeynolds, Roy. Sac. Proe. xxii, 1874, p. 531 ; Lord Eayleigh,
Theory of Sound, 1878, Vol. ii, §§ 289-290.]

ON THE EFFECT OF WIND ON THE INTENSITY OF SOUND. Ill

strata are retarded by friction against the earth, and by the
various obstacles they meet with ; the upper by friction against
the lower, and so on. Hence the velocity increases from the
ground upwards, conformably with observation. This difference
of velocity disturbs the spherical form of the sound-wave, tending
to make it somewhat of the form of an ellipsoid, the section of
which by a vertical diametral plane parallel to the direction of the
wind is an ellipse meeting the ground at an obtuse angle on the
side towards which the wind is blowing, and an acute angle on
the opposite side. Now, sound tends to propagate itself in a
direction perpendicular to the sound-wave ; and if a portion of the
wave is intercepted by an obstacle of large size, the space behind
is left in a sort of sound-shadow, and the only sound there heard
is what diverges from the general wave after passing the obstacle.
Hence, near the earth, in a direction contrary to the wind, the
sound continually tends to be propagated upwards, and conse-
quently there is a continual tendency for an observer in that
direction to be left in a sort of sound-shadow. Hence, at a
sufficient distance, the sound ought to be very much enfeebled ;
but near the source of disturbance this cause has not yet had time
to operate, and therefore the wind produces no sensible effect,
except what arises from the augmentation in the radius of the
sound-wave, and this is too small to be perceptible. In the con-
trary direction, that is, in the direction towards which the wind is
blowing, the sound tends to propagate itself downwards, and to be
reflected from the surface of the earth ; and both the direct and
reflected waves contribute to the effect perceived. The two waves
assist each other so much the better, as the angle between them is
less, and this angle vanishes in a direction perpendicular to the
wind. Hence, in the latter direction the sound ought to be propa-
gated a little better than even in the direction of the wind, which
agrees with the experiments of M. Delaroche. Thus the effect is
referred to two known causes, — the increased velocity of the air in
ascending, and the diffraction of sound.

On the Existence of a Second Crystallizable

Fluorescent Substance (Paviin) in the Bark of the

Horse-Chestnut.

[From the Quarterly Jotirnal of the Chemical- Society, xi, 1859,
pp. 17-21 : also Pogg. Ann. cxiv, 1861, pp. 646-51.]

On examining, a good while ago, infusions of the barks of
various species of ^sculus, and the closely allied genus Pavia,
I found that the remarkably strong fluorescence shown by the
horse-chestnut ran through the whole family. The tint of the
fluorescent light was, however, different in different cases, being as
a general rule blue throughout the genus jEscuIus, and a blue-
green throughout Pavia. This alone rendered it evident, either
that there were at least two fluorescent substances present, one in
one bark and another in another, or, which appeared more probable,
that there were two (or possibly more) fluorescent substances present
in different proportions in different barks.

On examining, under a deep violet glass, a freshly cut section
of a young shoot, of at least two years' growth, of these various
trees, the sap which oozed out from different parts of the bark or
pith was found to emit a differently coloured fluorescent light.
Hence, even the same bark must have contained more than one
fluorescent substance ; and as the existence of two would account
for the fluorescent tints of the whole family, a family so closely
allied botanically, the second of the suppositions mentioned above
appeared by far the more probable.

I happened to put some small pieces of horse-chestnut bark
with a little ether into a bottle, which was laid aside, imperfectly
corked. On examining the bottle after some time, the ether was
found to have evaporated, and had left behind a substance crystal-
lized in delicate radiating crystals. This substance, which I will
call paviin, when dissolved in water, yields, like sesculin, a highly

(paviin) in the bark of the horse-chestnut. 113

fluorescent solution, and the fluorescence is in both cases destroyed
(comparatively speaking) by acids, and restored by alkalies. The
tint, however, of the fluorescent light is decidedly different from
that given by pure sesculin, for a specimen of which I am indebted
to the kindness of the Prince of Salm-Horstmar, being a blue-
green in place of a sky-blue. The fluorescent tint of an infusion
of horse-chestnut bark is intermediate between the two, but much
nearer to sesculin than to paviin.

In all probability, the fluorescence of the infusions of barks from
the closely allied genera ^sculus and Pavia, is due to sesculin and
paviin present in different proportions, sesculin predominating
generally in the genus iEsculus, and paviin in Pavia.

iEsculin and paviin are extremely similar in their properties, so
far as they have yet been observed. They are most easily distin-
guished by the different colour of the fluorescent light of their
solutions, a character which is especially trustworthy, as it does
not require for its observation that the solutions should be pure.
Paviin, as appears from the way in which it was first obtained,
must be much more soluble than sesculin in ether. .^Esculin is
indeed described as insoluble in ether, but it is sufficieTitly soluble
to render the ether fluorescent. Paviin, like sesculin, is withdrawn
from its ethereal solution by agitation with water. Though of
feeble affinities, it is rather more disposed than sesculin to combine
with oxide of lead. If a decoction of horse-chestnut bark be
purified by adding a sufficient quantity of a salt of peroxide of iron
or of alumina, precipitating by ammonia, and filtering, and the
ammoniacal filtrate be partially precipitated by very dilute acetate
of lead, the whole redissolved by acetic acid, reprecipitated by
ammonia, and filtered, the fluorescent tint of the filtrate will be
found to be a deeper blue than that of the original solution ; while,
if the fluorescent substances combined with oxide of lead (the
compound itself is not fluorescent) be again obtained in alkaline
solution, the tint, as compared with the original, will be found
to verge towards green. The required solution is most easily
obtained from the lead-compounds by means of an alkaline
bicarbonate, which plays the double part of an acid and an
alkali, yielding carbonic acid to the oxide of lead, and ensuring
the alkalinity of the filtrate from carbonate of lead. It is very
easy in this way, by repeating the process, if necessary, on the
s. IV. 8

114 ON A SECOND CRYSTALLIZABLE FLUORESCENT SUBSTANCE

filtrate from the first precipitate, to obtain a solution which will
serve as a standard for the fluorescent tint of pure sesculin. A
solution, serving nearly enough as a standard of comparison in this
respect for pure paviin, may be had by making a decoction of a
little ash bark, adding a considerable quantity of a salt of alumina,
precipitating by ammonia, and filtering. By partial precipitation
in the manner explained, it is very easy to prove a mixture of
assculin and paviin to be a mixture, even when operating on
extremely small quantities.

It must be carefully borne in mind, that the characteristic fluor-
escent tint of a solution is that of the fluorescent light coming
from the solution directly to the eye. Even should a solution
of the pure substance be nearly colourless by transmitted light,
though strong enough to develope the fluorescence to perfection,
if the solution be impure it is liable to be coloured, most com-
monly yellow of some kind, which would make a blue seen through
it appear green. To depend upon the fluorescent tint, as seen
through and modified by a coloured solution, would be like
depending on the analysis, not of the substance to be investigated,
but of a mixture containing it. Yet in solutions obtained from
the horse-chestnut, and in similar cases, the true fluorescent tint
can be observed very well, in spite of considerable colour in the
solution.

The best method of observing the true fluorescent tint is
to dilute the fluid greatly, and to pass into it a beam of sun-
light, condensed by a lens fixed in a board, in such a manner that
as small a thickness of the fluid as may be shall intervene between
the fluorescent beam and the eye. If a stratum of this thickness
of the dilute solution be sensibly colourless, the tint of the fluor-
escent light will not be sensibly modified by subsequent absorp-
tion. This, however, requires sunlight, which is not always to
be had. Another excellent method, requiring only daylight, and
capable of practically superseding the former in the examination
of horse-chestnut bark, is the following, in using which it is best
that the solutions should be pretty strong, or at least not extremely
dilute.

A glass vessel with water is placed at a window, the vessel
being blackened internally at the bottom by sinking a piece of black

(paviin) in the bark of the horse-chestnut. 115

cloth or velvet in the water, or otherwise. The solutions to be
compared as to their fluorescent tint are placed in two test-tubes,
which are held nearly vertically in the water, their tops slightly
inclining from the window, and the observer regards the fluor-
escent light from above, looking outside the test-tubes. Since by
far the greater part of the fluorescent light comes from a very thin
stratum of fluid next the surface by which the light enters, the
fluorescent rays have mostly to traverse only a very small thickness
of the coloured fluid before reaching the eye ; the water permits
the escape of those fluorescent rays which would otherwise be
internally reflected at the external surface of the test tubes ; and
the intensity of the light of which the tint is to be observed is
increased by foreshortening. The observer would do well to
practise with a fluorescent fluid purposely made yellow by intro-
ducing some non-fluorescent indifferent substance ; thus, a portion
of the standard solution of sesculin mentioned above may be
rendered yellow by ferrid-cyanide of potassium. The more com-
pletely the fluorescent tints of the yellow and the nearly colourless
solution agree, the more nearly perfect is the method of observa-
tion. If ferro-cyanide of potassium be used in the experiment
suggested, instead of ferrid-cyanide, the most marked effect is
a diminution in the intensity of the fluorescent light, the cause
of which is that the absorption by this salt takes place more upon
the active, or fluorogenic, than upon the fluorescent rays. Since
substances of a similar character may be present in an impure
solution, the observer must not always infer poverty with regard
to fluorescent substances from a want of brilliancy in the fluor-
escent light.

The existence of paviin may perhaps account for the dis-
crepancies between the analyses of sesculin given by different
chemists. I should mention, however, that I have met with three
specimens of sesculin, and they all appeared to be free from paviin.
The reason why sesculin was obtained pure from a decoction con-
taining paviin also, is probably that the former greatly preponde-
rates over the latter in the bark of the horse-chestnut. A
decoction of this bark yielded to me a copious crop of crystals of
sesculin, while the paviin, together with a quantity of sesculin still
apparently in excess, remained in the mother-liquor. I may,
perhaps, on some future occasion communicate to the Society the

116 ON A SECOND CRYSTALLIZABLE FLUORESCENT SUBSTANCE, ETC.

method employed, when I have leisure to examine it further;
I will merely state for the present that it enabled me to obtain
crystallized sesculin in a few hours, without employing any other
solvent than water. In the method commonly employed," the
first crystallization of sesculin is described as requiring some
fourteen days.

On account of the small quantity, apparently, of paviin, as
compared with sesculin, present in the bark of the horse-chestnut,
a chemist who wished to obtain the substance for analysis would
probably do well to examine a bark from the genus Pavia, if such
could be procured. The richness of the bark in paviin, as com-
pared with sesculin, may be judged of by boiling a small portion
with water in a test tube; those barks in which the substance
presumed to be paviin abounds yield a decoction having almost
exactly the same fluorescent tint as that of a decoction of ash
bark.

A crystallizable substance, giving a highly fluorescent solution,,
has been discovered in the bark of the ash, by the Prince of Salm-
Horstmar*, who has favoured me with a specimen. This substance,
which has been, named fraxin by its discoverer, is so similar in
its optical characters to paviin that the two can hardly, if at all,
be distinguished thereby ; but as fraxin is stated to be insoluble
in ether, it can hardly be identical with paviin, which was left
in a crystallized state by that solvent. I find, however, that,
fraxin is sufficiently soluble in ether to render the fluid fluorescent,
so that after all it is only a question of degree, which cannot be
satisfactorily settled till paviin shall have been prepared in greater-
quantity.

* Poggendurff's Annalen, Vol. c, (1857), p. 607.

On the beaking of the Phenomena of Diffraction on the
Direction of the Vibrations of Polarized Light, with
Remarks on the Paper of Professor F. Eisenlohr.

[From the Philosophical Magazine, xvni, 1859, pp. 426-7.]

The appearance in the Philosophical Magazine for September
of a translation of Professor F. Eisenlohr's paper in the 104th
volume of Poggendorff's Annalen, induces me to offer some
remarks on the subject there treated of.

Had my paper "On the D3mamical Theory of Diffraction*"
been accessible to M. Eisenlohr at the time when he wrote, he
would have seen that I did not content myself with merely
resolving the vibrations of the incident light in directions parallel
and perpendicular to the diffracted ray, and neglecting the former
component, as competent to produce only normal vibrations, but
that I gave a rigorous dynamical solution of the problem, in
which the normal vibrations, or their imaginary representatives,
as well as the transversal vibrations, were fully taken into account,
though the result of the investigation showed that, in case of
diffraction in one and the same medium (the only case investi-
gated), the state of polarization of the diffracted ray was
independent of the normal vibrations. M. Eisenlohr's result, on
the other hand, confessedly rests on the assumption that the
diffracted ray may be regarded as produced by an incident ray
agreeing in direction of propagation with an incident ray which
would produce the diffracted ray by regular refraction, but in
direction of vibration (in the immediate neighbourhood of the
surface at which the diffraction takes place) with the actual inci-
dent ray. This assumption, though plausible at first sight, is
altogether precarious ; and since in the particular case of diffrac-
tion in one and the same medium it leads to a result at variance
with that of a rigorous investigation, it cannot be admitted.

* Cambridge Philosophical Transactions, Vol. ix, p. 1. [Ante, Vol. n, p. 243.]

118 ON THE DIFFRACTION OF POLARIZED LIGHT.

M. Eisenlohr's formula agrees no doubt very well with
M. Holtzmann's experiments; but then it must be recollected
that the formula contains a disposable constant, whereby such
an agreement can in good measure be brought about. But in
agreeing with these experiments, it is necessarily at variance
with mine, in passing to which it is not allowable to change' the
value of the disposable constant. I can no more ignore the
uniform result of my own experiments, than I am disposed to
dispute the accuracy of M. Holtzmann's, made under different
experimental circumstances. Whether the circumstances of his
experiments or of mine made the nearer approach to the sim-
plicity assumed in theory, or whether in both there did not exist
experimental conditions sensibly influencing the result, but of
such a nature that it would be impracticable to take account of
them in theory, is a question which at present I think it would
be premature to discuss. I still adhere to the opinion I formerly
expressed*, that the whole question must be subjected to a
thoroughly searching experimental investigation before physical
conclusions can safely be drawn from the phenomena.

* Phil. Mag. Ser. 4, Vol. xiii, p. 159. [Ante, p. 74 ; see also footnote.]

Note on Paviin.

[From the Quarterly Journal of the Chemical Society, xil, 1860,
pp. 126—128.]

The crystallizable substance, the existence of which in the
bark of the horse-chestnut I noticed in a former communication to
this Society*, and to which I gave the name paviin, together with
sesculin which it closely resembles in its properties, may be thus
prepared.

A decoction of the bark having been made, and allowed to
grow cold, there is added a persalt of iron, such as pernitrate,
until on testing a sample by the addition of ammonia, the pre-
cipitate separates at once in distinct flocks, leaving a bright pale
yellow highly fluorescent fluid, giving a mixture which filters
readily. The whole is then precipitated by ammonia and filtered ;
about one-fourth of the ammoniacal filtrate is precipitated with
acetate of lead, avoiding an excess, ammonia being added if
required ; the mixture is restored to solution by the addition of
acetic or dilute nitric acid, added to the remainder of the first
filtrate pre\'iously acidulated ; the whole precipitated by ammonia
and filtered ; and the filtrate precipitated by ammoniacal acetate
of lead and filtered. The two precipitates are collected separately,
and treated with acetic acid until they are dissolved, or at least
wholly broken up, filtered if need be, and set ' aside in a cool
place. The second precipitate yields sesculin, which makes its
appearance as a light precipitate, appearing under the microscope
to consist wholly of minute needles. The first precipitate yields
paviin, which ordinarily crystallizes in tufts of very long slender
silky crystals. When the solution is comparatively impure, it
sometimes crystallizes very slowly in shorter and far thicker
crystals. The crystallization of paviin is sometimes remarkably

* Chem. Sac. Quarterly Journal, Vol. xi, p. 17. [Ante, p. 112.]

120 NOTE ON PAVIIN.

facilitated by dropping in a very minute portion of the substance
from a previous preparation. When the crystallization, whether
of sesculin or of paviin, ceases to progress, the mass of crystals is
thrown on a filter, drained, and pressed out. The mother-liquors
being similarly treated by partial precipitation, yield additional
quantities of the substances.

The sesculin thus obtained, after merely pressing out the
crystals, without washing, is usually snow-white. The paviin,
which appears to be naturally slightly yellow, is often mixed with
a brown product of decomposition of other substances, which how-
ever, being insoluble in water, is of little consequence.

The properties, especially the optical properties, of paviin, so
far as I have observed, appear to be absolutely identical with those
of fraxin, a ciystallizable substance discovered in the bark of the
ash by the Prince of Salm-Horstmar*, who has favoured me
with a specimen. The substances would seem to be identical f,
but analyses are yet wanting. Accordingly, it is unnecessary to
describe the properties of paviin. I will limit myself to two
observations relative to horse-chestnut bark, in which optics come
in aid of chemistry.

The quantity of paviin contained in horse-chestnut bark is
larger than I had at first imagined. I find that in order to match
the tint of the fluorescent light of a decoction of horse-chestnut
bark by a mixed solution of aesculin and paviin, the substances
must be present in the proportion by weight of about 3 to 2. The
relative proportion of paviin as compared with sesculin actually
obtained is liable to be much less than this, which arises, I believe,
from the circumstance that paviin, from its somewhat stronger
affinities, is less easily separated than sesculin from certain readily
decomposed substances present in the decoction.

The production of sesculetin from sesculin may be elegantly
followed by combining the optical and chemical properties of the
substances. A solution liable to contain sesculetin is acidulated,
if not already acid, and agitated with about an equal volume of
ether. The ether withdraws the sesculetin, and when the whole
is examined by daylight transmitted through a deep manganese

* Poggendorff's Atmalen, Vol. o, p. 607.

t I find fraxin, like paviin, to be slightly soluble in ether, at least washed ether.

NOTE ON PAVIIN. 121

purple glass {Phil. Trans., 1853, p. 385*), the sesculetin shows itself
by the strong fluorescence of the ethereal solution. In this way it
is easy to tell what acids give rise to the formation of sesculetin by
boiling a teaspoonful of water containing, say, the hundredth of
a grain of sesculin with the acid to be tried. It is easy, too, to
demonstrate the absence of sesculetin in the bark itself, and its
presence in a decoction which has stood some time.

In describing an analysis of horse-chestnut bark not yet com-
plete, Eochleder mentions a nearly neutral f crystallizable substance
which accompanies cesculin in small quantity. If this be paviin, as
seems probable, the subject is at present in the hands of Eochleder,
who has also undertaken the analysis of fraxin.

[* Ante, p. 1.]

t Gmelin's Handbuch, Vol. viii, (1858), p. 25.

On the Colouring Matters of Madder.
By Dr E. Schunck. (Extract.)

[From the Qvurterly Journal of the Chemical Society, xil, 1860, pp. 218-222.]

Before concluding, it remains for me to say a few words in
regard to purpurine, the colouring matter which has by some
chemists been supposed to be, in addition to alizarine, essential
to the production of madder colours. The two chief properties,
whereby, according to those chemists who have examined it, it is
distinguished from alizarine are: 1. That it dissolves in alkalies
with a cherry-red or bright red colour, alizarine giving with
alkalies beautiful violet solutions. 2. That it is entirely soluble
in boiling alum-liquor, forming a solution of a beautiful pink
colour with a yellow fluorescence, whereas alizarine is almost
insoluble in the same menstruum. These properties are, how-
ever, not of a sufficiently decided character to entitle it to rank
as a distinct substance, as they might possibly be produced by an
admixture of alizarine with some foreign substance. Having con-
vinced myself by numerous experiments that almost all madder
colours may be produced by means of alizarine only, and that the
finer madder colours of the dyer contain little besides alizarine in
combination with the mordants, I made some attempts to prove
that purpurine contains ready-formed alizarine, to which its
tinctorial power may be supposed to be due. The optical pheno-
mena exhibited by solutions of purpurine are however so peculiar,
as to lead Professor Stokes, who has carefully examined them, to
the conclusion that they cannot be produced by any compound of
alizarine, or by a mixture of alizarine with any other substance
hitherto obtained from madder. Nevertheless it is certain that
alizarine and purpurine are nearly allied substances, since both of

ON THE COLOURING MATTERS OF MADDER. 123

them yield phthalic acid when decomposed by nitric acid, a pro-
perty which belongs, as far as is known, to no other substance
with the exception of naphthaline. There is one property by
which purpurine may be easily distinguished from alizarine, viz.
that of being decomposed when its solution in caustic alkali is
exposed to the air. The bright red colour of the solution, when
left to stand in an open vessel, soon changes to reddish-yellow,
and at length almost the whole of the colour disappears, after
which the purpurine can no longer be discovered in the solution.
This is probably the cause of the disappearance of purpurine,
when the method given by me for the preparation of alizarine
from madder and its separation from the impurities with which it
is associated, is adopted. This method, which depends on the
employment of caustic alkalies, is an imitation of that to which
dyers have recourse for the purpose of improving and beautifying
ordinary madder colours, and it is certain that during this process
the purpurine is either decomposed or by some means disappears.
The only advantage which purpurine presents over alizarine in
dyeing is that it imparts to the alumina-mordant a fiery red tint
which in some cases is preferred to the purplish-red colour from
alizarine. To the iron mordant it communicates a very unsightly
reddish-purple colour, presenting a disagreeable contrast with the
lovely purple from alizarine. In all madder colours which have
been subjected to a long course of after-treatment, the purpurine
is found to have almost entirely disappeared.

Professor Stokes has had the kindness to draw up, for the
purpose of being appended to this paper, the following account of
the optical characters of purpurine and alizarine, containing the
results, obtained by him on a renewed examination of their action
on light.

Optical Characters of Purpurine and Alizarine.

The optical characters of purpurine are distinctive in the very
highest degree ; those of alizarine are also very distinctive. The
characters here referred to consist in the mode of absorption of
light by certain solutions of the bodies, and occasionally in the
powerful fluorescence of a solution. They are specially valuable
because their observation is independent of more than a moderate
degree of purity of the specimens, and requires no apparatus

124 OPTICAL CHARACTERS OF PURPURINE AND ALIZARINE.

beyond a test-tube, a slit, and a small prism, a little instrument
which ought to be in the hands of every chemist.

Alkaline solution of purpurine. — If purpurine be dissolved in a
solution of carbonate of potash or soda, (it is easily decomposed
by caustic alkalies,) the solution obtained absorbs with greatest
energy the green part of the spectrum. In this and similar cases
it is necessary to take care either to use a sufficiently small
quantity of the substance, or else to dilute sufficiently the solu-
tion, or view it through a sufficiently small thickness ; otherwise
a broad region of the spectrum is absorbed, and the peculiar
characters of the substance depending on its mode of absorbing
light are not perceived. If the solution be contained in a wedge-
shaped vessel, the effect of different thicknesses is seen at a glance ;
but a test-tube will answer perfectly well if two or three different
degrees of dilution be tried in succession. When the light trans-
mitted through an alkaline solution of purpurine of suitable

Flo. 1. — Solution of purpurine in
carbonate of soda or potash, or in
alum -liquor.

Fio. 2. — Solution of purpurine in
bisulphide of carbon.

Fig. 3. — Solution of purpurine in
ether.

Fig. 4. — Alkaline solution of
alizarine.

strength, after being limited by a slit, is viewed through a prism,
two remarkable dark bands of absorption (Fig. 1) are seen about
the green part of the spectrum, comprising between them a band
of green light, which, though much weakened in comparison with
the same part of the unabsorbed spectrum, is bright compared
with the two dark bands, which latter in a sufficiently strong
solution appear perfectly black. The places of the dark bands,
estimated with reference to the principal fixed lines of the
spectrum, are given in the figure.

Solution in a solution of alum. — This solution has the same
peculiar mode of absorption, and (Fig. 1) will serve equally well
for it. But it has the further property of being eminently fluor-

OPTICAL CHAEACTERS OF PURPURINE AND ALIZARINE. 125

escent, ■which the alkaline solution is not at all. The fluorescent
light is yellow, but ordinarily appears orange from being seen
through the fluid. The difference between the alkaline and alum-
liquor solutions as to fluorescence does not depend on the acid
reaction of the latter, but on the alumina. A solution exhibiting
to perfection the peculiar properties of the alum-liquor solution
may be obtained by adding to a solution of purpurine in carbonate
of soda a solution of alum to which enough tartaric acid to prevent
precipitation, and then carbonate of soda, has previously been
added; and in this case the fluorescent solution is obtained at
once and in the cold. This forms a very striking reaction in a
dark room, according to the method described in the Philosophical
Transactions for 1853, p. 385*, with the combination, solution of
nitrate of copper and a red (Cu,0) glass. Some other colourless
oxides besides alumina develop in this manner fluorescence, though
to a less degree.

Solution in bisulphide of carbon. — This solution gives the highly
characteristic spectrum (Fig. 2) exhibiting four bands of absorption,
of which the first is narrower than the others, and the fourth is
very inconspicuous, hardly standing out from the general absorp-
tion which takes place in that region of the spectrum. The second
and third bands are the most conspicuous of the set.

Solution in ether. — This gives the characteristic spectrum
(Fig. 3) exhibiting two bands of absorption. The solution is
fluorescent, but not enough so to be perceptible by common
observation.

The spectra of the solutions of purpurine in other solvents
might be mentioned, but these are more than sufficient. In an
optical point of view purpurine is remarkable for the general
similarity of character, combined with diversity as to detail, which
its various solutions exhibit as to their mode of absorbing light.

Alkaline solution of alizarine. — The solution of alizarine in
caustic or carbonate of potash or soda, or in ammonia, exhibits
on analysis the characteristic spectrum (Fig. 4) having a band of
absorption in the yellow, and another narrower one between the
red and the orange. There is a third very inconspicuous band
at E, almost lost in the general darkening of that part of the
spectrum.

[* Ante, p. 1.]

126 OPTICAL CHARACTERS OF PURPURINE AND ALIZARINE.

Other solutions. — The solution of alizarine in ether or in bisul-
phide of carbon shows nothing particular. There is a general
absorption of the more refrangible part of the spectrum, but there
are none of those remarkable alternations of comparative trans-
parency and opacity which characterize purpurine. Alizarine is
hardly soluble in alum-liquor ; and in the case of the red solution
of mixed alizarine and verantine mentioned by Dr Schunck at
page 451 of the Philosophical Transactions for 1851, the absence
of the remarkable absorption-bands (Fig. 1) and the absence of
fluorescence, show instantly and independently of each other that
it is distinct from purpurine.

Optical detection of purpurine and alizarine. — The characters of
these substances are so marked that I do not know any substance
with which either of them could be confounded, even if we
restricted ourselves to any one of the solutions yielding the
peculiar spectra. Not only so, but these properties enable us to
detect small quantities, in the case of purpurine the merest trace,
of the substance, present in the midst of a quantity of impurities.
In the case of purpurine, a solution of alum is specially convenient
for use, because the impurities liable to be present do not with
this solvent absorb the part of the spectrum in which the bands
occur. In this way I was able, though operating on only a very
minute quantity of the root, to detect purpurine in more than
twenty species of the family Rubiaceae which were examined with
this view, comprising the genera Ruhia, Asperula, Galium, Cru-
cianella, and Sherardia. The detection of alizarine by means of
the characters of its alkaline solution is much less delicate, because
many of the impurities liable to be present absorb the part of the
spectrum in which all but the least refrangible of the absorption-
bands occur ; and as this band is not that which corresponds to
the most intense absorption, a larger quantity of the substance
must be present in order that the band may be perceived.

EXTEACTS RELATING TO THE EaELY HiSTORY

OF Spectrum Analysis.

On the Simultaneous Emission and Absorption of Rays of the
same definite Refrangihility ; being a translation of a portion
of a paper by M. LifeoN Foucault, and of a paper by
Professor Kirchhoff.

[From the Philosophical Magazine, xix, 1860, pp. 196-7.]

Gentlemen,

Some years ago M. Foucault mentioned to me in conver-
sation a most remarkable phaenomenon which he had observed
in the course of some researches on the voltaic arc, but which,
though published in L'Institut, does not seem to have attracted
the attention which it deserves. Having recently received from
Prof Kirchhoff a copy of a very important communication to the
Academy of Sciences at Berlin, I take the liberty of sending you
translations of the two, which I doubt not will prove highly
interesting to many of your readers.

I am. Gentlemen, Yours sincerely,
G. G. Stokes.

M. Foucault 's discovery is mentioned in the course of a paper
published in L'Institut of Feb. 7, 1849, having been brought
forward at a meeting of the Philomathic Society on the 20th of
January preceding. In describing the result of a prismatic
analysis of the voltaic arc formed between charcoal poles,
M. Foucault writes as follows (p. 45): —

" Its spectrum is marked, as is known, in its whole extent by
a multitude of irregularly grouped luminous lines ; but among
these may be remarked a double line situated at the boundary
of the yellow and orange. As this double line recalled by its
form and situation the line D of the solar spectrum, I wished to
try if it corresponded to it ; and in default of instruments for
measuring the angles, I had recourse to a particular process.

" I caused an image of the sun, formed by a converging lens,
to fall on the arc itself, which allowed me to observe at the same
time the electric and the solar spectrum superposed ; I convinced

128 FOUCAULT ON THE ABSORPTION OF SODIUM HAYS.

myself in this way that the double bright line of the arc coin-
cides exactly with the double dark line of the solar spectrum.

" This process of investigation furnished me matter for some
unexpected observations. It proved to me in the first instance
the extreme transparency of the arc, which occasions only a faint
shadow in the solar light. It showed me that this arc, placed in
the path of a beam of solar light, absorbs the rays D, so that the
above-mentioned line D of the solar light is considerably strength-
ened when the two spectra are exactly superposed. When, on
the contrary, they jut out one beyond the other, the line D
appears darker than usual in the solar light, and stands out bright
in the electric spectrum, which allows one easily to judge of their
perfect coincidence. Thus the arc presents us with a medium
which emits the rays D on its own account, and which at the
same time absorbs them when they come from another quarter.

" To make the experiment in a manner still more decisive, I
projected on the arc the reflected image of one of the charcoal
points, which, like all solid bodies in ignition, gives no lines;
and under these circumstances the line JD appeared to me as in
the solar spectrum."

Professor Kirchhoff s communication " On Fraunhofer's Lines,"
dated Heidelberg,' 20th of October, 1859, was brought before the
Berlin Academy on the 27th of that month, and is printed in
the Monatsbericht, p. 662.

" On the occasion of an examination of the spectra of coloured
flames not yet published, conducted by Bunsen and myself in
common, by which it has become possible for us to recognise the
qualitative composition of complicated mixtures from the appear-
ance of the spectrum of their blowpipe-flame, I made some
observations which disclose an unexpected explanation of the
origin of Fraunhofer's lines, and authorise conclusions therefrom
respecting the material constitution of the atmosphere of the sun,
and pei'haps also of the brighter fixed stars.

" Fraunhofer had remarked that in the spectrum of the flame
of a candle there appear two bright lines, which coincide with the
two dark lines D of the solar spectrum. The same bright lines
are obtained of greater intensity from a flame into which some
common salt is put. I formed a solar spectrum by projection,
and allowed the solar rays concerned, before they fell on the slit,
to pass through a powerful salt-flame. If the sunlight were
sufficiently reduced, there appeared in place of the two dark
lines D two bright lines; if, on the other hand, its intensity
surpassed a certain limit, the two dark lines D showed themselves
in much greater distinctness than without the employment of
the salt-flame.

KIRCHHOFF ON THE PRODUCTION OF DARK SPECTRAL LINES. 129

" The spectrum of the Drummond light contains, as a general
rule, the two bright lines of sodium, if the luminous spot of the
cylinder of lime has not long been exposed to the white heat ;
if the cylinder remains unmoved these lines become weaker, and
finally vanish altogether. If they have vanished, or only faintly
appear, an alcohol flame into which salt has been put, and which
is placed between the cylinder of lime and the slit, causes two
dark lines of remarkable sharpness and fineness, which in that
respect agree with the lines D of the solar spectrum, to show
themselves in their stead. Thus the lines D of the solar spectrum
are artificially evoked in a spectrum in which naturally they are
not present.

'' If chloride of lithium is brought into the flame of Bunser)»s
gas-lamp, the spectrum of the flame shows a very bright sharply
defined line, which lies midway between Fraunhofer's lines B and
C. If, now, solar rays of moderate intensity are allowed to fall
through the flarne on the slit, the line at the place pointed out is
seen bright on a darker ground; but with greater strength of
sunlight there appears in its place a dark line, which has quite
the same character as Fraunhofer's lines. If the flame be taken
away, the line disappears, as far as I have been able to see, com-
pletely

" I conclude from these observations, that coloured flames, in
the spectra of which bright sharp lines present themselves, so
weaken rays of the colour of these lines, when such rays pass
through the flames, that in place of the bright lines dark ones
appear as soon as there is brought behind the flame a source of
light of sufiicient intensity, in the spectrum of which these lines
are otherwise wanting. I conclude further, that the dark lines of
the solar spectrum which are not evoked by the atmosphere of the
earth, exist in consequence of the presence, in the incandescent
atmosphere of the sun, of those substances which in the spectrum
of a flame produce bright lines at the same place. We may
assume that the bright lines agreeing with D in the spectrum of
a flame always arise from sodium contained in it; the dark line
D in the solar spectrum allows us, therefore, to conclude that
there exists sodium in the sun's atmosphere. Brewster has
found bright lines in the spectrum of the flame of saltpeter at
the place of Fraunhofer's lines A,a,B] these lines point to the
existence of potassium in the sun's atmosphere*. From my ob-
servation, according to which no dark line in the solar spectrum
answers to the red line of lithium, it would follow with probability
that in the atmosphere of the sun lithium is either absent, or is
present in comparatively small quantity.

"The examination of the spectra of coloured flames has

[* See however p. 135 infra.']
S. IV. 9

130 KIRCHHOFF ON THE PRODUCTION OF DARK SPECTRAL LINES.

accordingly acquired a new and high interest; I will carry it
out in conjunction with Bunsen as far as our means allow. In
connexion therewith we will investigate the weakening of rays
of light in flames that has been established by my observations.
In the course of the experiments which have at present been insti-
tuted by us in. this direction, a fact has already shown itself
which seems to us to be of great importance. The Drummond
light requires, in order that the lines D should come out in it
dark, a salt-flame of lower temperature. The flame of alcohol
containing water is fitted for this, but the flame of Bunsen's gas-
lamp is not. With the latter the smallest mixture of common
salt, as soon as it makes itself generally perceptible, causes the
bright lines of sodium to show themselves. We reserve to our-
selves to develojDe the consequences which may be connected with
this fact."

Note. — The remarkable phenomenon discovered by Foucault,
and rediscovered and extended by Kirchhoff, that a body may be
at the same time a source of light giving out rays of a definite
refrangibility, and an absorbing medium extinguishing rays of
that same refrangibility which traverse it, seems readily to admit
of a dynamical illustration borrowed from sound.

We know that a stretched string which on being struck gives
out a certain note (suppose its fundamental note) is capable of
being thrown into the same state of vibration by aerial vibra-
tions corresponding to the same note. Suppose now a portion
of space to contain a great number of such stretched strings,
forming thus the analogue of a "medium." It is evident that
such a medium on being agitated would give out the note above
mentioned, while on the other hand, if that note were sounded
in air at a distance, the incident vibrations would throw the
strings into vibration, and consequently would themselves be
gradually extinguished, since otherwise there would be a creation
of vis viva. The optical application of this illustration is too
obvious to need comment

DYNAMICAL ILLUSTRATION OF SELECTIVE ABSORPTION. 131

On the Relation between the Radiating and Absorbing Powers
of different Bodies for Light and Heat. By G. KiRCHHOFF.
(Postscript.)

[Fogg. Ann. cix, 1860, p. 275 : trans, by F. Guthrie, Philosophical Magazine,
XX, Jiily 1860. This postscript communicated by author to Phil. Mag.,
pp. 19-21.]

1. Since the appearance of the above paper in Poggendorff's
Annalen, I have received information of a prior communication
closely related to mine. The communication in question is by
Mr Balfour Stewart, and appeared in the Transactions of the
Royal Society of Edinburgh, Vol. xxii, 1858. Mr Stewart has
made the interesting observation that a plate of rock-salt is much
less diathermic for rays emitted by a mass of the same substance
heated to 100° C, than for rays emitted by any other black body
of the same temperature. From this circumstance he draws
certain conclusions, and is led to a result similar to that which
I have established concerning the connexion between the powers
of absorption and emission. The principle enunciated by Mr
Stewart is, however, less distinctly expressed, less general, and
not altogether so strictly proved as mine. It is as follows : —
" The absorption of a plate equals its radiation, and that for
every description of heat."

2. The fact that the bright lines of the spectra of sodium-
and lithium-flames may be reversed, was first published by me
in a communication to the Berlin Academy, October 27, 1859.
This communication is noticed by M. Verdet in the February
Number of the Ann. de Chim. et de Phys. of the following year,
and is translated by Professor Stokes in the March Number of
the Philosophical Magazine. The latter gentleman calls attention
to a similar observation made by M. Leon Foucault eleven years
ago, and which was unknown to me, as it seems to have been
to most physicists. This observation was to the effect that the
electric arch between charcoal points behaves, with respect to
the emission and absorption of rays of refrangibility answering
to Fraunhofer's line D, precisely as the sodium-flame does ac-
cording to my experiments. The communication made on this
subject by M. Foucault to the Soc. Philom. in 1849 is reproduced
by M. Verdet, from the Journal de VInstitut, in the April Number
of the Ann. de Chim. et de Phys.

M. Foucault's observation appears to be regarded as essen-
tially the same as mine; and for this reason I take the liberty
of drawing attention to the difference between the two. The
observation of M. Foucault relates to the electric arch between
charcoal points, a phenomenon attended by circumstances which

9—2

132 KIRGHHOFF ON EARLY HISTORY OF SELECTIVE ABSORPTION.

are in many respects extremely enigmatical. My observation
relates to ordinary flames into which vapours of certain chemical
substances have been introduced. By the aid of my observation,
the other may be accounted for on the ground of the presence
of sodium in the charcoal, and indeed might even have been
foreseen. M. Foucault's observation does not afford any expla-
nation of mine, and could not have led to its anticipation. My
observation leads necessarily to the law which I have announced
with reference to the relation between the powers of absorption
and emission ; it explains the existence of Fraunhofer's lines, and
leads the way to the chemical analysis of the atmosphere of the
sun and the fixed stars. All this M. Foucault's observation did not
and could not accomplish, since it related to a too complicated
phenomenon, and since there was no means of determining how
much of the result was due to electricity, and how much to the
presence of sodium. If I had been earlier acquainted with this
observation, I should not have neglected to introduce some notice
of it into my communication, but I should nevertheless have
considered myself justified in representing my observation as
essentially new.

3. Since the above communication was printed in Poggen-
dorff's Annalen, I have learned in the course of a written
correspondence with Professor Thomson, that the idea was some
years ago thrown out, if not published, that it might be possible,
by comparing the spectra of various chemical flames with that
of the sun and fixed stars, in the manner I have described, to
become acquainted with the chemical constitution of the latter
bodies (an idea now demonstrated to be correct by the obser-
vations and theoretical considerations above set forth). Prof
Thomson writes : —

"Professor Stokes mentioned to me at Cambridge some time
ago, probably about ten years, that Professor Miller* had made
an experiment testing to a very high degree of accuracy the
agreement of the double dark line D of the solar spectrum with
the double bright line constituting the spectrum of the spirit-
lamp burning with salt. I remarked that there must be some
physical connexion between two agencies presenting so marked
a characteristic in common. He assented, and said he believed
a mechanical explanation of the cause was to be had on some
such principles as the following : — Vapour of sodium must possess
by its molecular structure a tendency to vibrate in the periods
corresponding to the degrees of refrangibility of the double

[* W. H. Miller, of Cambridge.]

KELVIN ON EARLY HISTORY OF SELECTIVE ABSORPTION. 133

line D. Hence the presence of sodium in a source of light
must tend to originate light of that quality. On the other hand,
vapour of sodium in an atmosphere round a source, must have
a great tendency to retain in itself, i.e. to absorb and to have
its temperature raised by, light from the source, of the precise
quality in question. In the atmosphere around the sun, therefore,
there must be present vapour of sodium, which, according to the
mechanical explanation thus suggested, being particularly opaque
for light of that quality, prevents such of it as is emitted from
the sun from penetrating to any considerable distance through
the surrounding atmosphere. The test of this theory must be
had in ascertaining whether or not vapour of sodium has the
special absorbing power anticipated. I have the impression that
some Frenchman did make this out by experiment, but I can
find no reference on the point.

"I am not sure whether Professor Stokes's suggestion of
a mechanical theory has ever appeared in print. I have given
it in my lectures regularly for many years, always pointing out
along with it that solar and stellar chemistry were to be studied
by investigating terrestrial substances giving bright lines in the
spectra of artificial flames corresponding to the dark lines of
the solar and stellar spectra."

On the Early History of Spectrum Analysis.

[From Nature, Vol. xn, 1876, pp. 188-9.]

The following extract from a letter*, relating to the early
history of spectrum analysis, from our highest English authority
on physical optics, cannot fail to interest, apart from its intrinsic
importance, a wide circle of readers. I have therefore obtained
permission from Professor Stokes to forward it to Nature.

C. T. L. Whitmell.

"Cambridge, Dec. 23, 1875.

" I felt that the coincidence between the dark D of the

solar spectrum and the bright Z) of a spirit-lamp with salted wick
could not be a matter of chance ; and knowing as I did that the
latter was specially produced by salts of soda, and believiug as

[* This letter was elicited by a syllabus of a course of optical lectures sent by
Mr WhitmeU to Prof. Stokes.]

134 ON THE EARLY HISTORY OF SPECTRUM ANALYSIS.

I did that even when such were not ostensibly present, they were
present ia a trace (thus alcohol burnt on a watch-glass and a candle
snuffed close, so that the wick does not project into the incandescent
envelope, do not show bright D), I concluded in my own mind that
dark D was due to absorption by sodium in some shape. In what
shape ? I knew that such narrow absorption-bands were only
observed ia vapours; I knew that as a rule vapours agree in
a general way with their liquids or solutions as to absorption, save
that in lieu of the capricious absorption • of the vapour, we have
a general absorption attacking those regions of the spectrum in
which the vapour-bands are chiefly found. Hence as the sodium
compounds, chloride, oxide, etc., are transparent, I concluded that
the absorbing vapour was that of sodium itself. Knowing the
powerful affinities of sodium, I did not dream of its being present
in a free state in the flame of a spirit-lamp ; and so I supposed
that the emitting body in the case of a spirit-lamp with salted
wick was volatilised chloride of sodium, capable of vibrating in
a specific time, or rather two specific and nearly equal periods, by
virtue of its sodium constituent ; but that to produce absorption
the sodium must be free. I never thought of the extension of
Prevost's law of exchanges from radiation as a whole to radiation
of each particular refrangibility by itself, afterwards made by
B. Stewart; and so I failed to perceive that a soda flame which
emits bright D must on that very account absorb light of the same
refrangibility.

" When Foucault, whom I met at dinner at Dr Neil Arnott's
when he came to receive the Copley Medal in 1855, told me of his
discovery of the absorption and emission of i) by a voltaic arc,
I was greatly struck with it. But though I had pictured to my
mind the possibility of emitting and absorbing light of the same
refrangibility by the mechanism of a system of piano strings tuned
to the same pitch, which would, if struck, give out a particular
note, or would take it up from the air at the expense of the aerial
vibrations, I did not think of the extension of Prevost's theory,
afterwards discovered by Stewart*, nor perceive that the emission
of light of definite refrangibility necessitated (and not merely
permitted) absorption of light of the same refrangibility.

[* Trans. R. S. Edin., xxii, March 1858; of. Lord Eayleigh, Phil. Mag. i,
1901, pp. 98-100, or Scientific Papers, Vol. iv, p. 494.]

ON THE EARLY HISTORY OF SPECTRUM ANALYSIS. 135

"Reviewing my then thoughts by the light of our present
knowledge, I see that my error lay in the erroneous chemical
assumption that sodium could not be free in the flame of a spirit-
lamp ; I failed to perceive the extension of Prevost's theory, which
would have come in conflict with that error*. — Yours sincerely,

{Signed) " G. G. Stokes.
"To Chas. Whitmell, Esq."

" P.S., Dec. 31. — As Sir Wm. Thomson has referred in print to
a conversation I had long ago with him on the subject, I take the
opportunity of describing my recollection of the matter.

"I mentioned to him the perfect coincidence of bright and
dark D, and a part at least of the reasons I had for attributing the
latter to the vapour of sodium, using I think the dynamical illus-
tration of the piano strings. I mentioned also, on the authority
of Sir David Brewster, another case of coincidence (as was then
supposed, though it has since been shown to be only a casual near
agreement) of a series of bright lines in an artificial source of light
with dark lines in the solar spectrum, from which it appeared to
follow that potassium was present in the sun's atmosphere. On
hearing this Thomson said something to this effect : ' Oh then, the
way to find what substances are present in the sun and stars is to
find what substances give bright lines coincident with the dark
lines of those todies.' I thought he was generalising too fast ; for
though some dark lines might thus be accounted for, I was dis-
posed to think that the greater part of the non-terrestrial lines of
the solar spectrum were due to the vapours of compound bodies
existing in the higher and comparatively cool regions of the sun's
atmosphere, and having (as we know is the case with peroxide of
nitrogen and other coloured gases) the power of selective absorption
changing rapidly and apparently capriciously with the refrangi-
bility of the light.

" If (as I take for granted) Sir William Thomson is right as to
the date [1852] when he began to introduce the subject into his
lectures at Glasgow (Address at the Edinburgh Meeting of the
British Association [1871], page xcv.f), he must be mistaken as to

[* For another statement, see the author's Burnett Lectures on Light, second
series, 1885, pp. 42-50.]

[t This Presidential Address is reprinted in Lord Kelvin's Popular Lectures
and Addresses, Vol. ii, see pp. 169-175.]

136 ON THE EARLY HISTORY OF SPECTRUM ANALYSIS.

the time when I talked with him about Foucault's discovery, for
I feel sure I did not know it till 1855. Besides, when I heard
it from Foucault's mouth, it fell in completely with my previous
thoughts*.

"I have never attempted to claim for' myself any part of
Kirchhoff's admirable discovery, and cannot help thinking that
some of my friends have been over zealous in my cause. As, how-
ever, my name has frequently appeared in print in connexion with
it, I have been induced to put on paper a statement of the views
I entertained and talked about, though without publishing.

"In ascribing to Stewart the discovery of the extension of
Prevost's law of exchanges f, I do not forget that it was re-discovered
by Kirchhoff, who, indeed, was the first to publish it in relation to
light, though the transition from radiant heat to light is so obvious
that it could hardly fail to have been made, as in fact it was made,
by Stewart himself (see Proceedings of the Royal Society, Vol. x,
p. 385 X)- Nor do I forget that it is to Kirchhoff that we owe the
admirable application of this extended law to the lines of the solar
spectrum."

[* This is borne out by four existing letters, Feb. 24 — Mar. 28, 1854, of Stokes
to Ttiomson, with replies : also letters of Nov. 26 and Deo. 6, 1855, relating to
the Fouoault experiment. (Printed in Appendix to this voluine.)]

[+ See Kirchhoff's criticism on this and other matters in Ijfgg. Ann. cxviii,
1862 ; Ges. Abhandl. pp. 625, 641 ; translated in Phil. Mag. xxv, 1863, pp. 250-262,
as " Contributions towards the History of Spectrum Analysis and of the Analysis
of the Solar Atmosphere"; and Stewart's rejoinder, ibid, pp. 354-360. Also
Eayleigh, loc. cit. supra, and a rejoinder by Kayser, Handbuch der Spectroscopic,
Vol. II, 1902, p. 10.]

[t " On the Light radiated by Heated Bodies," Feb. 7, 1860. It is convenient to
record here that in a supplementary paper, received by the Royal Society on May 22,
1860, " On the Nature of the Light emitted from Heated Tourmaline," Balfour
Stewart expresses his obligations to Prof. Stokes for suggesting " an apparatus " by
which an investigation of the polarization of the light was successfully made ;
namely " a thick spherical cast-iron bomb, about 5 inches in external and 3 inches
in internal diameter — the thickness of the shell being therefore 1 inch. It has a
cover removable at pleasure. There is a small stand in the inside, upon which the
substance under examination is placed, and when so placed it is precisely at the
centre of the bomb. Two small round holes, opposite to one another, viz. at the
two extremities of a diameter, are bored in the substance of the shell. ...Let the
bomb with the substance on the stand be heated to a good red heat, and then
withdrawn from the fire and allowed to cool...." SeeProc.Roy. Soc. x, 1860, p. 503;
Phil. Mag. xxi, 1861, p. 891. Kirchhoff had already made similar observations on
a tourmaline heated in a Bunsen flame : cf. Phil. Mag. xx, 1860, p. 18.]

Note on Internal Radiation.

[From the Proceedings of the Royal Society, xi, 1861, pp. 537-45.
Seoeived Dec. 28, 1861.]

In the eleventh volume of the Proceedings of the Royal
Society, p. 193, is the abstract of a paper by Mr Balfour Stewart,
in which he deduces an expression for the internal radiation in any
direction within a uniaxal crystal from an equation between the
radiations incident upon and emerging from a unit of area of a
plane surface, having an arbitrary direction, by which the crystal
is supposed to be bounded. With reference to this determination
he remarks (p. 196), " But the internal radiation, if the law of
exchanges be true, is clearly independent of the position of this
surface, which is indeed merely employed as an expedient. This
is equivalent to saying that the constants which define the
position of the bounding surface must ultimately disappear from
the expression for the internal radiation." This anticipation he
shows is verified in the case of the expression deduced, according
to his principles, for the internal radiation within a uniaxal crystal,
on the assumption that the wave-surface* is the sphere and
spheroid of Huygens.

In the case of an uncrystallized medium, the following is the
equation obtained by Mr Stewart in the first instance.

* To prevent possible misapprehension, it may be well to state that I use this
term to denote the surface, whatever it may be, which is the locus of the points
reached in a given time by a disturbance propagated in all directions from a given
point ; I do not use it as a name for the surface defined analytically by the equation
(a:2 + 2,2 + 22) (a2a;2 + J2j^2 + eH^^ _ „« ( js + ^2) x^ - b^ {c^ + a^) y' - c^ {a^ + b^) z^ + a'b^c^ = 0.

As the term wave-surface in its physical signification is much wanted in optics, the
surface defined by the above equation should, I think, be called Fresnel's surface,
or the wave-surface of Fresnel.

138 NOTE ON INTERNAL RADIATION.

Let R, R' be the external and internal radiations in directions
OP, OP', which are connected as being those of an incident and
refracted ray, the medium being supposed to be bounded by a
plane surface passing through 0. Let OP describe an elementary
conical circuit enclosing the solid angle S^, and let Bcj)' be the
elementary solid angle enclosed by the circuit described by OP'.
Let i, i' be the angles of incidence and refraction. Of a radiation
proceeding along PO, let the fraction A be reflected and the rest
transmitted; and of a radiation proceeding internally along P'O
let the fraction A' be reflected and the rest transmitted. Then
by equating the radiation incident externally on a unit of surface,
in the directions of lines lying within the conical circuit described
by OP, with the radiation proceeding in a contrary direction, and
made up partly of a refracted and partly of an externally reflected
radiation, we obtain

R cos i^ = {\- A') R' cos i' h^' + AR cos i B(f>,

or (1-A)R cos i Sc/) = (1 - A') R' cos i' B4>' (1).

In the case of a crystal there are two internal directions of
refraction, OPi, OP^, corresponding to a given direction PO oi
incidence, the rays along OPi, OP^ being each polarized in a par-
ticular manner. Conversely, there are two directions, PiO, P^O,
in which a ray may be incident internally so as to furnish a ray
refracted along OP, and in each case no second refracted ray will
be produced, provided the incident ray be polarized in the same
manner as the refracted ray OPi or OP 2- In the case of a crystal,
then, equation (1) must be replaced by

(1 - 4) E cos iS0 = (1 - -4i)-Ki cos ii S^i + (1 - A^)Ri cos ijS^j . . .(2).

In the most general case it does not appear in what manner,
if at all, equation (2) would split into two equations, involving
respectively R^ and R^. For if an incident ray PO were so
polarized as to furnish only one refracted ray, say OPx, a ray inci-
dent along P^O and polarized in the same manner as OP^ would
furnish indeed only one refracted ray, in the direction OP, but that
would be polarized differently from PO ; so that the two systems
are mixed up together.

But if the plane of incidence be a principal plane, and if we
may assume that such a plane is a plane of symmetry as regards

NOTE ON INTERNAL RADIATION. 139

the optical properties of the medium*, the system of rays polarized
in and the system polarized perpendicularly to the plane of inci-
dence will be quite independent of each other, and the equality
between the radiation incident externally and that proceeding in
the contrary direction, and made up partly of a refracted and
partly of an externally reflected radiation, must hold good for each
system separately. In this case, then, (2) will split into two
equations, each of the form (1), E now standing for half the whole
radiation, and R', A', etc. standing for R^, A^, etc., or R^, A^, etc.,
as the case may be. It need hardly be remarked that the value
of A is different in the two cases, and that R' has a value which
is no longer, as in the case of an isotropic medium, alike in all
directions. In determining according to Mr Stewart's principles
the internal radiation in any given direction within a uniaxal
crystal, no limitation is introduced by the restriction of equation
(1) to a principal plane, since we are at liberty to imagine the
crystal bounded by a plane perpendicular to that containing the
direction in question and the axis of the crystal.

Mr Stewart further reduces equation (1) by remarking that in
an isotropic medium, as we have reason to believe. A' = A, and
that the same law probably holds good in a crystal also, so that
the equal factors 1 — ^4, 1 — A' may he struck out. Arago long
ago showed experimentally that light is reflected in the same
proportion externally and internally from a plate of glass bounded
by parallel surfaces ; and the formulae which Fresnel has given to
express, for the case of an isotropic medium, the intensity of
reflected light, whether polarized in a plane parallel or perpen-
dicular to the plane of incidence, are consistent with this law. In
a paper published in the fourth volume of the Cambridge and

* According to Sir David Brewster (Report of the British Association for 1836,
Part II, p. 13, and for 1842, Part n, p. 13), when light is incident on a plane
surface of Iceland, spar in a plane parallel to the axis, the plane of incidence, which
is a principal plane, is not in general a plane of , optical, any more than of crystal-
line symmetry as regards the phenomena of reflexion, although, as is well known,
all planes passing through the axis are alike as regards internal propagation and
the polarization of the refracted rays. Hence, strictly speaking, the statement as
to the independence of the two systems of rays should be confined to the case in
which the principal plane is also a plane of crystalline symmetry. As, however,
the unsymmetrical phenomena were only brought out when the ordinary reflexion
was weakened, almost annihilated, by the use of oil of cassia, we may conclude that
under common circumstances they would be insensible.

140 NOTE ON INTERNAL RADIATION.

Dublin Mathematical Journal (p. 1)*, I have given a very simple
demonstration of Arago's law, based on the sole hypothesis that
the forces acting depend only on the positions of the particles.
This demonstration, I may here remark, applies without change to
the case of a crystal whenever the plane of incidence is a plane
of optical symmetry. It may be rendered still more general by
supposing that the forces acting depend, not solely on the positions
of the particles, but also on any differential coefficients of the
coordinates which are of an even order with respect to the time, —
a generalization which appears not unimportant, as it is ap-
plicable to that view of the mutual relation of the ether and
ponderable matter, according to which the ether is compared to a
fluid in which a number of solids are immersed, and which in
moving as a whole is obliged to undergo local dislocations to make
way for the solids.

On striking out the factors 1 — A and 1 — A', equation (1) is
reduced to

R' cos i B<j} ,„,

R " cos i' S<p' ^ ^'

In the case of an isotropic medium, R and R' are alike in all
directions, and therefore the ratio of cos i B^ to cos i' S(p' ought to
be independent of i, as it is very easily proved to bef. The same
applies to a uniaxal crystal, so far as regards the ordinary ray.
But as regards the extraordinary, it is by no means obvious that
the ratio should be expressible in the form indicated — as a quantity
depending only on the direction OP'. Mr Stewart has, however,
proved that this is the case, independently of any restriction as to
the plane of incidence being a principal plane, on the assumption
that the wave-surface has the form assigned to it by Huygens.

It might seem at first sight that this verification was fairly
adducible in confirmation of the truth of the whole theory,
including the assumed form of the wave-surface. But a little
consideration will show that such a view cannot be maintained.
Huygens's construction links together the law of refraction and
the form of the wave-surface, in a manner depending for its
validity only on the most fundamental principles of the theory of
undulations. The construction which Huygens applied to the

[* Ante, Vol. ii, p. 91.]

[t It is that of /j.'^ to fi?, S(j> being a conical angle.]

XOTr ox rSTEEXAL RADIATIOX. 141

ellipsoid ij equally applicable to any orJier srjfeee : it ^ ;.s a mere
giiess on his put that the extiaordinary ■wavc-sririSice in Iceland

spix ^vas an ellipjoid : ;\nd although the ehipsr-idal form resv.l-s
from the imperfect dynamical theory of FresneL it is certain that
rigorous dynamical theorios lead to different forms oi the wiwe-
smface. aocoiding to the suppisiiions made as to the existic;:: state
of thdr.;:^. For eveiy sttoh pos*ihle form the ratio expresso-d by
the right-hand member of equation (^3> oughx to come out in the
form indicated by the left-hand member, and not to involve
esplioitly the direction of the refracting plane : and a* it seemed
evident that it oonld not be possible, merely by stiA coneitii
considerations as those adduced by Mr Ste"wart, to distincaish
between those snrfaees which were and those which were not
dynamically possible fonns of the wave-surfiioe, I was led to
anticipate that the possability of expressing the r;jtio in qnestion
under the form indicated fras a general property of snrtaees. The
object of the present Note is to give a demonstration of the trr.th
of this anticipation, and thereby remoTe ti-om the verification the
really irrelevant conside:T>tion of a particnlar form of wave-s u^t^l!^e :
but it vras necessary in the tirst insi^ince to supply some steps of
Mr Stew;\rt's investigi^tion which are omitted in tiie published
abstiact.

The proposition tc* be proved may be somewhat ceneralited.

in a manner suggested by the consideration of internal reflexion
within a crystal, or refinction oitt of one crystalJited niediam into
another in optic;U contact with it. Thus genarali»ed it stands :is

follows :—

Imagine any two surfaces whatsoever, and also a fixed point 0:
imagine likewise a plane 11 passini: thjonirh 0. Let two points-
P, jP', situated on the two srjfi^ces respectively, and so relateii
that the t;uigent planes at those pc^ints intersect each other in the
plane IT, be called corresponding r'3ir>ts with -tfpod to the piane U.
Let P describe, on the stirf;»ce on whici it lies, an infinitesimal
close-d circuit, and P the ■'corresponding" circuit; let t^o. ^6' be
the solid imgles sitbtended at by these circuits resp-ectively. and
i. i' the inclinations of OP, OP' to the normal to D. Then shall
the ratio of cos iSa to cos i" (?<i) be of the form [P] : [P"]. where P
depends only on the lirst stirtoce and the position of P, and [P"]
onlv on the sec'ond surface and the position of P*. Moreover, if

142 NOTE ON INTERNAL RADIATION.

either surface be a sphere having its centre at 0, the corresponding
quantity [P] or [P'] shall be constant.

It may be remarked that the two surfaces may be merely two
sheets of the same surface, or even two different parts of the same
sheet.

Instead of comparing the surfaces directly with each other,
it will be sufficient to compare them both with the same third
surface ; for it is evident that if the points P, P' correspond to
the same point Pj, on the third surface, they will also correspond
to each other. For the third surface it will be convenient to take
a sphere described round as centre with an arbitrary radius,
which we may take for the unit of length. The letters Pj, ii, (^i
will be used with reference to the sphere.

Let the surface and sphere be referred to rectangular co-
ordinates, being the origin, and II the plane of aoy. Let x, y, z
be the coordinates of P; f , 77, f those of Pj. Then x, y, z will
be connected by the equation of the surface, and ^, »;, ^ by the
equation

^^ + 17'' 4-^^=1.
According to the usual notation, let

dz__ dz__ d?z^_ d?z _ d'z _
dx~P' dy"^' dx^~'^' d^~^' dp"*'

The equations of the tangent planes at P, Pj, X, Y, Z being the
current coordinates, are

Z-z = p{X-x) + q{Y-y),
^X + r,Y+^Z^l;
and those of their traces on the plane of xy are
pX +qY = px + qy—z,
^X + r,Y=l;
and in order that these may represent the same line, we must have

P P r,= g (4,)

px + qy-z' px + qy — z ^ ^'

To the element dxdy of the projection on the plane of xy
of a superficial element at P, belongs the superficial element
dS = Vl +p'' + q^ dxdy, and to this again belongs the elementary

solid angle — , where p = OP, and v is the angle between the

r

NOTE ON INTERNAL RADIATION. 143

normal at P and the radius vector. Hence the total solid angle
within a small contour is — ~ */! + p^ + q^ I \dxdy, the double in-
tegral being taken within the projection of that small contour.
Also cos i = zjp. Hence

cos i B^ = ^ — ^— Vl +p^ + (f I jdxdy ;

and applying this formula to the sphere by replacing 2 '^1 + p'' + q'
by 1, V by 0, and p by 1, we have

cosiiS^i= jjd^dr],

the double integral being taken over the projection of the corre-
sponding small area of the sphere.

Now by the well-known formula for the transformation of
multiple integrals we have

Ih^'Sm-Zih'^--

and therefore

cosiBcj) _zcosv'\/l + p'+ q'
cos ii B<j}i ^fd^dr) _^d^ dr)\ '
\da; dy dy dx)

But the first of equations (4) gives

jf. _ {px + qy — z)dp—p(xdp + ydq)
^~ {fx + qy-zf

^ {{qy -z)r- pys] dx + {{qy -z)s- pyt] dy ^
{px+qy-zy
Similarly,

_ {{px -z)t- qxs] dy + {{px -z)s- qxr] dx
^ ~ {px + qy- zY

Hence

d^ dr, d^dr, _ V

dx dy dy dx {px + qy — zY '
where

V= {{qy -z)r- pys] {{px -z)t- qxs]

- {{qy -z)s- pyt] {{px -z)s- qxr]

= {{ly - z) (p^ - ^) - Pi'^y] (*■* - *')

= z{z-px- qy) {rt — s^).

144 NOTE ON INTERNAL RADIATION.

Hence

cos iBff) 2 cos V*/ 1 + p'' + q'^ (z — px — qyy
cos t'l 8(^1 p* z{rt-s')

But if in- be the perpendicular let fall from on the tangent
plaAe at P,

z — px — qy = 'Jl + p'^ + q^ . ■^^t
and therefore

cos iB4> _ cos 1^ ■ CT^ (1 + p' + q^y
cosiiScf)^ p^ rt — s^

But ■sT = pcosv. Also the quadratic determining the principal
radii of curvature at P is

(H-s^)v' + (&c.)v + {l+p' + qJ=0;
and therefore if Vi, n, denote the principal radii of curvature,

(l+p' + qj
rt — s'
Hence

cos iB4>

— COS'' V . v-iV^ (5),

..(6),

cos ii 8^1
and

cosiS<^ _ cosiB<p cosiiB(j}i- cos^i'-'WiW

cos i' B(j)' cos I'l B(j)i ' cos i' Bip' cos-* v' . «/ t^a'
which proves the proposition enunciated.

In the particular case of an ellipsoid of revolution of which n
is the axial and m the equatorial semi-axis, compared with a
sphere of radius unity, both having their centres at 0', one of the
principal radii of curvature is the normal of the elliptic section,

which by the properties of the ellipse is equal to — m', m' denoting

the semi-conjugate diameter; and the other is the radius of curva-

HI/ ^

ture of the elliptic section, or — . Also ot is the perpendicular
■^ mn ^ ^

let fall from the centre on the tangent line of the section. Hence

from (5) or (6)

coszS</> _'nr' mm' m'^ _'!!7*m'* m*n^
cos i'B<f)' p* ' n ' mn n'p* ~ p* '

since tn-m' = mn. This agrees with Mr Stewart's result (p. 197),

since the Be and ^J? of Mr Stewart are the same as the R' and R

of equation (3).

On the Intensity of the Light reflected from

OR TRANSMITTED THROUGH A PiLE OF PlATES.

[From the Proceedings of the Royal Society, January 23, 1862.]

The frequent employment of a pile of plates in experiments
relating to polarization suggests, as a mathematical problem of
some interest, the determination of the mode in which the intensity
of the reflected light, and the intensity and degree of polarization
of the transmitted light, are related to the number of the plates,
and, in case they be not perfectly transparent, to their defect of
transparency.

The plates are supposed to be bounded by parallel surfaces,
and to be placed parallel to one another. They will also be
supposed to be formed of the same material, and to be of equal
thickness, except in the case of perfect transparency, in which case
the thickness does not come into account. The plates themselves
and the interposed plates of air will be supposed, as is usually the
case, to be sufficiently thick to prevent the occurrence of the
colours of thin plates, so that we shall have to deal with intensities
only.

On account of the different proportions in which light is
reflected at a single surface according as the light is polarized
in or perpendicularly to the plane of incidence, we must take
account separately of light polarized in these two ways. Also,
since the rate at which light is absorbed varies with its refrangi-
bility, we must take account separately of the different constituents
of white light. If, however, the plates be perfectly transparent, we
may treat white light as a whole, neglecting as insignificant the
chromatic variations of reflecting power. Let p be the fraction of
the incident light reflected at the first surface of a plate. Then
s. IV. 10

146 ON THE INTENSITY OF THE LIGHT REFLECTED FROM

1 — p may be taken as the intensity of the transmitted light*.

Also, since we know that light is reflected in the same proportion

externally and internally at the two surfaces of a plate bounded

by parallel surfaces, the same expressions p and I — p will serve

to denote the fractions reflected and transmitted at the second

surface. We may calculate p in accordance with Fresnel's formulae

from the expressions

. ., sini , ,

sm I = (1 ),

_ sitf {i — i') _ tan^ {i — i') ,„,

^~sinHi + 0' °^ ~ UvL^ipiT) ^ ''

according as the light is polarized in or perpendicularly to the
plane of incidence.

In the case of perfect transparency, we may in imagination
make abstraction of the substance of the plates, and state the
problem as follows : — There are 2m parallel surfaces {m being the
number of plates) on which light is incident, and at each of which
a given fraction p of the light incident upon it is reflected, the
remainder being transmitted; it is required to determine the
intensity of the light reflected from or transmitted through the
system, taking account of the reflexions, infinite in number, which
can occur in all possible ways.

This problem, the solution of which is of a simpler form than
that of the general case of imperfect transparency, might be solved
by a particular method. As, however, the solution is comprised in
that of the problem which arises when the light is supposed to be
partially absorbed, I shall at once pass on to the latter.

In consequence of absorption, let the intensity of light travers-
ing a plate be reduced in the proportion of 1 to 1 — qdx in passing
over the elementary distance dx within the plate. Let T be the
thickness of a plate, and therefore Tseci' the length of the path of
the light within it. Then, putting for shortness

g-qTseci'^g (3)^

* In order that the intensity may be measured in this simple way, which saves
rouble in the problem before us, we must define the intensity of the light trans-
mitted across the first surface to mean what would be the intensity if the light
were to emerge again into air across the second surface without suffering loss by
absorption, or by reflexion at that surface.

OR TRANSMITTED THROUGH A PILE OF PLATES. 147

1 to g will be the proportion in which the intensity is reduced by
absorption in a single transit. The light reflected by a plate will
be made up o£ that which is reflected at the first surface, and that
which suffers 1, 3, 5, etc. internal reflexions. If the intensity of
the incident light be taken as unity, the intensities of these various
portions will be

p,(l-p)V£r^(l-p)VY,etc.

and if r be the intensity of the reflected light, we have, by summing
a geometric series,

'--<-\^ «■

Similarly, if t be the intensity of the transmitted light,
and we easily find

1-P5'
which is in general less than 1, but becomes equal to 1 in the
limiting case of perfect transparency, in which case g = I.

The values of fi, i, and q in any case being supposed known, the
formulae (1), (2), (3), (4), (5) determine r and t, which may now
therefore be supposed known. The problem therefore is reduced
to the following: — There are m parallel plates of which each reflects
and transmits given fractions r, t of the light incident upon it :
light of intensity unity being incident on the system, it is required
to find the intensities of the reflected and refracted light.

Let these be denoted by <f> (m), \jr (m): Consider a system of
in-h n plates, and imagine these grouped into two systems, of vi
and n plates respectively. The incident light being represented
by unity, the light <f> (m) will be reflected fi^om the first group, and
yjr (m) will be transmitted. Of the latter the fraction i/r (m) will be
transmitted by the second group, and ^ (n) reflected. Of the latter
the fi'action •^|r (m) will be transmitted by the first group, and (/> (m)
reflected, and so on. Hence we get for the light reflected by the
whole system,

• (^ (m) + {yjrmy <f) (n) + {■fmY ^ (m) {cfmy + ...,
and for the light transmitted,

■\}r{m) a/t (w) + -i|r (m) (w) (/) (m) •v|r(M) + i/f (m) {<fmy {j)mf-^{n) + . . . ,

10—2

148 ON THE INTENSITY OF THE LIGHT BEFLECTED FEOM

which gives, by summing the two geometric series,

'^<^ + ^)='/'(^)+ l-^(m)^W ' ^^^'

^^^ + w\-^tMtW_ (7).

We get from (6)

^ (m + n) {1 - ^ (m) ^ (w)} = ^ (m) + (/> (n) [{-^mf - {j>my\ ;

and the first member of this equation being symmetrical with
respect to m and n, we get, by interchanging m and n and equating
the results,

(m) + ^ (n) [{-^mf - (^m)^} = <^ (w) + ^ (m) l(-«|m)2 - {^nf] ;

which is therefore constant. Denoting this constant for conveni-
ence by 2 cos a, we have

(^frmy= 1 -2 cos a. 4>{m) + (<l)my (8).

Squaring (7), and eliminating the function yjr by means of (8),
we find

{1 - ^ (in) ^ {n)Y {1 - 2 cos a . ^ (m + «) + [^ (m + n)J]

= {1 - 2 cos a . ^ (to) + {<i>my] {1-2 cos a . ^ (w) + ((^w)^}. . .(9).

From the nature of the problem, to and n are positive integers,
and it is only in that case that the functions <^, ■>{r, as hitherto
defined, have any meaning. We may, however, contemplate func-
tions (j), yjr oi a, continuously changing variable, which are defined
by the equations (6) and (7) ; and it is evident that if we can find
such functions, they will in the particular case of a positive integral
value of the variable be the functions which we are seeking.

In order that equations (6), (7) may hold good for a value zero
of one of the variables, suppose n, we must have (0) = 0, i/r (0) = 1.
The former of these equations reduces (9) for w = to an identical
equation. Differentiating (9) with respect to n, and after differen-
tiation putting « = 0, we find

(j)' (0) (/) (to) {1 - 2 cos a . ^ (to) + (<f)my] -I- cos a . </>' (to) - ^ (to) ^' (to)
= cos a . (j)' (0) {1 - 2 cos a . ^ (to) -I- (tpmy},

OR TRANSMITTED THROUGH A PILE OF PLATES. 149

or dividing out by <j) (m) — cos a (for (f> (m) = cos a would only lead
to <j) (m) = cos a = 0, i|r (m) = C),

<f>' (m) = 4>' (0) (1 - 2 cos a . (m) + (^my) (10).

Integrating this equation, determining the arbitrary constant
by the condition that (j)(m) = when m=,0, and writing /3 for
sin a . <j)' (0), we have

,, . sinm/3 ,,^,

6{m) = -^-. ^. (11).

^^ ■' sm (a + m/3)

Substituting in (8) and reducing, we find

r^rny = ^^^^^^^ (12).

But (8) was derived, not from (7) directly, but from (7) squared ;
and on extracting the square root of both sides of (12), we must
choose that sign which shall satisfy (7), and therefore we must
take the sign +, as we see at once on putting m = n = 0. The
equation (12) on taking the proper root, and (11), may be put
imder the form

<^(m) ^ ■<|r(m) ^ 1 ,^g,.

sin(m/S) sina sin(a + m/3)

and to determine the arbitrary constants a, y8 we have, putting
m = 1, and <^ (m) = r, ^jr (m) = t,

r t 1

sin/3 sina sin(a + ;S)
We readily get from equations (13),

1 J / \ J / \ sin a sin {« + (m + n) /3}
1 — * (m) m («) = -. — 7 r^. . — 7 -—^ ;

,. , X , / v sin a sin w/3

<f) {m + n) — (p (m) = -r-

.(14).

sin {« + (m + n) /3} sin (a + mB) '

whence the equations (6), (7) are easily verified. This verification
seems necessary in logical strictness, because we have no right to
assume d priori that it is possible to satisfy (6) and (7) for general
values of the variables; and in deriving the equation (10), the
equations (6) and (7) were only assumed to hold good for general
values of m and infinitely small values of n.

150 ON THE INTENSITY OF THE LIGHT REFLECTED FROM

The equations (13), (14) give the following gMCisi-geometrical
construction for solving the problem : — Construct a triangle of
which the sides represent in magnitude the intensity of the
incident, reflected, and refracted light in the case of a single
plate, and then, leaving the first side and the angle opposite to
the third unchanged, multiply the angle opposite to the second
by the number of plates ; the sides of the new triangle will repre-
sent the corresponding intensities in the case of the system of
plates. I say quasi- geometriced, because the construction cannot
actually be effected, inasmuch as the first side of our triangle
is greater than the sum of the two others, and the angles are
imaginary.

To adapt the formulae (13), (14) to numerical calculation, it will
be convenient to get rid of the imaginary quantities. Putting

^[(l+r + t)(l+r-t)(l + t-r){l-r-t)}=A...(15), ■■

we -have by the common formulae of trigonometry,

l+r^-t^ . ± V-TA

I a =

whence, putting

cos a = ^ : sm a =

Zr 2r

^(l+r'-t^ + A)^a (16),

we have

gv/-ia _ cQg g( ^ ,^_ 1 sin a = a^^.

It is a matter of indifference which sign be taken: choosing the
under signs, we have

2r sin a = — V— lA, e^"'i'' = a.
We have also

„ 1+f-f' . - r . V^A
cos /3 = 2^ , sm /3 = - sm a = ^-^ ,

no fresh ambiguity of sign being introduced. Putting therefore

^^(l+f-f^+A) = b (17),

we have

es/~i? = b ;
and equations (13) now give

<f>(m) _ ■\lr(m) _ 1
S" - 6-™ ~ a - a-' ~ ah"" - a-'b-"^ ^^^^-

OR TRANSMITTED THROUGH A PILE OF PLATES. 151

In the case of perfect transparency these expressions take a
simpler form, li r + t differ indefinitely little from 1, a and /3 will
be indefinitely small. Making a and /3 indefinitely small in (13)
and (14), and putting 1 — r for t, we find

4>(m)^ ylr(m) ^ 1

mr 1 — r l+(m — l)r

In this case it is evident that each of the 2m reflecting surfaces
might be regarded as a separate plate reflecting light in the pro-
portion of p to 1, and therefore we ought also to have, writing 2m
for m and p for r in the denominators of the equations (19),

<t>(m)_y}r(m)_ 1
2mp l-p l+(2m-l)p ^^

It is easy to verify that when ^ = 1 (4) reduces (19) to (20).

The following Table gives the intensity of the light reflected
from or transmitted through a pile of m plates for the values
1, 2, 4, 8, 16, 32, and co of m, for three degrees of transparency,
and for certain selected angles of incidence. The assumed refrac-
tive index fj, is 1-52. B = l— e-'^ is the loss by absorption in a
single transit of a plate at a perpendicular incidence, so that 8 =
corresponds to perfect transparency. The most interesting angles
of incidence to select appeared to be zero and the polarizing angle
w = tan~V ; but in the case of perfect transparency the result has
also been calculated for an angle of incidence a little (2°) greater
than the polarizing angle. <j) denotes the intensity of the reflected
and yjr that of the transmitted light, the intensity of the incident
light being taken at 1000. For oblique incidences it was necessary
to -distinguish between light polarized in and light polarized per-
pendicularly to the plane of incidence ; the suffixes 1, 2 refer to
these two kinds respectively. For oblique incidences a column is
added giving the ratio of -yjri to ■\jr^, which may be taken as a
measure of the defect of polarization of the transmitted light.
No such column was required for S = and i = -sr, because in
this case yjr^ = 1000.

* From a paper by M. Wild in Pogyendorff's Annalen [Vol. ix, 1856, p. 240],
I find that the formulffi for the particular case of perfect transparency have already
been given by M. Neumann. His demonstration does not appear to have been
published. [The question of the efficiency of a pile of plates was also considered by
Fresnel himself; his solution, which neglects opacity, was published posthumously,
(Euvres ComplUes, Vol. ii, 1868, pp. 789-92.]

152 ON THE INTENSITY OF THE LIGHT REFLECTED FROM

o

•^

i-H
00
00

1>

g

■^

05
CO
CD

r-l O

O O

o o

Til Tj

o
II

■a-

■s-

^H I— I (>)

-9-l-S-

I-H CD

(jq cx>
cq o

^ o

.-I o

o o

n

CD

rH O

CD

^ ^

CO

O

T— (

O

o
en

+
II

•a-l-a-
■^

o o

00
Oi
05

CD
OS

00
CD

l-H O

I-H O

M O
05 O

<sq

^

05

00

s

1

00

n

00

I-H

g

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I> o

OR TRANSMITTED THROUGH A PILE OF PLATES. 153

The intensity of the light reflected from an infinite number of
plates, as we see from (18), is a~^; and since a is changed into a~^
by changing the sign of a or of A,

a-^ = ^^(l+r^-t^-A) (21),

which is equal to 1 in the case of perfect transparency. Accordingly
a substance which is at the same time finely divided, so as to present
numerous reflecting surfaces, and which is of such a nature as
to be transparent in mass, is brilliantly white by reflected light,
— for example snow, and colourless substances thrown down as
precipitates in chemical processes.

The intensity of the light reflected from a pile consisting of an
infinite number of similar plates falls off rapidly with the trans-
parency of the material of which the plates are composed, especially
at small incidence. Thus at a perpendicular incidence we see from
the above Table that the reflected light is reduced to little more
than one half when 2 per cent, is absorbed in a single transit, and
to less than a quarter when 10 per cent, is absorbed.

With imperfectly transparent plates, little is gained by multi-
plying the plates beyond a very limited number, if the object be to
obtain light, as bright as may be, polarized by reflexion. Thus the
Table shows that 4 plates of the less defective kind reflect 79 per
cent., and 4 plates of the more defective as much as 94 per cent.,
of the light that could be reflected by a greater number, whereas
4 plates of the perfectly transparent kind reflect only 60 per cent.

The Table shows that while the amount of light transmitted at
the polarizing angle by a pile of a considerable number of plates is
materially reduced by a defect of transparency, its sbate of polari-
zation is somewhat improved. This result might be seen without
calculation. For while no part of the transmitted light which is
polarized perpendicularly to the plane of incidence underwent
reflexion, a large part of the transmitted light polarized the other
way was reflected an even number of times ; and since the length
of path of the light within the absorbing medium is necessarily
increased by reflexion, it follows that a defect of transparency must
operate more powerfully in reducing the intensity of light polarized
in, than of light polarized perpendicularly to the plane of polari-
zation. But the Table also shows that a far better result can be

154 ON THE INTENSITY OF THE LIGHT REFLECTED FROM

obtained, as to the perfection of the polarization of the transmitted
light, without any greater loss of illumination, by employing a larger
number of plates of a more transparent kind.

Let us now confine our attention to perfectly transparent plates,
and consider the manner in which the degree of polarization of the
transmitted light varies with the angle of incidence.

The degree of polarization is expressed by the ratio of -v/ti to ■yjr^,
which for brevity will be denoted by x- When % = 1 there is no
polarization ; when % = the polarization is perfect, in a plane
perpendicular to the plane of incidence. Now yjr (which is used
to denote t/ti or •v/r^ as the case may be) is given in terms of p by
one of the equations (20), and p is given in terms of i — i' and i + i'
by Fresnel's formulae (2). Put

i — i' = 6, i + i' = cr\
then, from (1),

di di' dd da . ., , •

- - . = 7 V = I ■■ — I ■, = 7 ■ — 7 ■, = cos I cos I da, suppose,

tan I tan i tan i — tan i tan i + tan i > cc '

whence

dO = &V!\dda>, da = avn.(j-da3 (22);

and we see that i and to increase together from i = to i = ^ir.
We have also

sin^^
^'~sinV^

dp^ = -^— — (sin 0- cos 6d6 - sin 6 cos crda)

'^ am' a- ^ '

2sin2^, . .
= ^ — (cos p — COS cr) da :

tan^d"
''^"tanV

dp'i = ~, — : — (tan a sec^ 6dQ — tan 6 sec^ crdcr)
"^ tan''(7 '

2 sin^ ^ cos 0- , -, , cos <r

...,„.. ( nC\'S rr - C^CW H\ n r.\ —

cos* p sm'^ cr ' cos" V

dpi.

Now cos 6 — cos (7 or 2 sin i sin i' is positive ; and cos cr is positive
from 1 = to i = ct, and negative from i = ra- to i = \tt. But (20)
shows that y^ decreases as p increases. From i = to i — -sr, p^ in-
creases and p2 decreases, and therefore -v/fi decreases and y^r^ increases,
and therefore on both accounts % decreases. When i = ot, dpjdi is
still positive, and therefore d-^jr^di negative, but yfr^ has its maximum
value 1, so that on passing through the polarizing angle % still

OR TRANSMITTED THROUGH A PILE OF PLATES. 155

decreases, or the polarization improves. When the plates are very
numerous, i|f2 = 1 at the polarizing angle, and on both sides of it
decreases rapidly, whereas tlr^, which is always small, suffers no
particular change about the polarizing angle. Hence in this case
X must be a minimum a little beyond the polarizing angle. Let
us then seek the angle of incidence which makes x a minimum in
the case of an arbitrary number of plates.

We have from (20) and (2),

_ sin' o- - sin^ 6 sin- a cos" 6 + (2m - 1) sin' cos' g-

^ ~ sin' (7 + (2m - 1) sin' 6 ' sin' a cos' ff - sin' 9 cos' o-
_ sin' cr cos' 6 + (2m - 1) sin' 6 cos' cr
sin'o- + (2m-l)sin'^

= 1 ^ (23)

cosec'W + (2m — l)cosec'o- ^'

Hence % is a minimum along with cosec' ^ + (2m — 1) cosec'o-.
Differentiating, and taking account of the formulae (22), we find,
to determine the angle of maximum polarization, the very simple
equation

cos sin' o- + (2m - 1) cos o- sin' -= (24).

For any assumed value of i from w to ^tt, this equation gives at
once the value of to, that is, the number of, plates of which a pile
must be composed in order that the assumed incidence may be that
of maximum polarization of the transmitted light. The equation
may be put under the form

- _ tan cr sin cr _ 1

tan ' sin ^ '^Pip^ '
Now we have seen that both pi and pa continually increase, and
therefore m continually decreases, from i = rs- to i = ^ir. At the
first of these limits p^ — 0, and therefore m = oo . At the second
Pj = P2=1, and therefore m = 1. Hence with a single plate the
polarization of the transmitted light continually improves up to a
grazing incidence, but with a pile of plates the polarization attains
a maximum at an angle of incidence which approaches indefinitely
to the polarizing angle as the number of plates is indefinitely
increased.

Eliminating m from (23) and (24), we find

X = ~ cos 0COS (T (25),

156 THE LIGHT EEFLECTED FROM A PILE OF PLATES.

■which determines for any pile %i, the defect of maximum polari-
zation of the transmitted light, in terms of the angle of incidence
for which the polarization is a maximum. We have, from (25),
(22), and (24),

c^%i = (sitf Q cos o- + sin^ a cos 0) cZw = — 2 («i — 1) cos o- sitf Odm,

and cos o- is negative. Hence %i decreases as « (and therefore %)
decreases, or as m increases. For m=\,i — ^tt and Xi = f^~^> fo^
m = 00 , cos o- = 0, and therefore Xi = 0, or the maximum polarization
tends indefinitely to become perfect as the number of plates is
indefinitely increased.

For a given number of plates the angle of maximum polari-
zation may be readily found from (24) by the method of trial and
error. But for merely examining the progress of the functions,
instead of tabulating i for assumed values of m, it will serve
equally well to tabulate m for assumed values of i. The following
Table gives for assumed angles of incidence, decreasing by 6° from
90°, the number of plates required to make these angles the angles
of maximum polarization of the transmitted light, and the value of
Xi , which determines the defect of polarization.

i= 90° 85° 80° 75° 70° 65° 60° 5^40' (=i!r)
m= 1 1-330 1-944 2-913 4-921 9-775 30-372 oo
Xi=-433 -422 -390 -337 -265 -177 -075

Report on Double Refraction.

[From the Report of the British Association for the Advancement
of Science, Cambridge, 1862, pp. 253—282.]

I regret to say that in consequence of other occupations the
materials for a complete report on Physical Optics, which the
British Association have requested me to prepare, are not yet
collected and digested. Meanwhile, instead of requesting longer
time for preparation, I have thought it would be well to take up
a single branch of the subject, and offer a report on that alone.
I have accordingly taken the subject of double refraction, having
mainly in view a consideration of the various dynamical theories
which have been advanced to account for the phenomenon on the
principle of transversal vibrations, and an indication of the experi-
mental measurements which seem to me most needed to advance
this branch of optical science. As the greater part of what has
been done towards placing the theory of double refraction on a
rigorous dynamical basis is subsequent to the date of Dr Lloyd's
admirable report on " Physical Optics,' I have thought it best to
take a review of the whole subject, though at the risk of repeating
a little of what is already contained in that report.

The celebrated theory of Fresnel was defective in rigour in two
respects, as Fresnel himself clearly perceived. The first is that the
expression for the force of restitution is obtained on the supposition
of the absolute displacement of a molecule, whereas in undulations
of all kinds the forces of restitution with which we are concerned
are those due to relative displacements. Fresnel endeavoured to
show, by reasoning professedly only probable, that while the
magnitude of the force of restitution is altered in passing from
absolute to relative displacements, the law of the force as to its
dependence on the direction of vibration remains the same. The
other point relates to the neglect of the component of the force in
a direction perpendicular to the front of a wave. In the state of
things supposed in the calculation of the forces of restitution called

158 REPORT ON DOUBLE REFRACTION.

into play by absolute displacements, there is no immediate recog-
nition of a wave at all, and a molecule is supposed to be as free to
move in one direction as in another. But a displacement in a
direction perpendicular to the front of a wave would call into play
new forces of restitution having a resultant not in general in the
direction of displacement ; so that even the component of the force
of restitution in a direction parallel to the front of a wave would
have an expression altogether different from that determined by
the theory of Fresnel. But the absolute displacements are only
considered for the sake of obtaining results to be afterwards applied
to relative displacements ; and Fresnel distinctly makes the suppo-
sition that the ether is incompressible, or at least is sensibly so
under the action of forces comparable with those with which we
are concerned in the propagation of light. This supposition
removes the difficulty; and though it increases the number of
hypotheses as to the existing state of things, it cannot be objected
to in point of rigour, unless it be that a demonstration might be
required that incompressibility is not inconsistent with the assumed
constitution of the ether, according to which it is regarded as con-
sisting of distinct material points, symmetrically arranged, and
acting on one another with forces depending, for a given pair,
only on the distance. Hence the neglect of the force perpendicular
to the fronts of the waves is not so much a new defect of rigour, as
the former defect appearing under a new aspect.

I have mentioned these points because sometimes they are
slurred over, and Fresnel's theory spoken of as if it had been
rigorous throughout, to the injury of students and the retardation
of the real progress of science ; and sometimes, on the other hand,
the grand advance made by Fresnel is depreciated on account of
his theory not being everywhere perfectly rigorous. If we reflect
on the state of the subject as Fresnel found it, and as he left it, the
wonder is, not that he failed to give a rigorous dynamical theory,
but that a single mind was capable of effecting so much.

The first deduction of the laws of double refraction, or at least
of an approximation to the true laws, from a rigorous theory is due
to Cauchy*, though Neumannf independently, and almost simul-
taneously, arrived at the same results. In the theory of Cauchy

* Mfmoires de V Academic, torn, x, p. 293.

t Poggendorff's Annalen, Vol. xxv, p. 418 (1832).

REPORT ON DOUBLE REFRACTION.

159

and Neumann the ether is supposed to consist of distinct particles,
regarded as material points, acting on one another by forces in the
line joining them which vary as some function of the distances,
and the arrangement of these particles is supposed to be different
in different directions. The medium is further supposed to possess
three rectangular planes of symmetry, the double refraction of
crystals, so far as has been observed, being symmetrical with
respect to three such planes. The equations of motion of the
medium are deduced by a method similar to that employed by
Navier in the case of an isotropic medium. The equations arrived
at by Cauchy, the medium being referred to planes of symmetry,
contain nine arbitrary constants, three of which express the
pressures in the principal directions in the state of equilibrium.
Those employed by Neumann contain only six suCh constants, the
medium in its natural state being supposed free from pressure.

In the theory of double refraction, whatever be the particular
dynamical conditions assumed, everything is reduced to the deter-
mination of the velocity of propagation of a plane wave propagated
in any given direction, and the mode of vibration of the particles
in such a wave which must exist in order that the wave may be
propagated with a unique velocity. In the theory of Cauchy now
under consideration, the direction of vibration and the reciprocal
of the velocity of propagation are given in direction and magnitude
respectively by the principal axes of a certain ellipsoid, the equation
of which contains the nine arbitrary constants, and likewise the
dfrection-cosines of the wave-normal. Cauchy adduces reasons for
supposing that the three constants G, H, I, which express the
pressures in the state of equilibrium, vanish, which leaves only
six constants. For waves perpendicular to the principal axes, the
squared velocities of propagation and the corresponding directions
of vibration are given by the following Table -.-^

W^flvp-normal *

X

y

z

Direction of vibration... -

a;

L

R

Q

y

R

M

P

z

Q

P

N

160 REPORT ON DOUBLE REFRACTION.

For waves in these directions, then, the vibrations are either
wholly normal or wholly transversal. The latter are those with
which we have to deal in the theory of light. Now, according
to observation, in any one of the principal planes of a doubly
refracting crystal, that ray which is polarized in the principal
plane obeys the ordinary law of refraction. In order therefore
that the conclusions of this theory should at all agree with obser-
vation, we must suppose that in polarized light the vibrations are
parallel, not perpendicular, to the plane of polarization.

Let I, m, n be the direction-cosines of the wave-normal. In the
theory of Cauchy and Neumann, the square v^ of the velocity of
propagation is given by a cubic of the form

if -I- OL^v^ + a.iV'^ + a, = 0,

where a.^, 0.4,, Wg are homogeneous functions of the 1st order as
regards L, M, N, P, Q, R, and homogeneous functions of the orders
2, 4, 6 as regards I, m, n, involving even powers only of these
quantities. For a wave perpendicular to one of the principal
planes, that of yz suppose, the cubic splits into two rational
factors, of which that which is of the first degree in v^, namely,

iP-m^R-n^'Q,

corresponds to vibrations perpendicular to the principal plane.
This is the same expression as results from Fresnel's theory,
and accordingly the section, by the principal plane, of one sheet
of the wave-surface, which in this theory is a surface of three
sheets, is an ellipse, and the law of refraction of that ray which
is polarized perpendicularly to the principal plane agrees exactly
with that given by the theory of Fresnel.

For the two remaining waves, the squared velocities of propa-
gation are given by the quadratic

(f _ mm- n'P) (v^ - mT - v^K) - ^rri'w'P'' = . . .(1) ;
but according to observation the ray polarized in the principal
plane obeys the ordinary law of refraction. Hence (1) ought to
be satisfied by v^ - {m? + nF) P = 0, which requires that

(ilf-P)(i\r-P) = 4P^

on which supposition the remaining factor must evidently be linear
as regards m^, n^, and therefore must be

tf-ni'M-n-N,

REPORT ON DOUBLE REFRACTION. 161

since it gives when equated to zero v^=M, or v^=^N for w = 1,
or w=l. And since the same must hold good for each of the
principal planes, we must have the three following relations
between the six constants,

{M-P){N-P) = ^P^; (N-Q)(L-Q) = 4>Q';

{L-R){M-R) = ^m (2).

The existence of six constants, of which only three are wanted
to satisfy the numerical values of the principal velocities of propa-
gation in a biaxal crystal, permits of satisfying these equations;
so that the law that the ray polarized in the plane of incidence,
when that is a principal plane, obeys the ordinary law of refraction
is not inconsistent with Cauchy's theory. This simple law is, how-
ever, not in the slightest degree predicted by the theory, nor even
rendered probable, nor have any physical conditions been pointed
out which would lead to the relations (2) ; and, indeed, from the
form of these equations, it seems hard to conceive what physical
relations they could express. Hence an important desideratum
would be left, even if the theory were satisfactory in all other
respects.

The equation for determining v^ virtually contains the theo-
retical laws of double refraction, which are embodied in the form
of the wave-surface. The wave-surface of Cauchy and Neumann
does not agree with that of Fresnel, except as to the sections of two
of its sheets by the principal planes, the third sheet being that
which relates to nearly normal vibrations. Nevertheless the first
two sheets, being forced to agree in their principal sections with
Fresnel's surface, differ from it elsewhere extremely little. In
Arragonite, for instance, in a direction equally inclined to the
principal axes, assuming Rudberg's indices* for the line D, I find
that the velocities of propagation of the two polarized waves,
according to the theory of Cauchy and Neumann, differ from
those resulting from the theory of Fresnel only in the tenth place
of decimals, the velocity in air being taken as unity. Such a
difference as this would of course be utterly insensible in experi-
ment. In like manner the directions of the planes of polarization
according to the two theories, though not rigorously, are extremely
nearly the same, the plane of polarization of a wave in which the

* Annales de Ghimie, torn. xLviii, p. 254 (1831).
S. IV. 11

162 REPORT ON DOUBLE REFRACTION.

vibrations are nearly transversal being defined as that containing
the direction of propagation and the direction of vibration, in
harmony with the previously established definition for the case
of strictly transversal vibrations.

Hence as far as regards the laws of double refraction of tho two
waves which alone are sujiposed to relate to tho visible pheno-
menon, and of the accompanying jiolarization, this theory, hy the
aid of the forced relations (2), is very successful. I am not now
discussing the generality, or, on the contrary, the artificially
restricted- nature, of tho fundamental suppositions as to the state
of things, but only the degree to which the njsults an; in accord-
ance with observed facts. But as regards the third wave the case
is very different. That theory should jjoint tn the ncccissary
existence of such a wave consisting of strictly normal vibrations,
and yet to which no known phenomenon can be referred, is bad
enough; but in the present theory the vibrations are not (.'ven
strictly normal, except for waves in a direction ])erpendicular to
any one of the principal axes. In Iceland spar, for instance, for
waves propagated in a direction inclined 45° to the axis, it follows
from the numerical values of the refractive indices for the fixed
line D given by Rudberg that the two vibrations in the principal
plane which can be propagated independently of each oth(;r are
inclined at angles of 9° 50' and 80° 10', or say 10° and 80°, to the
wave-normal. We can hardly suppose that a mere change of
inclination in the direction of vibration <jf' from 10° to 80" with
the wave-front makes all the difference whether the wave belongs
to a long-known and evident phenomenon, no other than the
ordinary refraction in Iceland spar, or not to any visible pheno-
menon at all.

It is true that before there can be any question of the third
wave's being perceived it must be supposed excited, and the means
of exciting it consist in the incident vibrations in air, which by
hypothesis are strictly transversal. Hence we have to inquire
whether the intensity of the third wave is such as to lead us
to expect a sensible phenomenon answering to it. This leads
us to the still more uncertain subject of the intensity of light
reflected or refracted at the surface of a crystal — more uncertain
because it not only depends on the laws of int(;iTial propagation,
and involves all the hypotheses on which these laws are theoreti-

REPORT ON DOUBLE REFRACTION. 163

cally deduced, but requires fresh hypotheses as to the state of
things at the confines of two media, introducing thereby fresh
elements of uncertainty. But for our present purpose no exact
calculation of intensities is required; a rough estimate of the
intensity of the nearly normal vibrations is quite sufficient.

In order to introduce as little as possible relating to the theory
of the intensity of reflected and refracted light, suppose the incident
light to fall perpendicularly on the surface of a crystal, and let this
be a surface of Iceland spar cut at an inclination of 45° to the axis.
For a cleavage plane the result would be nearly the same. Let
the incident light be polarized, and the vibrations be in the principal
plane, which therefore, according to the theory now under considera-
tion, must be the plane of polarization. The incident vibrations
are parallel to the surface, and accordingly inclined at angles of
9° 50' and 80° 10' to the directions of the nearly transversal and
nearly normal vibrations, respectively, within the crystal. Hence
it seems evident that the amplitude of the latter must be of the
order of magnitude of sin 9° 50', or about ^, the amplitude of
^dbration in the incident light being taken as unity. The velocity
of propagation of the nearly normal vibrations being to that of the
nearly transversal roughly as V3 to 1, as will immediately be shown,
it follows that the vis viva of the nearly normal would be to that
of the nearly transversal vibrations in a ratio comparable with that
of V3 X sin^ 9° 50' to 1, or about ^ij to 1. Hence the intensity of
the nearly normal vibrations is by no means insignificant, and
therefore it is "a very serious objection to the theory that no
corresponding phenomenon should have been discovered. It has
been suggested by some of the advocates of this theory that the
normal vibrations may correspond to heat. But the fact of the
polarization of heat at once negatives such a supposition, even
without insisting on the accumulation of evidence in favour of the
identity of radiant heat and light of the same refrangibility.

But the objections to the theory on the ground of the absence
of some unknown phenomenon corresponding with the third ray, to
which the theory necessarily conducts, are not the only ones which
may be urged against it in connexion with that ray. The existence of
normal or nearly normal vibrations entails consequences respecting
the transversal which could hardly fail to have been detected by

11—2

164 REPORT ON DOUBLE REFRACTION.

observation. In the first place, the vis viva belonging to the
normal vibrations is so much abstracted from the transversal, which
alone by hypothesis constitute light, so that there is a loss of light
inherent in the very act of passage from air into the crystal, or
conversely, from the crystal into air. About J^tli of the whole
might thus be expected to be lost at a single surface of Iceland
spar, the surface being inclined 45° to the axis, and the light being
incident perpendicularly, and being polarized in the principal plane;
and the loss would amount to somewhere about J^th in passage
across a plate bounded by parallel surfaces, by which amount the
sum of the reflected and transmitted light ought to fall short of
the incident. And it is evident that something of the same kind
must take place at other inclinations to the axis and at other
incidences. The loss thus occasioned in multiplied reflexions
could hardly have escaped observation, though it is not quite so
great as might at first sight appear, as the transversal vibrations
produced back again by the normal would presently become
sensible.

But the most fatal objection of all is that urged by Green*
against the supposition that normal vibrations could be propagated
with a velocity comparable with those of transversal. As trans-
versal vibrations are capable (according to the suppositions here
combated) of giving rise at incidence on a medium to normal or
nearly normal vibrations within it, so conversely the latter on
arriving at the second surface are capable of giving rise to
emergent transversal vibrations; so that not only would normal
vibrations entail a loss of light in the quarter in which light is
looked for, but would give rise to light (of small intensity it is
true, but by no means imperceptible) in a quarter in which other-
wise there would have been none at all. Thus in the case supposed
above, the intensity of the light produced by nearly normal vibra-
tions giving rise on emergence to transversal vibrations would be
somewhere about the {^^f or the ^ of the incident light. In the
case of light transmitted through a plate, the rays thus produced
would be parallel to the incident, or to the emergent rays of the
kind usually considered ; but if the plate were wedge-shaped the
two would come out in different directions, and with sunlight the
former could not fail to be perceived. The only way apparently of

* Gamb. Phil. Trans., Vol. vii, p. 2. [Green's Math. Papers, p. 243.]

EEPOET OX DOUBLE REFRACTION. 165

getting over this difficulty, is by making the perfectly gratuitous
assumption that the medium, though perfectly transparent for
the more nearly transversal vibrations, is intensely opaque for
those more nearly normal.

Lastly, Green's argument respecting the necessity of suppos-
ing the velocity of propagation of normal vibrations very great
has here full force as an objection against this theorj^ The
constants P, Q, R are the squared reciprocals of the three principal
indices of refraction, which are given by observation, and L, 21, iV
are determined in terms of P, Q, R by the equations (2), by the
solution of a quadratic equation. In the case of a uniaxal crystal
everything is symmetrical about one of the axes, suppose that of z,
which requires, as Cauchy has shown, that L =M= SR, and P = Q;
and of the equations (2) one is now satisfied identically, and the
two others are identical with each other, and give

For an isotropic medium we must have L = M=N = 3P = 3Q = 3R,
and the three equations (2) are satisfied identically. The velocity
of propagation of normal must be to that of transversal vibrations
as V3 to 1, and cannot therefore be assumed to be what may be
convenient for explaining the law of intensity of reflected light.

The theory which has just been discussed is essentially bound
up with the supposition that in polarized light the vibrations are
parallel, not perpendicular, to the plane of polarization. In prose-
cuting the study of light, Cauchy saw reason to change his views
in this respect, and was induced to examine whether his theory
could not be modified so as to be in accordance with the latter
alternative. The result, constituting what may be called Cauchy's
second theory, is contained in a memoir read before the Academy,
May 20, 1839*. In this he refers to his memoir on dispersion, in
which the fundamental equations are obtained in a manner some-
what difi'erent from that given in his " Exercices," but based on the
same suppositions as to the constitution of the ether. In the
new theory Cauchy retains the three constants G, H, I, expressing
the pressures in equilibrium, which formerly he made vanish, the

* " Sur la Polarisation reptiligne, et la double Refraction," Mem. de VAcademie,
torn, xvm, p. 153.

166

REPORT ON DOUBLE REFRACTION.

medium being supposed as before to be symmetrical with respect
to three rectangular planes. The squares of the velocities of
propagation, and the corresponding directions of vibration for the
three waves which can be propagated in the direction of each
of the principal axes, are given by the following Table.

Wave-normal

X

y

z

Direction of vibration... ■

X

L + G

R + H

Q+I

y

R+0

M+H

P+I

z

Q+G

P + H

N+I

According to observation, in each of the principal planes the
ray polarized in that plane obeys the ordinary law of refraction,
and therefore if we suppose that in polarized light the vibrations,
at least when strictly transversal, are perpendicular to the plane
of polarization, we must assume that R +H= Q + I, P + I = R + G,
Q + G = P + H, which are equivalent to only two distinct relations,
namely,

P-G = Q-H=R-T (3).

For a wave parallel to one of the principal axes, as that of x,
the direction of that axis is one of the three rectangular directions
of vibration of the waves which are propagated independently. For
such vibrations the velocity (v) of propagation is given by the
formula

v' = vi'(R + H) + n''(Q + I),

which by (3) is reduced to

v' = R + H=Q + I,

so that on the assumption that the velocity of propagation is the
same for a wave perpendicular to the axis of y as for one perpen-
dicular to the axis of z when the vibrations are parallel to the axis
of X, the law of ordinary refraction in the plane of yz follows from
theory.

For the two remaining waves which can be propagated inde-
pendently in a given direction perpendicular to the axis of x, the

REPORT ON DOUBLE REFRACTION. 167

vibrations are only approximately normal and transversal respec-
tively. In fact, for the three waves which can travel independently
in any given direction, the directions of vibration are not affected
by the introduction of the constants expressing equilibrium-
pressures, but only the velocities of propagation. The squares
of the velocities of propagation of the two waves above mentioned
are given as before by a quadratic ; and in order that the velocity
of propagation of the nearly transversal vibrations may be expressed
by the formula

v'' = c''m'' + b''n'' (4),

in conformity with the ellipsoidal form of the extraordinary wave
surface in a uniaxal crystal, and the assumed elliptic form of the
section of one sheet of the wave-surface in a biaxal crystal by a
principal plane, the quadratic in question must split into two
rational factors, which leads to precisely the same condition as
before, namely that expressed by the first of equations (2); and
by equating to zero the corresponding factor, we get

v' = (P + H)m^ + {P + 1)71^,

which is in fact of the form (4). Applying the same to each of the
other principal axes, we find again the three relations (2).

Hence Cauchy's second theory, in which it is supposed that in
polarized light the vibrations (in air or in an isotropic medium) are
perpendicular to the plane of polarization, leads like the first to
laws of double refraction, and of the accompanying polarization,
differing from those of Fresnel only by quantities which may be
deemed insensible. This result is, however, in the present case
only attained by the aid of two sets of forced relations, namely (2)
and (3), that is, relations which there is nothing cb priori to indicate,
and which are not the expression of any simple physical idea, but
are obtained hy forcing the theory, which in its original state is
of a highly plastic nature from the number of arbitrary constants
which it contains, to agree with observation in some particulars,
which beiag done, theory by itself makes known the rest. As
regards the third ray by which this theory like its predecessor is
hampered, there is nearly as much to be urged against the present
theory as the former. There is, however, this difference, that, as
there are only five relations, (2) and (3), between nine arbitrary
constants, there remains one arbitrary constant in the expressions

168 REPORT ON DOUBLE REFRACTION.

for the velocities of propagation after satisfying the numerical
values of the three principal indices of refraction, by a proper
disposal of which the objections which have been mentioned may
to a certain extent be lessened, but by no means wholly overcome.

I come now to Green's theory, contained in a very remarkable
memoir " On the Propagation of Light in Crystallized Media," read
before the Cambridge Philosophical Society, May 20, 1839*, and
accordingly, by a curious coincidence, the very day that Cauchy's
second theory was presented to the French Academy. Besides the
great interest of the memoir in relation to the theory of light,
Green has in it, as I conceive, given for the first time the true
equations of equilibrium and motion of a homogeneous elastic solid
slightly disturbed from its position of equilibrium, which is one of
constraint under a uniform pressure different in different directions.
In a former memoirf he had given the equations for the case in
which the undisturbed state is one free from pressure^. When I
speak of the true equations, I mean the equations which belong
to the problem when not restricted in generality by arbitrarily
assumed hypotheses, and yet not containing constants which are
incompatible with any well-ascertained physical principle. It is
right ,to mention, however, that on this point mathematicians are
not agreed; M. de Saint- Venant, for instance, maintains the justice
of the more restricted equations given by Cauchy§, though even
he would not conceive the latter equations applicable to such
solids as caoutchouc or jelly.

In these papers Green introduced into the treatment of the
subject, with the greatest advantage, the method of Lagrange, in
which the partial differential equations of motion are obtained from
the variation of a single force-function, on the discovery of the
proper form of which everything turns. Green's principle is thus
enunciated by him: — "In whatever manner the elements of any
material system may act on each other, if all the internal forces be
multiplied by the elements of their respective directions, the total
sum for any assigned portion of the mass will always be the exact

* Gamb. Phil. Trans., Vol. tii, p. 120. [Green's Math. Papers, p. 291.]
t " On the Reflexion and Refraction of Light," Gamb. Phil. Trans. Vol. vii, p. 1.
Bead Dee. 11, 1837. [Green's Math. Papers, pp. 243, 280.]

J They are virtually given, though not actually written down at length.
§ Gomptes Bendus, torn, liii, p. 1105 (1861).

REPORT ON DOUBLE KEFRACTION. 169

differential of some function." In accordance with this principle,
the general equation may be put under the form

jjjpdxdydz (^^hu + -^J-hv + ~ Sw) = jjjdxdydzB(l>.. .(5),

where x, y, z are the equilibrium coordinates of any particle, p the
density in equilibrium, u, v, w the displacements parallel to x, y, z,
and (^ the function in question. <^ is in fact the function the
variation of which in passing from one state of the medium to
another, when multiplied by dxdydz, expresses the work given
out by the portion of the medium occupying in equilibrium the
elementary parallelepiped dxdydz, in passing from the first state
to the second. The portion of the medium which in the state of
equilibrium occupied the elementary parallelepiped becomes in the
changed state an oblique-angled parallelepiped, whose edges may
be represented by dx (1 + Si), dy (1 + s^, c^^ (1 + S3), and the cosines
of the angles between the second and third, third and first, and first
and second of these edges by «, /3, 7, which in case the disturbance
be small will be small quantities only. It is manifest that the
function </> must be independent of any linear or angular displace-
ment of the element dxdydz, and depend only on the change of
form of the element, and therefore on the six quantities Si, So, S3,
«, /S> 7. which may be expressed by means of the nine differential
coefficients of u, v, w with respect to x, y, z, of which therefore ^ is
a function, but not any function, since it involves not nine, but only
six independent variables. If the disturbance be small, the six
quantities Si, Sj, S3, a, /3, 7 will be small likewise, and <^ may be
expressed in a convergent series of the form

'P — 4'o + 'pi + 4>2 + 4>s + ■ ■ ■ >
where <j)o, ^1, 4>i, 4>s, etc. are homogeneous functions of the six
quantities, of the orders 0, 1, 2, 3, etc. ; and if the motion be
regarded as indefinitely small, the functions cpg, (f>i... will be
insensible, the left-hand member of equation (5) being of the
second order as regards u, v, w. (po, being a constant, will not
appear in equation (5), and ^1 will be equal to zero in case the
medium in its undisturbed state be free from internal pressure,
but not otherwise. The function (p^, being a homogeneous function
of six independent variables of the second order, contains in its
most general shape twenty-one arbitrary constants, and <pi which
is of the first order introduces six more, so that the most general

170 REPORT ON DOUBLE REFRACTION.

expression for ^ contains no less than twenty-seven arbitrary-
constants, all which appear in the expressions for the internal
pressures and in the partial differential equations of motion*-

The general expressions for the internal tensions in an elastic
medium and the general equations of equilibrium or motion which
were given by Cauchy, and which are written at length in the 4th
volume of the Exercices de MatMmatiques, contain twenty-one
arbitrary constants when the undisturbed state of the medium is
one of uniform constraint, and fifteen when it is one of freedom
from pressure. In the latter case. Green's twenty-one constants
are reduced to two, and Cauchy's fifteen to only one, when the
medium is isotropic. Green's equations comprise Cauchy's as a
particular case, as will be shown more at length further on. It
becomes an important question to inquire whether Cauchy's equa-
tions involve some restrictive hypothesis as to the constitution of
the medium, so as to be in fact of insufficient generality, or whether,
on the other hand, Green's equations are reducible to Cauchy's by
the introduction of some well-ascertained physical principle, and
therefore contain redundant constants.

In the formation of Cauchy's equations, not only is the medium
supposed to consist of material points acting on one another by
forces which depend on the distance only (a supposition which,
at least when coupled with the next, excludes the idea of molecular
polarity), but it is assumed that the displacements of the individual
molecules vary from molecule to molecule according to the variation
of some continuous function of the coordinates; and accordingly the
displacements u, v, w of the molecule whose coordinates in equi-
librium are x -\- Ax, y + Ay, z + Az are expanded by Taylor's theorem
in powers of Ax, Ay, Az, and the differential coefficients dujdx, etc.
are put outside the sign of summation. The motion, varying from

* The twenty-seven arbitrary constants enter the equations of motion in such
a manner as to be there equivalent to only twenty-six distinct constants, the
physical interpretation of which analytical result will be found to be that a uniform
pressure alike in all directions, in the undisturbed state of the medium, produces
the same effect on the internal movements when the medium is disturbed as
a certain internal elasticity, alike in all directions, and of a very simple kind,
which is possible in a medium unconstrained in its natural state. The twenty-one
arbitrary constants belonging to ». medium unconstrained in its natural state are
not reducible in the equations of motion, any more than in the expressions for the
internal tensions, to a smaller number.

REPORT ON DOUBLE REFRACTION. 171

point to point, of the medium taken as a whole, or in other words
the mean motion, in any direction, of the molecules in the neigh-
bourhood of a given point, must not be confounded with the
motion of the molecules taken individually. The medium being
continuous, so far as anything relating to observation is concerned,
the former will vary continuously from point to point. But it by
no means follows that the motion of the molecules considered
individually should vary from one to another according to some
function of the coordinates. The motion of the individual mole-
cules is only considered for the sake of deducing results from
hypotheses as to the molecular constitution and molecular forces
of the medium, and in it we are concerned only with the relative
motion of molecules situated so close as to act sensibly on each
other. It would seem to be very probable, a priori, that a portion
by no means negligible of the relative displacement of a pair of
neighbouring molecules should vary in an irregular manner from
pair to pair ; and indeed if the medium tends to relieve itself from
a state of constrained distortion, this must necessarily be the case ;
and such a rearrangement must assuredly take place in fluids.
The insufficient generality of Cauchy's equations is further shown
by their being absolutely incompatible with the idea of incompressi-
bility. We may evidently conceive a solid which resists compression
of volume by a force incomparably greater than that by which it
resists distortion of figure, and such a conception is actually realized
in such a solid as caoutchouc or jelly.

I have not mentioned the hypothesis of what may be called,
from the analogy of surfaces of the second order, a central arrange-
ment of the molecules, that is, an arrangement such that each
molecule is a centre with respect to which the others are arranged
in pairs at equal distances in opposite directions, because the
hypothesis was merely casually introduced as one mode of making
certain terms vanish which are of a form that clearly ought not to
appear in the expressions relating to the mean motion, with which
alone we are ultimately concerned.

The arguments in favour of the existence of ultimate molecules
in the case of ponderable matter appear to rest chiefly on the
chemical law of definite proportions, and on 'the laws of crystallo-
graphy, neither of which of course can be assumed to apply to the
mysterious ether, of the very existence of which we have no direct

172 REPORT ON DOUBLE REFRACTION.

evidence. If, for aught we know to the contrary, the very suppo-
sition of the existence of ultimate molecules as applied to the ether
may entail consequences at variance with its real constitution, much
more must the. accessory hypotheses be deemed precarious which
Cauchy found necessary in order to be able to deduce any results
at all in proceeding by his method. There appears, therefore, no
sufficient reason d priori for preferring the more limited equations
of Cauchy to the more general equations of Green.

Green, on the other hand, takes his stand on the impossibility
of perpetual motion, or in other words, on the principle of the
conservation of work, which we have the strongest reasons for
believing to be a general physical principle*- The number of
arbitrary constants thus furnished in the case in which the undis-
turbed state of the medium is one of freedom from pressure is, as
has been stated, twenty-one. Professor Thomson has recently put
this result in a form which indicates more clearly the signification
of the constants f, and at the end of his memoir promises to show
how an elastic solid, which as a whole should possess this number
of arbitrary constants, could be built up of isotropic matter.

Green supposes, in the first instance, that the medium is
symmetrical with respect to planes in three rectangular directions,
which simplifies the investigation and reduces the twenty-seven
or twenty-one arbitrary constants to twelve (entering the partial
differential equations of motion in such a manner as to be there
equivalent to only eleven) or nine. It may be useful to give a
Table of the constants employed by Green, with their equivalents
in the theories of Cauchy and Neumann, the density of the medium
at rest being taken equal to unity for the sake of simplicity. The
Table is as follows : —

Green ABC OHI LMN PQR

Cauchy GHI LMN PQR PQR

Neumann 00 BOB A,AA„ A,AA„;

so that Green's equations are reduced to Cauchy's by making

L = P, M=Q, N=R (6).

* Whether vital phenomena are subject to this law is a question which we are
not here called upon to discuss.

t "Elements of a Mathematical Theory of Elasticity," Phil. Trans, for 1856,
p. 481. Read April 24, 1856. [Lord Kelvin's Math, and Phys. Papers, Vol. iii,
p. 84.]

REPORT ON DOUBLE REFRACTION. 173

For a plane wave propagated in any given direction there are
three velocities of propagation, and three corresponding directions
of vibration, which are determined by the directions of the principal
axes of a certain ellipsoid U=l, which he proposes to call the
ellipsoid of elasticity, the semiaxes at the same time representing
in magnitude the squared reciprocals of the corresponding velocities
of propagation; and Green has shown that U may be at once
obtained from the function — 2(j) by taking that part only which
is of the second order in u, v, w, and replacing u, v, w by x, y, z, and

the symbols of differentiation -j- , -j- , -j- by the cosines of the

angles which the wave-normal makes with the axes. This applies
whether the medium be symmetrical or not with respect to the
coordinate planes. Green then examines the consequences of
supposing that for two of the three waves the vibrations are
strictly in the front of the wave, as was supposed by Fresnel,
and consequently that the vibrations belonging to the third wave
are strictly normal. This hypothesis leads to five relations between
the twelve constants, namely

(?=ir=/= /.I suppose, P = iJ,-2L, Q = /ji-2M, i?=/i-2iV...(7);

and gives for the form of the fundamental function

- 2(6 = 24 -y~ + 2-B -J- + 2(7 -5-
^ ax ay as

^ (/duy fdvV fdwy] fdu dv dwV
+ ^tUJ +y +[-rs)\+ndx + dy^T.)
J {(dv dwV' dv dw\ ^(/dw duV dw du
\\dz dy) dy dz\ [\dx dz) dz dx

jn-{(du dvV du dv] .„.

\\dy dx) dx dy]

fi-om which the equations of motion, the expressions for the internal
pressures, and the equation of the ellipsoid of elasticity may be at
once written down.

The simpler case in which the medium in its natural state is
supposed free from pressure is first considered*- Green shows

* The results obtained for this case remain the same if we suppose the medium
in its undisturbed state to be subject to a pressure alike in all directions.

174 , EEPORT ON DOUBLE REFRACTION.

that the ellipse which is the section of the ellipsoid of elasticity
by a diametral plane, parallel to the wave's front, if turned 90° in
its own plane, belongs to a fixed ellipsoid, which gives at once
Fresnel's elegant construction for the velocity of propagation and
direction of the plane of polarization; but it is necessary to suppose
that in polarized light the vibrations are parallel, not perpendicular,
to the plane of polarization.

The general case in which the medium is not assumed to be
symmetrical with respect to three rectangular planes, and in which
therefore <p contains twenty-one arbitrary constants, is afterwards
considered ; and it is shown that the hypothesis of strict transver-
sality leads to fourteen relations between them, leaving only seven
constants arbitrary. But the function obtained on the assumption
of planes of symmetry contains no fewer, for the four constants
relating to these planes would be increased by three when the
medium was referred to general axes. Hence therefore the
existence of planes of symmetry is not an independent assump-
tion, as in Cauchy's theory, but follows as a result.

In this beautiful theory, therefore, we are presented with no
forced relations like Cauchy's equations; the result follows from
the hypothesis of strictly transversal vibrations, to which Fresnel
was led by physical considerations. The constant /a remains
arbitrary, and it is easy to see that this constant expresses the
square of the. velocity of propagation of normal vibrations. Were
this velocity comparable with the velocity of propagation of trans-
versal vibrations, theory would lead us still to expect normal
vibrations to be produced by light incident obliquely, though not
by light incident perpendicularly, on the surface of a crystal, and
the theory would still be exposed to many of the objections which
have been already brought forward. But nothing hinders us from
supposing, in accordance with the argument contained in Green's
former paper, that fi is very great or sensibly infinite, which
removes all the difficulty, since the motion corresponding to this
term in the expression for —24> would not be sensible except at
a distance from the surface comparable with the length of a wave
of light. Hence, although it might be .said, so long as fi was
supposed arbitrary, that the supposition of rigorous transversality
had still something in it of the nature of a forced relation between
constants, we se'e that the single supposition of incompressibility

REPORT OX DOUBLE EEFRACTIOX.

175

(under the action of forces at least comparable with those actiag
in the propagation of light) — the original supposition of Fresnel —
introduced into the general equations, suffices to lead to the
complete laws of double refraction as given by Fresnel. Were
it not that other phenomena of light lead us rather to the con-
clusion that the vibrations are perpendicular, than that they are
parallel to the plane of polarization, this theory would seem to
leave us nothing to desire, except to prove that we had a right
to neglect the direct action of the ponderable molecules, and to
treat the ether within a crystal as a single elastic medium, of which
the elasticity was different in different directions.

In his paper on Reflexion, Green had adopted the supposition
of Fresnel, that the vibrations are perpendicular to the plane of
polarization. He was naturally led to examine whether the laws
of double refraction could be explained on this hypothesis. When
the medium in its imdisturbed state is exposed to pressure diifering
in different directions, six additional constants are introduced into
the fiinction ^, or three in case of the existence of planes of
symmetry to which the medium is referred. For waves perpen-
dicular to the principal axes, the directions of vibration and squared
velocities of propagation are as follows : —

^VaTe-normal

X

y

Direction of vibration... -

1 -f"

G+A

y+B

il+C

\

S+A

H+B

L+C

_.

M+A

L+B

I+C

Green assumes, in accordance with Fresnel's theory, and with
observation if the vibrations in polarized light are supposed
perpendicular to the plane of polarization, that for waves perpen-
dicular to any two of the principal axes, and propagated by
vibrations in the direction of the third axis, the velocity of
propagation is the same. This gives three, equivalent to two,
relations among the constants, namely,

A-L = B-M=C-N = vswpTpose (9),

176 REPORT ON DOUBLE REFRACTION.

which are equivalent to Cauchy's equations (3). The conditions
that the vibrations are strictly transversal and normal respectively
do not involve the six constants expressing the pressures in equi-
librium, and therefore remain the same as before, namely (7).
Adopting the relations (7) and (9), Green proves that for the two
transversal waves the velocities of propagation and the azimuths of
the planes of polarization are precisely those given by the theory
of Fresnel, the vibrations in polarized light being now supposed
perpendicular to the plane of polarization.

As to the wave propagated by normal vibrations, the square of
its velocity of propagation is easily shown to be equal to

fi + AV' + Bvt' + Cii';

and as the constant //, does not enter into the expression for the
velocity of propagation of transversal vibrations, the same suppo-
sition as before, namely that the medium is rigorously or sensibly
incompressible, removes all difficulty arising from the absence of
any observed phenomenon answering to this wave.

The existence of planes of symmetry is here in part assumed.
I say in part, because Green shows that the six constants, expressing
the pressures in equilibrium, enter the equation of the ellipsoid of
elasticity under the form K{ar+y^+z'), where ^ is a homogeneous
function of the six constants of the first order, and involves likewise
the cosines I, m, n. Hence the directions of vibration are the same
as when the six constants vanish; the velocities of propagation
alone are changed ; and as the existence of planes of symmetry for
the case in which the six constants vanish was demonstrated, it is
only requisite to make the very natural supposition that the planes
of symmetry which must exist as regards the directions of vibration,
are also planes of symmetry as regards the pressure in equilibrium.

We ^e then that this theory, which may be called Green's
second theory, is in most respects as satisfactory (assuming for
the present that Fresnel's construction does represent the laws of
double refraction) as the former. I say in most respects, because,
although the theory is perfectly rigorous, like the former, the
equations (9) are of the nature of forced relations between the
constants, not expressing anything which could have been foreseen,
or even conveying when pointed out the expression of any simple
physical relation.

REPORT ON DOUBLE REFRACTION. 177

The year_1839 was fertile in theories of double refraction, and
on the 9th of December Prof MacCullagh presented his theory to
the Royal Irish Academy. It is contained in "An Essay towards a
Dynamical Theory of Crystalline Reflexion and Refraction*." As
indicated by the title, the determination of the intensities of the
light reflected and refracted at the surface of a crystal is what the
author had chiefly in view, but his previous researches had led him
to observe that this determination was intimately connected with
the laws of double refraction, and to seek to link together these laws
as parts of the same system. He was led to apply to the problem
the general equation of dynamics under the form (5), to seek to
determine the form of the function (^ (Fin his notation), and then to
form the partial differential equations of motion, and the conditions
to be satisfied at the boundaries of the medium, by the method of
Lagrange. He does not appear to have been aware at the time
that this method had previously been adopted by Green. Like
his predecessors, he treats the ether within a crystallized body
as a single medium unequally elastic in different directions, thus
ignoring any direct influence of the ponderable molecules in the
vibrations. He assumes that the density of the ether is a constant
quantity, that is, both unchanged during vibration, and the same
within all bodies as in free space. We are not concerned with the
latter of these suppositions in deducing the laws of internal vibra-
tions, but only in investigating those which regulate the intensity
of reflected and refracted light. He assumes further that the
vibrations in plane waves, propagated within a crystal, are recti-
linear, and that while the plane of the wave moves parallel to itself
the vibrations continue parallel to a fixed right line, the direction
of this right line and the direction of a normal to the wave being
functions of each other, — a supposition which doubtless applies to
all crystals except quartz, and those which possess a similar property.

In this method everything depends on the correct determina-
tion of the form of the function V. From the assumption that
the density of the ether is unchanged by vibration, it is readily
shown that the vibrations are entirely transversal. Imagine a
system of plane waves, in which the vibrations are parallel to
a fixed line in the plane of a wave, to be propagated in the crystal,

* Memoirs of the Royal Irish Academy, Vol. xxi, p. 17. [MacCuUagh's Collected
Works, pp. 145—193.]

S. IV. 1^

178 REPORT ON DOUBLE REFRACTION.

and refer the crystal for a moment to the rectangular axes of
x', y', /, the plane of x'y' being parallel to the planes of the
waves, and the axis of y' to the direction of vibration; and let
K be the angle whose tangent is dTi'jdz'. With respect to the form
of F, MacCullagh reasons thus: — "The function V can only depend
upon the directions of the axes of x, y', z' with respect to fixed
lines in the crystal, and upon the angle which measures the change
of form produced in the parallelepiped by vibration. This is the
most general supposition which can be made concerning it. Since,
however, by our second supposition, any one of these directions,
suppose that of x, determines the other two, we may regard V as
depending on the angle k and the direction of the axis of x' alone,"
from whence he shows that V must be a function of the quantities
X, Y, Z, defined by the equations

dz dy ' dx dz' dy dx '

This reasoning, which is somewhat obscure, seems to me to
involve a fallacy. If the form of V were known, the rectilinearity
of vibration and the constancy in the direction of vibration for a
system of plane waves travelling in any given direction would
follow as a result of the solution of the problem. But in using
equation (5) we are not at liberty to substitute for V (or (f>) an
expression which represents that function only on the condition that
the motion be what it actually is, for we have occasion to take the
variation 8F of V, and this variation must be the most general
that is geometrically possible though it be dynamically impossible.
That the form of V, arrived at by MacCullagh, is inadmissible, is,
I conceive, proved by its incompatibility with the form deduced by
Green from the very same supposition of the perfect transversality
of the transversal vibrations; for Green's reasoning is perfectly
straightforward and irreproachable. Besides, MacCullagh's form
leads to consequences absolutely at variance with dynamical
principles*.

. But waiving for the present the objection to the conclusion
that F is a function of the quantities X, Y, Z, let us follow the
consequences of the theory. The disturbance being supposed
small, the quantities X, Y, Z will also be small, and F may
be expanded in a series according to powers of these quantities ;

* See Appendix.

REPORT ON DOUBLE REFRACTION. 179

and, as before, we need only proceed to the second order if we
regard the disturbance as indefinitely small. The first term,
being merely a constant, may be omitted. The terms of the
first order MacCullagh concludes must vanish. This, however,
it must be observed, is only true on the supposition that the
medium in its undisturbed state is free from pressure. The terms
of the second order are six in number, involving squares and
products of X, Y, Z. The terms involving YZ, ZX, XY may be
got rid of by a transformation of coordinates, when V will be
reduced to the form

F=-i(a2Z2 + 6=P + c=^0 (10).

the constant term being omitted, and the arbitrary constants being
denoted by — |a-, — 16^ — \c^. Thus on this theory the existence
of principal axes is proved, not assumed. If MacCuUagh's ex-
pression for V (10) be compared with Green's expression for
(^ (8) for the case of no pressure in equilibrium, so that A=0,
B = 0, = 0, it will be seen that the two will become identical,

, „ . , /du dv dwy . „ ,

provided first we omit the term /a ( t- -f 3- + -i- I in Greens

expression, and secondly, we treat the symbols of differentiation as

„ 1 J. ■ , dv dw J dv dm

literal coefficients, so as to confound, tor mstance, ^ -^ and ^ -^ .

The term involving fi does not appear in the expressions for trans-
versal vibrations, since for these j" + j~ + j" = 0, and therefore

does not affect the laws of the propagation of such vibrations,
although it would appear in the problem of calculating the
intensity of reflected and refracted light ; and be that as it may,
it follows from Green's rule for forming the equation of the
ellipsoid of elasticity, that the laws of the propagation of trans-
versal vibrations will be precisely the same whether we adopt
his form of (^ or F (for the case of no pressure in equilibrium) or

f (JlJ fit) (I'tJ)\ ^

MacCuUagh's. Indeed, if we omit the term /^ (^^ + ^ + "^ j -

the partial differential equations of motion, on which alone depend
the laws of internal propagation, would be just the same in the
two theories* Accordingly MacCullagh obtained, though inde-

* See Appendix. MacCuUagh's reasoning appears to be so far correct as to
have led to correct equations, although through a form of V which may, I conceive,

12—2

180 REPORT ON DOUBLE REFRACTION.

pendently of, and in a different manner from Green, precisely
Fresnel's laws of double refraction and the accompanying polariz-
ation, on the condition, however, that in polarized light the
vibrations are parallel to the plane of polarization.

It is remarkable that in the previous year MacCullagh, in a
letter to Sir David Brewster*, published expressions for the
internal pressures identical with those which result from Green's
first theory, provided that in the latter the terms be omitted which
arise from that term in <^ which contains /^, a term which vanishes
in the case of transversal vibrations propagated within a crystal.
It does not appear how these expressions were obtained by
MacCullagh ; it was probably by a tentative process.

The various theories which have just been reviewed have this
one feature in common, that in all, the direct action of the ponder-
able molecules is neglected, and the ether treated as a single
vibrating medium. It was, doubtless, the extreme difficulty of
determining the motion of one of two mutually penetrating media
that led mathematicians to adopt this, at first sight, unnatural
supposition; but the conviction seems by some to have been
entertained from the first, and to have forced itself upon the
minds of others, that the ponderable molecules must be taken
into account in a far more direct manner. Some investigations
were made in this direction by Dr Lloyd as long as twenty-five
years ago-f. Cauchy's later papers show that he was dissatisfied
with the method, adopted in his earlier ones, of treating the ether
within a ponderable body as a single vibrating medium|; but he

be shown to be inadmissible. [The objection, however, is based on the hypothesis
that the potential energy ( - F) is due entirely to elastic deformation of the medium.
If the medium were a complex one containing kinetic molecules of permanent
gyrostatic type, simple rotation would excite dynamical reaction, so that the
objection would be evaded. The form of V adopted by MacCullagh is in fact
analytically identical with the one appropriate to Clerk Maxwell's electric theory of
light, of which, accordingly, MacCuUagh's analysis constituted a development in
advance. It was afterwards recognised (cf. p. 197 infra) by Sir George Stokes, that
an explicit dynamical basis might possibly be found for this theory by passing
beyond a hypothesis of elasticity of simple deformation. In his exposition of the
theory MacCullagh very fully admitted that a dynamical groundwork for it had yet
to be found. Cf. Larmor, Phil. Trans., 1894.]

* Phil. Mag. for 1836, Vol. viii, p. 103. [MacCuUagh's Collected Works, p. 75.]

+ Proceedings of the Royal Irish Academy, Vol. i, p. 10.

X See his optical memoirs published in the 22nd Volume of the M&moires
de I'Acadeviie.

REPORT ON DOUBLE REFRACTION. 181

does not seem to have advanced beyond a few barren generalities,
towards a theory of double refraction founded on a calculation of
the vibrations of one of two mutually penetrating media. In the
theory of double refraction advanced by Professor Challis*, the
ether is assimilated to an ordinary elastic fluid, the vibrations of
which are modified by resisting masses ; and his theory leads him
at once to Fresnel's elegant construction of the wave-surface by
points. The theory, however, rests upon principles which have not
received the general assent of mathematicians. In a work entitled
"Light explained on the Hypothesis of the Ethereal Medium being
a Viscous Fluid"f , Mr Moon has put in a clear form some of the
more serious objections which may be raised against Fresnel's
theory ; but that which he has substituted is itself open to formid-
able objections, some of which the author himself seems to have
perceived.

In concluding this part of the subject, I may perhaps be
permitted to express my own belief that the true dynamical
theory of double refraction has yet to be found.

In the present state of the theory of double refraction, it
appears to be of especial importance to attend to a rigorous
comparison of its laws with actual observation. I have not now
in view the two great laws giving the planes of polarization, and
the difference of the squared velocities of propagation, of the two
waves which can be propagated independently of each other in any
given direction within a crystal. These laws, or at least laws
differing from them only by quantities which may be deemed
negligible in observation, had previously been discovered by ex-
periment; and the deduction of these laws by Fresnel from his
theory, combined with the verification of the law, which his theory,
correcting in this respect previous notions, first pointed out, that
in each principal plane of a biaxal crystal the ray polarized in that
plane obeys the ordinary law of refraction, leaves no reasonable
doubt that Fresnel's construction contains the true laws of double
refraction, at least in their broad features. But regarding this
point as established, I have rather in view a verification of those
laws which admit of being put to the test of experiment with
extreme precision; for such verifications might often enable the

* Cambridge Philosophical Transactions, Vol. viii, p. 524.
+ Macmillau & Co., Cambridge, 1853.

182 REPORT ON DOUBLE REFRACTION.

mathematician, in groping after the true theory, to discard at
once, as not agreeing with observation, theories which might
present themselves to his mind, and on which otherwise he might
have spent much fruitless labour.

To make my meaning clearer, I will refer to Fresnel's con-
struction, in which the laws of polarization and wave-velocity are
determined by the sections, by a diametral plane parallel to the
wave-front, of the ellipsoid*

a;'!x;^ + b''y'' + c''z'=l (11),

where a, b, c denote the principal wave-velocities. The principal
semiaxes of the section determine by their direction the normals
to the two planes of polarization, and by their magnitude the
reciprocals of the corresponding wave-velocities. Now a certain
other physical theory which might be proposed")- leads to a
construction differing from Fresnel's only in this, that the planes
of polarization and wave-velocities are determined by the section,
by a diametral plane parallel to the wave-front, of the ellipsoid

^' + S + -; = 1 (12),

a^ b^ c^

the principal semiaxes of the section determining by their direction
the normals to the two planes of polarization, and by their magni-
tudes the corresponding wave-velocities. The law that the planes
of polarization of the two waves propagated in a given direction
bisect respectively the two supplemental dihedral angles made by
planes passing through the wave-normal and the two optic axes,
remains the same as before, but the positions of the optic axes
themselves, as determined by the principal indices of refraction,
are somewhat different ; the difference, however, is but small if the
differences between a^, &^, & are a good deal smaller than the
quantities themselves. Each principal section of the wave-surface,
instead of being a circle and an ellipse, is a circle and an oval, to
which an ellipse is a near approximation j. The difference between

* It would seem to be just as well to omit the surface of elasticity altogether,
and refer the construction directly to the ellipsoid (11).

[t This theory suggested itself independently to Lord Rayleigh, and is discussed
by him in Phil. Mag. xli, 1871, p. 519; Scientific Papers, i, p. 111.]

X The equation of the surface of wave-slowness in this and similar cases may be
readily obtained by the method given by Professor Haughton in a paper " On the
Equilibrium and Motion of Sohd and Fluid Bodies," Transactions of the Royal
Irish Academy, Vol. xxi, p. 172.

REPOKT ON DOUBLE REFRACTION. 183

the inclinations of the optic axes, and between the amounts of
extraordinary refraction in the principal planes, on the two theories,
though small, are quite sensible in observation, but only on con-
dition that the observations are made with great precision. We
see from this example of what great advantage for the advancement
of theory observations of this character may be*.

One law which admits of receiving, and which has received,
this searching comparison with observation, is that according to
which, in each principal plane of a biaxal crystal, the ray which is
polarized in that plane obeys the ordinary law of refraction, and
accordingly in a uniaxal crystal, in which every plane parallel to
the axis is a principal plane, the so-called ordinary ray follows
rigorously the law of ordinary refraction. This law was carefully
verified by Fresnel himself in the case of topaz, by the method of
cutting plates parallel to the same principal axis, or axis of
elasticity, carefully working them to the same thickness, and then
interposing them in the paths of two streams of light proceeding
to interfere, as well as by the method of prismatic refraction ; and
he states as the result of his observations that he can affirm the
law to be, at least in the case of topaz, mathematically exact. The
same result follows from the observations by which Rudberg so
accurately determined the principal indices of Arragonite and
topaz f, for the principal fixed lines of the spectrum. Professor
MacCullagh having been led by theoretical considerations to doubt
whether, in Iceland spar for instance, the so-called ordinary ray
rigorously obeyed the ordinary law of refraction, whether the
refractive indices in the axial and equatorial directions were
strictly the same, Sir David Brewster was induced to put the
question to the test of a crucial experiment, by forming a com-
pound prism consisting of two pieces of spar cemented together
in the direction of the length of the prism, and so cut from the
crystal that at a minimum deviation one piece was traversed
axially and the other equatorial ly J. . The prism having been
polished after cementing, so as to ensure the perfect equality of
angle of the two parts, on viewing a slit through it the bright
line D was seen unbroken in passing from one half to the other.

[* For Sir George Stokes' own observations on Iceland spar whicli rendered this
theory inadmissible see Roy. Soc. Proc. xx, 1872, p. 443, reprinted infra.]
t Annales de Chimie, torn, xlviii, p. 225 (1831).
J Report of the British Association for 1843, Trans, of Sect. p. 7.

184 EEPOKT ON DOUBLE REFRACTION.

More recently Professor Swan has made a very precise examination
of the ordinary refraction in various directions in Iceland spar by
the method of prismatic refraction*, from whencQ it results that
for homogeneous light of any refrangibility the ordinary ray follows
strictly the ordinary law of refraction.

It is remarkable that this simple law, which ought, one would
expect, to lie on the very surface as it were of the true theory of
double refraction, is not indicated a priori by most of the rigorous
theories which have been advanced to account for the phenomenon.
Neither of the two theories of Cauchy, nor the second theory of
Green, lead us to expect such a result, though they furnish arbitrary
constants which may be so determined as to bring it about.

The curious and unexpected phenomenon of conical refraction
has justly been regarded as one of the most striking proofs of the
general correctness of the conclusions resulting from the theory of
Fresnel. But I wish to point out that the phenomenon is not
competent to decide between several theories leading to Fresnel's
construction as a near approximation. Let us take first internal
conical refraction. The existence of this phenomenon depends
upon the existence of a tangent plane touching the wave surface
along a plane curve. At first sight this might seem to be a
speciality of the wave-surface of Fresnel ; but a little consideration
will show that it must be a property of the wave-surface resulting
from any reasonable theory. For, if possible, let the nearest
approach to a plane curve of contact be a curve of double curva-
ture. Let a plane be drawn touching the rim (as it may be called)
of the surface, that is, the part where the surface turns over,- in two
points, on opposite sides of the rim ; and then, after having been
sHghtly tilted by turning about one of the points of contact, let it
move parallel to itself towards the centre. The successive sections
of the wave-surface by this plane will evidently be of the general
character represented in the annexed figures, and in four positions

^234 6 6

==5

Transactions of the Royal Society of Edinburgh, Vol. xvi, p. 375.

REPOET ON DOUBLE REFRACTION. 185

the plane will touch the surface in one point, as represented in
Figs. 1, 2, 4, 5. Should the contacts represented in Figs. 4 and 5 take
place simultaneously, they maj- be rendered successive by slightly
altering the inclination of the plane. Hence in certain directions
there would be four possible wave-velocities. Now the general
principle of the superposition of small motions makes the laws
of double refraction depend on those of the propagation of plane
waves. But all theories respecting the propagation of a series of
plane waves having a given direction, and in which the disturbance
of the particles is arbitrary, but the same all over the front of a
wave, agree in this, that they lead us to decompose the disturbance
into three disturbances in three particular directions, to each of
which corresponds a series of plane waves which are propagated
with a determinate velocity. If the medium be incompressible,
one of the wave-velocities becomes infinite, and one sheet of the
wave-surface moves off to infinity. The most general disturbance,
subject to the condition of incompressibility, which requires that
there be no displacements perpendicular to the fronts of the waves,
may now be expressed as the resultant of tiuo disturbances, corre-
sponding to displacements in particular directions lying in planes
parallel to that of the waves, to each of which corresponds a
determinate velocity of propagation. We see, therefore, that the
limitation of the number of tangent planes to the wave-surface,
which can be drawn in a given direction on one side of the centre,
to two, or at the most three, is intimately bound up with the
number of dimensions of space ; so that the existence of the
phenomenon of internal conical refraction is no proof of the • truth
of the particular form of wave-surface assigned by Fresnel rather
than that to which some other theory would conduct. Were the
law of wave-velocit}- expressed, for example, by the construction
ah-eady mentioned liaving reference to the ellipsoid (12), the wave-
surface (in this case a surface of the 16th degree) would still have
plane curves of contact with the tangent plane, which in this case
also, as in the wave-surface of Fresnel, are, as I find, circles, though
that they should be circles could not have been foreseen.

The existence of external conical refi-action depends upon the
existence of a conical point in the wave-surface, by which the
interior sheet passes to the exterior. The existence of a conical
point is not, like that of a plane curve of contact, a necessary

186 REPORT ON DOUBLE REFRACTION.

property of a wave-surface. Still it will readily be conceived that
if Fresnel's wave-surface be, as it undoubtedly is, at least a near
approximation to the true wave-surface, and if the latter have,
moreover, plane curves of contact with the tangent plane, the
mode by which the exterior sheet passes within one of these plane
curves into the interior will be very approximately by a conical
point ; so that in the impossibility of operating experimentally on
mere rays the phenomena will not be sensibly different from what
they would have been had the transition been made rigorously by
a conical point.

There is one direction within a biaxal crystal marked by a
visible phenomenon of such a nature as to permit of observing
the. direction with precision, while it can also be calculated, on any
particular theory of double refraction, in terms of the principal
indices of refraction ; I refer to the direction of either optic axis.
Rudberg himself measured the inclination of the optic axes of
Arragonite, probably with a piece of the same crystal from which
his prisms were cut, and found it a little more than 32° as observed
in air, but he speaks of the difficulty of measuring the angle with
precision. The inclination within the crystal thence deduced is
really a little greater than that given by Fresnel's theory; but
in making the comparison Rudberg used the formula for the ray-
axes instead of that for the wave-axes, which made the theoretical
inclination in air appear about 2° greater than the observed*.
A very exact measure of the angle between the optic axes of
Arragonite for homogeneous light corresponding to the principal
fixed lines of the spectrum has recently been executed by Professor
Kirchhofff , by a method which has the advantage of not making
any supposition as to the direction in which the crystal is cut.
The angle observed in air was reduced by calculation to the angle
within the crystal, by means of Rudberg's indices for the principal
axis of mean elasticity; and the result was compared with the
angle calculated from the formula of Fresnel, on substituting for
the constants therein contained the numerical values determined
by Rudberg for all the three principal axes. The angle reduced
from that observed in air proved to be from 13' to 20' greater than
that calculated from Fresnel's formula. This small difference seems

* Annates de Chimie, tome xlviii, p. 258 (1831).
t Foggendorff's Annalen, Vol. cviii, p. 567 (1859).

REPORT OX DOUBLE REFRACTION. 187

to be fairly attributable to errors in the indices, arising from errors
in the direction of cutting of the prisms employed by Rudberg.
The angle measured by Earchhoff would seem to have been trust-
worthy to within a minute or less.

It is doubtful, however, how far we may trust to the identity of
the principal refractive indices in different specimens of the mineral.
Chemical analysis shows that Arragonite is not pure carbonate of
lime, but contains a variable though small proportion of other
ingredients. To these variations doubtless correspond variations
in the refractive indices; and De Senarmont has shown how the
inchnation of the optic axes of minerals is liable to be changed by
the substitution one for another of isomorphous elements*. More-
over, M. Des Cloizeaux has recently sho^\^l that in felspar and
some other minerals, which bear a high temperature without
apparent change, the inclination of the optic axes is changed in
a permanent manner by heatf; so that even perfect identity of
chemical composition is not an absolute guarantee of optical
identity in two specimens of a mineral of a given kind.

The exactness of the spheroidal form assigned by Huygens to
the sheet of the wave-surface within Iceland spar corresponding
to the extraordinary ray, does not seem to have been tested to the
same degree of rigour as the ordinary refraction of the ordinary-
ray; for the methods employed by Wollaston+ and Malus§ for
observing the extraordinary refraction can hardly bear comparison
for exactness with the method of prismatic refraction which has
been applied to the ordinary ray ; and observations on the absolute
velocities of propagation in different directions withia biaxal crystals
are still almost wholly wanting. This has long been recognised as
a desideratum, and it has been suggested to employ for the purpose
the displacement of fringes of interference. It seems to me that a
slight modification of the ordinary method of prismatic refraction
would be more convenient and exact ||.

Let the crystal to be examined be cut, unless natural faces or
cleavage planes answer the purpose, so as to have two planes
inclined at an angle suitable for the measure of refractions ; there

* Annales de Chimie, tome xxxiii, p. 391 (1851).

+ Annales de Mines, tome ii, p. 327 (1862).

X " On the Oblique Refraction of Iceland Spar," Phil. Tram, for 1802, p. 381.

§ Memoires de I'Imtitat ; Sav. Etrangers, tome ii, p. 803 (1811).

[II Cf. ante, Vol. i, p. 149 (1846).]

188 REPORT ON DOUBLE REFRACTION.

being at least two natural faces or cleavage-planes left undestroyed,
so as to permit of an exact measure of the directions of any artificial
faces. The prism thus formed having been mounted as usual, and
placed in any azimuth, let the angle of incidence or emergence
(according as the prism remains fixed or turns round with the
telescope) be measured, by observing the light reflected from the
surface, and likewise the deviation for several standard fixed lines
in the spectrum of each refracted pencil. Let the prism be now
turned into a different azimuth, and the deviations again observed,
and so on. Each observation furnishes accurately an angle of inci-
dence and the corresponding angle of emergence ; for if <p be the
angle of incidence, i the angle of the prism, D the deviation, and
■v/r the angle of emergence, D — 4) + ■yfr — i. But without making
any supposition as to the law of double refraction, or assuming
anything heyond the truth of Huygens's principle, which, following
directly from the general principle of the superposition of small
motions, lies at the very foundation of the whole theory of undu-
lations, we may at once deduce from the angles of incidence and
emergence the direction and velocity of projoagation of the wave
within the prism. For if a plane wave be incident on a plane
surface bounding a medium of any kind, either ordinary or doubly
refracting, it follows directly from Huygens's principle that the
refracted wave or waves will be plane, and that if </> be the angle
of incidence, <j)' the inclination of a refracted wave to the surface,
V the velocity of propagation in air, v the wave-velocity within the
medium,

sin (i> sin cj)'

Hence if ^', •yfr' be the inclinations of the refracted wave to the
faces of our prism, we shall have the equations

t)sin<^= Fsin^' (13),

?;sim/r= Fsini|/-' (14),

4>' + ^{r' = i (15).

The equations (13) and (14) give, on taking account of (15),
vsin^^ti:cos^-^=Fsin|cos-^-^' (16),

t;cos V-g-^sm ^--~-!-= Fcos^sm ^ (17),

REPORT ON DOUBLE REFRACTION. 189

whence by division

tan "^1-^' = tan i tan ^ cot ^ (18).

The equations (15) and (18) determine </>' and ->|r', and then (16)
gives V. Hence we know accurately the velocity of propagation of
a wave, the normal to which lies in a plane perpendicular to the
faces of the prism, and makes known angles with the faces, and is
therefore known in direction with reference to the crystallographic
axes. A single prism would enable the observer to explore the
crystal in a series of directions lying in a plane perpendicular to
its edge ; but as these directions are practically confined to limits
making no very great angles with a normal to the plane bisecting
the dihedral angle of the prism, more than one prism would be
required to enable him to explore the crystal in the most important
directions; and it would be necessary for him to assure himself that
the specimens of crystal, of which the different prisms are made,
were strictly comparable with each other. It would be best, as far
as practicable, to cut them from the same block.

The existence of principal planes, or planes of optical symmetry,
for light of any given refrangibility, in those cases in which they
are not determined by being at the same time planes of crystallo-
graphic symmetry, is a matter needing experimental verification.
However, as no anomaly, so far as I am aware, has been discovered
in the systems of rings seen with homogeneous light around the
optic axes of crystals of the oblique or anorthic system, there is no
reason for supposing that such. planes do not exist.

Appendix.

Further Comparison of the Theories of Green, MacCullagh,
and Cauchy.

In a paper "On a Classification of Elastic Media and the
Laws of Plane Waves propagated through them," read before the
Royal Irish Academy on the 8th of January, 1849*, Professor
Haughton has made a comparative examination of different theories
which have been advanced for determining the motion of elastic

* Transactions of the Royal Irish Academy, Vol. xxii, p. 97.

190 REPORT ON DOUBLE REFRACTION.

media, more especially those which have been applied to the
explanation of the phenomena of light. Some of the results
contained in this Appendix have already been given by Professor
Haughton ; in other instances I have arrived at different con-
clusions. In such cases I have been careful to give my reasons
in detail.

Consider a homogeneous elastic medium, the parts of which act
on one another only with forces which are insensible at sensible
distances, and which in its undisturbed state is either free from
pressure, or else subject to a pressure or tension which is the same
at all points, though varying with the direction of the plane surface
with reference to which it is estimated. Let x, y, z be the coordi-
nates of any particle in the undisturbed state, x-\-u, y + v, z + w
the coordinates in the disturbed state, and for simplicity take the
density in the undisturbed state as the unit of density. Then,
according to the method followed both by Green and MacCullagh,
the motion of the medium will be determined by the equation

f[f/(Pu (Pv <Pw \ fff

Jjj\df ^^ ^ dt^^^ '^ IJP ^^) '^«"^y'^^=]] I Hdxdydz...{\^),

where ^ is the function due to the elastic forces. To this equation
must be added, in case the medium be not unlimited, the terms
relative to its boundaries.

The function ^ multiplied by dxdydz expresses the work given
out by the element dxdydz in passing from the initial to the actual
state if we assume, as we may, the initial state for that in which
^ = 0. According to the supposition with which we started, that
the internal forces are insensible at sensible distances, the value of
<^ at any point must depend on the relative displacements in the
immediate neighbourhood of that point, as expressed by the differ-
ential coefficients of u, v, w with respect to x, y, z. For the present
let us make no other supposition concerning <p than this, that it is
some function (— /) of those nine differential coefficients; and let
us apply the equation (19) to a limited portion of the medium
bounded initially by the closed surface S. We must previously
add the terms due to the action of the surrounding portion of the
medium, which will evidently be of the form of a double integral
having reference to the surface S, an element of which we may

REPORT ON DOUBLE REFRACTION.

191

denote by dS. Hence we must add to the right-hand side of
equation (19)

jJEdS,
the expression for E having yet to be found.

Denoting for shortness the partial differential coefficients of

du du

du

d'l

■4, with respect to ^, •=, &c. by /' (^^j , /' ^^J , &c., we have

dyj dy
J., (du\ d8u ., (du\ dSu „

whence

-jjJB.I,d^dyd. =////' (g) ^ d^dyd.

d J,, idu
dx'' \dx) ' """ dy" \dy

+ &o.[- dxdydz.

We must now equate to zero separately the terms in our equation
involving triple and those involving double integrals. The result
obtained from the former further requires that the coefficient of
each of the independent quantities Sw, hv, Sw under the sign

shall vanish separately, whence

fdv\
{d^J
fdv\
ids

III

d^u _ _^ /•/ /^^ , d^
dP ~dx-' \dx) dy

d'v

di-

d^w
d¥

^ Kdyj^dz' \

d'v _ d J,, /dv\ ,d_., /dv\ ,§_ .,
d¥~dx^ \dx) dy^ \dyj dz^

'\ d J., Idw\

_d_r, (dw\ ,d_ J,, /dw\
~ dx-' \dxl dy^ \dyj

•(20)*,

These agree with Professor Haughton's equations (5).

192

REPORT ON DOUBLE REFRACTION.

dni.

'df

d'v

'df

d'w

'd<f)'

dt' "

du

' df

dv

' d¥~

dw

equations which may be written in an abbreviated form as
follows : —

•(21),

where the expressions within crotchets denote differential coeffi-
cients taken in a conventional sense, namely by treating in the

differentiation the symbols -^ , -y

coefficients, and prefixing to the whole term, and now regarding as
a real symbol of differentiation, whichever of these three symbols
was attached to the u, v, or w that disappeared by differentiation.

-J- as if they were mere literal

The equating of the double integrals gives
fj^dS = jjf' (g Budyd. +/// (g) Budzdx + &c.

If

du

+ »w./'

du

\dy,

/, I CiVj

8m + [&c.]Sw + [&c.]SwUS,

\dxj • \dyj \dzj

where I, m, n are the direction-cosines of the element dS of the
surface which bounded the portion of the medium under con-
sideration when it was in its undisturbed state. This expression
leads us to contemplate the action of the surrounding medium as
a tension having a certain value refeiTed to a unit of surface in the
undisturbed state. If P, Q, R be the components of this tension
parallel to the axes of x, y, z, they must be the coefficients of

hu, hv, Sw under the sign 1 1 , so that

du,\\
dz,

dv\ j.,fdv

^'<£V-r

dy
dw
dy

y

-/'O,

.(22).

These formulae give, in terms of the function (/>, the components
of the tension on a small plane which in its original position had
any arbitrary direction. If we wish for the expressions for the
components of the tensions on planes originally perpendicular to
the axes of x, y, z, we have only to put in succession Z= 1, m = 1,
n= 1, the other two cosines each time being equal to zero. If then

KEPORT OX DOUBLE REFRACTION. 193

Fx' Tyx, T„ denote the components in the direction of the axis of
X of the tension on planes originally perpendicular to the axes of
X. y, z, with similar notation in the other cases, we shall have

(d\i\ _, ., / dw 1 ,„ ., ldv\ \

'-/'©. ^'-=.'l|i' ^-=/'©

-.'■'(I)' ^-/'(S' '■'-/•(£)

•')S\*

^-=.'-(S)' ^-.=/'(S' ^»=/-*

dijl]

.(23y

The formnlse hitherto employed are just the same whether we
suppose the disturbance small or not ; and we might express in
terms of F^, Tj/?. "J^*-'- I'^nd therefore in terms of ^), and of the
differential coetScients of u, v. w with respect to x, y, and .-, the
components of the tension referred to a surface given in the actual
instead of the undistui-bed state of the medium, without supposing
the disturbance small. As, however, the investigation is meant to
be applied only to small distvirbances, it would only complicate the
formula- to no purpose to treat the disturbance as of arbitrary
magnitude, and I shall therefore regard it henceforth as indefi-
nitely small.

On this supposition we may expand <^ according to powers of the
small quantities diiidx, kc proceeding as far as the second order,
the left-hand member of (19) being of the second order as regards
V, V, «'. The formula? {22) or (23) show that <p will or will not
contain terms of the first order according as the undisturbed state
of the medium is one of uniform constraint, or of freedom from
pressure.

In Green's fii-st theory, and in the theory of MacCuUagh, <^ is
supposed not to contain terms of the first order. Accordingly in
considering the point with respect to which these two theories are
at issue. I shall suppose the medium in its vmdisturbed state to be
free from pressure. The tensions P, Q. R, Fx, kc. will now be small
quantities of the fii-st order, so that in the formulae (22) and (23)
we may suppose the tensions referred to a unit of surface in the
actual or the undisturbed state of the medium indifferently, and

* These agree with Professor Haughton's equations at p. 100, but are obtained
in a different manner.

S. IV. 13

194 KEPORT ON DOUBLE REFRACTION.

may moreover in these formulae, and in the expression for (p, take
X, y, z for the actual or the original coordinates of a particle.

Green assumes as self-evident that the value of ^ for any
element, suppose that which originally occupied the rectangular
parallelepiped dxdydz, must depend only on the change of form
of the element, and not on any mere change of position in space.
Any displacement which varies continuously from point to point
must change an elementary rectangular parallelepiped into one
which is oblique-angled, and the change of form is expressed by
the ratios of the lengths of the edges to the original lengths, and
by the angles which the edges make with one another or by their
cosines. If the medium were originally in a state of constraint,
<^ would contain terms of the first order, and the expressions for
the extensions of the edges and the cosines of the angles would be
wanted to the second order, but when ^ is wholly of the second
■order, those quantities need only be found to the first order. It is
€asy to see that to this order the extensions are expressed by

du dv dw , ,

di' d^' lb ^ ''

and the cosines of the inclinations of the edges two and two by

dv dw dw du du dv ,_ ,

dz dy' dx dz' dy dx

and ^ being a function of these six quantities, we have from (23)

■'-yz^' -t-zyi J-zx^^ J- xzt •'■xy^^ ■'-yx (-^b).

These are the relations pointed out by Cauchy between the nine
components of the three tensions in three rectangular directions,
whereby they are reduced to six. The necessity of these relations
is admitted by most mathematicians.

Conversely, if we start with Cauchy's three relations (26), we
have from (23)

/'©-/'©• /'(£)=/'©• /'©=/^(|)-(^')-

The integration of the first of these partial differential equations
gives /= a function of j- + j^ ^nd of the seven other differential
coefficients.

REPORT ON DOUBLE REFRACTION. 195

Substituting in the second of equations (27) and integrating, and
substituting the result in the third and integrating again, we

readily find

»

./ = a functio^i of the six quantities (24) and (25).

We see then that Green's axiom that the function ^ depends
only on the change of form of the element, and Cauchy's relations
(26), are but different ways of expressing the same condition ; so
that either follows if the truth of the other be admitted.

Cauchy's equations were proved by applyiag the statical
equations of moments of a rigid body to an elementary parallel-
epiped of the medium, and taking the limit when the dimensions
of the element vanish. The demonstration is just the same
whether the medium be at rest or iu motion, since in the latter
case we have merelj' to apply d'Alembert's principle. It need
hardly be remarked that the employment of equations of equi-
librium of a rigid body iu the demonstration by no means limits
the truth of the theorem to rigid bodies; for the equations of
equilibrium of a rigid body are true of any material system. In
the latter case they are not sufficient for the equilibrium, but all
that we are concerned with in the demonstration of equations (26)
is that they should be true.

On the other hand, the form of V ov <^ to which MacCuUagh
was led is that of a homogeneous function, of the second order, of
the three quantities

dw dv du dw dv du , ,

dy dz' dz doo' dx dy \-' )'

which, as is well known, are linear functions of the similarly
expressed quantities referring to any other sj-stem of rectangular
axes. On substituting in (23), we see that the normal tensions
on planes parallel to the coordinate planes, and therefore on any
plane since the axes are arbitrary', vanish, while the tangential
tensions satisfy the three relations

SO that the equations of moment of an element are violated. The
relative motion in the neighbourhood of a given point may be
resolved, as is known, into three extensions (positive or negative)
in three rectangular directions and three rotations. The directions

13—2

196 REPORT ON DOUBLE REFRACTION.

of the axes of extension, and the magnitudes of the extensions, are
determined by the six quantities (24) and (25), while the rotations
or angular displacements are expressed by the halves of the three
quantities (28). In this theory, then, the work stored up in an
element of the medium would depend, not upon the change of form
of the element, but upon its angular displacement in space.

It may be shown without difficulty that, according to the form
of </) assumed by MacCullagh, the equations of moments are violated
for a finite portion of the mass, and not merely for an element.
Supposing for simplicity that the medium in its undisturbed state
is free from pressure or tension, let us leave the form of tf) open for
the present, except that it i.s supposed to be a function of the
differential coefficients of the first order of u, v, w with respect
to X, y, z, and let us form the equation of moments round one of
the axes, as that of x, for the portion of the medium comprised
within the closed surface fS'. This equation is

the double integrals belonging to the surface. Since all the terms
in this equation are small, we may take x, y, z for the actual or the
equilibrium coordinates indifferently. Substituting from equations
(20), and integrating by parts, we find

J /{/■ (a) ' - /■ O v\ ''"'« * Ih -«')■'«

The double integrals in this equation destroy each other by virtue
of (22), so that there remains

But this equation cannot be satisfied, since the surface S within
which the integration is to be performed is perfectly arbitrary,

unless /'(tt") =/' (t- ) at all points. We are thus led back to

the equations (27), which are violated in the theory of MacCullagh.

+

+

REPORT ON DOUBLE REFRACTION. 197

The form of the equations such as (30) is instructive, as point-
ing out the mode in which the condition of moments is violated.
It is not that the resultant of the forces acting on an element of
the medium does not produce its proper momentum in changing
the motion of translation of the element ; that is secured by the
equations (20); but that a couple is supposed to act on each
element to which there is no corresponding reacting couple.

The only way of escaping from these conclusions is by denying
that the mutual action of two adjacent portions of the medium
separated by a small ideal surface is capable of being represented
by a pressure or tension, and saying that we must also take into
account a couple ; not, it is to be observed, a couple depending on
variations of the tension (for that would be of a higher order and
would vanish in the limit), but a couple ultimately proportional to
the element of surface. But it would require a function ^ of a
totally different form to take into account the work of such couples;
and indeed the method by which the expressions for the components
of the tension have been here deduced seems to show that in the
case of a function <^ which depends only on the differential coeffi-
cients of the first order of u, v, w with respect to x, y, z, the mutual
action of two contiguous portions of a medium is fully represented
by a tension or pressure.

Indeed MacCullagh himself expressly disclaimed having given
a mechanical theory of double refraction*. His methods have been
characterized as a sort of mathematical induction, and led him to
the discovery of the mathematical laws of certain highly important
optical phenomena. The discovery of such laws can hardly fail
to be a great assistance towards the future establishment of a
complete mechanical theory.

I proceed now to form the function ^ for Cauchy's most general
equations.

If we have given the expressions for -j- , -y-^ , --r— in terms of

* Transactions of the Royal Irish Academy, Vol. xxi, p. 50. It would seem,
however, that he rather felt the want of a mechanical theory from which to deduce
his form of the function tf> or F, than doubted the correctness of that form itself.
[In other words, his abstract theory fitted into the fundamental dynamical equation,
that of least action ; but concrete mechanical illustrations of it had not been found.
The distinction thus indicated is now more widely recognised since its application
by Kelvin and especially by Maxwell to electrical phenomena.]

198 REPORT ON DOUBLE REFRACTION.

the differential coefficients of u, v, w with respect to x, y, z, they do
not suffice for the complete determination of the function <^, as
appears from the equations (20) or (21) ; but if we have given the
expressions for the tensions P^, Pyx, &c., ^ is completely determin-
ate, as appears from equations (23). In using these equations, it
must be remembered that the tensions are measured with reference
to surfaces in the undisturbed state of the medium-; and therefore,
should the expressions be given with reference to surfaces in the
actual state, they must undergo a preliminary transformation to
make them refer to surfaces in the undisturbed state.

Supposing then the tensions expressed as required, in order to
find we have only to integrate the total differential

-d4>=^P.d^^ + P,d^^+P.d-^ + T,A^^- + T,xd^^

^ + T d^ + T d^ + T rf —
' dx^ ^'""^ dz^ ■^''''^ dx^ -^"'''^ dy-

+ Tx.yd'^ + r,,d'^ + Tx.d'2 + Tyxd2 (31),

the nine differential coefficients, of which c^ is a function, being
regarded as independent variables. Should the three equations
(27) be satisfied, the expression (31) will be simplified, becoming

+v(S+£) + r,.(|4;) (32),

where T^ denotes Ty^ or T^y, and similarly for Ty, T^.

The general expressions for the tensions resulting from Cauchy's
method are written at length in the equations numbered 17 and
18, pp. 133, 134 of the 4th volume of his Exercices de Maihi-
matiques, where the normal and tangential tensions, referred to
surfaces in the actual state of the medium, are denoted hy A, B,
C, D, E, F. These expressions contain 21 arbitrary constants, of
which six, ^, 23, ffi, 1), CE, jp, denote the tensions in the state of
equilibrium. If these be for the present omitted, the remaining
terms will be wholly small quantities of the first order, and there-
fore the tensions may be supposed to be referred to a unit of
surface in the actual, or in the undisturbed state of the medium
indifferently. On substituting now for P^, Py, Pz, T^,Ty, T^ in

REPORT OX DOUBLE REFRACTIOX. 199

(32) the remaining parts o{ A, B, C. D, E, F (observing that the
f, -q, f in Cauchy's notation are the same as u, v, w), it will be seen
that the right-hand member of the equation is a perfect differential,
integrable at once bv inspection, and giving

_,„(/» dii^' du\ _^,^.dv hht_ dv\ ^ dw fdv dw\
rfx\(ir dz' ~ di/\dtf dx) ~ ds \dz dy)

+ -^W— (— + — 1 + -^ r'— f '^' + — '^-l--^ V" — ' (~ + ~)
dx \dy das) ~ dy \ds dy) ~ d:\dx dz;

(33),

the arbitrar}- constant being omitted as nnnecess;\rv. We see that
this is a homogeneous function of the second degree of the six
quantities (24^) and (25), but not the most general function of that
nature, containing only 15 instead of 21 arbitrary constants.

Let us now form the part, of the expression for <^ involving the
constants which express the pressures in the state of equilibrium.
It will be convenient to-etiect the requisite transformation in the
expressions for the tensions bv two steps, first referring them to
surtaces of the actual extent, but in the original position, and then
to surfaces in the original state altogether.

Let P'x, T\-. izc. denote the tensions estimated with reference
to the actual extent but original direction of a surface, so that
P'j. (f .S, for instance, denotes the component, in a direction parallel
to the axis of a: of the tension on an elementary plane passing
through the point {or, y, :) in such a direction that in the undis-
turbed state of the medium the same plane of particles was

200

REPORT ON DOUBLE REFRACTION.

perpendicular to the axis of x, dS denoting the actual area of
the element. Consider the equilibrium of an elementary tetra-
hedron of the medium, the sides of which are perpendicular to the
axes of X, y, z, and the base in the direction of a plane which was
perpendicular to the axis of x; and let I, m, n be the direction-cosines
of the base ; then

P'^=lA+mF+nE, T'^=lF+mB + nD, T'^,= lE+mD+nC...(S4^y,
but to the first order of small quantities

, , du du

substituting in (34), and writing down the other corresponding
equations, we have

P'^^A-Fp-E'^
dy dz

\

P'=B-D

dv „ dv

dz

dx
dw

dx dy

ry.= D-G%.

E

dv
dx

r^.^E-A'^^-F^
'" dx dy

r„, = F-B~-D~

■" dy dz

,dw

^'--^-^Tx-^dy

T', = E-D

dy

G

dw
dy
du
dz

y ...(36).

^'--^-4:-4:/

Lastly, since an elementary area dS originally perpendicular to the

axis of X becomes by extension ( 1 + j +~j^) ^^> ^^^ similarly
with regard to y and z, we have

P ■ P' =T ■ T' = T ■ T' - 1 4-

'-xy ■

xy

dv dw'
dy dz

T) , ipi rp . rp/ rp . rn/ -i . ^^ , CtlA

■^y-'^y— -'■yz--'- yz — J-yx- ■>■ ya; — J- T ^^"^ ^^

.. d,w dv
dx dy

P. p/ , rp ^ rpi rp , rp/
z • -^ z ■'- zx * -'- zx ^ zy • -^ zy '

.(36).

Expressing P^, T^y, &c. in terms of P'^, T'^y, &c. by (36), then
P'^, T'^y, &c. in terms oi A, B, G, D, E, F by (35), and lastly sub-
stituting for ^, i? ... i'' the expressions, given by Cauchy, we find

REPORT ON DOUBLE REFRACTION.

V dzJ ax ay

dw\ cpdw '^dw

dzJ -"^ dx dy

du\ _. du ,^ du

201

T..

,.= i3(l + ;^) + J^+i3-

\ dxj dy dz

dv
dx

2'^=iF(i + $U@? + i

1 +

dyj
dv
dy.
dw\

dz

7

dx

dz

^^rz}+^ry+^Tx

dx.

dz

.(37).

<^2/ ;

Substituting now these expressions in (31) and iategrating, we have

/dwy

-20 = a

+ 33

+ GC

+ 21B

+ 2(B

du /duV I'dvV /dwy
dx \dxj \dx) \dx) J

\

2? +
dz

dv /duy fdvy fdwyi
%^[dyJ ^\dy) ^Kd^Ji

'dv\

4y)

'dv"-
Kdz! ■

dw\^
dil

\ ...(38),

dv dw du du dv dv dw dw)

dz dy dy dz dy dz dy dz)

dw du du du dv dv dw dw

dx dz dz dx dz dx dz dx

jjj- jdw dv dudu dv dv dw dw)

^ \dy dx dxdy dx dy dx dy\l

which is exactly Green's expression*, Green's constants A, B ... F
answering to Cauchy's ^, 13 . . . jp. The sum of the right-hand
members of equations (33) and (38) gives the complete expression
for — 20 which belongs to Cauchy's formulee. It contains, as we

* Cambridge Philosophical Transactions, Vol. vn, p. 127. [Green's Math.
Papers, p. 298.]

202 REPORT ON DOUBLE REFRACTION.

see, 21 arbitrary constants, and is a particular case of the general
form used by Green, which latter contains 27 arbitrary constants.

I have been thus particular in deducing the form of Green's
function which belongs to Cauchy's expressions, partly because it
has been erroneously asserted that Green's function does not apply
to a system of attracting and repelling molecules, partly because,
when once the function 4> is formed, the short and elegant methods
of Green may be applied to obtain the results of Cauchy's theory,
and a comparison of the different theories of Green and Cauchy is
greatly facilitated.

Ox THE LOXG SrETRUM OF ElXClKlC LlGHT.
[Frwu tie Pij".'.v>/jij>.i." Tra >::•«* for lv"i. Eeceivied J...nf 19, '.:^-::'

^^Al^^tr.w:, jnwa :ho jfVtwwcKii^ of tie F. y.il o- •■.:.-_. sn. pjx l-:-^.~

T'riK ;ir.:'-.or"# res^Arvhes oa r..:orv*.\-:-.v-^ had led him to peroeine cist ;'...;.?
■was (^>«tqo» fiir the iiko^ rvrrArc:! lo :-..v^r. '.e -;v< i^f :':.; s^kur <\*otrum, and
thst ekotrio l:c"r.: c ■-T,.:'..;v. n>y> of <ull iicbtr nfirai';:'; ;!it_v, n'aioh vnfie
qiliw interofptevi bv c'as^ Iwt t"::,.: qi;,.r:j ;r,v*:u:T:;o. ti-.e<»' rivs freelv.
AvWrdii-.c'v he w-a* led :? jwoeure -or:*:..* aud a "e:.< of quarri which, irh«a
«^^>tied u> the ej^aininittioii of the v h..:,- irv, imt . : :he disoh.-.r^ o: a Levden
;.-4r. by fiiartnii.i i. piire >y'evtriini i-vi rec«YV,.; :: C4i a h.;V-Iy f,-,-.,rv;^,irr.T
sv.h^tdi-vx', reviealed the c^^'srei.ee o: r..\> f.^miitu a sj-ieotium no '.-:>< than six
or ei^ -t t;u:o5 i^ Ic: .: -■.* :he visible sv-evrrv-u:. This Ior^ s-.vv:r'_v_.. .is fconied
by the vcliaic arv '.v.:h cojipM- eketrvxi;^ was exh;li:txi at a lecture r ;:; at
the Royal i-s::: -.ti :: ir. 1S-'v» : b..; :he aittbcn-. fta- rt.is.. :.s he nier.tior.cvK. did
noT :';jen r.-.rther rirsr.e the s -.h-e^-:. Havr; s-.";.s^-.r.".e:,:iy fiHsnd that the
svGrk of an .-AAv..:ie'i-vvii with a Leyder. ' .r in cv>nneilon with the s-:,vnd&rr
tenains^s yirfdevi a s- ■; ,:r..:i: quite I r ;"nt eee -.ih to wirk by, fce rvsv.-.i.^i the
iv.vtsticAt^v^a, aavi esaiuined the sievtrA exh:":-.:e<i by s. v;ir;ety o: inet.Cs as
elev-tr^v.es, as '«reU as the mode c: Ah.^'.^rl^ti^ ;• of the rAys of h ;b, tvirAiii/eility
by vaii AS >Aris:vi;-..v^ The s-jivtr;» of the :i:;t.h> tiiAy be vicvned a: - ie.isvj??
by ;r.v,i^-.s of tiu tvsjxiKe, and the uivvie of a1>s^ ri ti.u of the ■■.vis. . '.e rays by
a ci-. ;-.. s>. "--.itioM may be At once obs^-rvei ; it: theare ate h:riv-.-.-ties Atter..tit'.;
the ^wejiiTAtiot. in this way e: SAieiet.tly Aoe trAte majs of the ntetciHh" hues :
arxi ti-re ctwit liability of the t-Ays of high r^?fr.c.j:thh:ty t be alis.rVe.i by
imj«.-«h.Ks ■, ri;><'at in vwy ■.Al,..;te quai.t :ty t\-i..i;rs the ^vrtain determinativMtt
of the oi>:iv^4l oh^vrAoter. ja t;:is r-js-.v-vt, of sAhstAr.>-es which .^re otily
niovier-iteiy oxvsor.e a tiiAtter o: consivierAhle dittiovhty. HAvir.^ fbond thiit
Pr Miller had leer. er.rAjTsxi iK-detvr..iet:tiy .At the s-iVA; s.tbi-.vt, ".vcriif.; by
phcto;r.i-jhy, the ar.th.r deemed :t ■•-r.v.rvvscs.iry t.^ Attejipi a delir.eAtion of
the njetailio .it.es for wh..,h., however, he h.is reeentiy vievised ti praetiosl
rAitlAvi thAt -vis fomid to wv^rk s-itis;Ao:oriIy , t-r to examine fLtrther the
Ais-.r-.tion f r,iys of hiA r^angihility by solAti rs .: rtetihie stilts, i..

Tie -. rcsiit papa" oor.t^iir.s thwrfon? tn^irhy rvsvhts .htAirevi in ther
directi*. r.s in the s^ime wide field :' rcs-.ir.h. Am»-.c the r-.it^Cs examined,
the Ar.th. r had f.'Ar.d Ah.irAinium the richest ir. inv s: ie rsy^s f extr^tne
ivfrai -^lii-tv; and a«>iT*«.i:r.ciy ahuuininm eirvtr.>.i-s weie employed whe:i
th-e deportment of s'.ch rAys htid tv> be stvoiAliy exArAir.eh. As the br:cht
sIr.r.-hrLi-.:rA Kaes of h:;h refrsr.^hoii.ty lo r, t AV-tvAr t hate been tAi<-:A by
ph t.crsv-hy, a .irA^irj of the AhroAiriAr; sy>r\.t— .t:ii is ci-re-i. '"ith zinc and
o-.»dtii:v.ni f.^r ov^vAVcirtS'. r..

The .Attthc* h.As sls».- doscr.iod aju-o, h^.-.re-.i the mode of Ahts.arptier .: the
inTis:hie rA-.s br s».'h;tio; s f v.vr.oAs A-.v.J.^t-.ts ani.i ci"Ao<..s. a^s. Kochefs irf

204 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

these classes, he finds, are usually intensely opaque, acting on the invisible
spectrum with an intensity comparable to that with which colouring matters
act on the visible. This intensity of action causes the effect of minute
impurities to disappear, and thereby increases the value of the, characters
observed. It very often happens that at some part or other of the long
spectrum a band of absorption, or maximum of opacity, occurs; and the
position of this band affords a highly distinctive character of the substance
which produced it.

Among natural crystals, besides the previously known yellow uranite, the
author found that in adularia, and felspar generally, a strong fluorescence is
produced under the action of the rays of high refrangibility, referable not to
impurities, but to the essential constituents of the crystal. A particular
variety of fluor-spar shows also an interesting feature, though in this case
referable to an impurity, exhibiting a well-marked reddish fluorescence under
the exclusive influence of rays of the very highest refrangibility. This
property renders such a crystal a useful instrument of research.

With some metals broad, slightly convex electrodes were found to have a
great advantage over wires, exhibiting the invisible lines far more strongly,
while with some metals the difference was not great.

The blue negative light formed when the jar is removed, and the electrodes
are close together, was found to be exceedingly rich in invisible rays, especially
invisible rays of moderate refrangibility. These exhibited lines independent
of the electrodes, and therefore referable to the air. This blue light has a
very appreciable duration, and is formed by what the author calls an arc
discharge.

The paper concludes with some speculations as to the cause of the
superiority of broad electrodes, and of the heating of the negative electrode.

Introduot

■ion.

The experimental researches described in a former paper* led
me indirectly to the conclusion that the electric spark, whether
obtained directly from the prime conductor of an ordinary electri-
fying machine, or from the discharge of a Leyden jar, emits rays of
very high refrangibility, surpassing in this respect any that reach
us from the sun — and that these rays pass freely through quartz,
while glass absorbs them, as it does also the most refrangible of
the solar rays. I was induced in consequence to procure prisms
and a lens of quartz, which were applied in the first instance to
the examination of the solar spectrum, and which immediately
revealed the existence of an invisible region extending as far
beyond that previously known as the latter extends beyond the
visible .spectrum, and exhibiting a continuation of Fraunhofer's

* " On the Change of Refrangibility of Light," Phil. Tram, for 18.52, p. 463.
[Ante, Vol. in, p. 267 and Vol. iv, p. 1.]

ox THE LOXG SPECTRUM OF ELECTRIC LIGHT. 205

lines*. A map of the new lines was exhibited at an evening
lecture delivered before the British Association at their Meeting
in Belfiist in the antumn of the same year : and I then stated that
I conceived we had obtained evidence that the limit of the solar
spectrum in the more refi-angible direction had been reached. In
tact, the very same arrangement which revealed, by means of
fluorescence, the e.xistence of what were evidently rays of higher
refrangibility coming from the electric spark failed to show any-
thing of the kind when applied to the solar spectrum. At least,
the only link in the chain of evidence which remained to be sup-
plied by direct experiment related to the reflecting power, for rays
of high refrangibility, of the metallic speculum of the heHostat which
was employed to reflect the sun's rays into a convenient direction ;
and this was shortly afterwards tested by direct experiment, on
rays from an electric discharge separated by prismatic refraction.

In making preparations for a lecture on the subject delivered
at the Koyal Institution in February 1853, in which I had the
benefit of the kind assistance of Mr Faraday, recoirrse was naturally
had to electric light, on account of the extraordinary richness
which it had been fotind to possess in rays of high refrangibility.
Although fully prepared to expect rays of much higher refrangi-
bility than were found in the solar spectrum, I was perfectly
astonished, on subjecting a powerful discharge from a Leyden jar
to prismatic analysis with quartz apparatus, to find a spectrum
extending no less than six or eight times the length of the visible
spectrum, and could not help at fii-st suspecting that it was a
mistake arising fi-om the reflexion of stray light. A similarly
extensive spectrum was obtained from the voltaic arc, and this was
sufliciently bright to be exhibited to the audience, the arc passing
between copper electrodes, and the piu'e spectrum formed by
quartz apparatus being received on a piece of ttranium glass cut
for the purjiose. The spectram thus formed was found to consist
entirely of bright lines-f^, whereas the spectritm of the discharge of
a Levden jar had appeared (perhaps from not having been truly
in focus) to be continuous, or at least not wholly discontinuous.

The mode of absorption of light by coloured solutions, as
observed by the prism, affords in many cases most valuable
chaKicters of particular substances, which, strange to say. though so

* Ibid. p. 559.

t Proceedings of the Bcyal InstiUition, Vol. i, p. 264. [Ante, p. 28.]

206 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

easily observed, have till very lately been almost wholly neglected
by chemists. Haying obtained the long spectrum above mentioned,
I could not fail to be interested with the manner in which sub-
stances, especially pure but otherwise imperfectly known organic
substances, might behave as to their abeorption of the rays of
high refrangibility. But the difficulties attending the habitual
use of a nitric-acid battery of 30 or 40 cells deterred me from
entering on this investigation, and I determined to confine myself
to the solar spectrum.

On account of some inconvenience attending the tarnishing of
the speculum of my heliostat, I was induced to order a quartz
plate, intended to be either silvered or coated with the usual
amalgam of tin. On trying on a small scale the reflecting power
of such plates with respect to the invisible rays, which may be
done by means of fluorescence almost as easily as if those rays
were visible*, I noticed a remarkable falling off in the reflecting
power of the silvered plate for the most refrangible of the solar
rays, which I readily found was due to a peculiarity of the metal
silver. This metal is highly reflective for the invisible as it is for
the visible rays up to about the fixed line S-f, when its reflecting
power falls off, with remarkable rapidity, and for the more re-
frangible rays of the solar spectrum is comparable with that of a
vitreous substance rather than with that of a metal. Steel, gold,
tin, &c. showed nothing of the kind, but copiously reflected the
invisible rays.

A few years ago, as Dr Robinson was showing me some experi-
ments with the induction coil, it seemed worth while to try
whether the spark obtained when a Leyden jar has its coatings
connected with the secondary terminals might not be sufiiciently
strong to exhibit by projection the long spectrum shown by
electric light. On projecting a spectrum formed by a prism and
lens of quartz on a piece of uranium glass, the long spectrum

* Philosophical Transactions for 1852, p. 537.

t According to the notation employed in the Map published in the Philo-
sophical Transactions for 1859, Plate xlvii. In this Plate the group S should
have been represented as three lines, of which the middle (specially named S)
divides the interval between the 1st and 3rd in the proportion of 8 to 2 nearly,
the spaces between the lines being a little darkened by shading. [This map was
prepared by Prof. Stokes, and published by Bunsen and Roscoe as a base-line for
their curve of photo-chemical intensity.

Kayaer points out {Handbuch der Spectroscopic , i, p. 103) that the fundamental
observation in the text is not quite accurate, the quasi- vitreous reflection extending
only a limited distance beyond S.]

ox THE LONG SPKCTRUM OF ELECTRIC UGHT. 207

was in fact exhibited. It was not, indeed, so bright as when formed
by means of a powerful voltaic batter\-, but nevertheless was quite
bright enough to work by. It was discontinuous, consisting of
bright lines. On changing the metals between which the spark
passed, we found that the lines were changed, which showed clearly
that they were due to the particular metals.

A wide field of research was thus thrown open to anv one
taking the very moderate trouble attending the use of an in-
duction coil. It remained to study the lines given by different
metals and gases, and the absorbing action of various substances
with respect to the in-^-isible rays of different refrangibilities.

Various observations were made from time to time in this sub-
ject. As regards the metalhc lines, it is perfectly easy to view them
at pleasure ; but to obtain faithful delineations of them is another
matter. Even an accomplished artist would find difficulty in
obtaining by mere eye-sketching a faithful representation of an
object which requires to be seen in the dark. I tried different
methods without being able to satisfy m3'self as to the accui-acy of
the drawings which could be thus obtained, and irequently thought
of resorting to photography.

Meanwhile the mode of absorption of the rays of high refiangi-
bility by a good number of substances was observed. Xothing is
easier, to a person provided T\"ith a cell mth parallel faces of quartz,
than to observe by means of fluorescence the mode of absorption of
these rays by a given solution ; but to draw safe conclusions as to
the optical character in this respect of the substance deemed to be
in solution is not so easy as it might appear ; for the ravs of high
reirangibility are liable to be absorbed by an exceedingly small
amount of an impurity which may chance to be present without
the observer's knowledge. Thus I found that about a quarter of
a square inch of clean filtering paper sufficiently contaminated the
water contained in a small cell to interfere sensibly with its trans-
parency. Should the solution be transparent there would be no
difficulty, for the effect of an impui-ity would not be to render trans-
parent a solution which otherwise would be opaque. Should it,
on the other hand, absorb the in^'isible rays, or some of them,
with great energy, or in a peculiar manner, we might again
conclude that we had obtained the true character of the sub-
stance deemed to be observed. The most remarkable example of
this kind which I met with among inorganic colourless solutions

208 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

was in the case of nitric acid and its salts, such as nitrate of
potash, soda, ammonia, baryta, which absorb the rays of high
refrangibility with great energy and in a pecuUar manner, exhibit-
ing a maximum of opacity followed by a maximum of transparency,
beyond which the absorption becomes still more energetic than
before. But if the solution should be found to absorb the rays of
high refrangibility with only moderate energy, it would be left
doubtful whether the observed absorption might not be due to
some impurity ; and I did not see how this doubt could be solved
otherwise than by a laborious system of recrystallizations.

After having obtained these results, I found by conversation with
my friend Dr [W. A.] Miller that he also had been engaged at the
same subject, working by photography, and had prepared a number
of photographs of metallic spectra, and studied by the same means
the absorption of the rays of high refrangibility by a great variety
of substances, chiefly inorganic acids, bases, and salts, and the
commoner organic bodies. Although a large part of the task
which I had proposed to myself has thus been accomplished in
another way, there are many results which I have met with which
are not likely to have been obtained by one working by photo-
graphy, and I have therefore thought it well to draw up a paper
embodying these results, and thus forming, as it were, a supple-
ment to the paper by Dr Miller.

Preparation of a Screen hy means of a Salt of Uranium.

Few substances are more powerfully fluorescent than several of
the salts of sesquioxide of uranium ; and a piece of glass coloured
by uranium and polished along at least two planes at right
angles to each other is exceedingly convenient, from its powerful
fluorescence and its permanence, for a screen on which to receive
a spectrum. Nevertheless such a screen, which must be viewed in
particular directions in order to get the strongest effect, is in
many cases less convenient than a screen would be which was
prepared by means of a highly fluorescent powder treated like a
water colour, which could be viewed in all directions indifferently.
This is especially the case in taking measures by a method which
will be mentioned presently. Besides, I find an excellent piece of
such glass defective in fluorescent power as regards the extreme
lines shown by aluminium; and some specimens are defective
to a much greater extent, which is doubtless due to impurities.
Accordingly I have long regarded it as a desideratum to obtain by

ox THE LONG SrE'TRVM OF ELECTKIC UGHT. 209

pivoipitation an insoluble or ver\- spvriuciy soiiible s^ilt of ses«,]iii-
oxide of luanmui which should be as fluorescent as the Ivst sciks
of that base, and which might be treated like a water colour.
I have no\v succeeded in prepurini: such a s^iIt. thou^jh not by
direct precipitation.

The ordinary phosphsrte obtained by precipitation, the coni-
|vsi:lon of which, independently -of waiter of hydration, is
PO; \^U,OjVHO*. is only sliclirly fluorescent. If. however, this
sc'»i:. with as much wa:er as remains when it is washed by de-
cantation. be pur into a scw.oer. a little tree phosphoric or sulphuric
acid added, and then cr\ s:als of phosphate of s.xia. phosphate of
ammonia. mieivvi.\>-.i'-ic salt, or borax be added in excess, rhe
',^ri^iru;V. sell" is grv.dually chimced into one which is powerfully
d.io'.Ysoont. The ohaiiire seems T'^ take place most rapidly with
K^rax ; b,i: as an excess of rhis sciilt is liable slowly ro decompose
rhe fl.iorescenr sal: rli-sr formed, ir is betrer to employ a phosphate.
The quantity of acid she ild be s irrioien: to leave a decided
add reaction when the liquid is fully saturated by the alkaline
ph. srhate employ e-d. The chrun^e may be watched by observing
fiv>m time to rime the d r itscenee of rhe Svi.lr by dayhght, with
the aid of abs.-rbirig media. I: is cy^mplete in a few days at
furthesr. when the sai: is readv ro be collected.

This rei| rii-es precaution, as the s^ilr is quickly decomposed by
dilute acids (and accordr..j:iy by i:s own mother-Uquor if dilutedX
and even, rh -.^h mor>e s, ■■vl\ , by pure water, with the formation
apparently of the oriciiiai phosphate. I: is als.^ de\-vmp.^sed. at
lectsr in time, by aikaliae carbonates, with the formation of a
beautiful yellow non-duorescent s^iit '.vseniblin.^ the precipitate
iri^er, bv alkaline ccvrb-.uiates in salts of sesquioxide of nianinni.
The sc-,!t may be collected by adding at once, instvad of water,
a sc^irr.rateel s-:»luriou of borax, in tjuantiry at .east surticient
t? destv v the acid leacticaL The salt is then jviiw, off in
s'.tsix" r.si':-n firom a"v -.indiss'-^ived crystals of the alkaline ph'.''sphate
emploved. and collected on a filler. A pressed c^ike of this salt, or
■, per. IS tile on which the s,'.lt is spread, having: been moistened
with a s>?lurion •:'i borax, t'.nns an admirable scrten and is what
I have chietiv employed of late. It sh. w^ of ■, irse. the visible ;is

f Tiis fixmnla »as ecajtmeted on the 5::vjosiTioa. kbaa c:irrsn:, that the
atoniiv- v»^t of os:.-:Kii U ~. and ijf tus r^ii-ti 130, Cf. also p. il6 tK-n*.]

5. IV. 14

210 , ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

well as the invisible rays — the former by ordinary scattering, the
latter by fluorescence.

From the circumstances of its formation, the salt is probably
(abstraction being made of the water of hydration) the original
phosphate with the equivalent of constitutional water replaced by
an equivalent of an alkali, which would make it analogous to the
highly fluorescent natural yellow uranite. At any rate this hypo-
thesis guides us to its successful preparation, the conditions of
which it would not have been easy to make out by observation
alone. Without the use of free acid the fluorescence is not fully
developed, which is accounted for by the insolubility of the
original phosphate and the fluorescent salt, which presents an
obstacle to the complete conversion of the one into the other.

Metallic Lines.
These may be viewed, as already mentioned, by passing the
spark of an induction coil between two electrodes formed of the
metal to be examined (the secondary terminals being respectively
in connexion with the coatings of a jar of suitable size), forming
a pure spectrum by a prism and lens of quartz, the faces of the
prism being equally inclined to the axis of the crystal, and the
lens being cut perpendicular to the axis, and receiving the spectrum
on a suitable screen, for which, if a fluorescent liquid be employed,
it is to be placed in a quartz-faced vessel, in default of which a
piece of filtering paper may be saturated with the liquid.

If the visible spectrum and the very beginning of the invisible
be excepted, the lines thus seen vary from metal to metal^ and
therefore are to be referred to the metal and not to the air. They
are further distinguished from air lines by being formed only at an
almost insensible distance from the tips of the electrodes, whereas
air lines would extend right across. The spectrum is far too
extended to allow us to regard the whole at once as in the
position of minimum deviation; and if the prism be placed at
all near the electrodes, without which we should have com-
paratively little light to work with, the effect of the different
divergency, converted by the lens into convergency, of the rays
in the primary and secondary planes is very great. In order to
obtain a pure spectrum, the screen must be in focus as regards
the primary plane; and if a particular point P of the spectrum
be at a minimum deviation, the lines immediately about P are
reduced almost to points, which are the images, for light of that

ox THE LONG SPECTEUM OF ELECTRIC LIGHT. 211

refrangibility, of the tips of the electrodes, or, to speak more
exactly, of the part of the spark just outside the tips. But in the
secondary plane the rays on one side of P haye not yet reached
their focus, and on the other side have passed it; so that the
image of a point is a line, the primary focal line, of a length
increasing on receding from P in either direction, and accordingly
the spectral image of either tip, assumed to be a mere point,
"would be a pair of slender triangles vertically opposite, and having
their common vertex at P, their lengths lying in the plane of
refraction. The invisible spectrum is in fact made up of two such
pairs of triangles con-esponding to the two tips respectively, as
may be readily seen when the electrodes are not too close. At
a distance from P at which the length of the primary focal line
becomes equal to that of the image of the spark, the two lines
which are the images, for rays of the refrangibility answering to
that distance, of the tips of the electrodes meet in the middle of
the spectrum, and beyond that distance they overlap, so that a
line appears to run across the spectrum, though it relates to rays
which emanated only from the immediate neighbourhood of the tips
of the electrodes, as may be seen by turning the prism till that part
of the spectrum is at a minimum deviation, and focusing afresh.

Besides the bright lines, evidently due to metals, which have
been mentioned, other weaker light is perceptible, too faint for
precise observation. A portion of this is probably due to the aii-.

The chief part of the visible spectrum as seen by projection
appears plainly to belong to the air ; for the lines stretch across
the interval separa,ting the electrodes, while the lines belonging to
the metals extend but a little way, even in the \asible spectrum,
and the former reappear when the electrodes are changed. With
some metals, however, lines belonging to the metal appear in the
visible spectrum which are comparable in strength with the
invisible lines of high refrangibility; but in genera,l it is rather
remarkable how poor is the visible spectrum, and e^en the in-
visible region for a good distance beyond, compared with the part
of the spectrum of still higher refrangibility, with respect to strong
lines characteristic of the metal.

I have lately adopted a mode of laying down positions in the
invisible spectrum which is extremely simple and convenient, and
yields results agreeing well with one another. It might be applied

14—2

212 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

to the formation of maps of the metallic lines; but this is un-
necessary, as the subject has been worked out by Dr Miller. It is
still useful, however, for laying down the positions of bands of
absorption, being more convenient and exact than estimating their
place with reference to the known metallic lines.

The method is as follows. The quartz prism is placed on
a block, raising it to a convenient height above a long drawing-
board, to which the block is screwed, and is fixed at pleasure by a
screw pressing upon it from above. The lens is fixed in a blackened
board screwed edgeways to the drawing-board near the prism, so
as to be ready to receive the rays of all refrangibilities after
refraction through the prism. The focal length of the lens actually
used was about 12 inches, and its diameter 1^ inch. A convenient
distance of the spark from the prism having been selected (I chose
30 inches), the drawing-board was turned round till it attained
such a position that, on placing the prism in the position of
minimum deviation for the middle of the long spectrum, the
rays belonging to that part fell perpendicularly, or nearly so, on
the lens, which had previously been placed so that this should be
a convenient position relatively to the drawing-board. The prism
was then fixed by its screw, and to mark the angle of incidence
a pin was placed at the edge of the shadow of one of the blocks.
On account of the increasing refraction by the lens of rays of
increasing refrangibility, the locus of the foci of the different rays
formed an arc of a curve, or nearly a straight line, lying very
obliquely to the axes of the pencils coming through the lens.
The projection of this line on the board having been marked,
a line was drawn bisecting this at right angles, and at a point in
the latter line situated 11| inches from the former*, the board was
pierced for the insertion of a pivot, which carried two wooden
rulers, which could be clamped together at any convenient angle.
The shorter of these carried a vertical needle, which as the ruler
was turned moved in fi-ont of the focus of the different rays at
the distance of about a quarter of an inch. The longer ruler
carried a pricker, destined to mark on a sheet of paper, temporarily
fastened to the drawing-board, the position of any object observed.
Thus the prism, the lens, the axis of motion of the needle and
pricker, and the pin for fixing the angle of incidence retained
an invariable relative position when the drawing-board was inoved.-

* A longer distance would have been better.

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 213

In observing, the electrodes were placed at the proper distance,
and the board turned till the edge of the shadow fell on the pin.
The rulers were then turned together till any bright line or other
object was eclipsed by the needle, and its place was then pricked
down. To obtain a fixed point of reference, I generally pricked
down the position of the extreme red visible on a screen, such as
a piece of paper ; but if great accuracy were required, it might be
better to employ a well-marked green air line.

The metals the spectra of which I have observed are Platinum,
Palladium, Gold, Silver, Mercury, Antimony, Bismuth, Copper,
Lead, Tin, Nickel, Cobalt, Iron, -Cadmium, Zinc, Aluminium,
Magnesium. Several of these show invisible lines of extraordinary
strength, which is especially the case with zinc, cadmium, mag-
nesium, aluminium, and lead, which last, in a spectrum not
generally remarkable, contains one line surpassing perhaps all
the other metals. Other metals exhibit lines which in certain
parts of the spectrum are both bright and numerous ; so that, in
taking a rough view of the whole, certain parts of the spectrum
are bright and tolerably continuous, while other parts are compara-
tively weak. This grouping of the lines is especially remarkable
in copper, nickel, cobalt, iron, and tin. Of the metals mentioned,
magnesium gives by far the shortest spectrum, ending in a very
bright line, beyond which, however, excessively faint light may be
perceived to a distance about as great as the extent of the longer
spectra. Aluminium, on the other hand, stands at the head of the
above metals for richness in rays of the very highest refrangibility ;
and it is to this part of the spectrum that the strong lines above
mentioned belong. In calling these lines strong, it must be
understood that some allowance is made for their very high
refrangibility; for when observed as above described they do not
appear absolutely quite so strong as the bold lines of zinc or
cadmium. This is fartly due to the defective transparency of
quartz, which for this part of the spectrum shows itself by no
means perfect ; and indeed the highest aluminium line, which is
a double line, can only be seen by rays which pass through the
prism near its edge.

The following figure exhibits the principal lines of aluminium,
with zinc and cadmium for comparison. In the first of the
aluminium lines represented, I could not make out the division
into two parts corresponding to the tips of the electrodes.

214

ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

R denotes the extreme red visible on a screen; the lines in the
visible spectrum are omitted, as this has been made the subject of
elaborate researches by others. The
horizontal distances are proportional
to the distances of the several pricks
from that belonging to the extreme
red, and therefore vary as the
chords of the arcs described by the
pricker. This tends to correct to
a certain extent the exaggeration
of the more refrangible end of the
spectrum arising from the mode
adopted of laying down the posi-
tions of the lines. The lowest row
of lines in the figure, which is
placed here for the sake of com-
parison, will be referred to further
on.

Besides the lens above men-
tioned, I sometimes employ in a
different manner another of ^-inch
diameter and 2| inches focal length,
and accordingly large for its focal
length. This is used for forming
an image of the spark, which is
received on the substance that is
to be examined, or that is used for
examining the spark. The differ-
ence of focal length for the different
rays is so enormous that, while one
part of the spectrum is in focus,
other parts are utterly out of focus,
and thus we may judge in a general
way of the refrangibility of the rays
by which any particular effect is
produced. In this way such con-
centration of the rays is obtained,
that effects may be studied which

would not bear examination by prismatic analysis. In speaking
of this lens I shall call it the 2'5-inch lens, from its focal length.

ox THE LOXG SPECTKITI OF ELXCTRIC LIGHT. 215

Al\<o'j'C''oH of vte mpisible m^/s by Alhaloidf. Gl'^cosides. iv.

Before examining these ?ubcftanoes it i? i\-].iisire to dissolve
them, and we must lli^: inquire into the tninsparencv of the
s^^lvent. Fortunately the mt\?r useful of all solvents, w-ater, is
transparent when pure: and as to reas^^nts. we may employ
suiph inc or hydrochloric acid for an acid, these acids bemi: trans-
parent, and aiumorii.i. supjv>se. for tua alkali. In speaking: of a
suCvst^uice as transparent. I wish ir only to he imderstood that it
ts of a transpjuenoy eoiaparable with quartz. As to ammonia.
although It absorbs the more refirangible rays when in quantity
(tinless the observed absorption were due to s.^ine impttiityV it
:nay be deemed transparent in the small qu^mtity which alone
it is reqrdsite to employ. Even alcohol, which in the state in
which it IS to be had is defective in transparency, is sutiiciently
transparent to be cutployed as a solvent for such stibstanoes as
those tinder consideration, provided it be used in small tkickness
only.

The alkaloids and iriue-.'^sides which I have examined sure almost
without exception intensely opiqiie for a portion at legist of the
invisible rays, absorbing them with an energy comparable for the
most part to that with which ooloiirini: matters (such as alizanne,
Xo.) absorb the visible rays. The mode of absorption also is
frequently. I nui^ht almost say generally, highly characteristic :
so that by this single property they might be distinguished one
fivm another. It frequently happens t'.x> that the mode of
absorption decidedly changes acc^riini: ;is the s-.^lution is acid
or alkaline, which assists still fivrther in the discrimination.

In the examination I s.^metimes employ a small cell with
parallel feces of quartz, s.^metimes a wedge-shaped vessel, having:

its inclined feces als'.^ of quartz, but more commonly the former.
The cell being filled with the solvent, a minute quantitv- of the sub-
stance is introducevi. and the pr.^gress of the abs..^rption is watched
as the substan.v ^rradually diss^.^lves. the fluid meantime being
of course stirivd tip. Iti this way it is easy t'l seize the most
characteristic phase of the abs"-^rption. which may be then reiristered
bv the pricking instrument. When minima of opacity '.xviir. it is
best to seiaDe that sta^re of the abs'-^rpti.^n at which they are well
developed. When no minima occur, a greater or less pcirt of the

216 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

more refrangible region is quickly absorbed, after which the
absorption creeps on towards the less refrangible side. When
once it has become tolerably stationary, the limit of the rays
transmitted may be marked. It seems desirable not to go beyond
this point in the absorption, lest some possible impurity in the
substance examined, which if it had formed the whole of the
specimen would have absorbed rays of lower refrangibility, should
begin to make itself perceived, and its mode of absorption
should be mistaken for that of the substance professed to be
examined.

All the metallic spectra are discontinuous, which prevents the
mode of absorption of even a solid or liquid from being observed
quite so well as in the solar spectrum, even independently of the
greater intensity of the latter, and would greatly interfere with
the observation of narrow bands like those shown by the absorption
of certain gases in the visible spectrum, and of which chlorous
acid gas (CIO4) shows a splendid system in the invisible part of
the solar spectrum. Should a general absorption take place in
a part of the spectrum where previously a bright group of lines
was seen, with weaker light for some distance on both sides, it is
evident that at a certain stage of the absorption the bright group
would be left isolated, and the effect might be mistaken for a
maximum of transparency. In doubtful cases of this kind it is
requisite to change the electrodes, so as to use the spectrum of
some other metal; but practically the difficulty is not so great
as might be supposed.

It is desirable to choose a metal which gives a spectrum that
is bright and tolerably continuous in the region in which the
distinctive features of the absorption are most likely to occur. For
general use in the examination of substances such as here con-
sidered, I prefer tin — the electrodes (or one of them at least) being
broad, for a reason which will be mentioned presently. Tin, indeed,
is weak in the most refrangible region, though after a long
interval of weakness it shows one pretty strong line between'
the second and third of the strong aluminium lines; but with
these substances the distinctive features of the absorption hardly
ever occur so late. For combined strength and continuity, copper
answers well for the highly refrangible region in which tin is
weak; while mercury, which may be employed in the form of

ON THE LONG SPECTRUJI OF ELECTRIC LI(3HT.

217

amalgamated zinc, is the richest metal for the invisible region
just beyond the visible spectrum ; but I have employed tin almost
exclusively.

The following figure gives the bands of absorption observed in

Principal lines of
zino

Strychnine, in dilute
snlphuric acid

Brncine, in dilute
sulphuric acid

Morphine, in dilute
sulphuric acid

Codeine, in dilute
sulphuric acid

Narcotine, in dilute
sulphuric acid

Naiceine, in dUnte
sulphuric acid

Papaverine, in dilute
sulphuric acid

Caffeine, in dilute
sulphuric acid

Corydaline, in dilute
sulphuric acid

Piperine, in alcohol

^souUne, in dilute
ammonia

Phlorizine, in dilute
anunonia

Phlorizine, in dilute
sulphuric acid

Salicine, in water ...

Arbutiue, in water..

p

HHH

1

^HH

■

1

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solutions of several alkaloids and glucosides. The bold lines of
zinc are given as points of reference ; but the observations Avere

218 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

made with electrodes of tin. The border on the left is the limit of
the red light visible on a screen.

Although the central part of the maxima of transparency in
this figure is generally left white to save trouble, the reader must
not suppose that that part of the spectrum suffers no absorption.
On the contrary, it is more or less weakened when the solution
has the strength to which the figure corresponds, and disappears
altogether when the quantity of substance in solution is increased,
while at the same time the edge of the first band of absorption
creeps on a little towards the red, the absorption being usually
pretty definite at this edge. The measurements were taken from
the points where the light ceased to be sensible, which are repre-
sented in the figure by the junction of plain black and shaded
white. The shading merely represents the general effect, the
gradation of illumination not having been registered. It extends
in the figure, as a general rule, too far to the left of the edge
of the first black band, and accordingly does not represent the
absorption at that limit as sufficiently definite.

A glance at the figure will show how distinctive is the mode
of absorption of the rays of high refrangibility by these different
substances. Indeed this one character would serve to distinguish
all these substances one from another, unless it be morphine from
codeine, and caffeine from salicine. The dotted line in the figure
for sesculine denotes the commencement of the fluorescence, which
is situated near the line G of the solar spectrum. A solution of
brucine cuts off the invisible end of the solar spectrum about
midway between the lines 8 and T, and accordingly not far from
the end of the region which it requires a quartz prism and lens to
see. Accordingly, when these substances are examined by solar
light their distinctive characters are almost wholly unperceived, the
solutions of some appearing quite transparent, and those of others
merely cutting off the extreme rays to a greater or less distance.
With ajsculine alone the maximum of opacity lies within the solar
spectrum ; but even in this case we should have little idea of the
great increase of transparency about to take place.

The effect of acids and alkalies on all the glucosides referred
to in the figure presents one uniform feature. When a previously
neutral solution is rendered alkaline, the absorption begins some-

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 219'

what earlier, when rendered acid somewhat later. With salicine
there is merely an indication of this change, falling within the
limits of errors of observation; but in the other cases it is quite
perceptible, and with phlorizine the shifting of the band of absorp-
tion produced by an acid is very large. Fraxine (or paviine)
agrees remarkably with sesculine in all its optical characters ; the
maximum of absorption is merely situated a little nearer to the
red, and the tint of the fluorescent light corresponds to a slightly
lower mean refrangibility.

Quinine presents no decided maximum of transparency. With
this and the other bases observed, with one exception, the absorp-
tion, if changed at all, is changed in an opposite manner to the
glucosides when the base is set free by ammonia.

Bands of absorption occur also with neutral substances, for
example coumarine and paranaphthaline, which last exhibits a
system of such bands in the invisible part of the solar spectrum.

Aconitine, atropine, and solanine exhibit no bands of absorp-
tion, but merely a general opacity for the more refrangible rays..
The last, indeed, when dissolved in dilute sulphuric acid, is, for
this class of bodies, remarkably transparent ; while when the base-
is set free the solution, contrary to what takes place with the
other bases, becomes much more opaque, but the absorption is-
vague. I am not sure, however, how far the purity of the specimen
examined may be trusted, though it was white, and regularly
crystallized. It would be easy to examine more such substances ;
but what precedes is sufficient to show the value of the study of
the absorption of the rays of high refrangibility, as affording:
distinctive characters of substances little known.

Minerals.

I have examined a large number of minerals by the rays from
the induction spark, both as to their transparency and as to their
fluorescence. The transparency of those crystals which were of
such a form as to permit it, was examined by holding them in
front of a pure spectrum formed on a fluorescent screen. The
fluorescence was sought for by forming an image of the spark, for
which aluminium electrodes were employed, by the 2-5-inch lens,,
holding the mineral first at the focus of the visible rays, and then

220 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

moving it up towards the lens, and watching for any image which
might be formed by the rays of higher refrangibility. Should such
be observed, its nature was further demonstrated by interposing
in the path of the rays a very thin piece of mica. This .cut off
the image by intercepting the invisible rays, with respect to
which, except a small portion of the lowest refrangibility, mica
is intensely opaque.

Carbonate of lime, the sulphates of lime, baryta, and strontia,
and colourless fluor-spar, were found transparent (sulphate of
strontia less so), at least in the qualified sense above mentioned,
thus demonstrating the transparency of carbonic, sulphuric, and
probably hydrofluoric acid, and of the bases, lime, baryta, strontia *.
But this subject would be better followed out by salts artificially
prepared, and has been investigated by Dr Miller. In two cases
results of considerable interest were obtained with reference to
fluorescence.

At the time of writing my first paper, on the change of re-
frangibility of light, I had found but one mineral, yellow uranite,
to the essential constituents of which the property of fluorescence
plainly belongs f. In many other cases, both before and since that
time, I have observed with solar light fluorescence in minerals, but
always apparently having reference to unknown impurities, and
therefore to my mind of much inferior interest. By means of the
induction spark, employed as above described, I have found one
more fluorescent mineral;!:.

On receiving the image on adularia, and focusing it for the
rays of highest refrangibility, a pair of bluish dots were seen,
which were the images of the tips of the electrodes exhibited by
fluorescence. As the appearance was everywhere the same, on
natural faces and cleavage planes alike, and the same was observed
with colourless felspars generally from different localities, it is

[* This inference seems to be open to question, as also the ascription of the
opacity of mica to peroxide of iron.]

t Philosophical Transactions for 1852, p. 524. [Ante, Vol. iii, p. 352.]
X The method by which M. Edmond Becquerel has examined the fluorescence
■of minerals (Annales de 'Chimie, s6r. iii, torn. Lvii, p. iS) does not permit of
•distinct vision of the specimen from the distance of a few inches, which seems to
me necessary to allow the observer to judge whether the fluorescence which may
be observed is due to the essential constituents of the crystal or to accidental
impurities.

ox THE LONG SPECTRUM OF ELECTRIC LIGHT. 221

doubtless a property of the silicate of alumina and potash con-
stituting the crystal. Some specimens, it is true, did not show
the effect so strongly as adularia or moonstone ; but this is easily
explained by the greater purity of the latter varieties. For the
fluorescence extended to a very sensible though small depth
within the crystal, and yet the rays producing it were cut off'
by a film of mica much thinner than paper. The intense opacity
of mica is doubtless due to peroxide of iron, which nevertheless
forms no more than perhaps 5 per cent of the mineral. Hence a
very small percentage of peroxide of iron, or any other impurity
having a similar absorbing action, would suffice greatly to reduce
the quantity of fluorescent light emitted.

In a concentrated solar beam passed through a suitable ab-
sorbing medium, adularia did not show the least sign of fluor-
escence, in which respect it notably differs from common glass, such
as window-glass.

The other case of interest relates to a particular variety of
fluor-spar found at Alston Moor in Cumberland. This variety is
very pale by transmitted Ught, being in part of a brownish purple
colour, shows a strong blue fluorescence, and is eminently phos-
phorescent on exposure to the electric spark. On presenting such
a crystal to the spark passing between aluminium electrodes,
besides the usual blue fluorescence there is seen another of a
reddish colour, extending not near so far into the crystal. On
receiving on the crystal the image of the spark, and moving the
crystal fi-om the focus of the invisible rays towards the lens, it was-
soon in best focus for the rays producing the blue fluorescence. It
had to be moved much nearer to the lens before it came into focus
for the rays producing the reddish fluorescence, and was then at
the distance at which a well-defined image of the tips of the
electrodes is formed on the uranium salt; which proves that the
reddish fluorescence was produced by the rays belonging to the
bright lines (considered as a whole) of aluminium of extreme
refi-angibility.

The crvstal which showed this effect best was externally
colourless for about the ^th of an inch, which stratum showed
no fluorescence when examined in this way. Then came one or
two strata, parallel to the faces of the cube, showing the mddy

222 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

fluorescence, and exhausting apparently the rays capable of pro-
ducing that effect. The blue fluorescence extended much deeper,
and presented a stratified appearance, as Sir David Brewster long
ago observed.

On admitting a pencil concentrated by a quartz lens parallel to
and almost grazing a face of the cube, so that the rays traversed
the colourless stratum, the reddish fluorescence was observed in
the stratum which produced it to a long distance from the face by
which the rays were admitted, which demonstrates the trans-
parency of fluoride of calcium for the rays of very high refrangi-
bility.

The property of exhibiting such a well-marked effect under
the exclusive influence of rays of extreme refrangibility, renders
such a crystal a useful instrument of research. Several other
metals besides aluminium show the reddish fluorescence; but
none of those examined showed it so well, partly because it is
evidently produced more copiously by aluminium electrodes, and
partly because it is less masked by the blue fluorescence, the spec-
trum of aluminium being rather wanting in brightness until the
region of extreme refrangibility is reached.

If the crystal be held near the electrodes, and observed while
their distance changes, it will be found that on passing from the
greatest striking distance the reddish fluorescence decidedly im-
proves. On still further diminishing the distance between the
■electrodes, the reddish fluorescence appears still to increase ; though
whether this is a real absolute increase or only an increasing pre-
ponderance over the blue, it is not easy in this way to say for
certain. Hence the copiousness of rays of high refrangibility
increases at first, and continues to increase relatively if not abso-
lutely. It is supposed that the jar is sufficiently large to prevent
the discharge from degenerating into what will be presently
described as the arc discharge.

If the crystal be held close to the contact-breaker when the
secondary terminals are separated, and the effect be compared
with that of the secondary discharge (a jar being in connexion,
as has been supposed all along), the electrodes being of platinum
for fairness of comparison, it will be found that the proportion of

ox THE LOXG SPECTRUM OF ELECTEIC LIGHT. 223

rays of extremely high refi-angibility is decidedly greater for the
spark at the contact-breaker than for the secondary discharge.

On forming by the 2'5-Lnch lens an image of the spark from
aluminium electrodes, and placing a crystal, such as that above
mentioned, in the focus of the rays producing the reddish fluor-
escence, it is easy to determine the ti'ansparenc\- or opacity of
substances for those rays, the alteration of the focus by the intro-
duction of a thick plate being of course borne in mind, and the
crystal moved accordingly. The rays forming the image have had
to pass only through air, and through a very small thickness of
quartz, before reaching the crystal. In this -svay I have found
that even quartz itself in very moderate thickness is opaque for
these rays ; but difterent specimens, or different parts of the same
specimen, vary in this respect. I possess a large plate 042 inch
thick, cut perpendicular to the axis of the crystal, which is
generally transparent, but is slightly brownish on one side, to
the distance of about half an inch from the face of the hexagonal
prism. The coloxirless part of this plate, beyond a Httle distance
from the brownish part, is opaque for the rays in question*,
while the brownish part is nearly transparent. It may be inferred
that the colourless part contains a minute quantity of some im-
purity capable of absorbing these rays, which does not exist, at
least to the same extent, in the brownish part, although the latter
is not perfectly pure silica, as is shown by its colour. On the
whole, I am disposed to think that quartz, if it were rigorously
pure, would be transparent. We see at any rate how difficult it is
to draw certain conclusions respecting the transparency or opacity
of a substance which, in the state of purity in which it may 1^
obtained, shows only a slight defect of transparency.

I tried reflecting the rays from the spark by a fine Munich
grating, but the light was far too faint to be of any use. Possibly
a large and verj- closely ruled plane speculum, with a concave
speculum instead of a lens, might give light which it would be
possible to observe. But at present I have not found any sufficiently

* It shonld be mentioned that this part contains those delicate, definitely
directed, elongated laminae or crvstals. haidly visible except in a beam of sonlight,
which are called by practical opticians • bine shoots.' An examination of a number
of cnt pieces of qaartz lent me by Mr Darker confirms me in the suspicion that
snch crystals are more defective in transparency than other colonrless specimens
for the rays of extreme refrangibility. July 1S62.

224 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

marked effects referable to rays of still higher refrangibility to
make it worth trying.

The same crystal which showed the reddish fluorescence w'as
eminently phosphorescent, with a blue colour. The phosphor-
escence, like the fluorescence, was arranged in strata parallel to
the faces of the cube, and, like the reddish but unlike the blue
fluorescence, was not perceptible beyond a moderate distance from
the surface at which the exciting rays had entered. On forming
:in image of the discharge by the 2"5-inch lens, focusing the
crystal for the rays producing the reddish fluorescence, fixing it
there, and breaking the circuit after the induction coil had
worked for a little while, a dart of blue phosphorescent light
was seen in the crystal at the focus of the lens. On focusing
for the rays most efficient in producing the blue fluorescence, the
reddish was diffused over a broad portion of the strata producing
it ; and on repeating the above experiment in this position of the
crystal, the blue phosphorescence was seen similarly diffused. This
shows that the rays of extremely high refrangibility are those most
efficient in producing the blue phosphorescence.

[We may suppose that the blue fluorescence, the reddish fluor-
escence, and the blue phosphorescence are due to the action of
the assemblage of heterogeneous exciting rays on the same sub-
stance (doubtless some impurity taken up during crystallization),
or on two or three distinct substances. The blue fluorescence is
produced abundantly at a depth within the crystal at which the
two other effects are invisible ; but this alone is no proof of a
diversity in the nature, of the substaiice acted on, because the
rays producing the two latter effects would have been absorbed
before arriving at such a depth. Hence it is among the early
strata, in crossing which rays capable of producing each of the
three effects are still vigorous, that evidence must be sought, in
the coincidence or non-coincidence of the strata in which the three
effects are respectively perceived, of the probable identity or certain
diversity of nature of the substance acted on. At the time when
this paper was read I fancied I had observed slight discrepancies
as to coincidence in the strata. But a renewed examination, in
which a larger number of specimens were observed, leads me to
regard the fancied discrepancies as too doubtful to rely upon and
to overpower the increasing weight of evidence on the other side.

ox THE LONG SPECTBrM OF ELECTRIC LIGHT. 22-3

The blue fluorescence may be observed in the early strata
(Mrhich ordinarily, at le;\st \rith electrodes of aluminium and several
other metals, show a red) hy absorbing the more refitangible of
the exciting rays by a stiitable plate of qitarrz, or else by substi-
tuting for aluminium some metal, such as magnesium, which is
poor in rays of extreme refrangibiHty. On the other hand, the
red fluorescence really existing in the early strata, when it is
OTerpowert?d by the blue, may be seen by viewing the crystal
through a soltttion of chromate of pot;v?h. which greatly enfeebles
the blue fluorescence, while at the ssune time it transmits enough
of the spectrum to allow the unabsorbed residue to be at once
distinifuishable by its colour ( green) fiwm the red fluorescence.
In this wav the red fluorescence may be readily perceived even
with electrvties of magnesium. Acain, a pvrticular stratum
•which showed a blue fluorescence when acted on by i-ays which
entered bv a fiice of the cube, and before reaching it had to
traverse some other strata showing fluorescence, exhibited a red
fluorescence when acted on by rays which fell on it directly, having
been admitted through an octahedral fece.

It is more diflicidt to decide ;is to the identity or diversity
of the strata showiiit: respectively red fluorescence and blue phos-
phorescence, because the two effects are ■■bserved in a ditierent
wav; but as fer as I corJd decide, the strata appe;vred to corre-
spond.

On the whole, then, I am disposed to rbink it probable that
it is the same substance which, in ccnsequrncr of the action ■>!
ravs beginnini: with a part of the violet and extending ftom
thence onwards, exhibits a blue duortscence. which, in conse-
quence of the action of rays of extreme refrangibility, exhibits
a red fluorescence, and which, in cc'Dsequence of the action oi
ravs of a similar rtfningibihty. exhibits a powerful blue phos-
phorescence. At least, if the substances be diflerent they would
appear to have coexisted in solution, and so to have been taken
up totrether in the crystallization of the mineral. I should
mention, however, that it is contrary to all my experience that
the fluorescence of a single substance (i.e. not a mixture) should
thus, as it were, take a fir\?sh st^vrt with a f fji'/y dijferei,t ::lour on
proceeding onwards in the spectrum ; but then my experience is
s. iv." 1-5

226 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

derived mainly from the examination of substances in the com-
paratively short solar spectrum. — July, 1862.]

I have said that the phosphorescence was produced in certain
strata within the crystal. These strata were in some, places
sharply terminated, so as to be foreshortened into well-defined
lines. On watching the phosphorescence, there was nothing to
be seen at all like conduction ; the strata remained .sharply de-
fined as long as the light was strong enough to enable one to
judge. This is at variance with one of the two results which,
on the authority of others, I formerly mentioned as indicating
a distinction between phosphorescence and fluorescence*. On
trying shortly afterwards along with Mr Faraday, I could not
obtain either of these results. One of them, that i-elating to
apparent conduction, which was obtained by MM. Biot and
A. C. Becquerel, has since been explained by M. Edmond Bccquerel
as an illusion of observation f. The other, that relating to the
production of phosphorescence in Canton's phosphorus by rays
which had traversed a strong solution of bichromate of potash,
I am, after a conversation with Dr Draper, still unable to explain.

Advantage of Broad Electrodes.

At first I employed by preference wires or sharp pieces of
metal for electrodes, in consequence of the greater facility with
which the discharge passed, and the larger quantity of light given
out by the spark. Certain considerations, however, led me to try
broad electrodes ; and I accordingly procured electrodes of the
common metals shaped like small watch-glasses, about an inch
in diameter. These showed in some cases a most marked
superiority over thin wires, exhibiting the invisible metallic lines
in far greater strength, while with some metals there was not
much difference. With copper, for example, the superiority was
very great, with iron it was comparatively small.

Instead of electrodes of this shape, it is sufficient to take two
pieces of thick foil, make them slightly cylindrical by means of a
round ruler, or a pencil, and mount them with their convexities
opposed and the axes of the cylinders crossed.

* Philosophical Transactions for 1852, p. 547. [Ante, Vol. iii, p. 385,]
t Annales de Chimie, torn, lv, (1859), p. 112.

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 227

Besides copper, silver, tin, and aluminium show a great ad-
vantage of flat electrodes, and lead a moderate advantage, while
with zinc, as with iron, sharp electrodes are nearly as good. Brass
agrees in this respect with zinc, and not with copper, though it
shows the copper lines very strongly.

With such electrodes, however, the spark dances about ; and its
unsteadiness is objectionable in some experiments. A good part of
the advantage of flat electrodes is however retained if one only be
flat, especially if this be negative, and the spark is now steadier.
Instead of using the end of a wire to combine with a flat
electrode, it seems rather better, according to a plan suggested to
me by Dr Miller, to bend a wire to a gentle curve lying in a
vertical plane passing through the prism; or the edge of a flat
piece of metal may be similarly employed.

On forming an image of the spark between a sharp and a flat
electrode of copper, and receiving it on a fluorescent screen, the
flat electrode gave the brighter of the two images already mentioned,
and that, whether the electrode were positive or negative.

On similarly forming an image of the spark between two very
broad electrodes, and focusing for the rays of highest refrangi-
bility, the image did not, as usual, consist of two separate dots ;
but, whether it was, that, from the shortness of the spark, the two
ran into one, or that the rays belonging to the metallic lines of
high refrangibility were emitted throughout the whole length of
the spark, I am not quite certain ; but I incline to the latter
opinion, as a separation of the discharge into two portions,
corresponding to the immediate neighbourhood of the two elec-
trodes respectively, could hardly have escaped detection had it
existed.

Arc Discharge, and Lines of Blue Negative Light.

On diminishing the distance between the electrodes, formed
, suppose of copper wires, the brightness of the metallic lines at first
improves, and afterwards changes but little, or, if anything, rather
falls off. On still further diminishing the distance, so that the
electrodes almost touch, and the discharge passes with little noise,
a new set of strong lines make their appearance in the invisible
region of moderate refi-angibility. In this mode of discharge, in

15—2

228 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

which the negative electrode, if at all thin, quickly becomes red-
hot and fuses, the jar has not much influence, and the lines in
question are still better seen when it is suppressed altogether.
To show them to perfection, it is best to take a flat negative
electrode, so as to carry off the heat, and not to hide from the
prism any part of the blue negative light, and a sharp positive
electrode almost touching the former. In this way the visible
discharge is reduced almost wholly to an insignificant-looking star
of blue light; but it is wonderful how strong an effect it is
capable of producing in the invisible region. The most striking
part of the invisible spectrum consists of four bright lines, num-
bered 1, 2, 3, 4 in Fig. 1 [p. 214], situated not far from the visible
spectrum. These are followed, after a nearly dark interval, by light
arranged in masses resembling in its general aspect the groups
of copper lines (from which, however, it differs), but not strong
enough to be resolved or accurately measured. The figure repre-
sents also a couple of blue bands (b, b') seen by projection. These
are not seen on looking at the blue light directly with a flint-
glass prism of 60°, because everything is seen in too great detail.
Most of the air-lines in the invisible spectrum, especially the
bands beyond line 4, have an ill-defined look, and would probably
be resolved did the intensity of the light permit.

The appearance just described is independent of the nature
of the electrodes, and therefore is to be referred to the air,
and not to the metal. On viewing in a moving mirror the star
of light producing this effect, it is found to have a considerable
duration.

On slightly separating the electrodes, forming an image of the
discharge with the 2'5-inch lens, and receiving it on a cake of the
uranium salt, a very strong fluorescence was seen over the image
of the blue disk when the lens was focused for a point a little
beyond the visible spectrum. On moving the lens onwards, the
fluorescence produced by the rays belonging to this image spread
out into a ring ; and on moving still further, a tolerably well-
defined image of the whole discharge was perceived. Of this the
part belonging to the blue disk was the brightest, and was
surrounded concentrically by the ring before mentioned, now still
further widened. The image of the remainder of the discharge
was brightest where it was most contracted at the positive

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 229

electrode. The discharge generally was perhaps of slightly higher
refrangibility than the blue disk, even excluding from the latter
the rays belonging to the. ring. It thus appears that the four
bright lines figured were produced mainly by the blue negative
light.

The mode of transition of the discharge may be studied by
placing the electrodes at the greatest striking-distance and making
them gradually approach. At first there passes a clean bright
spark making a sharp report, and not resolved by a revolving
mirror. The invisible spectrum which this shows is too faint for
precise observation ; the visible spectrum shows chiefly air-lines.
As the electrodes approach, the spark becomes clothed by the
well-known yellowish envelope capable of being blown aside, and
the blue negative light begins to appear. A moving mirror, as
M. Lissajous has already observed*, shows an instantaneous spark
at the commencement, in point of time, of the envelope and blue
negative light, both which are drawn out, indicating a very
appreciable duration. On making the electrodes approach some-
what nearer, the spark diminishes, and the envelope is formed in
perfection, especially with broad electrodes. The air-lines now
begin to show themselves well, but are brightest on the side of the
spectrum answering to the blue negative light.

It might be supposed at first sight that the permanence of the
yellowish and of the blue light only indicated a glow of appreciable
duration left by a sensibly instantaneous discharge ; but several
circumstances indicate that the discharge itself lasts, and that
it is under its action that the glow takes place f. The action,
I am persuaded, is this : a spark first passes ; and this enables
a continuous discharge to pass, which is due, in part at least,
to the inductive action of the still falling magnetism, just as
a voltaic arc may be started in a powerful battery by passing
an electric spark between the sUghtly separated electrodes; and
the glowing of the air under the action of this discharge produces
the yellowish envelope and blue negative light. Thus, when the

* See Du Monoel, Recherchee sur la non-homogeneite de Vetincelle d'induction,
p. 107.

t Although this view may be considered already established (see the work by
the Vicomte Du Moncel just quoted), the observations here mentioned will not,
I hope, be altogether useless.

230 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

electrodes are nearly at the greatest distance at which this sort
of discharge takes place, the blue negative light is seen pretty
sharply terminated in a moving mirror. Were it a dying glow,
it ought to fade away ; but if produced under a discharge, it ought
to cease almost abruptly, inasmuch as at this distance of the
electrodes a continuous discharge is unable to pass when the
tension has sunk much below that under which it was first
produced.

The same conclusion may be drawn from an effect which
I once obtained, the exact conditions for the production of which
it is not easy to hit off. With a jar in connexion, each discharge
due to a single breach of contact appeared in a moving mirror as
a bright spark joined to a spark less bright by the blue negative
light, and also by the yellowish or reddish light, brightest close to
the positive electrode. Were the blue light due to a glow, it
ought to be reinforced instead of being put out by the second
spark, whereas the explanation of the result is easy on the
supposition of a continuous discharge. The first spark started
a continuous discharge, which emptied the jar less fast than it
was filled by the secondary coil ; so that presently another dis-
charge took place, which emptied the jar so that a continuous
discharge could no longer pass.

On viewing the broad discharge formed without a jar when
the electrodes are at a moderate distance, through a revolving disk
of black paper with a single hole near the circumference, while
the envelope was being blown aside, so as to get a succession of
momentary views of the discharge, the envelope was seen extra-
vagantly bent, as a flexible conductor might have been — not torn
across, as a column might have been which was heated by a
previous spark. The central spark, of course, was usuallj^ missing,
as it is sensibly instantaneous.

I have spoken of the arc, and especially the blue negative light,
as exhibiting air-lines. The arc, however, is liable to be coloured
not only by casual dust (as when it passes partly through the
flame of a spirit-lamp with a salted wick, when it is coloured
yellow by sodium), but also by matter torn from the positive
electrode. This is well seen with electrodes of aluminium, when
the arc or a portion of it is frequently coloured green. This green
light has a very sensible duration, and a distinctive prismatic

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 231

composition, and is brighter towards the positive than towards the
negative electrode, but is not confined to the immediate neighbour-
hood of that electrode (extending indeed sometimes over almost
the whole length of the arc), in which respect, and in its duration,
it differs from the light of the spark proper*. With aluminium
opposed to another metal, as copper or iron, the green light is seen
only when the aluminium is positive. Even with aluminium this
light may generally be got rid of by making the electrodes
approach; and it is the arc in what may thus be deemed its
normal state that was observed for the construction of the last line
of Fig. 1, though I have not at present noticed variations in the
invisible corresponding with those in the visible spectrum of the
arc discharge.

On the Cause of the Advantage of Broad Electrodes ; and on
the Heating of the Negative Electrode.

Although the spark appears instantaneous when viewed in a
moving mirror, it must yet occupy a certain time ; so that we have
in fact a brief electric current, to which we may apply Ohm's laws.
The electromotive force is here the difference of tensions of the
coatings of the jar. As to the resistance, the short metallic part
of the circuit may be neglected, and we need only attend to the
place of the discharge. The resistance here may be divided into
that due to the air and that due to the parts of the electrodes
close to the points of discharge. That the latter is by no means
insignificant, may be inferred from the enormous temperature to
which minute portions of the electrodes are raised, as indicated
by the excessively high refrangibility of the rays emitted by the
metals, in the state doubtless of vapour. By the use of flat
electrodes the striking-distance is materially diminished, without
any change in the difference of tension of the coatings of the jar.
Hence the electricity which it contains passes at a higher velocity,
and therefore produces a more powerful effect on the metals.

The injurious effect of the introduction of a small resistance
was very strikingly shown with broad, slightly curved copper

* The outer part of the jar-spark between aluminium electrodes has the same
green colour and prismatic composition, though in this case the green light is
sensibly instantaneous. July, 1862.

232 ON THE LONG SPECTRUM OF ELECTRIC LIGHT.

electrodes, three inches in diameter, by leading wires from a
coating of the jar into a tumbler of water, and from thence to the
corresponding electrode, when the spark became quite insignificant
in comparison to what it had been.

With one sharp and one flat electrode placed near together,
bright sparks passed when the connexion was metallic, and the
invisible spectrum then showed the copper lines, with one or two
air-lines not conspicuous; but when water was interposed the
spark was greatly reduced, and the invisible spectrum showed the
air-lines. In both cases the spark was followed by an arc dis-
charge, as might be seen in a moving mirror ; and in the latter
case the arc discharge was increased in consequence of the
diminution of the spark, which, though necessary to start it, was
formed at its expense ; and as in the arc discharge the jar was
idle, the increase of resistance in a circuit already comprising
the secondary coil was unimportant.

The fact that the blue negative light which appears when the
arc discharge is formed shows air-lines, points to the air as the seat
of the intense action which there takes place ; and the very high
refrangibility of some of the rays emitted, and the copiousness of
those rays, indicate how intense that action is. The heating of the
negative electrode seems to be a secondary effect, not due to the
direct passage of the electricity through the metal (for the section
through which it passes is not by any means small), but to the
heat communicated from the film of air investing it. Small as is
the mass of the film compared with that of the portion of the
electrode adjacent to it, the rate at which heat is communicated
is enormous. Thus with a positive point nearly touching under-
neath a negative electrode of platinum foil containing water, the
foil is kept red-hot under the water, though the mere passage of
electricity through the metal would be quite inadequate to
produce that effect. Corresponding to the heating of the elec-
trode by the air is the cooling of the air by the electrode ; and
such a powerful abstraction of heat can hardly take place without
altering the state of the film of air in relation to its power of
conducting electricity. This would seem to be the reason why the
film of air in contact with the negative electrode behaves so
differently from any arbitrary section of the column along which
the discharge takes place, and from offering greater resistance

ON THE LONG SPECTRUM OF ELECTRIC LIGHT. 233

becomes the seat of a more intense emission of highly refrangible
rays. At the positive electrode, at which, for whatever reason,
the issue of electricity is confined almost to a point, nothing of this
kind takes place ; but, from the contraction of the section through
which the electricity has to pass in the electrode, a minute portion
of the metal of which it is composed is so highly acted on that
matter belonging to the electrode is liable to appear in the arc.

These views lead to curious speculations respecting the negative
light in highly exhausted tubes, and respecting the remarkable
reversion of heating-effect which Mr Gassiot has obtained according
as the discharge is intermittent or continuous*, but I forbear to
speculate further.

* Proceedings of the Royal Society, Vol. xi, p. 329.

On the Change of Form assumed by Wrought Iron and
other metals when heated and then cooled by partial
Immersion in Water. By Lieut.-Col. H. Clark, E.A., F.R.S.
Note appended by Prof. Stokes.

[From the Proceedings of the Royal Society, xni, pp. 471-2. Eeceived

Feb. 9, 1863.]

The cause of the curious phenomenon described by Colonel
Clark in the preceding paper seems to be indicated by some of the
figures, especially those relating to hollow cylinders of wrought
iron, which are very instructive.

Imagine such a cylinder divided into two parts by a horizontal
plane at the water-line, and in this state immersed after heating.
The under part, being in contact with water, would rapidly cool
and contract, while the upper part would cool but slowly. Con-
sequently by the time the under part had pretty well cooled, the
upper part would be left jutting out; but when both parts had
cooled, their diameters would again agree. Now in the actual
experiment this independent motion of the two parts is impossible,
on account of the continuity of the metal ; the under part tends to
pull in the upper, and the upper to pull out the under. In this
contest the cooler metal, being the stronger, prevails, and so the
upper part gets pulled in, a little above the water-line, while still
hot. But it has still to contract on cooling ; and this it will do to
the full extent due to its temperature, except in so far as it may
be prevented by its connexion with the rest. Hence, on the whole,
the effect of this cause is to leave a permanent contraction a little
above the water-line; and it is easy to see that the con1;raction
must be so much nearer to the water-line as the thickness of the
metal is less, the other dimensions of the hollow cylinder and the
nature of the metal being given. When the hollow cylinder is
very short, so as to be reduced to a mere hoop, the same cause
operates ; but there is not room for more than a general in-
clination of the surface, leaving the hoop bevelled.

DEFORMATIONS DUE TO RAPID COOLING. 235

But there is another cause of deformation at work, the opera-
tion of which is well seen in Figs. 2 and 3. Imagine a mass of
metal heated so as to be slightly plastic, and then rapidly cooled
over a large part of its surface. In cooling, the skin at the same
time contracts and becomes stronger, and thereby tends to squeeze
out its contents. This accounts for the bulging of the ends of the
solid cylinders of wrought iron and the rents seen in their cylin-
drical surface. The skin at the bottom is of course as strong as at
the sides in the part below the water-line ; but a surface which
resists extension far more than bending has far less power to resist
pressure of the nature of a fluid pressure when plane than when
convex. The effect of the cause first explained is also manifest in
these cylinders, although it is less marked than in the case of the
hollow cylinders, as might have been expected.

The tendency of the cooled skin of a heated metallic mass to
squeeze out its contents appears to be what gives rise to the
bulging seen near the water-line in the hollow cylinder of brass.
Wrought iron, being highly tenacious even at a comparatively
high temperature, resists with great force the sliding motion of
the particles which must take place in order that the tendency of
the cooled skin to squeeze out its contents may take effect ; but
brass, approaching in its hotter parts more nearly to the state of
a molten mass, exhibits the effect more strongly. It seems pro-
bable that even in the case of brass a very thin hollow cylinder
would exhibit a contraction just above the water-line. Should
there be a metal or alloy which about the temperatures with
which we have to deal was stronger hot than cold, the efi'ect of the
cause first referred to would be to produce an expansion a little
below the water-line.

On the supposed Identity of Biliverdin with Chlorophyll,
with remarks on the constitution of chlorophyll.

[From the Proceedings of the Royal Society, xiii, pp. 144-5, Feb. 25, 1864.]

I HAVE lately been enabled to examine a specimen, prepared
by Professor Harley, of the green substance obtained from the
bile, which has been named biliverdin, and which was supposed by
Berzelius to be identical with chlorophyll. The latter substance
yields with alcohol, ether, chloroform, &c., solutions which are
characterized by a peculiar and highly distinctive system of bands
of absorption, and by a strong fluorescence of a blood-red colour.
In solutions of biliverdin these characters are wholly wanting.
There is, indeed, a vague minimum of transparency in the red;
but it is totally unlike the intensely sharp absorption-band of
chlorophyll, nor are the other bands of chlorophyll seen in bili-
verdin. In fact, no one who is in the habit of using a prism could
suppose for a moment that the two were identical ; for an obser-
vation which can be made in a few seconds, which requires no
apparatus beyond a small prism, to be used with the naked eye,
and which as a matter of course would be made by any chemist
working at the subject, had the use of the prism made its way
into the chemical world, is sufficient to show that chlorophyll and
biliverdin are quite distinct.

I may take this opportunity of mentioning that I have been
for a good while engaged at intervals with an optica-chemical
examination of chlorophyll. I find the chlorophyll of land-plants
to be a mixture of four substances, two green and two yellow, all
possessing highly distinctive optical properties. The green sub-
stances yield solutions exhibiting a strong red fluorescence; the
yellow substances do not. The four substances are soluble in the
same solvents, and three of them are extremely easily decomposed

ox THE COXSTITrXIOX OF CHLOBOPHYLL. 23 (

by acids or even acid salts, such as binoxalate of potash ; but by
pi-oper treatment each may be obtained in a state of veiy approxi-
mate isolation, so tar at least as colom-ed substances are concerned.
The phi/llocyanine of Fremj* is mainly the product of decom-
positioit by acids of one of the green bodies, and is naturally a
substance of a nearly neuti^al tint, shovring however extremely
sh;trp bands of absorption in its neutral sohuions. btit dissolves in
certain acids and acid solutions with a green or blue colour.
Fremys phyUoianthine differe according to the mode of prepara-
tion. When prepared by removing the green bodies by hydrate
of alumina and a little water, it is mainly one of the yellow
bodies : but when prep\red by hydrochloric acid and ether, it is
mainly a mixture of the same yellow body (partly, it may be,
decomposed") with the product of decomposition by acids of the
second green body. As the mode of prejiaration of phi/Uoa:aii-
tJiine is rather hinted at than described, I can only conjecttu-e
what the subst;tnce is ; but I suppose it to be a mixture of the
second yellow substance with the products of decomposition of
the other three bodies. Green sea-weeds {Chi oroij^ermea'') agree
with land-plants, except as to the relative proportion of the sub-
stances present; but in olive-coloured sea-weeds {^elanospennea)
the second green substance is replaced by a thuxi green substance.
;\nd the first yellow substance by a third yellow substance, to the
presence of which the dull colour of those plants is due. The
red colotiring-matter of the red sea-weeds {Ehodospermei.v), which
the plants contain in addition to chlorophyll, is altogether dif-
ferent in its nature from chlorophyll, as is already known, and
would appear to be an albuminous substance. I hope, before
lono;. to present to the Eoyal Society the details of these
reseaivhes.

* Compttf Iiendt.>. torn, l, p. 40-5.

On the Discrimination of Organic Bodies by their
Optical Properties.

[Discourse delivered at the Royal Institution of Great Britain, Mar. 4, 1864,
Roy. Inst. Proc. IV, pp. 223—231 : also Phil. Mag., xxvii, 1864, pp. 388—
95, Ann. der Phys., cxxvi, 1865, pp. 619 — 23, Journ. de Pharm., I, 1865,
pp. 292—8.]

The chemist who deals with the chemistry of inorganic sub-
stances has ordinarily under his hands bodies endowed with very
definite reactions, and possessing great stability, so as to permit
of the employment of energetic reagents. Accordingly he may
afford to dispense with the aids supplied by the optical properties
of bodies, though even to him they might be of material assistance.
The properties alluded to are such as can be applied to the
scrutiny of organic substances ; and therefore the examination of
the bright lines in flames and incandescent vapours is not con-
sidered. This application of optical observation, though not new
in principle (for it was clearly enunciated by Mr Fox Talbot more
than thirty years ago), was hardly followed out in relation to
chemistry, and remained almost unknown to chemists until the
publication of the researches of Professors Bunsen and Kirchhoff,
in consequence of which it has now become universal.

But while the chemist who attends to inorganic compounds
may confine himself without much loss to the generally recognized
modes of research, it is to his cost that the organic chemist,
especially one who occupies himself with proximate analysis,
neglects the immense assistance which in many cases would be
afforded him by optical examination of the substances under his
hands. It is true that the method is of limited application, for
a great number of substances possess no marked optical characters;
but when such substances do present themselves, their optical
characters afford facilities for their chemical study of which
chemists generally have at present little conception.

OPTICAL DISCRIMINATION OF ORGANIC BODIES. 239

ISvo distinct objects may be had in view in seeking for such
information as optics can supply relative to the characters of
a chemical substance. Among the vast number of substances
which chemists have now succeeded in isolating or preparing, and
which in many cases have been but little studied, it often becomes
a question whether two substances, obtained in different ways, are
or are not identical. In such cases an optical comparison of the
bodies will either add to the evidence of their identity, the force
of the additional evidence being greater or less according as their
optical characters are more or less marked, or will establish a
difference between substances which might otherwise erroneously
have been supposed to be identical.

The second object is that of enabling us to follow a particular
substance through mixtures containing it, and thereby to deter-
mine its principal reactions before it has been isolated, or even
when there is small hope of being able to isolate it ; and to
demonstrate the existence of a common proximate element in
mixtures obtained from two different soui-ces. Under this head
should be classed the detection of mixtures in what were supposed
to be solutions of single substances*

Setting aside the labour of quantitative determinations carried
out by well-recognized methods, the second object is that the
attainment of which is by far the more difficult. It involves the
methods of examination required for the first object, and more
besides; and it is that which is chiefly kept in view in the present
discourse.

The optical properties of bodies, properly speaking, include
eveiy relation of the bodies to light ; but it is by no means every
such relation that is available for the object in view. Refractive
power, for instance, though constituting, Like specific gra^it}-, Szc,
one of the characters of any particular pure substance, is useless
for the purpose of following a substance in a mixture containing
it. The same may be said of dispersive power. The properties
Avhich are of most use for our object are, first absorption, and
secondly fluorescence.

* The detection of mixtures by the microscopic examination of intermingled
crystals properly belongs to the first head, the question which the observer
proposes to himself being, in fact, whether the pure substances forming the
individual crystals are or are not identical.

240 ON THE DISCEIMINATION OF ORGANIC BODIES

Colour has long been employed as a distinctive character of
bodies ; as, for example, we say that the salts of oxide of copper
are mostly blue. The colour, however, of a body gives but very
imperfect information respecting that property on which the colour
depends ; for the same tint may be made up in an infinite number
of ways from the constituents of white light. In order to observe
what it is that the body does to each constituent, we must examine
it in a pure spectrum. [The formation of a pure spectrum was
then explained, and such a spectrum was formed on a screen by
aid of the electric light. On holding a cell containing a salt of
copper in front of the screen, and moving it from the red to the
violet, it was shown to cast a shadow in the red as if the fluid had
been ink, while in the blue rays it might have been supposed to
have been water. Ghromate of potash similarly treated gave the
reverse effect, being transparent in the red, and opaque in the
blue. Of course the transition from transparency to opacity was
not abrupt ; and for intermediate colours the fluids caused a partial
darkening. Indeed, to speak with mathematical rigour, the
darkening is not absolute even when it appears the greatest ; but
the light let through is so feeble that it eludes our senses. In this
way the behaviour of the substance may be examined with refer-
ence to the various kinds of light one after another ; but in order
to see at one glance its behaviour with respect to all kinds, it is
merely requisite to hold the body so as to intercept the whole beam
which forms the spectrum — to place it, for instance, immediately in
front of the slit.]

To judge from the two examples just given, it might be sup-
posed that the observation of the colour 'would give almost as
much information as analysis by the prism. [To show how far this
is from being the case, two fluids very similar in colour, port-wine
and a solution of blood, were next examined. The former merely
caused " a general absorption of the more refrangible rays ; the
latter exhibited two well-marked dark bands in the yellow and
green.] These bands, first noticed by Hoppe, are eminently
characteristic of blood, and afford a good example of the facilities
which optical examination affords for following a substance which
possesses distinctive characters of this nature. On adding to
a solution of blood a particular salt of copper (any ordinary copper
salt, with the addition of a tartrate to prevent precipitation, and

BY THEIR OPTICAL PROPERTIES. 241

then carbonate of soda), a fluid is obtained utterly unlike blood in
colour, but showing the characteristic bands of blood, while at the
same time a good deal of the red is absorbed, as it would be by
the copper salt alone. On adding, on the other hand, acetic acid
to a solution of blood, the colour is merely changed to a browner
red, without any precipitate being produced. Nevertheless, in the
spectrum of this fluid the bands of blood have wholly vanished,
while another set of bands less intense, but still very character-
istic, make their appearance. This alone, however, does not decide
whether the colouring matter is decomposed or not by the acid ;
for as blood is an alkaline fluid, the change might be supposed to
be merely analogous to the reddening of litmus. To decide the
question, we must examine the spectrum when the fluid is again
rendered alkaline, suppose by ammonia, which does not affect the
absorption bands of blood. The direct addition of ammonia to
the acid mixture causes a dense precipitate, which contains the
colouring matter, which may, however, be separated by the use
merely of acetic acid and ether, of which the former has been
already used, and the latter does not affect the colouring matter
of blood. This solution gives the same characteristic spectrum as
blood to which acetic acid has been added ; but now there is no
difficulty in obtaining the colouring matter in an ammoniacal
solution. In the spectrum of this solution, the sharp absorption
bands of blood do not appear, but instead thereof there is a single
band a little nearer to the red, and comparatively vague [this
was shown on a screen]. This difference of spectra decides
the question, and proves that hsematine (the colouring matter
prepared b}- acid, &c.) is, as Hoppe stated, a product of decom-
position.

The spectrum of blood may be turned to account still further
in relation to the chemical nature of that substance. The colouring
matter contains, as is well known, a large quantity of iron ; and it
might be supposed that the colour was due to some salt of iron,
more especially as some salts of peroxide of iron, sulphocyanide for
instance, have a blood-red colour. But there is found a strong
general resemblance between salts of the same metallic oxide as
regards the character of their absorption. Thus the salts of sesqui-
oxide of uranium show a remarkable system of bands of absorption
in the more refrangible part of the- spectrum. The number and
s. IV. 16

242 ON THE DISCRIMINATION OF ORGANIC BODIES

position of the bands differ a little from one salt to another ; but
there is the strongest family likeness between the different salts.
Salts of sesquioxide of iron in a similar manner have a family like-
ness in the vagueness of the absorption, which creeps on from one
part of the spectrum to another without presenting any rapid transi-
tions from comparative transparency to opacity and the converse.
[The spectrum of sulphocyanide of peroxide of iron was shown, for
the sake of contrasting with blood.] Hence the appearance of
such a peculiar system of bands of absorption in blood would
negative the supposition that its colour is due to a salt of iron as.
such, even had we no other means of deciding. The assemblage
of the facts with which we are acquainted seems to show that the
colouring matter is some complex compound of the five elements,
oxygen, hydrogen, carbon, nitrogen, and iron, which, under the
action of acids and otherwise, splits into hsematine and an albu-
minous substance.

This example was dwelt on, not for its own sake, but because
general methods are most readily apprehended in their appli-
cation to particular examples. To show one example of the
discrimination which may be effected by the prism, the spectra
were exhibited of the two kinds of red glass which (not to-
mention certain inferior kinds) are in common use, and which
are coloured, one by gold, and the other by suboxide of copper.
Both kinds exhibit a single band of absorption near the yellow or
green; but the band of the gold glass is situated very sensibly
nearer to the blue end of the spectrum than that of the copper
glass.

In the experiments actually shown, a battery of fifty cells and
complex apparatus were employed, involving much trouble and
expense. But this was only required for projecting the spectra on
a screen, so as to be visible to a whole audience. To see them,
nothing more is required than to place the fluid to be examined
(contained, suppose, in a test-tube) behind a slit, and to view it
through a small prism applied to the naked eye, different strengths
of solution being tried in succession. In this way the bands may
be seen by anyone in far greater perfection than when, for the
purpose of a lecture, they are thrown on a screen.

In order to be able to examine the peculiarities which a sub-
stance may possess in the mode in which it absorbs light, it is not.

BY THEIR OPTICAL PROPERTIES. 243

essential that the substance should be in solution, and viewed by
transmission. Thus, for example, when a pure spectrum is thrown
on a sheet of paper painted with blood, the same bands are seen in
the yellow and green region as when the light is transmitted
through a solution of blood and the spectrum thrown on a white
screen. This indicates that the colour of such a paper is in fact
due to absorption, although the paper is viewed by reflected light.
Indeed, by far the greater number of coloured objects which are
presented to us, such as green leaves, flowers, dyed cloths, though
ordinarily seen by reflexion, owe their colour to absorption. The
light by which they are seen is, it is true, reflected, but it is
not in reflexion that the preferential selection of certain kinds of
rays is made which causes the objects to appear coloured. Take,
for example, red cloth. A small portion of the incident light is
reflected at the outer surfaces of the fibres, and this portion, if it
could be observed alone, would be found to be colourless. The
greater part of the light penetrates into the fibres, when it im-
mediately begins to suffer absorption on the part of the colouring
matter. On arriving at the second surface of the fibre, a portion
is reflected and a portion passes on, to be afterwards reflected from,
or absorbed by, fibres lying more deeply. At each reflexion the
various kinds of light are reflected in as nearly as possible the
same proportion; but in passing across the fibres, in going and
returning, they suffer very unequal absorption on the part of the
colouriug matter, so that in the aggregate of the light perceived
the different components of white light are present in proportions
widely difi'erent from those they bear to each other in white light
itself, and the result is a vivid colouring.

There are, however, cases in which the different components of
white light are reflected with diiferent degrees of intensity, and
the light becomes coloured by regular reflexion. Gold and copper
may be referred to as examples. In ordinary language we speak
of a soldier's coat as red, and gold as yellow. But these colours
belong to the substances in two totally different senses. In the
former case the colouring is due to absorption, in the latter case to
reflexion. In the same sense, physically speaking, in which a
soldier's coat is red, gold is not yellow but blue or green. Such is,
in fact, the colour of gold by transmission, and therefore as the
result of absorption, as is seen in the case of gold-leaf, which

16—2

244 ON THE DISCRIMINATION OF ORGANIC BODIES

transmits a bluish-green light, or of a weak solution of chloride of
gold after the addition of protosulphate of iron, when the pre-
cipitated metallic gold remains in suspension in a finely-divided
state, and causes the mixture to have a blue appearance when
seen by transmitted light. In this case we see that while the
substance copiously reflects and intensely absorbs rays of all kinds,
it more copiously reflects the less refrangible rays, with, respect to
which it is more intensely opaque.

All metals are, however, highly opaque with regard to ray^ of
all colours. But certain non-metallic substances present them-
selves which are at the same time intensely opaque with regard to
one part of the spectrum, and only moderately opaque or even
pretty transparent with regard to another part. Carthamine,
murexide, and platinocyanide of magnesium may be mentioned as
examples. Such substances reflect copiously, like a metal, those
rays with respect to which they are intensely opaque, but more'
feebly, like a vitreous substance, those rays for which they are
tolerably transparent. Hence, when white light is incident upon
them the regularly-reflected light is coloured, often vividly, those
colours preponderating which the substance is capable of absorbing
with intense avidity. But perhaps the most remarkable example
known of the connexion between intense absorption and copious
reflexion occurs in the case of crystals of permanganate of potash.
These crystals have a metallic appearance, and reflect a greenish
light. They are too dark to allow the transmitted light to be
examined; and even when they are pulverized, the fine purple
powder they yield is too dark for convenient analysis of the trans-
mitted light. But the splendid purple solution which they yield
may be diluted at pleasure, and the analysis of the light trans-
mitted by it presents no difficulty. The solution absorbs prin-
cipally the green part of the spectrum ; and when it is not too
strong, or used in too great thickness, five bands of absoi-ption,
indicating minima of transparency, make their appearance [these
were shown on a screen]. Now, when the green light reflected
from the crystals is analyzed by a prism, there are observed bright
bands, indicating maxima of reflecting power, corresponding in
position to the dark bands in the light transmitted by the solu-
tion. The fifth bright band, indeed, can hardly, if at all,. be made
out; but the corresponding dark band is both less^ strong than the

BY THEIR OPTICAL PROPERTIES. 245

others and occurs in a fainter part of the spectrum. When the
light is reflected at a suitable angle, and is analyzed both by
a Nicol's prism, placed with its principal section in the plane of
incidence, and by an ordinary prism, the whole spectrum is reduced
to the bands just mentioned. The Nicol's prism would under
these circumstances extinguish the light reflected from a vitreous
substance, and transmit a large part of the light reflected from
a metal. Hence we see that as the refrangibility of the light
gradually increases, the substance changes repeatedly, as regards
the character of its reflecting power, from vitreous to metallic and
back again, as the solution (and therefore it may be presumed the
substance itself) changes from moderately to intensely opaque, and
conversely.

These considerations leave little doubt as to the chemical state
of the copper present in a certain glass which was exhibited. This
glass was coloured only in a very thin stratum on one face.
By transmission it cut off a great deal of light, and was bluish.
' By reflexion, especially when the colourless face was next the eye,
it showed a reddish light visible in all directions, and having the
appearance of comiag from a fine precipitate, though it was not
resolved by the microscope, at least with the power tried. It
evidently came from a failure in an attempt to make one of the
ordinary red glasses coloured by suboxide of copper, and the only
question was as to the state in which the copper was present. It
could not be oxide, for the quantity was too small to account for
the blueness, and in fact the glass became sensibly colourless in
the outer flame of a blowpipe. Analysis of the transmitted light
by the prism showed a small band of absorption in the place of the
band seen in those copper-red glasses which are not too deep ; and
therefore a small portion of copper was present in the state of
suboxide, i.e. a silicate of that base. The rest was doubtless
present as metallic copper, arising from over-reduction in the
manufacture ; and accordingly the blue colour, which would have
been purer if the suboxide had been away, indicates the true
colour of copper by transmitted light, quite in conformity with
what we have seen in the case of gold. Hence, in both metals
alike, the absorbing and the reflecting powers are, on the whole,
greater for the less than for the more refrangible colours, the law
of variation with refrangibility being of course somewhat different
in the two cases.

246 ON THE DISCRIMINATION OF ORGANIC BODIES

Time would not permit of more than a very brief reference to
the second property to which the speaker had referred as useful in
tracing substances in impure solutions — that of fluorescence. The
phenomenon of fluorescence consists in this, that certain substances,
when placed in rays of one refrangibility, emit during the time of
exposure compound light of lower refrangibility. When a pure
fluorescent substance (as distinguished from a mixture) is ex-
amined in a pure spectrum, it is found that on passing from the
extreme red to the violet and beyond, the fluorescence commences
at a certain point of the spectrum, varying from one substance to
another, and continues thence onwards, more or less strongly in
one part or another according to the particular substance. The
colour of the fluorescent light is found to be nearly constant
throughout the spectrum. Hence, when in a solution presented
to us, and examined in a pure spectrum, we notice the fluorescence
taking, as it were, a fresh start, with a different colour, we may be
pretty sure that we have to deal with a mixture of two fluorescent
substances.

It might be inferred db priori, that fluorescence at any particular
part of the spectrum would necessarily be accompanied by ab-
sorption, since otherwise there would be a creation of vis viva ; and
experience shows that rapid absorption (such as corresponds to
a well-marked minimum of transparency indicated by a deter-
minate band of absorption in the transmitted light) is accompanied
by copious fluorescence. But experience has hitherto also shown,
what could not have been predicted, and may not be universally
true*, that conversely, absorption is accompanied, in the case of
a fluorescent substance, by fluorescence.

* Fluorescent substances, like others, doubtless absorb the invisible heat-rays
lying beyond the extreme red, in a manner varying from one substance to another.
Hence, if we include such rays in the incident spectrum, we have an example of
absorption not accompanied by fluorescence. But the invisible heat-rays differ
from those of the visible spectrum (as there is every reason to believe) only in the
way that the visible rays of one part of the spectrum differ from those of another,
that is, by wave-length, and consequently by refrangibility, which depends on wave-
length. Hence it is not improbable that substances may be discovered which
absorb the visible rays in some parts of the spectrum less refrangible than that at
which the fluorescence commences ; and mixtures possessing this property may be
made at pleasure. Nevertheless the speaker has not yet met with a pure fluorescent
substance which exhibits this phenomenon. [See ante, Vol. iii, p. 411, note added
in 1901.]

BY THEIR OPTICAL PEOPEETIES. 247

From what precedes it follows that the colour of the fluorescent
light of a solution, even when the incident light is white, or merely-
sifted by absorption, may be a useful character. To illustrate this,
the electric light, after transmission through a deep-blue glass,
was thrown on solutions in weak ammonia of two crystallized
substances, sesculine and fraxine, obtained from the bark of the
horse-chestnut; the latter of the two occurs also in the bark of
the ash, in which, indeed, it was first discovered. Both solutions
exhibited a lively fluorescence ; but the colour was different, being
blue in the case of sesculine, and bluish-green in the case of fraxine.
A purified solution obtained from the bark exhibits a fluorescence
of an intermediate colour, which would suffice to show that sescu-
line would not alone account for the fluorescence of the solution of
the bark.

When a substance possesses well-marked optical properties, it
is in general nearly as easy to follow it in a mixture as in a pure
solution. But if the problem which the observer proposes to
himself be. Given a solution of unknown substances which presents
well-marked characters with reference to different parts of the
spectrum, to determine what portion of these characters belongs
to one substance, and what portion to another, it presents much
greater difficulties. It was with reference to this subject that the
second of the objects mentioned at the beginning of the discourse
had been spoken of as that the attainment of which was by far
the more difficult. The problem can, in general, be solved only
by combining processes of chemical separation, especially frac-
tional separation, with optical observation. When a solution has
thus been sufficiently tested, those characters which are found
always to accompany one another, in, as nearly as can be judged,
a constant proportion, may, with the highest probability, be re-
garded as belonging to one and the same substance. But while
a combination of chemistry and optics is in general required,
important information may sometimes be obtained from optics
alone. This is especially the case when one at least of the sub-
stances present is at the same time fluorescent and peculiar in its
mode of absorption.

To illustrate this the case of chlorophyll was referred to. An
eminent French chemist, M. Fremy, proposed to himself to ex-
amine whether the green colour were due to a single substance, or

248 OPTICAL DISCRIMINATION OF ORGANIC BODIES.

to a mixture of a yellow and a blue substance. By the use of
merely neutral bodies, he succeeded in separating chlorophyll into
a yellow substance, and another which was green, but inclining
a little to blue ; but he could not in this way get further in the
direction of blue. He conceived, however, that he had attained
his object by dissolving chlorophyll in a mechanical mixture of
ether and hydrochloric acid, the acid on separation showing a fine
blue colour, while the ether was yellow. Now solutions of chloro-
phyll in neutral solvents, such as alcohol, ether, &c., show a lively
fluorescence of a blood-red colour; and when the solution is
examined in a pure spectrum, the red fluorescence, very copious
in parts of the red, comparatively feeble in most of the green,
is found to be very lively again in the blue and violet. Now-
a substance of a pure yellow colour, and exercising its absorption
therefore, as such substances do, on the more refrangible rays,
would not show a pure red fluorescence. Either it would be
non-fluorescent, or the fluorescence of its solution would contain
(as experience shows) rays of refrangibilities reaching, or nearly
so, to the part of the spectrum at which the -fluorescence, and
therefore the absorption, commences ; and therefore the fluorescent
light could not be pure red, as that of chlorophyll is found to
be even in the blue and violet. The yellow substance separated by
M. Fr^my, by the aid of neutral reagents, is, in fact, non-fluor-
escent. Hence the powerful red fluorescence in the blue and violet
can only be attributed to the substance exercising the well-known
powerful absorption in the red, which substance must therefore
powerfully absorb the blue and violet. We can affirm, therefore,
a, priori, that if this substance were isolated it would not be blue,
but only a somewhat bluer green. The blue solution obtained by
M. Fr^my owes in fact its colour to a product of decomposition,
which when dissolved in neutral solvents is not blue at all, but of
a nearly neutral tint, showing, however, in its spectrum extremely
sharp bands of absorption.

On the Application of the Optical Properties of Bodies
TO THE Detection and Discrimination of Organic
Substances.

[A Discourse delivered before the Fellows of the Chemical Society,
June 2, 1864 : Chem. Soc. Journal, II, 1864, pp. 303—18.]

The optical properties of bodies, properly speaking, include
every phenomenon in which ponderable matter is related to light
by virtue of its molecular constitution, and not merely of its
external form. Many of these, however, are of no use in helping
us to follow a particular substance through mixtures or solutions
containing it, though they may be useful as additional characters
of substances which have been obtained in a state of isolation.
Take for example refractive power. The refractive power of a
pure substance, like its specific gravity, is one of the characters,
the assemblage of which serves to distinguish it from other bodies ;
but as all bodies in nature refract light, solvents and body dis-
solved alike, though not to the same extent, the observation of
the refractive power of a mixture would help us in disentangling
its constituents. The same may be said of dispersive power.
Circular polarization again belongs to the same class of properties ;
though from the fact that all inorganic, and a number of organic
solvents are destitute of it, it is sometimes employed to trace a
substance, but of course only in those cases in which we know
or may presume that there is but one substance present which'
possesses the property in any marked degree.

If we exclude the emission of definite rays by flames and
electric discharges, which is made known by spectral analysis, and
which, though most valuable for the detection of elementary bodies,

250 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

is of little or no avail for even the simplest and most stable com-
pounds, and cannot of course be applied to organic analysis, there
remain three phenomena in which bodies are related to light in
a manner varying from one ray to another, not in a gradual,
regular way like the refractive index, but in a way depending on
something in the molecular constitution of the particular body in
question, and changing sometimes in an apparently capricious way
from one ray to another. These are (1) absorption, (2) fluorescence,
(3) coloured reflexion.

I. The colour, of substances has long been used as an important
character ; thus for example it is a character of the salts of oxide
of copper, to yield in general blue solutions. In all cases in which
colour is presented to us, we must, in considering the physical
cause of the phenomenon, revert, in the first instance, to Newton's
discovery of the compound nature of white light, and inquire how
it comes to pass, that the homogeneous constituents of white light
are presented to us in different proportions from those in which
they occur in white light itself Now, if a coloured solution be
examined in homogeneous light of any kind, i.e., light of definite
refrangibility, it is found that the transmitted light, retaining all
the properties of the incident light*, becomes feebler and feebler,
as the thickness of the stratum through which it passes is increased.
A portion of the incident light continually disappears as the rays
traverse the solution, and is said to be absorbed. The incident
light being by hjrpothesis homogeneous, and the quantity of light
which is absorbed in passing across a given stratum being, as
experiment shows, proportional to the quantity which falls upon it,
it readily follows, that the intensity of the light which escapes
absorption decreases in geometrical progression, as the length of
path of the rays within the solution increases in arithmetical. The
rate of absorption, changes in general from one set of homogeneous
rays to another. For one part of the spectrum, the absorption
produced by the solution may be very powerful, for another part

* Sir David Brewster indeed conceived that he had succeeded in analysing by
absorption light which was homogeneous as regards refrangibility. But though the
direct judgment of the senses, tiihca the experiment is made in Sir David Brewster's
manner, is in accordance with his statenient, as might be expected from his well
known accuracy, the inference that a real analysis has taken place may be con-
sidered to have been disproved by subsequent researches.

AND DISCRIMINATION OF ORGANIC SUBSTANCES. 251

very weak, while for another part again the solution m,ay be
sensibly transparent like water.

To determine the absolute rate of absorption for homogeneous
light of a given kind would be useless, unless the body to be
examined were isolated; for not only would foreign substances
present contribute to the observed absorption, unless, indeed, they
happened to be perfectly transparent with respect to the part of the
spectrum selected, but even in an otherwise colourless solution, it
would be necessary to estimate the quantity of substance present,
since the rate of absorption would depend on the strength of the
solution. Hence, absolute absorbing power is of no more avail for
our purpose than refractive power.

But the relative fibsorption of different parts of the spectrum
is what may be observed at a glance (of course, qualitatively only,
not quantitatively); it is in general independent of the degree of
dilution of the solution, the solvents being supposed colourless, and
when the substance to be observed has in this respect well marked
characters, may be observed to a very great extent independently
of coloured impurities, even though they may be sufficient to change
the colour very greatly.

For this purpose, nothing more is required than to form in any
manner a pure spectrum, and interpose the coloured solution any-
where in the path of the rays forming it. The simplest mode of
obtaining a pure spectrum, when we have no occasion to place
objects in it, is to view a slit held against a luminous background,
through a prism applied to the eye. Hence the following simple
arrangement may be adopted.

A small prism is to be chosen, which may be made of rather
dense flint glass ground to an angle of about 60°, and need not be
larger than is sufficient just to cover the eye comfortably. The
top and bottom should be flat, for convenience of holding the prism
between the thumb and fore-finger, and of laying it down on an
end, so as not to scratch or dirty the faces. This forms the only
apparatus required beyond what the observer may readily make
for himself The ?lit may conveniently be made by taking a board
6 inches square, or a little longer in a horizontal direction, making
an oblong aperture in it in a vertical direction, and adapting to
the aperture two pieces of thin metal to form clean cheeks to the

252 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

slit. One of the metal pieces, should be moveable, to allow of
altering the breadth of the slit. About the fiftieth of an inch is
a suitable breadth for ordinary purposes. The board and metal
pieces should be well blackened.

On holding the board at arm's length against the sky or a
luminous flame, the slit being, we will suppose, in a vertical
direction, and viewing the line of light thus formed through the
prism held close to the eye, with its edge vertical, a pure spectrum
is obtained at a proper azimuth of the prism. Turning the prism
round its axis alters the focus, and the proper focus is got by
trial. The whole of the spectrum is not, indeed, in perfect focus
at once, so that in scrutinizing one part after another it is requisite
to turn the prism a little. When daylight is used, the spectrum is
known to be pure by its showing the principal fixed lines ; in other
cases the focus is got by the condition of seeing distinctly the
other objects, whatever they may be, which are presented in the
spectrum. The use of a prism in this way is at the first moment
a little puzzling, but soon becomes perfectly easy. To observe the
absorption-spectrum of a liquid, an elastic band is put round the
board near the top, and a test-tube containing the liquid is slipped
under the band, which holds it in its place behind the slit. The
spectrum is then observed just as before, the test-tube being turned
from the eye.

To observe the whole progress of the absorption, different
degrees of strength must be used in succession, beginning with a
strength which does not render any part of the spectrum absolutely
black, unless it be one or more very narrow bands, as otherwise
the most distinctive features of the absorption might be missed.
If the solution be contained in a wedge-shaped vessel instead of
a test-tube, the progress of the absorption may be watched in
a continuous manner by sliding the vessel before the eye ; but
for actual work this is an unnecessary luxury. Some observers
prefer using a wedge-shaped vessel in combination with the slit,
the slit being perpendicular to the edge of the wedge. In this
case each element of the slit forms an elementary spectrum corre-
sponding to a thickness of the solution, which increases in a
continuous manner from the edge of the wedge, where it vanishes.

In many cases nothing is observed, beyond a general absorption
of one or other end of the spectrum, or of its middle part, and

AXD DISCKIMIXATIOX OF ORGANIC SCBSTAJNCES. 253

the prism gives little information beyond what is got by the
eye. by observing the succession of colours produced by ditierent
thicknesses of the liquid. And here it may be remarked in
passing, with retei-ence to the description of pure substances, that
in specityiiig only one colour, that corresponding to a considerable
thickness, as is commonly done by chemists, the peculiar features
of the absorption are left almost wholly undescribed. Thus of two
solutions, one might be pink when dilute, passing on to red with
increase of strength or thickness, another yellow, passing through
orange to red. These woidd commonly be described as red. yet
the series of tints indicates an utter difference in the mode of
absoi-ption, the middle of the spectrum in the one case, and the most
refrangible end in the other, being the most powerfully attacked.

But in some cases, especially with substances of intense colorific
power, the mode of absorption is eminently characteristic. Two
or more dark bands are seen in the spectrum, indicating maxima
of absorption; and the positions of these bands, their relative
intensity, and their other featmvs. tbrm altogether a series of
characters the distinctive nature of which is such as those who
have neglected the use of the prism have little conception of
Thev render it perfectly easy in many eases to foUow a particular
substance among a host of impurities. For each coloured sub-
stance prodtices its own absorption, independently of the othere
(supposing the sitbstances do not chemically react on each other).
so that, imless the pirt of the spectrum in which the distinctive
bands, or most of them, occur, is wholly absorbed by the impurities,
the presence of the substance can still be recogmsed. Such a
complete obliteration is the less likely to occiu-, for this reason, that
when the characters of the solution are so strongly marked, it
almost always happens that a comparatively small quantity of the
substance suffices to produce the effect, and the solution must
consequentlv be so much diluted that the effect of the impurities
comparatively disappears.

Xor is this alL \Mien a substance exhibits marked characters
of one kind ia one solvent, it often happens that it shows different
and no less marked characters in a solvent of a different nature.
Xot only does this fiu-nish additional charactei^ by which the
substance can be distinguished firom others, but it is valuable for
following the stibstance when involved in impurities ; tor the

254 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

nature of the impurities may be such as to mask the substance in
one solvent and not in another. This is especially the case where
one solvent is alkaline and the other acid; but differences are
sometimes observed even with two neutral solvents.

To illustrate these principles, we may refer to the colouring
matters of madder. Alizarin and purpurin both yield highly dis-
tinctive spectra, the former, however, only in the case of solutions
containing caustic alkali*, whereas most solutions of the latter are
highly distinctive. Madder itself contains, either directly or as
the result of decomposition, a number of substances which in
alkaline solution absorb that part of the spectrum in which the
distinctive bands of purpurin occur. Hence, in a mixture obtained
from madder, and containing, we will suppose, purpurin in com-
paratively small quantity, the presence of purpurin would be
masked by the other substances in an alkaline solution. But in
ether or acidulated alcohol the other substances yield spectra
showing nothing particular, and interfering comparatively little
with the distinctive bands of purpurin ; while in an alum-liquor
solution made by boiling, not only are the purpurin-bands, which
in this solvent occur at a lower refrangibility than with ether,
more effectually separated from the absorption produced by the
associated substances, but those substances themselves are also in
good measure excluded.

For an example of the necessity of attending to the nature of
the solvent, even in the case of different neutral solvents, we may
refer to a yellow substance which is one of the constituents of the
green colouring matter of leaves. The alcoholic solution of this
substance exhibits two characteristic bands of absorption, the
first of which is situated immediately adjacent to the Im e F on
the more refrangible side. The solution in bisulphide o^,carbon
exhibits two similar bands, but much less refrangible, the ISie^ "
now nearly bisecting the bright interval between the first and
second dark bands. The substance is very easily decomposed bjr
acids, and even by acid salts, yielding a product of decomposition
which, in alcoholic solution, exhibits two bands of absorption like
the parent substance, but a good deal more refrangible. There is

* On boiling with an alkaline carbonate the same spectrum is obtained, but not
perfectly developed. An alcoholic solution with a little caustic potash introduced
gives it in perfection.

AND DISCRIMINATION OF ORGANIC SUBSTANCES. 255

the same change of position as in the former case, in passing from
alcohol to bisulphide of carbon, so that the solution of the product
of decomposition in bisulphide of carbon agrees almost exactly,
in colour and spectrum, with that of the parent substance in
alcohol.

Not only is an examination of the absorption-spectrum of a
substance useful for enabling us to follow the substance through
mixed solutions, but it sometimes reveals relationships in cases in
which they might not be suspected, if the origin of the substances
were unknown. Thus, the purpurein of Dr Stenhouse dissolves
in ether or acidulated alcohol with a red colour, while that of the
same solutions of purpurin is yellow. But the prism reveals in
both cases alike the existence of three bands of absorption, of
similar breadth, while the purpurein bands are situated nearer the
red than those of purpurin, by about one interval. This example
shows how deeply seated in the molecular constitution of a body
may in some cases be the cause which produces the bands.

Hitherto it has been supposed that the peculiarities of absorp-
tion of a substance were known, and applied to the detection of
that substance in a mixture. But the question may arise : —
Given a mixture of an unknown number of unknown substances,
which as a whole presents peculiarities of absorption, to determine
whether the whole of these peculiarities are due to the same
substance, and if not, what portion are due to one substance, and
what portion to another. Little can be done towards the solution
of this problem by the mere observation of absorption ; we can
only say, that some modes of grouping of bands of absorption
are common in solutions of pure substances, while others are
uncommon, and give rise to the suspicion of a mixture. The
phenomena of fluorescence give in some cases material assistance ;
but in general it is only by combining spectral analysis with pro-
cesses of chemical separation, especially fractional separation, that
a satisfactory conclusion can be arrived at. When a mixture is
thus tested in various ways, a conviction, at last approaching
certainty, is gradually arrived at, that those bands of absorption
which are always found accompanying one another belong to one
and the same substance.

For convenience and rapidity of manipulation, especially in
the examination of very minute 'quantities, there is no method

256 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

of separation equal to that of partition between solvents which
separate after agitation. Ether combined with water, either pure
or rendered acid or alkaline, is the most generally useful, and the
separation, if not otherwise fractional, may be rendered so by
introducing the acid or alkali by minute quantities at a time ; but
other solvents are useful in particular cases. Bisulphide of carbon
in conjunction with alcohol enabled the lecturer to disentangle the
coloured substances which are mixed together in the green colour-
ing matter of leaves. Solutions of various metallic oxides which
are naturally precipitable by an alkali or alkaline carbonate, but
are retained in solution by means of a tartrate, are very useful
in the examination of the true colouring matters,' not merely for
producing changes of colour and spectrum without precipitation,
but even, in conjunction with ether, for effecting chemical separa-
tions ; and fractional separation may be effected by making the
solution deviate very slightly from perfect neutrality. By com-
bining with ether such a solution of alumina, it was found possible
to separate and detect the purpurin, alizarin, and rubiacin present
in a portion of powder not exceeding in bulk a fraction of the head
of a pin.

II. The phenomenon of fluorescence consists in this, that
certain substances in solution, or even in the solid state, when
exposed to rays of one kind, emit for the time being rays of
another kind, as if they were self-luminous. The following law
appears to be universal : — The emitted rays are of lower refrangi-
hility than the exciting rays. The light emitted is heterogeneous,
even though the active rays be homogeneous, and it is remarkable
that it shows no traces of its parentage.

When a solution of a pure fluorescent substance is examined in
a pure spectrum formed in the usual way by means of sunlight, it
is found that, in passing in the direction from the red to the violet,
the fluorescence begins with more or less abruptness at a certain
part of the spectrum varying from one substance to another, and
continues from thence onwards, though often with considerable
fluctuations of intensity. The fluctuations are found to corre-
spond with fluctuations in transparency, so that, in a region of
the spectrum where a band of absorption occurs, the fluorescence
is unusually strong, while the rays exciting fluorescence are more

AND DISCRIMINATION OF OEGANIC SUBSTANCES. 257

quickly spent, and accordingly the fluorescence does not extend so
far into the solution. Experience shows that with solutions of
a pure substance, the tint of the fluorescent light is almost perfectly
constant throughout the spectrum.

In consequence of this law, the tint of the fluorescent light
emitted by a solution becomes a character of importance. This
tint, it must be remembered, is that of the light as emitted, not as
subsequently modified by absorption on the part of the solution, in
case the solution be sensibly coloured, and some precautions are
required in order to observe it correctly*- The fluorescence ob-
served in solutions from the barks of the horse-chestnut, ash, &c.,
was formerly attributed indiscriminately to the presence of sesculin,
whereas a purified solution from the bark of the horse-chestnut
exhibits a fluorescence very sensibly different from that of sesculin,
which observation alone would suffice to show that the bark miast
contain some other fluorescent substance besides sesculin.

As in the case of absorption, the nature of the solvent must
be attended to. The colour of the fluorescent light is liable to
change, not merely in passing from an alkaline to a neutral or
acid solution, but even occasionally in passing from one neutral
solvent to another. The lecturer has received from Dr Miiller
a specimen of a substance which in water exhibits a green, but
in ether a blue fluorescence.

The composition of fluorescent light, as revealed by the prism,
occasionally presents peculiarities, but in such cases they are found
to be connected with peculiarities in the mode of absorption, so
that the two are not to be regarded as independent characters of
a substance ; and as the peculiarities in the absorption are, as
a general rule, the more easily observed, it is only rarely that the
analysis of the fluorescent light is of much use.

The distribution of fluorescence in the spectrum often affords
valuable information, but its observation is not of that perfectly
simple character, requiring hardly any apparatus, that constitutes
one great advantage, for chemical application, of the observation
of absorption or of the tint of fluorescent light. The observation
is restricted to times when the sun is shining pretty steadily
(unless the observer has recourse to electric light, or at least lime-

* See Quarterly Journal of the Chemical Society, Vol. xi, p. 19. {Ante, p. 114.]
S. IV. 17

258 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

light); it is requisite to reflect the sun's light horizontally, without
which the observation would be most troublesome ; and unless the
reflexion be made by the mirror of a heliostat, the continual change
in the direction of the reflected light is most inconvenient. It is
requisite to use at least one good prism, better two or three, which
must be of tolerable size, in order to have light of sufficient
intensity, and the prisms must be combined with a lens, which
need not however be achromatic. Hence these observations are
not, like the former, adapted to the daily use of every chemist.

It has already been stated, as the result of experience, that the
colour of the fluorescent light of a single substande is constant
throughout the spectrum, or very nearly so. If, therefore, on
examining a solution in a pure spectrum thus formed by projection,
we find the fluorescence taking a fresh start with a different colour,
we may be almost certain that we have to do with a mixture of
two different fluorescent substances, the presence of which is
thus revealed without any chemical process. If, however, the
fluorescence of two fluorescent substances, which may be mixed
together, begins at nearly the same point in the spectrum (as
commonly happens when there is merely a slight difference of
tint in the colour of the fluorescent light of the two substances),
the coexistence of the two substances may escape detection when
the mixed solution is merely examined in a pure spectrum ; and in
such cases a combination of processes of fractional separation with
the easy observation of the tint of the fluorescent light is more
searching. This is the case, for instance, with the mixture of
sesculin and fraxin contained in a solution from the bark of the
horse-chestnut.

Experience has also indicated a most intimate connexion be-
tween the spectral distribution of fluorescence and that of opacity
in the case of solutions of pure substances. There are, indeed,
theoretical reasons for regarding it as not improbable that in-
stances may yet be found in which absorption, unaccompanied by
fluorescence, takes place, in the case of solutions of fluorescent
substances, in that part of the visible spectrum which is less
refrangible than the point at which the fluorescence commences,
but no such instance has yet been observed. Hence from the
distribution of the fluorescence, we may infer the character of the
absorption belonging to the fluorescent body. For this purpose it.

AND DISCRIMINATION OF OEGANIC SUBSTANCES. 259

is best to make the solution extremely dilute, when any bands of
absorption will have their positions indicated by beams of fluor-
escent light, while in the intermediate parts of relatively great
transparency, the fluorescence, in the case of such weak solutions,
is almost insensible.

As the occurrence of a decided difference of colour in the
fluorescent light seen at two different parts of the spectrum
implies, almost to a certainty, the presence of two different
fluorescent substances, so, conversely, the exhibition of the same
colour is an argument in favour of the identity of the substance
producing the fluorescence at the two parts. We cannot indeed
say that there may not be two substances present, the fluorescence
of which commences at nearly the same part of the spectrum ;
but assuredly, two different substances, the fluorescence of which
commenced at two widely different parts of the spectrum, would
reveal themselves by the difference of colour. For experience
shows that the refrangibility of the light emitted, at any part
of the incident spectrum, by the solution of a pure substance,
extends nearly up to that of the point of the incident spectrum
at which the fluorescence commences, but not much beyond ; and
though, in passing from one pure substance to another, variations
do occur in the relative brightness of the rays of less refrangibility
which compose the fluorescent light, yet, on the whole, there is so
close a connexion between the colour of the fluorescent light and
the refrangibility of the rays by which the fluorescence is first
produced, that no great variation in the one is compatible with
constancy or a mere trifling variation in the other.

For an example of the application of these principles, we may
refer to the green colouring matter of leaves. The alcoholic solu-
tion of this substance exhibits a lively fluorescence of a blood-red
colour, and shows also a certain system of bands of absorption.
Different chemists in different ways have obtained from it a yellow
substance, and M. Fremy, having obtained a yellow substance by
the aid of merely neutral reagents, has proved that such a sub-
stance pre-exists. The yellow solution was obtained by him in the
attempt to divide the green colouring matter into a yellow and
a blue ; but, by using neutral reagents only, he did not get further
in the direction of blue than a green of a bluer shade than at first,
which he supposed to be due to the imperfection of the modes of

17—2

260 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

separation. He conceived, however, that he had attained his object
by dissolving chlorophyll in a mechanical mixture of ether and
hydrochloric acid, the acid stratum, when the fluids separated after
agitation, exhibiting a blue colour.

Now, when an alcoholic solution of chlorophyll is examined in
a pure spectrum formed by sunlight, in the part of the spectrum
extending from the extreme red to the junction of the green and
blue the fluorescence exhibits remarkable fluctuations of intensity,
corresponding precisely to the bands of absorption in the trans-
mitted light. The red fluorescence is extremely lively in nearly
the whole of the blue and in the violet. This proves that the
main absorption of these colours cannot be due to the yellow body;
for the yellow substance would be either non-fluorescent, or its
fluorescence would be of some shade of green or yellow. On the
former supposition, the fluorescence of the chlorophyll solution in
the blue and violet would be dull, and on the latter would be of
some colour different from blood-red, unless the main absorption of
that part of the spectrum were due to the substance producing
the red fluorescence. In fact, when the yellow body is nearly
isolated, it exhibits two characteristic bands of absorption in the
blue, to which no fluorescence corresponds, which demonstrates
that the slight red fluorescence which the solution may still exhibit
is due to the remaining impurity. Hence, if the yellow body
were wholly eliminated from chlorophyll, the residue, containing
the substance showing the red fluorescence, would still powerfully
absorb the blue and violet, and therefore could not be blue, but
only a bluer green; so that in seeking to separate chlorophyll
into a blue and a yellow substance, M. Fremy was aiming at
an impossibility. His phyllocyanin is, in fact, a product of de-
composition, and is not blue at all, but merely dissolves in certain
acids with a blue colour. It may be mentioned in passing, that
the green fluorescent residue is still a mixture, consisting of
two different substances, both green, and both exhibiting a red
fluorescence.

III. The instances in which substances appear coloured by
reflexion are comparatively rare. It is very common in chemical
descriptions to read of a solution appearing of such a colour by
transmitted, and such a colour by reflected light. In many cases.

AND DISCRIMINATION OF ORGANIC SUBSTANCES. 261

that is a positive mistake, and the colour described as due to
reflexion is really due to transmission. A chemist views a solution
contained in a test-tube by transmission, and then by reflexion ;
and seeing, perhaps, some perfectly difierent colour in the latter
case, describes it as the colour of the solution by reflexion, whereas
it is merely the colour by transmission due to a greater thickness,
the light having been reflected at the back or bottom of the test-
tube, and so having twice passed through the solution. In other
cases the colour described as due to reflexion really arises from
fluorescence ; and though the statement may be true in the sense
intended, it seems objectionable to apply the term reflexion to
a process so utterly different. It is only in the case of metals,
such as gold and copper, and of certain other substances such as
murexide, platinocyanide of magnesium, &c., that colour is really
seen as the result of reflexion.

When this takes place in the case of non-metallic substances,
they are found to be endowed, for the colours so reflected, with
an intense opacity, comparable with that of metals ; while for other
parts of the spectrum they may be comparatively transparent, and
these parts they reflect with an energy conjparable to that of
a vitreous substance only. The variations of absorbing power in
passing from one part of the spectrum to another, and consequently
the variations in reflecting energy, are frequently much more
considerable, and accordingly the colour by reflexion is much richer
than in the case of metals.

An excellent example of the intimate connexion between
metallic reflexion and intense absorption is afforded by crystals
of permanganate of potash. These crystals exhibit a green metallic
reflexion, and when crushed yield a powder of an intense purple
colour by transmitted light. The colour is too intense for spectral
analysis, but the solution has a similar colour, merely less intense
as corresponds with its smaller concentration, and the analysis of
the light transmitted by the solution presents no difliculty. The
green is quickly absorbed, bui; when the solution is sufficiently
dilute, five eminently characteristic bands of absorption are seen
in that part of the spectrum. A sixth band comes out with a
greater thickness or else strength of solution, but even the fifth
is somewhat less strong than the others. When the light reflected
from a crystal is analysed, four bright bands are seen standing out

262 APPLICATION OF OPTICAL PROPERTIES TO DETECTION

on a generally luminous ground of inferior brightness. These bright
bands correspond in position with the principal dark bands in the
light transmitted by the solution, and therefore, it may be pre-
sumed, by the crystals themselves. When the angle of incidence
has a suitable value, and the reflected light is analysed by a Nicol's
prism, with its principal plane in the plane of incidence, and then
by a common prism, the spectrum is reduced to these four bright
bands. A fifth bright band could perhaps be made out, in the
case of a fine crystal with a fresh surface. Under the circum-
stances described, the Nicol's prism would extinguish the light
reflected from a vitreous substance, and transmit much of that
reflected from a metal. We see, therefore, that, as regards its
relations to light, the crystallized body passes repeatedly from
the condition of a vitreous to that of a metallic substance and
back again, as the refrangibility of the rays, in relation to which
it is considered, is continuously increased by a small amount.

The same relation between intense absorption and metallic
reflexion exists generally, though it cannot be always studied
by means of a solution. The platinocyanides, for example, yield
colourless solutions, so that the intense absorption which most of
them exercise for certain parts of the spectrum must be attributed
to the mode in which the molecules are built up in forming the
crystals ; but by attending to the colour of the light transmitted
by thin crystals, the law is found to be obeyed. Gold can only be
obtained, in solution, as gold, by means of the opaque solvent
mercury; but its colour by transmission may be studied in gold
leaf, or in a chemically deposited film, and is then found to be con-
formable to the law mentioned, the less refrangible colours, which
are those which are the more copiously reflected, being also those
which are the more intensely absorbed.

When a body endowed with the property of coloured reflexion,
such as permanganate of potash, is dissolved, in consequence of
the necessary dilution the opacity of the medium ceases to be, for
any part of the spectrum, of that intense kind which is necessary
for quasi-metallic reflexion ; and accordingly the light reflected by
the solution is colourless. Hence coloured reflexion is not avail-
able for following a substance through mixtures containing it.
The chemist ought, however, to be acquainted with its laws, in
order to understand the changes of colour which a substance

ASD PISOKIMIXATIOX OF OKGaXIC SISSTAXCES. _0o

}v>sj< ssiiii: the prvipem- is capable o: exhibi:inir in the ~>iid cod-

diriv^n. acvxwiiiis' to ;:s #:a:e of ;^ccrt C->::jn.

In Older that the colour due to reflexion jhcv.Id appear, i: :#
necessary that the s.ilxstaritV jho^-d have a certain amount of
coheience. Thus rdic-^ in :he f.rra of a fine IcN^je povrder i? bl.;e.
<; vcv. when viewed br leflexion. It would be irrorLcV is. however,
to diicrilv :ho bodv a* blue bv reflexion, if we were speak;.. c of
the prv^jvr:it:s of the sv-bsrcvaoe. and not the mere cmde rtsv..:*
of ob>cr\ ;-,::::: ir.r.de Mr.dir iriveii oirciimstancaSs For :hov.j:h it :5
true that the .icht bv which the blue colour is seen h;\5 ttnde^J^ re
reflexion (with, ttt whidi i; would not have i>eached the eye) it is
not i« r«f<s*iom thst the chromatic selection is made by virtue of
which the powder ;.p\x\.r^ blue, but :V.."i';j trar.f ^isfion. In n\ot
it is only a str.ali portion of the li^ht thst is redeoted .>t the ouier
ime^v...-.t sv.rt,-..^ of the -..-.riSiS : the creator part peaetr.^tes s little
way, and ;s reflected at v;\ri v.s depths, and in passing thro.^h
the iv.rti.'ts. in c.in.cand rvtnvninjr- snnc r^ absorption ou the yart
of the cv^lo.trcxi s.hstcut.Y. ^Vene the s.hstan.v intense iy opaque
tor ;3r the . ..-.~s :i the sy-c\.tn..n the powder would be not blue
but bl:\ci<. ;".s we se-i in tit; c-ase of piatinum-black. By PttrrLtshinir.
the powder is reduced to the stat: of a s t.iewhat coherent ma;ss.
and it now K^rins to exhibit the cpper colour dae to reflexion.
The internal reticxi ns are at the s^.nie time cr^a^.y weakened, so
that the iv\rt of the ii^ht which is rettecied from beneath and
nnd-vTc >,s absvq^^tion is much reduced. A yressec, iii:-,s^ :s not,
howev^^^r, an opti.vJly h:n. c:n-:ns medium., so that the colour by
rvnexion obtained by hnrn.sh.nc cannot ir. ctn;ra- "^t .,.tite part.
In the state of a tine .-.ystaiiin, p^^wder, indie tshihits a mixnu^
of the copper colo.tr d.tf to :\~.:xi n, and th; blue coloiu- due to
tTansffliissicai. thtach chs;rvod in the lt;.~.: rtht otert ftom the tnriScs
.IS a whole; while if the snhstan.v coidd be obtained in .arj:-:
ervstas, the colour by reflection would be s: n in perfection, suid
the ocJo.tr by tt^ansn.tssijn w:.n!d a:s:~.yy-:ar. tne .rysta.s betn^r
s^ n.sihiy oyt\ que.

On the Reduction and Oxidation of the Colouring
Matter of the Blood.

[From the Proceedings of the Royal Society, June 16, 1864.]

1. Some time ago my attention was called to a paper by-
Professor Hoppe*, in which he has pointed out the remarkable
spectrum produced by the absorption of light by a very dilute
solution of blood, and applied the observation to elucidate the
chemical nature of the colouring matter. I had no sooner looked
at the spectrum, than the extreme sharpness and beauty of the
absorption-bands of blood excited a lively interest in my mind,
and I proceeded to try the effect of various reagents. The obser-
vation is perfectly simple, since nothing more is required than to
place the solution to be tried, which may be contained in a test-
tube, behind a slit, and view it through a prism applied to the
eye. In this way it is easy to verify Hoppe's statement, that the
colouring matter (as may be presumed at least from the retention
of its peculiar spectrum) is unaffected by alkaline carbonates and
caustic ammonia, but is almost immediately decomposed by acids,
and also, but more slowly, by caustic fixed alkalies, the coloured
product of decomposition being the hsematine of Lecanu, which
is easily identified by its peculiar spectra. But it seemed to me
to be a point of special interest to inquire whether we could
imitate the change of colour of arterial into that of venous blood,
on the supposition that it arises from reduction.

2. In my experiments I generally employed the blood of
sheep or oxen obtained from a butcher; but Hoppe has shown
that the blood of animals in general exhibits just the same bands.
To obtain the colouring matter in true solution, and at the same
time to get rid of a part of the associated matters, I generally
allowed the blood to coagulate, cut the clot small, rinsed it well,
and extracted it with water. This, however, is not essential, and

* Virchow's ArcUv, Vol. xxiii, p. 446 (1862). [Cf. ante, p. 240.]

ON THE REDUCTION AND OXIDATION OF THE BLOOD.

265

blood merely diluted with a large quantity of water may be used ;
but in what follows it is to be understood that the watery extract
is used unless the contrary be stated.

3. Since the colouring matter is changed by acids, we must
employ reducing agents which are compatible with an alkaline
solution. If to a solution of protosulphate of iron enough tartaric
acid be added to prevent precipitation by alkalies, and a small
quantity of the solution, previously rendered alkaline by either
ammonia or carbonate of soda, be added to a solution of blood,
the colour is almost instantly changed to a much more purple red
as seen in small thicknesses, and a much darker red than before
as seen in greater thickness. The change of colour, which recalls
the difference between arterial and venous blood, is striking
enough, but the change in the absorption spectrum is far more
decisive. The two highly characteristic dark bands seen before
are now replaced by a single band, somewhat broader and less
sharply defined at its edges than either of the former, and occupy-
ing nearly the position of the bright band separating the dark
bands of the original solution. The fluid is more transparent for
the blue, and less so for the green than it was before. If the
thickness be increased till the whole of the spectrum more re-
frangible than the red be on the point of disappearing, the last
part to remain is green, a little beyond the fixed line b, in the
case of the original solution, and blue, some way beyond F, in the

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

B D

1

E I

1 B'i'

Iji:

1

1 i^HI

1

1

n'''i'!Fi||^H

266 ON THE REDUCTION AND OXIDATION

case of the modified fluid. Figs. 1 and 2 in the accompanying
woodcut represent the bands seen in these two solutions re-
spectively.

4. If the purple solution be exposed to the air in a shallow
vessel, it quickly returns to its original condition, showing the two
characteristic bands the same as before; and this change takes
place immediately, provided a small quantity only of the reducing
agent were employed, when the solution is shaken up with air. If
an additional quantity of the reagent be now added, the same
effect is produced as at first, and the solution may thus be made
to go through its changes any number of times.

5. The change produced by the action of the air (that is, of
course, by the absorption of oxygen) may be seen in an instructive
form on partly filling a test-tube with a solution of blood suitably
diluted, mixing with a little of the reducing agent, and leaving
the tube at rest for some time in a vertical position. The upper or
oxidized portion of the solution is readily distinguished by its
colour ; and if the tube be now placed behind a slit and viewed
through a prism, a dark band is seen, having the general form of
a tuning-fork, like Figs. 1 and 2, regarded now as a single figure,
the line of separation being supposed removed.

6. Of course it. is necessary to assure oneself that the single
band in the green is not due to absorption produced merely by
the reagent, as is readily done by direct observation of its spec-
trum, not to mention that in the region of the previous dark
bands, or at least the outer portions of it, the solution is actually
more transparent than before, which could not be occasioned by
an additional absorption. Indeed the absorption due to the re-
agent itself in its different stages of oxidation, unless it be
employed in most unnecessary excess, may almost be regarded as
evanescent in comparison with the absorption due to the colouring
matter; though if the solution be repeatedly put through its
changes, the accumulation of the persalt of iron will presently tell
on the colour, making it sensibly yellower than at first for small
thicknesses of the solution.

7. That the change which the iron salt produces in the spec-
trum is due to a simple reduction of the colouring matter, and not
to the formation of some compound of the colouring matter with
the reagent, is shown by the fact that a variety of reducing agents

OF THE COLOURING MATTER OF THE BLOOD. 267

of very different nature produce just the same effect. If proto-
chloride of tin be substituted for protosulphate of iron in the
experiment above described, the same changes take place as with
the iron salt. The tin solution has the advantage of being
colourless, and leaving the visible spectrum quite unaffected, both
before and after oxidation, and accordingly of not interfering in
the slightest degree with the optical examination of the solutions,
but permitting them to be seen with exactly their true tints.
The action of this reagent, however, takes some little time at
ordinary temperatures, though it is very rapid if previously the
solution be gently warmed. Hydrosulphate of ammonia again
produces the same change, though a small fraction of the colour-
ing matter is liable to undergo some different modification, as is
shown by the occurrence of a slender band in the red, variable in
its amount of development, which did not previously exist. In
this case, as with the tin salt, the action is somewhat slow, requir-
ing a few minutes unless it be assisted by gentle heat. Other
reagents might be mentioned, but these will suffice.

8. We may infer from the facts above mentioned that the
colouring matter of blood, like indigo, is capable of existing in two
states of oxidation, distinguishable by a differ ejice of colour and a
fundamental difference in the action on the spectrum. It may be
made to pass from, the m,ore to the less oxidized state by the action
of suitable reducing agents, and recovers its oxygen by absorption
from the air.

As the term hcematine has been appropriated to a product of
decomposition, some other name must be given to the original
colouring matter. As it has not been named by Hoppe *, I propose
to call it cruorine, as suggested to me by Dr Sharpey ; and in its
two states of oxidation it may conveniently be named scarlet
cruorin£ and purple cruorine respectively, though the former is
slightly purplish at a certain small thickness, and the latter is of
a. very red purple colour, becoming red at a moderate increase of
thickness.

9. When the watery extract from blood-clots is left aside in a
corked bottle, or even in a tall narrow vessel open at the top, it

[* It was in the year 1864, the date of the present paper, that Hoppe proposed
the name haemoglobin, now universally employed, Yirohow's Archiv, xxix, p. 223.]

268 ON THE REDUCTION AND OXIDATION

presently changes in colour from a bright to a dark red, decidedly
purple in small thicknesses. This change is perceived even
before the solution has begun to stink in the least perceptible
degree. The tint agrees with that of the purple cruorine obtained
immediately by reducing agents; and if a little of the solution
be sucked up from the bottom into a quill-tube drawn to a capillary
point, and the tube be then placed behind a slit, so as to admit of
analyzing the transmitted light without exposing the fluid to the
air, the spectrum will be found to agree with that of purple
cruorine. On shaking the solution with air it immediately
becomes bright red, and now presents the optical characters of
scarlet cruorine. It thus appears that scarlet cruorine is capable
of being reduced by certain substances, derived from the blood,
present in the solution, which must themselves be oxidized at its
expense.

10. When the alkaline tartaric solution of protoxide of tin is
added in moderate quantity to a solution of scarlet cruorine, the
latter is presently reduced. If the solution is now shaken with
air, the cruorine is almost instantly oxidized, as is shown by the
colour of the solution and its spectrum by transmitted light. On
standing for a little time, a couple of minutes or so, the cruorine
is again reduced, and the solution may be made to go through
these changes a great number of times, though not of course
indefinitely, as the tin must at last become completely oxidized.
It thus appears that purple cruorine absorbs /ree oxygen with
much greater avidity than the tin solution, notwithstanding that
the oxidized cruorine is itself reduced by the tin salt. I shall
return to this experiment presently.

11. When a little acid, suppose acetic or tartaric acid, which
does not produce a precipitate, is added to a solution of blood, the
colour is quickly changed from red to brownish red, and in place
of the original bands (Fig. 1) we have a different system, nearly
that of Fig. 3. This system is highly characteristic ; but in order
to bring it out a larger quantity of substance is requisite than in
the case of scarlet cruorine. The figure does not exactly corre-
spond to any one thickness, for the bands in the blue are best
seen while the band in the red is still rather narrow and ill-defined
at its edges, while the narrow inconspicuous band in the yellow
hardly comes out till the whole of the blue and violet, and a good

OF THE COLOURING MATTER OF THE BLOOD. 269

part of the green, are absorbed. The difference in the spectra
Figs. 1 and 3 does not alone prove that the colouring matter is
decomposed by the acid (though the fact that the change is not
instantaneous favours that supposition), for the one solution is
alkaline, though it may be only slightly so, while the other is acid,
and the difference of spectra might be due merely to this circum-
stance. As the direct addition of either ammonia or carbonate
of soda to the acid liquid causes a precipitate, it is requisite in the
first instance to separate the colouring matter from the substance
so precipitated.

This may be easily effected on a small scale by adding to the
watery extract from blood-clots about an equal volume of ether,
and then some glacial acetic acid, and gently mixing, but not
violently shaking for fear of forming an emulsion. When enough
acetic acid has been added, the acid ether rises charged with
nearly the whole of the colouring matter, while the substance
which caused the precipitate remains in the acid watery layer
below*. The acid ether solution shows in perfection the charac-
teristic spectrum Fig. 3. When most of the acid is washed out the
substance falls, remaining in the ether near the common surface.
If after removing the wash-water a solution, even a weak one, of
ammonia or carbonate of soda be added, the colouring matter
readily dissolves in the alkali. The spectrum of the transmitted
light is quite different from that of scarlet cruorine, and by no
means so remarkable. It presents a single band of absorption,
very obscurely divided into two, the centre of which nearly co-
incides with the fixed line D, so that the band is decidedly less
refrangible than the pair of bands of scarlet cruorine. The rela-
tive proportion of the two parts of the band is liable to vary. The
presence of alcohol, perhaps even of dissolved ether, seems to
favour the first part, and an excess of caustic alkali the second,
the fluid at the same time becoming more decidedly dichroitic:
The blue end of the spectrum is at the same time absorbed. The
band of absorption is by no means so definite at its edges as those

* If I may judge from the results obtained with the precipitate given by acetic
acid and a neutral salt, a promising mode of separation of the proximate con-
stituents of blood-crystals would be to dissolve the crystals in glacial acetic acid
and add ether, which precipitates a white albuminous substance, leaving the
hsematine in solution.

270 ON THE REDUCTION AND OXIDATION

of scarlet cruorine, and a far larger quantity of the substance is
required to develope it.

This difference of spectra shows that the colouring matter
(hsematine) obtained by acids is a product of the decomposition,
or metamorphosis of some kind, of the original colouring matter.

When hsematine is dissolved in alcohol containing acid, the
spectrum nearly agrees with that represented in Fig. 3.

12. Hajmatine is capable of reduction and oxidation like
cruorine. If it be dissolved in a solution of ammonia or of car-
bonate of soda, and a little of the iron salt already mentioned, or
else of hydrosulphate of ammonia, be added, a pair of very intense
bands of absorption is immediately developed (Fig. 4). These
bands are situated at about the same distance apart as those of
scarlet cruorine, and are no less sharp and distinctive. They are
a little more refrangible, a clear though narrow interval interven-
ing between the first of them and the line D. They differ much
from the bands of cruorine in the relative strength of the first
and second band. With cruorine the second band appears almost
as soon as the first, on increasing the strength or thickness of the
solution from zero onwards, and when both bands are well
developed, the second band is decidedly broader than the first.
With reduced hsematine, on the other hand, the first band is
already black and intense by the time the second begins to
appear; then both bands increase, the first retaining its superiority
until the two are on the point of merging into one by the ab-
sorption of the intervening bright band, when the two appear
about equal.

Like cruorine, reduced hsematine is oxidized by shaking up its
solution with air. I have not yet obtained hsematine in an acid
solution in more than one form, that which gives the spectrum
Fig. 3, and which I have little doubt contains hasmatine in its
oxidized form ; for when it is withdrawn from acid ether by an
alkali, I have not seen any traces of reduced hsematine, even on
taking some precautions against the absorption of oxygen. As the
alkaline solution of ordinary hsematine passes, with increase of
thickness, through yellow, green, and brown to red, while that of
reduced hsematine is red throughout, the two kinds may be con-
veniently distinguished as brown hwmatine and red hcematine

OF THE COLOURING MATTER OF THE BLOOD. 271

respectively*, the former or oxidized substance being the hsematine
of chemists.

13. Although the spectrum of scarlet cruorine is not affected
by the addition to the solution of either ammonia or carbonate of
soda, yet if after such addition the solution be either heated or
alcohol be added, although there is no precipitation decomposition
takes place. The coloured product of decomposition is brown
hsematine, as may be inferred from its spectrum. Since, however,
the spectrum of an alkaline solution of brown hsematine is only
moderately distinctive, and is somewhat variable according to the
nature of the solvent, it is well to add hydrosulphate of ammonia,
which immediately developes the remarkable bands of red hsema-
tine. This is the easiest way to obtain them; but the less
refrangible edge of the first band as obtained in this way is liable
to be not quite clean, in consequence of the presence of a small
quantity of cruorine which escaped decomposition.

Some very curious reactions are produced in a solution of
cruorine by gallic acid combined with other reagents, but these
require further study.

14. Hoppe proposed to employ the highly characteristic ab-
sorption-bands of scarlet cruorine in forensic inquiries. Since,
however, cruorine is very easily decomposed, as by hot water,
alcohol, weak acids, &c., the method would often be inapplicable.
But as in such cases the coloured product of decomposition is
hsematine, which is a very stable substance, the absorption-bands
of red hsematine in alkaline solution, which in sharpness, distinc-
tive character and sensibility rival those of scarlet cruorine itself,
may be employed instead of the latter. The absorption-bands of
brown hsematine dissolved in a mixture of ether and acetic acid,
or in acetic acid alone, are hardly less characteristic, but are not
quite so sensitive, requiring a somewhat larger quantity of the
substance.

15. I have purposely abstained from physiological speculations
until I should have finished the chemico-optical part of the sub-
ject ; but as the facts which have been adduced seem calculated to

[* In present-day nomenclature alkaline hamatin and hiemochromogen
respectively.]

272 ON THE REDUCTION AND OXIDATION

throw considerable light on the function of cruorine in the animal
economy, I may perhaps be permitted to make a few remarks on
this subject.

It has been a disputed point whether the oxygen introduced
into the blood in its passage through the lungs is simply dissolved
or is chemically combined with some constituent of the blood.
The latter and more natural view seems for a time to have given
place to the former in consequence of the experiments of Magnus.
But Liebig and others have since adduced arguments to show
that the oxygen absorbed is, mainly at least, chemically combined,
be it only in such a loose way, like a portion of the carbonic acid
in bicarbonate of soda, that it is capable of being expelled by
indifferent gases. It is known, too, that it is the red corpuscles in
which the faculty of absorbing oxygen mainly resides.

Now it has been shown in this paper that we have in cruorine
a substance capable of undergoing reduction and oxidation, more
especially oxidation, so that if we may assume the presence of
purple cruorine in venous blood, we have all that is necessary to
account for the absorption and chemical combination of the inspired
oxygen.

16. It is stated- by Hoppe that venous as well as arterial
blood shows the two bands which are characteristic of what has
been called in this paper scarlet cruorine. As the precautions
taken to prevent the absorption of oxygen are not mentioned, it
seemed desirable to repeat the experiment, which Dr Harley and
Dr Sharpey have kindly done. A pipette adapted to a syringe
was filled with water which had been boiled and cooled without
exposure to the air, and the point having been introduced into
the jugular vein of a live dog, a little blood was drawn into the
bulb. Without the water the blood would have been too dark for
spectral analysis. The colour did not much differ from that of
scarlet cruorine ; certainly it was much nearer the scarlet than
the purple substance. The spectrum showed the bands of scarlet
cruorine.

This, however, does not by any means prove the absence of
purple cruorine, but only shows that the colouring matter present
was chiefly scarlet cruorine. Indeed the relative proportions of
the two present in a mixture of them with one another and with

OF THE COLOURING MATTER OF THE BLOOD. 273

colourless substances, can be better judged of by the tint than
by the use of the prism. With the prism the extreme sharpness
of the bands of scarlet cruorine is apt to mislead, and to induce
the observer greatly to exaggerate the relative proportion of
that substance.

Seeing then that the change of colour from arterial to venous
blood as far as it goes is in the direction of the change from scarlet
to purple cruorine, that scarlet cruorine is capable of reduction
even in the cold by substances present in the blood (| 9), and that
the action of reducing agents upon it is greatly assisted by warmth
(§ 7), we have every reason to believe that a portion of the cruorine
present in venous blood exists in the state of purple cruorine, and
is reoxidized in passing through the lungs.

17. That it is only a rather small proportion of the cruorine
present in venous blood which exists in the state of purple cruorine
under normal conditions of life and health, may be inferred, not
only from the colour, but directly from the results of the most re-
cent experiments*. Were it otherwise, any extensive haemorrhage
could hardly fail to be fatal, if, as there is reason to believe, cruo-
rine be the substance on which the function of respiration mainly
depends ; nor could chlorotic persons exhale as much carbonic acid
as healthy subjects, as is found to be the case.

But after death there is every reason to think that the process
of reduction still goes on, especially in the case of warm-blooded
animals, while the body is still warm. Hence the blood found in
the veins of an animal some time after death can hardly be taken
as a fair specimen as to colour of the venous blood in the living
animal. Moreover the blood of an animal which has been sub-
jected to abnormal conditions before death is of course liable to
be altered thereby. The terms in which Lehmann has described
the colour of the blood of frogs which had been slowly asphyxiated
by being made to breathe a mixture of air and carbonic acid seem
unraistakeably to point to purple cruorine f.

18. The effect of various indifferent reagents in changing
the colour of defibrinated blood has been much studied, but not
always with due regard to optical principles. The brightening of

* Funk's Lehrhuch der Physiologie, 1863, Vol. i, § 108.
t Physiological Ghe.mwtry, Vol. ii, p. 178.
S. IV. 18

274 ON THE REDUCTION AND OXIDATION

the colour, as seen by reflexion, produced by the first action of
neutral salts, and the darkening caused by the addition of a little
water, are, I conceive, easily explained ; but I have not seen stated
what I feel satisfied is the true explanation. In the former case
the corpuscles lose water by exosmose, and become thereby more
highly refractive, in consequence of which a more copious reflexion
takes place at the common surface of the corpuscles and the
surrounding fluid. In the latter case they gain water by endos-
mose, which makes their refractive power more nearly equal to
that of the fluid in which they are contained, and the reflexion
is consequently diminished. There is nothing in these cases to
indicate any change in the mode in which light is absorbed by the
colouring matter, although a change of tint to a certain extent,
and not merely a change of intensity, may accompany the change
of conditions under which the turbid mixture is seen, as I have
elsewhere more fiilly explained *.

No doubt the form of the corpuscles is changed by the action
of the reagents introduced ; but to attribute the change of colour
to this is, I apprehend, to mistake a concomitant for a cause, and
to attribute, moreover, the change of colour to a cause inadequate
to produce it.

19. Very different is the effect of carbonic acid. In this case
the existence of a fundamental change in the mode of absorption
cannot be questioned, especially when the fluid is squeezed thin
between two glasses and viewed by transmitted light. I took
two portions of defibrinated blood ; to one I added a little of the
reducing iron solution, and passed carbonic acid into the other,
and then compared them. They were as nearly as possible alike.
We must not attribute these apparently identical changes to two
totally different causes if one will siiffice. Now in the case of the
iron salt, the change of colour is plainly due to a deoxidation of
the cruorine. On the other hand, Magnus removed as much as
10 or 12 per cent, by volume of oxygen from arterialized blood by
shaking the blood with carbonic acid. If, as we have reason to
believe, this oxygen was for the most part chemically combined,
it follows that carbonic acid acts as if it were a reducing agent.
We are led to regard the change of colour not as a direct effect of
the presence of carbonic acid, but a consequence of the removal of

* Philosophical Transactions, 1852, p. 527. [Ante, Vol. iii, p. 359.]

OF THE COLOURING MATTER OF THE BLOOD. 275

oxygen. There is this difference between carbonic acid and the
real reducing agents, that the former no longer acts on a dilute
and comparatively pure solution of scarlet cruorine, while the
latter act just as before.

If even in the case of blood exposed to an atmosphere of
carbonic acid we are not to attribute the change of colour to the
direct presence of the gas, much less should we attempt to
account for the darker colour of venous than arterial blood by the
small additional percentage of carbonic acid which the former
contains. The ascertained properties of cruorine furnish us with
a ready explanation, namely that it is due to a partial reduction of
scarlet cruorine in supplying the wants of the system.

20. I am indebted to Dr Akin for calling my attention to
a very interesting pamphlet by A. Schmidt on the existence of
ozone in the blood*. The author uses throughout the language
of the ozone theory. If by ozone be meant the substance, be it
allotropic oxygen or teroxide of hydrogen, which is formed by
electric discharges in air, there is absolutely nothing to prove its
existence in blood; for all attempts to obtain an oxidizing gas
from blood failed. But if by ozone be merely meant oxygen in
any such state, of combination or otherwise, as to be capable of
producing certain oxidizing effects, such as turning guaiacum
blue, the experiments of Schmidt have completely established its
existence, and have connected it, too, with the colouring matter.
Now in cruorine we have a substance admitting of easy oxidation
and reduction; and connecting this with Schmidt's results, we
may infer that scarlet cruorine is not merely a greedy absorber
and a carrier of oxygen, but also an oxidizing agent, and that it is
by its means that the substances which enter the blood from the
food, setting aside those which are either assimilated or excreted
by the kidneys, are reduced to the ultimate forms of carbonic acid
a,nd water, as if they had been burnt in oxygen.

21. In illustration of the functions of cruorine, I would refer,
in conclusion, to the experiment mentioned in § 10. As the purple
cruorine in the solution was oxidized almost instantaneously on
being presented with free oxygen by shaking with air, while the
tin-salt remained in an unoxidized state, so the purple cruorine of

* Veher Ozon im Blute, Dorpat, 1862.

18—2

276 A PROPERTY OF EQUIPOTENTIAL CURVES.

the veins is oxidized during the time, brief though it may be,
during which it is exposed to air in the lungs, while the substances
derived from the food may have little disposition to combine with
free oxygen. As the scarlet cruorine is gradually reduced, oxidizing
thereby a portion of the tin-salt, so part of the scarlet cruorine is
gradually reduced in the course of the circulation, oxidizing a
portion of the substances derived from the food or of the tissues.
The purplish colour now assumed by the solution illustrates the
tinge of venous blood, and a fresh shake represents a fresh passage
through the lungs.

On a Property of Curves which fulfil the condition
-y^ + -j^ = 0. By W. J. Macquorn Rankine. (Supplement.)

[From the Proceedings of the Royal Society, May 9, 1867.]

Professor Stokes has pointed out to me an extension
of the preceding theorem, viz. that at every multiple point in

a plane curve which fulfils the condition that -j-^ + -~ = 0, the

QjtJC dilj''

branches make equal angles with each other, so that, for example^
if n branches cut each other at a multiple point, they make with
each other 2n equal angles of tt/w.

Ox THE Internal Distribution of Matter which shall
produce a given potential at the surface of a
Gravitating Mass.

[From the Proceedings of the Royal Society, May 16, 1867.]

It is kiio-4vn that if either the potential of the attraction of
a mass attracting according to the law of the inverse square of the
distance, or the normal component of the attraction, be given all
over the surface of the mass, or anj- surface enclosing it (whiph
latter case may be included in the farmer by regarding the
internal density as null between the assumed enclosing surface
and the actual surface), the potential and consequently the attrac-
tion at all points external to the surface and at the surface itself
is determinate. This proposition leads to results of particular
interest when applied to the Earth, as I showed in two papers
published in 1849*, where among other things I proved that if
the surface be assumed to be, in accordance with observation, of
the form of an ellipsoid of revolution, Clairaut's Theorem follows
independently of the adoption of the hypothesis of original fluidity,
or even of that of an internal arrangement in nearly spherical
strata of equal density.

But though the law of the variation of gravity which was
origfinally obtained as a consequence of the hypothesis of primitive
fluidity, and was afterwards found by Laplace to hold good, on the
condition that the surface be an ellipsoid of revolution as well as a
surface of equilibrium, provided only the mass be arranged in
nearly spherical strata of equal density, be thus proved to be true
whatever be the internal distribution, the question may naturally
be asked. Does not the condition that the potential at the surface

* " On Attractions and on Clairaut's Theorem," Cambridge and Dublin Mathe-
matical Journal, Vol. iv, p. 194 [ante, Vol. n, pp. 104—130]; and "On the
Variation of Gravity at the Surface of the Earth," Cambridge Philosophical Trans-
actions, Vol. vni, p. 672 [ante. Vol. ii, pp. 131—171].

278 ON THE INTERNAL DISTRIBUTION OF MATTER

shall have its actual value require that the internal distribution
shall be compatible with that of a fluid mass, or at any rate shall
be such that the whole mass shall be arranged in nearly spherical
strata of equal density ? Such a question was in fact asked me by
an eminent mathematician at the time to which I have alluded.
I replied by referring to the well-known property of a sphere,
according to which a central mass may be distributed uniformly
over its surface without affecting the external attraction, by apply-
ing which proposition to a mass such as the Earth we may
evidently, without affecting the external attraction, leave a large
excentrically situated cavity absolutely vacuous, the matter pre-
viously within it having been distributed outside it. It is known
further that the mass of a particle may be distributed over any
surface whatsoever enclosing the particle without affecting the
external attraction, and in this way we see at once that we may
leave any internal space we please, however excentrically situated,
wholly vacuous ; nor is it necessary in doing so to introduce an
infinite density, by distributing the whole mass previously within
that space over its surface, since that mass may be conceived to be
divided into an infinite number of infinitely small parts, which are
respectively distributed over an infinite number of surfaces sur-
rounding the space in question. These considerations, however,
though they readily show that the internal distribution may be
widely different from any that is compatible with the hypothesis of
primitive fluidity, do not lead to the general expression for the
internal density. Circumstances have recently recalled my atten-
tion to the subject, and I can now indicate the mode of obtaining
the general expression required in the case of any given surface.

Let the mass be referred to the rectangular axes of x, y, z, and
let p be the density, V be the potential of the attraction. Then
for any internal point V satisfies, as is well known, the partial
differential equation

(PV d'V d'V , ,,,

'5^^ + "^^ + ^^="*'"'' ^^^'

or as it. may be written for brevity VF=0. This equation may
be extended to all space by imagining the body continued infi-
nitely, but having a density which is null outside the limits of the
actual body ; and by adopting this convention we need not trouble
ourselves about those limits. Conversely, if V be a continuously

WHICH SHALL PRODUCE A GIVKX EXTERXAL POTENTIAL. 279

varj-ing function of x, y, -, which ^-anishes at an infinite distance,
and s;itisties the partial differential equation (1). Fis the jxnemiiU
of the attraction of the mass whose density at the point (j-, y. .) is
p ; or. in other wonis.

• • • /

P

V^ll\'^,d^-d!,'d: (iX

where r is the distance between the points (^.r, y, -) and t,.i', y', :').
p the density at (.r'. y'. r'), and the limits are — x to + x , is the
c«^^mplete integral of {!) subject to the condition that T'shall vanish
at jtn infinite distance.

This may be proved in different ways : most directly perhaps
by taking the expression for the potential {U suppose) which
forms the richi-hand member of {'2i. substituting for p' its equiva-
lent — V 1", V being the same fimction of j', y', . that V is of

r. y, r, and n-aosfoiining the integral in the manner done by
Green*, whc-n we readily find C= V.

Supjvse now that we have a given closed surthce 5 containing
within it all the attracting matter, and that the potential has
a given, in general variable, value 1', at the surt'ace. For the
p(.>rtion of sp;iee external to S. I' is to be determined by the
general equation Vr = 0. subjeor to the conditions 1'= V at the
stirii»ce. and V=0 at an infinite disr;mce. We know that the
problem of determining Timder these circumstances admits of one
and but one solution, though it is only for a very limited number
of forms of the surfeice 5 that the solution can actually be effected.
Conceive the problem, however, solved, and firom the solution let the
value ot dV dr at the surthce be found, r being measured outwards
alouij the normah Now complete 1' for infinite spioe by assiirning
to the s}v\oe within ^ any arbititxry but couriiiiunis-'- function we

* "Es^sav on tbe Arplieasiou of Maihematical Anisivs;; to the Theories of
Eieo-.rio ;v and Magnetism." Xottingham, ISis. Art, 3; or the reprint in T'-t'.'ft';.-
■" ;.'«<i/. Vol. XLTv. V. 360.

T To SToid proliiitT. I include in •■ o,^::rinuv»'.;5 the requirement that the
differeiuis". cvi^iSiieuss of the function, to snv oraer required, shall \-arv eou-
i:n'.;o'..s>- What that order msy be it is perfectly easj in anr esse to see. We
mar of course imagine distrioutiens in \rhieh the density becx^mes infinite ii; one or
more ivir.:s, lines, or < -.rrsees. bnt so that a finite volume contains only a finite
niiss. But such dsaibutions may be regarded ss limiting. ,\na therefore psrtieuiar.
c-ases of a a;s:ribution in which the density is finite : .^nvi therefore ti.e surp-osjtion
thii! f is riuiie d.^s no: in effect limit the generality of our rtsr.'.is.

280 ON THE INTERNAL DISTRIBUTION OF MATTER

please, subject to the two conditions, 1st, that at the surface it is
equal to the given function Fj ; 2ndly, that it gives for the value
of dVjdv at the surface that already got from the solution of the
problem referred to in this paragraph. This of course may be
done in an infinite number of ways, just as we may in an infinite
number of ways join two points in a plane by a continuous curve
starting from the two points respectively in given directions, which
curve may be either expressed by some algebraical or transcen-
dental equation, or conceived as drawn libera vianu, and thought
of independently of any idea of algebraical expression. The
function V having been thus assigned to the space internal to S,
the equation (1) gives, according to what we have seen, the most
general expression for the density of the internal matter.

There is, however, no distinction made in this between positive
and negative matter, and if we wish to avoid introducing negative
matter we must restrict the function V for the space internal to S
to satisfy the imparity

dx^ dy^ dz^

It is easy from the general expression to show, what is already
known, that the matter may be distributed in an infinitely thin,
and consequently infinitely dense stratum over the surface S, and
that such a distribution is determinate.

We know that there exists one and but one continuous function
applying to the space within iS which satisfies the equation V F = 0,
and is equal to Vg at the surface. Call this function V^. It is to be
remarked that the value of dV^/dv at the surface is not the same
as that of dVjdv, V being the external potential, though Fj and F
are there each equal to Fq. The argument, it is to be observed,
does not assume that the two are different; it merely avoids
assuming that they are the same ; the result will prove that they
cannot be the same all over S unless the density,' and consequently
the potential, be everywhere null, and therefore F,, = 0. Now
attribute to the interior oi S a. function F which is equal to Fj
except over a narrow stratum adjacent to S, the thickness of which
will in the end be supposed to vanish, within which F is made to
deviate from Fj in such a manner as to render the variation of
dV/dv continuous and rapid instead of abrupt. On applying equa-
tion (1), we see that the density is everywhere null except within

WHICH SHALL PRODUCE A GIVEN EXTERNAL POTENTIAL. 281

this stratum, in which it is very great, and in the limit infinite.
For the total quantity of matter contained in any portion of the
stratum, we have from (1)

-^llhvdxdydz,

the integration extending over that portion. Let the portion in
question be that corresponding to a very small area A of the
surface 8 ; we may suppose it bounded laterally by the ultimately
cylindrical surface generated by a normal to S which travels round
the perimeter of A. Taking now rectangular coordinates X, fi, v,
of which the last is parallel to the normal at one point of A, since
V is not changed in form by referring it to a new set of rectangular
axes, we have for the mass required

1 [f[(d'V d'V d'V] ,, , ,

47r

Of the differential coefficients within brackets, the last alone
becomes infinite when the thickness of the stratum, and conse-
quently the range of integration relatively to X, becomes infinitely
small. We have in the limit

dv' dv dv

both differential coefficients having their values belonging to the
surface. Hence we have ultimately for the mass

A^fdV,_djV\

^sirKdv dv J'

Hence, if -nr be the superficial density, defined as the limit of the
mass corresponding to any small portion of the surface divided by
the area of that portion,

1 fdV, dV\ , ,

^ = 4^1^ ~^j ^'^^'

which is the known expression.

In assigning arbitrarily a function V to the interior of S, in

order to get the internal density by the application of the formula

(1), we may if we please discard the second of the conditions

dV dV
which V had to satisfy at the surface, namely, that ~ = -y- ; but

in bhat case to the mass, of finite density, determined by (1) must

282 INTERNAL DISTRIBUTIONS HAVING GIVEN POTENTIAL.

be added an infinitely dense and infinitely thin stratum extending
over the surface, the finite superficial density of this stratum being
given by (3).

We have seen that the determination of the most general
internal arrangement requires the solution of the problem, To
determine the potential for space external to S, supposed free from
attracting matter, in terms of the given potential at the surface ;
and the determination of that particular arrangement in which the
matter is wholly distributed over the surface, requires further the
solution of the same problem for space internal to S. If, however,
instead of having merely the potential given at the surface S we
had given a particular arrangement of matter within S, and sought
the most general rearrangement which should not alter the poten-
tial at S, there would have been no preliminary problem to solve,
since V, and therefore its differential coefficients, are known for
space generally, and therefore for the surface S, being expressed by
triple integrals.

Instead of having the attracting matter contained within a
closed surface S, and the attraction considered for space external
to 8, it might have been the reverse, and the same methods would
still have been applicable. The problem in this form is more
interesting with reference to electricity than gravitation.

Supplement to a papee on the Discontinuity of Arbitrary
Constants which appear in Divergent Developments.

[From the Transactions of the Cambridge Philosophical Society, Vol. XI,
Part II. Read May 25, 1868.]

In a paper "On the Numerical Calculation of a Class of
Definite Integrals and Infinite Series," printed in the ixth
volume of the Transactions of this Society *, I gave a method by
which a definite integral, to which Mr Airy was led in calcu-
lating the intensity of light in the neighbourhood of a caustic,
may be readily calculated for large values, whether positive or
negative, of a certain variable which appears as a constant under
the sign of integration. The method consists in forming a
differential equation of which the definite integral is a particular
solution, obtaining the complete integral of the equation under
a form, indicated by the equation itself, involving series according
to descending powers of the variable, and determining the
arbitrary constants. The equation admits also of integration by
means of ascending series multiplied by other arbitrary constants.
The ascending series are always convergent, but when the variable
is large begin by diverging rapidly : the descending series, on the
other hand, are always divergent, but when the variable is large
begin by converging rapidly.

The same method was found to apply to several other definite
integrals which occur in physical investigations, as well as to
differential equations of frequent occurrence. The ascending and
descending series are usually both required, the one for application
to small, the other to large values of the variable; and it is
necessary to connect the arbitrary constants in the descending
with those in the ascending series. The analytical determination
of the arbitrary constants by which the divergent series are
multiplied forms the chief difficulty, a difficulty only partially

[* Ante, Vol. ii, p. 329.]

284 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

overcome in the paper to which I allude. Some years later
I resumed the subject, and I then succeeded in overcoming the
remaining difficulty. The result is given in the paper* to which
the present is a supplement. It opens out some very curious
points of analysis, and seems to me to throw new light on the
theory of divergent series.

In the latter paper it is shewn that the constants in the
descending series are in reality discontinuous, retaining the same
value between values of the amplitude of the imaginary variable
comprised within certain limits, and then suddenly assuming
different values ; and a method is given of determining the critical
values of the amplitude at which the transition takes place. As
shewn in the paper, it is not essential for the application of the
method that we should be able to determine analytically the
relations between the constants in the ascending and descending
series ; we may determine the numerical relations by numerical
calculation, by calculating from the ascending and descending
series separately, and equating the results. With the exception of
the comparatively simple integral first discussed, with a view to
paving the way to the application of the method to more difficult
examples, all the integrals discussed in the paper give rise to
differential equations of the second order; and it often happens
that it is the differential equations themselves, and not the definite
integrals which are particular solutions of them, that present
themselves for treatment. As there are two arbitrary constants
regarded as unknown which have to be determined in terms
of two regarded as known, we require two equations. These
would be obtained by calculating the dependent variable for two
different values of the independent variable from the ascending
and descending series separately, and equating the results. We
should then have to solve two simple equations by which the two
unknown constants are determined.

The object of the present supplement is to shew that by a
simple application of the principles laid down in the paper itself
the equation expressing the equivalence of the two different
developments (by ascending and descending series) of the de-
pendent variable may be split into two previously to the deter-

* Cambridije Philosophical Transactions, Vol. x, p. 105. [Ante, p. 77 ; see
footnote, p. 80.]

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 285

mination of the arbitrary constants; and accordingly, in order
to determine the two arbitrary constants which are regarded as
unknown, that it is sufificient to calculate the dependent variable
for one value of the independent, from the ascending and descend-
ing series separately, and equate the results. Moreover, the
necessity of eliminating between two simple equations will thus
be avoided, which involves a saving of numerical calculation (in
case we should be unable to determine the unknown constants
analytically), which is not to be despised, seeing that the coefficients
involved are complex imaginaries.

With the exception of one comparatively simple case, all the
differential equations considered in my former paper, whether they
present themselves directly, claiming the consideration of their
complete integrals, or as equations which,certain definite integrals
that we have to deal with satisfy, and of which those integrals are
particular solutions, are particular cases of the differential equa-
tion

J + ^i + ^^^^^^2/ = o (1),

which in the present supplement I propose to consider in its
generality.

The four constants A, B, G, a in this equation are easily re-
duced to one. By writing aj^/t^+o) for x (the case a. = — 2 need not
be considered) we may get rid of *■ in the last term. By then
writing ex for x we may convert the last coefficient into any
constant we please. Lastly, by writing x'^y for y we may deter-
mine a so as to convert the coefficient of either the second or the
third term into zero or any constant we please. I will take as the
canonical form of the equation

d'y 1 dy n"" ,.,

di'^xTx-x^y^y • (')■

As n may be supposed to be a general constant, capable of
being imaginary, no generality is lost by assuming the above as
the canonical form. Nevertheless in all the applications of equa-
tion (1) that I know of, the coefficient of the third term when the
equation is reduced to the form (2) is either zero or negative, so
that n is either zero or a finite real quantity, which may be taken

286 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

positive. Accordingly in discussing the equation (2) I will mainly
consider the case in which n is real. The equation (1) being now
done with, the letters A, B, 0, a are set free for fresh use.

The complete integral of (2) in ascending series, obtained by
the usual methods, is

2/ = ^*" 1 + 979X9;^ +

2(2 + 2?i) 2 . 4 (2 + 2w)(4 + 2??.)

"^ 2 . 4 . 6 (2 + 2ji)(4 + 2n)(6 + 2w)

+ .

+ £«-» \l + ^r-7^ — s-x +

2 (2 - 2w) 2 . 4 (2 - 2m) (4 - 2n)

X 2T476~(2 - 2n) (4 - 2w)(6 - 2n) "^ " ' "

,(3).

When n is zero or any integer this integral takes a particular
form, which however, being a limiting and therefore particular
case of the integral for general values of n, need not at present
be separately considered.

The complete integral of (2) expressed in a form which is
convenient for calculation when x is large, that is, has a large
modulus, obtained as in my former papers, is

r, w(, P-4n2 (P-4nn(32-49in

(12 _ 4^2) (32 _ 4;^^) (52 _ 4^2)
"*" 1.2.3 (SscY '^'

(12 _ 4^2) (-32 _ 4n.2)(52 _ 4^2) I

1.2.S(8xy +...|...(4).

This expression takes no peculiar form when n is integral.
The series terminate whenever 2m is an odd integer 2i + l, in
which case (2) is the transformation of the well-known integrable
form

- dx' X dx x' "^ ^'

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 287

To render everything definite it will be necessary to specify the
values of a;"^" and a~^ which are supposed to be taken. Let*

a: = p (cos 6 + I sin 6) (5),

and starting from a real positive value of x, for which 6 is taken =0,
let any other value be deemed to be arrived at by continuous
variations of p and 6 without suffering p to vanish. The ampli-
tude being thus defined, and accordingly amplitudes such as a and
277 -f a being distinguished, notwithstanding that the numerical
value of w is the same in the two cases, I will take «" to be

jo" (cos nd + I sin nO),
and not

/a" {cos (n6 + 2imr) + i sin (nd + 2imr)}

where i is an integer different from zero ; and similarly with
respect to a;~" and ari. And the problem, to effect a slight im-
provement in the solution of which is the object of the present
paper, consists in finding the two relations which connect C, D with
A,B.

It was shewn in my former paper that the constants G, D are
in general discontinuous, changing their values when the ampli-
tude of a; passes, for the first through an odd, for the second
through an even multiple of tr. Let C, G" ... be what G becomes
when 6 passes through 17,^17 ... and D', D"... what D becomes
when 6 passes through 27r, 4sir .... Let 6' be an angle lying
between and tt, and for 6=6' let U, V, u, v denote the functions
multiplied hy A, B, C, D in (3), (4). We see from (3), (4) that
when 6 is increased by tt the functions multiplied hy A, B are
reproduced, except as to the constant multiplier e*"^' or e~^'"-
introduced, and the functions multiplied by G, D reproduce
each other, except as to the multiplier e"*"' introduced. Hence
supposing 6 = 6' + iir we have

for 0<6'<7r, y = AU+BV =Gu+Dv, \

for Tr<e<2'7r, y = Ae'^'^U +Be-"'"-V ^G'e-i'^v +I>e-i'"-u,\ (6),
for 27r< 0< Stt, y=u4e2"'C/-|-5e-2«"F= C'e-'"-u + D'e-"-v,]

and so on indefinitely, the expressions admitting of being continued
backwards for negative values of in a perfectly similar manner,
the constants G, D being changed alternately.

[* J -\ has been replaced by i throughout.]

288 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

On account of the linearity of (2), the constants G, D must be
linear functions of A,B, vanishing with them. Let then

G=pA^-qB, D=rA + sB v-W-

Now it follows fi'om (6) that De-i^^, C'e-*"" are composed of
^e»", ^e-""' as C,D are QiA,B\ that O'e""', D'e-"" are composed
of A e'^""', Be-^'^" as C, D oi A, B, and so on. Hence we have

Be^i^' = pAe"""- + qBe-""", \
C'e-i'' = rAe"""- + sBe-"-"', ( .o\

C'e-"' = pAe^^"' + qBe-^"""', >
B'e""- = r.ie^""' + s^e'^""' 1
and so on. Comparing these equations with (7), we have

J) = ^e(i+»)'" A + ge'*-")"" B = rA-^ sB,
which since A and B are independent gives

r = e(*+">'"p, s= et^""''"^ (9).

The comparison of the second and third of equations (8) leads
to the same result, and we have

C=pA + qB,

D =jpe(i+»)'" A + gett""'"'5,

and so on, an additional factor e'^+™"" or e'^~^'"'^' being introduced
whenever the increase of through an odd or even multiple of tt
changes C or D. It is easily seen that the same law applies when
6 increases negatively, the factor e"'!*™'"' being in this case intro-
duced. The determination of the series of constants C, C, C", : . .
D, D', £>", ... in terms of the arbitrary constants A, B is thus
reduced to that of the two constants p, q, which depend on n only.
In the integrable case, namely, that in which 2n is an odd integer,
we may see a priori that there can be no discontinuity in the
constants G, D, and accordingly in this case the factor e(i*«)"-'
is equal to 1.

It was shewn in my former paper that the discontinuity in the
expression of a continuous function by means of divergent series is
reconciled with the continuity of the function expressed, by the
circumstance that the part of the function which contains a

.(10),

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 289

constant coefficient that alters discontinuously, which part may be
expressed by a divergent series or in finite terms, is accompanied
by another part, expressed by means of a divergent series, which
in the neighbourhood of the critical value of the amplitude of the
variable becomes subject to a certain amount of vagueness, so that
its numerical value can only be obtained subject to a certain
amount of uncertainty, comparable in amount with the whole
value of the part of the function which contains the constant that
alters discontinuously*. Hence if there be no such accompanying
function the constant that is liable to change discontinuously as 6
passes through its critical value cannot do so. This conclusion is
easily verified by equations (10), which give

C — G = 2i cos mr.D,

so that C =0 if D = 0. We may notice that if D be real, the
discontinuity in the constant C as 6 passes through tt affects only
its imaginary part.

The first of equations (6) combined with the first two of (10)
gives, on account of the independence oi A, B,

!/■ = p (it +e<i+"i "'?))]

F=g(M + e<i-^'-^)j ^ ''■

Hence previously to the determination, whether analytical or
numerical, of the arbitrary constants in the descending series, we
are able to tell what combination of functions u, v represents,
except as to a constant multiplier, each of the functions U, V.
Moreover p, q, on which the complete determination of the arbitrary
constants has now been made to depend, are given separately by
the two equations (11), and may be determined by calculating the
four functions U, V. u, v for one value of the variable x.

[* This explanation may perhaps be put more concisely in the form that each
of the ' semi-convergent series ' or ' asymptotic expansions,' above referred to, ceases
to be ' uniformly ' semi-convergent at certain loci on the diagram of the complex
variable, in crossing which its value may in consequence change discontinuously.
The special relation in which asymptotic solutions stand to the regular solutions of
linear differential equations is of course involved in this remark. The nature of
the general phenomenon of non-uniform convergence, described in the text, had
been elucidated by Prof. Stokes, as is now well-known, as early as 1847. ' Cf.
■ante, Vol. i, pp. 279—286; also Vol. iv, p. 80.]

s. IV. 19

290 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

If Xi be the (i+l)th term in the first series within brackets

m

(3)

y _ a^ 1.3.5...(2i'-l)

*~1.2.3...2i^ (2 + 2n)(4 + 2«) ... (2i + 2^)

■•c" T(n+l)V(i + ^) _
1.2...2iT{^)r{i + n + l)'

IT

and ^±+M-(i±I) = 2 f\sin 4,r {cos <l>rd<p ...(12),

(i+n + l) Jo ^

which gives for the first part of y

The second part may be got by writing -n for n and J5 for ^,
provided w<4, so that equation (12) may hold good. Hence
subject to this restriction

IT

j^ B Ia^ — "^ 01-^ (sin <^)-2»[ (e^c^s* + g-rtcos*) ^^^ _ . .(13),

r(j — w) )

where tt^ has been written for V (^). This form of the integi-al
of (2) is already known.

We have now to connect the constants in the ascending and
descending series, which may be done by the intervention of the
integral (13) expressed in the form of definite integrals. This
may be effected precisely as in my former papers by seeking the
limit of the right-hand member of (13), when x increases in-
definitely ; only, it will suffice to do this for one amplitude of x.
Let then ^ = 0, and suppose x to increase indefinitely. Then
ultimately the difference between the whole integral from to
\-rr and the part from to <^i, where ^i is a positive quantity as
small as we please, vanishes compared with either ; and writing ^
for sin ^, 1 — | <^^ for cos ^, we shall have for the limit required
that of

Jo i r(^-l- n) ^ V (y\-n) ^ J ^

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 291

Putting ^xcj}^= <^'2, we shall ultimately have and oo for the limits
of the integral, however small ^, may have been taken, and we
shall have for the limit of y

+ B Efc^. x-i 2-«+J .^'-^"^ I e-*'^ d(f>',

i (,2 — »J) J

or 2/= ^^e^{^2"r(l+?i)+52^r(l-«)j.

V27ra;

Comparing with (4), in which ultimately

y = Cx~i e",
we have

C=~{2''T{l + n)A+2-«T(l-n)B} (14);

v27r

whence by comparison with the first of equations (10),

and then from the second of equations (10),

D = -^ e -i" {e""' 2«r(l + n)A +e-'"^' 2-»r(l - n)B} . . .(16).
V27r

C and D being thus determined from ^ = to d = ir, the rule
already given determines the constants in the descending series for
all amplitudes. It must be remembered however that in the
establishment of the formulae (14), (16) n has been restricted to be
less than ^. The series (3), (4) are of course subject to no such
restriction.

With the view of determining the arbitrary constants in the
divergent series when n" > ^, let us seek to make the integration
of (2) depend on that of a similar equation with a different value
of n. Assume

y = x'^Jx^zdx,

where a, /3 are disposable constants. Substituting in (2) we
have

a;a+p ^ + (2a + /3 + 1) ai-'+^-i z + (a.'- n') a^-^Jx^zdx = x^Jx^zdx.
dx

19—2

292 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

Assume o? — t? = .....(17),

divide by «", differentiate, and then divide by aP, and we have

^,+ (2« + 2^+l)-^+(/3-l)(2a+/3 + l)-, = .;

• and in order that this may be of the form (2) we must have

2a + 2/3 + 1 = 1, or /3 = - a,

when the last equation becomes

d'z ^ \dz (1 + ay .^, .jgs

dx^ (c dx a? "" ^'

and the relation between y and z is

y = x'^^x~'^zda, or z = af--rX~°-y (19).

Since from (17) a.= ±n, (18) differs from (2) in having n increased
or diminished by 1; and by (19) the integral of (2) is deduced
from that of (18) by integration, or the integral of (18) from that
of (2) by differentiation. Suppose we choose differentiation, and
wish n to be lowered in the deduced equation. Then we must
take (18) for the deduced equation, and put a. = — n. Suppose n
to lie between i — ^ and i + \, where i is a positive integer, and
repeat the process i times. Then we shall have

dx ' dx ' '" dx "'

or ^ = *""''' £0''"^ ^^^^'

z being the complete integral of

d^ \dz {n-if

dx^ X dx x^

The/orai of the integral of the deduced equation (21) may be
got at once from that of (2) by writing n — i for n, and in deducing
the integral by the formula (20), with a view to connect the
arbitrary constants A, B, G, D with those in the integral of a
similar equation in which n falls within the previously prescribed
limits, the two relations among which have been already found, we
need attend only to the leading terms.

The formula (20) applied to the leading term of the first series
in (3) gives

2w (2w - 2) (2w - 4) . . . (2w - 2i + 2) Aaf'-\

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 293

or ^'-^(l+^Ax'^'.

1 (1 +n — i)

On applying the formula to the second series, the initial terms
successively disappear on differentiation. The first which remains
is that which contains x~^+^, which produces

fir or 2-' ^ -^ 7?a!-i»-*i

(2-2»)(4-2n)...(2t--2«)' V{\-n+i)

provided we take V{m) for m negative to be defined by the
equation

r(m + l) = mr(m),

which being repeated a sufficient number of times reduces the
r of a negative quantity to the previously defined F of a positive
quantity.

On applying (19) to (4), we shall evidently get the leading
terms by differentiating the exponentials only. We thus get

Gx-i^ and D{-iy'x-ie-'';

and therefore we have for the complete integral of (21)

T{\+n — i) ^ ' r(l — w+*)

= Gx-i^ (1 +...} + D(- Vfx-ie-^il -...].

Since n—i lies between the limits — ^ and + \, the formulae
already investigated will give the two relations between the

constants. We have merely to write 2*=nT^ tvA, etc. in

place of A, etc., and n — iin place of n. We get thus from (14)

G = ~ |2»-T(1 + n-i).2i X?^'^'''' ■\ ^
V27rl V{l + n-i)

+ 2-«+T(l-n + ^•).2-JiXL_^5}

= -^ {2»r(l + n)A+ 2-»r(l - w) B],

so that (14) and the equations (15), (16) which follow from it,
which were at first proved subject to the restriction 1 > 2?i > — 1,
are now set free from that restriction, and shown to hold good
for all real values of n however great.

294 ON THE DISCONTINUITY OF ARBITRARY CONSTANTS

When n is imaginary, when for example the coefficient of the
third term in (2) is real but positive, we see that the attempt to
connect analytically G, D with A,B would lead to the introduction
of r functions of imaginary variables. As the function T has not
been investigated and tabulated for imaginary values of the
variable, we see that nothing would be gained by the attempt, and
we should be obliged to have recourse to numerical calculation.
In such case the formulae (10), (11) would render important
service. However in all actual investigations, so far as I am aware,
the coefficient is either zero or a negative quantity, so that n
is real*.

The solution of the problem proposed may now be deemed
complete, but as equations of the form (2) with n integral are of
frequent occurrence, it may be well actually to work out the special
forms assumed by several of our equations in this case.

Take first the case of n=0. Kepresent for shortness equa-
tion (3) by

y = Af(n) + Bf(-n),

and expand /(«) and/(— n) according to powers of n. Then
y = A{f(0)+f'(0)n + ...}+B{f(0)-f'iO)n + ...].

Put A+B = Ai, (A-B)n = B, (22),

so that

2V^'+¥j' ^ = 2l^-¥j (2^)'

and then make n vanish. We get y = Aif(0)+Bjf'(0), or putting
for/(0) andy"'(0) their values,

y = (4, + i^ilog*)(l + |+2^ + 2j^j-g^+...)

-A(|s. + 2^,5f. + 2r|J-g,'Ss+-) (24),

where

s=i+|+i..+}.

12 3 I

Substituting in (14) from (23), and then making n vanish, we get

C = -^{A, + {\og2-y)B,} (25),

[* Functions of complex order occurred in the problem of a cylindrical pendulum
ante, Vol. iii, p. 38.]

WHICH APPEAR IN DIVERGENT DEVELOPMENTS. 295

where 7 is Euler's constant "57721566 . . . the limit of Sx — log so for
x='cc, to which as we know — r'(l) is equal. The values of
D, C, B', etc., may be obtained in a similar manner as limits, or
deduced at once from C in the manner before explained. The
latter course appears to be the shorter. If TJ, V be the functions
multiplied by A-^, B^ in (24) for < ^ < tt, we see that when 6 is
increased by tt, U recurs, and V becomes V+inU. Hence from
(6), of which the right-hand members remain unchanged, we see
that D is composed of i (^i + ttiBi), iBi as C of A^, B^. ' Hence

Z)=-|=M,+ [<(log2--y)-,r]5a (26).

v27r

Equations (25), (26) agree with the equations (41) of my former
paper, the constants G, D having been interchanged in the present
supplement to make the law of progress of equations (10) more
evident. The agreement is proved by the known relation

7r-ir'(i) + log4 + 7 = 0.

There would be no difficulty after what precedes in forming
the expressions for C, G" ... U, D"..., but I forbear to do so, as it
is a matter of curiosity rather than utility.

Take now the case in which n is any positive integer i, and
first suppose n = i + S, where S is a small quantity which in the
end will be made to vanish. Let (3) be denoted by

y = Af{n) + p{n),

and putting n = i + 8 expand /(n), F{n) according to powers of S.
Then observing that

^'^^^ " 2T2M[rT7(2i^^2)^2i-^^*'''
we have

y=A\f(i)+f'ii)h + ...]

^^ ■X2)^2/ (-) + ^'(-)^-^-}-

S (2.2^4^..(2^-2)^2^•'
S 2.2^4^..(2i-2)22i

Let ^ + l.r^.r#^^t-^^wT7 = ^„ (27),

296 ON THE DISCONTINUITY OF AEBITEARY CONSTANTS

and after eliminating A make S vanish. Then

We have

■/■(^>^'{l + 2pT2i) + 2.4(2/2tX4 + 2i)--j-"^^^'*'
f'(i) = log ^ ./(i) - x^ y^^^ (Si+. - Si)

I + 2.4(2+2i)(4+2t) *^^'+^ -^i)-\

= (log a; + Si)f(i) - x' ^Si + 2^2^20 ^'+'

2.4(2 + 2t)(4+20'

+ ^ ^ /t> . f^-x/^ , c^/\ 'S'i+2 +

■■■■^2.4...(2i-2)(2m-2)(2w-4)...(2n-2i + 2)J

1 (-l)'a;^-"'

"^ 2 2 . 4 ... 2i; (2n - 2i + 2)(2w- 2i + 4) ... (2m - 2)

h

+ 7?r-

( (2i + 2)(2t + 2 - 2??.)

1^ )

■•■ (2i + 2)(2?; + 4)(2i+ 2 - 2n){2i + 4 - 2?i)"'"""j '

Hence differentiating with respect to n, and then putting n = i, we
have

F'{i)=ar-\l---——r + ...+ ■ ^ ^

2(2i-2) 2.4...(2i-2)(2i-2)(2i-4)...2

(- 1)'"*"'
+ 2.2^4^..(2t-2y 27 ^^°S ^ + '^^-)'^(^')

2 . 2^ . . (2t - 2)= 2i (2 (2 + 2t) ' ^ 2 . 4 (2 + 2i)(4 + 2t)

WHICH APPEAR IN DIVERGENT DEVELOPMENTS.

We have therefore finally

297

y = AJ{i)+Ba

.{.-

+

+ B

{-If

i+i

2.2^4^.,(2i:-2)^2i

2(2i-2) ' 2.4(2i-2)(2i-4)""

(- 1)^' a?''-"

"*" 2.4 ... (2t - 2)(2i - 2){2i - 4) ... 2

x"
^°^2(2 + 2i)'

+ ;

-(81+2—82) + .

..(30),

2.4(2 + 2i)(4+,2t)'+'

where So(= 0) has been inserted merely for the sake of regularity.
Such is the special form assumed by (3) in this case,/(i) being
defined by (29). It should be observed that when i = l the de-
nominator 2 . 2=* . . . (2i — 2)^ 2i is simply 2 x 2 or 4.

To connect C with 4„, B we must eliminate A from (14) by
means of (27), and then pass to the limit by making n = i. We
have

^=vfc^'"^^'+"^^" +

2-"r(l ::) (-l)^2"r(l + n)
^ ' S.2.2^..(2^■-2)=2^_

B\.

Now

V{l-n):

T{l+i-n)

'(\-n){2-n)...{i-n)

{-lfT{l + i-n)

~ h {n + \ - i){n + 2 - i) ... {11 -1)
(- 1)^ r (1 4- ^^ - w) r(l + w - i)
Sr{n)
which gives for the coefi&cient of B within the parentheses

(- 1)' ( 2-"r(l + i - v)T(\ +n-i) _ 2-"'^-"'r(l + w) ]
~S~| I>) T(i)r(i + 1) )

or ultimately

JZ^lfT" ,. limit i [T(i} r(i + 1) 2-« r(i - 8) r (1 + S)

{r(i)YTii + i)

-2«^(^ + S)^(^• + l + S)}

■ 2T(i)

■ 2T(i)

{log4+A'(t) + A'(i + l)}
{log4 + -Sfi_i + .Si-27},

298 ON DISCONTINUITY OF ARBITRARY CONSTANTS.

A'(^■) being the derivative of A(i) or logr(i), the value of which is
Si-i — 7. We have then in the limit

C = ^ |2'T(1 + i) A„ + ^^^ (log 4 +5f,_. +8,- 27) S} (31).

To get D let 6 be first comprised between and tt, and be
then increased by tt. We see from (29), (30) that setting aside
the augmentation of log x by tti the functions multiplied by J.,,,
B will be reproduced, with or without change of sign, according as
i is odd or even. And the augmentation of log a; will have the
same effect as increasing A,, by

(- 1)*+1 2-("-l) TTl

T{i)ni + \)

Hence — t-D will be composed of

B.

as is of A,,, B. Hence

//>

+ ^ [27r - (log 4 + ;Sf,_, + Si - 27) ^1 . . .(32).

Equations (30), (31), (32) may be simplified a little by in-
cluding

(- \y+^ 2-^'

in A.,.

{8i_, + Si)B

Postscript. — After the above paper was read, my attention
was called by Professor Cayley to a paper by Professor Kummer,
published in Crelle's Journal*, containing remarkable general
transformations of which those of the present paper, considered
apart from the question of the discontinuity of the constants
involved, are merely particular cases. The discontinuity of the
constants, however, the investigation of the nature and conse-
quences of which forms the main object of the present paper and
of that to which it is a supplement, has not been treated of,
perhaps not even perceived, by the eminent German mathema-
tician.

* Vol. XV, (1836), pp. 39 and 127. [This analysis of Kummer was utilized
by Kirchhoff, in his memoir on induced magnetism in cylinders, Crelle's Journal
fur Math., xlviii, 1853, Ges. Abhandl. p. 196, two years subsequent to Prof. Stokes'
memoir on pendulums {ante. Vol. iii, p. 1), to connect the various expansions of
Bessel functions of real argument. See also supra, p. 80, footnote.]

On the Communication of Vibration from a Vibrating
Body to a surrounding Gas.

[From the Philosophical Transactions for 1868. Received and read June 18.]

[Abstract, from Proceedings of the Royal Society, xvi, pp. 470—471.]

In the first vohime of the Transactions of the Cambridge Philosophical
Society will be found a paper by the late Professor John Leslie, describing
some curious experiments which show the singular incapacity of hydrogen
either pure or mixed with air, for receiving and conveying vibrations from
a bell rung in the gas. The facts elicited by these experiments seem not
hitherto to have received a satisfactory explanation.

It occurred to the author of the present paper that they admitted of
a ready explanation as a consequence of the high velocity of propagation of
sound in hydrogen gas operating in a peculiar way. When a body is slowly
moved to and fro in any gas, the gas behaves almost exactly like an incom-
pressible fluid, and there is merely a local reciprocating motion of the gas
from the anterior to the posterior region, and back again in the opposite
phase of the body's motion, in which the region that had been anterior
becomes posterior. If the rate of alternation of the body's motion be taken
greater and greater, or, in other words, the periodic time less and less, the
condensation and rarefaction of the gas, which in the first instance was
utterly insensible, presently becomes sensible, and sound-waves (or waves of
the same nature in case the periodic time be beyond the limits of audibility)
are produced, and exist along with the local reciprocating flow. As the
periodic time is diminished, more and more of the encroachment of the
vibrating body on the gas goes to produce a true sound-wave, less and less
a mere local reciprocating flow. For a given periodic time, and given size,
form, and mode of vibration of the vibrating body, the gas behaves so much
the more nearly like an incompressible fluid as the velocity of propagation of
sound in it is greater ; and on this account the intensity of the sonorous
vibrations excited in air as compared with hydrogen may be vastly greater
than corresponds merely with the difference of density of the two gases.

It is only for a few simple geometrical forms of the vibrating body that
the solution of the problem of determining the motion produced in the gas
can actually be efiected. The author has given the solution in the two cases
of a vibrating sphere and of an infinite cylinder, the motion in the latter case

300 ON THE COMMUNICATION OF VIBRATION

being supposed to take place in two dimensions. The former is taken as the
representative of a bell ; the latter is applied to the case of a vibrating string
or wire. In the case of the sphere, the numerical results amply establish the
adequacy of the cause here considered to account for the results obtained by
Leslie. In the case of the cyhnder they give an exalted idea of the necessity
of sounding-boards in stringed instruments ; and the theory is further appUed
to the explanation of one or two interesting phenomena.

In. the first volume of the Transactions of the Cambridge
Philosophical Society is a short paper by Professor John Leslie,
" On Sounds excited in Hydrogen Gas," iu which the author
mentions some remarkable experiments indicating the singular
incapacity of hydrogen for becoming the vehicle of the transmis-
sion of sound when a bell is struck in that gas, either pure or
mixed with air. With reference to the most striking of his ex-
periments the author observes (p. 267), "The most remarkable
fact is, that the admixture of hydrogen gas with atmospheric air
has a predominant influence in blunting or stifling sound. If one
half of the volume of atmospheric air be extracted [from the
receiver of the air-pump], and hydrogen gas be admitted to fill
the vacant space, the sound will now become scarcely audible."

No definite explanation of the results is given, but with
reference to the feebleness of sound in hydrogen the author ob-
serves, "These facts, I think, depend partly on the tenuity of
hydrogen gas, and partly on the rapidity with which the pulsations
of sound are conveyed through this very elastic medium"; and he
states that, according to his view, he " should expect the intensity
of sound to be diminished 100 times, or in the compound ratio
of its tenuity and of the square of the velocity with which it
conveys the vibratory impressions." With reference to the effect
of the admixture of hydrogen with air he says, " When hydrogen
gas is mixed with common air, it probably does not intimately
combine, but dissipates the pulsatory impressions before the sound
is vigorously formed."

In referring to Leslie's experiment in which a half-exhausted
receiver is filled up with hydrogen. Sir John Herschel suggests a
possible explanation founded on Dalton's hypothesis that every
gas acts as a vacuum towards every other*. According to this

* Encyclopaedia Metropolitana, Vol. iv, Art. Sound, § 108.

FROM A VIBEATING BODY TO A SURROUNDING GAS. 301

view there is a constant tendency for sound-waves to be propa-
gated with different velocities in the air and hydrogen of which
the mixture consists, but this tendency is constantly checked by
the resistance which one gas opposes to the passage of another,
calling into play something analogous to internal friction, whereby
the sound-vibration though at first produced is rapidly stifled.
Air itself indeed is a mixture; but the velocities of propaga-
tion of sound in nitrogen and oxygen are so nearly equal that the
effect is supposed not to be sensible in this case.

This explanation never satisfied me, believing, as I always have
done, for reasons which it would take too long here to explain, that
for purely hydrodynamical phenomena (such as those of sound) an
intimate mixture of gases was equivalent to a single homogeneous
medium. I had some idea of repeating the experiment, thinking
that possibly Leslie might not have allowed sufficient time for the
gases to be perfectly mixed, though that did not appear likely,
when another explanation occurred to me, which immediately
struck me as being in all probability the true one.

In reading some years ago an investigation of Mr Earnshaw's,
in which a certain result relating to the propagation of sound in
a straight tube was expressed in terms among other things of the
velocity of propagation, the idea occurred to me that the high
velocity of propagation of sound in hydrogen would account for
the result of Leslie's experiment, though in a manner altogether
different from anything relating to the propagation of sound in
one dimension only.

Suppose a person to move his hand to and fro through a small
space. The motion which is occasioned in the air is almost
exactly the same as it would have been if the air had been an
incompressible fluid. There is a mere local reciprocating motion,
in which the air immediately in front is pushed forwards, and that
immediately behind impelled after the moving body, while in the
anterior space generally the air recedes from the encroachment of
the moving body, and in the posterior space generally flows in
from all sides, to supply the vacuum which tends to be created ;
so that in lateral directions the motion of the fluid is backwards,
a portion of the excess of fluid in the front going to supply the
deficiency behind. Now conceive the periodic time of the motion
to be continually diminished. Gradually the alternation of move-

302 ON THE COMMUNICATION OF VIBEATION

meiit becomes too rapid to permit of the full establishment of the
merely local reciprocating flow ; the air is sensibly compressed and
rarefied, and a sensible sound-wave (or wave of the same nature,
in case the periodic time be beyond the limits suitable to hearing)
is propagated to a distance. The same takes place in any gas ;
and the more rapid be the propagation of condensations and rare-
factions in the gas, the more nearly will it approach, in relation
to the motions we have under consideration, to the condition of
an incompressible fluid; the more nearly will the conditions of
the displacement of the gas at the surface of the solid be satisfied
by a merely local reciprocating flow.

This explanation when once it suggested itself seemed so
simple and obvious that I could not doubt that it afforded the
true mode of accounting for the phenomenon. It remained only
to test the accuracy of the assigned cause by actual numerical
calculation in some case or cases sufficiently simple to permit of
precise analytical determination. The result of calculations of
the kind applied to a sphere proved that the assigned cause was
abundantly sufficient to account for the observed result. I have
not hitherto published these results ; but as the phenomenon has
not to my knowledge been satisfactorily explained by others,
I venture to hope that the explanation I have to offer, simple as
it is in principle, may not be unworthy of the notice of the Royal
Society.

For the purpose of exact analytical investigation I have taken
the two cases of a vibrating sphere and a long vibrating cylinder,
the motion of the fluid in the latter case being supposed to be in
two dimensions. The sphere is chosen as the best representative
of a bell, among the few geometrical forms of body for which the
problem can be solved. The cylinder is chosen as the representa-
tive of a vibrating string. In the case of the sphere the problem
is identical with that solved by Poisson in his memoir " Sur les
mouvements simultands d'un pendule et de I'air environnant," * but
the solution is discussed with a totally diff'erent object in view,
and is obtained from the beginning, to avoid the needless complexity
introduced by taking account of the initial circumstances, instead
of supposing the motion already going on.

* Memoires de VAcadSmie des Sciences, t. xi, p. 521.

FROM A VIBRATING BODY TO A SURROUNDING GAS. 303

A. Solution of the Problem in the case of a Vibrating Sphere.

Suppose an elastic solid, spherical externally in its undisturbed
position, to vibrate in the manner of a bell, the amplitude of
vibration being very small. Suppose it surrounded by a homo-
geneous gas, which is at rest except in so far as it is set in motion
by the sphere ; and let it be required to determine the motion of
the gas in terms of that of the sphere supposed given. We may
evidently for the purposes of the present problem suppose the gas
not to be subject to the action of external forces.

Let the gas be referred to the rectangular axes of x, y, z, and
let u, V, w be the components of the velocity. Since the gas is at
rest except as to the disturbance communicated to it from the
sphere, u, v, w are by a well-known theorem the partial differential
coefficients with respect to x, y, 2 oi a, function (f> of the coordi-
nates ; and if a^ be the constant expressing the ratio of the small
variations of pressure to the corresponding small variations of
density, we must have

dt'~ [dx^'^dy^^ dz'J ^^'

and if s be the small condensation,

-_1 #

^ - a' df

As we have to deal with a sphere, it will be convenient to refer
the gas to polar coordinates r, 6, a, the origin being in the cen|;re.
In terms of these coordinates, equation (1) becomes

d^d) „ [d^A 2dd) 1 d ( . ^d4>\ ^ 1 d^4>

df ~ '^ \dr^ r dr ' r' sin e d0 v"^ dd ) '^ r' sin' d(o']

(2);

and if u, v', w be the components of the velocity along the
radius vector and in two directions perpendicular to the radius
vector, the first in and the second perpendicular to the plane in
which Q is measured,

, di> , 1 dd} , 1 d^

dr r do rsmU dco ^ ^

304 ON THE COMMUNICATION OF VIBRATION

Let c be the radius of the sphere, and V the velocity of any
point of its surface resolved in a direction normal to the surface,
V being a given function of t, 6, and co ; then we must have

-,- = V, when r=c (4).

dr

Another condition, arising from what takes place at a great
distance from the sphere, will be considered presently.

The sphere vibrating under the action of its elastic forces, its
motion will be periodic, expressed so far as the time is concerned
partly by the sine and partly by the cosine of an angle proportional
to the time, suppose mat. Actually the vibrations will slowly die
away, in consequence partly of the imperfect elasticity of the
sphere, partly of communication of motion to tho gas, but for our
present purpose this need not be taken into account. Moreover
there will in general be a series of periodic disturbances co-
existing, corresponding to different periodic times, but these may
be considered separately. We might therefore assume

V= Usin mat + V cos inat,

but it will materially shorten the investigation to employ an
imaginary exponential instead of circular functions. If we take

V= fTe™"* (5),

where i is an abbreviation for V- 1, and determine by the con-
ditions of the problem, the real and imaginary parts of <j) and V
must satisfy all those conditions separately ; and therefore we may
take the real parts alone, or the coefficients of i or V — 1 in the
imaginary parts, or any linear combination of these even after
having changed the arbitrary constants which enter into the
expression of the motion of the sphere, as the solution of the
problem, according to the way in which we conceive the given
quantity V expressed.

The function ^ will be periodic in a similar manner to V, so
that we may take

^=-i/fe™»' (6).

As regards the periodicity merely, wo might have had a term
involving g-'™"* as well as that written above; but it will })(;
readily seen that in order to satisfy the equation of condition (4)

FROM A VIBRATING BODY TO A SURROUNDING GAS. 305

the sign of the index of the exponential in 4> must be the same
as in F.

On substituting in (2) the expression for <j) given by (6), the
factor involving t will divide out, and we shall get for the deter-
mination of •>|r a partial differential equation free from t. Now yjr
may be expanded in a series of Laplace's Functions so that

'ilr = y{r„ + y{r, + ^jr,+ (7).

Substituting in the differential equation just mentioned, taking
account of the fundamental equation

sm da \ do J sm-" 6 dco-

and equating to zero the sum of the Laplace's Functions of the
same order, we find

d'tn 2d±_^_n(n + l) _

dr' +,. dr r' tn + mtn-^.

This equation belongs to a known integrable form. The
integral is

^ [ 2 . tmr 2 . 4 {iviry

+ " ^ r 2 :ivir' + ' 2.4 {imrf ' ' " -J '

iin and u„' being evidently Laplace's Functions of the order n,
since that is the case with ^jr^.

It will be convenient to take next the condition which has to
be satisfied at a great distance from the sphere. When r is very
large the series within braces may be reduced to their first terms
1, and we shall have

The first of these terms denotes a disturbance travelling out-
wards from the centre, the second a disturbance travelling towards
the centre, the amplitude of vibration in both cases, for a given
phase, varying inversely as the distance from the centre. In the
problem before us there is no disturbance travelling towards
the centre, and therefore S«„' = 0, which requires that each
s. IV. 20

306 ON THE COMMUNICATION OF VIBRATION

function «„' should separately be equal to zero. We have there-
fore simply

_wfi n(n + l) (n-l)...(n + 2)
[ Z . imr 2 . 4 (%mrf

1.2.3 ...2W \ ,„-

"'" 2 . 4 . 6 ... 271 (imr)»j ^^ ''

or, if we choose to reverse the series,

.^ 1.3.5. ..(2n-l) r 2?i .
^ {mirp [ 1 .2n

(2n-2)2w ,. ,^ 2.4.6...2?i,. ,J ,_,

+ 1.2(2.-l)2n (^"^'•)^ + - + 1.2.3...2J "'">1 -^^^

Putting for shortness /„ (r) for the multiplier of Une'^"^ in the
right-hand member of (8) or (9), we shall have

= -e»'»"«^-)SM„/„(r).

It remains to satisfy the equation of condition (4). Put for
shortness

d \l _..™„ . . J 1

so that

F^{r) = {l+ imT)U i'r)-rf^{r) (10),

and suppose U expanded in a series of Laplace's Functions,

then substituting and equating the functions of the same order on
the two sides of the equation, we have

c
and therefore

^=__e^»^(at-r+o,2-^^/„(r) (11).

This expression contains the solution of the problem, and it
remains only to discuss it.

At a great distance from the sphere the function _/„ (r) becomes
ultimately equal to 1, and we have

^= _ ^%i™ ,..-.+. 2 _^^^ (12).

FROM A VIBRATIXG BODY TO A SUEROrXDIXG GAS. 307

It appears from (3) that the component of the velocity along the
radius vector is of the order 7-""^ and that in any direction per-
pendicular to the radius vector of the order r~'^, so that the lateral
motion may be disregarded except in the neighbourhood of the
sphere.

In order to examine the influence of the lateral motion in the
neighbourhood of the sphere, let us compare the actual dis-
turbance at a great distance with what it would have been if all
lateral motion had been prevented, suppose by infinitely thin
conical partitions dividing the fluid into elementary canals, each
bounded by a conical surface having its vertex at the centre.

On this supposition the motion in an}' canal would evidently
be the same as it would be in all directions if the sphere vibrated
by contraction and expansion of the surface, the same all round,
and such that the normal velocity of the surface was the same as
it is at the particular point at which the canal in question abuts
on the surface. Now if TJ were constant the expansion of U
would be reduced to its first term f/o, and seeing that yj, (r) = 1 we
should have from (11)

4>

pirn iat—y-rc)

u.

This expression will apply to any particular canal if we take Ua
to denote the normal velocity at the sphere's surface for that
particular canal ; and therefore to obtain an expression applicable
at once to all the canals we have merely to write U for &„. To
facilitate a comparison with (11) and (12) I shall, however, write
S Un for U. We have then

J,^_'^gimiat-r+c,:=^ (13).

^ r Foic)

It must be remembered that this is merely an expression
applicable at once to all the canals, the motion in each of which
takes place wholly along the radius vector, and accordingly the
expression is not to be differentiated with respect to or to wth
the view of applying the formulae (3).

On comparing (13) with the expression for the function (f> in
the actual motion at a great distance from the sphere (12), we see
that the two are identical with the exception that U„ is divided

20—2

308 ON THE COMMUNICATION OF VIBRATION

by two different constants, namely Ff,(c) in the former case and
Fn (c) in the latter. The same will be true of the leading terms
(or those of the order r""') in the expressions for the condensation
and velocity*. Hence if the mode of vibration of the sphere is
such that the normal velocity of its surface is expressed by a
Laplace's Function of any one order, the disturbance at a great
distance from the sphere will vary from one direction to another
according to the same law as if lateral motion had been prevented,
the amplitude of excursion at a given distance from the centre
varying in both cases as the amplitude of excursion, in a normal
direction, of the surface of the sphere itself. The only difference
is that expressed by the symbolic ratio Fn(c) : Fo(c). If we
suppose Fn (c) reduced to the form /j,„ (cos a» + V — 1 sin a„), the
amplitude of vibration in the actual case will be to that in the
supposed case as ij-„ to /i„^ and the phase in the two cases will
differ by «„— a„.

If the normal velocity of the surface of the sphere be not
expressible by a single Laplace's Function, but only by a series,
finite or infinite, of such functions, the disturbance at a given
great distance from the centre will no longer vary from one
direction to another according to the same law as the normal
velocity of the surface of the sphere, since the modulus /x,, and
likewise the amplitude «„ of the imaginary quantity Fn (c) vary
with the order of the function.

Let us now suppose the disturbance expressed by a Laplace's
Function of some one order, and seek the numerical value of the
alteration of intensity at a distance, produced by the lateral
motion which actually exists.

The intensity will be measured by the vis viva produced in a
given time, and consequently will vary as the density multiplied
by the velocity of propagation multiplied by the square of the
amplitude of vibration. It is the last factor alone that is
different from what it would have been if there had been no lateral

* Of course it would be true if the complete diiierential coefficients with respect
to r of the right-hand members of (12) and (13) were taken, but then the former
does not give the velocity u' except as to its leading term, since (12) has been
deduced from the exact expression (11) by reducing /„ (r) to its first term 1 ; nor
again is it true, except as to terms of the order r~', of the actual motion of the
unimpeded fluid that the whole velocity is in the direction of the radius vector.

FROM A VIBRATIXG BODY TO A JUKKOUXPIXG GAj. 309

motion. The amplitude is altered in rhe proportion of ^l vo u .
>o that if

/, -< rhe quantity by which the inreiisiry which wotild have
exured if the ttaid had been hiadered fi\»m lateral motion hsis ro
be divided.

For the fc: tive :\i , -.-s the values of the function /"» (^.-^ ;vre as

f V [.;.WS :_

/';.(C)=»MC+ 1.

.I

r,(e)=jm'.-4- 2 + v^,
tmc

Q

-f i U" ' = *""-" — 4 — ^ f- —

i;(e^ = iV;.-- :-. — +

60 60

imc {imcY ' {imcY'

r- - . , , 65 240 -52.3 .?2.5

Fac) = tmc- II-- . — . ^ — r, + -.

i»ic (:mo>^ (imcr (ttnc!^

I: \ be the .c'.ic:h of the sc-uud-w.xve c:rrtspoiidini: to the
period of the vibration, m = 2- \, so that nio is the ratio of the
circumreren.'o of the sphere to the !eiic:a of a -nave. If vre
SJ-pp-tse the c-is to be air and \ ro be 2 feet, which would oorre-
spvnd ro a'tx^ur 550 vibrari.ns in a second, and the circumference
2—- ro be 1 r'>?r i a size ;vnd pitch which wotdd correspond with
rhe oase ef a common hcv.se beU), we shah have mc=i. The
follov.hnir Table ^ivcs the V3,lr;es of the sjiuuv of the modulus
a:, i of the r.-.rio /. for the r.rucr:. ns F,,{c) of rhe rirsr five erviers.
for each of the vai.rcs 4 2. 1. ^. and J of mc. It wUl presenrly
appear why rhe Table h;is been exrended further ia the direction
of valv.es ;rrearer than ^ rh;m it has in the oppvvsire duvctioiL
Five si^riii£e;inr f.^rres at leasr are retained.

When mc = x we irer from the ar.ah-tic:\l expressions /, = 1.
We see frvim the Table rhar when mc is s>uiewhar I;irce /, is
liable ro be tx .'.'rr.r less than 1. ;vad i.vnse'.p.ienrly the so.rnd to be
cj li~le rr. ,>re inreirse rhaii if lateral motion had been prevented.
The p.^ssrbilirv of this is erplained by c\.i'asiderinir that the waves
or o--'.iderLSOiriori sp>readi:ic fivm rh'.^sc comparrmenrs of rhe sphere

310

ON THE COMMUNICATION OF VIBRATION

which at a given moment are vibrating positively, i.e. outwards,
after the lapse of a half period may have spread over the neigh-
bouring compartments, which are now in their turn vibrating

mc

n=0

n = l

n = i

H = 3

H=4

■^s

4

17

16-25

14-879

13-848

iO-177

2

5

5

9-3125

80

1495-8

i

1

2

5

89

3965

300137

o

S

0-5

1-25

16-25

1330-2

236191

72086371

3
/«

0-25

1-0625

64062

20878

14837899

18160x10"

>

4
2

1

1

0-95588
1

0-87523
1-8625

0-87459
16

1-1869
299-16

1

1

2-5

44-5

1982-5

150068

01

0-5

1

13

1064-2

188953

57669097

0-25

1

60-294

19650

13965 X 103

17092 X 10«

positively, so that these latter compartments in their outward
motion work against a somewhat greater pressure than if each
compartment had opposite to it only the vibration of the gas
which it had itself occasioned ; and the same explanation applies
mutatis mutandis to the waves of rarefaction. However, the
increase of sound thus occasioned by the existence of lateral
motion is but small in any case, whereas when mc is somewhat
small /„ increases enormously, and the sound becomes a more
nothing compared with what it would have been had lateral motion
been prevented.

The higher be the order of the function, the greater will be
the number of compartments, alternately positive and negative as
to their mode of vibration at a given moment, into which the
surface of the sphere will be divided. We see from the Table
that for a given periodic time as well as radius the value of /„
becomes considerable when n is somewhat high. However practi-
cally vibrations of this kind are produced when the elastic sphere
executes, not its principal, but one of its subordinate vibrations,
the pitch corresponding to which rises with the order of the vibra-
tion, so that m increases with that order. It was for this reason

FEOM A VIBRATiyG BODT TO A SUEEOUXDIXG GAS. 311

that the Table was extended from mc = 0"5 farther in the direction
of high pitch than low pitch, namely, to three octaves higher and
only one octave lower.

When the sphere vibrates symmetrically about the centre,
i.e. so that any two opposite points of the surface are at a given
moment moving with equal velocities in opposite directions, or
more generally when the mode of vibration is such that there is
no change of position of the centre of gravity of the volume,
there is no term of the order 1. For a sphere \-ibrating in the
manner of a bell the principal vibration is that expressed bj- a
term of the order 2. to which I shall now more particularly
attend.

Putting, for shortness, m^c- = q, we have

K-?+l, ^^ = (9*-93-i)^+(4--: =3-2 + - + -,

-~ 9^(9 + 1)

The minimum value of /, is determined by

^_ 6^-84^-54 = 0,

giving approximately

9=12-859, »«;= 3-586, /i:r= 13-859, ^/= 12049. 7, = 86941:

so that the utmost increase of sound produced by lateral motion
amounts to about 15 per cent.

I come now more particularly to Leslie's experiments. Xothing
is stated as to the form, size, or pitch of his bell ; and even if these
had been accurately described, there would have been a good deal
of guesswork in fixing on the size of the sphere which should be
considered the best representative of the bell. Hence all we can
do is to choose such values for m and c as are compai-able with the
probable conditions of the experiment.

I possess a bell, belonging to an old bell-in-air apparatus,
which may probably be somewhat similar to that used by Leslie.
It is nearly hemispherical, the diameter is 196 inch, and the pitch
an octave above the middle C of a piano. Taking the number of
vibrations 1056 per second, and the velocity of soimd in air 1100
feet per second, we have X=12-5 inches. To represent the bell

312 ON THE COMMUNICATION OF VIBRATION

by a sphere of the same radius would be very greatly to under-
rate the influence of local circulation, since near the mouth the
gas has but a little way to get round from the outside to the
inside, or the reverse. To represent it by a sphere of half the
radius would still apparently be to underrate the effect. Neverthe-
less for the sake of rather underestimating than exaggerating the
influence of the cause here investigated, I will make these two
suppositions successively, giving respectively c = '98 and c = '49,
mc = 4926, and mc = '2463 for air.

If it were not for lateral motion the intensity would vary from
gas to gas in the proportion of the density into the velocity of
propagation, and therefore as the pressure into the square root of
the density under a standard pressure, if we take the factor
depending on the development of heat as sensibly the same for
the gases and gaseous mixtures with which we have to deal. In
the following Table the first column gives the gas, the second the
pressure p, in atmospheres, the third the density I) under the
pressure p, referred to the density of air at the atmospheric
pressure as unity, the fourth, Qr, what would have been the
intensity had the motion been wholly radial, referred to the
interisity in air at atmospheric pressure as unity, or, in other
words, a quantity varying as ^ x (the density at pressure l)i
Then follow the values of q, I^, and Q, the last being the actual
intensity referring to air as before.

An inspection of the numbers contained in the columns headed
Q will show that the cause here investigated is amply sufficient to
account for the facts mentioned by Leslie.

It may be noticed that while g" is 4 times smaller, and I^ is 16
or 18 times larger, for c = '49 than for c = '98, there is no great
difference in the values of Q in the two cases for hydrogen and
mixtures of hydrogen and air in given proportions. This arises
from the circumstance that q is sufficiently small to make the
last terms in /a,^ and /a/, namely, 1 and Sl^'^^ the most important,
so that /„ does not greatly differ from 81g'"=. If this result had
been exact instead of approximate, the intensity in different'
gases, supposed for simplicity to be at a common pressure, would
have varied as I)*-; and it will be found that for the cases in
which p = l the values of Q in the Table, especially those in the
last column, do not greatly deviate from this proportion. But

FROM A VIBRATING BODY TO A SURROUNDING GAS. 313

r—t

1^

O!

«

r-H

CO

CO

OS

cy

I— 1

00

CO

o ^

o

r-

o

p p

p

p

O CN 1

■-1

as

o o o

o

o

o o

^

OJ O <35

a> a>

•

H"

00 O OO

^^

00

00 00

II

O ^ O

oq

(^

o ^

w

CM O CN

i^-

cq

(M 1:^

CO

lO

Ttl

«

CD

r— 1

i> 00 I>

lO

r-

r^

1^ ^ ^

J>-

CD

co ^

&^

o ^ c

^

o oi

CO O CD

o

CO

CO CO

9 P p

p

°

9 9

00

^

^

^^

o

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n

rM

o*

T— 1

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oq

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o

1:-

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p p

p

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■P r-i 1

00

'-'

to o CO

o

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CD (N

CS

mom

o

n

M CM

■

t-r

rH 1^- i-H

1— t

I-H CO

II

I— 1 ^ rH

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1— (

rH Tl<

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00

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o

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o

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1:^ I^

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Cq CD Cq

05

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Tt< 1— 1 -Tf

1— t

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tc oq

c<i p cq

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qq T*

Ir^

00

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p

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p

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m

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cn

00

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^

p

CO rH

t^

t^

CO

P P

P

p

"P

Ip

""•

mi

00

El,

rH

r^

p

9

u:;

^

r-t

T-H

(Q

qj

T3

-+

V, -2

s

"S

tn

<D

T3

S

2

i

o

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;.

-a

>

^ .jf

^

^H

h

-S

<

K

<j

H

^

^

t^

314 ON THE COMMUNICATION OF VIBRATION

the simplicity of this result depends on two things. First,
the vibration must be expressed by a Laplace's function of the
order 2; for a different order the power of D would have been
different ; and this is just one of the points respecting which we
cannot infer what would be true of a bell of the ordinary shape
from what we have proved for a sphere. Secondly, the radius
must be sufficiently small, or the pitch sufficiently low, to make q
small ; at the other extremity of the scale, in which c is supposed
to be very large, or X very small, Q varies nearly as D^ instead of
D', whatever be the order of the Laplace's function. Hence no
simple relation can be expected between the numbers furnished
by experiment and the numerical constants of the gas in such
experiments as those of M. Perolle*, in which the same bell was
rung in succession in different gases.

B. Solution of the Problem in the case of a Vibrating Cylinder.

I will here suppose the motion to be in two dimensions only.
In the case of a vibrating string, which I have mainly in view,
it is true that the amplitude of excursion of the string varies
sensibly on proceeding even a moderate distance along it, and
that the propagation of the sound-wave produced by no means
takes place in two dimensions only. But the question how far a
sound-wave is produced at all, and how far the displacement of
the gas by the cylinder merely produces a local motion to and
fro, is decided by what takes place in the immediate neighbour-
hood of the string, such as within a distance of a few diameters ;
and thougli the sound-wave, when once produced, in its subse-
quent progress diverges in three dimensions, the same takes place
with the hypothetical sound-wave which would be produced if
lateral motion were prevented, so that the comparison which it
is the object of the investigation to institute is not affected
thereby.

Assuming, then, the motion to be in two dimensions, and re-
ferring the fluid to polar coordinates, r, 6, r being measured from
the axis of the undisturbed cylinder, we shall have for the funda-
mental equation derived from (1)

clt'~ [dr'^r dr^r'de^-} ^^^^ '

* Memoires de I'Acadgmie des Sciences de Turin, t. iii, (1786-7) ; Mem. des
Gorrespondans, p. 1.

FROM A VIBRATING BODY TO A SURROUNDING GAS. 315

and if u', v' be the components of the velocity along and perpen-
dicular to the radius vector,

, dd) , 1 d(i)

If c be the radius of the cylinder, and F the normal
component of the' velocity of the surface of the cylinder, we
must have

-^ =V, when r = c.
dr

As before, I will suppose the motion of the cylinder, and conse-
quently of the fluid, to be regularly periodic, but instead of using
circular functions directly I will employ the imaginary exponential
e™"', i denoting as before V — 1, and will put accordingly
Y^^mat u^ jj being a given function of 0, and ^ = i|fe*™"*. For
a given value of r, yfr may by a known theorem be developed in a
series of sines and cosines of d and its multiples, and therefore
for general values of r can be so developed, the coefficients being
functions of r. If i/r^ be the coefficient of cosn^ or sinw^,
we find

-d;r + -^-^fn + m-.}rr, = (15).

If we suppose the normal velocity of the surface of the
cylinder to vary in a given manner from one generating line to
another, so that U is a. given function of 0, we may expand U in
a series of the form

U=U„+ Ui cos0+U^ cos2d+...

+ U,' sin 61 -I- U^' sin 2^+...

On applying now the equation of condition which has to be
satisfied at the surface of the cylinder, we see that a term
Un COS nd or Un sin nd of the nth order in the expression for U
will introduce a function yjrn of the same order in the general
expression for 0. Now the only case of interest relating to an
infinite cylinder is that of a vibrating string, in which the cylinder
moves as a whole. The vibration may be regarded as compounded
of the vibrations in any two rectangular planes passing through
the 5ixis, the phases of the component vibrations, it may be, being
different. These component vibrations may be treated separately

316 ON THE COMMUNICATION OF VIBRATION

and thus it will suffice to suppose the vibration confined to one
plane, which we may take to be that from which 6 is measured.
We shall accordingly have

U=U, cos 0,

Ui being a given constant, and the only function ■x/rji which will
appear in the general expression for <^ will be that of the order 1.
Besides this we shall have to investigate, for the sake of com-
parison, an ideal vibration in which the cylinder alternately con-
tracts and expands in all directions alike, and for which accordingly
Z7 is a constant U^. Hence the equation (15) need only be con-
sidered for the values and 1 of n.

For general values of ?! the equation (15) is easily integrated
in the form of infinite series according to ascending powers of r.
The result is

„ 1' m-r- m^r* ^ '

2(2-2?i)^2.4(2-2k)(4-2«)

When n is any integer the integral as it stands becomes illusory ;
but the complete integral, which in this case assumes a special
form, is readily obtained as a limiting case of the complete integral
for general values of n.

The series in (16) are convergent for any value of r however
great, but they give us no information of what becomes of the
functions for very large values of ?•.

When r is very large, the equation (15) becomes approximately

dr'

-+m'ylrn = 0,

the integral of which is ■«|r„ = iJe-*""" -I- jRV'™', where R and E' are
constant. This suggests putting the complete integral of (15)
under the same form, R and R' being now functions of r, which,
when r is large, vary but slowly, that is, remain nearly constant when
r is altered by only a small multiple of X. Assuming for R and R'
series of the form A7-^ -\- £r^ + Cry . . . , where a, ^S, y... are in
decreasing order algebraically, and determining the indices and

FROM A VIBRATING BODY Td A SURROUNDING GAS. 317

coefficients so as to satisfy (15), we get for another form of the
complete integral

1= - iir {V - 4)1°-) (3= - 4)!-)'

.^„ = (7(.»n-)-ie--'|l-J^: +

y...(i7).

1 . 2 {8imrf
_ (l--4»^) (3— 4y)(5-^-4»-)

l.'-2.S{Simry '^ "

^,. , , . f , (P - 4)r) (1= - 4(i-') (3^ - in*)
( l.iiDnr 1.2{b<imr)-

(l^-4» -) (3;- - -in") (5-j-4n=)
■•" l727S{8iinry "*" '

These series, though ultimately divergent, begin by converg-
ing rapidly when r is large, and may be employed with great
advantage when 7- is large, if we confine ourselves to the converg-
ing part. Moreover we have at once Z* = as the condition to be
satisfied at a great distance from the cylinder. If inc were large
we might employ the second form of integral in satisfying the
condition at the surface of the cylinder, and the problem would
present no further difficulty. But practically in the case of vibrat-
ing strings mc is a very small fraction ; the series (16) are rapidly
convergent, and the series (17) cannot be employed. To complete
the solution of the problem therefore it is essential to express the
constants A and B in terms of C and D, or at any rate to find the
relation between A and B imposed by the condition i) = 0.

This may be effected by means of the complete integral of
(15) expressed in the form of a definite integral. For «=0 we
know that

IT

■Jr„= {' \E + Flog {r sin' ^)} COS (mr COS ^d^ (18)

Jo

is a third form of the integral of (15). It is not difficult to deduce

from this the integral of (15) in a similar form for any integi-al

value of n. Assuming

and substituting in (15), we have
^+P^" + {2a + 13 + 1) r«+P-' %„

■ (a= - n'O r'-- jr^Xn<i'' + "^''"' I'^Xn^r = 0.

318 ON THE COMMUNICATION OF VIBRATION

Assume

a'-n^=:0 (19),

divide the equation by r", differentiate with respect to r, and then
divide by j*^. The result is

^.^2«+|^+l^„_^^2a + ^ + l)(^-l)f + m^X« = 0.

This equation will be of the same forai as (15) provided

a + ^ = 0,

which reduces the coefficient of the last term but one to — (a+ 1)^.
In order that this coefficient may be increased we must choose
the positive root of (19), namely n, which I will suppose positive.
Hence

^^^r-^r-^Xndr (20)

gives

d'Xn 1 dxn {n + 1)^
dr^ +r dr r^ Xn + ^Xn ^.

the same equation as that for the determination of •\/r„+i. Hence
expressing %„ in terms of ->|r„ from (20), writing n — 1 for n, and
replacing Xn-i by ■^n, we have

a formula of reduction which when n is integral serves to express
i|r„ in terms of yfri). We have

^»=^"C-rJ^» (^i>'

an equation which when applied to (18) gives the complete
integral of (15) for integral values of n in the form of a definite
integral.

Let us attend now more particularly to the case of m = 0.
The equation (16) is of the form -v^n = Af(n) + Bf{— n), f(n)
containing r as well as n. Expanding by Maclaurin's Theorem,
we have

yjr^ = (A+B)f{0) + (^-B)f'(0)n + {A+B)f"(0)-^^ + ....

FEOM A VIBRATING BODY TO A SURROUNDING GAS. 319

Writing A tor A + B, rr^B for A—B, and then making n vanish,
we have

t„ = ^/(0) + i?/'(0),
or

.|.„ = (^ + 51ogr)(l- — + 2^-2^-^^,+ ...)]

where

8n = 1-' + 2-^ + 3-' + . . . + n-\

The integral in the form (17) assumes no peculiar shape when n is
integral, and we have at once

( 12 1 2 ^2 1 2 02 K2 ) \

^" ^ ^ 1 1.8iTOr^l.2(8imr)^ 1.2.3(8tmr)'^"- j

( 12 1 2 Q2 1 2 Q2 K2 1

+ 7)fVmrWe^""-l I I ^ ■'^ • " I

^ ' \ l.Simr 1.2{8imrf 1.2.3(8mr)='^""J

(23).

I have explained at length the mode of dealing with such
functions, and especially of connecting the arbitrary constants in
the ascending and descending series, in two papers published in
the Transactions of the Cambridge Philosophical Society*, in the
second of which the connexion of the constants is worked out in
this very example. To these I will refer, merely observing that
while it is perfectly easy to connect A, B with E, F, the connexion
of C, D with E, F involves some extremely curious points of
analysis. The result of eliminating E, F between the two
equations connecting A, B with E, F and the two connecting C, D
with E, F is given, except as to notation, in the two equations
(41) of my second paper. To render the [present] notation identical
with that of the former paper, it will be sufficient to write

A — B log (tm) + B log {imr) for A+B log r,

and X for imr. The equations referred to may be simplified by

* " On the Numerical Calculation of a Class of Definite Integrals and Infinite
Series," Vol. ix, p. 166 [ante, Vol. ii, p. 329], and " On the Discontinuity of
Arbitrary Constants which appear in Divergent Developments," Vol. x, p. 105
[supra, p. 77 : see pp. 100 — 103]. A supplement to the latter paper has recently
been read before the Cambridge Philosophical Society [supra, p. 283 : see pp. 294—5
and 298].

320

ON THE COMMUNICATION OF VIBRATION

the introduction of Euler's constant 7, the value of which is
"57721666 etc., since it is known that

7r-ir'(i) + log4 +7 = 0,

T' (w) denoting the derivative of the function T(x). Putting

A — Blog im = A',

we have [here] by equations (41) of the second paper referred to

G = (27r)-J [iA' + [(log 2 - 7) i - tt] 5} (24),

i) = (27r)-^{^' + (log2-7)5} (25),

i being written for V— 1. It is shown in that paper that these
values of G, D hold good when the amplitude of the imaginary
variable x lies between the limits and tt, or that of r (supposed
to be imaginary) between the limits — ^tt and ^tt, but in crossing
either of these limits one or other of the constants G, D is
changed. In the investigation of the present paper r is of course
real.

We have now

A'=A-B\og im = (7 - log 2) B

for the relation between A and B arising from the condition that
the motion is propagated outwards from the cylinder; and sub-
stituting in (22), we have for the value of i/r„ subject to this
condition

...(26);

or expressed by means of the descending series,

. ._ 1^

\2tmr,

^^ = -Bi^W-i^^\l-

+

1 . 8imr
1^3' P.3^5^

1 . 2 {Simrf 1.2.3 {Simrf

...(27).

We have from (21)

^=

dr '

FROM A VIBRATING BODY TO A SURROUXDING GAS.

321

from which the complete integral of (15) for n = l may be s;ot
from that for n =0. In the form (17) of the integral the parts
arising from differentiation of the p\rts containing e~'""' and e'""'
respectively ■will cont;iin those same exponentials, and therefore
the complete integral of (^15) for n = 1, subject to the condition
that the pirt containing {'"'"■ shall disappear, will be i^ot bv
ditierentiating the complete integi-;\l for n — subject to that same
condition. The form of the integral in the shape of a descending
series is given at once by (17V Hence we get by differentiating
(2li~) !vnd (27). and at the same time changing the ;\rbitrary
constant by writing J?i«i~' for B,

+ ^^b-'''-2^^=+-:4M5^^--)

^ \2mr' ( l.Smr

-1.1.3.5 - l.l.y.5.7 I

1.2(8;/(r)= l.2.-S(Smrf ^■•■J---V-^ '■

To determine the arbitrary const;\nts Bi and B, the first
belonffincr to the acttial motion, the second to the motion which
would take place if the fluid were confined by an infinite number
of planes passing through the axis, we must have^ as before, for

iU

dr

— I- ij

dr - *- '•

whence
B ~ 2-2= 4^'

/ , mc . ■7r\ in-tr «iV \

-. - (7 + log Y + • o j i^ -0— 2^71 + -j

+

w^c

m\"

m\"

= Ft {inc) -r t ^ .^ i>nc\ suppose

S. IV.

.(30):
21

322

ON THE COMMUNICATION OF VIBRATION

cUi_ 1 3mc 7m'c^ llmV

, . mc .7r\ /mc 3»iV ,

mc „ 3m 'c^ „ 5m^(f „
~ "2"^~ 2^¥ '■'■2^¥76 '"■■

= — \Fi (mc) + i -fi {mc)[ , suppose

.(31).

If I be the ratio of the intensities at a distance in the supposed
and in the actual case, we see from (30) and (31) that / will be
equal to the ratio of the squares of the moduli of H and 5i, and
we shall therefore have

.(32).

{4,F,{mc)Y+ir^[Mmc)Y
m^c»[4{i?;(mc)}^+7r^|/„(mc)}^] ■"

For a piano string corresponding to the middle C, c may be
about '02 inch, and X is about 25 inches. This gives mc = '005027.
For such small values of mc, I does not sensibly differ from {;mc)~'\
which in the present case is 39571, so that the sound is nearly
40000 times weaker than it would have been if the motion of the
particles of air had taken place in planes passing through the axis
of the string. This shows the vital importance of sounding-
boards in stringed instruments. Although the amplitude of
vibration of the particles of the sounding-board is extremely small
compared with that of the particles of the string, yet as it presents
a broad surface to the air it is able to excite loud sonorous vibra-
tions, whereas were the string supported in an absolutely rigid
manner, the vibrations which it could excite directly in the air
would be so small as to be almost or altogether inaudible.

I may here mention a phenomenon which fell under my notice,
and which is readily explained by the principles laid down in
this paper. As I was walking one windy day on a road near
Cambridge, on the other side of which ran a line of telegraph, my
attention was attracted by a peculiar sound of extremely high
pitch, which seemed to come from the opposite side of the road.
On going over to ascertain the cause, I found that it came directly
through air from the telegraph wires. On standing near a tele-

FROM A VIBRATING BODY TO A SURROUNDING GAS. 323

graph post, the ordinary comparatively bass sound with which we
are so familiar was heard, appearing to emanate from the post.
On receding from the post the bass sound became feebler, and
midway between two posts was quite inaudible. Nothing was
then heai-d but the peculiar high-pitched sound, which appeared to
emanate from the wires overhead. It had a peculiar metallic ring
about it which the ear distinguished from the whistling of the wind
in the twigs of a bush. Although the telegraph ran for miles, it
was only at one spot that the peculiar sound was noticed, and even
there only in certain states of the wind. The wires seemed to be
less curved than usual at the place in question, from which it
may be inferred that they were there subject to an unusually
great tension.

The explanation of the phenomenon is easy after what pre-
cedes. The wires were thrown into vibration by the wind, and a
number of different vibrations, having different periodic times,
coexisted. As regards the vibrations of comparatively long
period, the air around the wires behaved nearly like an incom-
pressible fluid, and no sonorous vibrations of sensible amount
were produced. These \'ibrations of the wires were, however,
communicated to the posts, which being broad acted as sounding-
boards, and thus sonorous vibrations of corresponding period were
indirectly excited in the air. But as regards the vibrations of
extremely short periodic time, the ^^Tres in spite of their narrowness
were able by acting directly on the air to produce condensations
and rarefactions of sensible amount.

The diameter of the telegraph wire was about 'leS inch : and
if we take the C below the middle C of a piano for the representa-
tive of the pitch of the lower note, and a note five octaves higher
for that of the higher, we have in the first case \ = 50 inches
nearly, and in the second X = 50 x 2-^ giving in the former case
mc = -01043, and in the latter mc = -3338. The former of these
values is so small that we may take / = (mc)--; in the latter case
the formula (32) gives for I a value a little less than {mc)--.
We find in the two cases / = 9192 and 7 = 7-202 respectively, so
that in the former case the sound is more than 9000 times feebler
than that corresponding to the amplitude of vibration of the wire
on the supposition of the absence of lateral motion, whereas in

21 2

324

ON THE COMMUNICATION OF VIBRATION TO A GAS.

the latter case the actual iatensity is nearly one-seventh of the
full intensity corresponding to the amplitude.

The increase of sound produced by the stoppage of lateral
motion may be prettily exhibited by a very simple experiment.
Take a , tuning-fork, and holding it
in the fingers after it has been made
to vibrate, place a sheet of paper
or the blade of a broad knife with
its edge parallel to the axis of the
fork, and as near to the fork as
conveniently may be without touch- _
ing. If the plane of the obstacle I
coincide with either of the planes of
symmetry of the fork, as represented in section at A or B, no
effect is produced ; but if it be placed in an intermediate position,
such as C, the sound becomes much stronger.

Account of Observations of the Total Eclipse of the
Sun.... By J. Pope Hennessy. Note added by Prof. Stokes.

[From the Proceedings of the Royal Society, xvii, 1868, pp. 88-89.]

The phenomenon of the sun's crescent reflected on to
the disk of the moon would seem to have been something
accidental, perhaps (if seen by the writer only) a mere ghost,
depending on a double reflection between the glasses of his
instrument. The figure represents the "reflected" image as in
the same position as the crescent itself, not reversed, indicating
either a refraction or a double reflection.

The slender beams of light or shade shooting out from the
horns of the crescent would seem to admit of easy explanation,
supposing them to have been of the nature of sunbeams, depend-
ing upon the illumination of the atmosphere of the earth by the
sun's rays. The perfect shadow, or umbra, would be a cone
circumscribing both sun and moon, and having its vertex far
below the observer's horizon. Within this cone there would be
no illumination of the atmosphere, but outside it a portion of the
sun's rays would be scattered in their progress through the air,
giving rise to a faint illumination. When the total phase drew
near, the nearer surface of the shadow would be at no great
distance from the observer; the further surface would be remote.
Attend in the first instance to some one plane passing through
the eye and cutting the shadow transversely, and in this plane
draw a straight line through the eye, touching the section of the
cone which bounds the shadow; and then imagine other lines
drawn through the eye a little inside and outside this. In the
former case the greater part of the line, while it lay within the
lower regions of the atmosphere, would be in shadow, the only
part in sunshine being that reaching from the eye to the nearer
surface of the shadow; but in the latter case the line would be
in sunshine all along. In the direction of the former line, there-
fore, there would be but little illumination arising from scattered

326 PHENOMENA AT ECLIPSE OF THE SUN.

light, while in the direction of the latter the illumination would,
comparatively speaking, be considerable. In crossing the tangent
there would be a rapid change of illumination. Now pass on to
three dimensions. Instead of a tangent line we shall have a
tangent plane, and there will of course be two such planes,
touching the two sides of the cone respectively. Each of these
will be projected on the visual sphere into a great circle, a common
tangent to the two small circles, which are the projections of the
sun and moon. In crossing either of these there will be a rapid
change of illumination (feeble though it be at best) which will
be noticed. According as the observer rnentally regards darkness
as the rule and illumination as the feature, or illumination as the
rule and darkness as the feature, he will describe what he sees
as a beam or a shadow. The direction of these beams or shadows
given by theory, as just explained, agrees very well with the
drawing sent by Governor Hennessy, which does not represent
the left-hand beam so distinctly divided as it appears in the
woodcut.

On a certain Reaction of Quinine.

[From the Journal of the Chemical Society, May, 1869 : read Mar. 18.]

In the course of two papers on optical subjects, published
in the Philosophical Transactions, I have mentioned a peculiar
reaction of quinine having relation to its fluorescence*. About
that time I followed out the subject further, and obtained results
which were interesting to myself, especially in relation to a
classification of acids which they seemed to afford. Not being
a chemist, I did not venture to lay the results before the chemical
world. I have, however, recently been encouraged by a chemical
friend to think that a further statement of the results might prove
of some interest to chemists.

The reaxjtion is best observed by diffused dajdight entering a
darkened roomf through a hole in the shutter, which may be
four or five inches square, and which is covered with a deep
violet glass, coloured by manganese :|:. In front of the hole is

* Phil. Tram, for 1852, p. 541, and for 1853, p. 394. [Ante, Vol. ni, p. 267,
Vol. IV, p. 1.]

t In default of a darkened room, a common box, such as an old packing-case,
may be readily altered so as to answer very well. The box is sawn obliquely across,
the cutting plane being parallel to one edge, as
indicated by the figure, which denotes >■• vertical
section. The aperture thus made is covered by a
board nailed on, containing a hole, H, destined
to be covered by the glass plate, which is kept from
slipping down by a small ledge, L. A portion of
the upper covering of the box at E is removed to
allow the observer to see and manipulate. In

observing, the box is placed near a window, with its slant side turned towards the
light ; the hole is covered with its glass ; the object is placed at ; and the observer
looks in through E, covering his head with a dark cloth, to exclude stray light.

J Flint glasses answer best, the colour given by manganese to crown glass being
generally somewhat brownish. I have, however, seen one specimen of crown glass
coloured by manganese, the colour of which was as fine a purple as that of the flint
glasses.

A^

328 ON A CERTAIN REACTION OF QUININE.

placed a white porcelain tablet, or else one of the porcelain slabs
with shallow depressions used for colour tests. A solution of
quinine in very weak alcohol is strong enough for the observa-
tions, or else very minute fragments may be used. In some cases,
as for example with valerianic or benzoic acid, the presence of
alcohol interferes with the reaction.

It will conduce to brevity and clearness to describe in the
first instance, in a little detail, the phenomena exhibited by two
"particular acids, say sulphuric and hydrochloric.

Let a series of drops of the quinine solution be deposited on
the porcelain. If one of these be touched by a rod dipped in
dilute sulphuric acid, the beautiful fluorescence of the quinine
is instantly developed. If another drop be similarly touched by
a rod moistened with dilute hydrochloric acid, no apparent effect
is produced* Nor is this all. If a little hydrochloric acid be
introduced by a moistened rod into the fluorescing drop, the
fluorescence is immediately destroyed. If a little of the sulphuric
acid be introduced into the drop containing only hydrochloric
acid, no effect is produced t.

If a series of drops of a solution of quinine in dilute sulphuric
acid be deposited, and a little solution of chloride of potassium,
sodium, or ammonium be added, the fluorescence is immediately
destroyed. The action of sulphate of potassium, etc., on a solution
of quinine in water acidulated, whether with hydrochloric or
sulphuric acid, is in each case merely negative |.

Now, on trying a variety of acids, I found that with hardly
an exception, unless when the acid character of the acid was only

* It is true that a solution of quinine in dilute hydrochloric acid is fluorescent,
and with concentrated sunlight, or with sunlight uncondensed, but analysed by
absorption or dispersion, the fluorescence comes out strongly. It is, however,
notably inferior to that produced by sulphuric acid ; and for our immediate object
a mode of observation in which it hardly, if at all, appears, is even better than one
adapted to bring out comparatively feeble degrees of fluorescence. When I speak
in the text of fluorescence being destroyed, the expression must be understood in
this qualified sense.

t It must be understood that I am not here dealing with concentrated acids,
nor with any very great preponderance of one kind over another. I suppose all the
solutions to be dilute, and the quantity of acid employed, of whatever kind, to be
many times that merely required to combine with the quinine.

{ See the preceding note.

ON A CERTAIN REACTION OF QUININE.

329

indistinct, the acids ranged themselves with perfect definiteness
into two classes, which I will call class A and class B. Those of
class A developed fluorescence in a solution of quinine in water
just like sulphuric acid, the amount of fluorescence being com-
parable with that produced with sulphuric acid, and the tint
the same. Those of class B not only did not produce it, but
destroyed it when produced by acids of the class A. This de-
struction is produced by the alkaline salts of the acids, as well
as by the free acids themselves, and thus we are able to classify
acids without having specimens of the free acids at hand.

In the following lists, those acids which were tried only in-
directly, by means of one or more of their alkaline salts, are
distinguished by an asterisk : —

Class A.

Class B.

Acetic

Hydriodic*

Arsenic

Hydrobromic*

Benzoic

Hydrochloric

Chloric

Hydroferrocyanic*

Citric

Hydropalladiocyanic*

Formic

Hydroplatinocyanic*

Hyposulphuric*

Hydrosulphocyanic*

Iodic

Hyposulphurous*

Malic

Nitric

Oxalic

Perchloric

Phosphoric

Silico-fluoric

Succinic

Sulphuric

Tartaric

Valerianic

Unless a quinine solution be sufficiently dilute, when alcohol is
used, iodic acid produces a precipitate. In what precedes, it
must be understood that I refer in all cases to solutions. The
character of the fluorescence of the salts of quinine in the solid
state varies from salt to salt. The solid iodate obtained by

330 ON A CERTAIN REACTION OF QUININE.

precipitation is strongly fluorescent, with a blue considerably
deeper than that of the solutions.

It is not in all cases possible to try all the reactions stated to
belong to the acids above mentioned. Thus, in the case of iodic
acid, the solution cannot be tested by ferrocyanide of potassium,
which instantly decomposes the iodic acid. But the strong fluor-
escence of the iodic solution, and the immediate destruction of
the fluorescence by chloride of sodium, etc., alone suffice to show
definitely to which class iodic acid belongs.

The absorption of the fluorogenic* rays by the yellow ferro-
cyanide of potassium, would itself alone account for the apparent
destruction of the fluorescence if the salt were present in sufficient
quantity. Actually, however, the quantity which suffices to destroy
the fluorescence is much less than what would be required to
prevent its exhibition by the absorption either of the fluorogenic
or of the fluorescent rays, or of both. When the experiment is
properly tried, there cannot be a moment's hesitation that the
removal of the fluorescence is a true chemical reaction, and not
a mere optical effect. This may be further proved by spreading
a comparatively large quantity of the ferrocyanide solution in the
form of a broad drop on glass, and holding it immediately over
the gleaming drop of the quinine solution, when, though the
fluorogenic rays entering, and the fluorescent rays leaving the
drop, have both to pass through the whole thickness of the ab-
sorbing solution, the fluorescence observed is only somewhat
reduced. The absorption of the fluorescent rays in this case
goes indeed for little ; it is the absorption of the fluorogenic rays
that we have to consider. That there is a real reaction, and not
a mere optical effect, I have further proved by experiments in
a pure spectrum, which it would take too long to describe.

Hyposulphurous acid is, it is true, rather easily decomposed,
but the destruction of fluorescence by hyposulphite of soda is
quite independent of this circumstance. It takes place, for

* By this term I merely mean rays considered in their capacity of producing
fluorescence ; the introduction of such a term prevents circumlocution. It is con-
venient also to have a name for the rays emitted by a fluorescent body. If these
be called, as they are a little further on, fluorescent, no confusion can practically
result, though the term has of course a totally different signification as applied to
the rays or to the body emitting them.

ON A CERTAIN REACTION OF QUININE. 331

instance, at once on introducing a very dilute solution of hypo-
sulphite of soda into a drop in which the fluorescence had been
excited by very dilute citric or acetic acid.

After these remarks, which were necessary to prevent possible
misconception, we may return to our lists. A glance at these
lists shows that the classification made by the quinine reaction
agrees almost exactly with the old distinction of ox-acids and
hydracids. There is, however, one acid, the hyposulphurous,
which in the quinine reaction ranges itself with perfect definite-
ness in class B, but which is not, I believe, usually ranked by
chemists with the hydracids, except in so far as acids in general
have been regarded from this point of view. This led me to
seek whether there might not be other analogies between hypo-
sulphurous acid and the hydracids. I have noticed the two
following : —

1. It is known that a solution of chloride of mercury reddens
litmus, but the blue colour is restored by an alkaline chloride,
though itself neutral to colour tests. Now the very same effect
is produced by hyposulphite of soda. This salt and chloride of
mercury very readily decompose each other ; but if very dilute
solutions be used, the solution of chloride of mercury having
been coloured by a little litmus, it is easy to observe that the
first effect of the introduction of the hyposulphite, an effect
which takes place immediately, prior to the formation of any
precipitate, is to restore the blue colour.

2. It is known that cyanide of mercury is hardly decomposed
by ox-acids, so as simply to displace the hydrocyanic acid, but
readily by hydracids. Now, if a solution of hyposulphite of soda
be added to one of cyanide of mercury, the smell of hydrocyanic
acid is immediately perceived.

These circumstances bear out, as to hyposulphurous acid, the
classification afforded by the quinine reaction. If there be a real
difference of constitution between say sulphuric and hydrochloric
acids, expressed in symbols by writing the former (according to
the old equivalents) SO3 . HO, and not SO4 . H, and in words by
calling it an ox-acid, the mere fact that the radical of hypo-
sulphurous acid, regarded as a hydracid, contains oxygen, does
not, of course, oblige us to regard it as an ox-acid. It is that

332 ON A CERTAIN REACTION OF QUININE.

difference of constitution, whatever it may be, which must decide,
and if we may trust the quinine reaction, the isolation of S2O3,
the analogue of chlorine, would seem to be less improbable than
that of S2O2 , the analogue of sulphuric anhydride.

With hydrocyanic and hydrofluoric acids the reaction seemed
doubtful. These acids seemed to belong to class B as regards
the feeble amount of fluorescence which they developed, but not
to prevent the development of strong fluorescence by acetic,
sulphuric, etc., acids. Ferridcyanide of potassium had certainly
no such action as chloride of sodium, or even ferrocyanide of
potassium, in destroying the fluorescence ; but the deep colour
of the salt prevented a satisfactory decision whether the acid
really belonged to class A, or resembled hydrocyanic acid in its
action on quinine.

The destruction of the fluorescence of a solution of quinine
in a dilute ox-acid on the introduction of a hydracid or its salt,
would seem to indicate that the quinine combined by preference
with the hydracid. It seemed to me that it would be interesting
to restore, if possible, the fluorescence, without precipitation, by
the introduction of a substance which should have a preferential
affinity for the hydracid. In the case of hydrochloric acid, this
may be effected by a salt such as the sulphate or nitrate of the
red oxide of mercury. In trying the experiment it is convenient
to use only a little hydrochloric acid, or chloride of sodium, etc.,
otherwise the sparingly soluble chloride of mercury and quinine
is precipitated, which, however, redissolves on the addition of
more of the mercuric salt. That the restoration of fluorescence
is not a mere effect of the acid introduced with the mercuric
salt, may be proved by varying the experiment. Let quinine
be dissolved with more nitric or sulphuric acid than would other-
wise have been necessary ; add a little hydrochloric acid so as
barely to destroy the fluorescence, and then introduce a little
precipitate of oxide of mercury, stirring it up. As the oxide
dissolves the fluorescence returns.

Chloride of mercury does not impair the fluorescence of a
solution of quinine in an ox-acid ; if anything it sometimes seems
slightly to increase it. Chloride of barium, strontium, calcium,
magnesium, manganese, or zinc, acts like chloride of sodium.

ON A CERTAIN REACTION OF QUININE. 333

The fluorescence destroyed by a chloride is in some measure
restored by nitrate of cadmium.

When the fluorescence is destroyed by bromide or iodide of
potassium, it may be restored by oxide of mercury, j ust as in the
case of an alkaline chloride, and under the same conditions.

The precipitate of chloride of mercury and quinine, which is
liable to be produced in trying the above reactions, is strongly
fluorescent, with a blue which seems to be a trifle greener than
that of the solutions. The corresponding precipitate which may
be obtained with bromide of potassium, doubtless a bromide of
mercury and quinine, is white, and shows a pretty strong orange
fluorescence, a very unusual colour for the fluorescence of a
white substance. The iodide is pale yellow, and not sensibly
fluorescent, at least as examined by daylight with coloured
glasses.

Explanation of a Dynamical Paradox.

[From the Messenger of Mathematics, New Series, I, 1872, pp. 1 — 3.]

The answer to the following question, proposed in the Smith's
Prize Examination, 1871, is sent in compliance with a request
from one of the Editors :

" In a compound pendulum consisting of masses m, m' attached
to strings of length I, I', in which of course the most general small
motion in one plane consists of two harmonic vibrations superposed,
if the upper mass m be very large compared with the under ma^s m ,
it is clear that one of the two periodic times {that corresponding to
the mode of vibration in which m is nearly at rest) must be very
nearly the same as in a simple pendulum of length I', and the other
very nearly the same as in a simple pendulu/m of length I. By a
continuous variation of I', the former may be made to pass con-
tinuously from less to greater than the latter, and therefore for
some value of I' nearly equal to I the two must be equal. But when
a system is in stable equilibrium (as is clearly the case here) the
equation the roots of which give the times of vibration cannot have
equal roots, for that would imply the transitional condition between
stable and unstable.

" Point out precisely the fallacy which leads to the above
contradiction."

The fallacy lies in the tacit assumption that it is the same
root of the quadratic which determiaes the times of vibration
that correspond throughout to the same approximate physical
state, i.e. the state in which (not considering the special case
in which I, I' are nearly equal) the upper maSs is nearly at rest,
or the two masses move through comparable spaces, as the case
may be. Let T, T' be the two times of vibration (or half periods),
T, T the times of vibration of simple pendulums of lengths I, I';
and suppose m, m', I, V to change continuously, yet so that I
always remains distinctly greater than or distinctly less than I ;
i.e. so that the ratio of Z - Z' to I or V , though absolutely, it may be,
small, remains finite while m! : vi may be taken as small as we

EXPLANATION OF A DYNAMICAL PARADOX. 335

please. Of the two T, T, let T be that which for one set of
values of the constants m, m', I, I' is nearly equal to t ; then must
the same root T remain throughout that which is nearly equal
to T ; for it is obliged to be nearly equal to one of the two t, t'
which do not become nearly equal to each other. But if we
suppose I, I' to change continuously, so that I, from having been
distinctly less, becomes distinctly greater than I', and if T be that
root which for l<l' is, nearly equal to t, since t, t' pass through
equality, and T is merely known to be nearly equal to one of
them, there is nothing to shew which of the two t, t' it is that T
is nearly equal to when I > V. By the general principle referred
to in the second part of the question, we see that it must be t'.

The same thing may of course be shewn by the direct solution
of the problem. Putting n for tt/T, we find by the usual methods

mll'n* - (m + m) {I + I') giv" + (m + m) g' = 0,
g being gravity, and the roots of this quadratic in n'^ are

2in

^\j+h\/{Q-'T.

m 4
m + m W

If m be very small, and I, I' not very nearly equal, the radical
becomes very nearly Ijl ~ IjU ; and denoting by n^, n'^ the roots
corresponding to the signs +, — , respectively, we have very
nearly

"■=f{r4H7-r)}."-il74-G-f,

If Z < V, we have

but if I > r,

n^ = gll, n'^^gjl';
n- = g/l', n' = g/l.

When I, I' are nearly equal, we can no longer distinguish the
two harmonic vibrations by the character, that from one of them
m is nearly at rest, while the small mass m' moves considerably,
since the motion of m as compared with on' is comparable in the
two. In fact, the harmonic vibrations, which, when I, I' are dis-
tinctly different, are characterized by the properties above referred
to, have their properties interchanged when I : V passes through 1 *.

[* The two roots found above obviously cannot be equal. The limitations to
equality of roots being an indication of instability have been determined by
Weierstrasa and by Eouth; cf. Thomson and Tait, Nat. Phil. ed. ii, § 343.]

On the Law of Extraordinary Refraction in
Iceland Spar.

[From Proceedings of Royal Society, xx, pp. 44Z-4t. Received June 20, 1872.]

It is now some years since I carried out, in the case of Iceland
spar, the method of examination of the law of refraction which I
described in my report on Double Refraction, published in the
Report of the British Association for the year 1862, p. 272*. A
prism, approximately right-angled isosceles, was cut in such
a direction as to admit of scrutiny, across the two acute angles,
in directions of the wave-normal within the crystal comprising
respectively inclinations of 90° and 45° to the axis. The directions
of the cut faces were referred by reflection to the cleavage-planes,
and thereby to the axis. The light observed was the bright D of
a soda-flame.

The result obtained was, that Huygens's construction gives
the true law of double refraction within the limits of errors of
observation. The error, if any, could hardly exceed a unit in the
fourth place of decimals of the index or reciprocal of the wave-
velocity, the velocity in air being taken as unity. This result is
sufficient absolutely to disprove the law resulting from the theory
which makes double refraction depend on a difference of inertia in
different directions "f".

I intend to present to the Royal Society a detailed account of
the observations; but in the mean time the publication of this
preliminary notice of the result obtained may possibly be useful
to those engaged in the theory of double refraction ;J:.

[* Supra, p. 187.] [t Cf. supra, p. 182.]

[J This work was followed up by an investigation by Glazebrook, Phil. Trans.
CLXxi, 1880, pp. 421 — 449, with Prof. Stoljes' apparatus, which verified the same degree
of accuracy over a wide range of measurements, a limit being imposed by imperfec-
tions in the lenses employed, and by some measures by Abria and by Kohlrausoh also
of the same degree of accuracy. Finally C. S. Hastings, American Journal of
Science, cxxxv, Jan. 1888, pp. 60 — 73, pushed the verification within two units in
the sixth place of decimals in two measurements on Iceland spar. In the work of
Abria, referred to in the next paper, the uncertainty was about one per cent.]

Sue l'emploi du prisme dans la vitrification de la loi
de la double refraction.

[From Comptes Rendus, Vol. lxxvii, Nov. 17, 1873, pp. 1150-1152.]

La Communication de M. Abria {Comptes rendus, stance du
13 octobre, p. 814 de ce volume) me determine k appeler I'atten-
tion de I'Acad^mie sur une m^thode que j'ai propos^e pour le
mgme objet dans un travail sur la double refraction*, et que
j'ai appliqu^e plus tard au spath calcairef. Cette m^thode me
parait plus facile, plus gen^rale et plus exacte que celle de
M. Abria.

Quand on veut mesurer I'indice de refraction d'une substance
ordinaire, on emploie le plus souvent la m^thode de la deviation
minimum: Mais il y a une autre methode, aussi exacte et presque
aussi facile, qui consiste k mesurer la deviation pour un azimut
arbitrafre du prisme, et en outre Tangle d'incidence ou Tangle
d'6mergence, suivant que le prisme demeure en repos quand on
d^place la lunette, ou qu'il Taccompagne dans son mouvement.
Cette methode n'est pas nouvelle : elle a d^ja ^te employee par
M. Swan dans sa verification de la loi de Snellius pour le rayon
ordinaire du spath calcairej; mais on n'avait pas, a ma connais-
sance, indiqu^ le parti qu'on en pourrait tirer pour la recherche
de la loi de la refraction extraordinaire dans les cristaux. Le
phenomene que Ton observe dans le cas d'un cristal est le meme
que dans le cas d'une substance ordinaire, avec cette seule
difference que Ton obtient deux images au lieu d'une seule; on
pent encore mesurer la deviation de chacune des deux images, et
il ne s'agit que d'interpreter les resultats obtenus. Or, en
s'appuyant sur la demonstration qu'a donnee Huyghens pour la

* Report of the British Association for 1862, Part i, p. 272. [Supra, p. 187.]
t Proceedings of the Royal Society, Vol. xx, p. 443 (20 juin 1872). [Supra,
preceding page.]

X Tramactions of the Royal Society of Edinburgh, Vol. xvi, p. 375.

s. IV. 22

338 VERIFICATION OF THE LAW OF DOUBLE REFRACTION.

refraction en g^n^ral, demonstration qui, fondee sur le seul prin-
cipe de la coexistence des petits mouvements, n'exige aucune
hypothese sur la loi de variation des vitesses de propagation dans
diverses directions, on demontre facilement que les deux quantites
qui repr^sentent pour une substance ordinaire, 1° Tangle de
refraction, 2° I'indice de i-6fraction, et qui se deduisent des donn^es
d'observations par un calcul tres-facile, expriment pour un cristal,
1° I'inclinaison de I'onde refract^e k la suiface d'incidence, onde
qui est n^cessairement perpendiculaire au plan d'incidence, 2° le
rapport de la vitesse de propagation dans Fair k celle de I'onde
refract^e. La direction ainsi d^termin^e par rapport aux deux
faces du prisme est rapport^e ensuite, par le calcul, a des directions
fixes dans le cristal, I'orientation de chaque face artificielle ayant
et6 determin^e au moyen de la reflexion, par rapport k des faces,
soit naturelles, soit de clivage. On pent ainsi examiner un cristal
dans une s^rie de directions, au moyen d'un seul angle refringent,
et Ton pent faire tailler deux angles au moins sur un meme bloc
sans detruire les faces dont on a besoin pour la determination de
I'orientation des plans artificiels.

Je n'ai appliqu^ jusqu'ici cette m^thode qu'au spath calcaire,
cristal que j'ai choisi k cause de la facility avec laquelle on peut
s'en procurer de bons ^chantillons, et de I'^nergie de sa double
refraction, qui devrait rendre plus sensibles les hearts par rapport
a la loi d'Huyghens, s'il en existait. J'ai trouv^ que cette loi
repr^sente la refraction extraordinaire aussi exactement que la
loi de Snellius repr^sente la refraction ordinaire.

L'erreur moyenne de quinze observations du rayon extra-
ordinaire, faites dans des directions qui s'etendaient de 30 a.
60 degres environ de I'axe, et rapprochees de la formule deduite
de la construction d'Huyghens en y introduisant les indices
principaux, obtenus a 90 degres de I'axe, ne s'eievait qu'a 0,00013
de I'indice, quantite qui est de I'ordre des erreurs accidentelles de
mes observations, et qui correspond k y^'^^ environ de la difference
des indices principaux. L'erreur correspondante de deviation
dans un prisme de 45 degres est d'environ 25 secondes.

Notice of the Researches of the late Rev. W. Vernon
Harcourt on the Conditions of Transparency in Glass
AND THE Connexion between the Chemical Constitution
AND Optical Properties of Different Glasses.

[From the Report of the British Association for the Advancement of
Science, Edinburgh, 1871.]

The preparation and optical properties of glasses of various
compositions formed for nearly forty years a favourite subject
of study with the late Mr Harcourt. Having commenced in 1834
some experiments on vitrifaction, with the object stated in the
title of this notice, he was encouraged by a recommendation,
which is printed in the 4th volume of the Transactions of the
British Association, to pursue the subject further. A report
on a gas-furnace, the construction of which formed a preliminary
inquiry, in which was expended the pecuniary grant made by the
Association for this research in 1836, is printed in the Report
of the Association for 1844, but the results of the actual
experiments on glass have never yet been published.

My own connexion with these researches commenced at the
Meeting of the British Association at Cambridge in 1862, when
Mr Harcourt placed in my hands some prisms formed of the
glasses which he had prepared, to enable me to determine their
character as to fluorescence, which was of interest from the
circumstance that the composition of the glasses was known.
I was led indirectly to observe the fixed lines of the spectra
formed by means of them ; and as I used sunlight, which he had
not found it convenient to employ, I was enabled to see further
into the red and violet than he had done, which was favourable
to a more accurate measure of the dispersive powers. This inquiry,
being in furtherance of the original object of the experiments,
seemed far more important than that as to fluorescence, and
caused Mr Harcourt to resume his experiments with the liveliest
interest, an interest which he kept up to the last. Indeed it was
only a few days before his death that his last experiment was
made. To show the extent of the research, I may mention that

22—2

340 CONNEXION BETWEEN THE CHEMICAL CONSTITUTION AND

as many as 166 masses of glass were formed and cut into prisms,
each mass doubtless in many cases involving several preliminary
experiments, besides disks and masses for other purposes. Perhaps
I may be permitted here to refer to what I said to this Section on
a former occasion* as to the advantage of working in concert.
I may certainly say for myself, and I think it will not be deemed
at all derogatory to the memory of my esteemed friend and
fellow-labourer if I say of him, that I do not think that either
of us working singly could have obtained the results we arrived at
by working together.

It is well known how difficult it is, especially on a small scale,
to prepare homogeneous glass. Of the first group of prisms,
28 in number, 10 only were sufficiently good to show a few
of the principal dark lines of the solar spectrum ; the rest had
to be examined by the bright lines in artificial sources , of light.
These prisms appeared to have been cut at random by the optician
from the mass of glass supplied to him. Theory and observation
alike showed that striee interfere comparatively little with an
accurate determination of refractive indices when they lie in planes
perpendicular to the edge of the prism. Accordingly the prisms
used in the rest of the research were formed from the glass mass
that came out of the crucible by cutting two planes, passing
through the same horizontal line a little below the surface, and
inclined 22^° right and left of the vertical, and by polishing
the enclosed wedge of 45°. In the central portion of the mass the
striae have a tendency to arrange themselves in nearly vertical
lines, from the operation of currents of convection ; and by cutting
in the manner described, the most favourable direction of the
strise is secured for a good part of the prism.

This attention to the direction of cutting, combined no doubt
with increased experience in the manufacture of glass, was attended
with such good results that now it was quite the exception for
a prism not to show the more conspicuous dark lines.

On account of the inconvenience of working with silicates,
arising from the difficulty of fusion and the pasty character of the
fused glasses, Mr Harcourt's experiments were chiefly carried on

* Report of the British Association for 1862, Trans, of Sect. p. 1. [Introductory
remarks at the Cambridge meeting.]

OPTICAL PROPERTIES OF DIFFERENT GLASSES. 341

^'ith phosphates, combined in many cases with fluorides, and
sometimes with borates, tungstates, molybdates, or titanates.
The glasses formed involved the elements potassium, sodium,
lithium, barium, strontium, calcium, glucinum, magnesium,
aluminium, manganese, zinc, cadmium, tin, lead, thallium, bis-
muth, antimony, arsenic, tungsten, molybdenum, titanium, vana-
dium, nickel, chromium, uranium, phosphorus, fluorine, boron,
sulphur.

A very interesting subject of inquiry presented itself col-
laterally with the original object, namely, to inquire whether
glasses could be found which would achromatize each other so
as to exhibit no secondary spectrum, or a siugle glass which would
achromatize in that manner a combination of crown and flint.

This inquiry presented considerable difficulties. The dispersion
of a medium is small compared with its refraction; and if the
dispersive power be regarded as a small quantity of the first order,
the irrationality between two media must be regarded as depend-
ing on small quantities of the second order. If strias and
imperfections of the kind present an obstacle to a very accurate
determination of dispersive power, it will readily be understood
that the errors of observation which they occasion go far to swallow
up the small quantities on the observation of which the determina-
tion of irrationality depends. Accordingly, little success attended
the attempts to draw conclusions as to irrationality from the direct
observation of refractive indices ; but by a particular method of
compensation, in which the experimental prism was achromatized
by a prism built up of slender prisms of crown and flint, I was
enabled to draw trustworthy conclusions as to the character in this
respect of those prisms which were sufficiently good to show a few
of the principal dark lines of the solar spectrum*.

Theoretically any three different kinds of glass may be made
to form a combination achromatic as to secondary as well as
primary colour, but practically the character of dispersion is
usually connected with its amount, in such a manner that the
determinant of the system of three simple equations which must
be satisfied is very small, and the curvatures of the three lenses
required to form an achromatic combination are very great.

[* Rm). Soc. Proc. 1878, to be reprinted infra.]

342 CONNEXION BETWEEN THE CHEMICAL CONSTITUTION AND

For a long time little hope of a practical solution of the
problem seemed to present itself, in consequence of the general
prevalence pf the approximate law referred to above. A prism
containing molybdic acid was the first to give fair hopes of success.
Mr Harcourt warmly entered into this subject, and prosecuted his
experiments with unwearied zeal. The earlier molybdic glasses
prepared were many of them rather deeply coloured, and most of
them of a perishable nature. At last, after numerous experi-
ments, molybdic glasses were obtained pretty free from colour and
permanent. Titanium had not yet been tried, and about this
time a glass containing titanic acid was prepared and cut into
a prism. Titanic acid proved to be equal or superior to molybdic
in its power of extending the blue end of the spectrum more than
corresponds to the dispersive power of the glass ; while in every
other respect (freedom from colour, permanence of the glass,
greater abundance of the element) it had a decided advantage ;
and a great variety of titanic glasses were prepared, cut into
prisms, and measured. One of these led to the suspicion that
boracic acid had an opposite effect, to test which Mr Harcourt
formed some simple borates of lead, with varying proportions
of boracic acid. These fully bore out the expectation; the
terborate for instance, which in dispersive power nearly agrees
with flint glass, agrees on the other hand, in the relative extension
of the blue and red ends of the spectrum, with a combination of
about one part, by volume, of flint glass with two of crown.

By combining a negative or concave lens of terborate of lead
with positive lenses of crown and flint, or else a positive lens
of titanic glass with negative lenses of crown and flint, or even
with a negative of very low flint and a positive of crown, achromatic
triple combinations free from secondary colour may be formed
without encountering (at least in the case of the titanic glass)
formidable curvatures; and by substituting at the same time
a titanic glass for crown, and a borate of lead for flint, the
curvatures may be a Uttle further reduced.

There is no advantage in using three different kinds of glass
rather than two to form a fully achromatic combination, except
that the latter course might require the two kinds of glass to be
made expressly, whereas with three we may employ for two the
crown and flint of commerce. Enough titanium might, however

OPTICAL PROPERTIES OF DIFFERENT GLASSES. 343

be introduced into a glass to render it capable of being perfectly
achromatized by Chance's " light flint."

In a triple objective the middle lens may be made to fit both
the others, and be cemented. Terborate of lead, which is some-
what liable to tarnish, might thus be protected by being placed
in the middle. Even if two kinds only of glass are used, it is
desirable to divide the convex lens into two, for the sake of
diminishing the curvatures. On calculating the curvatures so as
to destroy spherical as well as chromatic aberration, and at the
same time to make the adjacent surfaces fit, very suitable forms
were obtained with the data furnished by Mr Harcourt's glasses.

After encountering great difficulties from strias, Mr Harcourt
at last succeeded in preparing disks of terborate of lead and of
a titanic glass which are fairly homogeneous, and with which it is
intended to attempt the construction of an actual objective which
shall give images free from secondary colour, or nearly so.

This notice has extended to a greater length than I had
intended, but it still gives only a meagre account of a research
extending over so many years. It is my intention to draw
up a full account for presentation to the scientific world in
some other form. I have already said that the grant made to
Mr Harcourt for these researches in 1836 has long since been
expended, as was stated in his Report of 1844; but it was his
wish, in recognition of that grant, that the first mention of the
results he obtained should be made to the British Association;
and I doubt not that the members will receive with satisfaction
this mark of consideration, which they will connect with the
memory of one to whom the Association as a body is so deeply
indebted.

On the Principles of the Chemical Correction
OF Object-glasses.

[Lecture to the Photographic Society. From the Photographic Journal,

Feb. 15, 1873.]

When I was invited by your Secretary to deliver a lecture
before you, I thought of different subjects, and one occurred to
me which I deemed might be of some interest to you. It is one
about which I have thought a good deal ; and I propose to entitle
the lecture " On the Principles of the Chemical Correction of
Object-glasses." I shall, however, leave for the present the
chemical rays out of consideration, and deal only with the visual
rays, because it is easier to explain the portion of the subject which
relates to the chemical rays by reference to the visual — since you
can more readily picture them to yourselves, and I can more readily
refer to them by naming the colours. When once the principles of
the mode of proceeding are understood with reference to the visual
rays, it is perfectly simple to apply them to the chemical rays.
As for the elementary principles of the subject, I shall not dwell
upon them, as I presume they are already familiar to you. You
well know that Newton discovered the compound nature of white
light, which, on passing through a prism, is bent round and
decomposed into different parts, producing different impressions
of colour. It was already known that, on passing light through
different substances, it was not bent round in the same, but in
different degrees. Thus, if two kinds of glass, flint and crown,
are formed into prisms of the same angle, and light be passed
through them, the flint will bend the rays more than the crown.
It is usually found that those substances which bend round the
rays more than others, also separate the different colours one from
another in a greater degree. Newton supposed that the separation
of the colours was in all cases proportional to the bending round
of the rays as a whole. He appears to have formed this idea

PRINCIPLES OF THE CHEMICAL CORRECTION OF OBJECT-GLASSES. 345

from the result of an experiment in which a glass prism was
compensated as to refraction by a hollow prism containing water,
the angle of which being unsuitable, he dissolved in the water
a little sugar of lead ; the consequence was that these two media
behaved very nearly alike; that is, they separated the different
colours in apparently the same proportion as they bent round
the light as a whole. W^re this generally true, as Newton sup-
posed, the construction of an achromatic telescope composed of
two kinds of glass would be impossible, — a very curious example
of a great man missing a great discovery through over hasty
generalization. Mr Hall, of Worcestershire, was the first to dis-
cover Newton's mistake, and he applied his discovery to the
construction of an achromatic telescope, and is said to have made
several ; but the subject was disregarded until the achromatic
telescope was rediscovered by Dollond. The possibility of forming
it depends upon the amount of separation of the colours produced
in the bending round of the light as a whole not bearing in all
substances the same proportion to the bending, as Newton sup-
posed. If I take a prism of crown glass and a prism of flint glass,
and bend the light passing through the crown glass so as to
oppose the flint glass, and imagine the angle of the latter altered
until the bending round of the light by the flint just compensates
the bending round by the crown, a great deal of colour is apparent,
and I must use a slenderer prism of flint if I wish to destroy the
separation of colours produced by the crown, which will leave an
outstanding deviation in the direction of that produced by the
crown glass prism by itself Now a ray of white light passing
through a compound lens consisting of a convex lens of crown
glass and a weaker concave of flint will be deflected very nearly
in the same way as if it passed through two slender prisms,
contained by the tangent planes drawn at the points where the
ray of mean refrangibility strikes the four surfaces ; and we can
thus readily understand how such a compound lens may be
achromatic.

I have here, on this diagram, the different lengths of spectra
which will be produced by slender prisms of the same angle, of
water, crown glass, flint glass, bisulphide of carbon, and oil of cassia.
The separation of the colours, you will see, differs enormously;
and the bending round of the rays is different also. On comparing

346 ON THE PRINCIPLES OF THE CHEMICAL CORRECTION

the oil of cassia with the water, it will be seen that the bending
round of the rays by the former is about twice as great as by
the latter, while the separation of the colours is about eight times
as great. Substances accordingly differ widely in their dispersive
poivers.

If you take a crown and a flint glass, the separation of colours
produced by the flint will be about double the separation produced
by the crown, supposing the angle to be small and of the same
size in each case ; whereas the deviation by the flint will be
greater than that by the crown in the ratio of only 6 to 5, or
thereabouts. But not only do different substances space out the
whole spectrum in different proportions as compared with the
bending round of the light as a whole ; the different parts are
not spaced out in the same relative proportion. In the upper
part of the figure the various spectra represented are all reduced
in the same proportion, so as to occupy the same length from B
to H ] and the places of the intermediate lines G, D, E, F, G are
laid down from the numbers given by Fraunhofer and by the late
Professor Powell. The intermediate parts of the spectra by no
means correspond in these various media. If we take the two
extremities, water and oil of cassia, the blue end of the spectrum
is spaced out far more in proportion by the oil of cassia, while
the red end is more spaced out by the water, in consequence of
the irrationality (or want of proportionality) of dispersion. It is
not possible, then, by a combination of two media such as crown
and flint glass, to construct an object-glass which shall be strictly
achromatic. For if the proportion of the whole spacing-out of the
colours between flint and crown be two to one (which is near
enough for explanation), the proportion at the blue end will be
more than two to one, and the proportion at the red end less, and
consequently the power of the flint glass which will be requisite
to compensate the crown glass, if we want to combine two colours
at the blue end, will be somewhat less than would be necessary if
we wanted to combine two colours at the red end. It is therefore
not so simple a matter as it might at first sight appear to construct
an object-glass as nearly achromatic as may be. We may measure
the refractive indices of both kinds of glass for various points of
the spectrum ; but the question is, when we have got them, what
are we to do with them ? for we may take the refractive indices

OF OBJECT-GLASSES. 347

for one pair of fixed lines, and by employing them for the determi-
nation of the dispersive ratio, get one result, while with another
pair we should get another result.

In this diagram the figures of the spectra produced by crown
and flint glass are reduced in proportion, so as to occupy the same
length from B to H ; and it will be seen that the intermediate
lines in the crown lie a little ahead of those in the flint, the green
part of the spectrum being more refracted by the former. The
consequence is that in an object-glass, as nearly achromatic as may
be, the green part of the spectrum is refracted more than the two
ends. The focal length of the combination is a minimum for a
certain point within the spectrum. The effect of this is seen quite
readily in even a small achromatic object-glass. Turn a small
telescope to an object, such as trees in the shade seen against the
sky, or a piece of perforated zinc, and focus the telescope on it.
Then put it out of focus both ways. When you push in the eye-
piece, you will observe the dark parts of the field invaded by a
purplish light, and when you draw it out by a green. The reason
is that the different parts of the spectrum are not brought strictly
to the same focus. The focal length being a minimum within the
spectrum, it results that when the eyepiece is put out of focus by
pushing in, the two ends of the spectrum are very determinately
out of focus, and the blue and red rays invade the dark parts of
the field ; but when it is drawn out, the green rays are out of focus,
and the dark parts of the field are then invaded by a green light.
In the case of a large object-glass, the sharpness of the image
depends materially upon the part of the spectrum where you turn
it back. It may not be, perhaps, very easy to follow with the
mind's eye the positions of the different foci of the various colours
when you have to conceive them all arranged along the same line,
nor is it easy to represent them without confusion on a diagram.
But I have here got a figure (Fig. 1) which is partly diagrammatic,
and which will serve the purpose. It is not the thing which you
actually see; but it is intended to illustrate what takes place.
Suppose a horizontal line, BB, representing the axis of an object-
glass. Draw vertical lines, and measure along one of them, BH,
the position of any particular colour in some standard spectrum.
Mark off distances representing the positions of the principal fixed
lines fi-om B to H. The lines drawn at these distances parallel to

848 ON THE PRINCIPLES OF THE CHEMICAL CORRECTION

BB represent and correspond to the lines of Fraunhofer. From
the actual place of the focus for any colour in the axis of the lens,
draw a line till it meets the horizontal line corresponding to that
colour. Do the same for all the other points of the spectrum;
then the position of the various foci can be represented to the
mind's eye by the curve which you get by joining all these points.
Having got the curve, if you want to find the position of the focus
of any particular colour in the axis of an object-glass, all you have
to do is this : — look out for the place of the colour in the vertical
line BH, from that place draw a horizontal line to meet the curve,
from the point of section let fall a perpendicular on the line BB ;
then the foot of the perpendicular will be the focus of the colour
in question.

H-

^- H—

f-

\

D

1 D

n

/

B^

1 ! ]L

-" B—

Fig. 1.

This figure, which is carefully drawn to scale, represents the
position of the foci of the different colours in an object-glass the
data for which are given by Fraunhofer. You see it is very far
indeed from being true that the two extremities of the spectrum
are collected at the same point in the axis of the lens. What then
is the condition that has to be satisfied ? According to the data
given by Fraunhofer, it appears that the bending back of the
spectrum takes place between the lines D and E, but much nearer
to D — that is, just at the part of the spectrum where it is brightest.
And that appears to be the condition giving the sharpest form
possible of the achromatic object-glass, namely to make the focal
length of the combination a minimum for the brightest part of the
spectrum, which is situated about one-third of the distance from
the line D towards E. The character of the compensation as to

OF OBJECT-GLASSES. 349

colour in an object-glass can be determined by a very simple test.
Let the telescope be directed to an object bright on one side and
dark on the other, as a chimney seen against the sky, or a piece of
white paper on black velvet. Let the edge separating the bright
from the dark part be brought into the centre of the field and
viewed in focus. Cover half the object-glass by a screen having
its edge parallel to the edge of the object, and then the other
half, alternately, and you will see the dark edge invaded by the
secondary colours — in the one case by the two ends of the spectrum,
and in the other case by the part where the spectrum is bent
back. When the compensation is such as to render vision sharpest,
the tint in the latter case lies about midway between yellow and
green. This is an exceedingly delicate test of achromatization.
If an object-glass of crown and flint glass be thus constructed, the
blue end of the spectrum will be greatly out of focus, and a glass
so achromatized will show a very considerable portion of blue.
But the subject of the optician is to construct a glass which shall
give the sharpest vision, never heeding if there should be some
blue outstanding.

In order to unite the neighbouring colours at the blue end,
the flint lens must be a little weaker than the above. Imagine
a crown glass lens compensated by a flint, the power of which
continuouslj' increases. The foci for all the colours will be thrown
outwards as the flint lens increases in strength ; but the blue end
will be thrown out faster than the red. An uncompensated crown
lens would have its focus for the blue nearer to the glass than the
red ; and if you oppose it by a flint concave glass of increasing
power, you will throw out both foci, as well as the foci for the
intermediate colours, though not exactly at the same rate for all.
Reverting now to the diagrammatic construction, you will see that
the distribution of the foci is represented at fu-st by a curve not
much differing fi-om a straight line. This curve will be drawn
outwards ; but the combination will begin to be achromatic at
the blue end of the spectrum ; and the curve having first been
nearly straight, will begin to turn at the end representing the
blue, and there will be a minimum of focal length at the point
of extreme violet. As the flint lens gets more powerful, the focus
is bent back near the middle of the spectrum, then near to the
red; and by bending it back altogether, you would again get

350 ON THE PRINCIPLES OF THE CHEMICAL COEEECTION

nearly a straight line, the blue foci being now further from the
glass than the red. The only option the optician has is to choose
what part of the spectrum it shall be where the focal length of
the combination is a minimum. If you wish to determine very
exactly what the requisite powers of the two lenses shall be, there
is a very simple way of effecting it. Suppose you take the crown
and flint glasses to be employed, and form little prisms of both
for the purpose of measurement. Measure the refractive indices
of the two glasses for each of three fixed lines (bright or dark) in
the spectrum. It does not matter what lines are taken, so that
they divide the spectrum pretty fairly; and bright lines, which
are easily produced artificially, will do just as well as dark. By
means of these six indices the ratio of the dispersive powers which
must be employed to give the best result may be simply deter-
mined, as I have shown in the Report of the British Association
for 1855, Part II, p. 14*. I have also employed another method
in determining the ratios of the dispersive powers of different
kinds of glass, which I devised in the course of some researches
made by the late Mr Vernon Harcourt and myself, and which is
no less accurate than that which involves the measurement of
fixed lines, and is also easier, as it can be used with common
daylight. This, however, I shall not trouble you with, and I shall
now pass on to speak of the chemical rays.

What do we mean by chemical rays ? We know that light
or, possibly, something which accompanies light, is capable of
producing chemical changes, such as, for instance, those effected
on salts of silver, on which the whole practice of photography is
founded. At one time it was supposed by many that certain rays
were distinctly chemical in their action, while others accompany-
ing them were simply luminous; but I suppose hardly anyone
imagines this now-a-days. When a pure solar spectrum is formed,
we see in it the fixed lines called Fraunhofer's lines. If the
spectrum, instead of being viewed, is received on a sensitized
plate, we get an impression photographed in which far more dark
lines are exhibited than seen directly by the eye, the lines in fact
extending a long way beyond the extreme violet. The spectrum
given by the electric discharge goes enormously beyond even this
— to the extent of six or eight times the length of the visible

[* Supra, p. 64.]

OF OBJECT-GLASSES. 351

spectrum. This, however, need not be considered as part of my
subject, and I shall simply confine myself to the solar spectrum.
Whether you see the spectrum with the ej'e or obtain an impres-
sion of it on a sensitized plate, you get the same series of dark
lines in all that part of the spectrum that affects both the eye
and the plate ; the only difference is that the part of the spectrum
towards the red end is more or less inefficient in producing
chemical changes (to what extent depends upon the particular
nature of the preparation), while at the other end of the spectrum
the chemical effects are produced far beyond the extreme violet.
There is no reason then for supposing there are distinct rays,
capable of producing chemical effects, mixed with the visual rays ;
but everything leads us to believe that excitement of the sensation
of light and the production of chemical changes are merely different
effects produced by the same agent. The spectral rays are also
capable of producing other effects, as when certain objects placed
in part of the spectrum give out light for the time being, as
though they were self-luminous, the light so emitted being
different altogether in its nature from those rays which are active
and which cause the objects to give out that light. The light
is in general of a different colour from that of the incident rays ;
and not only so, but it is a matter of perfect indifference whether
the rays which produce it are visible or not. Through the kind-
ness of Mr Spiller I am enabled to show you this property.
[Edperime)it.]

"When I spoke just now of chemical rays my meaning was,
rays contemplated with reference to their chemical and not to
their visual effects. I cannot, of course, speak absolutely of the
amount of such effects, as that would depend upon the nature
of the particular substance acted upon. In the case of chlorophyll,
for example. Sir J. Herschel found by experiment that it was
nearly the extreme red which acted most powerfully in bleaching-
the substance. On the other hand, salts of silver are influenced
chietlv b\- the rays at the violet or more refrangible end; and
even among the silver salts there is a difference of action, to a
certain extent, according to the salt. Thus the iodide of silver
is affected powerfully by the violet rays from G to H, while the
bromide appears to be affected a great deal lower down. I may
also add that Dr Draper found that the process which lies at the

352 ox THE PRINCIPLES OF THE CHEMICAL CORRECTION

foundation of all life, the decomposition of the carbonic acid of
the air % vegetables under the influence of the solar rays, is
produced chiefly by the brightest part of the spectrum. \ou
will understand why I call this process fundamental as regards
life, because the growth of vegetables depends upon it — with the
exception of the fungi, which may be called vegetables of prey ;
and animals live, either directly or indirectly, upon vegetables.
The chemical action, then, of rays depends upon the substance
acted upon ; and I will now confine myself to the salts of silver
used in photography, which are mainly affected by the violet end
of the spectrum, the mean centre of action being perhaps midway
between G and H, or perhaps nearer to G. Suppose you wanted
to construct an object-glass in which the most efficient chemical
rays should be as much as possible concentrated to one focus;
j'ou must proceed according to the very same principle as you
did with reference to the visual rays ; only instead of bending
back the spectrum (or making the focal length of the combination
a minimum) for the brightest part of the spectrum considered
visually, you must take the brightest part of the spectrum
considered chemically, bending it back somewhere between G
and H\ that is to say, the lens must be very determinately under-
corrected. If an objective be as nearly as possible chemically
achromatic, it will be inconvenient for common use in photography,
because the chemical and visual foci will not coincide. By the
visual focus of an imperfectly compensated lens I mean that
place where the best visual image is produced, which would lose
sharpness by either going outwards or inwards. If a lens so
achromatized were focused it would give you a sharp chemical
image ; but the chemical and visual foci would not coincide, the
chemical lying inside the visual, whereas in a lens corrected so
as to give sharpest vision the chemical focus would lie a little
outside the visual ; and of course for some intermediate correction
the two foci would coincide.

The distance between the chemical and visual foci, supposing
they do not coincide, will not be the same for objects at different
distances, but will vary nearly as the square of the distance
between the image and the object-glass. The inconvenience
thence arising in the ordinary practice of photography, combined
with the circumstance that for pleasing effects the last degree

OF OBJECT-GLASSES. . 353

of sharpness of image is not required, renders it undesirable, in
aiming at an ideal perfection, to surrender the convenience of
making the two foci coincide. If your object, however, were
celestial photography, the case would be different; for there the
object is always practically at a uniform distance, and therefore,
if the chemical and visual foci do not coincide, the distance
between the two will always be the same, and can easily be
allowed for; and in a large instrument our aim should be to
obtain the sharpest possible chemical definition. The part of the
spectrum where it must be doubled back in order to give the
sharpest chemical definition, or else a coincidence of the chemical
and visual foci, must be determined by experiments made once
for all ; and then an object-glass corrected in either of these ways
can be constructed by simple rules. I should suppose that for
the former the place of minimum focal length ought to lie about
midway between G and H, and for the latter about one-third of
the distance FG from F towards G.

The effect of irrationality constitutes the chief obstacle to the
perfection of a properly constructed achromatic telescope ; and it
becomes an interesting question whether it is possible to remove
this defect. The possibility of doing so was shown long ago by
Dr Blair in the case of lenses composed of at least one fluid. He
even constructed one in which the aperture was as much as one-
third of the focal length, and found there was no perceptible
colour ; but no solid media possessing similar properties had been
discovered. Mr Vernon Harcourt for a long period was engaged
on this very subject; and in the course of that research nearly
200 prisms, composed of different experimental glasses, were
formed and examined. For a long time there appeared to be
very little hope of getting rid of the secondary spectra; there
seemed to be a general law, connecting the relative extension of
the blue and red ends of the spectrum with the dispersive power
of the medium, which would render the destruction practically
impossible. Hardly anything can be done in this direction with
flint glasses of greater or less dispersive power; you must have
recourse to glass of different composition altogether. It was
found at length that molybdic and titanic acids have the power
of conferring on glass the property of extending the blue, as
compared with the red end of the spectrum, much more than
s. IV. 23

354 THE CHEMICAL CORRECTION OF OBJECT-GLASSES.

corresponds to the dispersive power; while on the other hand
terborate of lead, which in dispersive power nearly agrees with
flint glass, lies decidedly nearer to crown than to flint in point
of irrationality. By the use of such glasses it would be possible
to destroy the secondary dispersion in an object-glass. Terborate
of lead is slightly yellow, is too soft to be easily worked to a
correct form, and is rather liable to tarnish, though the last defect
does not much signify when the terborate is used for the middle
lens of a triple, of which the adjacent surfaces fit each other
and are cemented. The most promising combination, however,
appeared to be a titanic glass in lieu of crown, achromatized by
a light flint. In consequence of the elevation of dispersive power
produced by the introduction of titanic acid, the curvature would
be rather severe if the objective were constructed of only two
lenses ; but by making the objective a triple, the convex lens
being replaced by two convex lenses placed first and third, the
curvatures to be encountered would be no greater than in an
ordinary double.

On the Improvement of the Spectroscope. By Thomas
Grubb, F.R.S. Note appended by Prof. Stokes.

[From the Proceedings of the Royal Society, xxii, April 30, 1874, p. 309.]

If a ray of light be refracted in any manner through any
number of prisms arranged as in a spectroscope, undergoing, it
may be, any number of intermediate reflections at surfaces parallel
to the common direction of the edges of the prisms — or, more
generally, if a ray be thus refracted or reflected at the surfaces of
any number of media bounded by cylindrical surfaces in the most
general sense (including, of course, plane as a particular case), the
generating lines of which are parallel, and for brevity's sake will
be supposed vertical, and if a be the altitude of the ray in air,
a', a", ..., its altitudes in the media of which the refractive indices
are ya', fi,", ..., then

(1) The successive altitudes will be determined by the
equations

sin a = /u.' sin a' = //," sin a" = . . . ,
just as if the ray passed through a set of parallel plates.

(2) The course of the horizontal projection of the ray will be
the same aS would be that of an actual ray passing through a set

of media of refractive indices , — , ... instead of

cos a cos a

fi, fj," As a' < a, the fictitious index is greater than the

actual, and therefore the deviation of the projection is increased

by obliquity.

These two propositions, belonging to common optics, place the
justice of Mr Grubb's conclusions in a clear light*.

[* Namely, that the curvature of the lines, when the slit is straight, is a function
of (in fact proportional to) the dispersion. For developments, cf. Lord Kayleigh,
Phil. Mag. viii, 1879, Scientific Papers, i, p. 542, and Larmor, Proc. Gamb. Phil.
Soc. VII, 1890, p. 85.]

23—2

On the Construction of a perfectly Achromatic
Telescope.

[From the Report of the British Association for the AdvaticeTnent
of Science, Belfast, 1874.]

At the Meeting of the Association at Edinburgh in 1871, it
was stated that it was in contemplation actually to construct
a telescope, by means of disks of glass prepared by the late
Mr Vernon Harcourt, which should be achromatic as to secondary
as well as primary dispersion. This intention was subsequently
carried out, and the telescope, which was constructed by
Mr Howard Grubb, was now exhibited to the Section. The
original intention was to construct the objective of a phosphatic
glass containing a suitable percentage of titanic acid, achromatized
by a glass of terborate of lead*. As the curvature of the convex
lens would be rather severe if the whole convex power were
thrown into a single lens, it was intended to use two lenses of this
glass, one in front and one behind, with the concave terborate
of lead placed between them. It was found that, provided not
"more than about ^ of the convex power were thrown behind, the
adjacent surfaces might be made to fit, consistently with the
condition of destroying the spherical as well as the chromatic
aberration. This would render it possible to cement the glasses,
and thereby protect the terborate, which was rather liable to
tarnish.

At the time of Mr Harcourt's death, two disks of the titanic
glass had been prepared, which it was hoped would be good
enough for employment, as also two disks of terborate. These
were placed in Mr Grubb's hands. On polishing, one of the
titanic disks was found to be too badly striated to be employed ;
the other was pretty fair.- As it would have required a rather

* The percentage of titanic acid was so chosen that there should be no
irrationality of dispersion between the titanic glass and the terborate.

THE CONSTRUCTION OF A PERFECTLY ACHROMATIC TELESCOPE. 357

severe curvature of the first surface, and an unusual convexity of
the last, to throw the whole convex power into the first lens, using
a mere shell of crown glass behind to protect the terborate,
Prof Stokes thought it more prudent to throw about ^ of the
whole convex power into the third or crown-glass lens, though at
the sacrifice of an absolute destruction of secondary dispersion,
which by this change from the original design might be expected
to be just barely perceptible. Of the terborate disks, the less
striated happened to be slightly muddy, from some accident in the
preparation ; but as this signified less than the strise, Mr Grubb
deemed it better to employ this disk.

The telescope exhibited to the Meeting was of about 2|- inches
aperture and 21 inches focal length, and was provided with an
objective of the ordinary kind, by which the other could be
replaced, for contrasting the performance. When the telescope
was turned on to a chimney seen against the sky or other suitable
object, and half the object-glass covered by a screen with its edge
parallel to the edges of the object, in the case of the ordinary
objective vivid green and purple were seen about the two edges,
whereas with the Harcourt objective there was barely any per-
ceptible colour. It was not, of course, to be expected that the
performance of the telescope should be good, on account of the
difficulty of preparing glass free from strise, but it was quite
sufficient to show the possibility of destroying the secondary
colour, which was the object of the construction.

On the Optical Properties of a Titano-Silioic Glass. .

[From the Report of the British Association for the Advancement of Science,

Bristol, 1875.]

At the Meeting of the Association at Edinburgh in 1871,
Professor Stokes gave a preliminary account of a long series of
researches in which the late Mr Vernon Harcourt had been
engaged on the optical properties of glasses of a great variety of
composition, and in which, since 1862, Professor Stokes had
cooperated with him*. One object of the research was to obtain,
if possible, two glasses which should achromatize each other
without leaving a secondary spectrum, or a glass which should
form with two others a triple combination, an objective composed
of which should be free from defects of irrationality, without
requiring undue curvature in the individual lenses. Among
phosphatic glasses, the series in which Mr Harcourt's experiments
were for the most part carried on, the best solution of this
problem was offered by glasses in which a portion of the phos-
phoric was replaced by titanic acid. It was found, in fact, that
the substitution of titanic for phosphoric acid, while raising, it is
true, the dispersive power, at the same time produces a separation
of the colours at the blue as compared with that at the red end
of the spectrum, which ordinarily belongs only to glasses of a
much higher dispersive power. A telescope made of disks of
glass prepared by Mr Harcourt was, after his death, constructed
for Mrs Harcourt by Mr Howard Grubb, and was exhibited to the
Mathematical Section at the late Meeting in Belfast. This tele-
scope, which is briefly described in the ' Report 'f, was found fully
to answer the expectations that had been formed of it as to
destruction of secondary dispersion.

Several considerations seemed to make it probable that the
substitution of titanic acid for a portion of the silica in an
ordinary crown glass would have an effect similar to what had

* Report for 1871. Transactions of the Sections, p. 38 [supra, p. 339].
t Ihid. 1874, Trans. Sect. p. 26 [supra, p. 357].

ON THE OPTICAL PROPERTIES OF A TITANO-SILICIC GLASS. 359

been observed in the phosphatic series of glasses. Phosphatic
glasses are too soft for convenient employment in optical instru-
ments ; but should titano-silicic glasses prove to be to silicic what
titano-phosphatic glasses had been found to be to phosphatic, it
would be possible, without encountering any extravagant curva-
tures, to construct perfectly achromatic combinations out of glasses
having the hardness and permanence of silicic glasses ; in fact
the chief obstacle at present existing to the perfection of the
achromatic telescope would be removed, though naturally not
without some increase to the cost of the instrument. But it
would be beyond the resources of the laboratory to work with
silicic glasses on such a scale as to obtain them free from strise, or
even sufficiently free to permit of a trustworthy determination of
such a delicate matter as the irrationality of dispersion.

When the subject was brought to the notice of Mr Hopkinson,
he warmly entered into the investigation; and, thanks to the
liberality with which the means of conducting the experiment
were placed at his disposal by Messrs Chance Brothers, of
Birmingham, the question may perhaps be considered settled.
After some preliminary trials, a pot of glass free from strise was
prepared of titanate of potash mixed with the ordinary ingredients
of a crown glass. As the object of the experiment was merely
to determine,' in the first instance, whether titanic acid did or did
not confer on the glass the unu.sual property of separating the
colours at the blue end of the spectrum materially more, and at
the red end materially less, than corresponds to a similar disper-
sive power in ordinary glasses, it was not thought necessary to
employ pure titanic acid; and rutile fused with carbonate of
potash was used as titanate of potash. The glass contained about
7 per cent, of rutile ; and as rutile is mainly titanic acid, and
none was lost, the percentage of titanic acid cannot have been
much less. The glass was naturally greenish, from iron contained
in the rutile ; but this did not affect the observations, and the
quantity of iron would be too minute sensibly to affect the
irrationality.

Out of this glass two prisms were cut. One of these was
examined as to irrationality by Professor Stokes, by his method
of compensating prisms*, the other by Mr Hopkinson, by accurate

[* Supra, p. 341

360 ON THE OPTICAL PROPERTIES OF A TITANO-SILICIC GLASS.

measures of the refractive indices for several definite points in the
spectrum. These two perfectly distinct methods led to the same
result— namety, that the glass spaces out the more as compared
with the less refrangible part of the spectrum no more than an
ordinary glass of similar dispersive power. As in the phosphatic
series, the titanium reveals its presence by a considerable increase
of dispersive power ; but, unlike what was observed in that series,
it produces no sensible effect on the irrationality. The hopes,
therefore, that had been entertained of its utility in silicic glasses
prepared for optical purposes appear doomed to disappointment.

P.S. — Mr Augustus Vernon Harcourt has now completed an
analytical determination which he kindly undertook of the titanic
acid. From 2'171 grammes of the glass he obtained "IS gramme
of pure titanic acid, which is as nearly as possible 6 per cent.

On a Phenomenon of Metallic Reflection.

[From the Report of the British Association for the Advancement of Science,

Glasgow, 1876.]

The phenomenon which I am about to describe was observed
by me many years ago, and may not improbably have been seen
by others ; but as I have never seen any notice of it, and it is in
some respects very remarkable, I think that a description of it
will not be unacceptable.

When Newton's rings are formed between a lens and a plate of
metal, and are viewed by. light polarized perpendicularly to the
plane of incidence, we know that, as the angle of incidence is
increased, the rings, which are at first dark-centred, disappear on
passing the polarizing angle of the glass, and then reappear
white-centred, in which state they remain up to a grazing inci-
dence, when they can no longer be followed. At a high incidence
the first dark ring is much the most conspicuous of the series.

To follow the rings beyond the limit of total internal reflection
we must employ a prism. When the rings formed between glass
and glass are viewed in this way, we know that as the angle of
incidence is increased the rings one by one open out, uniting with
bands of the same respective orders which are seen beneath the
limit of total internal reflection ; the limit or boundary between
total and partial reflection passes down beneath the point of
contact, and the central dark spot is left isolated in a bright
field*.

Now when the rings are formed between a prism with a
slightly convex base and a plate of silver, and the angle of
incidence is increased so as to pass the critical angle, if common
light be used, in lieu of a simple spot we have a ring, which
becomes more conspicuous at a certain angle of incidence well
[* Ante, Vol. ii, p. 358.]

362 ON A PHENOMENON OF METALLIC KEFLECTION.

beyond the critical angle, after which it rapidly contracts and
passes into a spot.

As thus viewed the ring is, however, somewhat confused. To
study the phenomenon in its purity we must employ polarized
light, or, which is more convenient, analyze the reflected light by
means of a Nicol's prism.

When viewed by light polarized in the plane of incidence, the
rings show nothing remarkable. They are naturally weaker than
with glass, as the interfering streams are so unequal in intensity.
They are black-centred throughout, and, as with glass, they open
out one after another on approaching the limit of total reflection
and disappear, leaving the central spot isolated in the bright field
beyond the limit. The spot appears to be notably smaller than
with glass under like conditions.

With light polarized perpendicularly to the plane of incidence,
the rings pass from dark-centred to bright-centred on passing the
polarizing angle of the glass, and open out as they approach the
limit of total reflection. The last dark ring to disappear is not,
however, the first, but the second. The first, corresponding in
order to the first bright ring within the polarizing angle of the
glass, remains isolated in the bright field, enclosing a relatively,
though not absolutely, bright spot. At the centre of the spot the
glass and metal are in optical contact, and the reflection takes
place accordingly and is not total. The dark ring, too, is not
absolutely black. As the angle of internal incidence increases by
a few degrees, the dark ring undergoes a rapid and remarkable
change. Its intensity increases till (in the case of silver) the ring
becomes sensibly black ; then it rapidly contracts, squeezing out,
as it were, the bright central spot, and forming itself a dark spot,
larger than with glass, isolated in the bright field. When at its
best it is distinctly seen to be fringed with colour, blue outside,
red inside (especially the former), showing that the scale of the
ring depends on the wave-length, being greater for the less refran-
gible colours. This rapid alteration taking place well beyond the
critical angle is very remarkable. Clearly there is a rapid change
in the reflective properties of the metal, which takes place, so to
speak, in passing through a certain angle determined by a sine
greater than unity.

ON A PHENOMENON OF METALLIC REFLECTION. 363

I have described the phenomenon with silver, which shows it
l>est; but speculum-naetal, gold, and copper show it very well,
while with steel it is far less conspicuous. When the coloured
metals gold and copper are examined by the light of a pure
spectrum, the ring is seen to be better formed in the less than in
the more refrangible colours, being more intense when at its best ;
while with silver and speculum-metal there is little difference,
except as to size, in the different colours. Haematite and iron
pyrites, which approach the metals in opacity and in the change
of phase which they produce by reflection of light polarized
parallel relatively to light polarized perpendicularly to the plane
of incidence, do not exactly form a ring isolated in a bright field ;
but the spot seen with light polarized perpendicularly to the
plane of incidence is abnormally broad just about the limit of
total reflection, and rapidly contracts on increasing the angle
of incidence.

It seemed to me that a sequence may be traced from the
rapidly contracting rings of diamond seen in passing the polarizing
angle of that substance, through the abnormally broad and rapidly
contracting spot seen with iron pyrites just about the limit of
total reflection, and the somewhat inconspicuous ring of steel seen
a little beyond the limit, to the intense rapidly contracting ring of
silver seen considerably beyond the limit. If so, the full theory
of the ring will not be contained in the usually accepted formulae
for metallic reflection, modified, as in the case of transparent sub-
stances, in accordance with the circumstance that the incidence
on the first surface of the plate of air is beyond that of total
reflection.

MacCuUagh was the first to obtain the formulae for metallic
reflection, showing that they were to be deduced from Fresnel's
formulae by making the refractive index a mixed imaginary,
though they are usually attributed to Cauchy, who has given
formulae differing firom those of MacCullagh merely in algebraic
detail. As regards theory, Cauchy made an important advance
on what MacCullagh had done in connecting the peculiar optical
properties of metals with their intense absorbing power*. Now
Fresnel's formulae do not include the phenomena discovered by

* The apparent difference between MacCullagh and Cauchy as to the values of
the refractive indices of metals is merely a question of arbitrary nomenclature.

364 ON A PHENOMENON OF METALLIC REFLECTION.

Sir George Airy, which are seen in passing the polarizing angle
of diamond, and which have been more recently extended by
M. Jamin to the generality of transparent substances * ; and if
these pass by regular sequence to those I have described as seen
with metals beyond the limit of total internal reflection, it follows
that the latter would not be completely embraced in the appli-
cation of Fresnel's formulae, modified to suit an intensely absorbing
substance and an angle of incidence given by a sine greater than
unity t.

[* Traced by Lord Bayleigh for the case of water, mainly, but not entirely, to
the presence of a surface film of transition arising from contamination, Phil. Mag.
1892, Scientific Papers, iii, p. 496 ; reference is there made to similar observations
by Drude on recently formed cleavage planes in crystals.]

t It was long ago observed, both by Professor MaoCallagh and Dr Lloyd, that
when Newton's rings are formed between a glass lens and a metallic plate, the first
dark ring surrounding the central spot, which is comparatively bright, remains
constantly of the same size at high incidences, although the other rings, like
Newton's rings formed between two glass lenses, dilate greatly as the incidence
becomes more oblique. See Proceedings of the Royal Irish Academy, Vol. i, p. 6.

Prklimixaky Note ox the Compouxd Nature ok the Lixe-
Spectka of Elemestart Bodies. By J. N. Lockyek. FK.S.
(Exti-aci.^

[From the Proaetiii.ijf 0/ tAe Sciml Stvktu. xxiv, ISTo, p. :ioi]

MareA 'X ISTd

My bear Lockyek, — You miirht perh;ips like that I shoxild
put on paper the substance of the remarks I made las^t night as to
the evidence of the dissociation of calcium.

When a solid body such a* a platinum wire, traversed hy a
volt;vie current, is heated to iucandesoence. we know that as the
wmperataire increases, not only does the radiation of each par-
ticular refraugibility absolutely inci^ase, but the proportion of the
radiations of the different refcsmgibilities is changed, the prv>portion
of the higher to the lower increasing with the teiuperatm-e. It
w^uld be in accontuice with analogs t-o supp^^se that as a rule the
s;\me would take place in an ineandeseem sarface. thotigh ia this
case the spectrum would be discontinuous instead of continuous*
Thus if A, B, C. D. E denote conspicuous bright hnes, of in-
ore;vfin>j refi-angibility. in the spectrum of the vapour, it might
very well be that at a oompvraTively low tempei!\ture A should be
the brightest ivnd the most persistent; a: a higher temj)eratxu'e.
while all were brighter thtui before, the relative brightness might
be changed. ;uid C might be the brightest and the most j>ei^istent.
:uid at a still higher temperatv.re E. If. now, the quiuitity of
j>?rsis:ence were in each c;is^e reduced till all lines but one ilis-
appeiued. the outstanding line might be A at the lowest
temperature. C at the higher, E at the highest. If so. iu case
the vaix)ur showed its presence by absoi-ption but not emission,
it follows, from the con>?spondence between absorption and

[* KsTsex regarvfe tliij as stiii ims^tUed, Ha«*ti,-S d^-r 5rv,.-fros«>jw>. n, 1;>02,
p. 331.]

366 INFLUENCE OF TEMPERATURE ON LINE-SPECTRA.

emission, that at one temperature the dark line which would be
the most sensitive indication of the presence of the substance
would be A, at another C, at a third E. Hence, while I regard
the facts you mention as evidence of the high temperature of the
sun, I do not regard them as conclusive evidence of the dissociation
of the molecule of calcium.

Yours sincerely,

G. G. Stokes.

APPENDIX.

Correspondence of Prof. G. G. Stokes and Prof. W.
Thomson on the nature and possibilities of spectrum
analysis.

[The following letters from Lord Kelvin came to light in
arranging the scientific correspondence of Sir George Stokes. On
supplying their dates to Lord Kelvin, he was able to extend the
record. The parts relating to spectrum analysis are here printed,
with Lord Kelvin's permission, as supplementary to the extracts
contained in pp. 127 — 136 of this volume : cf p. 136. It may be
recalled that Prof Stokes became Secretary of the Royal Society
in 1854]

2 College, Glasgow
Feb. 20, 1854.

My dear Stokes

It is a long long time since I have either seen you or
heard from you, and I want you to write to me about yourself and
what you have been doing since ever so long. Have you made
any more revolutions in Science ? or done any of the exp' research
on the friction of air? I saw a notice of your lecture at the
R. I. Tell me any new discoveries you have made, &c. However
I do not mean to impose upon you by demanding all this, but if
there is anything short and good you can tell me I shall be glad
to hear it. I want to ask you about artif lights and the solar
dark lines. Is there any other substance than soda that is related
to -D ? Are bright lines corresponding to it to be seen where soda
is not present ? Have any terrestrial relations to any other of the
solar dark lines been discovered (or to the dark lines of any of the
stellar spectra)? Are all artificial lights subject to dark lines?
I should be greatly obliged by your telling me in a word or two
what is known on these questions, which I suppose you will easily do.

Yours always truly

William Thomson

368 CORRESPONDENCE OF G. G. STOKES AND W. THOMSON

Pembroke Coll. Cambridge
Feb. 24tk, 1854.

My dear Thomson

Now for your questions. I am not aware that
there is any pure substance known to produce the bright line Z)
except soda. See end * It would be extremely difficult to
prove, except in the case of gases or substances volatile at a not
very high temperature, that the bright line D, if observed in a
flame, was not due to soda, such an infinitesimal quantity of soda
would be competent to produce it. It is very common in ordinary
artificial flames (such as a candle &c.) but I think in such cases it
may be attributed with probability to soda. In a spirit lamp
I feel satisfied it is derived from the wick, for I find that alcohol
burnt in a clean saucer does not give it, except perhaps a flicker
now and then. Miller told me (and I have verified the observa-
tion) that it is not found in an oil lamp, and I find that when the
wick of a candle is cut short, so as to be surrounded by gas, and
not to project into the luminous envelope where the combustion
goes on, D disappears.

Sir D. Brewster states (Brit. Ass. Report, 1842, p. 15 of the
2nd part) that the flame of deflagrating nitre contains bright
lines corresponding to the dark lines A and B of Fraunhofer', and
implies that other of the bright lines of this flame correspond to
the dark lines of the solar spectrum. I saw somewhere a state-
ment, I think, by Sir D. B., that the flame, I think of burning
potassium, certainly some flame in which potash was concerned,
gave 7 bright lines corresponding to the dark lines forming
Fraunhofer's group a. These are the only cases I know of in
which identification has yet been established, but the subject has
barely yet been attacked. A vast deal of measurement has yet to
be gone through. I think it likely that very interesting results
will come out.

You will find in Moigno's Repertoire d'optique modeme (part III
p. 1237) much information on the subject of your questions.

You ask "Are all artificial lights subject to dark lines?" No,
it is quite the exception. When there are lines of any kind it is
usually bright lines. The flame of nitrate of strontia (i.e. a flame
coloured red by nitrate of strontia) shews such dark Unes in the
red, but then these same dark lines are found in the spectrum of

[' See however p. 135 supra.'\

ON THE POSSIBILITIES OF SPECTRUM ANALYSIS. 369

common light transmitted across the flame, so that they appear
to be due, not to the non-production of light of definite or almost
definite refi-angibility, but to its absorption by a certain gas or
gases produced by combustion.

*Moigno mentions (p. 1244) that Foucault in experiments
with the electric pile, as well as Miller in observations on the flame
of a spirit lamp to which various salts had been added, was struck
with the constant occurrence of the bright line D.

Yours very truly

G. G. Stokes

2 College, Glasgow
March 2, 1854.

My dear Stokes

Many thanks for your most satisfactory answers to all
my questions. It was by a "lapsus pennae" that I wrote dark
lines instead of bright lines, when I asked if all artif sources are
subject to them.

I think it is really a splendid field of investigation, that of the
relations betw. the bright lines of artif light and the dark lines
of the solar spectrum. Dont you think anyone who takes it up
might find a substance for almost each one of the principal dark
lines by examining the eflfects of all salts on the flame of burning
alcohol, or on other artif lights ? I think it will lead us to a
qualitative analysis of the sun's atmosphere. Do you think the
supposed tendency to vibrate D of other salts besides common
salt mentioned by Moigno sufliciently established to want no con-
firmation. I am much disposed to doubt it, from what you have
told me. I have tried a spirit lamp behind a slit through which
sun light is coming and the efl'ect is most decided. If I remember
right the magnifying power I used was sufficient to divide D ; but
certainly whatever it was the light of the sp. lamp corresponded
as exactly as could be observed with the dark of the solar D.
It was curious to observe, the dark line not sensibly illuminated
by the full light of the sp. lamp coming through it, (the brightness
on each side was so great) but a line of light above and below the
solar spectrum appeared as an exact continuation of the dark
line D. Will you not take up the whole subject of spectra of

s. IV. 24

370 COEEESPONDENCE OF G. G. STOKES AND W. THOMSON

solar and artif lights, since you have already done so much on it ?
I am quite impatient to get another undoubted substance besides
vapour of soda in the sun's atmosphere. What you tell me looks
very like as if there is potash too. I think copper &c. would be
hopeful. The galvanic arc shows rather broad bands of green for
copper than fine lines. Salts of copper sh* be tried on flame.
Do try iron too. There must be a great deal of that about the
sun, seeing we have so many iron meteors falling in, and there
must be immensely more such falling in to the sun. I find the
heat of combustion of a mass of iron w* be only about -g^^jyu of
the heat derived from potential energy of gravitation, in approach-
ing the sun. Yet it w'' take 2000 pounds of meteors per sq. foot
of the sun, falling annually to account for his heat by gravitation
alone.

Yours very truly

W. Thomson

Pembroke Coll. Cambridge
March 7th, 1854.

My deae Thomson

There are one or two points which occur to me with
reference to your last letter which I wish to mention.

Miller told me of an experiment of his which he performed
many years ago for testing the coincidence of the dark double
line I) of the solar spectrum with the bright line in the light of a
spirit lamp. The sun's light was introduced by a slit, and refracted
by 3 good prisms, and then viewed through a telescope with a
pretty high mag. power. The two lines forming the line D were
"like that" (as he said holding two of his fingers about 3 inches
apart) and he counted six fixed lines between them. The whole
apparatus was left untouched till dusk, and then a spirit lamp was
placed behind the slit. This gave two bright lines coinciding, as
near as measurement could give, with the two dark lines D.

Miller seemed disposed to regard this as an accidental coin-
cidence. It seemed to me that a plausible physical reason might
be assigned for it by supposing that a certain vibration capable of
existing among the ultimate molecules of certain ponderable
bodies, and having a certain periodic time belonging to it, might

ON THE POSSIBILITIES OF SPECTRUM ANALYSIS. 371

either be excited when the body was in a state of combtistion, and
thereby give rise to a bright line, or be excited by luminous
vibrations of the same period, and thereby give rise to a dark line
by absorption.

But we must not go on too fast. This explanation I have not
seen, so far as I remember, in any book, nor do I know a single
experiment to justify it. I am not aware that any absorption
bands seen in the spectrum of light transmitted across any vapour
that has been examined have been identified with D.

I gave you Moigno's statement about Foucault's exper' but
I confess I am sceptical. I want more explicit proof of the absence
of soda in some shape.

If I remember right chloride of copper on packthread, put
into the flame of a spirit lamp, gave two broadish green luminous
bands (besides two more refrangible) which are each resolved,
even by the naked eye with a highly dispersive prism of 60°, into
a system of lines. Metallic copper in the voltaic arc is I believe
somewhat different.

Yours very truly

G. G. Stokes

2 College, Glasgow
March 9, 1854.

My dear Stokes

It was Miller's experiment (w'' you told me about
a long time ago), which first convinced me there must be a
physical connection between agency going on in and near the sun,
and in the flame of a sp. lamp with salt on it. I never doubted
after I learned Miller's experiment that there must be such a
connection, nor can I conceive of any one knowing Miller's experi-
ment and doubting. There is I suppose something in Miller's
mind inconceivable to me. You told me too your mechanical
explanation, which struck me at the time, and for years has been
taking a deeper and deeper hold [on] me.

If it could only be made out that the bright line D never
occurs without soda, I sh* consider it as perfectly certain that
there is soda or sodium in some state in or about the sun. If

372 CORRESPONDENCE OF G. G. STOKEJS AND W. THOMSON

bright lines in any other flames can be traced as perfectly as
Miller did in his case, to agreement with dark lines in the solar
spectrum, the connection w*^ be equally certain, to my mind. I
quite expect a qualitati^-e analysis of the sun's atmosphere by
experiments like Miller's on other flames.

Could you make anything decided (mathematical investigation)
of your mechanical theory ? Can you investigate mechanically the
undulator^- theory of radiant heat, e.g. a hot black ball, in the
centre of a hollow black sphere, each of given temperature, what
is the wave length, or lengths, of the undulations ? The wave
lengths, as experiment shows, are less the higher the temperature
of the hot body. It is a splendid subject for mathematical in-
vestigation.

Yours always truly

W. Thomson

Senate-House Cambridge
March 8t/i, 1854.

My dear Thomson

Miller is now in the Senate-House examining for the
Natural Sciences Tripos. I spoke to him about the fixed line D.
I find he has not published the observation. I find I did him
wrong in supposing that he regarded the coincidence as accidental :
he supposes that there is some cause for it.

Yours very truly

G. G. Stoke.s

Pembroke Coll. Cambridge
March 28th, 1854.

My dear Thomson

Since I wrote last I have been somewhat of an invalid.
Between the sailing of the Baltic fleet one day, Westminster
Abbey the next, and one or two other things of a similar nature,
I caught cold which produced a severe inflammation in my left
eye. I was confined to my room for a week, and spent some days

ox THE Pt^SSlBlLlTlES OF SiPEOTRVM AXALYSIS. 37o

in darkness, uot however making erperiments on invisible light.
My eye i# now well, so that I can read aad write as usual.

I eertaiiilv think that Foucault's .Jc«. observations as to the
general occurrence of the lised line D require confirmation. I am
dispose!.! to suspect contamination with sodium, or some compound
of sodium. At the s;\me time, although the compounds of sodium
do produce this bright line in flames. I do not think that we can
necessaiily infer the pivsenoe of sodium or any compound of
siidium fi\»ni the appearance of this bright line. I will explain my
meaning presently.

As ti-' the augmentation of the refrangihility of the emitted
rays when the temperature of a body is more and more raised,
I would refer you to some exper'* of Drapers, ia case you should
not have seen them. They were published a few years asjo in the
Phil. Mag.

As to the cause of this ivsult I can state nothing positive as
you cjvn conceive. It fe\lls in very well with certain conjectures
which I have made in my paper On the Change of Eefirangibility
of Light (art. 2oO >.vc.') but these are only conjectures.

I was greatly struck with the enormous length of the spectrum
which I obtaineil with my qtiartz train with the powerfid battery
belonging to the Eoyal Institution. It fer surpassed in length
the solar spiectrum. I mean even taking in that highly refrangible
pvrt which a quartz train is ivquired to show. In the case of
metaUic points this spectrum consisted of isolated bright lines.
I cannot help thinking that decompositions of a very high order
may be c;oing on in snch an arc (the voltaic are I mean) and
that a careful examination of these lines may lead to remarkable
inferences respecting the bixiies at present regsu-ded as elementary.
There is nothiui; extravagant in this supposition : few chemists
I imatnne believe that the so-called elements are all really such.

Xow it is quite conceivable that chemically ptu-e metals should
ac:i>?e with compvninds of sodium in gndng the bright line D. If
this were made out I should s^iy that perhajis these metals were
ct.>mpounds of sodium, but more probably they and sodium were
compnuuis oi some substance yet more elementary.

Yours very truly

G. G. Stoker

374 CORRESPONDENCE OF G. G. STOKES AND W. THOMSON

69 Albert St. Regent's Park
London Nov 26, 1855

My dear Thomson

I write to refer you to a paper of Foucault's in which he
finds that the voltaic arc produces by absorption the fixed line D
in light in wh. it did not before exist, and that the bright line D
is of constant occurrence in the arc. L'Institut N° 788. Jan.
or Feb. 1849.

Yours very truly

G. G. Stokes

69 Albert Street Regent's Park
London Dec 6th 1855

My dear Thomson

I was speaking to Foucault about the artificial dark
line 2). It is easily produced by arranging so that the coke pole
shall be seen through the arc, the pole itself giving an uninter-
rupted spectrum in which the dark line D is seen by absorption
while in the less bright spectrum formed by the arc itself is seen
the bright line D as a continuation of the other.

Yours very truly

G. G. Stokes

Admiralty

London .hi/i/ 7/71

Dear Stokes

Many thanks for your letter and printed paper which
are most valuable to me. •

It was certainly prior to the summer of 1852 that you taught
me solar and stellar chemistry. I never was at Cambridge from
the siimmer of 1852 until we came there in 1866, and the con-
versation was certainly in Cambridge. I had begun teaching it
to my class in session 1852-3, (as an old student's note book

■^X THE K^SSIRIUTIES OF SPECTRUM ANALYSIS 375

which I have shows*) and probably earlier althouoh of rhis
I have not evidence. I used alw-ays to show a spirit lamp flame
with salt on it, behind a slit prv^longing the dark line D bv bright
continuation. I always gtwe your dvnamical explanation, alwsns
ass^erteti that certainly there was Siidiuni vapour in the suns
atmospheix? and in the atmospheres of stars which show pivsence
of the i)'s, and always pv^iuted out that the wav to find other
substajioes besides Sixlium in the sun and stai^ was to comp;vi\?
bright lines produced by them in artificial flames with dark lines
of the spectra of the lights of the distant bodies.

. You ver\- probably learned more fix>m Foucault of what he had
done when you met him in 1S55. but you cei-taiuly rold me of
a Frenchman (^I dont recollect the name you then gave me or
even that you mentioned the name though it is most probable
you did) having obtained the dark line by absorption.

Yours truly

W. Thomsox

P.S. I think of using almost all you s;iy of Hei-schel, in mv
address as from you.

* I spe^ik from memorv-. I believe it was of session 1S52-3.
I have it in Glasirow.

Cambridge llfA •Aiity ISTo
My peak Thomsox

You have pushed me fcu- too much forwiuxl as to
spectrum analysis. Whitmell sent me a syllabus of his lectures,
in wh. I was mentioned l\u- too prominently, ;ind this led me to
write to him my n?oollection of my ideas (and it is a distinct one)
when we talked about the matter. It was yoa who proposed to
extend the method so much. I ii^ther thought you were .Ulowing
your horse to run away with you.

Most of the absorbing ^^optieally not uientally) mattei^ that
then came acivss me were things that would uot stand the tire,
and I dont know that I ever asked myself the question Wnat
would fcvke place if a colom-^ glass were heated / Anyhow
Stewart s extension of Prevosts Theory of Exohangej in the
Pi-oc, R. S. X o>o form (^for I was not then acquainted with his
Edin. Phil. Ti-.ms. pipers though I might have been) came before
me with all the freshness of a new disco\ery. I don't know
whether vou had known it betoro.

376 CORRESPONDENCE OF G. G. STOKES AND W. THOMSON.

As to the bright line D of a spirit lamp with salted wick. I
thought the flame gave such a shake to the Na CI molecule as to
make the Na bell ring, but did not think the more ethereal vibra-
tions of D pitch could do it unless the Na were free, which I did
not think of its being in the flame of a spirit lamp, though I
thought the tremendous actions on the sun's surface might set it
free. When later (1855) I heard from Foucault's lips of his
remarkable discovery I still thought of the sodium being set free
by the tremendous action of the voltaic arc of a battery of 40 or
50 elements of Grove or Bunsen.

I thought from memory that it was wheyi Foucault came to
receive the Copley medal that he told me of it, though all I could
feel certain of from memory that it was in the evening at Dr Neil
Amott's. On referring to your Glasgow address*, I was going
to strike out the date, when I obtained proof positive that I was
right. I feel certain that I should have mentioned Foucault's
remarkable discovery in what I drew up for the President about
his researches if I had then known of it (Proc. R. S. vii, 571).

Yours sincerely

G. G. Stokes

[* Edinburgh address, 1871, loc. cit. p. 135 supra?]

INDEX TO VOLUME IV.

[The references are to pages.']

Absorption of light, mechanism of, 69 ;

Appendix, 376.
Achromatism of double object glass,

theory of, 63, 350.
Achromatism, practical investigations,

337 ; test of, in telescope, 357.
Acids, action on quinine, 328.
Attraction, discontinuity at surface layer,

281.
Bessel functions, relations between con-
stants in expansions by, 100, 286 ;

of integral order, 294, applied to

aereal waves, 316.
Biliverdin distinguished from chloro-
phyll, 296.
Blood, oxidation and reduction of, 264,

272.
Cauchy, A. L., on double refraction,

170 ; on metallic reflection, 363.
Chestnut bark, new constituent of, 112,

119.
Chlorophyll, constitution of, 237, 247,

260.
Chromatic aberration, correction of

secondary, 348.
Colours, nature of compound, 65 ; origin

of natural colour, 153, 243; body

colour of loose powder, 263.
Complex integration along different

paths, 93, 287.
Convergence, limited, of series, 80.
Cooling, deformations due to rapid, 234.
Degradation of light, 4.
Diamond, reflection from, 363.
Diffracted light, polarization of, 74;

determines direction of vibration, 117.
Discontinuity in constants occurring in

expansions in series, 80, 282.

Dispersion, false, of light by small
particles in glass, 244.

Divergent series, discontinuity in use of,
80, 289.

Double refraction, report on theories of,
157 ; experimental test of its laws,
336, 337.

Dynamical paradox explained, 334.

Elastic theories of the aether, 157.

Electric discharges, optical constitution
of, 226 ; difference between the two
electrodes, 231.

Electric telegraph, theory of signalling,
61.

Enclosure, radiation belonging to tem-
perature, 136.

Equilibrium of gases in blood, 274.

Equipotential curves cross at equal
angles, 276.

Eye, chromatic aberration of, 59.

Fluorescence, methods of observation,
1, 326 ; history of, 18 ; applied to
spectrum, 207 ; preparation of screen,
208 ; striated in minerals, 222 ; as
chemical test, 256 ; connexion with
absorption, 258 ; action of acids on
quinine, 13, 327 ; of chestnut bark,
110, 119; theoretical, 246.

Foucault, L., on absorption of D lines,
130, 134, 374.

Fraunhofer, J., his achromatic com-
binations, 63 ; on spectra, 368.

Fresnel, A., on double refraction, 157
scg.; theory of pile of plates, 145.

Glasses, relation of optical quality to
composition, 339 ; titano-phosphatic
suitable for achromatizing objectives,
356, but not titano-silicic, 358.

378

INDEX.

Gravity, theory of Hartou pit experi-
ment, 70.

Green, G., on elastic propagation and
double refraction, 172.

Haematin, 271.

Haemoglobin, 240 ; its reactions, 264.

Haidinger, W., letter to, 50 ; his brushes,
60.

Hankel, H., on Bessel functions, 80.

Haroourt, Eev. W. V., researches on
glasses, 339 seq.

Hydrogen atmosphere stifles sound, 300.

Hyposulphurous acid, nature of, tested
by quinine, 331.

Kelvin, Lord (Thomson, W.), on begin-
nings of spectrum analysis, 132, 135,
367 ; on cable telegraphy, 61.

Kirchhoff, G., on spectrum analysis,
128, 136.

Limit, mathematical, approached dif-
ferent ways, 88.

MacCuUagh, J., theory of double re-
fraction, 177 ; of metallic reflection,
363.

Madder, optical analysis of, 123.

Mass, general distribution producing
given external gravity, 277.

Metallic reflection of crystals, 16, 38,
244 ; complementary to absorption,
43, 248 ; verified on absorption bands
of permanganate, 46, 261.

Metallicreflection,peculiarityof Newton's
rings by, 361 ; silver transparent in
ultra-violet, 206.

Miller, W. H., on coincidence of D and
sodium lines, 370, 372.

Newton's rings by metallic reflection
beyond the critical angle, 361.

Object-glasses, chemical correction of,
344 ; triple perfect achromatic, 356,
358.

Optical properties as tests for organic
bodies, 238, 249 ; of glasses, 339.

Photographic lenses, correction of, 344.

Platinocyanide crystals, diohroic, polar-
ized fluorescence, 16 ; solutions in-
active, 17.

Polarization produced by pile of plates,
theory, 145.

Polarized optical rings, peculiarity in
photographs, 30.

Prisms, compensating, for investigating
dispersion, 341.

Quinine, metallic reflection of a salt of,
19 ; influence of, various acids on its
fluorescence, 13, 328.

Radiation, law of exchanges, 129 ; veri-
fied for polarized radiation of tour-
maline, 136 ; theoretical verification
for internal radiation in crystals, 137.

Bealization of constant-temperature
radiation, 136.

Series, semi-convergent, discontinuity
in constants, 77, 282.

Shadow-patterns, 55.

Solar eclipse, appearance at horns of
crescent, 325.

Sounding-boards, necessity for, 300.

Spectroscope, curvature of images in,
355.

Spectrum analysis, history of, 127 ;
Appendix, 367 ; may transcend chemi-
cal analysis, 373.

Spectrum, ultra-violet, 28 ; long, of
electric light, 203; of metals, 210;
absorption, of organic bodies, 216 ;
relation of intensities of lines to
temperature, 365.

Stewart, Balfour, on law of exchanges
of radiation, 136, 375 ; verified for
tourmaline in enclosure, 136.

Telegraphy, submarine, 61.

Transition-films, in reflection of light,
364.

Tuning-fork, mutual interference of
prongs, 324.

Uranium salts, fluorescent, 209, even
when pure, 220.

Vibrations, communication to a gas,
299, from a sphere, 303, from a
cylinder, 314.

Vibrations, direction of, in polarized
light, 50.

Wind, effect on sound, 110.

CAMBEIDGE : PlilNTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.

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