- re,
A„A, - rift
If we write C,A,S, - t, and denote the total deviation
of the wave-front by {„ we have
AtI> - AJ3 - A^ cos 8, = re - rw cos S„
3,Bl,ZEdhyG00gle
120 The Lnminiferous Medium,
and therefore (neglecting second-order terms in w/c)
ain A,S,D c - wcoaSi c w w B
Denoting by 8 the value of S. when w is zero, we have
Bin (i-B) c
sin t e.
Subtracting this equation from the preceding, we have
8 - 8, w
sin 8 c
Now the telescope by which the emergent wave-front B, D
is received is itself being carried forward by the earth's motion;
and we must therefore apply the usual correction for aberration
in order to find the apparent direction of the emergent ray. But
this correction is w sin B/c, and precisely counteracts the effect
which has been calculated as due to the motion of the prism.
So finally we see that the motion of the earth has no first-order
" influence on the refraction of light from the stars.
Fresnel inferred from his formula that if observations were
made with a telescope filled with water, the aberration would be
unaffected by the presence of the water — a result which was
verified by Airy* in 1871. He showed, moreover, that the
apparent positions of terrestrial objects, carried along with the
observer, are not displaced by the earth's motion ; that experi-
ments in refraction and interference are not influenced by any
motion which is common to the source, apparatus, and observer ;
and that light travels between given points of a moving material
system by the path of least time. These predictions have also been
confirmed by observation: Eespighifin 1861,andHoekJin 1868,
experimenting with a telescope filled with water and a terrestrial
source of light, found that no effect was produced on the
phenomena of reflexion and refraction by altering the orienta-
* Proo. Bo;. Soc., iz, p. 35. t Mem. Aocad. Sci. Bologna, ii, p. 279.
J Art. Niwb., Ixxiii, p. IBS.
3,Bl,ZEdhyG00gk
from Bradley to FresneL 121
tion of the apparatus relative to the direction of the earth's
motion. E. Maacart* in 1872 discussed experimentally the
question of the effect of motion of the Bource or recipient of
light in all its bearings, and showed that the light of the sun and
that derived from artificial sources are alike incapable of revealing
by diffraction-phenomena the translatory motion of the earth.
The greatest problem now confronting the investigators of
light was to reconcile the facts of polarization with the principles
of the wave-theory. Young had long been pondering over this,
but had hitherto been baffled by it In 1816 he received a
visit from Arago, who told him of a new experimental result
which he and Fresnel had lately obtained! — namely, that two
pencils of light, polarized in planes at right angles, do not
interfere with each other under circumstances in which ordinary
light shows interference-phenomena, but always give by their
reunion the same intensity of light, whatever he their difference
of path.
Arago had not long left him when Young, reflecting on the
new experiment, discovered the long-sought key to the myBtery :
it consisted in the very alternative which Bernoulli had rejected
eighty years before, of supposing that the vibrations of light are
executed at right angles to the direction of propagation.
Young's ideas were first embodied in a letter to AragoJ
dated Jan. 12, 1817. " I have been reflecting," he wrote, " on the
possibility of giving an imperfect explanation of the affection
of light which constitutes polarization, without departing from
the genuine doctrine of undulations. It is a principle in this
theory, that all undulations are simply propagated through
homogeneous mediums in concentric spherical surfaces like the
• Ann. de l'Jicole Noeraale, (2) i, p. 167.
I fit waanot publiibed until IH19, in Anniles d« Chimin, x; Freanel'e (Eurret,
i, p. 609. By meuns of this result, Fresnel was able to give a complete explana-
tion Of a clan of phenomena which Arngo had discovered in 1811, viz. that when
polarised light El transmitted through thin plates of sulphate of lime or mica, and
afterwards analysed by a prism of Iceland spar, beautiful complementary colours
ire displayed. Young had shown that these effect! are due essentially to inter-
ference, but had not made dear the part played by polarization.
J Young's fFerit, i., p. 380.
3,Bl,ZEdhyG00gle
122 The Lumini/erous Medium,
undulations of sound, consisting simply in the direct and retro-
grade motions of the particles in the direction of the radius,
with their concomitant condensation and rarefactions. And
yet it is possible to explain in this theory a transverse vibration,
propagated also in the direction of the radius, and with equal
velocity, the motions of the particles being in a certain constant
direction with respect to that radius ; and this is a polarization.."
In an article on " Chromatics," which was written in
September of the same year* for the supplement to the
Encyclopaedia Britannica, he says :f " If we assume as a mathe-
matical postulate, on the undulating theory, without attempting
to demonstrate its physical foundation, that a transverse motion
may be propagated in a direct line, we may derive from this
assumption a tolerable illustration of the subdivision of polarized
light by reflexion in an oblique plane," by " supposing the polar
motion to be resolved " into two constituents, which fare
differently at reflexion.
In a further letter to Arago, dated April 29th, 1818, Young
recurred to the subject of transverse vibrationB, comparing light
to the undulations of a cord agitated by one of its extremities-!
This letter was Bhown by Arago to Fresnel, who at once saw
that it presented the true explanation of the non-interference
of beams polarized in perpendicular planes, and that the latter
effect could even be made the basis of a proof of the correctness
of Young's hypothesis : for if the vibration of each beam be
supposed resolved into three components, one along the ray and
the other two at right angles to it, it is obvious from the Arago-
Fresnel experiment that the components in the direction of the
ray must vanish : in other words, that the vibrations which
constitute light are executed in the wave-front.
It must be remembered that the theory of the propagation
of waves in an elastic solid was as yet unknown, and light was
* Peacock's Lift of Young, p. 391. t Young's Work; i., p. 379.
J This (oology bad been given by Huoke in a communication to the Royal
Society on Feb. 16, 1671-2. But tbere seem* no nawm to suppose that Hook*
appreciated the point now advanced by Young.
3,Bl,ZEdhyG00gle
from Bradley to Fresnel. 123
BtSll always interpreted by the analogy with the vibrations of
sound in air, for which the direction of vibration is the same as
that of propagation. It was therefore necessary to give some
justification for the new departure. With wonderful insight
Fresnel indicated* the precise direction in which the theory of
vibrations in ponderable bodies needed to be extended in order
to allow of waves similar to those of light : " the geometers," he
wrote, " who have discussed the vibrations of elastic fluids hitherto
have taken account of no accelerating forces except those arising
from the difference of condensation or dilatation between conse-
cutive layers." He pointed out that if we alao suppose the
medium to possess a rigidity, or power of resisting distortion, such
as is manifested by all actual solid bodies, it will be capable of
transverse vibration. The absence of longitudinal waves in the
aether he accounted for by supposing that the forces which oppose
condensation are far more powerful than those which oppose
distortion, and that the velocity with which condensations are
propagated is so great compared with the speed of the oscillations
of light, that a practical equilibrium of pressure is maintained
perpetually.
The nature of ordinary non-polarized light was next discussed.
" If then," Fresnel wrote,t " the polarization of a ray of light
consists in this, that all its vibrations are executed in the same
direction, it results from any hypothesis on the generation of
light-waves, that a ray emanating from a single centre of dis-
turbance will always be polarized in a definite plane at any
instant. But an instant afterwards, the direction of the motion
changes, and with it the plane of polarization ; and these
variations follow each other as quickly as the perturbations of
the vibrations of the luminous particle : so that even if we could
•Annalee de Chimie, ivii (1821), p. ISO; (Event, i, p. 629. Young had
already drawn attention to this point. " It U difficult," ha eay* in hi* Ltetun* on
A'ttvrul Philosophy, ed. 1807, toI. i, p. 138, "to compare the lateral adheiion, or
the force which restate the detriurfon of the parta of a solid, with any form of direct
tobeiion. Thia force constitutes the rigidity or hardneaa of a aolid body, and ia
wholly abaent from liquida."
t lac. dt., p. 186.
3,Bl,ZEdhyG00gle
124 The Lumini/erous Medium,
isolate the light of this particular particle from that of other
Luminous particles, we should doubtless not recognize in it any
appearance of polarization. If we consider now the effect pro-
duced by the union of all the waves which emanate from the
different points of a luminous body, we see that at each instant,
at a definite point of the aether, the general resultant of all the
motions which commingle there will have a determinate
direction, but this direction will vary from one instant to the
next. So direct light can be considered as the union, or more
exactly as the rapid succession, of systems of waves polarized in
all directions. According to this way of looking at the matter,
the act of polarization consists not in creating these transverse
motions, but in decomposing them in two invariable directions,
and separating the components from each other; for then, in
each of them, the oscillatory motions take place always in the
same plane."
He then proceeded to consider the relation of the direction of
vibration to the plane of polarization. " Apply these ideas to
double refraction, and regard a uniaxal crystal as an elastic
medium in which the accelerating force which results from
the displacement of a row of molecules perpendicular to the
axis, relative to contiguous rows, is the same all round the
axis ; while the displacements parallel to the axis produce
accelerating forces of a different intensity, stronger if the
crystal is " repulsive," and weaker if it is " attractive." The
distinctive character of the rays which are ordinarily refracted
being that of propagating themselves with the same velocity
in all directions, we must admit that their oscillatory motions
are executed at right angles to the plane drawn through these
rays and the axis of the crystal; for then the displacements
which they occasion, always taking place along directions
perpendicular to the axis, will, by hypothesis, always give rise
to the same accelerating forces. But, with the conventional
meaning which is attached to the expression -plane of polarization,
the plane of polarization of the ordinary rays is the plane
through the axis: thus, t» q^pepsil- of polarized lights the
3,Bl,ZEdhyG00gle
from Bradley to Fresnel. 125
oscillatory motion is executed at right angles to the plane of
polarizatvm."
This result afforded Fresnel a foothold in dealing with the
problem which occupied the rest of hie life : henceforth hie aim
was to base the theory of light on the dynamical properties of
the luminiferous medium.
The first topic which he attacked from this point of view
was the propagation of light in crystalline bodies. Since
Brewster's discovery that many crystals do not conform to the
type to which Huygens' construction is applicable, the wave
theory had to some extent lost credit in this region. Fresnel,
now, by what was perhaps the most brilliant of all his efforts,*
not only reconquered the lost territory, but added a new domain
to science.
He had, as he tells us himself, never believed the doctrine
that in crystals there are two different luminiferous media,
one to transmit the ordinary, and the other the extraordinary
waves. The alternative to which he inclined was that the two
velocities of propagation were really the two rootB of a quadratic
equation, derivable in some way from the theory of a single
aether. Could this equation be obtained, he was confident of
finding the explanation, not only of double refraction, but also
of the polarization by which it is always accompanied.
The first step was to take the case of uniaxal crystals,
which had been discussed by Huygens, and to see whether
Huygens' sphere and spheroid could be replaced by, or made to
depend on, a single surface.t
Now a wave propagated in any direction through a uniaxal
■His flrat memoir on Doable Refraction m presented to the Aoademy on
Not. 19th, 1821, but ha* not bean published except in liii collected work*:
__ ■ matter of fact, satisfied by
the electric force in toe electro- magnetic theory of light. The continuity of
■curt e if eqniTaJent to the continuity of the magnetic vector acroai the interface,
and the continuity of ~de,ftiz leadj to the ma equation aa the continuity of
the component of electric force in the direction of the intersection of the
interface with the plane of incidence.
D,Bl,ZEdhyG00gle
1 48 The Aether as an Elastic Solid.
remained to supply the boundary -co nditione at an interface,
which are required in the discussion of reflexion, and the
relations between the elastic constants of the solid, which are
required in the optics of crystals. Cauchy seems to have con-
sidered the question from the purely analytical point of view.
Given certain differential equations, what supplementary con-
ditions must be adjoined to them in order to produce a given
analytical result ? The problem when stated in this form
admits of more than one solution ; and hence it is not surprising
that within the space of ten years the great French mathe-
matician produced two distinct theories of crystal-optics and
three distinct theories of reflexion* almost all yielding correct
or nearly correct final formulae, and yet mostly irreconcilable
with each other, and involving incorrect boundary-conditions
and improbable relations between elastic constants.
Cauchy's theories, then, resemble Fresuel s in postulating
types of elastic Bolid which do not exist, and for whose
assumed properties no dynamical justification is offered. The
same objection applies, though in a loss degree, to the original
form of a theory of reflexion and refraction which was
discovered about this timet almost simultaneously by James
MacCullagh (b. 1809, d. 1847), of Trinity College, Dublin,
and Franz Neumann (J. 1798, d. 1895), of Kbnigsberg. To
these authors is due the merit of having extended the laws
of reflexion to crystalline media; but the principles of the
theory were originally derived in connexion with the simpler
case of isotropic media, to which our attention will for the
present he confined.
' One yet remains to be mentioned.
+ The outlinns of the theory were published by MacCullagb, in Brit. Ateoc Bop.
1835 ; and hie results were given in Phil. Mug. i (Jan., 1837), and in Proc.
Royal Irish Acad, iviii. (Jan., 1837). Neumann's memoir waa presented to the
Berlin Academy towards the end of 1836, and published in 1837 in Abh. Berl.
Ak. Buadem Jahre 1835, Math. Klaate, p. 1. So far aa publication ie concerned,
the priority would seem to belong to MacCuUagh; but there are reoaona for
believing that (he priority of discovery really reata with Neumann, who bad
arrived at his equations a year before they were communicated to the Berlin
Academy.
dhyGoogle
. The Aether as an Elastic Solid. 140
MacCullagh and Neumann felt that the great objection
to Fresnel's theory of reflexion was its failure to provide for
the continuity of the normal component of displacement, at the
interface between two media ; it is obvious that a discontinuity
in this component could not exist in any true elastic-solid
theory, since it would imply that the two media do not remain
in contact. Accordingly, they made it a fundamental con-
dition that all three componente of the displacement must be
coutinuous- at the interface, and found that the sine-law and
tangent-law can be reconciled with this condition only by
supposing that the aether-vibrations are parallel to the plane of
polarization : which supposition they accordingly adopted. In
place of the remaining three true boundary- conditions, however,
they used only a single equation, derived by assuming that
transverse incident waves give rise only to transverse reflected
and refracted waves, and that the conservation of energy holds
for these — i.e. that the masses of aether put in motion,
multiplied by the squares of the amplitudes of vibration, are
the same before and after incidence. This is, of course, the
same device as had been used previously by Fresnel; it
must, however, be remarked that the principle is unsound as
applied to an ordinary elastic solid; for in such a body the
refracted and reflected energy would in part be carried away
by longitudinal waves.
In order to obtain the sine and tangent laws, MacCullagh
and Neumann found it necessary to assume that the inertia
of the luminiferous medium is everywhere the same, and
that the differences in behaviour of this medium in different
substances are due to differences in its elasticity. The two
laws may then be deduced in much the same way as in the
previous investigations of Fresnel and Gauchy.
Although to insist on continuity of displacement at the
interface was a decided advance, the theory of MacCullagh and
Neumann scarcely showed as yet much superiority over the
quasi mechanical theories of their predecessors. Indeed,
MacCullagh himself expressly disavowed any claim to regard
3,Bl,ZEdhyG00gle
150 The Aether as an Elastic Solid.
his theory, in the form to which it had then been brought, as a
final explanation of the properties of light. " If we are asked,"
he wrote, " what reasons can be assigned for the hypotheses on
Which the preceding theory is founded, we are far from being
able to give a satisfactory answer. We are obliged to confess
that, with the exception of the law of vis viva, the hypotheses
are nothing more than fortunate conjectures. These conjectures
are very probably right, since they have led to elegant laws
which are fully borne out by experiments ; but this is all we
can assert respecting them. We cannot attempt to deduce
them from first principles ; because, in the theory of light,
such principles are still to be sought for. It is certain, indeed,
that light is produced by undulations, propagated, with
transversal vibrations, through a highly elastic aether ; but the
constitution of this aether, and the laws of its connexion (if it
has any connexion) with the particles of bodies, are utterly
unknown."
The needful reformation of the elastic-solid theory of
reflexion was effected by Green, in a paper* read to the
Cambridge Philosophical Society in December, 1837. Green,
though inferior to Cauchy as an analyst, was his superior in
physical insight ; instead of designing boundary-equations for
the express purpose of yielding Freenel's sine and tangent
formulae, he set to work to determine the conditions which are
actually satisfied at the interfaces of real elastic solids.
These he obtained by means of general dynamical principles.
In an isotropic medium which is strained, the potential energy
per unit volume due to the state of stress is
_ If, 4 \ /3e, fc„ &,\' 1 l/de. fe-V fdem 3e,\*
where e denotes the displacement, and k and n denote the two
* Tram. C»mb. Phil. Soo., 1838 ; Green's Math. Paperi, p. its,
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid. 151
elastic constants already introduced; by substituting this value
of __

__S--l£ + -n] grad div e - n curl curl e ;
' (where p denotes the density), the equation of motion may be
deduced.
But this method does more than merely furnish the equation
of motion
K^Tijgrad
div e + »Ve,
which had already been obtained by Cauchy ; for it also yields
the boundary-conditions which must be satisfied at the interface
between two elastic media in contact ; these are, as might be
guessed by physical intuition, that the three components of the
displacement* and the three components of stress across the
interface are to be equal in the two media. If the axis of x
be taken normal to the interface, the latter three quantities
are
(.-J.)*.-*. ,(M), -a „£♦*).
The correct boundary-conditions being thus obtained, it was
a simple matter to discuss the reflexion and refraction of an
incident wave by the procedure of Fresnel and Cauchy. The
result found by Green was that if the vibration of the aethereal
molecules is executed at right angles to the plane of incidence,
the intensity of the reflected light obeys Fresnel 'b sine-law, pro-
vided the rigidity n is assumed to be the same for all media,
but the inertia p to vary from one medium to another. Since
the sine-law is known to be true for light polarized in the plane
of incidence, Green's conclusion confirmed the hypotheses of
* The** Aral three conditions ire of courae not dytifcmical but geometric*!.
3,Bl,ZEdhyG00gle
152 The Aether as an Elastic Solid.
Fresnel, that the vibrations are executed at right angles to the
plane of polarization, and that the optical differences between
media are due to the different densities of aether within them.
It now remained for Green to discuss the case in which the
incident light is polarized at right angles to the plane of inci-
dence, so that the motion of the aethereal particles is parallel to
the intersection of the plane of incidence with the front of the
wave. In this case it ia impossible to satisfy all the six
boundary-conditions without assuming that longitudinal vibra-
tions are generated by the act of reflexion. Taking the plane
of incidence to be the plane of yz, and the interface to be the
plane of xy, the incident wave may be represented by the
equations
5 3
Cy - A £■/(< + h + my) ; e„ - - A —f(t + lz + my) ;
oz oy
where, if i denote the angle of incidence, we have
I = J— cos i, m = - ./£- sin i.
There will be a transverse reflected wave,
and a transverse refracted wave,
where, since the velocity of transverse waves in the second
medium is \/n//>i> we can determine /i from the equation
(,- + »■-£;
ft
there will also be a longitudinal reflected wave,
ey - D g-/(( - A* + my); «, - D ^f(t - Xz + my),
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid: 153
where A is determined by the equation
kt + ;7l
and a longitudinal refracted wave,
t, - E -f(t + X& + my),
where A, ie determined by
*,■ + »•- 5^--
it, - /(M)'**(M)"^-&)'-
The usual variational equation
I'
^^ + ^i^ + Wt^dxdydzm-\\\S*dxd'A'
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid. 163
then yields the differential equations of motion, namely :
and two similar equations.
These differ from C&uchy's fundamental equations in having
greater generality: for Cauchy's medium was supposed to be
built up of point-centres of force attracting each other according
to some function of the distance ; and, as we have seen, there
are limitations in this method of construction, which render it
incompetent to represent the most general type of elastic solid.
Cauchy's equations for crystalline media are, in fact, exactly
analogous to the equations originally found by Navier for
isotropic media, which contain only one elastic constant instead
of two.
The number of constants in the above equations still exceeds
the three which are required to specify the properties of a
biaxal crystal : and Green now proceeds to consider how the
number may be reduced. The condition which he imposes for
this purpose is that for two of the three waves whose front is
parallel to a given plane, the vibration of the aethereal molecules
shall be accurately in the plane of the wave : in other words,
that two of the three waves shall be purely distortional, the
remaining one being consequently a normal vibration. This
condition gives five relations* which may be written : —
/' - fi - 2/ / - ft - 2g; V - ? - 2A;
where p denotes a new constant^
* A* Green showed, the hypotheaia of tianivnnality really involve* the exiitence
of plane* of lymmatry, eothat it alone ii Capable of giving 14 relatione between the
21 conatant* : and 3 of the remaining 7 constant* may be removed bj change of
axet, leaving only four.
t It wm afterward* ghown by Barre de Saint- Tenant (*. 1797, d. 1883),
Journal dt Math., vii (1868), p. 399, that if the initial atreuea be auppoaed to
vanish, the conditiona which must be aitie&ed among the remaining nine conatant*
M 2
3,Bl,ZEdhyG00gle
164 The Aether as an Elastic Solid.
Thus the potential energy per unit volume may be written
-)'■
At this point Green's two theoriofl of crystal-optics diverge
from each other. According to the first theory, the initial
j G, H, I are zero, so that
. , lie, 9a, fr.\
^m+w-'m^i^w-'m
>H%+m-
.3e. dfj)
". *i »>/> 9> k>ftfi *'i 'n urdei' >hat the wave-surface may be Frwnel't, an ths
folio wing : —
(3i -/)(&•-/) -{/+/•}'
{3«-j){M -*) = [> + •)'
1 (Sa - A) (3* - A) = (A + A")1
l(3a - j) (3d- A) (3» -/) + (3— A) (3» -/) (3« - f ) - 2(/+/) (? +j0 (A + A').
There reduce to Green's relation.! when the additional equation b — t ii inrame-i .
Saint-Tenant disputed the validity of Green's relations, assailing that they?ere
compatible only with ieotropy. On this controversy cf. E. T. G-laiebrook, Brit.
Aino. Report, 1886, p. 171, and Karl Pearson in Todhunter and Pearson's Sultry
of EUiticity, ii, } 147.
3,Bl,zEdhyG00gle
The Aether as an Elastic Solid. 165
This expression contains the correct number of constants,
namely, four: three of them represent the optical constants
of a biaxal crystal, and one (namely, p) represents the square of
the velocity of propagation of longitudinal waves. It is found
that the two sheets of the wave-surface which correspond to the
two distortions! waves form a Fresnel's wave-surface, the third
sheet, which corresponds to the longitudinal wave, being an
ellipsoid. The directions of polarization and the wave-velocities
of the distortional waves are identical with those assigned by
FresneL provided it is assumed that the direction of vibration
of the aether-particles is parallel to the plane of polarization ;
but this last assumption is of course inconsistent with Green's
theory of reflexion and refraction.
In his Second Theory, Green, like Cauchy, used the condition
that for the waves whose fronts are parallel to the coordinate
planes, the wave- velocity depends only on the plane of polariza-
tion, and not on the direction of propagation. He thus obtained
the equations already found by Cauchy —
G-f- H-g = I-k.
The wave-surface in this case also is Fresnel's, provided it
is assumed that the vibrations of the aether are executed at
right angles to the plane of polarization.
The principle which underlies the Second Theories of Green
aud Cauchy is that the aether in a. crystal resembles an elastic
solid which is unequally pressed or pulled in different directions
by the unmoved ponderable matter. This idea appealed strongly
to W. Thomson (Kelvin), who long afterwards developed it
further* arriving at the following interesting result :— Let an
incompressible solid, isotropic when unstrained, be such that its
potential energy per unit volume is
where q denotes its modulus of rigidity when unstrained, and
•Proc.B.S.Edin.XT(18S7),p.21: Phil. Mag. xxy (1888) p. 116: Baltimer*
Laturt, (ed. 1904), pp. 228-259.
3,Bl,ZEdhyG00gle
1 66 The Aether as an Elastic Solid.
o*. 0*> 7*. denote the proportions in which lines parallel to the
axes of strain are altered ; then if the solid be initially strained
in a way defined by given values of a, /3, y, by forces applied to
its surface, and if waves of distortion be superposed on this
initial strain, the transmission of these waves will follow exactly
the laws of Fresnel's theory of crystal-optics, the wave-surface
being
V-l £^_i ?*>.!
2 9$
There is some difficulty in picturing the manner in which
the molecules of ponderable matter act upon the aether so as to
produce the initial strain required by this theory. Lord
Kelvin utilized* the suggestion to which we have already
referred, namely, that the aether may pervade the atoms of
matter so as to occupy space jointly with them, and that its
interaction with them may consist in attractions and repulsions
exercised throughout the regions interior to the atoms. These
forces may be supposed to be so large in comparison with those
called into play in free aether that the resistance to compres-
sion may be overcome, and the aether may be (Bay) condensed
in the central region of an isolated atom, and rarefied in its
outer parts. A crystal may be supposed to consist of a group
of spherical atoms in which neighbouring spheres overlap each
other ; in the central regions of the spheres the aether will be
condensed, and within the lens-shaped regions of overlapping
it will he still more rarefied than in the outer parts of a solitary
atom, while in the interstices between the atoms its density
will be unaffected. In consequence of these rarefactions and
condensations, the reaction of the aether on the atoms tends
to draw inwards the outermost atoms of the group, which,
however, will be maintained in position by repulsions between
the atoms themselves; and thus we can account for the pull
which, according to the present hypothesis, is exerted on the
aether by the ponderable molecules of crystals.
* Baltimore Liuturei (*d. 1904), p. 263.
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid. 167
Analysis similar to that of Cauchy's and Green's Second
Theory of crystal-optics may be applied to explain the doubly
refracting property which is possessed by strained glass ; but
in this case the formulae derived are found to conflict with
the results of experiment. The discordance led Kelvin to
doubt the truth of the whole theory. "After earnest and
hopeful consideration of the stress theory of double refraction
during fourteen years," he said,* "lam unable to see how it
can give the true explanation either of the double refraction of
natural crystals, or of double refraction induced in isotropic
solids by the application of unequal pressures in different
directions."
It is impossible to avoid noticing throughout all Kelvin's
work evidences of the deep impression which was made
upon him by the writings of Green. The same may be said
of Kelvin's friend and contemporary Stokes; and, indeed, it
is no exaggeration to describe Green as the real founder of
that " Cambridge school " of natural philosophers, of which
Kelvin, Stokes, Lord Bayleigh, and Clerk Maxwell were the
most illustrious members in the latter half of the nineteenth
century, and which is now led by Sir Joseph Thomson and
Sir Joseph Larmor. In order to understand the peculiar
position occupied by Green, it is necessary to recall some-
thing of the history of mathematical studies at Cambridge.
The century which elapsed between the death of Newton
and the scientific activity of Green was the darkest in the
history of the University. It is true that Cavendish and
Young were educated at Cambridge; but they, after taking
undergraduate courses, removed to London. In the entire
period the only natural philosopher of distinction who lived
and taught at Cambridge was Miehell ; and for some reason
which at this distance of time it is difficult to understand
fully, MichelTs researches seem to have attracted little or no
attention among his collegiate contemporaries and successors,
* Baitimori Ledum (fit. 1904), p. 268.
3,Bl,ZEdhyG00gle
168 The Aether as an Elastic Solid.
who silently acquiesced when his discoveries were attributed to
others, and allowed his name to perish entirely from Cambridge
tradition.
A few years before Green published his first paper, a
notable revival of mathematical learning swept over the
University ; the fluxional symbolism, which Bince the time of
Newton had isolated Cambridge from the continental schools,
was abandoned in favour of the differential notation, and the
works of the groat French analysts were introduced and
eagerly read. Green undoubtedly received his own early
inspiration from this source ; but in clearness of physical
insight and conciseness of exposition he far excelled his
masters ; and the slight volume of his collected papers has
to this day a charm which is wanting to the voluminous
writings of Cauchy and Poisson. It was natural that such an
example should powerfully influence the youthful intellects of
Stokes — who was an undergraduate when Green read his memoir
on double refraction to the Cambridge Philosophical Society' — -
and of William Thomson (Kelvin), who came into residence two
years afterwards.*
In Bpite of the advances which were made in the great
memoirs of the year 1839, the fundamental question as to
whether the aether-particlea vibrate parallel or at right angles
to the plane of polarization was still unanswered. More light
was thrown on this problem ten years later by Stokes's inves-
tigation of Diffraction. t Stokes showed that on almost any
conceivable hypothesis regarding the aether, a disturbance in
which the vibrations are executed at right angles to the plane
of diffraction must be transmitted round the edge of an opaque
body with less diminution of intensity than a disturbance whose
vibrations are executed parallel to that plane. It follows that
when light, of which the vibrations are oblique to the plane of
* It iu in the year Thomson took hit degree (1816) that he bought, and read
with delight, the electrical memoir -which Given had published at Nottingham in
1828.
t Tram. Camb. Phil. Soc., ix (1849), p. 1. Stokei'a Melk. and Phy: Ftptr;
U, p. 218.
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid. 169
diffraction, is so transmitted, the plane of vibration will be more
nearly at right angles to the plane of diffraction in the diffracted
than in the incident light. Stokes himself performed experi-
ments to test the matter, using a grating in order to obtain
strong light diffracted at a large angle, and found that when
the plane of polarization of the incident light was oblique to the
plane of diffraction, the plane of polarization of the diffracted
light was more nearly parallel to the plane of diffraction. This
result, which was afterwards confirmed by L. Lorenz * appeared
to confirm decisively the hypothesis of Fresnel, that the vibra-
tions of the aethereal particles are executed at right angles to
the plane of polarization.
Three years afterwards Stokes indicated! a second line of
proof leading to the same conclusion. It had long been known
that the blue light of the sky, which is due to the scattering of
the sun's direct rays by small particles or molecules in the
atmosphere, is partly polarized. The polarization is most
marked when the light comes from a part of the sky distant 90°
from the sun, in which case it must have been scattered in a
direction perpendicular to that of the direct sunlight incident
on the small particles; and the polarization is in the plane
through the Bun.
If, then, the axis of y by taken parallel to the light incident
on a small particle at the origin, and the scattered light be
observed along the axis of x, this scattered light is found to be
polarized in the plane xy. Considering the matter from the
dynamical point of view, we may suppose the material particla
to possess so much inertia (compared to the aether) that it is " -
practically at rest. Its motion relative to the aether, which is
the cause of the disturbance it creates in the aether, will there-
fore be in the same line as the incident aethereal vibration,
but in the opposite direction. The disturbance must be
transversal, and must therefore be zero in a polar direction and
* Ana. d. Ph ji. rati (1880), p. 316. Phil. Mag. ui (1861), p. 321.
t Phil. Tram., 1852, p. 463. Stokw'i Math, and Fhyi. Paptrt, iii, p. 287.
Cf. the foot-note addad on p. 361 ot the Math, and Phyi. Papert.
3,Bl,ZEdhyG00gle
170 The Aether as an Elastic Solia.
a maximum in an equatorial direction, its amplitude being, in
fact, proportional to the Bine of the polar distance. The polar
line must, by considerations of symmetry, be the line of the
incident vibration. Thus we see that none of the light scattered
in the ^-direction can come from that constituent of the incident-
light which vibrates parallel to the x axis ; so the light observed
in this direction must consist of vibrations parallel to the z-axis.
But we have seen that the plane of polarization of the scattered
light is the plane of xy ; and therefore the vibration is at right
angles to the plane of polarization."
The phenomena of diffraction and of polarization by scatter-
ing thus agreed in confirming the result arrived at in Fresnel's
and Green's theory of reflexion. The chief difficulty in accepting
it arose in connexion with the optics of crystals. As we have
seen, Green and Cauchy were unable to reconcile the hypothesis
of aethereal vibrations at right angles to the plane of polariza-
tion with the correct formulae of crystal-optics, at any rate bo
long as the aether within crystals was supposed to be free from
initial stress. The underlying reason for this can be readily
seen. In a crystal, where the elasticity is different in different
directions, the resistance to distortion depends solely on the
orientation of the plane of distortion, which in the case of light
is the plane through the directions of propagation and vibration.
Now it is known that for light propagated parallel to one of the
axes of elasticity of a crystal, the velocity of propagation
depends only on the plane of polarization of the light, being the
same whichever of the two axes lying in that plane is the
direction of propagation. Comparing these results, we see that
the plane of polarization must be the plane of distortion, and
therefore the vibrations of the aether-particles must be executed
parallel to the plane of polarization, t
* The theory of polarization by snmil particles wnl afterwards investigated by
Lord Bayleigh, Phil. Meg. xli (1871).
t In Freansl'B theory of crystal -optica, in which the aether -Tib ration* are at
right angles to the plane of polarization, the velocity of propagation depend! only
on the direction of vibration, not on the plane through thin and the direction of
tranemiasion.
3,Bl,ZEdhyG00gle
The Aether as an Elastic Solid. 171
A way of escape from this conclusion suggested itself to
Stokes,* and later to Eankinet and Lord Kayleigh.J What if the
aether in a crystal, instead of having its elasticity different in >
different directions, were to have its rigidity invariable and its
inertia different in different directions ? This would bring the \
theory of crystal-optics into complete agreement with Fresnel's
and Green's theory of reflexion, in which the optical differences
between media are attributed to differences of inertia of the
aether contained within them. The only difficulty lies in
conceiving bow aelotropy of inertia can exist ; and all three
writers overcame this obstacle by pointing out that a solid
which is immersed in a fluid may have its effective inertia
different in different directions. For instance, a coin immersed
in water moves much more readily in its own plane than in the
direction at right angles to this.
Suppose then that twice the kinetic energy per uuit volume
of the aether within a crystal is represented by the expression
fW /&,Y /fc.V
and that the potential energy per unit volume has the same
value as in space void of ordinary matter. The aether is
assumed to be incompressible, so that div e is zero : the potential
energy per unit volume is therefore
*"** \\dy + dzj + \dz + to) + [ox + dy) ~ 3y &
_ . fade, .Sfj&vl
3* 3a; ~ &c dy)'
where n denotes as usual the rigidity.
* Stokes, in a Utter to Lord Hay! sigh, inserted in his Mimoir and Scitntijie
CorrapondiHct, ii, p. 09, explains that the idea presented itself to him while he
n> writing the paper on Fluid Motion which appeared in Trans. Gamh. Phil.
See., nil (1843), p. 10S. He loggeated the a-are-iiuface to which thia theorj
lead* in Brit. Aeeoe. Sep., 188!, p. 269.
t Phil. Meg. (4), i (1861), p. 441. I Phil. Hag. (4), xli (1871), p. 619.
dhyGoOgk
172 The Aether as an Elastic Solid.
The variational equation of motion is
-fllMlM*!)}^
where y denotes an undetermined function of (x, y, z) : the term
in p being introduced on account of the kinematics! constraint
expressed by the equation
div e - 0.
The equations of motion which result from this variational
equation are
&t, dp _.
r #* __