A History of the Theories of Aether and Electricity (1910) — part 11 of 29

The hesitation which was felt in accepting the rotationally elastic aether arose mainly from the want of any readily conceived example of a body endowed with such a property. This difficulty was removed in 1889 by Sir William Thomson (Lord Kelvin), who designed mechanical models possessed of rotational elasticity. Suppose, for example,* that a structure is formed of spheres, each sphere being in the centre of the tetrahedron formed by its four nearest neighbours. Let each sphere be joined to these four neighbours by rigid bars, which have spherical caps at their ends so as to slide freely on the spheres. Such a structure would, for small deformations, behave like an incompressible perfect fluid. Now attach to each bar a pair of gyroscopically-mounted flywheels, rotating with equal and opposite angular velocities, and having their axes in the line of the bar : a bar thus equipped will require a couple to hold it at rest in any position inclined to its original position, and the structure as a whole will possess that kind of quasi- elasticity which was first imagined by MacCullagh.

This particular representation is not perfect, since a system of forces would be required to hold the model in equilibrium if it were irrotationally distorted. Lord Kelvin subsequently invented another structure free from this defect. t

• Comptes Eendus, Sept. 16, 1889 : Kelvin's Math, and Phys. Papers, iii, p. 466.

tProc. Roy. Soc. Edinb., Mar. 17, 1890: Kelvin's Math, and Phys. Papers, iii, p. 468.

158 The Aether as an Elastic Solid.

The work of Green proved a stimulus not only to MacCullagh but to Cauchy, who now (1839) published yet a third theory of reflexion.* This appears to have owed its origin to a remark of Green's, f that the longitudinal wave might be avoided in either of two ways — namely, by supposing its velocity to be indefinitely great or indefinitely small. Green curtly dismissed the latter alternative and adopted the former, on the ground that the equilibrium of the medium would be unstable if its compressibility were negative (as it must be if the velocity of longitudinal waves is to vanish). Cauchy, without attempting to meet Green's objection, took up the study of a medium whose elastic constants are connected by the equation

k + ±n = 0,

so that the longitudinal vibrations have zero velocity; and showed that if the aethereal vibrations are supposed to be executed at right angles to the plane of polarization, and if the rigidity •of the aether is assumed to be the same in all media, a ray which is reflected will obey the sine-law and tangent-law of Fresnel. The boundary-conditions which he adopted in order to obtain this result were the continuity of the displacement e and -of its derivate 3e/9#, where the axis of x is taken at right angles to the interface.* These are not the true boundary-con- ditions for general elastic solids ; but in the particular case now under discussion, where the rigidity is the same in the two media, they yield the same equations as the conditions correctly given by Green.

The aether of Cauchy's third theory of reflexion is well worthy of some further study. It is generally known as they .contractile or labile^ aether, the names being due to William

• Comptes Rendus, ix, p. 676 (25 NOT., 1839), and p. 726 (2 Dec., 1839).

t Green's Math. Papers, p. 246.

J Comptes Eendus, x, p. 347 (March 2, 1840) : xxvii, p. 621 (1848) ; sxviii, p. 25 (1849). Mem. de 1'Acad., xxii (1848), pp. 17, 29.

§ Labile or neutral is a term used of such equilibrium as that of a rigid body oil -a perfectly smooth horizontal plane.

The Aether as an Elastic Solid. 159

Thomson (Lord Kelvin), who discussed it long afterwards.* It may be defined as an elastic medium of (negative) com- pressibility such as to make the velocity of the longitudinal wave zero : this implies that no work is required to be done in order to give the medium any small irrotational disturbance. An example is furnished by homogeneous foam free from air and held from collapse by adhesion to a containing vessel

Cauchy, as we have seen, did not attempt to refute Green's objection that such a medium would be unstable ; but, as Thomson remarked, every possible infinitesimal motion of the medium is, in the elementary dynamics of the subject, proved to be resolvable into coexistent wave-motions. If, then, the velocity of propagation for each of the two kinds of wave-motion is real, the equilibrium must be stable, provided the medium either extends through boundless space or has a fixed containing vessel as its boundary.

When the rigidity of the luminiferous medium is supposed to have the same value in all bodies, the conditions to be satisfied at an interface reduce to the continuity of the displacement e, of the tangential components of curl e, and of the scalar quantity (k + ^n) div e across the interface.

Now we have seen that when a transverse wave is incident on an interface, it gives rise in general to reflected and refracted waves of both the transverse ajid the longitudinal species. In the case of the contractile aether, for which the velocity of propagation of the longitudinal waves is very small, the ordinary construction for refracted waves shows that the directions of propagation of the reflected and refracted longitudinal waves will be almost normal to the interface. The longitudinal waves will therefore contribute only to the component of displacement normal to the interface, not to the tangential components : in other words, the only tangential components of displacement at the interface are those due to the three trans- verse waves — the incident, reflected, and refracted. Moreover, the longitudinal waves do not contribute at all to curl e ; and,

• Phil. Mag. xxvi (1888), p. 414.

160 The Aether as an Elastic Solid.

therefore, in the contractile aether, the conditions that the tangential components of e and of n curl e shall be continuous across an interface are satisfied by the distortional part of the disturbance taken alone. The condition that the component of e normal to the interface is to be continuous is not satisfied by the distortional part of the disturbance taken alone, but is satisfied when the distortional and congressional parts are taken together.

The energy carried away by the longitudinal waves is infinitesimal, as might be expected, since no work is required in order to generate an irrotational displacement. Hence, with this aether, the behaviour of the transverse waves at an interface may be specified without considering the irrotational part of the disturbance at all, by the conditions that the conservation of energy is to hold and that the tangential components of e and of n curl e are to be continuous. But if we identify these transverse waves with light, assuming that the displacement e is at right angles to the plane of polarization of the light, and assuming moreover that the rigidity n is the same in all media* (the differences between media depending on differences in the inertia p), we have exactly the assumptions of Fresnel's theory of light : whence it follows that transverse waves in the labile aether must obey in reflexion the sine-law and tangent-law of Fresnel.

The great advantage of the labile aether is that it overcomes the difficulty about securing continuity of the normal com- ponent of displacement at an interface between two media : the light-waves taken alone do not satisfy this condition of continuity ; but the total disturbance consisting of light- waves and irrotational disturbance taken together does satisfy it ; and this is ensured without allowing the irrotational disturbance to carry off any of the energy. f

• This condition is in any case necessary for stability, as was shown by R. T. Glazebrook : cf. Thomson, Phil. Mag. xxvi, p. 500.

f The labile-aether theory of light may be compared with the electro-magnetic theory, by interpreting the displacement e as the electric force, and pe as the electric displacement.

The Aether as an Elastic Solid. 161

William Thomson (Lord Kelvin, b. 1824, d. 1908), who devoted much attention to the labile aether, was at one time led to doubt the validity of this explanation of light* ; for when investigating the radiation of energy from a vibrating rigid globe embedded in an infinite elastic-solid aether, he found that in some cases the irrotational waves would carry away a considerable part of the energy if the aether were of the labile type. This difficulty, however, was removed by the observationf that it is sufficient for the fulfilment of Fresnel's laws if the velocity of the irrotational waves in one of the two media is very small, without regard to the other medium. Following up this idea, Thomson assumed that in space void of ponderable matter the aether is practically incompressible by the forces concerned in light-waves, but that in the space occupied by liquids and solids it has a negatiye_CQmpres^biIiljy , so as to give zero velocity for longitudinal aether- waves in these bodies. This assumption was based on the conception that material atoms move through space without displacing the aether: a conception which, as Thomson remarked, contradicts the old scholastic axiom that two different portions of matter cannot simultaneously occupy the same space.J He supposed the aether to be attracted and repelled by the atoms, and thereby to be condensed or rarefied. §

The year 1839, which saw the publication of MacCullagh's dynamical theory of light and Cauchy's theory of the labile aether, was memorable also for the appearance of a memoir by Green on crystal-bptics.H This really contains two distinct theories, which respectively resemble Cauchy's First and Second Theories : in one of them, the stresses in the undisturbed state

• Baltimore Lectures (edition 1904), p. 214.

t Ibid. (ed. 1904), p. 411.

^ Michell and Boscovich in the eighteenth century had taught the doctrine of the mutual penetration of matter, i.e. that two substances may be in the same place at the same time without excluding each other : cf. Priestley's History i., p. 392.

6 Cf. Baltimore Lectures (ed. 1904), pp. 413-14, 463, and Appendices A and E.

|j Cambridge Phil. Trans., 1839 ; Green's Math. Papers;?. 293.

M

162 The Aether as an Elastic Solid.

of the aether are supposed to vanish, and the vibrations of the aether are supposed to be executed parallel to the plane of polarization of the light ; in the other theory, the initial stresses are not supposed to vanish, and the aether- vibrations are at right angles to the plane of polarization. The two investigations are generally known as Green's First and Second Theories of crystal-optics.

The foundations of both theories are, however, the same. Green first of all determined the potential energy of a strained crystalline solid ; this in the most general case involves 27 constants, or 21 if there is no initial stress.* If, however, as is here assumed, the medium possesses three planes of symmetry at right angles to each other, the number of constants reduces to. 12, or to 9 if there is no initial stress; if e denote the dis- placement, the potential energy per unit volume may be written

fo \2 ("be \2 fr\f \i\ (ffo \2 /f)p \2 /a/, N

• $} * (£) 1 + ** ft) + (I) + (I

fty*.

7 ty tz 9

sf3ey 3g, •*• 4/1 ;? + s-

2</ \dz ty

dx

The usual variational equation

= - [[f

• For there are 21 terms in a homogeneous function of the second degree in six variables.

The Aether as an Elastic Solid. 163

then yields the differential equations of motion, namely :

8 / dex dey fe,\ a / aex a^ , a^\

• — a— + h ^ + g — + — a — + A — + 0 — ], acVSaj ty y dzj dx\ fa ty y fa)9

and two similar equations.

These differ from Cauchy'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 : —

a. « b = c = JJK;

/'-j»-2/ / = M-2<7; tf-M-2fc; where /z denotes a new constant, f

• As Green showed, the hypothesis of transversality really involves the existence of planes of symmetry, so that it alone is capable of giving 14 relations between the 21 constants : and 3 of the remaining 7 constants may be removed by change of axes, leaving only four.

t It was afterwards shown by Barre de Saint- Venant (b. 1797, d. 1886), Journal de Math., vii (1863), p. 399, that if the initial stresses be supposed to vanish, the conditions which must be satisfied among the remaining nine constants

M 2

164 The Aether as an Elastic Solid.

Thus the potential energy per unit volume may be written

. n^* ^ rrdey^. Tde*

d> = 6r — + ±L — + -I ^~

ox oy oz

i T ] Wx\ ^ ™y\ -L. oe* I1 Ur + Ur- + Ur

At this point Green's two theories of crystal-optics diverge from each other. According to the first theory, the initial stresses G-, H, I are zero, so that

", > «>/>#>>/' /, *'» in order that the wave-surface may be Fresnel's, are the following : —

(34 _/) (3c -/)=(/ + /')*

((3a-

(Za - h) (3£ - h) = (h + A')

These reduce to Green's relations when the additional equation b = c is assumed.

Saint-Venant disputed the validity of Green's relations, asserting that they?are compatible only with isotropy. On this controversy cf. E. T. Glazebrook, Brit. Assoc. Report, 1885, p. 171, and Karl Pearson in Todhunter and Pearson's History of Elasticity, ii, § 147.

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, ju) 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 distortional 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 —

O-f-H-g-I-h.

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 and 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

P 7 where q denotes its modulus of rigidity when unstrained, and

• Proc. R. S. Edin. xv (1887), p. 21 : Phil. Mag. xxv (1888) p. 116 : Baltimore Lectures (ed. 1904), pp. 228-259.

1 66 The Aether as an Elastic Solid.

«> j3> 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

q q

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 (say) 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 be 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 Lectures (ed. 1904), p. 253.

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,* " I am 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 Eayleigh, 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 Larrnor. 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 Michell ; and for some reason which at this distance of time it is difficult to understand fully, Michell's researches seem to have attracted little or no attention among his collegiate contemporaries and successors,

• Baltimore Lectures (ed. 1904), p. 258.

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 since the time of Newton had isolated Cambridge from the continental schools, was abandoned in favour of the differential notation, and the works of the great 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 spite of the advances which were made in the great memoirs of the year 1839, the fundamental question as to whether the aether-particles 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.f 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 was in the year Thomson took his degree (1845) that he bought, and read with delight, the electrical memoir which Green had published at Nottingham in 1828.

f Trans. Camb. Phil. Soc., ix (1849), p. 1. Stokes's Math, and Phys. Papers, ii, p. 243.

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 indicatedf 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 sun.

If, then, the axis of y be 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 particle 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

• Ann. d. Phj-s. exi (1860), p. 315. Phil. Mag. xxi (1861), p. 321. t Phil. Trans., 1852, p. 463. Stokes's Math, and Phys. Papers, iii, p. 267. €f. the foot-note added on p. 361 oi the Math, and Phys. Paper*.

170 The Aether as an Elastic Solia.

a maximum in an equatorial direction, its amplitude being, in fact, proportional to the sine 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 o>axis ; so the light observed in this direction must consist of vibrations parallel to the 2-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 so 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.f

• The theory of polarization by small particles was afterwards investigated by Lord Rayleigh, Phil. Mag. xli(187l).

fin Fresnel's theory of crystal-optics, in which the aether-vibrations are at right angles to the plane of polarization, the velocity of propagation depends only on the direction of vibration, not on the plane through this and the direction of transmission.

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- op tics 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 how 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 unit volume of the aether within a crystal is represented by the expression

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

---

dz dx d

where n denotes as usual the rigidity.

• Stokes, in a letter to Lord Rayleigh, inserted in his Memoir and Scientific Correspondence, ii, p. 99, explains that the idea presented itself to him while he was writing the paper on Fluid Motion which appeared in Trans. Camh. Phil. Soc., via (1843), p. 105. He suggested the wave-surface to which this theory leads in Brit. Assoc. Rep., 1862, p. 269.

t Phil. Mag. (4), i (1851), p. 441. J Phil. Mag. (4), xli (1871), p. 519.

1 72 The Aether as an Elastic Solid.

The variational equation of motion is

pi ^r/ dex + pz —% dey + pz — 2 $ez[ dx dy dz

where p denotes an undetermined function of (x, y, z) : the term in p being introduced on account of the kinematical constraint expressed by the equation

div e = 0.

The equations of motion which result from this variational equation are

<hw=-£+nV*e" ••'""•'

and two similar equations. It is evident that p resembles a hydrostatic pressure.

Substituting in these equations the analytical expression for a plane wave, we readily find that the velocity F of the wave is connected with the direction-cosines (X, ^t, z/) of its normal by the equation

A2 u* vz

n- PIV* r n-ptV" + n- pzV* = '

When this is compared with Fresnel's relation between the velocity and direction of a wave, it is seen that the new formula differs from his only in having the reciprocal of the velocity in place of the velocity. About 1867 Stokes carried out a series of experiments in order to determine which of the two theories was most nearly conformable to the facts : he found the con- struction of Huygens and Fresnel to be decidedly the more correct, the difference between the results of it and the rival construction being about 100 times the probable error of observation.*

• Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion (Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be found. On the accuracy of Fresnel's construction cf. Glazebrook, Phil. Trans, clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60.

The Aether as an Elastic Solid. 173

The hypothesis that in crystals the inertia depends on direction seemed therefore to be discredited when the theory based on it was compared with the results of observation. But when, in 1888, W. Thomson (Lord Kelvin) revived Cauchy's theory of the labile aether, the question naturally arose as to whether that theory could be extended so as to account for the optical properties of crystals : and it was shown by E. T. Glazebrook* that the correct formulae of crystal-optics ar& obtained when the Cauchy-Thomson hypothesis of zero velocity for the longitudinal wave is combined with the Stokes-Kankine- Rayleigh hypothesis of aelotropic inertia.

For on reference to the formulae which have been already given, it is obvious that the equation of motion of an aether having these properties must be

(pie*, pzey, p3O = -n curl curl e,

where e denotes the displacement, n the rigidity, and (plt p2, /o3) the inertia : and this equation leads by the usual analysis ta Fresnel's wave-surface. The displacement e of the aethereal particles is not, however, accurately in the wave-front, as in Fresnel's theory, but is at right angles to the direction of the ray, in the plane passing through the ray and the wave- normal, f

Having now traced the progress of the elastic-solid theory so far as it is concerned with the propagation of light in ordinary isotropic media and in crystals, we must consider the attempts which were made about this time to account for the optical properties of a more peculiar class of substances.

It was found by Arago in 181 IJ that the state of polarization of a beam of light is altered when the beam is passed through a plate of quartz along the optic axis. The

• Phil. Mag. xxvi (1888), p. 521 ; xxviii (1889), p. 110.

t This theory of crystal-optics may be assimilated to the electro-magnetic theory by interpreting the elastic displacement e as electric force, and the vector (pifx, p^y, ptfz) as electric displacement.

• Mem. de 1'Institut, 1811, Part I, p. 115, sqq.

174 The Aether as an Elastic Solid.

phenomenon was studied shortly afterwards by Biot,* who showed that the alteration consists in a rotation of the plane of polarization about the direction of propagation : the angle of rotation is proportional to the thickness of the plate and inversely proportional to the square of the wave-length.

In some specimens of quartz the rotation is from left to right, in others from right to left. This distinction was shown by Sir John Herschelf (b. 1792, d. 1871) in 1820 to be associated with differences in the crystalline form of the specimens, the two types bearing the same relation to each other as a right-handed and left-handed helix respectively. FresnelJ and W. Thomsong proposed the term helical to denote the property of rotating the plane of polarization, exhibited by such bodies as quartz : the less appropriate term natural rotatory polarization is, however, generally used.||

Biot showed that many liquid organic bodies, e.g. turpentine and sugar solutions, possess the natural rotatory property : we might be led to infer the presence of a helical structure in the molecules of such substances ; and this inference is sup- ported by the study of their chemical constitution; for they are invariably of the "mirror-image" or "enantiomorphous" type, in which one of the atoms (generally carbon) is asym- metrically linked to other atoms.