A History of the Theories of Aether and Electricity (1910) — part 9 of 29

In a further letter to Arago, dated April 29th, 1818, Young recurred to the subject of transverse vibrations, comparing light to the undulations of a cord agitated by one of its extremities.^ This letter was shown 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 Life of Young, p. 391. t Young's Works, i., p. 279.

JThis analogy had been given by Hooke in a communication to the Royal Society on Feb. 15, 1671-2. But there seems no reason to suppose that Hook e- appreciated the point now advanced by Young.

from Bradley to FresneL 123

still 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 also 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,f " 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

*Annales de Chiinie, xvii (1821), p. 180; (Eiwres, i, p. 629. Young had already drawn attention to this point. " It is difficult," he says in his Lectures on Natural Philosophy, ed. 1807, vol. i, p. 138, "to compare the lateral adhesion, or the force which resists the detrusion of the parts of a solid, with any form of direct cohesion. This force constitutes the rigidity or hardness of a solid body, and is wholly absent from liquids."

t Loc. cit, p. 185.

124 The Luminiferous 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, in a pencil of polarized light, the

from Bradley, to Fresnel. 125

oscillatory motion is executed at right angles to the plane of polarization"

This result afforded Fresnel a foothold in dealing with the problem which occupied the rest of his life : henceforth his 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 roots 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.f

Now a wave propagated in any direction through a uniaxal

*His first memoir on Double Refraction was presented to the Academy on Nov. 19th, 1821, but has not been published except in his collected works: (Eitvres, ii, p. 261. It was followed by other papers in 1822; and the results were finally collected in a memoir which was printed in 1827, Mem. de VAcad. vii, p. 45, (Euvres, ii, p. 479.

t In attempting to reconstruct Fresnel's course of thought at this period, the present writer has derived much help from the Life prefixed to the (Euvres de Fresnel. Both Fresnel and Young were singularly fortunate in their biographers : Peacock's Life of Young, and this notice of Fresnel, which was the last work of Verdet, are excellent reading.

126 The Luminiferous Medium,

crystal can be resolved into two plane-polarized components ; one of these, the " ordinary ray," is polarized in the principal section, and has a velocity vl9 which may be represented by the radius of Huygens' sphere — say,

Vi = &;

while the other, the " extraordinary ray," is polarized in a plane .at right angles to the principal section, and has a wave- velocity v9, which may be represented by the perpendicular drawn from the centre of Huygens' spheroid on the tangent-plane parallel to the plane of the wave. If the spheroid be represented by the equation

if + z'" x*

— + ^ = 1-

and if (I, m, n) denote the direction-cosines of the normal to the plane of the wave, we have therefore

v,~ = a*(m* + n*) + ?>2/2.

But the quantities 1/Vi and l/t?8, as given by these equations, are easily seen to be the lengths of the semi-axes of the ellipse in which the spheroid

62(?/3 4- z~) + arx- = 1

is intersected by the plane

Ix + my + nz = 0 ;

.and thus the construction in terms of Huygens' sphere and spheroid can be replaced by one which depends only on a single surface, namely the spheroid

Having achieved this reduction, Fresnel guessed that the

<?ase of biaxal crystals could be covered by substituting for the latter spheroid an ellipsoid with three unequal axes — say, xz if z* _+£+_ = If I/Vi and l/^ denote the lengths of the semi-axes of the .ellipse in which this ellipsoid is intersected by the plane Ix 4 my + nz - 0, from Bradley to Fresnel. 127 it is well known that #1 and vz are the roots of the equation in v ; --0; 1 . 1 ,1 tf tf v- ti £2 «3 and accordingly Fresnel conjectured that the roots of this equation represent the velocities, in a biaxal crystal, of the two plane-polarized waves whose normals are in the direction (I, m, n). Having thus arrived at his result by reasoning of a purely geometrical character, he now devised a dynamical scheme to suit it. The vibrating medium within a crystal he supposed to be ultimately constituted of particles subjected to mutual forces ; and on this assumption he showed that the elastic force of restitution when the system is disturbed must depend linearly on the displacement. In this first proposition a difference is apparent between Fresnel's and a true elastic-solid theory ; for in actual elastic solids the forces of restitution depend not on the absolute displacement, but on the strains, i.e., the relative displacements. In any crystal there will exist three directions at right angles to each other, for which the force of restitution acts in the same line as the displacement : the directions which possess this property are named axes of elasticity. Let these be taken as axes, and suppose that the elastic forces of restitution for unit displacements in these three directions are 1/5], l/c2, l/«s respectively. That the elasticity should vary with the direction of the molecular displacement seemed to Fresnel to suggest that the molecules of the material body either take part in the luminous vibration, or at any rate influence in some way the elasticity of the aether. A unit displacement in any arbitrary* direction (a, )3, 7) can be resolved into component displacements (cos a, cos /3, cos 7) parallel to the axes, and each of these produces its own effect 128 The Luminiferous Medium ^ independently ; so the components of the force of restitution are COS a COS )3 COS y €l ft £3 This resultant force has not in general the same direction as the displacement which produced it ; but it may always he decomposed into two other forces, one parallel and the other perpendicular to the direction of the displacement ; and the former of these is evidently The surface COS2 a COS2 )3 COS2 7 I {_ I £_ fl €2 £3 X2 V* £i £2 £3 will therefore have the property that the square of its radius vector in any direction is proportional to the component in that direction of the elastic force due to a unit displacement in that direction : it is called the surface of elasticity. Consider now a displacement along one of the axes of the section on which the surface of elasticity is intersected by the plane of the wave. It is easily seen that in this case the com- ponent of the elastic force at right angles to the displacement acts along the normal to the wave-front; and Fresnel assumes that it will be without influence on the propagation of the vibrations, on the ground of his fundamental hypothesis that the vibrations of light are performed solely in the wave-front. This step is evidently open to criticism ; for in a dynamical theory everything should be deduced from the laws of motion without special assumptions. But granting his contention, it follows that such a displacement will retain its direction, and will be propagated as a plane-polarized wave with a definite velocity. Now, in order that a stretched cord may vibrate with unchanged period, when its tension is varied, its length must be increased proportionally to the square root of its tension ; and similarly the wave-length of a luminous vibration of given period is proportional to the square root of the elastic force (per unit from Bradley to Fresnel. 129 displacement), which urges the molecules of the medium parallel to the wave-front. Hence the velocity of propagation of a wave, measured at right angles to its front, is proportional to the square root of the component, along the direction of dis- placement, of the elastic force per unit displacement ; and the velocity of propagation of such a plane-polarized wave as we have considered is proportional to the radius vector of the surface of elasticity in the direction of displacement. Moreover, any displacement in the given wave-front can be resolved into two, which are respectively parallel to the two axes of the diametral section of the surface of elasticity by a plane parallel to this wave-front ; and it follows from what has been said that each of these component displacements will be propagated as an independent plane-polarized wave, the velocities of propagation being proportional to the axes of the section,* and therefore inversely proportional to the axes of the section of the inverse surface of this with respect to the origin, which is the ellipsoid * + £ + *-i. £i £2 £3 But this is precisely the result to which, as we have seen, Fresnel had been led by purely geometrical considerations ; and thus his geometrical conjecture could now be regarded as substantiated by a study of the dynamics of the medium. It is easy to determine the wave-surface or locus at any instant —say, t = 1 — of a disturbance originated at some previous instant — say,£ = 0 — at some particular point — say, the origin. For this wave-surface will evidently be the envelope of plane waves emitted from the origin at the instant t = 0 — that is, it will be the envelope of planes Ix + my + nz - v = 0, where the constants /, m, n, v are connected by the identical equation I2 + m* + nz = 1, * It is evident from this that the optic axes, or lines of single wave-velocity, along which there is no double refraction, will be perpendicular to the two circular sections of the surface of elasticity. K 130 The Lumimferous Medium, and by the relation previously found — namely, /2 m2 n~ 1 By the usual procedure for determining envelopes, it may be shown that the locus in question is the surface of the fourth degree xz_ _f _fl_ _ n which is called Fresnel's wave-surface* It is a two-sheeted surface, as must evidently be the case from physical considerations. In uniaxal crystals, for which *2 and c 3 are equal, it degenerates into the sphere r2 = l/e>, and the spheroid ^ + fl (tf + Z2) = 1. It is to these two surfaces that tangent-planes are drawn in the construction given by Huygens for the ordinary and extraordinary refracted rays in Iceland spar. As Fresnel observed, exactly the same construction applies to biaxal crystals, when the two sheets of the wave-surface are substi- tuted for Huygens' sphere and spheroid. " The theory which I have adopted," says Fresnel at the end of this memorable paper, " and the simple constructions which I have deduced from it, have this remarkable character, that all the unknown quantities are determined together by the solution of the problem. We find at the same time the velocities of the ordinary ray and of the extraordinary ray, and their planes of polarization. Physicists who have studied attentively the laws of nature will feel that such simplicity and * Another construction for the wave-surface is the following, which is due to MacCullagh, Coll. Works, p. 1. Let the ellipsoid *ix~ + 62^" ~*~ *3~~ = * be intersected hy a plane through its centre, and on the perpendicular to that plane take lengths equal to the semi-axes of the section. The locus of these extremities is the wave-surface. from Bradley to FresneL 131 such close relations between the different elements of the phenomenon are conclusive in favour of the hypothesis on which they are based." The question as to the correctness of Fresnel's construction was discussed for many years afterwards. A striking conse- quence of it was pointed out in 1832 by William Kowan Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who remarked* that the surface defined by Fresnel's equation has four conical points, at each of which there is an infinite number of tangent planes ; consequently, a single ray, proceeding from a point within the crystal in the direction of one of these points, must be divided on emergence into an infinite number of rays, constituting a conical surface. Hamilton also showed that there are four planes, each of which touches the wave- surface in an infinite number of points, constituting a circle of contact : so that a corresponding ray incident externally should be divided within the crystal into an infinite number of refracted rays, again constituting a conical surface. These singular and unexpected consequences of the theory were shortly afterwards verified experimentally by Humphrey Lloyd,f and helped greatly to confirm belief in Fresnel's theory. It should, however, be observed that conical refraction only shows his form of the wave- surf ace to be correct in its general features, and is no test of its accuracy in all details. But it was shown experimentally by Stokes in 1872J Glazebrook in 1879,§ and Hastings in 1887,1 1 that the construction of Huygens and Fresnel is certainly correct to a very high degree of approximation; and Fresnel's final formulae have since been regarded as unassailable. The dynamical substructure on which he based them is, as we have seen, open to objection ; * Trans. Roy. Irish Acad., xvii (1833), p. 1. t Trans. Roy. Irish Acad., xvii (1833), p. 145. Strictly speaking, the bright oone which is usually observed arises from rays adjacent to the singular ray : the latter can, however, be observed, its enfeeblement by dispersion into the conical form causing it to appear dark. I Proc. R. S., xx, p. 443. § Phil. Trans., clxxi, p. 421. || Am. Jour. Sci. (3), xxxv, p. 60. K 2 132 The Luminiferous Medium, but, as Stokes observed*: "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." In a second supplement to his first memoir on Double Eefraction, presented to the Academy on November 26th, 1821,-]- Fresnel indicated the lines on which his theory might be extended so as to take account of dispersion. " The molecular groups, or the particles of bodies," he wrote, " may be separated by intervals which, though small, are certainly not altogether insensible relatively to the length of a wave." Such a coarse- grainedness of the medium would, as he foresaw, introduce into the equations terms by which dispersion might be explained ; indeed, the theory of dispersion which was afterwards given by Cauchy was actually based on this principle. It seems likely that, towards the close of his life, Fresnel was contemplating a great memoir on dispersion^ which was never completed. Fresnel had reason at first to be pleased with the reception of his work on the optics of crystals : for in August, 1822, Laplace spoke highly of it in public ; and when at the end of the year a seat in the Academy became vacant, he was encouraged to hope that the choice would fall on him. In this he was disappointed. §. Meanwhile his researches were steadily continued ; and in January, 1823, the very month of his rejection, he presented to- the Academy a theory in which reflexion and refraction] | are referred to the dynamical properties of the luminiferous media. *Brit. Assoc. Rep., 1862, p. 254. t (Euvres, ii, p. 438. J Cf. the biography in (Euvres de Fresnel, i, p. xcvi. § Writing to Young in the spring of 1823, he says : " Tous ces memoires, que dernierement j'ai pre'sentes coup sur coup a 1'Academie des Sciences, ne m'en ont pas cependant otivert la porte. C'est M. Dulong qui a ete nomine pour remplir la place vacante dans la section de physique. . . Vous voyez, Monsieur, que la theorie des ondulations ne m'a point porte honheur : mais cela ne m'en degoute pas : et je me console de ce malheur en m* occupant d'optique avec une nouvelle ardeur." || The MSS- was for some time believed to be lost, but was ultimately found among the papers of Fourier, and printed in Mem. de 1'Acad. xi (1832), p. 393 : (Euvres, i, p. 767. from Bradley to FresneL 133 As in his previous investigations, he assumes that the vibrations which constitute light are executed at right angles to the plane of polarization. He adopts Young's principle, that reflexion and refraction are due to differences in the inertia of the aether in different material bodies, and supposes (as in his memoir on Aberration) that the inertia is proportional to the inverse square of the velocity of propagation of light in the medium. The conditions which he proposes to satisfy at the interface between two media are that the displacements of the adjacent molecules, resolved parallel to this interface, shall be equal in the two media ; and that the energy of the reflected and refracted waves together shall be equal to that of the incident wave. On these assumptions the intensity of the reflected and refracted light may be obtained in the following way : — Consider first the case in which the incident light is polarized in the plane of incidence, so that the displacement is at right angles to the plane of incidence ; let the amplitude of the displacement at a given point of the interface be / for the incident ray, g for the reflected ray, and h for the refracted ray. The quantities of energy propagated per second across unit cross-section of the incident, reflected, and refracted beams are proportional respectively to where cb c2, denote the velocities of light, and pl} pz the densities of aether, in the two media ; and the cross-sections of the beams which meet the interface in unit area are cos i, cos i, cos r respectively. The principle of conservation of energy therefore gives c,p! cos i ./2 = c,/o! cos i . gz + c2/o2 cos r . h~. The equation of continuity of displacement at the interface is / + 9 = h. 134 The Luminiferous Medium, Eliminating li between these two equations, and using the formulae sin2 T Co2 pi sin2 i C* p2 ' we obtain the equation Z. _ sm (^ ~ r) g sin (i + r) Thus when the light is polarized in the plane of reflexion, the amplitude of the reflected wave is Q-l -T\ (ft ^ -0*\ - — \-. r x the amplitude of the incident vibration. sin pj + r) Fresnel shows in a similar way that when the light is polarized at right angles to the plane of reflexion, the ratio of the amplitudes of the reflected and incident waves is tan (i - r) tan (i + r) These formulae are generally known as Fresnel' s sine-law and FresneTs tangent-law respectively. They had, however, been discovered experimentally by Brewster some years previously. When the incidence is perpendicular, so that i and r are very small, the ratio of the amplitudes becomes Limit , ^ + r or where ju2 and //i denote the refractive indices of the media. This formula had been given previously by Young* and Poisson,f on the supposition that the elasticity of the aether is of the same kind as that of air in sound. When i + r = 90°, tan (i + r) becomes infinite : and thus a theoretical explanation is obtained for Brewster 's law, that if the incidence is such as to make the reflected and refracted rays- * Article Chromatics, Encycl. Britt. Suppl. t Mem. Inst. ii. (1817). fro vi Bradley to Fresnel. 135 perpendicular to each other, the reflected light will be wholly polarized in the plane of reflexion. Fre&nel's investigation can scarcely be called a dynamical theory in the strict sense, as the qualities of the medium are not defined. His method was to work backwards from the known properties of light, in the hope of arriving at a mechanism to which they could be attributed ; he succeeded in accounting for the phenomena in terms of a few simple principles, but was not able to specify an aether which would in turn account for these principles. The " displacement " of Fresnel could not be a displacement in an elastic solid of the usual type, since its normal component is not continuous across the interface between two media.* The theory of ordinary reflexion was completed by a dis- cussion of the case in which light is reflected totally. This had formed the subject of some of Fresnel's experimental researches several years before; and in two papersf presented to the Academy in November, 1817, and January, 1818, he had shown that light polarized in any plane inclined to the plane of reflexion is partly "depolarized" by total reflexion, and that this is due to differences of phase which are introduced between the components polarized in and perpendicular to the plane of reflexion. " When the reflexion is total," he said, " rays polarized in the plane of reflexion are reflected nearer the surface of the glass than those polarized at right angles to the same plane, so that there is a difference in the paths described." This change of phase he now deduced from the formulae already obtained for ordinary reflexion. Considering light polarized in the plane of reflexion, the ratio of the amplitudes of the reflected and incident light is, as we have seen, sin (i - r) sin (i + r) ' when the sine of the angle of incidence is greater than /i2/jui, * Fresnel's theory of reflexion can, however, he reconciled with the electro- magnetic theory of light, by identifying his "displacement" with the electric force. f (Euvres de Fresnel, i., pp. 441, 487. 136 The Luminiferous Medium. so that total reflexion takes place, this ratio may be written in the form where 6 denotes a real quantity defined by the equation tan cos ^ Fresnel interpreted this expression to mean that the amplitude of the reflected light is equal to that of the incident, but that the two waves differ in phase by an amount 0. The case of light polarized at right angles to the plane of reflexion may be treated in the same way, and the resulting formulae are completely confirmed by experiment. A few months after the memoir on reflexion had been presented, Fresnel was elected to a seat in the Academy ; and during the rest of his short life honours came to him both from France and abroad. In 1827 the Royal Society awarded him the Rumford medal ; but Arago, to whom Young had confided the mission of conveying the medal, found him dying ; and eight days afterwards he breathed his last. By the genius of Young and Fresnel the wave-theory of light was established in a position which has since remained unquestioned ; and it seemed almost a work of supererogation when, in 1850, Foucault* and Fizeau,f carrying out a plan long before imagined by Arago, directly measured the velocity of light in air and in water, and found that on the question so long debated between the rival schools the adherents of the undulatory theory had been in the right. * Comptes Rendus, xxx (1850), p. 551. t Ibid., p. 562. ( 137 ) CHAPTER V. THE AETHER AS AN ELASTIC SOLID. WHEN Young and Fresnel put forward the view that the vibrations of light are performed at right angles to its direction of propagation, they at the same time pointed out that this peculiarity might be explained by making a new hypothesis regarding the nature of the luminiferous medium ; namely, that it possesses the power of resisting attempts to distort its shape. It is by the possession of such a power that solid bodies are distinguished from fluids, which offer no resistance to distortion; the idea of Young and Fresnel may therefore be expressed by the simple statement that the aether behaves as an elastic solid. After the death of Fresnel this conception was developed in a brilliant series of memoirs to which our attention must now be directed. The elastic-solid theory meets with one obvious difficulty at the outset. If the aether has the qualities of a solid, how is it that the planets in their orbital motions are able to journey through it at immense speeds without encountering any perceptible resistance ? This objection was first satisfactorily answered by Sir George Gabriel Stokes* (b. 1819, d. 1903), who remarked that such substances as pitch and shoemaker's wax, though so rigid as to be capable of elastic vibration, are yet sufficiently plastic to permit other bodies to pass slowly through them. The aether, he suggested, may have this combination of qualities in an extreme degree, behaving like an elastic solid for vibrations so rapid as those of light, but yielding like a fluid to the much slower progressive motions of the planets. Stokes's explanation harmonizes in a curious way with Fresnel's hypothesis that the velocity of longitudinal waves in * Trans. Camb. Phil. Soc., viii, p. 287 (1845). 138 The Aether as an Elastic Solid. the aether is indefinitely great compared with that of the transverse waves ; for it is found by experiment with actual substances that the ratio of the velocity of propagation of longitudinal waves to that of transverse waves increases rapidly as the medium becomes softer and more plastic. In attempting to set forth a parallel between light and the vibrations of an elastic substance, the investigator is compelled more than once to make a choice between alternatives. He may, for instance, suppose that the vibrations of the aether are executed either parallel to the plane of polarization of the light or at right angles to it ; and he may suppose that the different refractive powers of different media are due either to differences in the inertia of the aether within the media, or to differences in its power of resisting distortion, or to both these causes combined. There are, moreover, several distinct methods for avoiding the difficulties caused by the presence of longitudinal vibrations ; and as, alas ! we shall see, a further source of diversity is to be found in that liability to error from which no man is free. It is therefore not surprising that the list of elastic-solid theories is a long one. At the time when the transversality of light was dis- covered, no general method had been developed for investi- gating mathematically the properties of elastic bodies; but under the stimulus of Fresnel's discoveries, some of the best intellects of the age were attracted to the subject. The volume of Memoirs of the Academy which contains Fresnel's theory of crystal-optics contains also a memoir by Claud Louis Marie Henri Navier* (&. 1785, d. 1836), at that time Professor of Mechanics in Paris, in which the correct equations of vibratory motion for a particular type of elastic solid were for the first time given. ISTavier supposed the medium to be ultimately constituted of an immense number of particles, which act on each other with forces directed along the lines joining them, and depending on their distances apart ; and showed that if e denote * Mem. de 1'Acad. vii, p. 375. The memoir was presented in 1821, and published in 1827. The Aether as an Elastic Solid. 139 the (vector) displacement of the particle whose undisturbed position is (x, y, z], and if p denote the density of the medium, the equation of motion is p — = - 3n grad div e - n curl curl e, ot where n denotes a constant which measures the rigidity, or power of resisting distortion, of the medium. All such elastic properties of the body as the velocity of propagation of waves in it must evidently depend on the ratio n/p. Among the referees of one of Navier's papers was Augustine Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of the nineteenth century,* who, becoming interested in the question, published in 1828f a discussion of it from an entirely different point of view. Instead of assuming, as Navier had done, that the medium is an aggregate of point-centres of force, and thus involving himself in doubtful molecular hypotheses, he devised a method of directly studying the elastic properties of matter in bulk, and by its means showed that the vibrations of an isotropic solid are determined by the equation 82e (1 4 \ • p — = - [fc + -n\ grad div e - n curl curl e ; here n denotes, as before, the constant of rigidity; and the constant &, which is called the modulus of compression^. denotes the ratio of a pressure to the cubical compression produced by it. Cauchy's equation evidently differs from Navier's in that