and Neumann the ether is supposed to consist of distinct particles, regarded as material points, acting on one another by forces in the line joining them which vary as some function of the distances, and the arrangement of these particles is supposed to be different in different directions. The medium is further supposed to possess three rectangular planes of symmetry, the double refraction of crystals, so far as has been observed, being symmetrical with respect to three such planes. The equations of motion of the medium are deduced by a method similar to that employed by Navier in the case of an isotropic medium. The equations arrived at by Cauchy, the medium being referred to planes of symmetry, contain nine arbitrary constants, three of which express the pressures in the principal directions in the state of equilibrium. Those employed by Neumann contain only six such constants, the medium in its natural state being supposed free from pressure.

In the theory of double refraction, whatever be the particular dynamical conditions assumed, everything is reduced to the determination of the velocity of propagation of a plane wave propagated in any given direction, and the mode of vibration of the particles in such a wave which must exist in order that the wave may be propagated with a unique velocity. In the theory of Cauchy now under consideration, the direction of vibration and the reciprocal of the velocity of propagation are given in direction and magnitude respectively by the principal axes of a certain ellipsoid, the equation of which contains the nine arbitrary constants, and likewise the direction-cosines of the wave-normal. Cauchy adduces reasons for supposing that the three constants G, H, I, which express the pressures in the state of equilibrium, vanish, which leaves only six constants. For waves perpendicular to the principal axes, the squared velocities of propagation and the corresponding directions of vibration are given by the following Table :--

For waves in these directions, then, the vibrations are either wholly normal or wholly transversal. The latter are those with which we have to deal in the theory of light. Now, according to observation, in any one of the principal planes of a doubly refracting crystal, that ray which is polarized in the principal plane obeys the ordinary law of refraction. In order therefore that the conclusions of this theory should at all agree with observation, we must suppose that in polarized light the vibrations are parallel, not perpendicular, to the plane of polarization.

Let l, m, n be the direction-cosines of the wave-normal. In the theory of Cauchy and Neumann, the square v2 of the velocity of propagation is given by a cubic of the form

6

where α2, α, α are homogeneous functions of the 1st order as regards L, M, N, P, Q, R, and homogeneous functions of the orders. 2, 4, 6 as regards l, m, n, involving even powers only of these quantities. For a wave perpendicular to one of the principal planes, that of yz suppose, the cubic splits into two rational factors, of which that which is of the first degree in v2, namely,

v2 — m2R — n2 Q,

corresponds to vibrations perpendicular to the principal plane. This is the same expression as results from Fresnel's theory, and accordingly the section, by the principal plane, of one sheet of the wave-surface, which in this theory is a surface of three sheets, is an ellipse, and the law of refraction of that ray which is polarized perpendicularly to the principal plane agrees exactly with that given by the theory of Fresnel.

For the two remaining waves, the squared velocities of propagation are given by the quadratic

(v2 — m3M — n2P) (v2 - m2P - n2N) - 4m2n2 P2=0 ...(1); but according to observation the ray polarized in the principal plane obeys the ordinary law of refraction. Hence (1) ought to be satisfied by v2 − (m2 + n2) P = 0, which requires that

(M − P) (N − P) = 4P2,

on which supposition the remaining factor must evidently be linear as regards m2, n2, and therefore must be

v2 — m2M — n2N,

ང

since it gives when equated to zero v2 M, or v2= N for m=1, or n 1. And since the same must hold good for each of the principal planes, we must have the three following relations between the six constants,

(M −P)(N − P)=4P2; (N−Q) (L − Q) = 4Q2 ;

The existence of six constants, of which only three are wanted to satisfy the numerical values of the principal velocities of propagation in a biaxal crystal, permits of satisfying these equations ; so that the law that the ray polarized in the plane of incidence, when that is a principal plane, obeys the ordinary law of refraction is not inconsistent with Cauchy's theory. This simple law is, however, not in the slightest degree predicted by the theory, nor even rendered probable, nor have any physical conditions been pointed out which would lead to the relations (2); and, indeed, from the form of these equations, it seems hard to conceive what physical relations they could express. Hence an important desideratum would be left, even if the theory were satisfactory in all other respects.

The equation for determining v2 virtually contains the theoretical laws of double refraction, which are embodied in the form of the wave-surface. The wave-surface of Cauchy and Neumann. does not agree with that of Fresnel, except as to the sections of two of its sheets by the principal planes, the third sheet being that which relates to nearly normal vibrations. Nevertheless the first two sheets, being forced to agree in their principal sections with Fresnel's surface, differ from it elsewhere extremely little. In Arragonite, for instance, in a direction equally inclined to the principal axes, assuming Rudberg's indices for the line D, I find that the velocities of propagation of the two polarized waves, according to the theory of Cauchy and Neumann, differ from those resulting from the theory of Fresnel only in the tenth place of decimals, the velocity in air being taken as unity. Such a difference as this would of course be utterly insensible in experiment. In like manner the directions of the planes of polarization according to the two theories, though not rigorously, are extremely nearly the same, the plane of polarization of a wave in which the * Annales de Chimie, tom. XLVIII, p. 254 (1831).

S. IV.

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vibrations are nearly transversal being defined as that containing the direction of propagation and the direction of vibration, in harmony with the previously established definition for the case of strictly transversal vibrations.

Hence as far as regards the laws of double refraction of the two waves which alone are supposed to relate to the visible phenomenon, and of the accompanying polarization, this theory, by the aid of the forced relations (2), is very successful. I am not now discussing the generality, or, on the contrary, the artificially restricted nature, of the fundamental suppositions as to the state of things, but only the degree to which the results are in accordance with observed facts. But as regards the third wave the case is very different. That theory should point to the necessary existence of such a wave consisting of strictly normal vibrations, and yet to which no known phenomenon can be referred, is bad enough; but in the present theory the vibrations are not even strictly normal, except for waves in a direction perpendicular to any one of the principal axes. In Iceland spar, for instance, for waves propagated in a direction inclined 45° to the axis, it follows from the numerical values of the refractive indices for the fixed line D given by Rudberg that the two vibrations in the principal plane which can be propagated independently of each other are inclined at angles of 9° 50′ and 80° 10′, or say 10° and 80°, to the wave-normal. We can hardly suppose that a mere change of inclination in the direction of vibration of from 10° to 80° with the wave-front makes all the difference whether the wave belongs to a long-known and evident phenomenon, no other than the ordinary refraction in Iceland spar, or not to any visible phenomenon at all.

It is true that before there can be any question of the third wave's being perceived it must be supposed excited, and the means of exciting it consist in the incident vibrations in air, which by hypothesis are strictly transversal. Hence we have to inquire whether the intensity of the third wave is such as to lead us to expect a sensible phenomenon answering to it. This leads us to the still more uncertain subject of the intensity of light reflected or refracted at the surface of a crystal-more uncertain because it not only depends on the laws of internal propagation, and involves all the hypotheses on which these laws are theoreti

cally deduced, but requires fresh hypotheses as to the state of things at the confines of two media, introducing thereby fresh elements of uncertainty. But for our present purpose no exact calculation of intensities is required; a rough estimate of the intensity of the nearly normal vibrations is quite sufficient.

1 5'9'

In order to introduce as little as possible relating to the theory of the intensity of reflected and refracted light, suppose the incident light to fall perpendicularly on the surface of a crystal, and let this be a surface of Iceland spar cut at an inclination of 45° to the axis. For a cleavage plane the result would be nearly the same. Let the incident light be polarized, and the vibrations be in the principal plane, which therefore, according to the theory now under consideration, must be the plane of polarization. The incident vibrations are parallel to the surface, and accordingly inclined at angles of 9° 50′ and 80° 10' to the directions of the nearly transversal and nearly normal vibrations, respectively, within the crystal. Hence it seems evident that the amplitude of the latter must be of the order of magnitude of sin 9° 50', or about the amplitude of vibration in the incident light being taken as unity. The velocity of propagation of the nearly normal vibrations being to that of the nearly transversal roughly as √3 to 1, as will immediately be shown, it follows that the vis viva of the nearly normal would be to that‍ of the nearly transversal vibrations in a ratio comparable with that of √/3 × sin2 9° 50' to 1, or about to 1. Hence the intensity of the nearly normal vibrations is by no means insignificant, and therefore it is a very serious objection to the theory that no corresponding phenomenon should have been discovered. It has been suggested by some of the advocates of this theory that the normal vibrations may correspond to heat. But the fact of the polarization of heat at once negatives such a supposition, even without insisting on the accumulation of evidence in favour of the identity of radiant heat and light of the same refrangibility.

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But the objections to the theory on the ground of the absence of some unknown phenomenon corresponding with the third ray, to which the theory necessarily conducts, are not the only ones which may be urged against it in connexion with that ray. The existence of normal or nearly normal vibrations entails consequences respecting the transversal which could hardly fail to have been detected by