is small, the exponential factor shows that the motion quickly diminishes with the distance from the refracting surface. The quantities p, q, r, s may now be obtained by substituting ,, 1, 1 from equations (42) and (43). Green*, to whom the above investigation is due, assumes n = n1, and Rayleigh† has put the solution for r in that case into the form If two media do not differ much in optical properties, so that the refractive index is nearly equal to one, we obtain for the ratio of amplitudes the expression tan (0-0') as required by experiment when the vibration takes place at right angles to the plane of incidence. As has been pointed out above, the tangent formula is only approximately correct, but the deviations are not so great as those which Green's formula would lead us to expect, and are sometimes in the other direction. The alternative according to which differences in optical properties are due to differences in elasticity, leads to results which can in no way be reconciled with observed facts. If we place ourselves on the standpoint of the elastic solid theory, we are therefore compelled to conclude *Collected Works, p. 245. + Collected Works, vol. 1. p. 129. that the rigidity of the æther is the same in all media. Even then we arrive at an unsatisfactory result so far as light polarized at right angles to the plane of incidence is concerned. 142. Lord Kelvin's theory of contractile æther. According to the most general equations of the motion of an elastic substance (Art. 132), a disturbance spreads in the form of two waves, the condensational longitudinal wave propagated with a velocity √(k+n)/p and the transverse distortional wave propagated with a velocity √np. The phenomena of light leave no room for a longitudinal wave propagated with finite velocity. It has been got rid of in the theory so far considered by taking the elastic body as incompressible. The coefficient k then is infinitely large, and the longitudinal disturbance is propagated with infinite velocity. This elastic solid theory of the æther, as discussed in the preceding investigations, does not, however, consistently lead to facts which are in agreement with observation. It fails to account for the laws of double refraction and for the observed amplitude of light reflected from transparent bodies. That theory was therefore considered dead, until Lord Kelvin* resuscitated it in a different form by showing how, dropping the hypothesis of "solidity," an elastic theory of the æther may still be a possible one. The characteristic distinction of the new theory lies in the bold assumption that the velocity of the longitudinal wave, instead of being infinitely large, is infinitely small. This requires that kn shall be zero, so that is negative. A medium in which there is a negative resistance to compression would at first sight appear to be essentially unstable, but Lord Kelvin shows that the instability cannot come into play, if the æther is rigidly attached to a bounding surface. So long as there is a finite propagational velocity for each of the two kinds of wave motion, any disturbance set up in the medium cannot lead to instability. Putting therefore the constant A of Article 131 equal to zero, and taking the rigidity to be equal in all media, Lord Kelvin has shown that the theory leads to Fresnel's tangent formula for the amplitude of light polarized in a plane perpendicular to the plane of incidence. Glazebrook† then showed that the consideration of double refraction leads to Fresnel's wave surface, while J. Willard Gibbs pointed out that the new form of elastic æther theory must always lead to the same equation as the electromagnetic theory, provided we replace the symbol which denotes 'displacement' in one theory by that which denotes 'force' in the other and vice versa. *Phil. Mag. xxvi. p. 414. 1888. If in equations (9) we write kn, and allow different values of p according as the displacements are in the direction x, y, or z, the equations become with our present notation These equations are identical with (24) provided that we replace P, Q, R in the latter by έ, n, , and μ, K1, K2, K, by 1/n, P1, P2, P3, respectively. As regards surface conditions, we must now remember that the resistance to compression being negative, there may be infinite compression or dilatation at any point or surface at which a condensational wave tends to start. The surface at which reflexion takes place gives rise according to the preceding article to condensational waves, hence disregarding this wave which can only be propagated with zero velocity, the conditions which hold in the general elastic theory, in which the condensational wave is considered, are not necessarily satisfied. They must be replaced by others, which, as J. W. Gibbs has shown, may be obtained directly from the equations of motion. Introduce new quantities defined by Performing the differentiation so as to obtain έ", dy ", " in terms of §, 7, 5, we arrive at the expressions which stand on the right-hand side of (47) with the sign reversed. We take the boundary to be normal to the axis of x. If ' and 'were discontinuous at such a boundary, dn' dy and would be infinitely large, and this would make ŋ" and ¿′′, dr dx and consequently also d2n d2 infinitely large, which is obviously not admissible. Hence we conclude that n' and ' are continuous. similar reasoning shews that as ' and 'must remain finite, ŋ and must be continuous. The conditions of continuity are therefore A But these are exactly the conditions which we have seen must hold in the electromagnetic theory if έ, n, are replaced by P, Q, R. The analogy is made complete if it is noticed that the continuity of έ' and " follows from the above equations, and that according to the first of equations (47) if v1 = √n/p, The continuity of έ" is seen to involve the continuity of the normal force and this corresponds to the continuity of electric displacement in the electromagnetic theory. The analogy has therefore been proved both for the equations regulating the motion of the medium and for the surface conditions. If this analogy is kept in view, all the results which have been found to hold in the electromagnetic theory may be translated at once into consequences of the contractile æther theory. Thus in the theory of double refraction, the displacements of the latter theory are not in the wave surface, but are normal to the ray as has been shown for the electric forces in Art. 133. Thus adopting the contractile æther theory we may conclude at once that when plane waves are propagated through a doubly refracting medium the elastic force and not the displacement is in the plane of the wave. I have given a statement of this theory on account of its mathematical interest, but it has now been abandoned by its author*. 143. Historical. AUGUSTIN LOUIS CAUCHY, born August 21st, 1789, in Paris, died May 23rd, 1857, at Sceaux, near Paris, was one of the large number of celebrated French mathematicians who, during the end of the 18th and the beginning of the 19th century, made the first serious advance in Mathematical Physics since Newton's time. Cauchy's contribution to the theory of light consisted in initiating the endeavour to deduce the differential equations for the motion of light from a theory of elasticity. This theory was based on definite assumptions of the actions between the ultimate particles of matter. The luminiferous æther like other matter was supposed to be made up of distinct centres of force acting upon each other according to some law depending on the distance. Cauchy explained the phenomena of dispersion by supposing that in the media in which dispersion takes place, the distance between the ultimate particles is no longer small compared with the wave-length. He thus arrived at a formula which for a long time was considered to represent satisfactorily the connexion between wave-length and refractive index (Art. 150). Cauchy also showed that metallic reflexion may be accounted for by a high *Baltimore Lectures, p. 214. coefficient of absorption: by interpreting Fresnel's sine and tangent formula, in the case where the index of refraction is imaginary, he obtained the equations for the elliptic polarization of light reflected from metallic surfaces, which are still adopted as correctly representing the facts. GEORGE GREEN was born at Sneinton, near Nottingham, in 1793, and only entered the University of Cambridge at the age of 40. Having graduated in 1837 as fourth wrangler, he was elected to a fellowship in Gonville and Caius College in 1839, and died in 1841. The following paragraph which stands at the head of his celebrated Memoir on the Reflexion and Refraction of Light will show the ideas which guided him in his work. "M. Cauchy seems to have been the first who saw fully the utility of applying to the Theory of Light those formula which represent the motions of a system of molecules acting on each other by mutually attractive and repulsive forces; supposing always that in the mutual action of any two particles, the particles may be regarded as points animated by forces directed along the right line which joins them. This last supposition, if applied to those compound particles, at least, which are separable by mechanical division, seems rather restrictive; as many phenomena, those of crystallisation for instance, seem to indicate certain polarities in these particles. If, however, this were not the case, we are so perfectly ignorant of the mode of action of the elements of the luminiferous ether on each other, that it would seem a safer method to take some general physical principle as the basis of our reasoning, rather than assume certain modes of action, which, after all, may be widely different from the mechanism employed by nature; more especially if this principle include in itself, as a particular case, those before used by M. Cauchy and others, and also lead to a much more simple process of calculation. The principle selected as the basis of the reasoning contained in the following paper is this: In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But, this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other. One of the advantages of this method, of great importance, is, that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are |