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Report on Double Refraction. By G. G. STOKES, M.A., D.C.L., Sec.R.S., Lucasian Professor of Mathematics in the University of Cambridge.

I REGRET to say that in consequence of other occupations the materials for a complete report on Physical Optics, which the British Association have requested me to prepare, are not yet collected and digested. Meanwhile, instead of requesting longer time for preparation, I have thought it would be well to take up a single branch of the subject, and offer a report on that alone. I have accordingly taken the subject of double refraction, having mainly in view a consideration of the various dynamical theories which have been advanced to account for the phenomenon on the principle of transversal vibrations, and an indication of the experimental measurements which seem to me most needed to advance this branch of optical science. As the greater part of what has been done towards placing the theory of double refraction on a rigorous dynamical basis is subsequent to the date of Dr. Lloyd's admirable report on "Physical Optics," I have thought it best to take a review of the whole subject, though at the risk of repeating a little of what is already contained in that report.

The celebrated theory of Fresnel was defective in rigour in two respects, as Fresnel himself clearly perceived. The first is that the expression for the force of restitution is obtained on the supposition of the absolute displacement of a molecule, whereas in undulations of all kinds the forces of restitution with which we are concerned are those due to relative displacements. Fresnel endeavoured to show, by reasoning professedly only probable, that while the magnitude of the force of restitution is altered in passing from absolute to relative displacements, the law of the force as to its dependence on the direction of vibration remains the same. The other point relates to the neglect of the component of the force in a direction perpendicular to the front of a wave. In the state of things supposed in the calculation of the forces of restitution called into play by absolute displacements, there is no immediate recognition of a wave at all, and a molecule is supposed to be as free to move in one direction as in another. But a displacement in a direction perpendicular to the front of a wave would call into play new forces of restitution having a resultant not in general in the direction of displacement; so that even the component of the force of restitution in a direction parallel to the front of a wave would have an expression altogether different from that determined by the theory of Fresnel. But the absolute displacements are only considered for the sake of obtaining results to be afterwards applied to relative displacements; and Fresnel distinctly makes the supposition that the ether is incompressible, or at least is sensibly so under the action of forces comparable with those with which we are concerned in the propagation of light. This supposition removes the difficulty; and though it increases the number of hypotheses as to the existing state of things, it cannot be objected to in point of rigour, unless it be that a demonstration might be required that incompressibility is not inconsistent with the assumed constitution of the ether, according to which it is regarded as consisting of distinct material points, symmetrically arranged, and acting on one another with forces depending, for a given pair, only on the distance. Hence the neglect of the force perpendicular to the fronts of the waves is not so much a new defect of rigour, as the former defect appearing under a new aspect.

I have mentioned these points because sometimes they are slurred over, and Fresnel's theory spoken of as if it had been rigorous throughout, to the injury of students and the retardation of the real progress of science; and

sometimes, on the other hand, the grand advance made by Fresnel is depreciated on account of his theory not being everywhere perfectly rigorous. 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.

The first deduction of the laws of double refraction, or at least of an approximation to the true laws, from a rigorous theory is due to Cauchy though Neumann† independently, and almost simultaneously, arrived at the same results. In the theory of Cauchy 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

*Mémoires de l'Académie, tom. x. p. 293

+ Poggendorff's Annalen, vol. xxv. p. 418 (1832).