﻿524 The Hon. J. W. Strutt on Double Refraction. 



of the velocities may be got rid of, so that 



2T=JM(p,P+^+M' 2 )*> 

 where p x) p yi p g are positive quantities representing the densities 



corresponding to the three coordinate axes. The expression of 

 the potential energy I suppose to be exactly the same as in va- 

 cuum ; and thus by Lagrange's general method* we find for the 

 equations of motion, 



d*% n dS 





a) 



On account of the incompressibility of the aether, 8 is very 

 small; but it does not follow that the terms containing it are to 

 be omitted, for a 2 is correspondingly great. We may, however, 

 write p for # 2 S, and p may then be compared to a hydrostatic 

 pressure. The problem of double refraction is solved so soon as 

 the laws are known which regulate the possible directions of vi- 

 bration and corresponding velocities of propagation for every 

 position of the wave- front. 



Let us consider a plane wave whose front is at any time given 



he + my + nz — Nt t 



so that /, m, n are the direction-cosines of the wave-normal, and 

 V the velocity of propagation. Also let denote the actual dis- 

 placement in the plane of the wave, and \/jlv its direction. Thus 



f=\0, v =p0, £=vd; 

 and 



d *Q d P , 72 r-724 



«dt 2 dy 

 _ dp 



(2) 



V P*-M2 



df dz 



+#vv 2 0. 



Now let 



Q—Q e i{lx-tmy+nz- 





p = p € i(lx+my + nz-Vt) J " " ' * " V / 



where O and p are complex constants. On substitution in (2), 

 * Thomson and Tait, p. 710. Green, Camb. Trans. 1838. 



