524 Mr. R. T. Grlazebrook on the Application of the 



contributes something to the energy. It follows, hence, that 

 we must have the condition B = B' satisfied, and optical 

 differences must arise from differences in the optical density 

 of the aether on either side of the surface. In the paper this 

 relation is assumed to simplify the formulae, the foregoing 

 considerations show that it is necessary. This point, however, 

 is dealt with by Sir William Thomson himself in a note in 

 this number of the Philosophical Magazine. 



To turn now to the problem of Double Refraction. Since, 

 according to our theory, the rigidity of the aether is the same 

 in all media, it is clear that we cannot explain double refrac- 

 tion by variation of rigidity in different directions in a crystal, 

 and we are driven to consider the hypothesis advanced by 

 Rankine, Stokes, and Rayleigh, and which has been shown 

 by the latter two to lead, if the aether be incompressible, to a 

 wave-surface other than that of Fresnel. 



According to this hypothesis the density of the aether is to 

 be treated as a function of the direction of displacement. 

 The kinetic energy will be a quadratic function of the dis- 

 placements, and for one set of axes may be written 



i 1ST (P-x ' U * + Py i 2 + Pz ™ 2 ) dx ^ dz ' 



We suppose that these axes coincide with the axes of the 

 crystal. The potential energy has, according to our suppo- 

 sition, its usual form, and the constants A, B are the same 

 as those for an isotropic medium. Thus, following Lord 

 Rayleigh's paper (Phil. Mag. June 1871), we have as the 

 equations of motion * : — 



^ = (A-B)^+B V 2 « 



d 2 v _ dd , -,3,0 



P ^ = (A-B)^+B V 2 , 



(3) 



, ^ du dv dw , A x 



where g= —+■-=-+ -=-• {±) 



dx ay dz 



Hence, differentiating with regard to x, y, z, and adding, 



d 2 / du dv , dw\ . «s, , K s 



M'-s+^+'-E-r^* (5) 



* A theory leading to equations practically the same as these has been 

 given by Sarrau, Liouville Journal, ser. ii. tomes xii. and xiii. (see 

 Glazebrook, " Report on Optical Theories," B. A. Report, 1885, p. 174), 

 and by Boussinesq, Liouville Journal, ser. ii. tome xiii. (Glazebrook, 

 u Report on Optical Theories," p. 213), and in these the same conclusions 

 as to the direction of vibration are arrived at. 



