﻿The Hon. J. W. Strutt on Double Refraction. 525 



Mo(y 2 p-b*) = -ip l,-) 



M0o<Tp,-n- W*,\ .... (4) 



and since /X + wz/-6 + wv=0, 

 I 2 rrv 



+ ™ To =0 



or if, as in the ordinary notation, a, 6, c are the principal velo- 

 cities of propagation *, 



The equations determining the directions of vibration are 



\\j*~7y^w~^) + v\tf~tf; 0i I . (6) 



/X + m^ + ?iv=0. J 



Equations (5) and (6) constitute the analytical solution of the 

 problem. I had originally expected to reproduce in their inte- 

 grity the beautiful laws of Fresnel ; but a slight examination 

 will show that, in order to reconcile (5) and (6) with Fresnel's 

 equations, w r e must write, for V 2 , « 2 , Z> 2 , c, 



Jl L Jl jl 



V 2 ' « 2 ' b*' c 2 



respectively. The directions of vibration are parallel to the axes 

 of the section of the ellipsoid 



by the plane of the wave ; and the velocities of propagation are 

 directly proportional to the lengths of the axes. Accordingly 

 the w r ave-suface is the envelope of planes drawn parallel to the 



CO XI 2i 



central sections of the ellipsoid -3 + t^ + -g = 1 at distances di- 

 rectly proportional to .the lengths of the axes. Fresnel's surface 

 is the locus of points situated on the normals to the sections of 

 the same ellipsoid and at the same distances. We see, therefore, 

 that the new surface is related to Fresnel's in the following 

 way : — Through any point of the latter draw a plane perpendi- 

 cular to the line joining it to the centre; the envelope of these 

 planes is the former surface. In the principal planes of a biaxal 

 crystal the new surface agrees with Fresnel's as regards the sec- 

 * The meaning of b is here changed. 



