98 



and suppose the axes of coordinates to be the principal axes 

 of the crystal, the equations, in question may be thus written : 



le -"" dy dz' 



"», , c/x „ rfz 



cW ~ " dz 



K2) 



d'Z__^,d_Y_^,.., 



dt^ ~ ' dx 

 and if we further put 



~ dz dy ^ dx 



dy' 



'Ik 



dz 



^-dy 



dm 

 dx 



they will take the following simple form : 



d% 



de 



— — a^x. 



dt- 



d% 

 dt' 



= — c'z, 



(3) 



(4) 



in which it is remarkable that the auxiliary quantities 

 ?b *Ji> ^i> ^^^ exactly, for an ordinary medium, the compo- 

 nents of the displacement in the theory of Fresnel. In a 

 doubly defracting crystal, the resultant of ^i, »ji, ^i is per- 

 pendicular to the ray, and comprised in a plane passing 

 through the ray and the wave normal. Its amplitude, or 

 greatest magnitude, is proportional to the amplitude of the 

 vibration itself, multiplied by the velocity of the ray. 



The conditions to be fulfilled at the separating surface of 

 two media were given in the abstract already referred to. 

 From these it follows, that the resultant of the quantities 

 sii >?ij X>\, projected on that surface, is the same in both 

 media ; but the part perpendicular to tlie surface is not the 

 same ; whereas the quantities ^, »), Z,, are identical in both. 

 These assertions, analytically expressed, would give five 

 equations, though four are sufficient ; but it can be shown 

 that any one of the equations is implied in the other four, 

 not only in the case of common, but of total reflexion ; which 



