of a Stream of Light passing through a Crystalline Plate. 81 

 The formulas for these transformations are 



x = Xi cos Y sin 0, ?/ = # t sin 0-f- Y cos 0, 



Xi = X cos x + Z sin x, z = X sin x + Z cos x 



from which we obtain 



x = X cos cos x~~Y sin 0+Z cos sin x> 

 ?/ = X sin cos x + Y cos + Z sin sin x> 

 2 = X sin + Z cos - 



Now the equation to the wave surface referred to the axes of 

 elasticity is 



, &, c being the principal wave velocities, and the condition that the 

 plane 



nz = 



should be a tangent plane to it, is obtained by eliminating p between 

 the equations 



Hence the condition that in the new system of co-ordinates the 

 plane 



mY+rcZ = 1 



should touch the wave surface is found by eliminating^ between the 

 equations 



(I cos 0cos x~ m sin 0-j-w cos sin 



(Z sin cos x + m cos + TO sin sin x) 2 (Zsin x~ _ 



*- 2 - 2 



The result of this elimination is a biquadratic in w, which, from 

 the nature of the problem, has two real positive and two real nega- 

 tive roots, and if n\, n 2 are the positive roots of the biquadratic, the 

 relative retardation required is 



3. Before proceeding with the general case, these results may be 

 applied to certain simple cases : 



u 2 



