152 Mr Glazebrook, On the general equations, <&c. [April 28, 



Hence, from (45) we obtain 



p s sin? = 1^ sinj; 



pr sin e r 2 sin e cos (e — £") ' " v '' 



We have also a third expression for a, viz. 



a = - 4tt F. 5. JOT (b 2 - c 2 ) r 2 sin e cot (e - £) 

 4<TrBMN(b 2 -c 2 ) sin £ 



Fsin(e-£) 



(49). 



This is as far as we can carry the theory with the equations 

 at present considered. But a, /3, 7 have to satisfy equations (31). 

 We must therefore investigate the conditions which this gives rise 

 to. 



Now it is clear that the value of a already found, will satisfy 

 (31), for the value in (42) is obtained from 



^-And 2 --? 6 ^) etc 



while the values in (44) come from 



_ cfo dy d 2 <& 

 dz doc dtdy ' '* 



and these are the equations used in forming (31). 



Thus the values found satisfy (31) without any fresh conditions. 

 So that without some other condition the problem of the propagation 

 of an electromagnetic disturbance in a crystal is indeterminate. 



For a given direction of displacement we have a definite velocity 

 and a definite magnetic force, but an infinite number of plane 

 wave fronts, all of which pass through the direction of the magnetic 

 force; for a given direction of wave propagation we have an in- 

 finite number of directions of electric displacement lying on a 

 certain cone, and to each of these there corresponds a definite 

 velocity and a definite direction for the magnetic force. 



Now experiments shew that the velocity of a plane wave of 

 light in a crystal has one of two definite values, and that these 

 values agree very closely with those given by Fresnel's theory. 



Let us assume that these conditions hold for the electro- 

 magnetic displacement ; then since we are to have two values each 

 for L, M, N, when I, m, n are given L, M, N lies on a cone of the 

 second degree with its vertex at the origin the additional relation 

 between L, M and N must be of the form 



PL + QM + RN=Q, 



