1884.] of the electromagnetic field. 149 



We have seen that in an isotropic medium a wave of electric 

 displacement is propagated with a velocity 1/y '/xK. Let us suppose 

 that in the crystal there can also be a wave of electric displace- 

 ment given by 



f=BL sin 8 (32), 



etc. 



4) 



where 8 = — (In + my + nz — Vt). 



Then 3> = - 2\B cos e cos 8 (33), 



and we require to find the relations between the quantities 

 V, L, M, N, I, on, n. 



Substituting in the equations for /, g, h we have 



L (m 2 + n 2 ) (V 2 -a 2 )-Mlm (V 2 -b 2 ) 



-Nln(V 2 -c 2 ) = (34), 



and two similar equations. On multiplying by I, m, n and adding 

 the sum is zero, there are therefore only two independent equations 

 and these are insufficient to determine V, L, M and N. Thus the 

 problem as it stands is indeterminate, another equation is required 

 for its solution. 



Let a 2 Ll + b 2 Mm + c 2 Nn = q 2 cos £ 



q 2 being a quantity to be further denned shortly. Then we find 



_ La 2 (m 2 + n*) - MbHm - Nc\l _ Lb? - q 2 cos £ 



L(m 2 + n 2 )-Mlm-Nnl ~ L-lcose ( '" 



The other two equations can be treated similarly ; and we find 

 finally 



La 2 — lq 2 cos f _ Mb 2 — mq 2 cos £_ Nc 2 — nq* cos £ 

 L — I cos 8 M— m cos 8 N — ncos8 ^ 



On eliminating q 2 cos £ from the first and second and first and 

 third respectively, we get 



(L - I cos 8) [IMN (b 2 - c 2 ) + mNL (c 2 - a 2 ) 



+ nLM(a 2 -b 2 )} = (37). 



Hence we find 



,. I m n _ 



C08S = I = 3? = J =1 ' 

 or !(6*_e*)+™(c'-a*) + |(««-S») = (38). 



