of Luminous Waves in Transparent Bodies. 209 



Now there is a very simple relation between P and N ; for, 



putting 5 a,' = ic cos 6, ^1/ ^ n sin 9, and therefore 8 ^'* = ~ 



o 



(3 + 4 COS 2 a + cos 4 6) and 8 x^ 8 ^ _ ^ _ cos 4 9), and 



8 



observing that S wi — y^'('") ""^cos 2 6 and 2 m ~f{r) ?<'*cos4 9 



are each zero, in consequence of the symmetry of the arrange- 

 ment of the particles with respect to the coordinate axes, it is 



evident that Sm — /' (r) 8.r't= 3Sm — /' (r) 8*- 8/, that is, 



P= 3N. 



•rr , • M + 3 N ^ M + N „ 



Hence, it for brevity we put — ~ = A, — = r>, 



and therefore N = A — B, our equation becomes 



dt- dx' \tiy «- / ' \dxdy \ 



and we may obtain, in the same manner exactly, similar 



d^B (^2 

 equations for -~ and -jii > ^Iso in the same way we find the 



following, viz. — 



^' - A — "' 4- R {'^1^ M ^-^\ 

 dt' ~ ' dx^ "^ ' \dy;' "^ dz'^ ) 



1 "\dx,dy, dx,dz,) ' 'J I ^2^^ 



+ (* - •*•-) (i/ - .Vi) /3 + (* - */) (~- -,) 7) J- 



1 • -I • c (^'^ iPyt 



and similar equations ror —j—r -r!, 



' dt'' d t' 



where A, B, C^ are quantities analogous to A B C, 



We shall now proceed to apply these equations to deter- 

 mine the circumstances of propagation plane waves in differ- 

 ent cases. 



The first case will be that in which the plane waves are 

 propagated with a uniform velocity, the vilirations being sup- 

 J'/al. Mail. S. '.',. V()i.20. No. 130. March 18 12. P '' 



