Electromagnetic Waves. 



147 



of £ only ; and that B/0 is a function of 77, f only. These 

 restrictions are satisfied in all the applications which have 

 been made hitherto. It may be noticed further that when 

 A depends only on f, we may reduce our work by making a 

 change of variable to f where dg f = Ad%; and so we may 

 effectively put A = l, without real loss of generality. 



'lhen, on making use of the facts that A = 1 and that 

 JB/O is independent of £, the last pair of equations (1*2) 

 give 



and 



m dv\c*'dt) 



m^ = -£^ 



-} 



(1-21) 



= +^(C/3). 



o , K oU 



B-dr, 



~dt 



)■ 



(1-3) 



It is therefore possible to write 

 . 1 a /K3U\ 1 



1. p 9L 



where r — -^— , 



at J 



and U is still left arbitrary. 



If we substitute for Y, Z and /3, 7 from equations (l'l) 

 and (1*3), the second and third of equations (M) become 

 (making use of the fact that A = 1) , 



ittl^ a*v~a?\ as; - a?v A ??/'| 



The equations (1*31) lead to the relation 

 B 2 U y^KB 2 U 



X = 



(1-4) 



df c 2 -dt 2 



Substitute next for ft, 7 from (1*3) in the first of 

 equations (E), and we obtain the result: 





(1-41) 



If we substitute for X from (1*4) in (1*41) we obtain 

 a differential equation for U: 



" 3f 



_1_ f 3/C3U\ .3/BBUU 



■ ap^Bola^B aw + a?\c ac/i ' 



fl-5) 



L 2 



