255] ELECTROMAGNETIC WAVES. 543 



and 



* 



1 3 rir r2 

 _ __ I I tF(x + at cos a) cos (ifa sin a cos %) sin adadx 



ZTT ot J Q J ^L 

 2 



7T 



i r* r * 



~ I tg(x + at cos a) cos (16^ sin a cos v) sin adadx- 



?r J o J -^ 



e 



2?r 



2 



Let us put 



t'& sin a = f . 

 The definite integral 



IT 

 1 



(16) /o(f) = - 



is one of the set 



(17) /,() = n^X cos(? cos 



which may be evaluated by developing the cosine into a series 

 and integrating each term. We thus obtain 



(i8) /(?) = - 2 



w g . ./-| 



an infinite power-series in f, the coefficient of f 2 ? being 



(2?)! 

 where 



IT 



(19) / M -i 



2 



an integral which we can evaluate. Integrating by parts, writing 

 the integrand (sin 2 ^ ^ cos %) cos 2 ^" 1 %, 



