204 



PIERCE. 



The third and fourth terms may be integrated directly, 

 other terms let us introduce a change of variable as follows: 

 Let 



u = cos 6 



— du 



In the 



dd 



sin 6 



then 





/2 dd 



sin d 



Ji 1 -u" 2 Jo [l -\- u'^ 1 - ur^' 



^1 n dv 1 p du ^ 1 r+^ du 



2 Jo l-\- u 2J-1 1 + u 2J-1 1 + ti 



(26) 



With this operation as a model, two of the other integrals of (25) 

 may be written, respectively 



r 



~ co s (2 A cos 6) dd 

 sin d 



- cos (A cos 6) dd 

 sin ^ 



1 r+i cos (2 ^w) C?M ,„„, 



= 2J_, l + « '■ (20 



_ 1 r+^ cos (^-lii) du , . 



~ 2 Xi l + « • ^^^ 



X 



Another of the integrals, examined in more detail, gives 

 '^/2 cos d sin (2 ^ cos 0) (/5 



'^ M sin (2 ^m) du 



sind 



I 



10 



= 9 r^' f r^— - Ar) s»^ (2 ^w) 



2 Jo \1 — U 1 + Uj 



^ _1 n sin {2 Au)du 1 T"^ sin (l 

 2 Jo 1 + M ^ 2 Jo "1 



^ _ 1 r+i sinO 

 2 J-1 1 



du 



(2 ylw) c?w 



2 ^w) rfw 



Similarly, the remaining integral becomes 



"'^/^ cos d sin (rl cos d) dd 1 /"+i sin (Au) du 



(29) 



f 



sin0 



2X1 1 



+ u 



(30) 



