204 PIERCE. 



The third and fourth terms may be integrated directly. In the 

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

 Let 



u = cos 9 



,„ — du 

 dd = -^—. 

 smp 



then 



r-/2 _dB ^ p - du ^ \ p f 1 1__V^ 



Jo sin 6 Ji 1 — u^ 2 Jo \1 -{- u I — u) 



_i n du 1 r° du ^ 1 r+i _ 



~ 2 Jo 1 + i< 2 J_i 1 + w ~ 2 J_i 1 



0?li 



+ ^/ 



(26) 



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

 may be WTitten, respectively 



r-/2 cos (2 A cos e) dd ^ 1 r+i cos (2 Au) du , 



Jo slnd "2J-1 1 + w ' ^ ^ 



c/w 



r^/-^ cos (A cos g) f/e __ 1 r+^ COS (yli/.) 

 Jo sin^ 2 J_i 1 + 2/ 



Another of the integrals, examined in more detail, gives 



(28) 





2 cos 6 sin (2 A cos 0) dd 



sin 



i 



*^ u sin (2 ylw) c?w 

 1 - w2 



_ _ 1 ri s in (2 ^m) rfi/ 1 r-i sin (2 ^^) f/t/ 



2 Jo 1 + w 2 J„ 1 + z/ 



1 r+i sin (2 Au) du , . 



~~2j-, \ + u ■ -^-^^ 



Similarly, the remaining integral becomes 



« 



r^/^ cos d sin (A cos 6) dd ^ _\_ T+i sin {Au) du 

 Jo sine ~ 2./_i 1 + 7^ ■ 



