230 



PIERCE. 



which by dropping the primes and substituting in (92) and (91) gives 



_ r^ cos 4> sin {B si 



Jo 1 — sin- 



si n d cos (f>) d4) 

 d cos2 4> ■ 



(93) 



Now expanding in series as follows : 



• / r. • « X T. • ^ B^ sin^ 6 cos^ (j> , 

 sin {B sin 6 cos cf)) = B sm 6 cos — r 



3! 



B^ sin^ 6 cos* 



5! 



and 



= 1 + sin2 e cos2 (j) + sin* 6 cos* c/> + 



1 — sin^ 6 cos^ (^ 



and by taking the product of these two series we obtain 



r/2 



(93a) 



'='£ 



d(t> 



B sin 6 cos^ </> 



+ \ B-^ [sin^^cos^c^ 



( B^ B' I 



+ j 5-.^+?7 ^sin50cos«<^ 



+ 



} 



(94) 



Integrating (94) by formula 483 of B. O. Peirce's Tables, we obtain 

 "1 



F=27r 



5sin^ 



1-3 \ 



, 1-3-5 \' B\ B'l . ,^ 



, 1-3-5W i p B'.B-' Sn . , 



(95) 



We shall next proceed to perform the second integration with 

 respect to cf) indicated in (89). For abbreviation let us wTite 



