230 



PIERCE. 



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



, C' COS (b sin (B sin 6 cos <^) dcf) 



'="1 — r^ 



sin^ 9 cos^ (f) 



Now expanding in series as follows: 



„ . i« B^ sIto? 6 cos^ <f> . 

 sin (B sin d cos ^) = B s,md cos ^ ?r; h 



(93) 



3! 



B^ sin^ d cos* (A 



and 



1 



= 1 + sin2 Q ^jos^ + sin* cos* 4> + 



1 — sin^ d cos^ <^ 



and by taking the product of these two series we obtain 



(93a) 



V 



d4> 



B sin 6 cos^ 4> 



+ \B 



B^ ] 

 — r-r |- sin^ 6 cos* <t> 



( B^ B^ ) 



+ \B-~ + ^,\ sin^ cos« <^ 



+ 



(94) 



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

 "1 



V=2t 



+ 



B sin e 



'■'\B-^'-.r^e 



2-4 ) 



3! \ 



+ 2:4:61^-37 + 5! .1""^ 



+ 2:4:6:81 ^-3! +5! -7! (^^^^^ 



(95) 



We shall next proceed to perform the second integration with 

 respect to (^ indicated in (89). For abbreviation let us write 



