232 



PIERCE. 



r 



de sin" d sin (A cos 6), 



(100) 



where ?i = 2, 4, 6, 8, 



I3 is the simplest of these integrals and will be considered first.^ By 

 expanding sin (A cos 6) in series we have 



-i: 



/2 ,„ . ^ ( ^ v43 cos3 , A^ cos^ 



rfe sin" d\ A cos 9 — 1 



( 3! 5! 



which by Byerly Int. Calc, Art. 99, Ex. 2, may be integrated in 

 Gamma Functions as follows: 



h = 



,3 r (2) r 



71+ 1 



.+ 1 3!2rfn + 3_^^ 



+ 



^.r(3)r(^ 

 ^'2rr^^ + i 



^'2rr^ + i 



+ 



(101) 



If we note that 



fn-}-_o I . A w+ 5 ?t+3 ??. + 1 /w + 1 



r (2) = 1 

 r (3) = 2! 

 r (4) = 3! 



we obtain 



, A .43 2 



is = 



ON + 



A^ 2^2! 



w+1 3! (n+ l)(w+3) ' 5! (w+ l)(n + 3)(n + 5) 



