232 PIERCE. 



ddsm'^dsmiAcosd), (100) 



r 



where 7i = 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 



h=l de,.n-e^Acose-—^^ + -^^ | 



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

 Gamma Functions as follows: 



r (2) r ^'' + -^ 





n+l .3!.2j,^»+_3^j 



A' ' ' \ 2 



5!2r(''l±i_^, 



+ (101) 



If we note that 



/ n+5 , , \ n+5 71+3 n+l /tH-J. 

 V2"^V 2 2 2V2 



r (2) = 1 



r (3) = 2! 



r (4) = 3! 

 we obtain 



J ^ A _ 43 2 A^ 2^2! 



' n+l 31 (n+ 1) (n + 3) "*" .5!" {n + 1) (ti + 3) (71 + 5) 



