31 



and since 



P"+l= (2n + 2)! sin"6> = (27i+ 1) P" sini 



" + l 2"+i(n + 1)! 



"+1 (2: 



1 ( Y^ _^\ 



71+ l)(2n + 2)"^ \ dy dz I 



Similarly 



"+1 (2n+ l)(2n+ 2) \ ^s 5 yV 



[78a] 



[78b] 



The special case, s = 0, is also easily solved by setting = 0. Since P°_^(cos 0) = 1, 



da° 



71+1 d X 



[79] 



The identities of [77a] and [77b] can be transformed by means of the recurrence formula (see 

 Reference 13, page 360) 



cos 6 P^ = P„^+i - (71 + s) sin Q P^' 



[80] 



We then have 



^.+l^%+l n,l dx I 



1 r r Ida'-'^ db'-'^\ da' 



= ^^ Jp^--isin0L(s-l) -i?- - -^ -2(n+s)— ^ 



2(71+ 1) \" L \ dy dz I Bx 



+ P'+^ sine -^^+ " > 

 \ dy dz '} 



[81a] 



"+1 \ "+!■ 71 + 1 da; / 2(71 + 1) \" L \ ^2 dy I dx \ 



[81b] 



,p-lsin.(-^+^n) 

 \ 52 dy 1} 



