30 



i- (n • Vfcl) (r .) = y^P^(cos 6')(a^ cos s A + b^ sin s X) 



[75] 



which permit the determination of a^ , b^. 



The solutions are most conveniently found in the form of recurrence formulas. Since 

 [75] are identities, and the a^, b^ have explicit values in terms of the derivatives of , we 

 can write 



n+l 



-,_A_ (n . V )"+! =^^„^i(cos e)K^^ COS sX + 6^^^ sinsA) 



s=o 



[76] 



= — i— yP„^(cos 61) [ COS sA(n • t^ )a^ + sin s \ (n • t7 ) 6^ ] 



When the operations are carried out in tljs second form, it can be reduced to a sum of terms of 

 the type A cos sk, B sin sX, which are linearly independent, so we may equate coefficients 

 of the two forms. We have then the further system of identities. 



"+l "+i 2{n+l) 



2P^cos0 -+r,{s - l)P^~^sin0f _IL_ - " I 



dx ^ d y dz ' 



177a] 



ps ^>s = ± 



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



(9 6^ /aa^~^ db^~'^\ 

 2P^cos0— ^+7,(5-1)/^-! sine :L + "—\ 



d 2 By 



P^+^ sit\d\- -^^ + " I 

 \ ;3c ;i ,, '-1 



(93 5 y 



The special case in which s = n + 1 is easily solved, 



p;; + l a-l\ = _J_ rjin) P" sin e( ^ - ^ ) 

 " + 1 " + i 2(n + 1) ' " \ dz dz I 



[77b] 



