14 



where fi = cos 6, and the P^iti) are the associated Legendre functions. In this expansion, the 

 first double summation, in which R appears to a positive power, represents the potential of the 

 undisturbed stream combined with all the other sineularities within S and outside S-, and the 

 second summation, involving R to negative powers, represents the potential of the singularity 

 at r.. The first summation is convergent for < ff < |r. - r . |, and the second is convergent for 

 all R >0. 



The functions of $ which appear in Equation [26] are: 



<x) n 



n = ^ ^'^^" Pn « COS s A + bl Sin s A) 

 n=o s=o 

 oo n 



- y^ y^^"^ + 1)/?-^"+! ) P„^(< cos sX + 7^ sin sA) 



[28a] 



n=o s=o 



n=o s=> o 



cos s A + 6^ sin s \) sin i 



cos sA + 6* sin s A) sin 6 



[28b] 



ra=o s=o 



s\ - 6^ cos S\) 



n=o 5=1 



~ ^ ^ sR'^""^^^ Pl(al^ sin sA - jT^cos sA) 



[28c] 



When these are substituted in Equation [26], the resulting expressions become quite cumber- 

 some. However, since F (i) is known to be independent of R^, it is evident that the net co- 

 efficient of R^. must vanish unless < = 0, so only the latter terms need be considered. A fur- 

 ther reduction can be made by taking account of the integrations with respect to \ since all 

 terms contain products of the type 





- 





- 



-. 



cos 







cos 





or 



sA 





or 



^A 



sin 







sin 





