With the substitution y = cos 6, integration with respect to y yields 



B 



a 



CO 



r ^ [(x^-R^)^ (-1 ^U2(xW)(|r-x|-R-x) 



J r2 L ^|r-x| \r-^\) 



- J (|R-x|)-^-(R-hc)^ I dR 



This is readily evaluated by considering separately the integrations from 

 a to X, and from x to «>, The former yields 



/-^ = -I? 



4 B [ ^.^ Ua , 2 2. 

 1^ = y -^ I RdR = 5- (x -a ) 



and the latter gives 





dR ^ _ Ua 

 2 X 



Hence 



R 



X 



3 

 , Ua 3Ua ,,„. 



" = "i + "2 = 7T - 2^ ^""^^ 



2x 



in agreement with the distrubance velocity in (1) when 6=0, given in (44). 



This result indicates that the doublet at the center of the sphere is 

 an irrotational equivalent of the negative of the internal vorticity, since 

 the sum of their fields is zero exterior to the sphere. 



Some Applications of the Vorticity Theorem 



We have previously investigated irrotational equivalents of the bound- 

 ary layer and wake such that the outer irrotational flow was preserved. It 



18 



