140, 141.] CIRCULAR VORTICES. 160 



where 



r ={(x- as )* + ^ + ^ - 2* cos (S- - * )} 

 Now if 



r 



we have - M sin S- + .A r cosS- ..... ........... -(45); 



and, since the integral 



/Y/Vsm^-^ ) , 7C v/ 7 7 

 I I - - - OT a^- dx d-& 



is identically zero, 



.(46). 



Combining (45) and (46), we find 



(47). 



If the variable of integration in (44) be changed from $ to e, 

 where e = ^- / -^ the limits of integration- for e are and 2?r; 

 and since 



cose 



{(a? - xj + OT 2 + -c/ 2 - 2W cos e}^ 



-x )* + (*r + &amp;lt;*y -,[1 __ r__ 

 2-BTw Jr 2 OTOT &quot; 



and 



we may write (44) in the form 



-Ji-*W ...... (48) 



Here J\, J^ denote the complete elliptic integrals of the first 

 and second kinds with respect to the modulus 



VIZ., 



