The complex potential can be written, from Equations [35a] and [40a], 

 w = - A In (2-a) + Vze~''^ + w^ 



where w is the partial potential due to the presence of the cylinder. Then 



dw A _,^ '^^c '^"^c 



= - + Ue y + , - u ^ + iv ^ = 



ds s-a dz "" "" dz 



from Equation [25i]. 



On S, l/{z-a) can be expanded by the binomial theorem: 



1 a a' 



-A\ 

 z-a \ z ,2 ,3 



Furthermore, dw /dz will vanish at infinity, so that, on the contour <S, u, ^ can be expanded 

 as in Equation [72d]: 



'^ 277 2 ^2 ' dz 2m ,2 ^3 



All but two of the resulting terms in the integral around S then give zero. 



On CT, which is traversed in the negative direction, the residue of [-2A/(z-a)] 

 [dw^/dz] contributes to the integral iniA [dw^/dz] = iniA {-u^ + iv ^), since dio /dz 



z ~ a - 



is analytic at 2 = 0; see Section 30. The term in AU cancels one obtained from S. 

 Thus Equation [74g] becomes 



X -iY = - pTVe~'y + 2npA (w„^ - iv^^). [75c] 



As was seen in Section 40, A is real for a source; hence Equations [75a, b] follow. 

 If there is a line vortex instead of a source at 2 = a, with circulation V around it, by 

 [40m] A = -i Tq/Sw; hence 



X = - pVU sin y - pFg v^^, Y = pFU cos y + pT^ u^^. [75d, e] 



169 



