w) = a„ + 



[76c] 



In the integral, all terms give zero except that ^ b^ dz/z = 2nib^; see Section 30. Also, 

 (/') (i 6^ e~^y) = (R) [b. e"'''], where (R) indicates as in Section 74 that only the real part of 

 what follows is to be taken. Hence 



Tj = — pV^S^, S^ = 27t{R) 



6j e-'y 



- S, 



[76de] 



or, if the cylinder is moving toward positive x, 



{R)b^ 



2n - S. 



U 



[76 f] 



Finally, let the shape of the cylinder be changed by means of a transformation which 

 is single valued and regular outside of the cylinder and which leaves the plane of z unchanged 

 at infinity. Toward infinity such a transformation from z to a new variable z' can be written 



'l' *2' 



[76g] 



If this series is substituted for z in Equation [76c] as it stands, however, the boundary 

 condition on the flow miay no longer be satisfied at the surface of the cylinder. To avoid 

 this difficulty, let the moving cylinder first be brought to rest by superposing uniform motion 

 in the opposite direction; then, from Equation [35a], tv becomes, in place of Equation [76c], 



^ . .1 



;j = Ue-'y z + a^ + .... = Ue-'yz'+a^ + (U b{e-'y + 6j) —^ 



The boundary condition on the cylinder is now, on the 3-plane, I'j = constant, and this con- 

 dition remains satisfied on the 2 'plane. Uemoval of the term U e~''y 3' then sets the cylinder 

 moving again, at speed U m a direction inclined at the angle y to the positive real axis of 

 s'; the corresponding flow is represented by 



= flg + (U b(e~'y + 6j) —J 



[76h] 



172 



