For the torque, Equation [74h] gives similarly 



N^- I p{R) \ (— +26jt/e-'A <f — 



Write 



6 = b/+ib'', 6' and 6 ''real. 

 Then 



N = 2npU (R) (ih^e~'y) = 2npU (6j'sin y - 6 "cos y). [74m] 



Thus the torque is independent of P. 



If the fluid is brought to rest at infinity by using a frame of reference moving with the 

 fluid, the cylinder is moving at velocity U, with components U cos y, U sin y, but both force 

 and torque are the same as before. 



The final remark may be made that, if dw/dz contains two or more poles inside the path 

 of integration, representing line sources, vortices, dipoles, or other singularities, these in- 

 cluded singularities in combination with themselves or each other contribute nothing on the 

 whole to the integral for the force. Consider, for example, two terms in dw/dz of the form 

 A/{z-(if., B/iz-b)'" where n and m are positive integers and A, B, a, b are constants. The 



contribution of these two terms to ^ (dw/dz) dz is 



2AB 



dz. 



(z-af" (2-< (s-6r {2-l>)_ 



Even the middle term here integrates to zero. For, the path of integration may be displaced 

 toward infinity without crossing any singularity of the integrand and hence without changing 

 the value of the integral; and toward infinity, using the binominal theorem, 



1 mb 



{z- a) (z- b) \z 2 J \z z 



This integrates to zero, since n + m > 1. 



In the same way it may be seen that the same product term contributes nothing to the 

 integral for the moment L provided n + m > 2. 



(See Reference 1, Article 72b, where V, V replace -U cos y, -V sin y, and a , fi 

 replace 6 ', 6''; Reference 2, Section 6.41.) 



167 



