Ill 



T^l = — P 4 'I' in'^^ = -^ P ^ ^ 4>l <is = — p V ^ cji dy, 



since I ds = dy. Here p is the density of the fluid and the integration is to be extended 

 around the surface of the cylinder. 

 Now, if 



w - (j) + iib and z = x + iy, 

 then 



<f) dy = (f ") (w dz) - iIj dx, 



where the symbol (/') signifies that only the imaginary part of what follows is to be taken, 

 and without including the factor i; thus 



(/') (wdz) = (/') [(</S + iip) (dx + idy)] = cfj dy + ip dx. 



Also, between two points on the surface ds apart, ip differs by dtp where dip = - g ds =- IVds 

 = - V dy\ whence, after integrating, ip = - Uy + C on the surface, where C is a constant. 

 Substituting, 



T^ = — pU[{l')(^hdz)- US] [76a] 



where S stands for - (^ydx and represents the cross-sectional area of the cylinder. Here C 

 has disappeared because ^ C dx = C <^ dx = 0. 



This result may now be generalized so as to allow the cylinder to be moving at speed 

 U in a. direction inclined at an angle y to the positive a;-axis. Both the cylinder and the flow 

 are rotated through an angle y about the origin if z is replaced by se"'^, as explained in 

 Sections 25 and 34. Thus Equation [76a] is replaced by 



1 

 T^ = — pU[il'){^we-'y dz)- US], [76b] 



since the rotation does not alter S. ' 



Another useful form for T^ may be obtained by displacing the contour of integration 

 toward infinity, which does not alter the value of the integral; see Section 29. Then w can 

 be expanded as in Equation [72d], but here the last two terms of that expansion vanish, so 

 . that ' 



171 



