The transfoTmed cylinder. Let the center of the circle of radius a, which represents a 

 circular cylinder, be located on the s-plane at the point 



he'"^ 



[77c] 



where h and 7/ are real positive constants, or at the point (A cos r], h sin r;); and let the fluid 

 at infinity have a velocity V inclined at an angle y to the negative a;-axis, with components 

 -V cos y, -U sin y, see Figure 115. Then, from Equation [69j], the complex potential is 



w= U 



is-he'^l) e-'y + 



ly 



ir z-he'1 



— In [77d] 



2?? a 



z-he''n 

 By substitution for 2 from Equation [77a], w can be found, if necessary, as a function of 2' 



Figure 115 - Diagram for Equation [77d]. 



The lift on either cylinder is given by the Kutta-Joukowski formula [73a]. A simple 

 formula for the torque on the transformed cylinder can be obtained from the Blasius theorem. 

 For this purpose, dw/dz' m\xst be expanded in descending powers of 2', as in Equation [74j], 

 but terms in higher negative powers than I/2' are not needed here. From Equations [77d] 

 and [77b] 



dw dw /dz' I c^ 

 dz' dz/ dz \ ,2 



ue-'y 



a^U 



(z-he'l) 



,'y 



ir 



2'^ z-he'^l. 



[77e] 



175 



