The motion of such a cylinder through fluid stationary at infinity, at velocity U 

 •parallel to the common chord of the arcs, can be represented by adding a term - Us in w. 

 Then, at points where ^ is small and hence z large, using Equation [33i] in Equations [88k] 

 and [89b], 



cU 



/2m C 



m 



^ L, 6ot 



(f^f-)' 



cV I 1 



32 V„2 / 



Thus, in Equations [76c,d,fl, (R) b^ = l>i ^ (c^fi/S) {1/m^ - 1), and the kinetic energy of the 

 fluid per unit length of the cylinder is 



r, = — p 



1 2 



2/7 / 1 



~1 c'-S 



U\ 



[89n] 



where S is the cross-sectional area of the cylinder or 



[2(1 - m) TT + sin 2mv]. 



[89o] 



The case of motion perpendicular to the chord is also easily treated by noting first 

 that the slightly modified transformation mz' - Z = c coth (^/2ot), with z = c coth (^/2) as 

 before, flattens the outline of the cylinder into the segment of the real 2-axis from Z =- c 

 to Z = c. The cylinder is thereby transformed into a lamina of width 2c. From Equation 

 [85b] for transverse flow on the Z-plane past such a lamina, w =- iU {Z^ - c^^^. Further- 

 more, at infinity, Z^2mc/^-*mz, so that uniform flow on the Z-plane transforms into similar 

 flow on the 2-plane but with the velocity multiplied by m (since w = UZ becomes w = mUz). 

 Hence, after multiplying w by 1/m in order to keep the stream velocity equal to f/ on the 

 2-plane, and substituting for Z in terms of s. 



icU 



sinh 



2m 



[89p] 



216 



