graphical transformation of that for the cylinder; a sufficient number of points on the curves 

 can be transferred into their new positions by the method described in Section 77 and illus- 

 trated in Figure 114. 



From Equations [78h] and [77a] 



dw 

 dz' 



dw 

 dz/ 



'dz' 

 dz 



Z - C L 



{z-ihY 



iT 



2n{z-ih) 



[78i] 



from which the components of velocity w' and v' \n the transformed flow can be found. 



On the arc itself, since the corresponding s-point lies on the initial circle, z is rep* 

 resented by s = zA + ae'* where c is a variable angle, shown in Figure 116. Hence 



{z-ihY 



-U ^^i(a+£)_g-i(a+0) = 2ze-'« sin(a+0, 



ir 



27T{z-ih) 2na 



and, since z^ - c = {z + c) (z - c) and \i e~ 



= 1, 



dw 

 dz' 



2 V sin ( a + t ) + 



277(2 



[78j] 



Here r = |3|, r^ = \z - c|, r^ = \z + c\\ and these quantities represent distances that can be 

 measured on a plot. The point on the arc at which q as calculated from Equation [78j] is the 

 velocity can be found by graphical transfer of the corresponding point on the circle. 



Streamlines for /3 = 11 deg,a = 25 deg and T = are shown in Figure 117; the diagram 

 has been tipped up to save space. Another case of streamlines about a flattish arc is shown 

 in Figure 118; see Reference 114. About a semicircular arc, streamlines for three cases of 

 non-circulatory flow are shown in Figure 119; see Reference 113. 



Figure 117 — Streamlines past a circular 

 arc without circulation. 



179 



