Now, it is clear from Equation [78a] that the points 2 = - c correspond to s'= i 2c. 

 As 3, starting from z = c, traverses the initial circle positively \x^ to z = - c, d^ - 6^ retains 

 a constant value, for a geometrical reason; hence by Equation [78b] d^- ^2' likewise remains 

 constant, and s ' therefore, traces a circular arc extending from s'= 2c to 2' = -2c. As z con- 

 tinues past 3 = - c, ^2 changes by n, and d^- O^hy 2tt\ this is easily seen to be equivalent 

 to no change at all in 9/- 6', so that 3 'must now retrace the arc, arriving back at s'= 2c 

 as z comes to c. 



The constant value of 6. - 9^ along the upper part of the circle can be written 



9, -d^ = — - B [78c] 



where the significance of j8 is shown in Figure 116a, and 



c = a cos ^, h - a sin fi = c tan j8. 



[78d,e] 



The angle between the tangent and the chord at each end of the arc on the 2 '-plane is 

 77 - (9/- 9^) or 2 6; thus the arc has a total angular length of 4/S. Its radius /?, and its 

 camber C, or the ratio of its maximum height above the chord to the length of the chord, are, 

 from the geometry of Figure 116a and Equation [78d,e], 



R = 



2c 



sin 2^ ^/^ 



, C = 



R (1 -cos 2fi) 

 2R sin 2^ 



1 

 — tan 



2 



[78f,g] 



The interior and the exterior of the circle are each mapped onto the entire 2 '-plane; 

 the mapping of the interior is to be ignored here. 



For an application, it will be convenient to assume the fluid to approach at infinity 

 from a direction inclined at an angle 9 below the positive x- or a;'-axis, with components of 

 velocity -V cos a, +U sin a . Since the flow at infinity remains unaltered, the angle of 

 approach is the same in the transformed as in the original flow. Then, in Equation [77d] for 

 the complex potential w, y = - ci'i and here 7; = n/2, e"? = i. Thus 



w=V 



{z-ih)e'-^ + 



z - ih 



iV z - ih 



In 



277 a 



[78h] 



Substitution for 2 in terms of 2' from Equation [77a] then yields mj as a function of 

 2', and from w = <j) + i\p, 2'= x' + iy', the potential 9 and stream function \[j can be found; 

 but the equations are complicated. The flow net is most easily constructed by direct 



178 



