whence A" = 1 and 



1/2, 



In C = In [< + (r - 1) ] - ^ 77 = cosh t 



[llle] 



9 1/2 1,1 



[luf.g] 



since e~' '^ = - 1. Here, when ;; is real and negative, (^ - 1) - - yjt - 1. 



Now, as 3 traces one of the streamlines ABC or A'B'C, t moves along its real axis 

 from - °o, or from + oo, toward 0; and i/f obviously has different values on these two stream- 

 lines, which bound a jet of fluid. Hence the flow on the i-plane resembles that toward a sink 

 located at the origin. This fact suggests the following trial assumption as to the complex 

 potential: 



- A, 



w = c \r\ t, t = e 



w /c 



where c is real; see Section 40. According to this assumption, the free streamlines extend 

 up to the point ;; = 0, or i^ = - ?'. 



Then dw = cdt/t and, from Equation [Ilia] and Equation [lllf], integrating, 



dz = - C dw = c [t + {r - 1) ] dt/t\ 



, 1/2 



« + (r - 1) + sin' 



[lllh] 



T 1 / Z 



For the significance of {t - 1) see Section 107. 

 To evaluate sin~ , consider, in general, sin~ z. Write 



sin z = V ^ i£,^ 



where v and ^ are real. Then z = x + iy - sin {v + i^) = sin v cosh ^ -^ i cos v sinh f 

 and 



X = sin V cosh ^, y = cos v sinh (f. 



[lllij] 



Thus ^ and v serve as elliptic coordinates on the s-plane and can be found for any point; 

 see Section 82, where 77 = {n/2) = v. 



In analogy with Equations [82e, f]. 



1 r O T 1/2 T o 1/2 



cosh ^= — ![(,'>'+ 1)2 + 2/2] + [(j; - 1)2 + y2] j 



[111k] 



274 



