On the free streamlines q = 1. On the side A'B' of the wall of the tank, t is real and t > 1; 

 hence 



„ 1/2 -^ 1/1 \ ^ 1/2 1/1 



Also, s is real and z = x. Thus, from Equation [lllq]. 



1 . -1 / 2? 

 — + sin ( 



? U^ . 1 



where sin is in the first quadrant. The side A'B'ls symmetrical to AB. 

 Along the median plane, or IJ in Figure 179, 



. , 9 1/2 , ,9 1/2 



t^ i \t\, {f - 1) =i (\t\^ + 1) 



[Ills] 



Thus 



q = -V ^ {\t\ + (\t\'^ + 1) 



1/2, 



and 



1*1 =|- -? A, (|^|2 + 1)'^' = — + 9 



Also, if s = sin 0-/t) - sin (l/z \t\, then V^l = i sin s = sinh (is), hence s =- z sinh' 

 (l/j^l). Finally, z = zy. Hence, from Equation [lllq]. 



2 a 



— - sinh" 



2q 



1-q^ 



[lilt] 



In describing the form of the free streamlines, a more convenient parameter than t is 

 the angle 6, actually negative, between the a;-axis and the tangent to the streamline; and a 

 fresh integration is less troublesome, because of the singularity at ^ = 0. On a free streamline 



e'^ cos 0= - {C+ C ) = -t. 



Hence, using Equations [Ilia] and [lllp]. 



[lllu,v] 



2a . sin 9 



dz - dx + idy = - ^ dw = (cos 6 + i sin 6) 



2 + 77 cos 6 



276 



