Furthermore, if t has a large negative real value, 1 < t < \fa\ if t is large and positive, 

 < T < 1. Hence, as ;; changes from a large negative to a large positive real value, all of 

 the four binomials whose logarithms occur in Equation [108c] remain real, but 1 - t changes 

 from negative to positive, and its amplitude from -77 to 0. The imaginary part of s is thereby 

 increased by -inK. But, on the diagram, 2 changes from A to F; hence the change in its 

 imaginary part is also -ih^. Thus -inK = -ih^. From this result and Equations [108f,g] 



K = — , c = 



[108h,i,j] 



Finally, at E or t - a, t = 0, s = L by Equation [108c]. Hence, if the origin is placed 

 at £, L = 0. Then, from Equations [108c], [108h,i,j], [108b] and [108d], 



1 / 1 + T 

 z = X ■¥ ly - — A, In + A„ In 



[108k] 



„ 77 w /(h, U) 



„ n w /(h, U) 

 _hl{e 1 -1)_ 



1/2 



+ iifj. 



[1081, m] 



These equations fix 9!) and 1// as functions of x and y, but their interpretation is involved. 

 For the velocity, from Equations [108b, e] and Equations [108h,i], 



dw 



dz 



= V 



t - a 



t - 1 



= U' 



[lOSn] 



On all walls parallel to U^ r is real and positive, so that, from Equation [108n], 

 T = q./U. On AB, t<Q,l< r <^^ h^/h^; on CD, < t <1, t > \fa= h^/h^; on EF, 

 t > a, T < 1. Hence, from Equation [108n] and Equation [108k], 



on AB: 



1 / q+U ^1 ^ - ^2 9 \ 



V < q < h. U/h^, X = — A, In + A„ In , 



1 2' ^11 ^_ fy 2 h^U + h^q I 



on CD: 



1 / q+U h^q-h^V 



q > h^ V/hr,, a; = — A, In + A„ In 



: 12' ^1 ^ _ fy 2 h^q + h^V 



268 



