Along the wall AB the fluid velocity is directed toward B and its magnitude g increases from 

 zero at A, or at infinity, to unity at B. Hence the .^vector, of magnitude 1/q, lies on its 

 positive real axis, and its end moves on the i^-plane from oo when s is at ^ to i^ = 1 at 6. 

 Along the curving free streamline BC, (^ then moves along the unit circle below its real axis; 

 the direction of the ^ vector is at each point that of the tangent to BC. A similar streamline 

 coming from the right transforms into the negative real axis of ^ from - c» to - 1, together with 

 another part of the unit circle below the axis. Finally, because of the obvious symmetry, 

 there must be a central streamline IJ which is straight throughout and becomes part of the 

 imaginary axis of ^. 



The boundary ABB' A' thus traced on the ^-plane, consisting of two segments of the 

 real axis and a semicircle, is next transformed into the entire real axis of another variable t. 

 For this purpose a transformation is first made to a new variable 



In ^= In 1^1 + i amp C [111b] 



where In \C\ stands for the ordinary real logarithm. This converts the boundary into a semi- 

 infinite rectangle. AB becomes the positive real axis of In t^. On the unit circle. In \t^\ = 

 and amp (granges from to -77; hence the semicircle below the real axis of (^becomes the 

 segmert of the imaginary axis from In ^ = to In i^ = - i tt. On A'B', amp ^ = - i 77, hence, on 

 the plane of In ^, A'B' becomes a line parallel to AB at a distance in below it. 



The Schwarz-Christoffel transformation, discussed in Section 31, is now used to con- 

 vert the rectangle ABB' A' on the plane of In ^ inta the real axis of i. The space between 

 AB and ^'6 'is to be regarded as the interior of the rectangle, since, as the boundary 

 ABCJC'B' A' on the z-plane is traced, the fluid lies on the left. Hence exterior angles of 

 77/2 occur at B and B'. Let the corresponding values of t be chosen as -1 and +1. 



Then, putting a^ = - 1, ot^ = 1/2, a = 1, a^ = 1/2 in Equation [31a], 



— \n C=K{t+l)~^^'^ U-iy^^'^ [111c] 



dt 



and, integrating. 



In C=K\n[t + {t^ -1) ] + L. [Hid] 



The choice of amplitudes here is as in Section 107. 



The constants K and L can now be adjusted to bring the corners B and ZJ'into the 

 correct position on the plane of In ^. Inserting into Equation [llld] il = - 1 for In ^ = 0, then 

 t = 1 for In (, = - in: 



= A' In (-1) + L = in K + L, -in = L, 



273 



