Numerical Solutions 



If the corner at b were removed first, and that at A second, the resulting 

 angle in the z" plane could be obtained from (7) by replacing y and y\ by their 

 complements and q by p . Thus we would obtain 



2p 



q 



(8) 



Comparison of (7) and (8) shows that y" < y". Since, in general, mapping into 

 a circle is accomplished more readily for nearly circular sections, it would be 

 desirable to select that order which makes 7" closer to 7t/4 . For example, if 

 y = 77-/8 (a/b = 0.445), a = 7r/4 , and /S = Stt/S, we have p = 3/2, q = 5/4 , and 

 then, from (7) and (8), we obtain y'/ = -n/e and y'j = 277/15 . In this case the 

 original order appears to be preferable. The difference between (7) and (8) is 

 independent of yj, viz.. 



71 - y2 = T V 



q 



Let us combine the transformations (3) and (6). Put 



(9) 



(10) 



We then also have, putting z = x + iy, 



2 , 2 2 , ^. 



X +y -a +2i ay 

 7 - 



( X + a) ^ + y 



and, with z" = cr + i-u, 



1 + z 



2T + i {a"- + T^ - 1) 

 {a- 1) ^ + T 2 



Solving both (3) and (6) for z' gives 



(11) 



(12) 



1 + z p , 1 + z 



= ib 



l/p 



1 - Z 



and then, by (4), 



iZ i -iZ 



1 i21 



ZP e'p 



(13) 



From (11) - (13) we can compute the real and imaginary parts of a point z" 

 from the coordinates of a point z . 



If, in accordance with the foregoing discussion, it is preferable to reverse 

 the order of the transformations, this can effectively be accomplished by ro- 

 tating the given profile through an angle of 77/2 rad, by 



1622 



