96 Samuel Karp, Jack Kotik, and Jerome Lurye 
and the dimensionless stream-furction is given by 
B, cos 6/2 
Guz ce nin oo (MES ayr'2 con Se .. - (56) 
r 
If we solve Eq. (54b) when &= 0 (the streamline passing through the stagnation point) for r 
as a function of 6 we obtain the following expansion of r in terms of 0 and €: 
2/3 € as 2 1/3 € - <3 
r=) (xaiwo7a) +38 Si (zeae) to OM 
where the successive terms involve ascending powers of €.* We conclude that for small € 
the shape at one end is determined mainly by the singularity of the dipole distribution at 
that end. The series Ay + A,x' +... will have a radius of convergence equal to L, the dis- 
tance to the other singularity. In order to use Eq. (57) to find the approximate shape at the 
ends we relate the coefficients B, and B, to the solution g(x) of the integral equation. Since 
€g(x) generates the desired unnormalized shape shrunk by a factor of L in the x-direction, 
and since the dipole density corresponding to €g(x) is - Lcég(x)/m, we have from Eq. (54b) 
Le _ Le h(x) Le hx" ) 
- — g(x) = -— € Sr =e ee 
7 ae ee 
Lec ! 
=- CB. + Bex ot yee) 
nxt 0 1 (58) 
from which 
h(x') = (By + Byx' +...) V1 - x’ (59) 
and therefore 
h(x'=0) = h(x=-i)=B 
(x'=20) = A(x=- 5)= 8B, (60a) 
and 
dh dh 4 1 
ane ax Gr Pad Bie (60b) 
x'2Q x=-1/2 
These equations determine By and B, in terms of A(x). Since we have only a digital approx- 
imation to h(x) we were only able to determine B, and B, approximately. Figure 7 shows the 
extent to which the approximate shape at the ends matched the linearized shape, and the 
way in which we joined the two. This example is for f = « but it is fairly typical. Figure 8 
shows the resulting shapes for a number of values of f. 
*The next term involves all the B i: 
