162 Transactions of the Boyal Society of South Africa. 
From this we get 
_ ^ f 1 
({(^ + a)2 + r2}^ {(z - a.y r^i 
S<p. X \ 2 + a)2 + 3r2 2 (z - a)2 + r2 
■ = - 4- «) . 7^ - - «) rr 
5r rM ' { 0^ + r?)^^ + r^}! { - a)2 + r^}! ) * 
From 9 we have 
8(p. ^1 8\l/. 
8z 7'^ 
Sr ' S^^ ' 
where -ify-^ is the stream-function of the motion. 
Substituting the values of ~, and integrating, we get 
I 2 (z + + 2 - + r"- ) 
Each of the terms of this ecLuation satisfies the differential equation for 
We again firstly draw the two families of curves 
2{z + a)- + r2 
~^'{(z^^'^^ 
2 (z - af + r2 
— -h . , 
and from these two families of curves find 
If we transfer the origin to the point z— — a, and then express the 
equation in the polar form, then we have 
V, = ^ (cos 26 + 3). 
Drawing this family of curves with the origin at (i) ^ = — a, 
(ii) 0 = -f a, and combining the two families by the vector law% we find 
the stream-lines for a uniform line distribution of sources. 
On Plate XIV the families A and B are given by 
^V=-|-(cos 26 + 3) (\= 1) 
with the origin at z = -{- ^ and z = — ^ respectively. Combining these, 
we find the family C, which will then be the stream-lines for a line source 
of strength A = 1. 
Superposing upon this, the uniform stream given by 
rPo= - 16r4 m 64) 
we find the stream-lines E, 
