of Edinburgh, Session 1869 - 70 . 
91 
of zero flow, P and P', lying in the line bisecting the angle C, and 
such that PC is a mean proportional to BC and AC. The directions 
of the orthogonal branches at P bisect the angle APB and its 
supplement. 
For the initial line is a tangent at the singular points if 
d-u 
dx 1 
_ V 
sin 20\ 
-7- ) 
= 0 
(19). 
C 
Let now APC = a, BPC = — - a = /3, and assume the bisector 
A 
of APB as initial line. Then 
. C / 1 
sin — 
( 
2 VPA 2 PB 2 
1 \ sin a - f3 sin 26 
+ 
) 
PC 2 
which since 
1 _ 1 sin 2 (3 
PA PC 2 - 2 C 
sm- — 
2 
1 1 sin 2 a 
PB 2 PC 2 . , C 
sm 2 — 
2 
becomes, 
0.2 
sin 2 /3 - sin 2 a - sin a - (3 . sin a + /3 = 0, 
which satisfies (20). 
The chief interest lies in the cases where the cubic breaks up 
into a straight line and a conic. . This takes place for one stream 
line of the system when all the sources lie on a straight line, or 
when they form an isosceles triangle with points of the same sign 
at the base. The cases are— 
