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 
, S ( ± I£«) 
- 0 
(19). 
Let now APC = a, BPC = — - a = j3, and assume the bisector 
u 
of APB as initial line. Then 
• C / 1 1 \ 
Sm 2 (pA 2 PB 2 ) 
sin a - (3 _ 2 sin 26 
which since 
*) + 
PC 2 
1 
PC 2 ' 
sin 2 (3 
2 
1 
PC 2 
sin 2 a 
' . 2 c 
sm 2 — . 
2 
becomes, 
sin 2 (3- - sin 2 a - sin a — (3 . sin a + j3 = 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 — 
