of Edinburgh, Session 1869-70. 
85 
The same result is easily found from the analytical condition for 
a singular point ^ ^ = 0. 
ax ay 
For - ^ ~ = velocity parallel to axis of y, 
ClCC 
= velocity parallel to axis of a?, 
ay ax 
or directly by differentiation. 
du 
dx 
(Lib 
dy 
) 
(»)■ 
The nature of the intersection of the branches of a stream line 
at a multiple point is easily determined. 
At an ra-point, the angles at which the branches cut the axis of 
x are the roots of the equation — 
(s + “ = 0 
( 10 ). 
TirK * d u d u 
Where, since — — = - — — 
dx 1 dy 2 
d m u 
dx m 
d m u 
d m u 
d m 
dx m 2 dy 1 dx m ~ *dy- 
d 1 
<fec., 
dx m ~ 1 dy dx m 3 dy i 
Whence (10) becomes 
m . m — 1 
&c. 
d m u A 
V 
c&c” 1 l dy 
[ w tan <p 
tan 2 <p 4- &c,^ + 
tan 3 <p + &c.^ — 0. 
1 . 2 
m . m — 1 . m — 2 
1.2.3 
We can choose the axes so that —7 = 0, and reduce the equa- 
dx m 
tion to 
, ^ m.m — l.m — 2, * ^ 
m tan p - - — g tan 3 ? + ■•• = 0 . (11), 
VOL. VII. 
M 
