of Edinburgh, Session 1869 - 70 . 
85 
The same result is easily found from the analytical condition for 
• i . i dzt du a 
a singular point — = — = 0. 
ax ciy 
For ~ ^ ~ = velocity parallel to axis of y, 
ct'OC ay 
du dv 
dy dx 
or directly by differentiation. 
= velocity parallel to axis of x, 
du _ ^ 
dx 
du 
dy 
- V 
y_-lf 
x — h 
( 9 )- 
The nature of the intersection of the branches of a stream line 
at a multiple point is easily determined. 
At an wi-point, the angles at which the branches cut the axis of 
x are the roots of the equation—■ 
' d d 
dx + tan ? 3^ 
u = 0 . 
( 10 ). 
Where, since 
d * 
d?u 
dx 2 
d m u 
dx™ 
'u 
d 2 u 
W 
d m u 
d x m ~ 2 dy' 2 dx m ~ i dy A 
d m u 
d m u 
&c. 
dx m l dy dx™ 3 dy d 
Whence (10) becomes 
d m u 
&c. 
dx r ‘ 
^1 - tan 2 <p + &c.^ + 
d m u / m . m -1 . m - 2 , 3 ^ , „ \ .. 
| m tan <p - --—-— -tan 3 <p + Ac. )=■(). 
dx™ l dy 
1.2.3 
d m u 
We can choose the axes so that AA = 0, and reduce the equa- 
dx m ’ 1 
tion to 
m tan © - 
m . m — 1 . m — 2 _ 3 
1.2.3 
tan ^ “}■ ... •— 0 
( 11 ). 
VOL. VII. 
M 
