of Edinburgh, Session 1869 - 70 . 
87 
This equation lias 2 m - n roots. 
a 
i ) 
oq + 
7 r 
2m — n 
cq + 
27T 
2m — n 
&c. 
So that each stream line has 2m-n asymptotes equally inclined 
to one another. 
Transforming to rectangular co-ordinates, and choosing the 
asymptote as axis of x, (8) reduces to 
When y = 0, x lias two roots = co if 
2(±&)=0 . . . • ( 14 ). 
Hence the asymptote is such that the algebraic sum of the per¬ 
pendiculars from the sources diminished by the sum of the perpen¬ 
diculars from the sinks is zero. It is obvious without analysis 
that this condition is necessary, that the velocity perpendicular to 
the asymptote, at its point of contact with the curve, may be absolute 
zero. If sinks weigh upward, all lines passing through the centre 
of gravity of the system are asymptotes, and 2m —n of these lines, 
equally inclined to each other, belong to one stream line. The 
system must have a centre of gravity, for by pairing sources and 
sinks we produce couples which will always give a single resultant 
when compounded with the weights of the extra sources. 
A complete system has no centre of gravity, but (14) is satisfied 
for all lines perpendicular to the axis of the resultant couple. If 
the axis of the couple formed by pairing a source and sink at dis¬ 
tance p m makes an angle \J/ m with the axis of the resultant couple 
5 (p sin if/) = 0 . . . (15), 
an equation with only one root to determine the direction of the 
asymptote. In this case the asymptote meets the curve in a double 
point, and has contact of the third order, or x has three roots = co . 
The condition for this is obviously— 
2(=b&7i) = 0 . . . (16), 
which since i£(rh&) = 0, does not depend on the point of the 
asymptote from which h is reckoned. 
If (15) is satisfied identically, the asymptote meets the curve in 
