86 
Proceedings of the Royal Society 
or tan m<p = 0 . . . (12), 
<p = — , where l is any integer from 1 to m. 
m 
Thus the branches make equal angles with each other. This 
proposition depends solely on the relation = 0. It is therefore 
true, also, for the equipotential lines, as is otherwise obvious.* 
The general nature of the stream lines will be different, accord- 
ing as the number of sinks is or is not equal to the number of 
sources. In the former case, 2(0) = 0 is satisfied at all points 
infinitely distant, the radii being all parallel, and the positive 
and negative angles equal in number. Hence one stream line 
has the straight line at infinity as a branch, or intersects the straight 
line at infinity at right angles, and therefore has an asymptote. 
This stream line will, in general, be of the n — 1 th degree. In some 
cases it may be of a lower degree ; as, for example, when the conic 
at infinity is its other branch. A case of this sort will be given 
below. The other stream lines of the system cannot meet the line 
at infinity, and cannot have asymptotes. However far they run 
out, they must therefore loop and return. 
When there are more sources than sinks, 20 becomes indeter- 
minate at an infinite distance, as might have been anticipated from 
the fact, that in this case there is a constant flow of electricity out- 
wards, implying a sink at an infinite distance. The line at infinity 
is not in this case a stream line, and will be cut by all the stream 
lines, which do not loop except at finite distances, and have all 
asymptotes. 
The asymptotes, in this case, may be easily constructed by the 
aid of equations (6) and (8). 
At the infinitely distant point of contact the velocities due to 
all sources are in the same direction, or the asymptote must be 
parallel to the radii. 
If there are m sources and n — m sinks, the stream line whose 
asymptote makes an angle a with the initial line is obviously 
2 9 = (2m-w)a = tan 0 (13). 
* I have since found that this result has been already proved for plane 
curves by Professor Rankine and Professor Stokes (Proc. R.S., 1867), and for 
spherical harmonics by Sir W. Thomson and Professor Tait, in their treatise 
on Natural Philosophy. 
