88 
Proceedings of the Royal Society 
a triple point. Two of the branches belong to the line at infinity, 
and the finite branch sinks to the n- 2 degree. 
In this case not only 2(db k) — 0, but 2(d= h) = 0. Hence (16) 
no longer gives a fixed point on the asymptote, but only fixes its 
direction. A further analytical condition is easily found, but is 
unnecessary. For in this case the centre of gravity of the sources 
coincides with the centre of gravity of the sinks. The stream lines 
due to the sources alone would have the same sets of asymptotes 
as those due to sinks. One of these sets is necessarily asymptotic 
in the complete system, which has always one line with real 
asymptotes. The set will consist of ^ rays, all passing through 
A 
the common centre of gravity of the sources and sinks, and equally 
inclined to one another. 
Rectilineal Branches are asymptotes coinciding with their curves. 
Hence, in an incomplete system, all straight lines pass through 
the centre of gravity of the system, and belong to one stream line, 
unless the centre of gravity be a source. In any case they are 
equally inclined to one another, for if not branches of one stream 
line, they would be so for the system got by removing the source 
at their intersection. 
In a complete system there can be only one rectilineal stream line, 
unless sinks and sources have a common centre of gravity. In the 
n 
latter case, there can be at most ^ straight lines, forming equally 
inclined rays through that point. 
The condition for a rectilineal branch is in general that the 
sources must be either on the line or be two by two, each other’s 
images on the line. For if not, remove all the sources on the line 
and all pairs of sources which are each other’s images in the line. 
Next, remove all sources on one side of the line by placing equal 
sources of opposite sign at the place of their images The straight 
line is still a stream line, and on one side of it there are no 
sources, and therefore constant potential, which is absurd. Simi- 
larly it can be shown that a circle is a possible stream line only 
when the sources are on the curve or image each other. From 
this it follows that no finite number of sources can give parallel 
rectilineal streams or non-intersecting circular streams. 
