90 
Proceedings of the Pioycd Society 
sources of the same or opposite signs. We will now proceed to 
consider the cases that arise when there are three or four sources. 
Three Sources .—In general the curves will he cubic passing 
through the three sources, and having asymptotes determined as 
above. The direction of flow at any point of the field may be 
found by observing that if <p be the angle between the tangent 
and a radius vector, 
2 ± fPl = 0 . 
r 
It will sometimes be possible to find the direction of flow geometri¬ 
cally by the following obvious theorem. 
If a circle be described touching a stream line at any point, and 
cutting off from the radii vectores of that point, fractions of their 
lengths, fx &c., wdiere /x is negative if the point of intersection 
is in the radius vector produced, and also negative if the radius 
vector is drawn from a sink, then— 
2 (/x) = 0 . 
When the number of sources is large this theorem is not in 
general convenient, but it is often applicable where there are only 
three points. 
The lines of flow can, however, be readily described with any 
degree of accuracy when there is one sink, by describing segments 
of circles with constant difference of angle through the sink and 
one source, and drawing through the other source straight lines 
with the same difference of angle. The stream lines will be 
diagonals of the quadrilaterals into which the field is thus divided. 
The process may be extended to the case of two sources and two 
sinks by taking the intersections of two sets of circles. 
When there are two sources and one sink, the singular points 
may be found by an easy geometrical method. Let A, B, be 
sources, C the sink, and P a point of zero velocity. The resultant 
velocity due to A and C is in the tangent to the circle PAC, and 
also—since P is a singular point—in the line PB. Therefore— 
BPC=PAC. 
Similarly 
APC = PBC. 
Hence PCA, BCP are similar triangles, and there are two points 
