92 
Proceedings of the Royal Society 
1. Two Sources and a Sink. — The conic is always a circle with 
the sink as centre. If the sink lies in the line of the sources pro- 
duced, the radius of the circle is a mean proportional to the dis- 
tances of the sink from the sources. If the sink lie between the 
sources, the circle is impossible. If the sink is the vertex of an 
isosceles triangle, the circle passes through both sources, and all 
asymptotes meet in the point of zero flow furthest from the sources. 
If the sink is half way between the sources, there are two straight 
lines and a real and impossible circle. 
2. Three Sources of the same Sign. — Every stream line has 
three asymptotes, meeting in the centre of gravity, and inclined at 
angles of . If one of these asymptotes becomes a branch, the 
other branch is a hyperbola, with centre of gravity as centre, and 
axes in ratio of fS to 1. If the points form an isosceles triangle, 
the hyperbola passes through the extremities of the base. If the 
triangle is equilateral, the hyperbola coincides with its asymptotes. 
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If the vertical angle is less than -g , the rectilineal branch is the 
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transverse axis ; if greater than -g- , it is the conjugate. If the 
points are all in a line, the vertices of the hyperbola lie on that 
line, and are the points of zero flow, which are easily found. If one 
point is half way between the other two, we have two rectilineal 
branches and two hyperbolas, the conjugate axis of the one being 
equal to the transverse axis of 1 the other. The hyperbolas are, 
therefore, confocal. 
Four Points . — Complete System . 
Singular Points. — If A and B are sources, C and D sinks, there 
is a singular point at P, if the circles APC, BPD, and also APD, 
BPC touch at P. Hence, there are no real singular points if the 
sides of the quadrilateral ACBD intersect, unless all the points be 
on a circle, which in this case contains all the singular points. 
Straight Lines. — The one stream line which has an asymptote is 
of the third degree. If a straight line is one factor, the other 
factor is a conic, which is always a circle. For if A, C are the 
images of B, D respectively in the straight line, a circle can be 
