84 Proceedings of the Royal Society 
must assume the same number of sources and sinks without the 
circle, and n — m sources at the centre. 
(e.) The straight line equidistant from two equal sources of the 
same sign is clearly a stream line for these points. Hence the 
image of any point in a straight line is an equal point, which is its 
optical image. 
I have constructed the equation 
29 = a 
on the assumption that all the sources are equal, because the degree 
of the stream line is equal to the number of equal sources (positive 
and negative) to which the system can be reduced. For if h , h be 
the co-ordinates of P, the equation becomes 
2±tan^=C, . . . (7). 
X—il 
If f y — denote the sum of all the combinations of expres- 
\h—xj m 
sions dtz y ~ — \ , taken m at a time, we may write this 
x-li ’ J 
1 - cY y -^\ - + A"f) -&c, =0 (8), 
\x — hj i \x - hJ -2 \x-hj-s \x — lij 4 
an equation of the n ih degree if there be in all n sources. 
The degree of the equipotential lines is also = n if there be an 
equal number of sources and sinks. In general, if there be m 
sources of one sign, and n — m of another, and m ]> n— m , 2 m is the 
degree of the equipotential lines. This is one of many features 
which make it more convenient to work with stream lines. 
It is obvious from equation (8), that every stream line must pass 
through all the sources. Thus, the circle in case (c), which passes 
through no source, is not a complete stream line, the other branch 
being the straight line APP', which passes through all the sources. 
Distinct stream lines can intersect only at a source, for at no other 
point can 2$ be indeterminate. Where two branches of the same 
stream line intersect the velocity is necessarily zero, changing sign 
in passing through the point. The physical meaning of a branch 
is that two streams impinge, and are thrown off with an abrupt 
change of direction. 
