98 
Proceedings of the Royal Society 
or — 
a 4 -f & 2 y 2 — h 2 x 2 — 2 ^ a 2 ky = 0 
— a series of similar hyperbolas, whose centres lie on the axis of y, 
whose axes are parallel to the co-ordinate axes, and inversely pro- 
portional to the co-ordinates of the source, and which all cut the 
a 2 
axis of x at points distant ± — from the origin. Obviously one 
of the lines is the great circle perpendicular to the line joining the 
sources. 
For the stream lines we have in this case, 
observing that 
<p — <p' = c, 
tan <p = 
tan <p ' = 
xk — hy 
— az 
xk + hy 
This equation becomes 
k 2 x 2 - Wy 2 - a 2 z 2 + \xz = 0 , 
a cone which intersects the tangent plane to the sphere at the 
extremity of the axis of zc, in a series of similar ellipses, having 
their centres on the intersection of the plane with the plane of xz, 
and passing through the points a, ri= ^ , 0. Two of the stream 
lines are manifestly great circles, whose equations are x = 0 and 
2 = 0 . 
If we divide the sphere along the former of these circles, we cut 
off the subsidiary source and sink, and get the case of a hemisphere, 
in which the source and sink are equidistant from the pole. A 
curious hemispherical case is got by dividing the sphere along the 
equipotential hemisphere. In this case we have two sources of 
the same sign within the hemisphere, one being the subsidiary 
source of the removed sink. But in order that the distribution 
may remain unchanged, we must have the potential maintained 
constant at the edge of the hemisphere. This may be effected by 
making the base a conductor with a sink at its centre, or, indeed, 
