Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 291
THE DISTRIBUTION OF ELECTRICITY ON A CONDUCTING BOWL UNDER THE INFLUENCE
OF AN EXTERNAL ELECTRIFIED POINT.
13. In order to adapt the method of this paper to obtain corresponding results
for the case of a spherical bowl, we must suppose the surface across which the passage
from the first space to the second takes place, to be a spherical bowl with the fundamental circle for its rim. If the angle of the bowl is /3, we must suppose that in
the first space 0 has values from /3-27r, on the negative side of the bowl, up to 8/
on the positive side, and that as we then pass through the bowl into the second
space, 0 increases from /3 up to 3+ 27r, when the positive side of the bowl has again
been reached. If the convexity of the bowl is upwards, 3 is less than 7r; if downwards, 3 is greater than 7t.
The image of a point P (po, 0o, s0) in the first space and above the bowl is the
point P' (po, 2/3 - o, fo) in the second space, and below the bowl.
The expression
U =, [+ +-sin- {cos 1 (0e - ) sech } ai
V cosh p, - cos (2/3 - 0) 2 ( 2 2 JJ
- \Xcosh po-Po cos0~ 02) P'Q + si- {cos 2( + 0o- 2/3) sech a....... (13)
corresponds to the expression in (2); it is a potential function which vanishes over
the disc, and of which the only infinity in the first space is at P; where it becomes
infinite as 1/PQ.
The Green's function GpQ is therefore given by the formula
GPQ = P - cos-l cos 2 (8 - 0) sech 2 c}
PQ?r
+ \ cosh %p-s cos 0(2)3 ~ f ) POQ s + - si 2( + 9- ) sech 2 a......(14).
cosh pL- cos 80- ) P'Q 2 9
By introducing an auxiliary point L whose coordinates are po, 0 F /, 0, this
expression may be thrown into a geometrical form corresponding to (4), and the
expressions obtained by Lord Kelvin for the density on either side of the disc may be
deduced; it is however hardly worth while to give the details of the process, as it
is precisely similar to that which has been carried out in the case of the circular
dise.
37-2