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.

296 MR MACDONALD, DEMONSTRATION OF GREEN'S FORMULA
when r' >r, and
In 9 P n( )
rn+l- ap aP '
(1 -/o2) ~ ~
an a~j
when r >r', where the summations extend to all the positive values of n which make
P, (cos a) vanish. When a =r/2
aP, aP,. 1,
an aLP
rn
and V= 2q 2 rn+ Pn (/),
when r'>r, where the summation extends to all the positive odd integers, that is
V = q q
/r2 + r'2 - 2rr' cos 0 N/r2 + r'2 + 2rr' cos 0
which agrees as it ought to with the expression for the potential due to a charge q
at a point distant r' from an infinite conducting plane at potential zero.
(2) To find the potential at any point due to the spindle formed by the revolution
of a segment of a circle about its chord, when its surface is freely charged.
This is immediately obtained by inversion from the above case. Let:0 be the angle
in the segment of the circle whose revolution describes the spindle, ~ the angle in
any other segment of a circle on the same chord, j = log -, where r1, r2 (r >r2) are
r2
the distances of a point on a segment from the extremities of the chord; then putting
q = - Vr' and observing that the cone of angle 0 in the dielectric inverts into the
spindle the generating segment of which contains an angle eo, the potential at any
point due to the spindle when charged to potential Vo is given by
e- *</______ P- (cos f)
V= Vo + 2 V0 /2 (cosh V - cos:)2 e- P (. )
(lL02) aLI-O
where zO =cos o and the summation extends to all the positive values of n which make
P,n (cos ef) vanish. The case of the sphere is that when = 7r/2. It may be verified
that the density of the distribution on the spindle near one of the conical points agrees
with that found ~ 1. For the density at any point on it is given by
_ _O {2 (cosh - cos ^)}2 _ _;;;