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. 289
now let po become very small, as P, P' move away from the origin, the expression
for the potential becomes, when higher powers of po than the first are omitted,
cos ) cosh cos sinh pcos
2q P 1 PO s- Pi C 2 -
7' CP+ ~ —
cosh 2 cP osh2 2 p-cos2 0 cosh2
^ /cos^ \ cosh 2 p ^cos sih pCOS
1 1 1 i1
2q p c! p cosh - p cos-0sinh 2
a.CP CpV keosu g p) / h pcOS2 P C osh2 1
now C(7P = a hnco b P = -, hence if q be made indefinitely great so that -p2 =,,we
cosh p - 1 po P2
find for the required potential
+ _ 2 - (, X /O S 1 1 1 s e c h I _\ 2/ S C O S2co s - 0 s i n h p
r cos- (cos 0 sech - p - a cos. / s
7r
-2osh2p/p </cosu p - cos 0)
2a /cosh - cos 0 a sinh p
now - = -- x = cos
ri + r2 1 cosh p - cos '
hence we find that the potential due to the induced electricity, in a field of force of
potential xLx, is
2 X s 2ai 2a 2a /(r - + r)2 - 4 (11
ri~' + r,cs(r~ r,)c
12. In order to find an expression for the potential of the induced electricity on
the dise, when it is placed in a given field of force, we apply the well-known theorem
that if a- is the surface density at the element dS of the surface of a conductor when
acted on by a unit charge placed at an external point Q, the potential function at Q
which has values V given at every point of the conductor is fVa-dS, the integration
being taken over the whole surface of the conductor. Suppose V(p, f) to be the given
potential function at the element p, <b whose area we denote by dS, on either side of
the disc; the potential function at the point (p,,,0, ) external to the dise which
on the dise takes the values V(p, ) is then, using the expressions found in Art. 8,
whi (1 + cosu p) cos 0, + t
|in t2akR o v e h hl tan-s u / coshcf cosd, V(p, /)dS,
VOL. XVIII. 37