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.

FOR ELECTRIC DENSITY NEAR THE VERTEX OF A RIGHT CONE. 295
the expression for V becomes
I\ X' M7X77-n,
4 e oo' (2n +1) sin m7 sin P (Pn)
aXo 1ai - a {n a nJ +m2w2 '
effecting the summation with respect to mn, this becomes
2 A cosh (n + 2) ( - + ) cosh (n + 2) (o - X - )
p 2q =qe 2 _2/
V=- a ----— a --- — - p, ----n — - pn)
~a sinh +n ) X(l (1 -
-!~o x ~o~) ~ /02)
when X > ', and
A+~' cosh (n + ) (Xo -X+ ) - cosh (n + ) (Xo - ')
'= ^ 2qe-2 E /p, (/,
when X < X'. Making X,\ o the space becomes that bounded by the cone =a and
the sphere r= a; and the potential inside an uninsulated hollow conductor of this form
under the influence of a charge q at the point r' = ae-' on the axis is given by
A+A'
2qe 2 e-(n+)(-À')-e- (n+) (À+À')
(1 - jo02) na ' 3n aS/ Pi
when >', and by
V= 2qe 2 e(n+t) (-X') - e-(n+i) (A+X)
a - aP),p, a *Pn
an2 al
when X< X', that is by
ri' rn ny'n\ P (.,
V=- 2q(r, n+l t n+l ' -P, aPn
r a7 c
when r'>r, and by
( r/n rar'n\ an ( a,
V=-2q\ a-^). -3P 3P
when r >r'. To obtain the potential in Mehler's case when the cone extends to infinity
put a = o and then
2 ___ P, (/)
V=- 2q,- r ~' (- p), dPf,