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.

MR MACDONALD, DEMONSTRATION OF GREEN'S FORMULA, ETC. 293
~ 1. Green's case.
With the usual notation, the expression V - Arn P, (t/) is a solution of Laplace's
equation in the neighbourhood of the vertex of the cone which is equal to V, on the
surface of the cone for which P (cosa) vanishes, where a is the semivertical angle of
the cone. That it may be the required solution Pn(,) must not vanish for any value of 0
between a and wr; for if it vanished for a value a', where a'> a, the expression would then
be the solution for the space between the two coaxal conducting cones whose semivertical
angles are a and a', or for some other space not entirely bounded by the cone whose semivertical angle is a. Hence n must be such that Pn (/u) does not vanish for a value of 0
which is greater than a; now the kth zero of Pn (/) considered as a function of n
diminishes as 0 increases, therefore n must be the least zero of P, (cos a). Therefore
the potential in the neighbourhood of the vertex of a right cone of semivertical angle a,
forming part of a conducting surface which is charged to potential Y0, is V0 - ArnPn (p),
where n is the least zero of P,(cosa) and A is a constant depending on the form and size
of the surface. Hencet the density of the distribution in the neighbourhood of the vertex
of the cone varies as r"-l, where r is the distance from the vertex and n is given by
n = o/c/a where x0 is the least zero of JO (x), when a is small, by (4z + 2) a = 37r, when a is
2
nearly 7r/2, and by 2nlog-=1, when a is nearly 7r and 7r- a =y. Thus Green's results
7
are verified.
~ 2. Mehler's cases.
(1) The distribution of electricity on a right cone under the influence of a charge
on its axis.
Let the space to be considered be the space bounded by the two concentric spheres
r = b, r = a and the cone 0 = a, where r, 0, > are polar coordinates, and let there be
a charge q at the point r=r', 0=0. The conditions to be satisfied by the potential are
V= 0, when r= a and a > 0 > 0,
V = 0, when r = b and a > 0 > 0,
V= 0, when = a and a > r > b,
and 2V 28V 1i a ( V
and arV + 2a - + 1 a (1 - 2) av- + 47rp =
ar" r ar +rL ( + 4 r=O
throughout the space. Put r= ae-À, then the equation to be satisfied by V becomes
ax2 a*a + 4Maa2e-2pld = 0;
Macdonald, " the zeros of the ha ic () considered as a function of Poc. Lon. M. Soc. 1899.
t Loc. cit.