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.

XIII. Demonstration of Green's Formula for Electric Density near the Vertex
of a Right Cone. By H. M. MACDONALD, M.A., Fellow of Clare College.
[Received 13 October 1899.]
IN a footnote in his Essay on Electricity Green makes the following statement*:
"Since this was written, I have obtained formulæe serving to express, generally, the law
of the distribution of the electric fluid near the apex O of a cone, which forms part
of a conducting surface of revolution having the same axis. From these formule it
results that, when the apex of the cone is directed inwards, the density of the electric
fluid at any point p, near to it, is proportional to rn-1; r being the distance Op, and
the exponent n very nearly such as would satisfy the simple equation (4n+2), =3=7r;
where 2/3 is the angle at the summit of the cone. If 2/3 exceeds 7r, this summit is
directed outwards, and when the excess is not very considerable, n will be given as
above: but 2/3 still increasing, until it becomes 27r-2 2y, the angle 2y at the summit
of the cone which is now directed outwards, being very small, n will be given by
2
2n log - = l." The method by which he obtained these results was never published and
the problem was not again attempted till 1870 when Mehler+ gave a solution for the
electrical distribution on a right cone under the influence of a point charge; but the
expression given by him for Green's function is so complicated as to make it difficult
to obtain results from it, and the form of the expression does not exhibit the fact that it
is discontinuous. In the following analysis a solution for the distribution near the vertex
of a right cone forming part of a surface of revolution freely charged (Green's case) is
obtained; also solutions for the distributions on a right cone, and on a surface whose form
is the spindle formed by the revolution of a segment of a circle about its chord, under the
influence of point charges on the axis. Solutions for both these latter problems have also
been given by Mehler~. The cases when the point charge is not on the axis can easily be
deduced, but present no special interest.
The solutions here given are examples of a general method, which depends for its
application on the fact that the writer has recently been able to determine the values of
n in terms of u for which the harmonie Pn (u) vanishes.
* Green, Essay on Electricity and Mlagnetism, 1828; + I have been unable to obtain Mehler's paper conMathematical Papers, p. 67. taining the results for the cone and have had to rely on
t Green's statement is quoted and applied by Max- Heine's account of it, Theorie der Kugelfunctionen, Vol. II.
well, Cavendish Papers, 1879, p. 385, with the' remark pp. 217-250.
that no proof had ever been given. ~ Cavendish Papers, loc. cit.