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.

XII. On Green's Function for a Circular Disc, with applications to Electrostatic Problems. By E. W. HoBsoN, Sc.D., F.R.S.
[Received 7 October 1899.]
THE main object of the present communication is to obtain the Green's function
for the circular disc, and for the spherical bowl. The function for these cases does not
appear to have been given before in an explicit form, although expressions for the
electric density on a conducting disc or bowl under the action of an influencing point
have been obtained by Lord Kelvin by means of a series of inversions. The method
employed is the powerful one devised by Sommerfeld and explained fully by him in
the paper referred to below. The application of this method given in the present paper
may serve as an example of the simplicity which the consideration of multiple spaces
introduces into the treatment of some potential problems which have hitherto only been
attacked by indirect and more ponderous methods.
THE SYSTEM OF PERI-POLAR COORDINATES.
1. The system of coordinates which we shall use is that known as peri-polar coordinates, and was introduced by C. Neumann* for the problem of electric distribution
in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate
system; in order to measure the position of any point P, let a plane PAB be drawn
through P containing the axis of the circle and intersecting the circumference of the
PA
circle in A and B; the coordinates of P are then taken to be p = log PB, which
is the angle APB, and b the angle made by the plane APB with a fixed plane
through the axis of the circle. In order that all points in space may be represented
uniquely by this system, we agree that 0 shall be restricted to have values between
-7r and 7r, a discontinuity in the value of 0 arising as we pass through the circle,
so that at points within the circumference of the circle, 0 is equal to 7r, on the upper
side of the circle, and to - r on the lower side of the circle, the value of 0 being
zero at all points in the plane of the circle which are outside its circumference. As
* Theorie der Elektricitdts- und Widrme-Vertheilung in einem Ringe. Halle, 1864.