G10 
MR. W. M. HICKS OK TOROIDAL FUNCTION'S. 
imaginary argument and (wlien the whole of space outside a tore is in question) of 
orders (2?i-j-l)/2. For space inside a tore we have a corresponding analogy with zonal 
harmonics of the second kind. The properties of these functions are found to have 
analogies with those of the ordinary spherical harmonics, but with essential differences. 
The space outside a tore is different from that outside a sphere in being cyclic ; in 
general, then, the functions for space outside will not be determinate from the surface 
conditions alone. The above functions are suitable only when there is no cyclic 
function : it is shown how to obtain a function which will complete the solution. 
The third section deals with tesseral toroidal functions, which come into use for the 
most general case of non-symmetrical conditions. It is shown how the different 
orders and ranks depend on each other, so that they may be calculated in terms of 
two. Integral expressions are also obtained, as in the second section, which are needed 
in finding the coefficients in expansions in series. 
The fourth section briefly notices the functions suitable for tores without a central 
opening. These functions bear the same relation to the foregoing functions that 
cylindric harmonics (Bessel’s functions) do to spherical harmonics. 
In the fifth section a few examples are given of the application of the method, such 
as the potential of a ring, the electric potential of a tore and its capacity, the electric 
potential of a tore and an electrified circular wire whose axis is the same as that of the 
tore, the potential under the influence of an electrified point arbitrarily placed, and 
the velocity potential for a tore moving parallel to its axis, as well as the energy of 
the motion. 
Of previous writings on the subject, or nearly connected therewith, I am only 
acquainted with two. In IIiemann’s ‘ Gesammelte Werke * (chap, xxiv.) is a short 
paper of six pages, “Ueber das Potential eines Binges.” He arrives at the same diffe¬ 
rential equation as (7) in this paper, points out that a solution can be expressed as 
a hypergeometric series in several ways, and that each function can be expressed in 
terms of two, which are elliptic integrals of the first and second kinds. The paper is 
a posthumous one and is not developed. There is a note on the same subject by 
W. D. Niven in the ‘Messenger of Mathematics’ for December, 1880. Though not 
bearing on the same subject, a paper may be mentioned by C. Neumann, “Allgemeine 
Losung des Problemes itber den stationaren Temperaturzustand eines homogenen 
Korpers welclrer von irgend zwei nichtconcentrischen Kugelflachen begrenzt wird ” 
(H. W. Schmidt, Halle). This is a pamphlet of about 150 pages. He uses co-ordinates 
analogous to those in the present paper, but the method of development is very 
different. The functions are spherical and cylindric harmonics of real argument, and 
those of the second kind do not enter. He considers the stationary temperature in a 
shell bounded by non-concentric spheres ; in an infinite medium in which are two 
spherical cavities ; and similar cases when the boundaries touch. Its interest in con- 
* The greater portion of the following pages was completed before I became acquainted with this 
paper of Riemann’s or with that of Neumann’s mentioned below. 
