[ G0 ‘ J ] 
On Toroidal Functions. 
XIV. By W. M. Hicks, M. A ., Fellow of St. John's College, Cambridge, 
Communicated by J. W. L. Glaisher, M.A., F.R.S. 
Received February 21,—Read March 3, 1881. 
The following investigation was originally undertaken as the foundation for certain 
researches on the theory of vortex rings, with especial reference to a theory of gravi¬ 
tation propounded by the author in the Proceedings of the Cambridge Philosophical 
Society (vol. iii., p. 276). As the results seemed interesting in themselves, and as 
they also serve as a basis for other investigations, more particularly in electricity and 
conduction of heat, I have thought it advisable to publish it as a separate paper, 
especially as I cannot hope to find leisure for some time to complete my original 
purpose. 
The word “ tore ” is used as a name for an anchor-ring, here restricted to a circular 
section, and lay “ toroidal functions ” are understood functions which satisfy Laplace’s 
equation and which are suitable for conditions given over the surfaces of tores. 
The first section is devoted to the general theory of the employment of two dimen¬ 
sional equipotential lines in certain cases as orthogonal co-ordinates in problems of 
three dimensions. From this we pass at once to the particular case where the two- 
dimensional lines are the system of circles through two fixed points and the system of 
circles orthogonal to them. It is shown that these satisfy the conditions of applica¬ 
bility. By revolution about the line through the two points we have functions suit¬ 
able for problems connected with two spheres. By revolution about the line bisecting 
at right angles the distance between the points we have functions associated with 
anchor-rings or tores. By the first system it is also possible to deduce functions for 
what may be called a self-intersecting tore, and by the second for two intersecting 
spheres. A second application is made for the particular case where the opening of a 
tore vanishes and there is a double cuspidal point at the centre. 
The second section is devoted to the development of zonal toroidal functions—that 
is, for conditions symmetrical about the axis' 5 ' of a tore. It is shown that for space 
not containing the critical axis these are the same as zonal spherical harmonics of 
* Throughout tbe paper tbe axis of a tore is taken to be tbe line perpendicular to its plane through its 
centre; the circle traced out by the centre of the generating circle of a tore will be called the circular 
axis, and the circle by the two points above mentioned the critical circle. 
MDCCCLXXXI, 4 K 
