﻿Self -Induction and Gravity-Potential of a Ring. 457 



functions which naturally come in are Toroidal Functions, 

 the properties of which are developed in two papers in the 

 Transactions of the Royal Society*. These papers are 

 referred to in the present pages as [T. F. i.] and [T. F. ii.]. 

 The special problems considered are : — 



(1) The force-flux function and self-induction for a ring in 



which the current-density varies inversely as the dis- 

 tance from the straight axis. This is the case of 

 current through a ring consisting of a single turn of 

 round wire. 



(2) The same quantities when the current-density is con- 

 stant. This leads to the case of a uniformly wound 

 coil of circular cross section. 



(3) The method is than exemplified by finding the gravity- 



potential of a finite ring — a problem which has been 

 recently very fully considered by Mr. Dyson f. 



1. The force is symmetrical round the straight axis of the 

 rino-. Take this axis for the axis of z and the origin at the 

 centre of the ring, and let p, z denote the cylindrical coordi- 

 nates of any point. Further, let R denote the radius of the 

 circular axis, r that of a cross section, and a that of the 

 critical circle in the curvilinear coordinates used in toroidal 

 functions — i. e. in the system of coaxal circles having Oz 

 for common radical axis and having one circle of the 

 system coinciding with the cross section. The length of a is 

 equal to the length of a tangent line from the centre to the 

 circle. It will be found convenient to express values later 

 in terms of R and the angle subtended at the centre by a 

 cross section : this angle will be denoted by a. 



The flux function y]r at a point P will be taken as denoting 



the total flux up through a circle, radius PN (see figure). 

 it is the same as that through a diaphragm stretching from 



* Phil. Trans. 1881, Part iii. p. 609, and 1884, Part i. p. 161 

 t Phil. Trans. 1893, A. pp. 43 & 1041. F 



Phil. Mag. S. 5. Vol. 38. No. 234. Nov. 1894. 2 I 



