MR, W. M. HICKS ON TOROIDAL FUNCTIONS. 
641 
For a point on the axis, u =0 P /; = 7r and the above become respectively 
2/T7T\/r—C.S' 1 _ 
/ o T n r — •= —j c and n\/l — cSA„ cos (nv+a„). 
v 2(C — cos v — v ) v ' 
It will therefore be more convenient to determine the A„, a lt from this simplified 
case. 
It is clear that 1/{C'— cos-v— v} 1 can be expanded in a series of powers of cosines 
of (v—v ) and therefore of multiples of the same. 
Hence 
a,,= — nv 
and 
2/^S' 
2(0' —cos 6) 
, cos n6 
Therefore 
A _2/z.S' |' 2ir cos nOdO 
77 "~(7JJ 0 V^'- cost? 
But 
=- 4/xS'Qh 
7tA 0 =2/xS , Q / 0 
Hence in general the potential for points outside the tore u is 
^= 4 ^- , (o C Zf)Ap..Q'. cos «(W)-4P„Q' 0 } 
Consequently the potential for points within the tore u is 
^=A(|A) ! » sp '' q " cos n-iP'»Q 
Both these series have been shown to be convergent. 
If M be the whole mass of the ring’ 
(36a) 
. . (36b) 
M= 27rb[i — - , 
It follows as a corollary that the potential for a mass M on the axis is, for all points 
not on the axis, 
2Ma/ 2 sin f 
(ITT 
\/Q-c{t.- ) Q, l coan(v—v)—l>Q 0 } • • • • (37) 
Also, putting M at O.A and —M at 0. —A, and making v zero and M infinite, the 
potential for a uniform field of force parallel to axis is 
//VC— ct*nQ lU sin nv 
MDCCCLXXX1. 
”0 
4 0 
( 33 ) 
