ME. W. M. HICKS OK TOROIDAL FUNCTIONS. 
649 
[ September , 1881.—At tlie suggestion of one of the Referees I give a few additional 
numerical illustrations. The first is the ratio of the density of electricity at a point 
on a tore furthest from the axis to that at a point nearest the axis. The potential 
due to the distribution of electricity on the tore is given by (39). The normal force 
at any point of the tore is 
b(f> bit _ C — c bcf) 
bio bn a bio 
whilst for points furthest from the axis u — u', v — 0, and for points nearest u—u, 
v—tt. Putting these in, remembering that 2 8 dP w / dot = ( 2n1)(P„ + [ — CP„) and 
( 2 ? 2 +l)(P n+ 1 Q il —P hQmu) = 277, it is easily shown that the above ratio is 
/C4l\i 27r fe“?; + ^“ ' ' ' ) _ i(Qo+Qi) — 2(C + 1)2( — )"nQ n 
C 1 27r (2Po+p^+p 2 + • • • )+i(Qo~Qi) — 2 (C— l)2wQ re 
If the first n sequence equations in Q be added together there results 
4(C— l)^AQ rt =(2w+ l)(Qa+i“Q li ) + Qo — Qi 
whence 4(C— l)2nQ rt =Q 0 — Q x 
Further, putting (— )"Q, ( =gq, the sequence equation for q is 
(2 n +1 )q n+1 + 4nCgq + (2n—l) q,^— 0 
whence as before 
4(C + l)^i —)"wQ )l — 4(C + l)2 hn/ (l —— (/ 0 — — (Qo+Qi) 
Finally then the ratio of the densities is 
1 (-Y 
— ix 1— L 
1 + P\32T 0 1 
-W 1 
*- 0 n 
If terms higher than P 2 be neglected this is 
M+P\3E-2PE 
1 —kj E + 2PF 
I have not been able to find a finite expression for %\/V n and — )"/P,„ but when 
the ratio of r to R is very small, the first two terms are sufficient. In any other case 
MDCCCLXXXI. 4 P 
