G42 
MR. W. M. HICKS OH TOROIDAL FUNCTIONS. 
17. Electric 'potential of a tore and its capacity. 
Let Y be the constant potential of the tore {u). Then (A,,, a„) must be determined, 
so that 
rf>= v/C-TrS" A„P„ cos n{y-\-ct,) 
maj=Y for all values of v when u — u'. 
Hence a„ = 0 and 
7rA„P / H .= 2y[ 
* cos n6cW 
and 
j 0 f O'—cos 6 
7rAoP'o= v/SVQ'o 
.-.<(>= 22,P„ cos m-+'A P, 
7T 
• (SS) 
(C + S> 
Tliis series is easily seen to be convergent, since (§ 10) it is less than % 7 Q/ + g/p» > 
1 b<f> 
where C+S < C'+Sh 
To find the capacity of the ring we must take the surface integral of 
So, q denoting the capacity, 
over it. 
1 I 2 *’ dn' , b(b du 
q = 4ttY J o L Up ~dd CA 'bu ' dn 
-Ahri r_A 
-dl\] Q' 
t r A J 0 C-'c\2 x /C-c P u +' //L ' C ~dv. I P 
COS 
nvdv 
or dropping the dashes, and writing 
du 2S 
( _2 x /2«SQ„r 
r ] p 9 n +1 
" -(P, + 1-CP„) 
d , 2%+1/P k+1 ^ 
Now 
dU 2S \ P; 
Ucos nvdv 
i„LCW = ^ 2Q * 
cos nv 
a/ 0 —c 
dv 
and 
wnence 
dQn _ 2?? + l /r\ pr\ \ 
f y=A(2»+l)(P (+1 Q,-P,Q,. + i)p- 
7T .1 
= 4o2,|+2a^by (24) . . . 
J- J- n 
( 40 ) 
