MR. F. W. DTSOISl ON THE POTENTIAL OF AN ANCHOR RING. 
67 
Section 111 .—Electric Potential of an Anchor Ring. 
§ 9. The potential at any external point may be taken to be 
Aj^l^ + + Ct^'Aglg + &c. 
For this is a solution of Laplace’s equation, finite at all points outside the ring, 
and vanishing at infinity. 
The constants are to be determined by the condition that at the surface of the ring 
the potential is the constant Vq. 
Using the expansions of § 3 (A.), we have that at any external point 
Vc — Ai -s Z -|-2 
2Z + 1 „ . 108/ + 27 
2/+ 3 , 12/+ 5 0 
o- 1 + ^28 - sq + ^ 
1+1 . 9/ + 3 A , * (7^ . 12/ + 11 
16 “ ' 2048 
+ cos X <1 Ai ( — - s d-ss'j _j_ _ j X d-s3 ) d- Agcr'^' 
36/ + 51 
-1 
4.sJ 
2 ..2 . m + 4^ , 
+ cos 2 x^Ai( 18 ■«" + ' 356 ’«‘) + A 20 -qi+ 8^ 
5/ + 2^ Q , » o 3s . . .1 
12/ + 9 
+ ^3^" h + I ) + A,o-' 
d- cos 3x I Ai S® d- Ago-- - d- Ago-^^ - d- A^o-s J 
d- cos 4x j Aj + Aoo-^ yIs s" d- Ago-^ + Ago-'^ ~ 
All terms have been retained which will, when R is put = a or s put = cr, be of 
the order cr\ 
(IN dY _ clY 
chi f/E a ds 
s being put = cr, after differentiation. 
Therefore 
ac 
dN_ 
dn 
= {A. (- 1 + + g .*) + A, (- J + + 
-f (r COS X -I A V ^ d" A cr^) d- Ag — 1 -}- —yA o-' ) ~ f Agcr- 
+ 
(P COS 2x |Aj 
6 A. + 1 . lOA — d .A , A 12X + 3 , 
16 
32 
crMd-A 
3 32 cr“ — 2 A 3 — lOA^o-' 
+ cos 3x -I Ad 2 A2 -A - A 3 i - A. 0 
, , , I , 140X + 104 . * . . * I 
d- o- cos 4x -j A, j ^^24 + ~ Aj. — 24 A 5 1- - 
K 2 
