1058 
MR. F. W. DYSOR ON THE POTENTIAL OF AN ANCHOR RING. 
§ 12, Now when Fv = p = a (1 +2ft COS Vj and Vg must have the same value. 
This will enable us to find the terms of the second order in V^. 
Remembering that for a solid ring of circular section, V and cZV/rZR are continuous 
at the surface, and cZ'^Yq/cZR^ = (rZ-Vi/cZR^) + 47r, and that for a distribution of 
surface density a cos ny 
cIY, clYi , , . „ 
we see that when R = /3 = a(l + S/3,( cos ny) 
V — V 
2TTCt“ 
^ (^3 cos 2x + . . . + A. cos nx + . . .f 
Jj'TTCt ^ 
47r« 2/3„ coBnx ,n o t ^ o i 
-^^ a (A cos 2x + . . . + A cos nx + 
+ 2 (% — o-A A+i) (log + 
loo--1 
^ a 
2 log _ _ 1 
^ (t 
+ N (in Ai +11 COS X + ^ o" 
a 
8c 
cr COS X 
4 
— A — B (cosy + ^ 
Since Va — Vj = 0, we find 
Therefore 
A I p ^ /T m 2L + 1 ^ p p 1 
A + B, — (L — 1)S^ — ^ cr 2 Aj Ai+i 
4 
B 
L - 1 
4 
cr 
B 2 
' Hn 
y 
j 
_Zi_ 
27rcd 
T L 1 L 1 . X 
+ 
/I IH+i 1 E«-A . 
— ■ -r COS In 
!i+l r.i - 1 — 1 / V 
n a 
lH + 1 
n (71 + 1) 
n — 1 a 
COS (n + 1) X 
i)x 
J_ /T 11 ^ 2L + 1 p p 
+ (B ~ 1 ) S ^ — , O- X Pn p,j + 1 
L - 1 
10 
0-" S Ar 
+ 
L - 1 ^ ^ , /'R . (T W 
o-SAn-cosx + ^;^) . . 
(15.) 
