92 
MR. P. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
23. At the surface of the ring 
VH- 
is constant. Therefore 
V + " (c-Kcosx)^ 
is constant at the surface of the ring, or 
V 
CO'^C^ 
+ -scoaxf 
27r«^ 
is constant, when 
Let 
be called f{s). 
Then 
5 = o- (1 + ^3 cos 2x d- ^3 cos 3x + ^4 cos 4x). 
n o 
M-'C 
i+ -scosxf 
2'7Ta^ 47ra‘ 
/W =/(^) +/ (o-) o- (A cos 2x +...) + 4/"(<^) o-"- A=L^!iix . 
Now 
and 
1 
1 . CO^C'^ 
/'(s) - A, ( - I + I«) - A,<r £ + 
+ COS X i Aj - — 
I A„a arc- 
§2 ■ 27ra2 r X 
+ |cos 2x \ A^ 
6Z + 1 2AgO-' , eo®C“ ^ j 
16 
^1 
o o 
CO^C' 
r (‘I = ? + S + ^ - 2x. 
8 — 
o o 
co-^C" 
Therefore, (jf (a) (/Sg cos 2 x + &c,) 
0 o 
0 )"^'" <J' 
= (A^.»-A3 + —."-lA 
A 
H ” 2 
+ ( 7 O' ~ 
+ 
Ao 
47ra^ 
3 o- /Sg cos X 
A 
A^ ( — 1 + - cr^ 
Ac, 
7“' +w 
+ Ai 
A 
4 
cr 
o o 
CO'^C'^ 
+ 1 1 - A - ^ - Ai ^3 ^ cos 3x 
4 “ 
CA+ 1 
Ag 
9 
A„ 
2 0 
47rffl" 
(T 
/^g i cos 2x 
■lira" 
2 O' ) - Ai ^4 i COS 4x, 
