1056 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
It is necessary to know U as far as the second order of the small quantities 
ySg, &c., Suppose V; is obtained in the form Vq + S'lL cos where are 
functions of R. Then it is necessary to know Vq and to the second order, but 
V 2 . . . Vn . . . only to the first order. 
Now 
27ra~ 
h , 8^’ , 1 L /log8c/a - 1 k Pd\ 
:;3 = log:; +i(i -:5) + <^(-5-7“iT*) ^ 
/ 
R\'* cos ny , a 
' +4 
n 
1 
n a 
+1 
n — 1 cv 
p«-i\ 
—1 cos {n-\)x 
. 1 , . .X 11 
H-;-—7T cos \n + 1 ) y 1 
n(^i + 1) '' 
+ A + B 
/E , (7 E^ 
- cos X + 7 — 
\a 4 «- 
where A and B are small quantities of the second order in . . . /8„ . . . , which must 
be found by comparing the value Vy at the surface of the ring with the value of Vq 
there. 
§11. We shall now find Vq the potential of the ring at an external point. 
Let Q be an external point whose coordinates are 7’ and 6. 
Let P be a ])oint on the axis, C be the centre of gravity of the cross-section. Let 
PCO = a and let CP = 7\. 
The potential at P is 
2. rr 
0 J 0 
iTT 
(c — E COS x) E f?E dy 
\/{rd + IP — 2Erj cos (% — «)} 
"I + cos(x- «)+ ;r3 7 ^2 [cos (x - a)] - |-cosx 
cp^ 
cos 
X cos (x - a) . . . I dx- 
