MR. F. W. DYSON ON THt: POTENTIAL OP AN ANCHOR RING. 1069 
As in § 11, we find that at the surface of the ring, 
Vo - V,- = 27ra2 2 —log - + f 2 {a„, sin ncf> + /3, cos n(f,y irr 
w cc ^ 
+ « 2 [a,, sin ncf) + /3„ cos 7i(j)) [— Ittw 2 (a« sin n(j) + /8« cos n(/))] — C. 
Since Vq and V^- are equal at the surface,^ 
C = 27ra3fW -^- - 
Now 
(32). 
u= i fv^c/m 
+ (log - 1) 2 — 1- (c - Pv cos x) p c^P dx d(f> 
= 27r%2c [ j(log ^ 1 ^ - £-3 + 2 2 [log ^ — 2/(n)] (a«sin n(fi + cos ncfi) 
+ (log V — 1^ ' 
a 
See- 
o\d<l> 
= 27r'V*^c {log - + i + [log — l) 2 {ay + ^y) 
\ CL ' Oj j 
+ (2 1og^' - 4/(«))2(a,/+ft^)|. 
Let CTo be the mean value of a. 
Then 
§ 20, When the ring is thin and n is small the change in the radius of the cross- 
section is slow, and an approximation to the disturbed motion is easily found by the 
method of parallel sections. 
TTpH = const. = C 
