1062 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Therefore 
= — ca 
cos 
X 
a IH 
Ac 
1 • • Pi" 
+ - S («A< + «A-) — cos nx- 
The term in may be omitted. 
It is easily seen, by § 4, that a more approximate solution is obtained by assuming 
(p 
• . fP ‘a P“ , a Pd 
— «cAi 1 - cos X + r ~ ^ "T X 
^ ^ Ac or 32c cd ^ 
P'^ n P"'*"^ 
+ SA.-|-cos»x + l)x 
, . ( 20 ). 
The constants are easily found to be 
Ai 
A. 
o -2 
OC7" 
32 
o /A, {n + 2) 
n An (% + 1) 
cAi+i 
At the surface of the ring, therefore, 
4 > = 
— ac\ 
ac I cos X ( 1 
^) + J } + «’2 (f - nx + J cos (» - 1) X 
= — ac 1 cos X (1 
— ) + - 
16 / ^ 4 
+ ^ 
'A. 
n 
'C^n +1 
'n{n + 1); 
cos nx 
and 
• / , o'\ , ^ n 
— = - C (cos X + ^ ) + Cl S A cos nx. 
The kinetic energy is given by 
2T = 
= 27rac I 4’ ^ (1 — O- COS x) f?x 
27rVo .| ci= (1 _ 
O' /3,i /3n +1 
n 
■ ■ (21). 
To this must be added the kinetic energy of the undisturbed motion, which is 
given by 
