MR. F. W. DTSON ON THE POTENTIAL OF AN ANCHOR RING. 1063 
2T = 77(^^ [ [ (c “ R COS v) 3R dR dy 
•'o •’0 
= (I + I cr" + f cr^/S^).(22), 
where ^ is the undisturbed angular velocity of the fluid. 
Therefore 
2T = M I 
dll 
9o-2\ . , /3d — Cr^ndi+\ 
16 
+ N 
n 
+ (1 +1 o-'^ + f r • (^^)' 
§ 15. Also 
U = 
7 
27rc 
1 I 1 a f 1 5 
log - + . _ &c. - — log - - A 
Aj;_1 o 2 (oh + l)(?i 1) ^ Q 
n 
2n(n + 1 ) 
(24), 
where a and c now denote the mean radii of the cross-section and of the ring respec¬ 
tively ; o- = ajc, and is small ; y is the constant of gravitation. 
Retain only the terms of the highest order; Lagrange’s equations give 
^ 0 /3a , M“ H — 1 ^ 
Ma- - + 7 = 0. 
n Ittc ' y.r 
in 
Thus the period of the oscillation of the type A» cos ny is given approximately by 
A + TT : 5 -— yAj = d. 
(25). 
V Stt 
-, and the time a fluted wave of 
{n — 1 ) 7 
this type would take to travel round the ring is 
n 
_8^ 
(h — 1) 7 
[The equation giving the period of the disturbance of type /3o cos 2y is 
u\aA-i ^ + q (log q - *)} = 0, 
Therefore the value of in the undisturbed motion is given by 
