1070 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Therefore 
2T = 
J 0 
2 cC 2 
3 <- 
a 
2 cC 2 
1 + f^(a/ + A.')} 
— ^ ( 1 + 2 S (a.,f -f ^,f') } 
a, 
0 
A 2 
— ^2 {1 + 2 S (a/ + 
Let A„ be the component of momentum corresponding to the coordinate a„. Then 
the whole kinetic may be expressed in the form 
W 
2Mc 
3 [ 1 + 2 S + Ai^)} + k A/ . 
• (34), 
where the terms in the summation are necessarily positive. Thus 
T _ U = i 5 :aa/ + 1 + 2 s (a/ + Ac)]. 
2M3 
-=';7[l^H-i + 2X[L-2/00-i](«/ + ;8/)} . . (35). 
Hamilton’s equations will give 
|F.A4 + -^«„-^“hL-2/(«)-U«» = 0 .... (36). 
The steady motion will be stable or unstable as 
2M3 
- ~^[L - 2 f{n) - i] IS > or < 0. 
“ [L - 4/(n) + f] is > or < 0. 
L-4/(») + i = log^"-3i. 
"O 
When c > 3«q, this is positive : therefore for any ring whose cross-section is smaller 
than this, beaded waves cause instability. 
§ 21. Next consider a disturbance which leaves the cross-section of the ring by any 
plane through the axis an exact circle of radius a, but the centre of the section is 
2h^ 
That is, according as 
Now, when n = 1 
