MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
1055 
Adding to this the value given in (9) for the potential inside the ring 11 = o, we 
have correct to the first power of the potential inside R = a (1 + ySg cos 2;^). 
The terms in of the exhaustion of potential energy of this ring are given by 
^a(l -r 
277%^ \ \ j or 
Jo Jo 
»2ir /»a(l +^2COS2 x) 
^L-1 E 
E-n 
2 a 
+ 0-^“ 
+ 2^a%f‘'[{ j(A _ fi cos X + 3°; 
P2 p4" 
_3_ /T _ __^ _ 
16 rl ^^2 9 6 ^^4 
COS 2y I (c — R COS y) R cZR f/y 
E^ 
32 V 2 cP 
— 2 — — 2L I Uc — R COS y) R dR c/y, 
all the other terms vanishing on integration. 
Therefore this part of the exhaustion of energy 
= 277^cdc^ 
J 0 
4L - 5 
“TlT" 
+ A (L — 4 ) ~ ImT cos^ 2 y + 2^ cos^ y — - 3-0 (L d" 1 (/y 
= - 27rVc yS^ I (L - 1^,-).(13). 
July 22, 1893.] 
Section II. 
§ 10. An annular form is possible for gravitating fluid rotating round an axis in 
relative equilibrium. When the cross-section of the annulus is small compared with 
its radius, the cross-section is nearly circular. 
Let the ring be disturbed so that the cross-section takes the form given by 
the equation 
= a [ 1 -f ^.3 cos 2y + . . . + cos ny + . . .}. 
We shall prove that the ring is stable for disturbances of this kind, and find the 
periods of the various oscillations. 
It is necessary to find the exhaustion of potential energy. Let this be called U. 
Then 
= 77 
Vy (c — R cos y) R c/R dx 
where Vy is the potential at an internal point. 
