MR. P. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
1073 
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M = III {c — X — ^') clx dz d(j)' 
= 2n^a^c. 
§ 23. The exhaustion of potential energy is therefore diminished when the ring 
is displaced so that the central circle does not remain circular. For this kind of 
disturbance therefore, the ring is stable, even when the fluid does not rotate. The 
effect of rotation is to increase tliis stability. 
We have therefore the following results. 
The annular form of equilibrium of rotating gravitating fluid is stable for disturbances 
symmetrical about the axis, and for disturbances which alter the shape of the central 
curve, but is unstable for long beaded disturbances. 
This result was, perhaps, to be expected, as by means of beaded waves, the mass 
would naturally be broken up into spheroidal masses. 
Section III. 
§ 24. The methods given in my paper, ante, may be used to find the potential of 
any ring whose cross-section does not deviate far from a circle. They will not, 
however, apply to a ring whose cross-section is very elliptic, any more than the 
potential of an elliptic cylinder can be obtained as an approximation from one 
whose section is circular. In this section, the potential of a ring of elliptic cross- 
section is obtained by taking the known result for an elliptic cylinder as a first 
approximation. The value of the potential obtained applies only to points not far 
from the surface of the ring. The potential at internal points may be derived from 
this, while the potential at other points may be found easily by other methods. 
Consider a ring, whose cross-section is elliptic, the major axis of the ellipse being 
perpendicular to the axis of the ring. 
Let the figure represent a section through the axis of the ring. 
MDCCCXCIJL—A. 
d A^c J 
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