[ lOil 1 
XX. The Potential of an Anchor Ring .—Part IT. 
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By F. W. DysoN; M.A., Fellow oj Trinity College, Cambridge, Isaac Newton Student 
in the University of Cambridge. 
Communicated by Professor J. J. Thomson, F.R.S. 
Received March 16,—Read April 20, 1893. 
Introduction. 
This paper is a continuation of that at pp. 43-95 supra, on “ The Potential of 
an Anchor Ping.” In that paper the potential of an anchor ring was found at 
all external points; in this, its value is determined at internal points. The annular 
form of rotating gravitating fluid was also discussed in that paper ; here the 
stability of such a ring is considered. In addition, the potential of a ring whose 
cross-section is elliptic, being of interest in connection with Saturn, is obtained. The 
similarity of the methods employed, as well as of the analysis, has led me to give in 
this paper also a determination of the steady motion of a single vortex-ring in an 
infinite fluid, and of several fine vortex rings on the same axis. 
In Section I. solutions of Laplace’s equation applicable to space inside an anchor 
ring are obtained. These results are applied to obtain the potential of a solid ring at 
internal points, and also of a distribution of matter on the surface of the ring. The 
work done in collecting the ring from infinity is obtained. 
In Section IT. the stability of an annulus of rotating gravitating fluid is considered 
for three kinds of disturbances. 
(1) Fluted: i.e., those in which the ring remains symmetrical about its axis,''" but 
the cross-section is deformed. 
(2) Twisted: i.e., those in which the cross-section remains circular, but the 
circular axis of the ring is deformed. 
(3) Beaded: i.e., those in which the circular axis of the ring is undisturbed, but 
the cross-section is a circle of variable radius. 
The ring is found to be stable for fluted and twisted waves, but is broken up by 
long beaded waves. 
* Axis of the ring throughout the paper means the axis of revolution; 
is called the circular axis. 
the central circle of the ring 
G R 
MDCCCXCIII.—A. 
30.12.93 
