1042 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Section III. is devoted to Saturn’s rings. In Laplace’s proof Mec. Cel.,’ Book 3 , 
c. 6 ) that the rings are not continuous fluid, he assumes the attraction of a ring of 
elliptic cross-section on a point at the surface to be the same as that of an elliptic 
cylinder. Mine. Kowalewski, in her memoir on the ring of Saturn (‘ Astronomische 
Nachrichten,’ No. 2643, vol. Ill, 1885) uses a method which applies only to rings of 
nearly circular section. Here J have attempted to find the potential of a ring of 
elliptic cross-section. Ai^plied to Saturn, the results obtained agree fairly with 
Laplace’s. 
In Section IV. the steady motion of a single vortex-ring of finite cross-section is 
discussed. If m be its strength, c its mean radius, and its cross-section be given by 
Pv = a { 1 + ^3 cos 2 x + ^3 cos 3x + cos 4x + . . .}, 
it is shown that ySo, ^ 83 , . . . are of the 2nd, 3rd, &c., orders in ctjc, and their values 
are found as far as (a/c)"^. 
The velocity of the ring 
m 
... r. 8c 
= log ■■ 
lire ® f 
a 
JL 
4 
12 log— — 15 
*= a 
32 
12L - 17 
32 
The results agree with those given by Mr. Hicks, obtained by means of Toroidal 
Functions. (‘Phil. Trans.,’ 1884-1885.) 
In Section V. the motion of a number of fine vortex rinofs on the same axis is 
o 
discussed. Equations are obtained giving the forward velocity and the rate of 
increase of the radius of each ring. Let wq be the strength, the mean radius, 
(Xj the radius of the cross-section, and the distance of the centre along the axis of 2 : 
for one of the rings. 
It is shown that the kinetic energy of the system is given by 
T = 8si'l“(log^;' 
, c,c., cos d) (16 
+ niurn.y , "o —L; 
'o\/{(% - A)" + V - 
cos (/) -f q~} 
The equations of motion are 
— JUjCj^Cj = 
Sir 0c 
Stt 0q J 
The momentum integral takes the simple form 
Y (vn jC^") = const. 
