374 



Mr. F. W. Dyson. 



[Apr. 20, 



2 7rafi n 



A distribution of matter on the ring of surface density /3 W cos w% 

 gives on the ring the potential V where 



cos%x_j_ a I" cos (?a -j- 1) x cos(w— ])xl 

 n 4ra \ n + 1 n—1 J 



a 2 f 2% + 3 . t . 2n— 3 . . 1 



H i t r~> r cos (n+2 v — — r- (cos n — 2) y > 



16 w I + (» + 2) ^ ^ A (>— 1) (n— 2) V ;X J 



+ &c. 



3. The stability of an annulus of rotating fluid is considered for three 

 kinds of disturbances : fluted, in which case the ring remains symme- 

 trical about its axis, but its cross-section ceases to be circular ; twisted, 

 in which case the central circle of the ring is deformed, but the cross- 

 section is undisturbed ; beaded, in which case the central line remains 

 circular, and the cross-section is a circle, but one of varying radius. 

 It is proved that when the cross- section is small compared with the 

 radius the annular form is stable for fluted and twisted disturbances, 

 but is broken up by beaded waves. 



4. In Laplace's proof that Saturn's rings cannot be continuous 

 fluid rotating in relative equilibrium, he assumes that the attraction 

 of the ring at a point on the surface is the same as that of a cylinder. 

 Madame Kowalewski, in her memoir, uses a method which applies 

 only to rings of nearly circular cross-section. Here I have found 

 the potential of a ring of elliptic cross- section. The results are 

 complicated. For a very flat ring of mass M of mean radius c, and 

 whose cross-section has a semi-major axis a, the exhaustion of 

 potential energy is if (a/c) 2 is neglected 



2ttc \ & a A ] 



As applied to Saturn, the result obtained is that for the ring to be 

 continuous fluid its density would have to be about 100 times the 

 density of the planet. 



5. Let m be the strength and c the mean radius of a vortex ring. 

 Let its cross-section be given by 



R = a{l + Acos2x + y8 3 cos3x + Acos4x-r- }• 



Then /3 2 , /3 4 , &c, are of the 2nd, 3rd, 4th.. . . . orders in ajc. Their 

 values are obtained as far as (a/c) 4 . 

 The velocity of the ring is found to be 



