338 ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 



22. Applying this result to the case of a ring, and putting s for x, and 



<r for f we have 



(r = A cos MS, and T= AirkA cos w, 



so that -RN=47rk, 



P- 



when 7/i is very large, and this is the greatest value of N. 



The value of L has little effect on the condition of stability. If L and 

 J/ are both neglected, that condition is 



o>'>27-856 (2irk) .............................. (78), 



and if L be as much as $N, then 



o> 3 >25-649 (2irk) .............................. (79), 



so that it is not important whether we calculate the value of L or not. 



The condition of stability is, that the average density must not exceed a 

 certain value. Let us ascertain the relation between the maximum density of 

 the ring and that of the planet. 



Let 6 be the radius of the planet, that of the ring being unity, then the 

 mass of Saturn is ^nb'k' = CD* if k' be the density of the planet. If we assume 

 that the radius of the ring is twice that of the planet, as Laplace has done, 

 then & = and 



= 334-2 to 3077 .............................. (80), 



so that the density of the ring cannot exceed -g-J^ of that of the planet. Now 

 Laplace has shewn that if the outer and inner parts of the ring have the same 

 angular velocity, the ring will not hold together if the ratio of the density of 

 the planet to that of the ring exceeds 1*3, so that in the first place, our ring 

 cannot have uniform angular velocity, and in the second place, Laplace's ring 

 cannot preserve its form, if it is composed of loose materials acting on each 

 other only by the attraction of gravitation, and moving with the same angular 

 velocity throughout. 



23. On the forces arising from inequalities of thickness in a thin stratum 

 of fluid of indefinite extent. 



The forces which act on any portion of a continuous fluid are of two kinds, 

 the pressures of contiguous portions of fluid, and the attractions of all portions of 

 the fluid whether near or distant. In the case of a thin stratum of fluid, not 



