344 ON THE STABILITY OP THE MOTION OF SATURN'S RINGS. 



The quantity 2mc (2mc - c"* 1 " 1), 



has a minimum when 27nc='607 (94), 



and the wave-length is 10*353 times the thickness of the stratum. 



In this case 2roc (2mc '""" 1)= -'509 (95), 



and X= f 50QirkA cos mx (96). 



24. Let us now conceive that the stratum of fluid, instead of being infinite 

 in extent, is limited in breadth to about 100 times the thickness. The pressures 

 and attractions will not be much altered by this removal of a distant part of 

 the stratum. Let us also suppose that this thin but broad strip is bent round 

 in its own plane into a circular ring whose radius is more than ten times the 

 breadth of the strip, and that the waves, instead of being exactly parallel to 

 each other, have their ridges in the direction of radii of the ring. We shall 

 then have transformed our stratum into one of Saturn's Rings, if we suppose 

 those rings to be liquid, and that a considerable breadth of the ring has the 

 same angular velocity. 



Let us now investigate the conditions of stability by putting 



x = - 2irkmc(2mc - e" Jme - 1) 



into the equation for n. We know that x must lie between and - to 



1 o J 



ensure stability. Now the greatest value of x in the fluid stratum is 509rr/:. 

 Taking Laplace's ratio of the diameter of the ring to that of the planet, this 

 gives 42 -5 as the minimum value of the density of the planet divided by that 

 of the fluid of the ring. 



Now Laplace has shewn that any value of this ratio greater than 1'3 is 

 inconsistent with the rotation of any considerable breadth of the fluid at the 

 same angular velocity, so that our hypothesis of a broad ring with uniform 

 velocity is untenable. 



But the stability of such a ring is impossible for another reason, namely, 

 that for waves in which 2wc> T147, x is negative, and the ring will be destroyed 

 by these short waves in the manner described at page (333). 



When the fluid ring is treated, not as a broad strip, but as a filament of 

 circular or elliptic section, the mathematical difficulties are very much increased, 



