118 ON THE RINGS OF SATURN. 



The attraction of the whole system, considering its mass to be uniformly distributed, 



I have next computed by quadratures. Breadth of whole system = 0.335. Radius of 

 outer edge = 1. 



Distance of particle within the oiter edge = 0.0075 Attraction = -\-4.o2 X masss of ring. 



.< u « u 0474 " 2.42 



" " «« " .0875 " 1.70 



" « " " .1275 " 1.16 



« " " " .1675 " 0.61 



" " " " .2075 " +0.04 



« " " " .2475 " —0.52 



« » " " .2875 " 1.32 



«« " " » .3275 " —3.53 



These two tables give the means of finding / and f with sufficient exactness. For 

 Saturn we have 



s = t^ (:^^y-^ '°S- « = 9-5367 ; log. mass of ring = 7.4848. 



The density of a ring, for/o,/„ and/^, is assumed = Saturn's, unless it be otherwise 

 stated. A change in the density affects only that part of the ring's attraction depending 

 on fo,fi, and f^. But/ + y will be changed very nearly in the direct ratio of the differ- 

 en densities when the rings are narrow. 



We will first suppose the case of but one ring without division. 



r = 0.665 

 r' = 1.000 

 r„ = 0.8325 

 r' — r = 0.335 



Upon a particle at a distance within the outer edge = 0.21, the attraction of the 

 whole ring becomes = 0. This gives for the time of rotation t = 0.43. The excess 

 of Saturn's attraction over the centrifugal force at the inner edge = 0.37. At the outer 

 edge the centrifugal force is in excess by 0.33. We must therefore have, — 



/>0.37 and /> 0.33 



But/ = 0.0040 and / = 0.0070 



Assumed value of r' — r = 0.335 

 Required " < 0.0058 



If there be but one ring, it will be necessary to increase its attractive force by sixty 

 times its probable value, in order to retain its particles on its surface. 



With a single division into two equal rings, we have for the inner of the two, giving 

 such a time of rotation as will retain particles on the middle from leaving their place, 

 « = 0.39 r= 0.665 />0.25 / > 0.19 



r' — 0.8325 /= 0.0050 / = 0.0042 



r' — r— 0.1675 r' — r computed = 0.0036 



