ON THE RINGS OF SATURN. 117 



we may then infer that the intervals, which reflect no light at all, cannot occupy an area 

 so large as one fourth of the average breadth of the rings ; that is, r' — r > 0.04. 



The above are very liberal allowances, but it is important to assume the intervals 

 as large as possible, so as to diminish the chances of a collision, which at best is almost 

 inevitable. 



We come now to consider the forces acting on the rings. 



Lety^ be the force with which a particle at the outer extremity of the major axis of a 

 ring is attracted to its surface by the sum of the attractions of all the rings,/ the same 

 force for the inner edge, s the mass of Saturn, and t the time of revolution of any ring in 

 days, the centrifugal force at the distance r will be = -s-j log. k = 9.1207. 



Then, in order that the particle should remain on the surface, we must have 



J ^ ^ — i^' J -^ f — r^- 

 Therefore, 



If we put F ^ -^ = the attraction of Saturn on the middle of any ring, we obtain 



the relation, 



f+f -. „ r' — r 

 F ^ n 

 From the smallness of the mass of the ring, as well as from its unfavorable distribu- 

 tion, it is easy to see that r' — r must be very small compared with r^. 



To obtain /and /, I have computed from Laplace's formula the following values : 

 / is the attraction of a single ring upon a particle on its surface, at the extremity of the 

 major axis of its base ; / and/ are the attractions of the two next adjacent. The inter- 

 val between = 0.01, 2 6 = -gw- The radius of the outer edge of the outer ring being ^ 1. 



Attractions of Three Narrow Rings. 



/o 



/. /. 



/o /o 



a = 0.01 -f0.00661 —0.284 -f0.134 



.02 679 .393 .150 



.03 685 .460 .154 



.04 689 .507 .157 



.09 691 .543 .160 



.06 692 .571 .162 



.07 693 .594 .164 



.08 693 .614 .165 



.09 694 .631 .166 



.10 +0.00694 —0.646 -f 0.166 

 VOL, v. NEW SERIES. 17 



