104 G, P. Bond on the Rings of Saturn. 



For the middle ring, 



/>00173 /'>0-0205 i=0'46 



/= 00046 /^ = 00075 



r'-?'<0'0U64 computed. 



r'-r =00200 assumed. 



For the inner ring, 



/>0'0415 J'y 00288 f=034 



/= 0113 /'=-00004 



r' — r< 00031 computed. 



r'-r = 0*0200 assumed. 



In order to preserve the mass as previously adopted, we must 

 suppose an average density about three times that of Saturn 



we obtain, 



/ and / 



/= +00263 /= +00091 



r'-r<0010l 

 r'-r = 0200 -^ 



3, 



A density six times that of Saturn would just suffice to retain 

 the particles on the surface of the inner ring. To effect this 

 without changing the mass, we must diminish h in the same pro- 

 portion. But the attraction of a thin and narrow ring upon a 

 particle at the extremity of its major axis varies nearly as h x den- 

 sity. Mecanique Celeste^ vol. ii, [2095]. Therefore / is not 

 increased when we increase the density by diminishing b. 



If a further diminution of width is attempted, a difficulty is 

 encountered in the width of the intervals. 



In the last case, suppose the area occupied by the intervals is 

 already double the limit previously assigned. If we lessen the 

 space occupied by the intervals, by bringing the adjacent rings 

 nearer together,/ decreases instead of increases. 



But there are still stronger objections to a large number of 

 small rings near to each other. 



It is known in the case of a single ring, that, if it were per- 

 fectly uniform in every part of its circumference, the slightest 

 exterior disturbance would precipitate it upon the body of the 

 planet. To avoid this catastrophe, we must suppose each ring 

 to be an irregular solid, its center of gravity not coinciding with 

 its center of figure, but having a motion of rotation about the 

 body of Saturn. In addition to this, a number of regular con- 

 centric rings are in a position of unstable equilibrium, by virtue 

 of their own mutual attractions. The slightest inequality in the 

 intervals would have the effect of'throwing the whole system 

 into confusion. 



Let us suppose, for instance, that the inner ring deviate by 

 ever so small an amount from an exact central position with ref- 



