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G. P. Bond on the Rings of Saturn. 101 



If we adopt for unity the radius of the outer edge of the outer 

 ring, we have from observation the thickness of the ring =2b <^5^ o* 

 Let r and r' be the radii of the inner and outer edgesj and i the 

 interval between two adjacent rings, 



r-\-r' ^ 



Any intervals permanently existing so large as one half, or even 

 one third, of that usually seen, could not escape observation. 

 Moreover, if the subdivisions are numerous, the width of the 

 mtervals must be proportionally diminished, because the whole 

 area occupied by them goes to diminish the amount of light re- 

 flected, and to increase the density of each ring, both of which 

 are already large. The light of the ring being sensibly brighter 

 than that from an equal area on the ball, it is not probable 

 that any considerable part of the light of the sun is transmitted 

 through intervals. And to preserve the same mass, if the in- 

 tervals are large, the matter must be compressed, as it is not 

 allowable to give a thickness greater than is indicated by obser- 

 vation. To avoid the hypothesis of a reflective power, and a 

 density greater than we are warranted in assuming, we must, 

 therefore, consider the intervals to be very narrow^ We may 

 take, then, the width of all but the known interval as certainly 

 less than 001, which is one half of the width of the known 

 interval. From the blackness of the shadow of the ring upon 

 the ball, which would be diminished in intensity were a consid- 

 erable part of the sun's rays transmitted, we may 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>0-04. 



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

 ^sume 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. 



If/ 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 sura 

 of the attractions of all the rings, / the same force for the inner 

 ^dge, s the mass of Saturn^ and t the time of revolution of any 



kr 

 ^^^g m days, the centrifugal force at the distance r will be =^— i 



log. A:=9a20r. 



Then, in order that the particle should remain on the surface, 

 ^e must have 



Therefore, 





r ^ 



^'"r<^(/+/> 



