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G. P. Bond on the Rings of Saturn. 103 



. B'lt f+f will be changed very nearly in the direct ratio of the 

 different densities when the rings are narrow. 



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



r=0-665 r' = 1000 



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 



^ t^n^e of rotation <=0-43. The excess of Saturn'' s attraction 



' I 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 there- 

 fore have, 



/>0-37 and /'>0-33 



But /= 0-0040 and /'= 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 /'=00042 

 r'-r =0-1675 r'-r computed =00036 



For the outer ring, 



/= 0-00 12 /'= 0-0081 



r'-r< 00066 computed, 

 r'— r=0*1675 assumed. 



As no change of mass or density within the limits of proba- 

 bility will account for so large differences, we must therefore still 

 lurther reduce the width of the rings. 



, By trying different values, it will be found necessary to dimin- 

 ish r'-r so far, that the intervals occupy nearly as much area as 

 the reflecting surface, which cannot be admitted, for reasons 

 Delore given. 



We will take r'-r=0-02, which corresponds to eleven equal 

 rings distant from each other by 001. 



For the outside ring, 



<=0-59 /> 0-0023 /'> 00202 



/ = - 0-0036, tendency is from the surface. 

 /'= 0-0144 



r'-r< 00097 computed. 

 r'-r= 00200 assumed. 



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