For the outer ring, 



ON THE RINGS OF SATURN. 119 



/= 0.0012 / = 0.0081 



r' — r<^ 0.0066 computed. 

 r' — r = 0.1675 assumed. 



As no change of mass or density within the limits of probability will account for so 

 large differences, we must therefore still further reduce the width of the rino^s. 



By trying different values, it will be found necessary to diminish r' — /• so far that 

 the intervals occupy nearly as much area as the reflecting surface, which cannot be ad- 

 mitted, for reasons before given. 



_ We will take r' — a = 0.02, which corresponds to eleven equal rings distant from 

 each other by 0.01. 

 For the outside ring, 



< = 0.59 /> 0.0023 /> 0.0202 



y = — 0.0036, tendency is from the surface. 

 / — 0.0144 



r' — r<i 0.0097 computed, 

 r' — r = 0.0200 assumed. 

 For the middle ring, 



/> 0.0172 /> 0.0205 / = 0.46 



/=: 0.0046 / = 0.0095 



r' — »• < 0.0064 computed. 

 r" — r = 0.0200 assumed. 

 For the inner ring, 



/> 0.0415 /> 0.0288 < = 0.34 



/= .0113 / = — 0.0004 



r' — r <:^ 0.0031 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. By recomputing / and f for the inner ring 

 with a density = 3, we obtain, 



/= +0.0263 / = +0.0091 



r' — r< 0.0101 

 r — r — 0.0200 



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 

 b in the same proportion. But the attraction of a thin and narrow ring upon a particle 

 at the extremity of its major axis varies nearly as 6 X density. Mkanique Celeste, 

 Vol. II. [2095]. Therefore / is not increased when we increase the density by dimin- 

 ishing b. 



