354 PROCEEDINGS OF THE AMERICAN ACADEMY 



When the point is very near the inner or outer circumference of the 

 ring, it is necessary to substitute for c-,„ the more exact value, 



in which 2 b = the thickness of the ring. 

 If we put 



cos t„ = 4 TT — ?,„ = ' -^~^rr ■ — = «'« 



' -r Pm 



this value gives, 

 cos J 



i^^c,,, = log -. 



"'m 



When the attracted point is near the plane of the ring, the attraction 

 parallel to this plane is given by the preceding formula, and that per- 

 pendicular to the ring is given by the formula, 



in which 



V — — — F^c ^ E^ c 



' + P ' — P 



and z = the distance of the attracted point from the plane of the ring. 



It appears from these formulae, that, if Saturn's ring were one solid 

 ring of uniform thickness, its tenacity must be sufficient to sustain, in 

 the form of a wire, on the surface of the earth, a weight equal to six 

 thousand miles of its own length ; that is, it must be six hundred times 

 stronger than the strongest iron wire. The demand for a strength 

 which so immensely surpasses all experience, is a powerful argument 

 against this constitution of the solid ring. 



If the ring were subdivided into smaller rings, and if the plane 

 of either of the secondary rings were not to pass through the centre 

 of Saturn, this ring would vibrate back and forth perpendicular to 

 its plane, and the whole time of oscillation would be the same as 

 that of its revolution about the primary. The different rings would 

 consequently have different times of vibration, so that they must con- 

 stantly be in opposite phases of vibration. The average extent of 

 vibration for all the rings could not then be materially different from 

 the average apparent thickness of the whole ring. 



