236 OUTLINES OF NATURAL PHILOSOPHY. 



c. The line NS, (fig. 24.) through which a body fall- 

 ing toward S will have, at any distance from S, the 

 same velocity that the revolving body has at that 

 same distance, is equal to the transverse axis of the 

 ellipsis ; and the velocity of the revolving body at 

 D, its mean distance, is equal to that which is ac- 

 quired by falling from N to K, the middle point 

 between N and S. If d be a given distance from 

 S 9 and/ the centripetal force at that distance, be- 



eause - 



But a F = c% therefore d*f= a c* and c = ^L-, 



a* 



242. If bodies describe different ellipses about 

 a centre to which they are urged by centripetal 

 forces in the inverse ratio of the squares of 

 the distances, the squares of the times of re- 

 volution will be as the cubes of the mean dis- 

 tances* 



df k 



The velocity c, at the mean distance a, is J . . , and 



a * 



therefore the sector described in a second by the 

 radius vector, when the body is at its mean dis- 

 tance, is = ~ y and as oft as this is con- 

 2a* 



tained 



