LAW OF THE UNIVERSE. 231 



V 2 V* 



=- : -. Now as in the circle V and E, v and r are both 

 E r 



constant quantities, the centripetal force is itself constant, 

 which retains a body by deflecting it towards the centre of 

 the circle. 



Secondly. The times in which the whole circles are described 

 (called the periodic times) are as the total circumferences or 

 peripheries ; T : t : : P : p. But the peripheries are as the 



P p 

 radii or : : E : r. Therefore T : t : : E : r ; also V :t :: = :-, 



JL t 



E r 



therefore inversely as the radii, or T : t : : -^ : -, and V s : 



V v 



E 2 r 2 V 2 v 2 



v 8 :::. But the centripetal forces F :/::=-: ; sub- 



_L 6 1 1 7* 



E 2 r 2 

 stituting for the ratio of V* : v 9 , its equal the ratio of : , 



E r 

 F :/::: ; or the centripetal forces are directly as the 



J. v 



distances and inversely as the squares of the periodic times ; 

 the forces being as the distances if the times are equal ; and 

 the times being equal if the forces are as the distances. It 

 also follows that if the periodic times are as the distances, 



then F : / : : - t : ; that is, :: :-, or inversely as the 



it T Jet T 



distances. In like manner if the periodic times are in pro- 

 portion to any power n, of the distance, or T : t : : E" : r", we 



shall have T 2 : F : : E 2 " : " and F : / : : ~ : -J- ; that is 



: : ; 2^i : Is^I ' an( ^ conversely if the centripetal force is in 



the inverse ratio of the (2 n l) tb power of the distance, the 

 periodic time is as the n th power of that distance. Likewise, 



E r 



as the velocities of the bodies in their orbits or V : v : : -= : -, 



i 6 



