36 NEWTON S PRINCIPIA. 



T) 2 2 



ratio of V 2 : v 2 , its equal the ratio of =- 2 : -^ , F : f : : 



~2 : ~2*j r the centripetal forces are directly as the 



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 : f : : =- : ; that is, 



E T 



:: :-, or inversely as the distances. In like man- 



Jbv T 



ner if the periodic times are in proportion to any power 

 n, of the distance, or T : t :: E&quot; : r&quot;, we shall have 



T 2 : P :: E 2 : r^ and F : / :: ^ : ~ n \ that is 



: : M _ t : ^n-i 5 an d conversely if the centripetal force 



is in the inverse ratio of the (2n l) th power of the dis 

 tance, the periodic time is as the n ih power of that dis 

 tance. Likewise, as the velocities of the bodies in their 



orbits or V :::: if we make T : t :: E w : r, 



T) 11 



then V : v :: g^ : -, or :: j^ : l . Thus, sup- 



o 



pose n is equal to ^ we have for the velocities V : v 

 :: ... _.. : - t or they are in the inverse subduplicate pro 



portion of the distances; and for the centripetal forces we 

 have F : f :: jpi * -3^1 :: t?2 : &quot;2? or the attraction 

 to the centre is inversely as the square of the distance. 



