NEWTON S PRINCIPIA. 



35 



portion of the radii ; therefore the centripetal forces are di 

 rectly as the squares of the arcs, and inversely as the radii. 



It is difficult to imagine a proposition more fruitful 

 in consequences than this ; and therefore it has been de 

 monstrated with adequate fulness. 



In the^rs^ place, the arcs described being as the velocities, 

 if F,/ are the centripetal forces, and V, v the velocities, and 

 R, r the radii, F :/ :: V 2 : y 2 ; and also :: r : R ; orF : 



V 2 v 2 

 f :: :- : . Now as in the circle V and R, v and r 



JAi T 



are both 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 cir 

 cumferences or peripheries ; T : t : : P : p : But the pe 

 ripheries are as the radii or :: R : r. Therefore T : 



p p 



t :: R : r; also V : v :: = : ^-, therefore inversely as 



j_ * 



the radii, or T :*::?:- and V 2 : v 2 :: ?* : -*. But 



V 2 v 2 

 the centripetal forces F :/:: ^ : 5 substituting for the 



D 2 





