54 NEWTON S PRINCIPIA. 



4 . P N 



P 2 x PN 3 



p2 



4 A o 6 x S P ; so that, neglecting the constant 



the centripetal force is as the distance directly 

 &quot;Trom hence it follows, conversely, that if the centripetal 

 force is as the distance, the orbit is elliptical or circular, 

 for by reversing the steps of the last demonstration we 

 arrive at an equation to the ellipse; or, in case of the two 

 axes being equal, to the circle. It also follows that if 

 bodies revolve in circular or elliptical orbits round the 

 same centre, the centre of the figures being the centre 

 of forces, and the force being as the distance, the periodic 

 time of all the bodies will be the same, and the spaces 

 through which they move, however differing in length 

 from each other, will all be described in the same time. 

 This proposition, which sometimes has appeared paradoxical 

 to those who did not sufficiently reflect on the subject, is 

 quite evident from considering that the force and velocity 

 being increased in proportion to the distance, and the 

 lengths of similar curvilinear and concentric figures being 

 in some proportion, and that always the same, to the radii, 

 the lengths are to each other as those radii, and conse 

 quently the velocity of the whole movement is increased 

 in the same proportion with the space moved through. 

 Hence the times taken for performing the whole motion 

 must be the same. Thus, if V and v are the velocities, 

 B. and r the radii, S and s the lines described in the 

 times T and t, by two such bodies round a common 

 centre, Y:v::E,:r, and S:s::R:r; and because 



V = ~ and v= 1, ~ :|:.R: 



