232 CENTRAL FORCES; 



if we make T : t : : E" : r", then V : v : : : -^, or : : -^ : 



1 3 



-j3j. Thus, suppose n is equal to - we have for the velocities 



1 1 



V : v : : =. : =, or they are in the inverse subduplicate 



proportion of the distances ; and for the centripetal forces we 

 have F : / : : ^^ : t : : : ; or the attraction to the 



-ti T -Ll T 



centre is inversely as the square of the distance. 



Now if n - f, T : $ :: B* : r , or T 2 : t* : : E 3 : r 3 ; in other 



2i 



words the squares of the periodic times are as the cubes of 

 the distances from the centre, which is the law discovered by 

 Kepler from observation actually to prevail in the case of the 

 planets. And as he also showed from observation that they 

 describe equal areas in equal times by their radii vectores 

 drawn to the sun, it follows from the fundamental proposi- 

 tion, first, that they are deflected from the tangents of their 

 orbits by a power tending towards the sun ; and then it 

 follows, secondly, from the last deduction respecting it, (namely, 



the proportion of F :/ : : ^ : -^,) that this central force acts 



it Ir 



inversely as the squares of the distances, always supposing 

 the bodies to move in circular orbits, to which our demon- 

 stration has hitherto been confined.* 



The extension, however, of the same important proposition 

 to the motion of bodies in other curves is easily made, that is 

 to the motion of bodies in different parts of the same curve 

 or in curves which are similar. For in evanescent portions 

 of the same curve, the osculating circle, or circle which has 

 the same curvature at any point, coincides with the curve at 

 that point; and if a line is drawn to the extremity of that 



* This sesquiplicate proportion only holds true on the supposition of 

 the bodies all moving without exerting any action on each other. 



