56 





NEWTON S PRINCIPIA. 



law of the force for the three curves (parabola, hyper 

 bola, and ellipse) severally. For this purpose a simple 

 reference to the formulas already stated will suffice; 

 indeed our illustration of those formulas has already anti 

 cipated this. 



If O P A be a conic section whose parameter is D, 

 S Y the perpendicular to the tangent T P, P R 

 the radius of curvature at P; then SY : SP :: D 



: P N (the normal), and S Y = ^ ^ ^ ; also P R = 



4 P N 3 



j-p Substitute these values of S Y and P R (p 



and R) in the expression formerly given for the central 



force 



SP 



x B 



, and we have D 3 . 8 P 3 4 P 



8 P 



or 



jj ^ p 2 , which is (D being invariable) as the inverse 



square of the distance. Therefore any body moving in 

 any of the conic sections by a force directed to the 

 focus, is attracted oy a centripetal force inversely as the 

 square of the distance from that focus. This demon- 



