LAW OF THE UNIVERSE. 241 



noulli (Mem. Acad. des Sciences, 1710), is, perhaps, the 



r 



most elegant of any, F = - ; and this results from 



substituting 2 E for its value , in the equation to F, 



dp 



dp 



deduced above from Newton's formula, namely, F = , . 



2p s dr 



But the proposition is so important, that it may be well to 

 prove it, and to show that it is almost in terms involved in 

 the third corollary to Prop. VI. Book I. of the Principia. 



By that corollary F = g (C being the osculating circle's 



chord which passes through the centre of forces). But draw- 

 ing S Y, the perpendicular to the tangent, and P C F through 



the centre of the circle, which is, therefore, parallel to Y S, 

 and joining VF, we have VP:PF :: S Y : SP or C : 2E ::p : r 



and C = - -^-, which substituted for C in the above equa- 

 tion, gives F = 



In all these cases p is to be found first, and the expres- 

 sion for it (because, pp. 286, 287, TP : PM :: TS : S Y and 



TS = 



dy 



and 



_ 



dy 



is p = 



