NEWTON S PRINCIPIA. 49 



To take an example in the parabola, where S being 

 the focus, and O S = , y 1 = 4 a x, and T M = 2 x 9 

 and;; = Y S = */\a + x) a ; r = S P = a + x, and 



E = ^- r 



= 2 (a + a?) A / &quot; + *; we have therefore F as 

 V 



M N 



x 2 (a 



meter) is constant, inversely as the square of the dis 

 tance : And the other formula F as 3 ^ gives the 



same result for the law of force, or 



Again, in the ellipse, if a be half the transverse axis, 

 and b half the conjugate, and r the radius vector, we have 



/ r a b d r f 



p = b \f - -- , and d p = - 3- therefore 

 V 2-r /-2- 



dp a b d r a 



the formula -~ becomes : 5 = Tr~5 or 



p*.d* ft* A / r x ^ x dr b 2 r 2 



the force is inversely as the square of the distance. 



* This result coincides with the synthetical solution of Sir Isaac Xewton 

 in Prop. XIII. 



E 



