LAW OF THE UNIVERSE. 245 



because by the property of the circle S X S B or (a + &) 

 (a - b) = a 8 - 8 - P S X S V ; therefore 



a* -I* 2a* + 2bx 



S V = ^^^=== and P V = . 



tj a* + 2 b x -f- b* \/ a* -\- 2 b x -\- b* 



Now by the formula already stated as Bernoulli's, but 

 really Sir Isaac Newton's, the centripetal force in P is as 



SP 



, K being the radius of curvature, and in the circle 

 fe A X " 



that is constant being = a, the semidiameter ; therefore the 

 force is as ^ . . . or as ~. 



8 a 3 



B 0' x S P B O 8 x S P 3 



8 r RS 



BO 8 2 a 



X P S* 



SP 3 



BO 2 



= P V. Therefore the central force is as -r- - ; or 



P V* X 8 P 2 



(because B 2 is constant) the central force is inversely as 

 the square of the distance and the cube of the chord jointly. 

 Of consequence, where S is in the centre of the circle and 

 b o, the force is constant, S P becoming the radius and P V 

 the diameter ; and if S is in the circumference of the circle as 

 at B, or a = 5, then the chord and radius vector coinciding, 

 the force is inversely as the fifth power of the distance, and is 

 also inversely as the fifth power of the cosine of the angle 

 PSO. 



By a similar process it is shown that in an ellipse the force 

 directed to the centre is as the distance. Indeed, a property 

 of the ellipse renders this proof very easy. For if S Y is the 

 perpendicular to the tangent T P, and N P (the normal) 



