NEWTON S PKINCIPIA. 



53 



BO 



force is as p V3 ^ p2 ; or (because O B 2 is constant) the 



-XT V X ^5 -L 



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 con 

 stant, SP becoming the radius and PV the diameter ; and 

 if S is in the circumference of the circle as at B, or a = b, 

 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) parallel to SY, and S A the semi-conjugate 



axis ; S A is a mean proportional between S Y and PN, 



A S 2 

 and therefore S Y = - ; also the radius of curva 



ture of the ellipse is (like that of all conic sections) 



4PN 3 



equal to - , P being the parameter. Therefore 



we have to substitute these values for S Y and the 

 radius of curvature, R, in the expression for the central 



E 3 



