LAW OF THE UNIVERSE. 249 



8 D* P X s 

 ' 



^ -r^o ^. ->- we obtain the expressions for the value of 

 8 D* . P IX 3 



S Y, the perpendicular, and for E, the radius of curvature. 

 But no curves can have the same value of S Y and K, except 

 the conic sections ; because there are no other curves of the 

 second order, and those values give quadratic equations 

 between the co-ordinates. 



By pursuing another course of the same kind algebraically, 

 we obtain an equation to the conic sections generally, accord- 

 ing as certain constants in it bear one or other proportion to 

 one another, The perpendicular S Y and the radius of cur- 

 vature are given in terms of the normal; and either one 



Q 



(d a^-f d y 2 ) 

 or the other will give the equation. Thus E = - -^- 



xa( 



\ 



4 P N 8 4 if 3 



= jp- = ^ dx , X (d oc* + dy*)* which gives D 2 d x 3 = 



4 y* X (d* y d x d* x d y) an equation to the co-ordinates. 

 Now whether this be resolvable or not, it -proves that only 

 one description of curves, of one order, can be such as to have 

 the property in question. The former operation of going 

 back from the expression of the central force, proves that the 

 conic sections answer this condition. Therefore no other 

 curves can be the trajectories of bodies moving by a centri- 

 petal force inversely as the square of the distance.* 



This truth, therefore, of the necessary connexion between 

 motion in a conic section and a centripetal force inversely as 

 the square of the distance from the focus, is fully established 

 by rigorous demonstration of various kinds. 



* The equation may be resolved and integrated ; there results, in the 

 first instance, the equation dx = ^ , and therefore the integral 



is this quadratic, c 2 ^ - 2 c>/ - 2cC^- r -C 2 -|-D 2 = 0. 



