NEWTON S PRINCIPIA. 57 



stration, therefore, is quite general in its application to all 

 the conic sections. 



It follows that if a body is impelled in a straight line 

 with any velocity whatever, from an instantaneous force, 

 and is at the same time constantly acted upon by a cen 

 tripetal force which is inversely as the square of the distance 

 from the centre, the path which the body describes will 

 be one or other of the conic sections. For if we take the 



expression ^ rrjj* and work backwards, multiplying the 



J . O -L 



numerator and denominator both by S P, and then mul- 



8 D 2 P N 3 

 tiplying the denominator by Q r\2 u \3&amp;gt; we obtain the 



expressions for the value of S Y, the perpendicular, and 

 for K, the radius of curvature. But no curves can have 

 the same value of S Y and R, 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 alge 

 braically, we obtain an equation to the conic sections 

 generally, according as certain constants in it bear one 

 or other proportion to one another. The perpendicular 

 S Y and the radius of curvature are given in terms of the 

 normal ; and either one or the other will give the equation. 



(dx^ + dy^ 4 P N 3 4v 3 



Thus =- =xdx+d 



, 



dxxd i 



which gives D 2 d x 3 = 4 ?/ 3 x (d 2 y d x d 2 x d y) an 

 equation to the co-ordinates. Now whether this be resol 

 vable 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 sec- 



