248 



CENTRAL FOBCES ; 



Next, where the centre of force is in the focus. If P A 

 be a conic section whose parameter is D, S Y the perpen- 

 dicular to the tangent T P, P E the radius of curvature at P ; 



T) S P 

 then SY:SP::iD:PN (the normal), and S Y = 



also P K = 





Substitute these values of S Y and P R 



(p and E) in the expression formerly given for the central 



SP 



- -, and we have U* . S P 3 4 P N 3 or T 



P X E -7T-F7-^r X ^ D X 



force 



which is (D being invariable) as the inverse square of the 

 distance. Therefore any body moving in any of the conic 

 sections by a force directed to the focus, is attracted by a, 

 centripetal force inversely as the square of the distance from 

 that focus. This demonstration, 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 centripetal 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 =r ?-=- 



D . S P* 



and work backwards, multiplying the numerator and deno- 

 minator both by S P, and then multiplying the denominator 



