136.] CENTRAL FORCES. 71 



the equation of kinetic energy is negative, zero, or positive. Owing to 

 the value of h given in (14), this criterion agrees with the form (13), 

 Art. 130. 



It should be observed that it follows from (13) that the nature of the 

 conic is independent of the direction of the initial velocity. 



134. The criterion (13) can be given the following interpretation. 

 Consider a particle attracted by a fixed centre according to Newton's 

 law. If it move in a straight line passing through the centre, the 

 principle of kinetic energy gives for its velocity, at the distance r, 



hence, if it start from rest at an infinite distance from the centre, it 

 would acquire the velocity V2/z/r at the distance r. The criterion (13) 

 is therefore equivalent to saying that the orbit is an ellipse, a parabola, 

 or a hyperbola, according as the velocity at any point is less than, equal 

 to, or greater than the velocity which the particle would have acquired at 

 that point by falling towards the centre from infinity (comp. Art. 57). 



135. For a central conic, whose axes are 20, 2b, we have /= $ /a, 

 e = V# 2 qp &l a (the upper sign relating to the ellipse, the lower to the 

 hyperbola), so that the equations (19) reduce to the following: 



h = ^. (20) 



a 



The latter relation, with the value of h from (14), gives for the major 

 or real semi-axis a : 



while the former, with the value of c as given in (5), Art. in, deter- 

 mines the minor or transverse axis b : 



(22) 



136. The magnitudes of the axes having thus been found, their 

 directions can be determined by a simple construction which furnishes 

 the second focus. 



