278 MOTION UNDER A VARIABLE FORCE 



Thus the class of conic described depends only on the velocity 

 of projection, and not on the direction : the conic is a hyperbola, 

 parabola, or ellipse, according as 



v 2 > = or < 

 r 



The actual eccentricity depends on both E and h, for if e is the 

 eccentricity, we have , 2 



224. In order that the particle may describe a circle we must 

 have e*= 0, and therefore * ^, 2 



Putting E = v 2 -- and h = pv (so that p is the perpendicular 



from the center of force on to the direction of projection), this 

 reduces to 2 o 



t:_*ff>V*-o, 



Since p is necessarily less than r, neither term in this equation 

 can be negative. Thus, in order that the equation may be satisfied, 

 both terms must vanish, and we must have 



p = r and v* = 

 r 



The first equation expresses that the projection must be at right 

 angles to the line joining the particle to the center - of force. The 

 second equation, which can be written 



7 2== 7' 



shows that the attractive force must just produce the acceleration 

 appropriate to motion in a circle of radius r. 



