KINEMATICS. [224. 



directly proportional to the distance from the fixed centre O, i.e. 

 f(r)=r. 



Another very important particular case is that of planetary 

 motion in which f(r)=/j,/r 2 ; this will be discussed below, 

 Arts. 236, 239. 



We proceed to establish the fundamental properties of central 

 motion. 



224. The motion is fully determined if in addition to the 

 f<>rm of the f unction /(r) we know the "initial conditions," say 

 the initial distance OP Q = r Q (Fig. 53) and the initial velocity v^ 

 of the point at the time /=o. As ^ must be given both in 

 magnitude and direction, the angle ^ between r and V Q must 

 be known. 



225. It is evident, geometrically, that the motion is confined 

 to the plane determined by O and V Q since the acceleration 



Fig. 53. 



always lies in this plane. This fact that the motion is plane 

 depends solely on the former of the two conditions of our 

 problem (Art. 223) ; that is, any motion in which the acceleration 

 passes constantly through a fixed point is plane. 



226. With O as origin, let x, y be the rectangular Cartesian 

 co-ordinates of the moving point P t and r, 6 its polar co-ordinates, 

 at any time/. Then cos#=;r/r, sin 0=y/r are the direction 

 cosines of OP=r, and, therefore, those of the acceleration/, 



