228.] PLANE MOTION. 12 $ 



provided the sense of j be away from the centre, i.e. provided 



the force causing the acceleration be repulsive. In the case of 



attraction^ the direction cosines of j are of course x/r, y/r. 



Thus the equations of motion are in the case of attraction, 



r/ \ / 



~ f(r} r (49) 



For repulsion, it would only be necessary to change the sign of 



227. To perform a first integration, multiply the equations 

 (49) by y, x and subtract when the left-hand member will be 

 found an exact derivative, while the right-hand member van- 

 ishes. Hence, integrating and denoting the constant of integra- 

 tion by h, we find 



, ,f-,f =,, ^ (50) 



or, introducing polar co-ordinates, 



These equations show that the sectorial velocity is constant, 

 and \h for our problem (see Art. 135 and Art. 163, Ex. (4)). 



228. Let 5 be the sector P OP described by the radius vector 

 r in the time t, so that dS=^r*d9. Then (5 1) can be written in 

 the form 



whence integrating 



S=\ht; (53) 



this expresses the fact that the sector is proportional to the time 

 in which it is described which is of course only another way of 

 stating the proposition of Art. 227. 



