62 KINETICS OF A PARTICLE. [117. 



117. It is easy to see that the methods here explained would 

 apply even to the more general problem when the force F, while 

 passing ahvays through a fixed centre, is not a function of the 

 distance r alone, but a function of both co-ordinates r and 6. The 

 principal difference will appear in the impossibility of perform- 

 ing directly the integration indicated in (7). 



With F=mf(r, 0), the equation of kinetic energy is, for 

 attraction, 



and substituting for v 2 the first of the values given in (8), we 

 find 



This equation shows that the motion relative to the radius 

 vector takes place as if the actual resulting force F=mf(r, 6} 

 were increased by an additional force m<?/r*. 



For the law of force we have, as in Art. 115: 



118. We proceed to the consideration of some special cases. 

 The most important of these are the case of a force directly 

 proportional to the distance, and that of a force inversely pro- 

 portional to the square of the distance. 



119, Force Proportional to the Distance : f(r) = K 2 r. The equations 

 of motions (2) are in this case 



, 



the upper sign holding for attraction, the lower for repulsion. Their 

 solution is very simple, because each equation can be integrated sepa- 

 rately. We find, in the case of attraction, 



x = #! cos K.t 4- a. 2 sin *t, y = b^ cos K/ -f- b% sin K/, 

 and in the case of repulsion, 



x = a lf + a#- Kt , y = V + V* ; 

 <*i a z> b, fit being the constants of integration. 





