6 4 



KINETICS OF A PARTICLE. 



[123- 



of force, if these centres be regarded as containing masses equal to K?. 

 It is evidently a fixed point. 



123. By introducing the co-ordinates of the mean centre, we can 

 now reduce the equations of motion to the simple form 



where K 2 = SK/. Finally, taking the mean centre as origin, we have 



\--ty, 



u * _ _ K 2~ 



*" 



^ = -A. 



,^u2 



It thus appears that the motion of the particle is the same as if there were 

 only a single centre of force, viz. the mean centre (x,y,z), attracting 

 with a force proportional to the distance from this centre. 



The plane of the orbit is, of course, determined by the mean centre 

 and the initial velocity. The equation of this plane can be found by 

 applying the principle of areas (Art. 94). 



124, It is easy to see that most of the considerations of Art. 122 

 apply even when some or all of the centres repel the particle with force 

 proportional to the distance. It may, however, happen in this case tha 

 the mean centre lies at infinity, in which case it can, of course, not be 

 taken as origin. 



Simple .geometrical considerations can also be used to solve the 



problem. Thus, in the case of two 

 attractive centres O 1} O 2 (Fig. 18) 

 of equal intensity * 2 , the forces can 

 evidently be represented by the dis 

 tances POi = r lt PO 2 = r 2 of the par 

 tide P from the centres. Thei 

 resultant is therefore = 2 PO, if O 

 denotes the point midway between O 

 and <9 2 ; and this resultant alway 

 passes through this fixed point O, and is proportional to the distance 

 PO from this point. 



Fig. 18. 



125. Exercises. 



(i) Determine the constants of integration in Art. 119, if x ,y are 

 the co-ordinates of the particle at the time /=o and x , j the com 



