232 MOTION OF SYSTEMS OF PARTICLES 



The first part may be taken to be 



^X^dx+^Y^dy +2^i, (82) 



while the second is 



^X l dx l 4-_2)r i %i +2X^i- ( 83 ) 



By equation (77), we have 



v d/H 



^ = M M' 



where M is the total mass of the system, expressing that the 

 center of gravity moves as though it were a particle of mass M 



acted on by a force of components 2^fi, 2)-^i> 2/^i- ^ ^ s a ^ once 

 clear that expression (82) represents the work done in the motion 

 of this imaginary particle, and this we know must be equal to the 

 increase in its kinetic energy. 



The total work done is the sum of expressions (82) and (83). 

 This total work is equal to the increase in the total kinetic energy 

 of the system (by 140), and this again (by 186) is equal to the 

 increase in the kinetic energy of motion relative to the center of 

 gravity of the particles, plus the increase in the kinetic energy of 

 the imaginary particle of mass M moving with the center of gravity. 



This latter increase, as we have just seen, is represented by 

 expression (82), so that the former must be represented by 

 expression (83). 



Thus the increase in kinetic energy relative to the center of 

 gravity is 





and is therefore equal to the work done by the forces, calculated 

 as though the center of gravity were at rest. 



188. Thus we see that, in the theorem that the increase in 

 kinetic energy is equal to the work done, it is legitimate to calcu- 

 late both the kinetic energy and the work done by considering 

 motion relative to the center of gravity only ; i.e. the system may 

 be treated as though its center of gravity remained at rest. 



