574 Prof. R. C. Tolman on the Relativity 1 lieory : 



If, now, we have a system consisting of a number of 

 particles m 1? m 2 , &c, we may write 



Fi+ii=^(^qi), 



F 2 + I 2 =^-(m 2 q 2 ), 



&G. 



where F^ F 2 , &c, are the external forces applied to the 

 particles from outside the system, and I l9 I 2 , &c, are the 

 internal forces arising from mutual reactions within the 

 system. 



If in addition to the particles, the system also contains 

 mass which is continuously distributed in space (for example 

 mass corresponding to potential energy) we may write 



Jf &,+ji*,-jij (pq L )dv = |Jpq dv, 



where f and i are the external and internal forces acting ou 

 the distributed mass per unit volume, p the density of the 

 mass, and the integrations extend over the whole volume v 

 of the system. 



Adding to the previous equations we obtain 



2F + 2I+ \fdv+ \idv=j t femq+\pqdi\ 



or since by the third law of motion (i. e. the equality of 

 action and reaction) the sum of the internal forces %l+\idv 

 is equal to zero, we may write 



2F + [fdv = j t (tmq + L dv). 



(3) 



This equation states that the rate of change in the total 

 momentum of a system of particles is equal to the sum of 

 the external forces applied. In case the system is unacted 

 upon by external forces the rate of change of the momentum 

 becomes zero, which is the principle of the conservation of 

 momentum *. 



* The methods of proof used here and in later paragraphs are 

 obviously modelled on those familiar in the classical mechanics. 



