592 Prof. R. C. Tolman on the Relativity Theory : 



Since we have shown that h is independent of the mass of 



the particles we see that the average value of — ° is the 



i/l — v 2 

 same for particles of all different masses. This is the principle 

 in relativity mechanics that corresponds to the law of the 

 equipartition of vis viva in the classical mechanics. Indeed, 

 for low velocities the above expression reduces to m v 2 the 

 vis viva of Newtonian mechanics, a fact which affords an 

 illustration of the general principle that the laws of Newtonian 

 mechanics are always the limiting form assumed at low 

 velocities by the more exact formulations of relativity 

 mechanics. 



We may now call attention in passing to the fact that this 



m v 

 quantity — ° , whose value is the same for particles of 



different masses, is not the relativity expression for kinetic 



energy which is given rather by the formula 1 ,- : 2 —m Q \ 



So that in relativity mechanics the principle of the equi- 

 partition of energy is merely an approximation. We shall 

 later return to this subject. 



Equality of Temperature. 



For a system of particles of masses m a} ?nj, &c, enclosed 

 in the volume V, and having the definite energy content E, 

 we have shown that 



and 4,7rV* b e ~ h ^ +m * \ 2 d^ 



are the respective probabilities that given particles of mass 

 m a or mass m& will have momenta between ty and ^r + d-dr. 

 Suppose now we consider a differently arranged system in 

 which we have N a particles of mass m a by themselves in a 

 space of volume V a and N& particles of mass m b in a con- 

 tiguous space of volume V&, separated from Y a by a partition 

 which permits a transfer of energy, and let the total energy 

 of the double system be, as before, a definite quantity E (the 

 energy content of the partition being taken as negligible). 

 Then by entirely similar reasoning to that just employed we 

 can obviously show that 



and 47rV t « s e~ A ^ 2+ms V 2 ^ 



