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



For simplicity let us denote the ratios^, -^ , &c. by 



the symbols w a , w a , &c. These quantities w a , w a , &c.,fare 

 evidently the probabilities in the case of this particular 

 macroscopic state, that any given particle m a will be found in 

 the respective regions Nos. la, 2a, &c. 

 We may now write 



log W= — Nc£ic a log iv a —'N b Zw b log iv b — , &c. 



where the summation extends over all the regions Nos. la, 

 2a, ...lb, 2b, &c. 



Equilibrium Relations. 



Let us now suppose that the system of particles is con- 

 tained in an enclosed space and has the definite energy 

 content E. Let us find the most probable distribution of the 

 particles. For this the necessary condition will be 



SlogW=-N«2(logw;a + l)Sti?«-N5S(log«; 6 + l)Sti?j...=0 . (3) 



In carrying out our variation, however, the number of 

 particles of each kind must remain constant so that we have 

 the added relations, 



%BlV a = 0, %Sw h = 0, &C. 



Finally, since the energy is to have a definite value E, it 

 must also remain constant in the variation, which will provide 

 still a further relation. Let us write the energy of the 

 system in the form 



E = N«2w«Ea + N 6 2nE 6 +... ? . . . (4). 



where E a is the energy of a particle in the region la, &c. 



Since in carrying out our variation the energy is to remain 

 constant we have the relation 



SE = N«2E a 8t0 a -hN 6 2E 6 Su; 6 + ... =0. . . (5) 



Solving the simultaneous equations (3), (4), and (5) by 

 familiar methods we obtain 



log W a + 1 + XEa + fla=0, 



log w b + 1 + XE 6 +[jL b = 0, 

 &c. 



where X, M«, /^&, &c, are undetermined constant multipliers. 

 (It should be specially noticed that X is the same constant in 

 each of the series of equations.) 



