THE NEW STATISTICAL MECHANICS 379 



in harmony with the "second alternative" aforesaid, wherein the total 

 entropy of two gases of identical P and T is the sum of their sep^arate entro- 

 pies! It is otherwise in the old statistics, and that was the source of the 

 troubles of that elder theory. But to proceed: 



In W = S In Wj = iVln C - S Nj In Nj + N (49) 



In the ordinary space and in the most probable state, N j is the same for all 

 the regions, as we have found already (page 363) and therefore is equal to TV 

 divided by F/T'o, or to NVo/V; here Vq is the volume of the region (not the 

 cell!). The next step is to put this into (49), realizing now that each term 

 becomes the same and the whole summation is F/Ko times the typical term. 

 One is agreeably surprised to fmd that T'''o tumbles out of the expression: 

 this is a feature of the new statistics — the regions have but an intermediate 

 and an auxiliary quality, the size assigned to them is gone from the final 

 equations. In its place appears the volume of the cell, which is Vo/C, 

 and which I denote by qc. For Sc we have: 



S, = Nk In V - Nk In .V - Nk In qc + N (50) 



Note the last three terms, for future comparison with the two last of (46); 

 but at this moment note especially the first, and compare it with the first 

 of (46). Entire agreement is attained by assigning to k the value, 



k = R/No (51) 



as in the old statistics. The "Boltzmann constant" k is the "gas-constant" 

 R divided by the "Avogadro number" Na. 



Seeking now the "contribution of temperature to entropy," Sm, we turn 

 to the momentum-space. Here the most probable distribution is given by 

 (21), and is to be inserted into (50): 



\nW = NlnC - Z Nj In Nj + N 



= NlnC - NlnA -\- NX ABEje'^^' -f N (52) 



It will be recalled from the earlier article, or failing this can easily be seen, 

 that, 



A^ 2 A'''^' = N, AS AEje""''' = U (53) 



U standing as heretofore for the total energy-of the gas. The expression 

 (52) is simplified of aspect, and multiplying it by k, we find for Sm, 



Sm =^ k\n W = kN In C - kN In A -f kBU + kiY (54) 



Though I have spoken of this as the contribution of temperature to entropy, 

 the temperature is nowhere to be seen! It is waiting on the doorstep; 

 but before allowing it in, I wish to operate on the quantity In .1. 



