MEMORIAL TO CLASSICAL STATISTICS 133 



for the number of cells in the former box. The thing miscalled entropy is 

 kNinM in the first case and (kxNlnM + ^.TA''ln.r) in the second case. It is 

 the term ^.vAHn.v which is the rift. 



Clearly we could abolish this term by allowing the volume of the cells to 

 swell in the ratio .v:l when going from the former case to the latter. This 

 is the same as making Ho proportional to the number of atoms in the sample 

 of gas which happens to be under study. Since in equation (31) the volumes 

 T'o and //o (of the elementary cells in ordinary space and in momentum- 

 space) are indissolubly bound together in the product F0//0 , this is the same 

 as making T'o-^o equal to some constant multiplied by the number of atoms 

 under study. 



Such, if I interpret correctly, was the idea proposed by Sackur in 1912. 

 While it does the task required, it is an "ad hoc" assumption of the most 

 barefaced character. If the gas under study is at first divided into two 

 parts by a partition and the partition is then abolished, the cells must be 

 supposed to swell up at the moment when the partition vanishes. 



We can also abolish the fatal term by going back to equation (1) for the 

 number of complexions, and removing the factor Nl in the numerator and 

 replacing it by unity. We then have unity divided by the original de- 

 nominator, which in the (most probable) case of the uniform distribution is 

 (AYM)! raised to the power M, as I remarked on page 121. Using the 

 super-Stirling approximation, we find that the logarithm of one fraction is 

 (NlnM — iVlniV). The factor Nl which we formerly had in the numerator 

 killed off the term (—NlnN), but now that we have taken it out, this term 

 survives. If now we say that k times the logarithm of W/Nl shall be the 

 picture of entropy in the classical statistics, then the term (— ^MnA^) 

 •comes over into the right-hand member of (31). It may be amalgamated 

 with the last term already standing there; and when this is done, we find 

 VqHq multiplied by A^ exactly as Sackur put it there, and with the same 

 wished-for result. 



This, if I interpret correctly, is the idea proposed in 1913 by Tetrode. It 

 does the task required of it, but its drawback is that the removal of the 

 factor Nl from the right-hand member of (1), a drastic piece of surgery as 

 it were, violates the system of the classical statistics.^ 



I was not, however, thinking merely of this achievement when on Page 

 132 I spoke of "a remarkably successful feat of tampering." To show the 



^ This may seem too strong a statement. We are, after all, only asked to accept k In 

 (W/Nl) as our picture of entropy, instead of ^InlF; why be reluctant? But in effect, as I 

 see it, we are asked first to accept k\nWf as our picture of entropy, / being an arbitrary 

 function of A''; and then we are asked so to choose/, that the dependence of k In Wf on N 

 shall conform to the actual behavior of entropy. This is different from and much less 

 impressive than our original procedure, which consisted in first realizing that W is the 

 number of complexions, and then discovering that k In W depends on volume and on 

 temperature in just the right ways for entropy. 



