120 BELL SYSTEM TECHNICAL JOURNAL 



under such conditions Nature rejects that distribution which so to speak is 

 "stable against" every conceivable variation, and elects that peculiar distri- 

 bution which is stable not against any conceivable variation but only against 

 the possible ones. Perhaps this is because the uniform distribution would 

 entail the consequences mentioned on page 117, or perhaps there is no sense 

 in saying that it is "because" of this or "because" of that. Anyhow, the 

 peculiar distribution is the one which the data sustain. 



However I have not really defined the peculiar distribution as yet, having 

 merely thrown the symbols A and B into equation (8) as though they stood 

 for completely disposable constants. It can readily be seen that at the 

 most there can be but one disposable constant, for A and B interlinked by 

 the obvious equation: 



S Wi = ^ S exp {-BE,) = 1 (10) 



But even B is not disposable, if the total energy U and the average energy 

 per molecule U are preassigned; for there is another obvious equation: 



17-2 EiWi = Ai: Eiexp (-BEi) (11) 



What with equations (10) and (11), there is no longer anything disposable 

 about the constants A and B. The peculiar distribution in the momentum- 

 space is completely defined. It is the Maxwell-Boltzmann distribution-law 

 obtained from the Maxwell statistics, and sometimes known as the "canoni- 

 cal" distribution. 



To summarize now the Boltzmann statistics as on page 113 the Maxwell 

 statistics was summarized: 



A gas is more likely to he found in its most probable state than in any other. 

 The probability of a state is found by imagining it as a distribution of numbered 

 molecules among cells, in the coordinate-space and in the momentum-space. 

 That of any distribution is measured by the number of inventories compatible 

 therewith. By this criterion the most probable distribution in coordinate-space 

 is the uniform one, and by this criterion carefully hedged about, the most probable 

 distribution in momentum-space is the Maxwell-Boltzmann or canonical One. 

 It is necessary to liken molecules of a single kind to numbered balls, differing in 

 no way except the numbering. 



This point was reached by statistical mechanics about fifty years ago. 

 Had it not been for Planck's wish and tenacious will to explain the black- 

 body radiation-law, it might have been the stopping-point. 



A Helpful and Troublesome Coincidence between Two 

 Different Quantities 



Let us return to the game with the sack, the balls and the basket, played 

 in the manner which led to good results when applied to the molecules in the 

 coordinate-space. 



