372 BELL SYSTEM TECHNICAL JOURNAL 



I review the situation. We began by dividing space (ordinary space, or 

 momentum-space, or ^i-space) into what were called cells in the earlier 

 article and are now designated "regions." We wanted to reach, as most- 

 probable-distribution of the atoms among the regions, the uniform spread 

 in ordinary space and the Maxwell-Boltzmann law in momentum-space. 

 To do this by the method of the new statistics, we divided each region into 

 many "cells." The first stage of the argument then consisted in taking a 

 typical region, and ascertaining the most probable distribution of the cells 

 among the populations. We then evaluated PFmax, the number of ways in 

 which this distribution could be realized. Inserting In PFmax into the argu- 

 ment we continued into the second stage, and attained the wanted result. 

 But now it turns out that in the first stage we might have omitted to ascertain 

 the most probable distribution of the cells among the populations in the 

 typical region. Anybody might win through to the same desirable outcome 

 without even suspecting that there is a most probable distribution of cells 

 among populations. All he needs is to evaluate Wtot, the number of ways 

 in which all possible distributions of cells among populations within the 

 region can be realized. He may then replace In PTmax by In PFtot, and pro- 

 ceed with the second stage as before. Since the two logarithms are 

 practically equal, the outcome is the same. 



There are accordingly two routes to the result, which do not merge until 

 the argument is carried partway to the conclusion. Is one of them right 

 and the other wrong? Or to ask a milder question: is either to be preferred 

 to the other? 



So far as I can see, neither can be proved wrong, and the question must 

 be asked in the milder form. For myself I stand by the preference exhibited 

 in this article, for the basic reason that along this route each of the stages of 

 the argument consists in finding a most probable distribution: first for the 

 cells among the populations of each region by itself, and then for the atoms 

 among the regions. By the other route the two stages are difi"erently 

 handled, since in the first stage one considers all the distributions (of cells 

 among populations in each region by itself) and then in the second stage the 

 most probable distribution (of atoms among regions). There is also the 

 minor advantage, that the value of TFmax is much easier to derive than the 

 value of TT^tot, or at least so it seems to me. However, many physicists 

 of eminence have preferred the second route. Anyone may say of course 

 that the question is foolish, since the number of complexions subsumed under 

 the most probable distribution is so large a fraction of the total number of 

 complexions altogether that no danger arises from confusing them. This is 

 what the equations have been saying, and now I have said it again in words. 



* It was the other way about in the somewhat similar case which was treated in the 

 earlier article. 



