368 BELL SYSTEM TECHNICAL JOURNAL 



most probable distribution of the C cells among the possible populations can 

 be realized, there being iV,- atoms in the region. Nothing of the sort ap- 

 peared in the old statistics. 



Now form the product of all of the quantities Wj. This is a "probability" 

 relating to the entirety of all the regions, therefore to the whole of the gas. 

 It is the total number of ways in which the most probable distribution can 

 simultaneously be achieved within each of the regions. It is taken to be the 

 total number of ways in which the most probable distribution of the gas- 

 as-a- whole can be realized; for in the new statistics we have no other way 

 of defining the most probable distribution of the gas-as-a-whole, than this 

 way of subdividing first into cells and then into regions comprising many 

 cells. The symbol for this product shall be W, for although I have already 

 used that symbol in this article, its former meaning is now taken over by 

 Wj, and it is free again. We have: 



In IF = Z In Wj (17) 



the quantities Wj being still those functions of iV",- which were shown in 

 (16). 



We seek now a set of values of Nj such that when it is realized. In W, 

 and therefore W also, shall have a value stationary with respect to all 

 variations 8Nj conforming to the sole condition that the total number of 

 atoms shall remain the same, which is to say, 25iV/ shall vanish. 



It may be recalled that a similar problem arose in the old statistics. I 

 treat it here in a more general and hardly less simple way, by writing the 

 self-evident equation: 



5lnW = z'^S^8Nj (18) 



dNj 



For the fulfilment of our wish it is a sufficient condition that all of the deri- 

 vatives on the right-hand side should have the same value; since than 8ln W 

 will be 28Nj multiplied by a constant, and when one vanishes so will the 

 other. For this it is in turn a sufficient condition that all of the independent 

 variables Nj should have the same value. 



Uniform spreading of the atoms among the regions, with equal numbers in 

 all regions of equal size, is therefore the condition in which In W has a 

 "stationary" value, which as always is assumed to be a maximum value. 

 With the new definition of probability, the state of uniform spreading be- 

 comes the most probable in the new statistics, as with the old definition of 

 probability it was in the old. 



We go into the momentum-space to see whether the Maxwell-Boltzmann 

 law results from the new statistics. 



The momentum-space is now to be divided into regions of equal size, 



