366 BELL SYSTEM TECHNICAL JOURNAL 



We seek a set of values d such that when it is realized, the quantity W 

 shall have a value stationary with respect to all variations bCi conforming 

 to two conditions: first, that the number of cells shall remain the same, which 

 is to say, 25Cf shall vanish; and second, that the number of atoms shall 

 remain the same, which is to say, 'Zibd shall vanish. 



Such a set is the following: 



Ci = Cae-'^ (7) 



a and /3 standing for constants yet to be determined; for taking the first 

 variation of InW from (6), we find: 



8{\n W) = -Z8Ci(l + hi Ci) 



= - 25Ci(l + In Ca) + /SSzSCi . (8) 



and the required condition is fulfilled. Assuming without proof that the 

 stationary value of W is also a maximum value, and referring to W as the 

 "probability" of the distribution of cells among populations, we have come 

 to the startling conclusion that the most probable distribution is the one 

 given by (7) ! 



I call this a startling conclusion, because it contravenes our inbred con- 

 viction that the natural distribution of a gas in ordinary space is the uniform 

 distribution. Of course, in the last two sentences I have used the word 

 "distribution" in two senses, and this must be rectified at once. What I 

 have just called "the uniform distribution" is the uniform distribution in the 

 old sense — the same number of particles in every cell. In the new sense of 

 the word, this is a distribution in which all of the cells have the same popula- 

 tion, and therefore in which one basket contains all of the balls. Definitely, 

 this is not, in the new statistics, the most probable distribution! Indeed 

 it is not even a conceivable distribution, for the number of cells is infinite. 



To mitigate this clash of theory with experience we can do nothing else 

 than assume our cells to be so tiny that in any region of the gas large enough 

 to be surveyed by observation, there is a mighty number of the cells. Then 

 at worst we can take it from experience that in the normal natural abiding 

 state of the gas the number of atoms in each region will be the same if all 

 the regions are of equal volume, while within each region we can distribute 

 the atoms among the cells as the new statistics tells us to. However, it 

 may yet be possible to come to this conclusion from the theory. In prepara- 

 tion for the effort, I sketch the procedure for evaluating the constants a 

 and /3 in the distribution (7). 



A similar task was set before us in the earlier article: that of evaluating 

 the constants of the Maxwell-Boltzmann law in terms of the total number 

 and the total energy of the atoms. Here for any region we are to evaluate 

 the constants a and /3 in terms of the number of cells C and the number of 



