THE NEW STATISTICAL MECHANICS 369 



each large enough to comprise a great number C of cells and small enough 

 so that the function E of equation (3) may be deemed sensibly constant 

 throughout it; Ej shall stand for the value of E appropriate to the region. 

 (It is convenient to imagine the regions as layers separated from one another 

 by concentric spheres having the origin for their common centre). For 

 each of the regions Wj, the number of ways in which the most probable 

 distribution of the cells among the possible populations can be realized, is 

 given again by (16) ; and W, the number of ways in which the most probable 

 distribution can simultaneously be achieved within each of the regions, is 

 given again by (17). 



We now seek a set of values of Nj such that when it is realized, \nW shall 

 have a value stationary with respect to all variations SNj conforming to two 

 conditions: first, that the total number of atoms shall remain the same, which 

 is to say, ZdNj shall vanish; and second, that the total energy of the gas 

 shall remain the same, which is to say, XEjSNj shall vanish. 



Referring back to (18), we see that for the fulfilment of our wish the 

 following is a sufficient condition: 



"W^ = P + QE< (19) 



P and Q standing for constants; for when these substitutions are made into 

 every term of the summation on the right of (18), the expression to which 

 6\nW is there equated may be regrouped into one term proportional to 

 ZSNj and one proportional to "^Ej-dNj, and vanishes when it ought to 

 vanish. 



Gone is the comfortable ease with which we disposed of the corresponding 

 problem in the ordinary space! There we did not even have to know what 

 sort of function Wj is of N'j] whatever it might be, we were able to conclude 

 that Nj must be the same for every region. Here the outcome must depend 

 upon the functional relation between Wj and Nj. There is, however, no 

 ground for apprehension, for though the function in question looks rather 

 involved in equation (16), its derivative is surprisingly simple, and we come 

 with ease to the condition which we seek: 



In (Nj + C) - \nNj= P-h QEj (19) 



which may be rewritten thus: 



£ = -1 +/+«^' (20) 



Nj 



This is not the Maxwell-Boltzmann law, but approaches that desired law in 

 what I will call the "limit of extreme rarefaction," where the number of 

 cells in the region exceeds manyfold the number of atoms. As C/Nj grows 



