TEE NEW STATISTICAL MECHANICS 367 



atoms N. The task is greatly eased by the opportunity of using two well- 

 known formulae: 



\ + x+x'+--- = {\- x)-' (9) 



1 + 2x + 3x' + • • • = (1 - xy^ (10) 



The start is made from the two self-evident equations: 



C = 2Ci = C^ae^'^ (11) 



^T = ziCi = CZiae~'^ (12) 



By putting x for e~^ and using (9) and (10), the student can easily win 

 through to the results, 



a = (1 - e-^), C/N - / - 1 (13) 



and then to the final form of the distribution-law (7): 



and finally, after consulting k5), to the expression for the number of ways 

 H nuix in which this the most probable distribution — ''most probable" in the 

 eyes of the new statistics — can be realized. Its logarithm is: 



InW^n... = Cln^^^+ iVln^^^ (15) 



This is the most important formula of the new statistics, as will presently 

 be clear. 



Divide now the space containing the gas into "regions" of equal size, 

 each comprising the same number C of cells, which number shall be great. 

 For the benefit of those to whom the memory of the previous article may 

 still be vivid, I say now that insofar as there is any correspondence of the 

 new to the old statistics, these "regions" correspond to the "cells" of the older 

 theory. This is the reason why, in my recent brief synopsis of the old 

 statistics, I used the word "region" to replace the word "cell" used in the 

 prior article. Let the subscript j be the marker for these regions, so that 

 Nj shall stand for the number of atoms in the 7th region. Put Nj for N 

 in {IS). Now each member of (15) refers explicitly to theyth region, and 

 on the left I should put (In rr(max) y), but for two purposes — one of which is 

 brevity, while the other will appear in due time — I put In Wj instead: 



lnP7,F= Cln^^^-±-?-f i\^yln^^^-±-? (16) 



\^ iV y 



The quantity IF,- is an odd sort of "probability" relating only to the 

 contents of the region j. It is, to repeat, the number of ways in which the 



