684 BELL SYSTEM TECHNICAL JOURNAL 



of the mathematical aids in Statistical theory. It is: 



x\ = {lirxyi^ixleY, (9) 



log x! = X log X — X + i log {2irx). 



The first two terms of this latter expression form an approximation 

 singularly good even when x is no greater than ten or thereabouts. 

 Using it we have: 



log W = const. - Y.Ni log Ni. (10) 



Denote by W^ the value of W for some particular distribution iVi", 

 Ni^, • • -NJ and by W = W^ + bW its value for some other only 

 slightly different distribution A^i" + 8Nu iVs" + SiVs, • --NJ + bN^. 

 The difference between the values of log W^for these two distributions 

 is to first order of approximation: 



b\ogW = 8WIW = - Z(l + log Ni'')8Ni. (11) 



If W^ is a maximum for the distribution iVi"- • -Nm^, then the difference 

 between log W° and the value of log W ior any other slightly different 

 or "slightly varied" distribution must vanish to first approximation. 

 The quantity on the right of (11), the "first variation" of log W, 

 must be zero for any permitted set of values of 8Ni, • • • 8Nm; meaning 

 by "permitted" any set of integer values adding up to zero, for we 

 consider an assemblage of an invariable number of particles. 



Now one sees immediately that the right-hand side of (11) does 

 vanish, if all the populations iVi° have the same value, say a; for then 



\ogW= - Z(l + log a)8Ni = const. E^A^i (12) 



i i 



and the permitted variations are precisely those, for which the sum- 

 mation X^iVi is zero. 



We do therefore reach the result which was desired. Failing it, 

 this mode of "counting the ways in which a distribution may be 

 realized" would have been unprofitable. As it is, the quantities W 

 and log W are greatest for the uniform distribution which seems 

 intuitively the most probable and is the rule for gases, and least for 

 the utterly non-uniform one which seems the least probable. Tenta- 

 tively the former is adopted as measure of the "probability" of a 

 distribution. 



I point out in passing that while the foregoing result is mathe- 

 matically valid for any value of the constant a, the total number of 

 particles prescribed for the assemblage determines the value of a 

 which is physically permissible: viz. N/m. 



