118 BELL SYSTEM TECHNICAL JOURNAL 



be greater than that for any non-uniform distribution, whether of the same 

 ^ or of a different U, confined within the sphere. However, by blocking off 

 the whole of the momentum-space beyond the sphere, we have barred a 

 whole lot of distributions corresponding to the same U and having some of 

 their dots beyond the sphere. By no means have we proved that the W- 

 value for the uniform distribution within the sphere is greater than that for 

 any and all of the barred distributions. Now if we can agree that the block- 

 ing-off of part of the momentum-space is a silly thing to do and unacceptable 

 to Nature, the argument for the uniform distribution is spoiled, and we have 

 to look for a new idea. 



At this point it seems best to go through the mathematical process for 

 finding the distribution of greatest W in the coordinate-space and the 

 momentum-space, just as that process is presented in the textbooks. 



We return to equation (1) and make it a manageable one by having re- 

 course to that godsend of statistical mechanics, the "Stirling approxima- 

 tion", which may be written thus: 



\nN\^ N\nN - N + In \/27riV (2) 



This is valid only for large values of N, though writers on S.M. never seem 

 to remember how large the values must be. For still larger values of N we 

 can drop ofif the last two terms, arriving at a sort of super-Stirling approxima- 

 tion which however itself is commonly called the Stirling approximation: 



\nN\ = N\nN (3) 



Putting (3) into (1), we find: 



\nW =- N\nN -i:Ni\n Ni (4) 



Defining some quantities Wi by the equations Ni = Nwj , we make this over 

 into: 



In T^ = - TV S Wi In Wi (5) 



having availed ourselves of the obvious fact that Zwi is equal to unity. 



We might now convert this into an equation for W, but this would be a 

 waste of time and energy, since whenever W has a maximum so also will 

 In W. With In W, therefore, we operate from now on. Making small varia- 

 tions in the quantities Ni , and making therefore small variations — call them 

 8Wt — in the quantities Wi , we find in first approximation for the ensuing 

 change in In W, 



d\nW = - iV S (1 + In Wi) 8 Wi (6) 



