MEMORIAL TO CLASSICAL STATISTICS 119 



Now we are restricting ourselves to variations in the quantities Ni which 

 leave unchanged the total number of molecules in the cells, or of balls in the 

 baskets — to variations, therefore, for which 



S iVi = iV = constant, 2 5 Wi = (7) 



This restriction being miroduced into (6), 8 In W proceeds to vanish if and only 

 ifu'i has the same value for all of the cells. Now, the vanishment of 8 In W is 

 a necessary condition for having a maximum of W at the situation in ques- 

 tion. I do not refer to it as a sufficient condition, because it admits of a 

 minimum or of what is technically known as a "stationary" value of W in the 

 situation in question. However it has already been shown, without the aid 

 of the Stirling approximation, that the expression to which we are approxi- 

 mating is greater for the uniform distribution than for the neighboring non- 

 uniform ones. It may therefore be accepted that here we have a maximum 

 of TI' for the uniform distribution, and have reached the old result in a new 

 way; an achievement nearly useless, were it not a prelude to the performance 

 in momentum-space. 



I continue to use the symbols W and N and Ni and Wi , but now with ref- 

 erence to the distribution of the representative dots in momentum-space. 

 A new symbol, E; , shall signify the energy of a molecule in the iih cell of the 

 momentum-space. We wish at all costs to avoid the conclusion that the 

 stable distribution in the momentum-space is the uniform one. Boltzmann 

 managed to avoid it, and his was the following way: 



Let us write, for the number of molecules in the ith cell, the expression: 



Nwi = NA exp (-BEi) (8) 



and insert it into (6). We shall find: 



8 In W = - A^ S (1 -f- In^ - BE,) 8wi . (9) 



Of the three terms on the right, two vanish for all variations in which the 

 total number of molecules remains the same. The third does not — but it 

 will vanish for a restricted class of these variations, to wit, those and those 

 only for which the total energy of all the molecules remains the same; for 

 XIWiEi is precisely that total energy. 



Some writers at this point ask the student to imagine a gas in a container 

 being completely cut oS from energy-interchange with the container-walls 

 and with the whob of the outside world, and therefore being limited to the 

 particular c'-value with which it started out. Others import the word "tem- 

 perature" which I am desperately (and vainly) trying to keep out until I am 

 ready to bring it formally into the discourse, and aver that the gas is nearly 

 or quite so limited if the walls of the container have the same temperature as 

 the gas itself. The student may take his choice, but must suppose that 



