370 BELL SYSTEM TECHNICAL JOURNAL 



greater and greater, the first term on the right recedes into relative insignif- 

 icance; and with an ever-increasing degree of approximation, we have: 



Nj = 7\Me"^^' (21) 



with NA put for Ce~^ and B for Q — which is the Maxwell-Boltzmann law 

 and the law confirmed by experiment. 



A helpful and troublesome coincidence between two different quantities 



When in the earlier article I used the section-heading repeated just above, 

 it referred to the near-equality between the logarithms of two different 

 numbers: one number being that of all the complexions compatible with the 

 most probable distribution (of numbered atoms sprinkled among numbered 

 cells in ordinary space) and the other being that of all the complexions 

 compatible with all conceivable distributions altogether. The most probable 

 distribution had so great a share of all conceivable complexions, that no 

 grave error was committed in pretending (so long as we were dealing with 

 In W) that it actually had them all without exception! 



A similar coincidence occurs in the new statistics, and will now be set 

 forth. 



Consider the jth region by itself. In (15) I have given the expression 

 for In PFmax, the logarithm of the number of ways in which the most probable 

 distribution of cells among populations can be realized. This is now to be 

 compared with In PFtot, the logarithm of the total number of ways in which 

 all possible distributions of cells among populations can be realized. Note 

 that I say "all possible" and not "all conceivable" distributions! The only 

 possible ones are those which are compatible with the fixed number iV,- 

 of atoms. This limitation prevents us from proceeding by the easy route 

 of the earlier article. Indeed in order to solve the problem "in how many 

 ways can all possible distributions of cells among populations be realized?" 

 it is necessary, or at any rate customary, to restate it in a very different 

 manner, which is the following: 



In how many different ways can iV, un-numbered balls be distributed among 

 C numbered baskets? Two ways are considered as different unless ni = ni 

 for every value of i (fii and Ui standing for the populations of the ith basket 

 in the two ways). 



Notice that again the balls stand for the atoms and the baskets for 

 compartments in space, as they did in the old statistics! We are playing 

 a new game with the old baskets and the old balls, instead of playing the 

 old game with new balls and new baskets as we have just finished doing. 

 It has to be a new game, for the numbers have been removed from the old 

 balls and the old game is therefore unplayable. 



This is, to put it mildly, one of the less perspicuous problems of the 



