116 BELL SYSTEM TECHNICAL JOURNAL 



number of ways in which the first {Ni + N2) of the balls might have come 

 out of the sack without changing the inventory. 



The process of proof need not be carried further. X has been evaluated. 

 It bears no earmark of whatever particular inventory the student may have 

 chosen to adopt at the beginning. It depends only upon the sequence of 

 numbers A^i , A^ , • • • , N u fixed by the caller, which sequence I will hereafter 

 term a "distribution". 



The number of inventories — or ''complexions", to use a commoner word — - 

 for the distribution iVi , • • • , N m is therefore given by the formula, 



W = Nl/Nil N2I • • • NmI = Nl/UNil (1) 



The theorem to which we are advancing afi&rms that this number has its 

 maximum value for the uniform distribution — the distribution in which the 

 caller assigns the same number of balls, N/M, to each of the baskets. 



The usual argument for this statement may be put as follows: Let us 

 assume the uniform distribution, with A = (N/M) balls in each basket, and 

 compare its value of W with that of one of the neighboring distributions 

 such as the one in which there are (A + 1) balls in the first of the baskets, 

 (A — 1) in the second and A in each of the rest. It is not even necessary 

 to get out a pencil and paper to see that W for the latter is less than W for 

 the former, being in fact just A /(A + 1) times as great. The same is 

 evidently true for disarrangements of the uniform distribution which involve 

 more than two baskets and more than one ball per basket. The conclusion 

 is clinched by the obvious fact that when all of the balls are in any one bas- 

 ket, W has its least possible value, viz. unity. (To unite this formally with 

 the previous statements, one must follow the mathematicians' practice of 

 using a symbol 0! or "zero-factorial" and giving it the value unity). 



We shall have to play this not so very entertaining game on several oc- 

 casions in S.M., altering the meaning of the balls and the meaning of the 

 baskets from one occasion to the next. The reader has probably guessed 

 that the balls stand for the molecules. The guess is right in the classical 

 statistics, wrong in the newer forms. To get at the meaning of the baskets, 

 suppose the gas contained in a box of volume V, the interior of which is 

 divided up by impalpable coordinate-planes into compartments or cells all 

 of the same volume Vo . The baskets stand for the cells. 



Now we have the theorem that W is greatest for the uniform distribution 

 of the balls in the baskets, and the assertion that the most probable state 

 of a gas is the state of uniform density, all ready to be fitted together. The 

 process of fitting-together is of the simplest. W is christened the "prob- 

 ability" of the state described by the "distribution" Ni, N2, • • • N m , the 

 quantities Ni now standing for the numbers of molecules in the various cells. 

 Not only is the state of uniformity the most probable one by this definition, 



