MEMORIAL TO CLASSICAL STATISTICS 129 



ing the stipulation that we will admit only such variations of the quantities 

 Wi and Wi as leave N' and N" and i/ie sum of the energies U' and U" — not how- 

 ever the individual energies U' and U" — unchanged. In acting this way 

 we are doing something which may be regarded as "considering a mixture", 

 since we are allowing for the possibility that energy may pass from the one 

 gas to the other and the other to the one. Equally well are we considering 

 the case of two gases separated by a partition through which energy may 

 pass, but not the atoms. Since in such a case we really ought to take into 

 account the atoms and the energy of the partition also, we must appease the 

 critics by providing that the partition shall be very thin. 



Choose any set of values of the quantities N i , which is to say, any particu- 

 lar distribution of the first gas; and choose any set of values of the quantities 

 Ni , which is to say, an}^ particular distribution of the second gas. Go back 

 to equation (1) and put primes on all the symbols N, Ni , N^ , • • • on the 

 right-hand side of that equation. The resulting expression gives the total 

 number of inventories or complexions of the first gas. Take oflf the primes 

 and affix double primes to each of these symbols. The resulting expression 

 gives the total number of inventories or complexions of the second gas. 

 Every complexion of either may coexist with any complexion of the other. 

 Therefore the total number of complexions of the pair of gases is the product 

 of the two expressions. It is this product which is TT for the pair of gases, be 

 they mixed or side-by-side. 



With use of the Stirling approximation, the logarithm of IF for the pair 

 is the sum of two such expressions as we have seen in (5) : 



In W = -N"Zwi In w- - N^Zw- In w'! (32) 



and its variation is: 



8 In W = -iV'2(l + In Wi)8wi - iV"2(l -f In w")8w7 (33) 



Let us now give a trial to the tentative distribution, 



w'i = A' exp (-B'Ei), w" = A" exp {~B"E>> (34) 



On substituting this into {33) we find that if B' is unequal to B", the dis- 

 tribution has a stationary value of IF with respect only to such variations 

 as leave the energies of the two gases separately unchanged — the result 

 which we had before. If however B' and B" are the same, then IF is 

 stationar>^ with respect to variations which leave the sum of the energies 

 unchanged, either being allowed to gain or lose so long as the other loses or 

 gains by an equal amount. Since each B is controlled by the corresponding 

 U/N, the distribution {33) has a stationary value of IF for variations of the 

 type in question if and only if the average energy of the atoms of each gas 

 is the same. Since each B controls the corresponding A , this condition of 



