SYSTEMS COMPOSED OF MOLECULES. 191 



is therefore the probability-coefficient for a phase specifically 

 defined. This has evidently the same value for all the 

 [iY . . . \v h phases obtained by interchanging the phases of 

 particles of the same kind. The probability-coefficient for a 

 generic phase will be \vi_. . . [z^ times as great, viz., 



e . (501) 



We shall say that such an ensemble as has been described 

 is canonically distributed, and shall call the constant its 

 modulus. It is evidently what we have called a grand ensem- 

 ble. The petit ensembles of which it is composed are 

 canonically distributed, according to the definitions of Chapter 

 IV, since the expression 



(502) 



is constant for each petit ensemble. The grand ensemble, 

 therefore, is in statistical equilibrium with respect to specific 

 phases. 



If an ensemble, whether grand or petit, is identical so far 

 as generic phases are concerned with one canonically distrib- 

 uted, we shall say that its distribution is canonical with 

 respect to generic phases. Such an ensemble is evidently in 

 statistical equilibrium with respect to generic phases, although 

 it may not be so with respect to specific phases. 



If we write H for the index of probability of a generic phase 

 in a grand ensemble, we have for the case of canonical 

 distribution 



H = + M.n + >**- _ (503) 



It will be observed that the H is a linear function of e and 

 v v . . . v h ; also that whenever the index of probability of 

 generic phases in a grand ensemble is a linear function of 

 e, j/j, . . . v h i the ensemble is canonically distributed with 

 respect to generic phases. 





