188 SYSTEMS COMPOSED OF MOLECULES. 



the indentification of any particular particle of the first system 

 with any particular particle of the second. And this would 

 be true, if the ensemble of systems had a simultaneous 

 objective existence. But it hardly applies to the creations 

 of the imagination. In the cases which we have been con- 

 sidering, and in those which we shall consider, it is not only 

 possible to conceive of the motion of an ensemble of similar 

 systems simply as possible cases of the motion of a single 

 system, but it is actually in large measure for the sake of 

 representing more clearly the possible cases of the motion of 

 a single system that we use the conception of an ensemble 

 of systems. The perfect similarity of several particles of a 

 system will not in the least interfere with the identification 

 of a particular particle in one case with a particular particle 

 in another. The question is one to be decided in accordance 

 with the requirements of practical convenience in the discus- 

 sion of the problems with which we are engaged. 



Our present purpose will often require us to use the terms 

 phase, density-in-phase, statistical equilibrium, and other con- 

 nected terms on the supposition that phases are not altered 

 by the exchange of places between similar particles. Some 

 of the most important questions with which we are concerned 

 have reference to phases thus defined. We shall call them 

 phases determined by generic definitions, or briefly, generic 

 phases. But we shall also be obliged to discuss phases de- 

 fined by the narrower definition (so that exchange of position 

 between similar particles is regarded as changing the phase), 

 which will be called phases determined by specific definitions, 

 or briefly, specific phases. For the analytical description of 

 a specific phase is more simple than that of a generic phase. 

 And it is a more simple matter to make a multiple integral 

 extend over all possible specific phases than to make one extend 

 without repetition over all possible generic phases. 



It is evident that if i>i, v z . . . v h , are the numbers of the dif- 

 ferent kinds of molecules in any system, the number of specific 

 phases embraced in one generic phase is represented by the 

 continued product [z^ [^ ]^ and the coefficient of probabil- 



