190 SYSTEMS COMPOSED OF MOLECULES. 



which the systems differ only in phase a petit ensemble. A 

 grand ensemble is therefore composed of a multitude of petit 

 ensembles. The ensembles which we have hitherto discussed 

 are petit ensembles. 



Let i>j, . . . v h9 etc., denote the numbers of the different 

 kinds of particles in a system, e its energy, and q l1 . . . q n , 

 p l , . . . p n its coordinates and momenta. If the particles are of 

 the nature of material points, the number of coordinates (n) 

 of the system will be equal to 3 v l . . . + 3 v h . But if the parti- 

 cles are less simple in their nature, if they are to be treated 

 as rigid solids, the orientation of which must be regarded, or 

 if they consist each of several atoms, so as to have more than 

 three degrees of freedom, the number of coordinates of the 

 system will be equal to the sum of v lt i> 2 , etc., multiplied 

 each by the number of degrees of freedom of the kind of 

 particle to which it relates. 



Let us consider an ensemble in which the number of 

 systems having v 19 . . . v h particles of the several kinds, and 

 having values of their coordinates and momenta lying between 

 the limits q l and q^ + dq 1 , p 1 and p l + dp l , etc., is represented 

 by the expression 



(498) 



where IV, O, , /^ , . . . p h are constants, N denoting the total 

 number of systems in the ensemble. The expression 



Q-f Wi - 



Ne (499) 



[vi."h 



evidently represents the density-in-phase of the ensemble 

 within the limits described, that is, for a phase specifically 

 defined. The expression 



e * (500) 



