ENSEMBLE OF SYSTEMS. 5 



v 



involving also the ^'s but not the a's ; that the potential energy, 

 when it exists, is function of the <?'s and a's ; and that the 

 total energy, when it exists, is function of the jt?'s (or ^s), the 

 9's, and the a's. In expressions like dejdq^ them's, and not 

 the q's, are to be taken as independent variables, as has already 

 been stated with respect to the kinetic energy. 



Lev us imagine a great number of independent systems, 

 identical in nature, but differing in phase, that is, in their 

 condition with respect to configuration and velocity. The 

 forces are supposed to be determined for every system by the 

 same law, being functions of the coordinates of the system 

 q 19 . . . q n , either alone or with the coordinates a 1? a 2 , etc. of 

 certain external bodies. It is not necessary that they should 

 be derivable from a force-function. The external coordinates 

 a 15 a 2 , etc. may vary with the time, but at any given time 

 have fixed values. In this they differ from the internal 

 coordinates q 1 , . . . q n , which at the same time have different 

 values in the different systems considered. 



Let us especially consider the number of systems which at a 

 given instant fall within specified limits of phase, viz., those 

 for which 



Pi <Pi< Pi", qi <qi< q", 



Pn <Pn< P", qn < q < 



the accented letters denoting constants. We shall suppose 

 the differences p^' p{, q^ q^, etc. to be infinitesimal, and 

 that the systems are distributed in phase in some continuous 

 manner,* so that the number having phases within the limits 

 specified may be represented by 



i') (?" - ?'), (10) 



* In strictness, a finite number of systems cannot be distributed contin- 

 uously in phase. But by increasing indefinitely the number of systems, we 

 may approximate to a continuous law of distribution, such as is here 

 described. To avoid tedious circumlocution, language like the above may 

 be allowed, although wanting in precision of expression, when the sense in 

 which it is to be taken appears sufficiently clear. 



