6 VARIATION OF THE 



or more briefly by 



. . . dp n dq l . . . dq n , (li) 



where D is a function of the p's and q's and in general of t alb 3, 



for as time goes on, and the individual systems change the\r 



phases, the distribution of the ensemble in phase will in gen- 



eral vary. In special cases, the distribution in phase will 



remain unchanged. These are cases of statistical equilibr turn. 



If we regard all possible phases as forming a sort oi exten- 



ision of 2 n dimensions, we may regard the product of differ- 



fentials in (11) as expressing an element of this extension, and 



\D as expressing the density of the systems in that element. 



We shall call the product 



dp l ... dp n dq lf . . dq n (12) 



an element of extensionrin-phase, and D the density-inr-phase 

 of the systems. 



It is evident that the changes which take place in the den- 

 sity of the systems in any given element of extension-in- 

 phase will depend on" the dynamical nature of the systems 

 and their distribution in phase at the time considered. 



In the case of conservative systems, with which we shall be 

 principally concerned, their dynamical nature is completely 

 determined by the function which expresses the energy (e) in 

 terms of the |?'s, <?'s, and a's (a function supposed identical 

 for all the systems) ; in the more general case which we are 

 considering, the dynamical nature of the systems is deter- 

 mined by the functions which express the kinetic energy (e p ) 

 in terms of the p's and <?'s, and the forces in terms of the 

 <?'s and 's. The distribution in phase is expressed for the 

 time considered by D as function of the p's and q's. To find 

 the value of dD/dt for the specified element of extension-in- 

 phase, we observe that the number of systems within the 

 limits can only be varied by systems passing the limits, which 

 may take place in 4 n different ways, viz., by the p l of a sys- 

 tem passing the limit p^, or the limit p/', or by the q l of a 

 system passing the limit q^ or the limit <?/', etc. Let us 

 consider these cases separately. 



