PROBABILITY OF PHASE. 17 



The condition of statistical equilibrium may be expressed 

 by equating to zero the second member of either of these 

 equations. 



The same substitutions in (22) give 



.,=' (43) 



(IX.... =- (44) 



That is, the values of P and rj, like those of D, are constant 

 in time for moving systems of the ensemble. From this point 

 of view, the principle which otherwise regarded has been 

 called the principle of conservation of density-in-phase or 

 conservation of extension-in-phase, may be called the prin- 

 ciple of conservation of the coefficient (or index) of proba- 

 bility of a phase varying according to dynamical laws, or 

 more briefly, the principle of conservation of probability of 

 phase. It is subject to the limitation that the forces must be 

 functions of the coordinates of the system either alone or with 

 the time. 



The application of this principle is not limited to cases in 

 which there is a formal and explicit reference to an ensemble of 

 systems. Yet the conception of such an ensemble may serve 

 to give precision to notions of probability. It is in fact cus- 

 tomary in the discussion of probabilities to describe anything 

 which is imperfectly known as something taken at random 

 from a great number of things which are completely described. 

 But if we prefer to avoid any reference to an ensemble 

 of systems, we may observe that the probability that the 

 phase of a system falls within certain limits at a certain time, 

 is equal to the probability that at some other time the phase 

 will fall within the limits formed by phases corresponding to 

 the first. For either occurrence necessitates the other. That 

 is, if we write P' for the coefficient of probability of the 

 phase pi, q n ' at the time ^, and P" for that of the phase 

 jp/', . . . q n " at the time tf', 



2 



