142 MOTION OF SYSTEMS AND ENSEMBLES 



that extension-in-phase, if they leave it, the exceptions, i. g., 

 the number which pass out of the extension-in-phase and do 

 not return to it, being less than any assignable fraction of the 

 whole number. In other words, the probability that a system 

 taken at random from the part of a canonical ensemble which 

 is contained within any given extension-in-phase, will pass out 

 of that extension and not return to it, is zero. 



A similar theorem may be enunciated with respect to a 

 roicrocanonical ensemble. Let us consider the fractional part 

 of such an ensemble which lies within any given limits of 

 phase. This fraction we shall denote by F. It is evidently 

 constant in time since the ensemble is in statistical equi- 

 librium. The systems within the limits will not in general 

 remain the same, but some will pass out in each unit of time 

 while an equal number come in. Some may pass out never 

 to return within the limits. But the number which in any 

 time however long pass out of the limits never to return will 

 not bear any finite ratio to the number within the limits at 

 a given instant. For, if it were otherwise, let / denote the 

 fraction representing such ratio for the tune T. Then, in 

 the time T, the number which pass out never to return will 

 bear the ratio f F to the whole number in the ensemble, and 

 in a time exceeding T/(fF) the number which pass out of 

 the limits never to return would exceed the total number 

 of systems in the ensemble. The proposition is therefore 

 proved. 



This proof will apply to the cases before considered, and 

 may be regarded as more simple than that which was given. 

 It may also be applied to any true case of statistical equilib- 

 rium. By a true case of statistical equilibrium is meant such 

 as may be described by giving the general value of the prob- 

 ability that an unspecified system of the ensemble is con- 

 tained within any given limits of phase.* 



* An ensemble in which the systems are material points constrained to 

 move in vertical circles, with just enough energy to carry them to the 

 highest points, cannot afford a true example of statistical equilibrium. For 

 any other value of the energy than the critical value mentioned, we might 



