148 MOTION OF SYSTEMS AND ENSEMBLES 



average value of the index of probability of phase, is consist- 

 ent with an approach to a limiting condition in which that 

 average value is less. We might perhaps fairly infer from 

 such considerations as have been adduced that an approach 

 to a limiting condition of statistical equilibrium is the general 

 rule, when the initial condition is not of that character. But 

 the subject is of such importance that it seems desirable to 

 give it farther consideration. 



Let us suppose that the total extension-in-phase for the 

 kind of system considered to be divided into equal elements 

 (D V) which are very small but not infinitely small. Let us 

 imagine an ensemble of systems distributed in this extension 

 in a manner represented by the index of probability 77, which 

 is an arbitrary function of the phase subject only to the re- 

 striction expressed by equation (46) of Chapter I. We shall 

 suppose the elements D V to be so small that rj may in gen- 

 eral be regarded as sensibly constant within any one of them 

 at the initial moment. Let the path of a system be defined as 

 the series of phases through which it passes. 



At the initial moment (') a certain system is in an element 

 of extension DV f . Subsequently, at the time ", the same 

 system is in the element DV". Other systems which were 

 at first in DV will at the time t" be in DV", but not all, 

 probably. The systems which were at first in DV 1 will at 

 the time t' f occupy an extension-in-phase exactly as large as at 

 first. But it will probably be distributed among a very great 

 number of the elements (DV) into which we have divided 

 the total extension-in-phase. If it is not so, we can generally 

 take a later time at which it will be so. There will be excep- 

 tions to this for particular laws of motion, but we will con- 

 fine ourselves to what may fairly be called the general case. 

 Only a very small part of the systems initially in D V will 

 be found in DV" at the time t", and those which are found in 

 DV" at that time were at the initial moment distributed 

 among a very large number of elements D V. 



What is important for our purpose is the value of 77, the 

 index of probability of phase in the element DV" at the time 



