140 MOTION OF SYSTEMS AND ENSEMBLES 



generate the volume through which it passes. In equal times 

 the front generates equal extensions in phase. This is an 

 immediate consequence of the principle of conservation of 

 extension-in-phase^ unless indeed we prefer to consider it as 

 a slight variation in the expression of that principle. For in 

 two equal short intervals of time let the extensions generated 

 be A and B. (We make the intervals short simply to avoid 

 the complications in the enunciation or interpretation of the 

 principle which would arise when the same extension-in-phase 

 is generated more than once in the interval considered.) Now 

 if we imagine that at a given instant systems are distributed 

 throughout the extension A, it is evident that the same 

 systems will after a certain tune occupy the extension B, 

 which is therefore equal to A in virtue of the principle cited. 

 The front of the ensemble, therefore, goes on generating 

 equal extensions in equal times. But these extensions are 

 included in a finite extension, viz., that bounded by certain 

 limiting values of the energy. Sooner or later, therefore, 

 the front must generate phases which it has before generated. 

 Such second generation of the same phases must commence 

 with the initial phases. Therefore a portion at least of the 

 front must return to the original extension-in-phase. The 

 same is of course true of the portion of the ensemble which 

 follows that portion of the front through the same phases at 

 a later time. 



It remains to consider how large the portion of the ensemble 

 is, which will return to the original extension-in-phase. There 

 can be no portion of the given extension-in-phase, the systems 

 of which leave the extension and do not return. For we can 

 prove for any portion of the extension as for the whole, that 

 at least a portion of the systems leaving it will return. 



We may divide the given extension-in-phase into parts as 

 follows. There may be parts such that the systems within 

 them will never pass out of them. These parts may indeed 

 constitute the whole of the given extension. But if the given 

 extension is very small, these parts will in general be non- 

 existent. There may be parts such that systems within them 



