THROUGH LONG PERIODS OF TIME. 141 



will all pass out of the given extension and all return within 

 it. The whole of the given extension-in-phase is made up of 

 parts of these two kinds. This does not exclude the possi- 

 bility of phases on the boundaries of such parts, such that 

 systems starting with those phases would leave the extension 

 and never return. But in the supposed distribution of an 

 ensemble of systems with a uniform density-in-phase, such 

 systems would not constitute any assignable fraction of the 

 whole number. 



These distinctions may be illustrated by a very simple 

 example. If we consider the motion of a rigid body of 

 which one point is fixed, and which is subject to no forces, 

 we find three cases. (1) The motion is periodic. (2) The 

 system will never return to its original phase, but will return 

 infinitely near to it. (3) The system will never return either 

 exactly or approximately to its original phase. But if we 

 consider any extension-in-phase, however small, a system 

 leaving that extension will return to it except in the case 

 called by Poinsot * singular,' viz., when the motion is a 

 rotation about an axis lying in one of two planes having 

 a fixed position relative to the rigid body. But all such 

 phases do not constitute any true extension-in-phase in the 

 sense in which we have defined and used the term.* 



In the same way it may be proved that the systems in a 

 canonical ensemble which at a given instant are contained 

 within any finite extension-in-phase will in general return to 



* An ensemble of systems distributed in phase is a less simple and ele- 

 mentary conception than a single system. But by the consideration of 

 suitable ensembles instead of single systems, we may get rid of the incon- 

 venience of having to consider exceptions formed by particular cases of the 

 integral equations of motion, these cases simply disappearing when the 

 ensemble is substituted for the single system as a subject of study. This 

 is especially true when the ensemble is distributed, as in the case called 

 canonical, throughout an extension-in-phase. In a less degree it is true of 

 the microcanonical ensemble, which does not occupy any extension-in-phase, 

 (in the sense in which we have used the term,) although it is convenient to 

 regard it as a limiting case with respect to ensembles which do, as we thus 

 gain for the subject some part of the analytical simplicity which belongs to 

 the theory of ensembles which occupy true extensions-in-phase. 



