91-93] Maxwell's Treatment of Equipartition 83 



returning to the initial phase. As soon as the initial phase is recovered, 

 a cycle is established and no new phases can be reached, however long the 

 motion may continue." 



It is, however, pretty clear that the assumption cannot be justified, if 

 taken quite literally. It is known that in connection with every dynamical 

 problem, there are an infinite number of re-entrant paths the "periodic 

 /orbits" of astronomy so that obviously a system on one of these paths 

 will never reach the phases outside the one particular path, while a system 

 not on one of these paths can never reach the phases represented by points 

 on them. 



This objection might be met by arguing that the re-entrant paths only 

 form an infinitesimal fraction of the whole, and that it is quite conceivable 

 that all the phases outside these re-entrant paths form a single path. If 

 this were so, it would be immaterial, for a system on this single path, 

 whether we averaged over the whole energy surface, or only over the path. 

 This defence, at any rate at first sight, does not seem very plausible. It 

 requires us to suppose that the paths are all re-entrant, but that one of 

 them is infinitely longer than all the others added together. It must also 

 be noticed that there are dynamical systems in which all the paths are 

 re-entrant and of finite length, as for example occurs in the case instanced 

 by Lord Rayleigh (I.e. ante) of a particle describing an orbit about a centre 

 of force, the law of force being /*r. 



93. An escape is perhaps made possible by assuming that the system 

 does not continually traverse a single path undisturbed, but that by the 

 agency of external forces, it is at times removed from one path to another. 

 If the action of these external agencies is sufficiently fortuitous it may be 

 that it is legitimate to suppose that the system passes through all phases 

 on the energy surface. A warning must, however, be entered as to the 

 nature of the agencies which may be regarded as fortuitous. The essential 

 elements of the question may all be represented by the simple case of a 

 billiard ball moving on a smooth billiard table. Here the impacts of the 

 ball on the cushions are not fortuitous. In fact the cushions may be 

 replaced by a field of repulsive force which becomes infinite at the cushions 

 and vanishes elsewhere, and the motion of the ball is now undisturbed motion 

 in this field of force. Again, if the system consists of two billiard balls 

 moving upon the same table, the collisions between them cannot be regarded 

 as fortuitous, because the impulsive forces between them at collisions can 

 be treated as a special case of a continuous system of forces acting between 

 them. Obviously the same consideration covers the case of a gas of the 

 most general kind moving undisturbed by external agencies, in a closed 

 vessel of any kind. 



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