82 The Law of Distribution [CH. v 



91. The result just reached is Maxwell's main result. If we wish it to 

 'apply to the motion of dynamical systems, we must suppose an assemblage 



of systems started with energies intermediate between the narrow limits 

 E and E Q + dE, in such a way that their density in the generalised space 

 is uniform, i.e., so that all values of the coordinates and momenta which are 

 consistent with the energy lying within the specified limits, are equally 

 probable. The separate systems have of course no interaction one with 

 another. It then follows that initially and throughout all time the mean 

 energies of the various momentoids are equal. 



By addition over all possible values of the energy, we can arrive at the 

 result that for an assemblage of systems having all possible values for the 

 coordinates and momenta, provided only they are started so that the initial 

 density in the generalised space is uniform, the mean energies of the various 

 momentoids are equal. 



We can, however, obtain a result more general than this. The motion 

 in the generalised space is confined to the loci E = constant, so that if we 

 take an initial distribution of density (pj in the generalised space such that 



pi = *C&) (203), 



where <J>(E) is any function of the energy, then this distribution is a 

 permanent distribution, i.e., equation (203) is satisfied through all time. 

 And by addition of the result obtained in the last section, it follows that in 

 this assemblage the mean values of the energies of the various momentoids 

 are equal. 



Continuity of path. 



92. Attempts have been made to extend this result j/ojthe time-average 

 of the energies of the various momentoids of a single system. It is obvious 

 that this cannot be done without further assumption of some kind. For 

 instance it may be that the path of the single system is entirely confined 

 to a certain definite region of the energy-surface on which it is moving, and 

 in this case it would obviously be fallacious to calculate the time-average 

 by integrating over the whole surface. The assumption which is usually 



ade, in order to make the extension to a -time-average possible, is that 

 generally known as the assumption of continuity of path. It is "that the 

 system, if left to itself, will, sooner or later, pass through every phase which 

 is consistent with the conservation of energy"*. Lord Rayleigh-f- points out 

 that " if we take it quite literally, the assumption is of a severely restrictive 

 character ; for it asserts that the systems, starting from any phase, will 

 traverse every other phase (consistent with the energy condition) before 



* Maxwell, Collected Works, u. p. 714. 



t "The law of partition of Kinetic Energy," Phil. Mag. [5] XLIX. p. Ill, 1900. 



