84 The Law of Distribution [CH. v 



Extension to time-averages. 



94. Even after agreeing to shelve these difficulties connected with 

 " Continuity of path," there is considerable disagreement among authorities 

 as to the legitimacy of the extension to time-averages. 



Lord Rayleigh writes as follows*: 



"The extension to time-averages, the aspect under which Lord Kelvin 

 has always considered the problem, is important, the more as some authors 

 appear to doubt the possibility of such extension. Thus Professor Bryan f, 

 speaking of Maxwell's assumption, writes : " To discover, if possible, a 

 "general class of dynamical systems satisfying the assumption, would form 

 " an interesting subject for future investigation. It is, however, doubtful 

 " how far Maxwell's law would be applicable to the time-averages of the 

 " energies in any such system. We shall see, in what follows, that the law of 

 " permanent distribution of a very large number of systems is in many cases 

 " not unique. Where there is more than one possible distribution it would 

 " be difficult to draw any inference with regard to the average distribution 

 " (taken with respect to the time) for one system." 



"The extension to time-averages appears to me" (we are still quoting 

 Lord Rayleigh) " to require nothing more than Maxwell's assumption, with- 

 out which the law of distribution itself is only an artificial arrangement, 

 sufficient indeed, but not necessary for steadiness. We shall still speak of 

 the particle moving in two dimensions, though the argument is general. 

 It has been shewn that at any moment the w-energy and the v-energy of 

 the groups of particles J is the same ; and it is evident that the equality 

 subsists if we integrate over any period of time. But if this period be suffi- 

 ciently prolonged, and if Maxwell's assumption be applicable, it makes no 

 difference whether we contemplate the whole group of particles or limit 

 ourselves to a single member of it. It follows that for a single particle the 

 time- averages of u 2 and v 2 are equal, provided the averages are taken over a 

 sufficient length of time. 



" On the other hand, if in any case Maxwell's assumption be untrue, not 

 only is the special distribution unnecessary for steadiness, but even if it be 

 artificially arranged, the law of equal time-averages does not follow as a 

 consequence." 



In the case in which Maxwell's assumption is true, Lord Rayleigh's 

 argument certainly seems (to the present author at least) to be incontro- 

 vertible, and it applies, as he states, to the most general system. The 



* Phil. Mag. [5] XLIX. p. 108. 

 t British Association Report, 1894, 11. 



t Lord Rayleigh's ' ' group of particles " is the same as our assemblage of systems represented 

 in the generalised space. 



