108 Lord Rayleigh on the Law of 



We have thus investigated for a particle in two dimensions 

 the law of steady distribution, and the equal partition of 

 energy which is its necessary consequence. And we see 

 that "the only assumption necessary to the direct proof is that 

 the system, if left to itself in its actual state of motion, will, 

 sooner or later, pass through every phase which is consistent 

 with the equation of energy " (Maxwell) . It will be observed 

 that so far nothing whatever has been said as to time- 

 averages for a single particle. The law of equal partition, as 

 hitherto stated, relates to a large number of particles and to a 

 single moment of time. 



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 Prof. Bryan (Report, § 11, 1894), 

 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 to require 

 nothing more than Maxwell's assumption, without 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 shown that at any moment the u- 

 energy and the v-energy of the group of particles is the same; 

 and it is evident that the equality subsists if we integrate over 

 any period of time. But if this period be sufficiently pro- 

 longed, 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 be 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. 



Having now considered the special problem at full — I hope 



