114 Lord Rayleigh on the Law of 



If the integration between -f- so be carried out completely, we 

 recover the value unity. 



The interest of the case where n is very great lies of course 

 in the application to a gas supposed to consist of an immense 

 number of similar molecules*, or of several sets of similar 

 molecules ; and the question arises whether (45) can be 

 applied to deduce the Maxwellian law of distribution of 

 velocities among the molecules of a single system at a given 

 instant of time. A caution may usefully be interposed here 

 as to the sense in which the Maxwellian distribution is to be 

 understood. It would be absurd to attempt to prove that the 

 distribution in a single system is necessarily such and such, for 

 we have already assumed that every phase, including every 

 distribution of velocities, is attainable, and indeed attained if 

 sufficient time be allowed. The most that can be proved is 

 that the distribution will approximate to a particular law for 

 the greater part of the time, and that if sensible deviations 

 occur they will be transitory. 



In applying (45) to a gas it will be convenient to suppose 

 in the first instance that all the molecules are similar. Each 

 molecule has several degrees of freedom, but we may fix our 

 attention upon one of them, say the #- velocity of the centre 

 of inertia, usually denoted by u. In (45) the whole system is 

 supposed to occupy a given configuration ; and the expression 

 gives us the distribution of velocity at a given time for a 

 single molecule among all the systems. The distribution of 

 velocity is the same for every other molecule, and thus the 

 expression applies to the statistics of all the molecules of all 

 the systems. Does it also apply to the statistics of all the 

 molecules of a single system ? In order to make this inference 

 we must assume that the statistics are the same (at the same 

 time) for all the systems, or, what comes to the same thing 

 (if Maxwell's assumption be allowed), that they are the same 

 for the same system at the various times when it passes through 

 a given configuration. 



Thus far the argument relates only to a single configuration. 

 If the configuration be changed, there will be in general a 

 change of potential energy and a corresponding change in the 

 kinetic energy to be distributed amongst the degrees of free- 

 dom. But in the case of a gas, of which the statistics are 

 assumed to be regular, the potential energy remains approxi- 

 mately constant when exclusion is made of exceptional 

 conditions. The same law of distribution of velocity then 

 applies to every configuration, that is, it may be asserted 



* The terms " gas " and " molecule " are introduced for the sake of 

 brevity. The question is still purely dynamical. 



