﻿the Partition of Energy, 477 



in unit volume, that is to say, Jt> = P/ Vlog(l/$). Since the 

 gas temperature is measured by the relation pV = P£T, we 

 are again led to the relation d = e~ lik . 



We may observe that it is now possible to drop the 

 assumption of commensurability, which was necessary in 

 the sections which dealt with quantized systems. It was 

 there essential, physically speaking, in order that it should 

 be possible that the whole of the energy should be held 

 somewhere; but as we now have molecules which can hold 

 energy in any amounts, it may be dispensed with, the modifi- 

 cation being justified on the same assumptions and by 

 the same sort of limiting process as have been used in this 

 section. Again, we can see that the correct results are 

 obtained if H(~) replaces h(z) in (12*2) and all the other 

 integrals, even though the interpretation as coefficient in a 

 power series is no longer possible, and though the integrand 

 is no longer single valued. In such many- valued integrands 

 the limiting process shows that we simply require to take 

 that value which is real on the positive side of the real axis. 



§ 13. The Maxwell Distribution Law. 



We have carried out the whole process so far with quantized 

 systems included in the assembly, but it may be observed 

 that it is immediately applicable to an assembly composed 

 solely of molecules. If this is done the value of c t in (12*51) 

 establishes at once the Maxwell distribution law, and its- 

 fluctuation in (12 # 55) proves that it is a normal property of 

 the assembly. This is probably the simplest complete proof 

 of the ordinary distribution law ; its special advantage is 

 that by means of the fluctuations it is easily established that 

 the actual distribution will hardly ever be far from the 

 average. 



The method can also be made to establish the distribution 

 law for a mixture of gases *, and indeed for a mixture of 

 any kind, provided that the systems can be considered to 

 have separate energies. 



It is also possible to extend the method to cases in which 

 the total momentum or angular momentum is conserved, by 

 constructing partition functions in more than one independent 

 variable. In fact, there will be as many independent 

 variables as there are uniform integrals of the dynamical 

 equations of the assembly. For simplicity we shall suppose 

 that the linear momentum in a given direction is conserved, 



* The effects of the semi-permeable membranes of thermodynamics- 

 can be conveniently treated by the partition function. 



