necessary for Equipartltion of Energy. 595 



§ 17. The ideal gas of the kinetic theory may be supposed 

 to be devoid of mass- velocity, both translational and rota- 

 tional, and to be fully defined, in its steady state, by its 

 density and temperature. For such a gas the only %'s which 

 can occur are the Xi aR d %2 of the last section, so that the 

 solution is 



log/) = A 1 Xi + A 2 x2. 

 Chano-ino- the constants this becomes 



P = Ae~^, (23) 



in which A, h are determined uniquely by the values of the 

 density and temperature. 



For such a gas we have, therefore, proved that there is 

 only one steady state, subject to the hypothesis that the gas 

 is " molekular-ungeordnet," and this steady state is that given 

 by the well-known Boltzmann law. 



To arrive at this result we have found it necessary to 

 suppose that there are only two invariable quantities — the 

 mass and energy of the gas (corresponding to the two varia- 

 bles density and temperature). The result may break down 

 for either of two reasons : 



(i.) It may be that there is some third invariable quantity 

 connected with the coordinates of the gas. If this is so, two 

 samples of a gas having the same temperature and density 

 will not in general possess the same physical properties. 

 The uniformity of the experimental results obtained from 

 different samples of gas, seems to supply an argument of 

 overwhelming strength against supposing this to be the case. 



(ii.) It may be that the two quantities (2%i and 2Xs) which 

 have been supposed to be invariable are not really so. This 

 is certainly the case with the gases of nature, in which the 

 aggregate energy of the molecules is subject to dissipation 

 into the aether. 



If we admit this latter objection it is at once obvious, on 

 physical grounds, that no steady state is possible. Mathe- 

 matically we are left with a solution in which p is propor- 

 tional to X\y an d is therefore constant in the 2n -dimensional 

 space. This solution fails because it does not give p — at 

 infinity. 



Conclusion. 



§ 18. To sum up, we have seen that for a gas of which the 

 molecules are of any kind whatever, the solution for the 

 steady state is unique when a steady state is possible. We 

 have found out how to determine this steady state when the 

 structure of the molecules is completely known. For an 



