Law of Partition of Energy. 591 



and asymptotic. Maxwell and Rayleigh on the other hand 

 assume it to be cyclic and reversible. If, therefore, Boltz- 

 mann is right, Maxwell and. Rayleigh are wrong. If 

 Maxwell and Rayleigh are right in their assumption, Boltz- 

 mann must be wrong — that is, his assumption must be 

 untrue, for the proof founded on it is irrefragable. This, I 

 think, is the true state of the case. But it will not make 

 Maxwell and Rayleigh right, either their assumption or the 

 proof founded on it. 



14. The law of equal partition in the form «m 3 = MU 2 has 

 been proved by many writers in many ways. But with the 

 exception of Maxwell and Rayleigh, who (I think) fail to 

 prove the law at all, everyone bases his proof expressly or by 

 implication on the assumption of independence, as does 

 Boltzmann. And all these proofs stand or fall with 

 Boltzmann's. No one has yet pointed out how the assump- 

 tion of the molecular ungeordnet state can be directly used in 

 argument, or how it differs from the implied assumption of 

 independence. 



15. The assumption of the independence of the chances 

 (art. 11) is, as I maintain, untrue. A motion is surely con- 

 ceivable in which molecules very near each other have on 

 average a certain velocity in common. I think this is a 

 necessary ' consequence of the existence of intermolecular 

 forces. It is probably true of a liquid. Why not in some 

 degree of a gas ? But Boltzmann by his fundamental 

 assumption excludes from consideration all cases of this kind. 

 He does not prove their non-existence, he takes it for granted. 



16. In order to express the possibility of such a motion, 

 we must represent the law of distribution of the velocities 

 u t V\W l . . . w n of our molecules (at any given level of poten- 

 tial if they be in a field of external force) by the exponential 

 £-~ h Qdui . . . dw n . Here Q contains not squares of the 

 velocities u x v x , . . w n only, as in Boltzmann's theory, but is 

 ,a quadratic function comprising also products of the form 

 n.m', wd, ww 1 . The object ought to be to keep these products 

 in, not to keep them out. 



.Ontlie Law of Equal Partition if Boltzmann' s assumption 

 be not made. 



17. I will now show what on this hypothesis becomes of 

 the law of equal partition of energy. I do this firstly to 

 obtain a more general result from which Boltzmann's can be 

 deduced as a particular case. Secondly, to show how little 

 vre .gain in ease of analysis by omitting the products from 



