FOUNDATIONS OF THE KINETIC THEORY OF GASES. 77 



IV. Clerk- Maxwell's Theorem. 



15. In the ardour of his research of 1859, # Maxwell here and there con- 

 tented himself with very incomplete proofs (we can scarcely call them more 

 than illustrations) of some of the most important of his results. This is 

 specially the case with the investigation of the law of ultimate partition of 

 energy in a mixture of smooth spherical particles of two different kinds. He 

 obtained, in accordance with the so-called Law of Avogadro, the result that the 

 average energy of translation is the same per particle in each system; and he 

 extended this in a Corollary to a mixture of any number of different systems. 

 This proposition, if true, is of fundamental importance. It was extended by 

 Maxwell himself to the case of rigid particles of any form, where rotations 

 perforce come in. And it appears that in such a case the whole energy is 

 ultimately divided equally among the various degrees of freedom. It has since 

 been extended by Boltzmann and others to cases in which the individual 

 particles are no longer supposed to be rigid, but are regarded as complex 

 systems having great numbers of degrees of freedom. And it is stated, as the 

 result of a process which, from the number and variety of the assumptions made 

 at almost every stage, is rather of the nature of playing with symbols than of 

 reasoning by consecutive steps, that in such groups of systems the ultimate 

 state will be a partition of the whole energy in equal shares among the classes 

 of degrees of freedom which the individual particle-systems possess. This, if 

 accepted as true, at once raises a formidable objection to the kinetic theory. 

 For there can be no doubt that each individual particle of a gas has a very 

 great number of degrees of freedom besides the six which it would have if it 

 were rigid : — the examination of its spectrum while incandescent proves this at 

 once. But if all these degrees of freedom are to share the whole energy (on the 

 average) equally among them, the results of theory will no longer be consistent 

 with our experimental knowledge of the two specific heats of a gas, and the 

 relations between them. 



16. Hence it is desirable that Clerk-Maxwell's proof of his fundamental 

 Theorem should be critically examined, and improved where it may be found 

 defective. If it be shown in this process that certain preliminary conditions 

 are absolutely necessary to the proof even of Clerk-Maxwell's Theorem, and 

 if these cannot be granted in the more general case treated by Boltzmann, it is 

 clear that Boltzmann's Theorem must be abandoned. 



17. The chief feature in respect of which Maxwell's investigation is to be 

 commended is its courageous recognition of the difficulties of the question. 

 In this respect it far transcends all other attempts which I have seen. Those 



* Phil. Mag., 1860. 



