344 Prof. Tait on the Foundations of 



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 has since been ex- 

 tended by Boltzmann and others to cases in which the individual 

 particles are no longer supposed to be hard smooth spheres, but 

 complex systems having great numbers of degrees of freedom. 

 And it is stated, as the result of a process which 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 rigid : — the examination of its spectrum 

 wdiile incandescent proves this at once. Bat 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 

 relations between the two specific heats of a gas. 



Hence it is desirable that 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 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. 



1. The chief features, besides too great conciseness, in 

 respect of which Maxwell's proof is objectionable are : — 



(a) He assumes that the transference of energy from one 

 system to the other can be calculated from the results of 

 a single impact between particles, one from each system, 

 each having the average translational energy of its system. 



Thus (so far as this step is concerned) the distribution of 

 energy in each system may be any whatever. 



(b) In this typical impact the velocities of the impinging 

 spheres are taken as at right angles to one another, so that 

 the relative speed may be that of mean square as between the 

 particles of the two systems. The result obtained is fallacious 

 because, in general, the directions of motion after impact are 

 found not to be at right angles to one another, as they would 

 certainly be (on account of the perfect reversibility of the 

 motions) were this really a typical impact. 



