FOUNDATIONS OF THE KINETIC THEORY OF GASES. 255 



except (is consequences of the mutual impacts of the particles in each of the two 

 systems separately. Professor Boltzmann himself, indirectly and without any 

 justification (such as I have at least attempted to give), assumes almost all that 

 he objects to as redundant in my assumptions, with a good deal more besides. 

 But he says nothing as to the relative numbers of the two kinds of particles. 

 Thus I need not, as yet, take up the question of the validity of Professor 

 Boltzmann's method of investigation (though, as hinted in § 31 of my first 

 paper, I intend eventually to do so) ; and this for the simple reason that, in 

 the present case, I cannot admit his premises. 



Mr Burbuky assumes the non-colliding particles to be in the "special 

 state," and proceeds to prove that the single additional particle will not disturb 

 it. But, supposing for a moment this to be true, it does not prove that the 

 solitary particle would (even after the lapse of ages) reduce any non-colliding 

 system, with positions at any instant, speeds, and lines of motion, distributed 

 absolutely at random (for here there cannot be so much as plausible grounds 

 for the introduction of Professor Boltzmann's assumptions) to the " special 

 state." If it could do so, the perfect reversibility of the motions, practically 

 limited in this case to the reversal of the motion of the single particle alone, 

 shows that the single particle would (for untold ages) continue to throw a 

 system of non-colliding particles further and further out of the " special" state; 

 thus expressly contradicting the previous proposition. In this consequence of 

 reversal we see the reason for postulating a very great number of particles of 

 each kind. If Mr Burbury's sole particle possessed the extraordinary powers 

 attributed to it, it would (except under circumstances of the most exact adjust- 

 ment) not only be capable of producing, but would produce, absolute confusion 

 among non-colliding particles already in the special state. Considering what 

 is said above, I do not yet see any reason to doubt that the assumption of 

 collisions among the particles of each kind, separately, is quite as essential to 

 a valid proof of Maxwell's Theorem as is that of collisions between any two 

 particles, one from each system. I have not yet seen any attempt to prove 

 that two sets of particles, which have no internal collisions, will by their 

 mutual collisions tend to the state assumed by Professor Boltzmann. Nor can 

 I see any ground for dispensing with my farther assumption that the number 

 of particles of one kind must not be overwhelmingly greater than that of the 

 other. A small minority of one kind must (on any admissible assumption) 

 have an average energy which will fluctuate, sometimes quickly sometimes very 

 slowly, within very wide and constantly varying limits. 



De Morgan* made an extremely important remark, which is thoroughly 

 applicable to many investigations connected with the present question. It is 

 to the effect that "no primary considerations connected with the subject of 



* Ennjc. Metropolitana. Art. Theory of Probabilities. 



