434 Assumptions required for the Proof of A vogadro's haw. 



be proved by the help of a different and much more compre- 

 hensive set of assumptions ! " c Et6/3&) ye rvcfxp, Ato^eve?" ! He 

 allows that my proof is correct ; and I am willing (without 

 reading it) to allow as much for his. The point at issue, then, 

 is : — Which of us has made the fewer, or the less sweeping, 

 assumptions ? Another question may even be : — Whose as- 

 sumptions are justifiable ? 



My assumptions are (formally) three, but the first two are 

 expressly regarded as consequences of the third, which is thus 

 my only one, viz. : — 



There is free access for collision between each pair of par- 

 ticles, whether of the same or of different systems ; and the 

 number of particles of one kind is not overwhelmingly greater 

 that that of the other. 



From this I conclude (by general reasoning as to the be- 

 haviour of communities) that the particles will ultimately 

 become thoroughly mixed, and that each system (in conse- 

 quence of its internal collisions) will assume the " special state." 



Prof. Boltzmann denies the necessity for internal collisions 

 in either system, and assumes that (merely by collisions of 

 particles of different kinds) uniform mixing, and distribution 

 of velocities symmetrically about every point, will follow ! 

 Surely this requires proof, if proof of it can be given. So 

 sweeping is the assumption that it makes no proviso as to 

 the relative numbers of the particles in the two systems ! The 

 character of this absolutely tremendous assumption is so totally 

 different from that of mine that it is impossible to compare 

 the two. My assumption has, to say the least, some justifica- 

 tion ; but I fail to see even plausible grounds for admitting 

 that of Prof. Boltzmann. There is no need to inquire as to its 

 truth, at present; for I am not now discussing his extension 

 of Maxwell's Theorem which, of course, is implied in it. The 

 question is : — Is Prof. Boltzmann's assumption, even if cor- 

 rect, sufficiently elementary and obvious to be admitted as an 

 axiom ? It is so wide-reaching as, in effect, to beg the whole 

 question ; and I venture to assert that, on grounds like these, 

 it cannot possibly be shown that any of my assumptions are 

 unnecessary. 



The objection raised in Prof Boltzmann's " Second Ap- 

 pendix" (which is not in my German copy) was made long 

 ago to me by Prof. Newcomb and by Messrs. Watson and 

 Burbury. * I have replied to this also in my Part II., and 

 I will not discuss it now. I need only say that Prof. Boltz- 

 mann, while causelessly attributing to me a silly mathema- 

 tical mistake, has evidently overlooked the special importance 

 which I attach to the assumed steadiness of the " average 

 behaviour of the various groups of a community. 33 



