Theory of Diminishing Entropy. 47 
the motion, and therefore impossible. It follows, since 
Maxwell’s law of distribution of velocities asserts the con- 
dition of independence and cannot exist without it, that 
Maxwell’s law and Boltzmann’s H theorem are alike impos- 
sible, at least as accurate propositions. It does not follow 
but that results obtained by their use may be true for rare 
gases to sufficient approximation. 
13. Inasmuch as Willard Gibbs has made great use of 
. . >) e p—e 
this same expression Ae~*2"@*+"+) in the form D=e'¢ ; 
it is right to point out that the objection, which is I think 
fatal to it in Boltzmann’s theorem, is wholly inapplicable to 
it as used by Willard Gibbs. Because Willard Gibbs’s 
systems, unlike Boltzmann’s molecules, do not collide with, 
or in any way influence, one another. That is evidently 
required by the argument of Chapter I. 
14. As a further example of an assumption in aid of our 
mathematics, we might assume the following. Consider the 
two equal elementary extensions in phase DV, and DY,. 
Assume that the number of systems which in time dé pass 
out of DV, into DV, is proportional to the number which for 
the time being are in DV,. That is, if «, denote the density 
in phase for DV,, the number in question is « a, DV, dt, 
where xj. is a positive constant. Similarly the number 
fimemepass from DV, to DV, is x,,% DV,dt... and 
Kpg=Kgp is a positive constant. That seems a reasonable 
assumption. If we now make H= ~ log 2, the summation 
including all the DYV’s, ae g 
oe a“ 1 ra 5 . 
—, and this is re- 
dt Bit db 
ducible, as in Boltzmann’s H theorem, to a series. of terms 
/ 
ly ; 
of the form «(«—.w’) log —, and is therefore negative. Our 
ae 
assumption has enabled us to remove the ambiguity of sign 
in the direction of motion. 
15. So Planck, in his Jrreversible Strahlungsvorgange, 
makes a similar assumption, viz., that the energy which his 
resonator emits into space per unit of time is proportional 
to the energy which for the time being it has. 
16. Again, in the diffusion of gases, let there be in a 
vessel a mixture of oxygen and nitrogen at uniform pressure 
and temperature of the combined gases, but the proportions 
of the respective gases being initially different at different 
points. Experience teaches that the mixture left to itself 
tends to become uniform, so far as this, that anv space large 
enough to be measured will ultimately contain oxygen and 
nitrogen in the same proportions, from whatever part of the 
