AG Mr. 8. H. Burbury on the 
volume of molecules of the class m whose velocities at a 
given instant lie between the limits 
Dy ae Uy Oy 0, 2. ee aD, 
And let /’ du’... dw,’ denote the number per unit volume 
of the class m’ whose velocities lie between the limits 
uy... 14’ +du,’ &e. 
Then the number per unit volume of pairs (m, m') whose 
velocities respectively lie within the limits aforesaid is 
i J! dy. 
That statement involves no physical assumption. Then he 
says the number of these pairs for which in time dt m col- 
lides in a given way with the m’ of the same pair is propor- 
tional to /. f’ di, ... dw,’. That statement involves a physical 
assumption of the most important and far-reaching character, 
namely (see Art. 11), that the chance of a molecule having 
velocities within the assigned limits is at every instant in- 
dependent of the positions and velocities of all the other 
molecules for the time being. For if not, the number of 
collisions between m and m’ is not necessarily proportional 
to the product of fdu,...dw, and f’ du’ ...dw,’, because 
there may exist a stream by virtue of which m and m/’ 
have on average velocities of the same sign, and that 
affects the frequency of such collisions. It may conceivably 
be true that there are no such streams, but it is not axiomatic. 
On this assumption Boltzmann proves rigorously that Max- 
well’s law of distribution of velocities, namely, that the chance 
of the molecules m, mg... m, having at any instant velocities 
between the limits uw and u,+du, ... w, and w,+dw, is pro- 
portional to e~*=m"+e+e) du, ... dwn, 18 a sufficient condition 
for stationary motion; and he further proves, by the H theorem, 
that it is a necessary condition, because H diminishes irre- 
versibly until that distribution is attained. 
11. It will be observed that Maxwell’s law expresses the 
condition of independence of Art. 10, and cannot, therefore, 
be true unless the condition of independence is true. Boltz- 
mann’s assumption, therefore, whatever its apparent form, 
as it is sufficient to prove Maxwell’s law, necessarily involves 
the condition of independence. 
12. Now it has been shown (see the paper by Mr. J. H. 
Jeans on the Kinetic Theory of Gases, Phil. Mag. May 
1903), and I think it is unquestionably true, that in a system 
of molecules which have finite dimensions, or act on each 
other at finite distances, the continued existence of the con- 
dition of independence is inconsistent with the continuity of 
