‘686 
cept at the instants of collision. Many ingeni- 
ous minds have since occupied themselves with 
this problem; and many discussions of it have 
been published with the purpose of improving 
upon the work of Maxwell, though none, so 
far as the reviewer is aware, has reached, 
for the case of a much rarefied gas, a different 
result. The especial defect of Maxwell’s argu- 
ment is his failure to show that the condition 
which he arrives at as the final condition of 
the gas is a necessary state, although he has 
shown it to be a possible state. Boltzmann es- 
pecially has undertaken to supply what was 
lacking in the demonstration of Maxwell. 
‘Starting with a gas which has not yet reached 
its condition of ‘stationary motion,’ and in 
which the particles influence each other only at 
impact, he made a very particular study of the 
possibilities and results of collisions, with the 
purpose of showing that these results would as 
a whole tend to bring about the state of Max- 
wellian distribution of velocities, which would 
therefore bea necessary and final state. Asan 
indispensable part of his argument he framed 
-and used the so-called H-theorem. To attempt 
here a definite statement of this theorem would 
be folly. Let it suffice that AH is a function 
based upon the laws of probability and that, 
according to Boltzmann, it necessarily de- 
creases, through collisions, with lapse of time 
and by its diminution marks the progress of 
the gas towards the Maxwellian state, which is 
attained when H becomes a minimum. But 
critics have objected, Why must the # function 
diminish? If we imagine the velocity of every 
particle of the gas reversed at any instant, the 
H function ought to increase. Are not the re- 
verse velocities as probable as those you im- 
agine? And should not the net effect of all 
collisions be to leave H unchanged? ‘To this 
Boltzmann replied that reverse velocities would 
indeed cause H to increase; but he urged 
that it was not allowable to imagine every ve- 
locity reversed. For example, in a case where 
a partial mixture of gases has come about by 
interdiffusion, a reversal of all velocities would 
cause the gases to separate from each other. 
This was an admirable and enlightening reply 
to the doubt raised, but the discussion is so 
beset with difficulties and possible obscurities 
SCIENCE. 
[N.S. Vou. X. No. 254. 
that Mr. Burbury has done students good ser- 
vice in examining with much care a funda- 
mental assumption upon which the argument of 
Boltzmann is based. Burbury’s statement of 
this ‘assumption A’ is as follows: 
‘““The chance of any molecule having velocity 
in « between u and w+ du is independent, not 
only of its position in space, but also of the 
v, w, which it has in directions y and z, and 
further except in the case mentioned below, 
it is independent of the positions and velocities 
at the instant of all the other molecules of the 
system. The excepted case is when the two 
molecules are so placed that they are, or very 
recently have been, within one another’s sphere 
of action. The force of this exception, and the 
necessity for it, will appear in the consideration 
of the H-theorem.’’ 
Some of the most salient facts of the situa- 
tion are these : 
1. Boltzmann in preparing his H-theorem 
treats the number of pairs of particles which 
are on the point of colliding, at given velocities 
and angles, as a function of these velocities and 
angles and of these alone; but he treats the 
number of pairs which are just parting from each 
other at the same velocities and angles as a 
function of the pre-collision velocities and 
angles of the now separating pairs, on the 
ground that their number is determined by the 
number of pairs which an instant before were 
on the point of colliding with each other at 
certain velocities and angles alone capable of 
producing the post-collision velocities and 
angles mentioned. This is a matter of prin- 
ciple, not merely of convenience; for if par- 
ticles just about to collide and particles just 
parting were numbered by like functions of 
velocities and angles, the number of particles 
leaving any class would be exactly equal to the 
number entering it, and there would be no H- 
theorem. 
2. The function which expresses the number 
of particles having velocities lying within cer- 
tain limits becomes the Maxwellian function 
when H has reached a minimum; and when 
this state is attained the exception noted in as- 
sumption A disappears. 
The close scrutiny of assumptions is character- 
istic of Burbury’s book. The fact that he has 
