48 Theory a Diminishing Entropy. 
containing vessel it is taken. As regards less than measurable 
spaces we do not know by experience that this is the case, 
nor have we any right to infer from analogy or otherwise 
that it is so. 
We might assume in explanation of this our experience— 
and it would be a reasonable assumption—that the number 
of oxygen resp. nitrogen molecules which pass out of any 
measurable space S per unit of time is proportional te the 
number which for the time being are within S. And that 
assumption would be made to lead to the application of 
Boltzmann’s logarithmic theorem above mentioned. 
17. The assumption that we have just made in the dif- 
fusion problem Art. 16, although it follows as a consequence 
of the impossible condition of independence, may be true 
while that condition is untrue. Again, the condition of in- 
dependence, although it cannot be true at every instant 
without violating the continuity of the motion, may be true 
at some one initial instant, if, for instance, the molecules 
have velocities attributed to them at that instant at haphazard. 
And the consequences which follow from the condition of 
independence may be expected to continue in force in a 
greater or less degree, and so entropy to diminish, for some 
time after the initial instant. After a sufficient lapse of time 
this would in an isolated system cease to be true. But if 
the system under consideration is not isolated, but continually 
receiving disturbances from without due to causes independent 
of its own state of motion for the time being, the effect of 
these will be to restore in whole or in part the initial hap- 
hazard character of the motion. Now any actual material 
system generally does from time to time experience external 
disturbances which will have this effect. So the condition 
of independence, though inadmissible as an accurate law, 
may be expected to have great influence in fact. 
18. I think the conclusions that may fairly be drawn at 
present are as follows :— 
(a) If a material system consist of parts, the motions of 
which are separately reversible, it is impossible to prove that 
the motion of the aggregate is irreversible without making 
some physical assumption to be used in support of the pure 
mathematical theorem, whether that be the logarithmic 
theorem employed by Boltzmann and by Willard Gibbs 
(Theorem IX.) orany other mathematical theorem independent 
of time. 
(b) Such assumption, to be admissible, must be mathema- 
tically possible, and must be at least not inconsistent with 
experience. 
