Variation of Entropy. Lk 
It is this problem (p. 148) which gives rise to Mr. Bur- 
bury’s final difficulty, and which leads him to the conclusion 
that the hypotheses made by Prof. Gibbs concerning the 
mechanical systems are not sufficient to serve as a basis of 
rational thermodynamics. Perhaps I may best show why 
Prof. Gibbs’ conclusions seem to me legitimate by restating 
his demonstration, as I understand it, in slightly different 
form and with special reference to the objections which have 
been brought against it. Let us consider an ensemble of 
systems not in statistical equilibrium. An ensemble is in 
statistical equilibrium if, during any interval of time, as 
many systems enter any fixed element of extension-in-phase 
as leave it during the same interval; the density-in-phase in 
any fixed element (finite or infinitesimal) does not change 
with the time. But when the ensemble is not in statistical 
equilibrium the density in fixed elements ot extension-in- 
phase will vary with the time, and therefore the value of » 
associated with a fixed finite element (as explained above) 
will obviously vary. ‘The question is whether the average 
value, 7, for the whole ensemble, as determined by the use 
of these elements, will increase or decrease with the time. 
At a certain initial instant t’ let the density be distributed in 
any arbitrary manner throughout the extension-in-phase, 
that is, at this instant, we may consider D (or 7) to be given 
as ‘an arbitrary function of the phase.” Later, of course, it 
will be a function of the phase and of the elapsed time. 
With this initial distribution given we may now choose a 
system of fixed finite elements of extension-in-phase, DV, 
small enough so that the density may be regarded as sensibly 
constant throughout any one of them. At a later time ¢” 
(anless the motion in phase is of a highly special and rela- 
tively improbable kind) the systems which were together in 
one element at ¢’ will not all be in a single element. Thus 
some of the systems which were at ¢’ in the element DV’ 
may now be in the element DV”, but they will have mixed 
with them systems which, at ¢’, occupied other elements, 
DV,’, DV.’, &. If, now, we ascertain the average density 
in DV” and take its logarithm (7), thus assuming that 7 has a 
constant value for all the systems in the element, we shall, 
by Theorem IX. of Chapter XI., get a less value than if we 
took the actual values of 7 which the separate systems have. 
But the actual values of 9 for the separate systems are those 
which they have brought with them into DV”, and are the 
same which they had in their scattered condition in DV/’, 
DV,/, &c. at the instant ¢’. Theretore the value of » which 
we have obtained by averaging the density in the element 
