pen” 
IV. On the Theory of Diminishing Entropy. 
by 8. A. Bursury, /..S.* 
ih. ITH regard to the first difficulty which I found in 
Willard Gibbs’s Chapter XII., I gladly accept 
Dr. Bumstead’s conclusion that the best way of meeting it 
is to define the density by finite elements of volume. I hope 
that Dr. Bumstead will consider the question what element 
of volume should, in given cases, be chosen for the definition. 
2. Adopting this course, we dispense with the theorems 
by which D for Willard Gibbs’s ensemble, or p for the 
coloured liquid of his illustration, were proved to be constant 
in time, these theorems being based on the unworkable de- 
finition of density as the limiting ratio of quantity to the 
containing volume. Our familiar experience that stirring 
the coloured liquid tends to reduce it to a uniform condition 
is thus accepted as an experimental law, and we have made 
our mathematics consistent with it. But from this point of 
view the title of Chapter XII., ‘“ On the motions of systems 
and ensembles of systems through long periods of time,” seems 
wholly misleading. For the tendency to uniformity does 
not depend on length of time in any especial manner. If 
ee be the rate of change of entropy with the time, it is 
not necessary that ¢ should be long in order to make 
WETS finite. 
3. Willard Gibbs’s explanation that this tendency depends 
on the order in which we proceed to the limits, if I rightly 
understand it, leaves the result after all indeterminate. I[ 
may be allowed to illustrate it, as I understand it, thus. Let 
x be a quantity for which I have to call a value, and I may 
eall it as great as I like without limit. Let y be another 
quantity for which Dr. Bumstead has to call a value, and he 
may call it as small as he likes without limit. Then what 
about the product zy? If I have to call first, then no matter 
what I call z, Dr. Bumstead can make «zy infinitely small. 
But if he calls first, then no matter what he calls y, I can 
make wy infinitely great. I think a solution of the problem 
in this form could hardly be considered satisfactory. 
4. My second difficulty was and is as to the value of 7 to 
be attributed to a system. Here Dr. Bumstead says I have 
misapprehended Willard Gibbs’s teaching, which has to do, 
not with single systems, but with large averages. I think 
however that, doubtless owing to my lack of perspicuity, 
* Communicated by the Author. 
