44 Mr. 8. H. Burbury on the 
Dr. Bumstead has misapprehended me. I will restate my 
difficulty in Willard Gibbs’s ipsisst¢ma verba. Twice at the 
head of p. 150 (which is the point where the difficulty arose) 
he speaks of “ the average value of 7 for” a group of systems. 
Now it is impossible for an average “value for” the group 
to exist, unless there exists a definable “value for” each 
system in the group. I give an example. The average 
‘chest measurement for”? the students now in Yale Uni- 
versity is, say, 36 inches. That implies that there exists a 
definite system of measurement for each individual student. 
We cannot avoid giving the definition by saying that our 
anthropometric investigations are “ statistical,” and have only 
to do with large averages, not with individual measurements. 
What then does Willard Gibbs mean by the * value of » for” 
a system ? 
). At time ¢’ a certain group of systems occupy the ele- 
mentary extension in phase DV’. Their number is Ne” DV’ 
by definition of 7 Chapter I. Then I suppose at that instant 
n is “the value of for” each system of the group, and is 
therefore the average value for the group. Another group 
at the same instant occupy DV,’, and their number is 
Ne’ DV,’.. Now Willard Gibbs says 7! is “an arbitrary 
function of the phase,” meaning, as Dr. Bumstead under- 
stands it, an arbitrary function of the phase at the instant ¢. 
At another instant ¢’”, which may, so far as we have yet gone, 
be before or after ¢’, the systems which at ¢ were all in DV’, 
are scattered over more than one elementary extension in 
phase. What at the second instant is “the value of » for” 
any system of the group? One possible answer is, that each 
system of the group always retains the same 7 which was 
“the value for it” at ¢’. But if that were adopted no system 
ever changes its 7. There can, therefore, be no question of 
the mean value 7 increasing or diminishing for the ensemble. 
Another possible answer is that by an “arbitrary function 
of the phase” is meant an arbitrary function of the phase in 
which the system is for the time being. According to this 
view, the number of systems in any elementary extension in 
phase is constant. The ensemble is then in statistical equi- 
librium. Willard Gibbs cannot have meant either this answer 
or the other. I think he has not quite explained his 
meaning. 
The Theory of Diminishing Entropy. 
6. The remaining difficulty is that the argument of pp. 150, 
151 will work either forwards or backwards. Entropy may, 
for all that appears, either increase or diminish. The 
