16 Dr. H. A. Bumstead on the 
uniform density of the coloured component may be produced 
by stirrmg.” And it is really this latter form which he 
uses when he comes to apply the principle, as in the third 
paragraph on page 154, in which the qualification, “if very 
small differences of phase are neglected,” is of course equi- 
valent to taking finite elements of extension-in-phase. The 
infinite time idea was, I believe, introduced merely as an 
alternative (and nota preferable) way of regarding the subject. 
Admitting then the possibility of variation of the density- 
in-phase D in the finite elements through which a moving 
system passes, Mr. Burbury finds a difficulty in the definition 
of 7 in the expression D=Ne’. He says, “so long as 9 
remains constant for the same system, we may define 7 to 
be the entropy which that system has. ... But being 
now supposed to be variable for the same system. we require 
a definition.” Here I think the whole trend and spirit of 
Prof. Gibbs’ method has been misapprehended ; unless I 
have mistaken his position, Prof. Gibbs would not have 
admitted that the 9 for a single system, although exactly 
determined, corresponds to what we call entropy in bodies 
met with in nature. So far as he applies his results to ther- 
modynamies, he regards the bodies of nature as corresponding, 
not to a definite system, but to a system chosen at random 
out of a properly distributed ensemble; so that it is certain 
average properties of the ensemble which we observe ex- 
perimentally, and not the properties of a single system. The 
average value of 7 for the whole ensemble (taken with the 
negative sign) corresponds to the entropy of any body which 
the ensemble is capable of representing, and we are no more 
concerned about the 7 of a single system (except in so far as 
a knowledge of it may be necessary to get a correct average) 
than we are concerned with the exact configuration of the 
system. But it is evident that we may get a sufficiently 
close value of the average for the whole ensemble by adding, 
not the » for every system, but the mean values ot for each 
one of a set of finite elements of extension-in-phase taken 
sufficiently small. Thus all we are concerned to know is the 
mean value of » in an element, and hence the equation 
D=WNe" may still serve as the definition of 9, for all necessary 
purposes, even though D is no longer the exact density at a 
point but is the mean density throughout an element. 
Prof. Gibbs’ statement in a succeeding paragraph (p. 148) 
that 4 “is an arbitrary function of the phase,” which Mr. 
Burbury takes to be a new definition, is, I think, not a defini- 
tion at all, but the statement of an assumed initial condition 
in the particular problem which he is then considering. 
