Variation of Entropy. a 
in which the mean square of the density has its minimum 
value, and the density is uniform.” It seems to me that this 
statement may be justified as follows :—If, after a certain 
amount of stirring, we should determine the density of the 
colouring matter in ‘the liquid, using finite elements of volume 
sufficiently large, we should find the density sensibly constant 
in the ditferent elements; but if the elements were chosen 
small enough (but still finite) some of them would be entirely 
within the coloured portions and some in the uncoloured 
portions, and the density in such an estimate would no longer 
appear to be uniform. If now the stirring were continued, 
a time would come when these smaller elements would all 
have the same average density, and so on indefinitely ; and 
no system of finite elements, however small, could be as- 
signed in advance in which the average density could not be 
made the same for all by a sufficient amount of stirring. In 
other words, if we are allowed to stir as long as we please, 
we may use elements (in the estimate of density) as small as 
we please. That this is what Prof. Gibbs means by his 
somewhat guarded statement about an infinite amount of 
stirring, seems plain in the light of the preceding paragraphs 
in which he discusses the effect of altering the order in which 
limits are taken. This latter consideration was one of which 
he not infrequently made use; I recall that he once em- 
ployed it to reconcile conflicting views as to the interpretation 
of Fourier’s series in a discussion which arose in the columns 
of ‘ Nature’ (vol. lix. p. 200). 
Although it seems to me that the statement can be thus 
justified, | nevertheless must agree with Mr. Burbury that 
the other way of escaping the difficulty, viz. by defining the 
density by finite elements of volume (or of extension-in- 
phase) is preferable. If I understand the matter correctly, 
this is not because there is anything in the structure of the 
ensemble of systems corresponding to a molecular structure 
in the liquid, for a system of n degrees of freedom occupies 
no finite extension in the 2n-fold space in which its possible 
phases lie; but it is because we are unable, owing to the 
finiteness of our perceptions, to recognize very small dif- 
ferences of phase, just as we are unable to recognize very small 
differences of position in the analogous case of the liquid. 
And it is certainly nearer the truth to base the doctrine of 
the increase of entropy upon the finiteness of our perceptions 
rather than upon the infiniteness of time. That this was also 
Prof. Gibbs’ opinion I believe is evidenced by the sentence 
following the one quoted (“ one may perhaps be allowed,” &c.) 
in which he says, ‘‘ We may certainly say that a sensibly 
