12 Dr. H. A. Bumstead on the 
DV” at ¢” is less than that based on the same systems as they 
were at ¢’; and, as this is true for every element, the average 
value of y for the whole ensemble so deteonatede is less at Y! 
than at ¢’.. This diminution in the average is wholly the 
result of mixing, in the elements, systems each of which 
preserves a constant value of 7. Tt may be well illustrated 
by the hydrodynamical case which Prof. Gibbs uses earlier 
in the same chapter (p. 146), in which a cylindrical mass of 
liquid is imagined, one sector of 90° being black and the rest 
white. If it is given a motion of rotation about the axis of 
the cylinder, in which the angular velocity is any function 
of the distance from the axis (except in the highly special 
case when this function is a constant), “the black and white 
portions would become drawn out into thin ribbons which 
would be wound spirally about the axis.”” At any given 
instant ¢t' we might choose a system of finite elements of 
volume so small that the density of the black colour in any 
one would be constant, either unity or zero; but at a later 
time the same elements would each contain a mixture of 
black and white, provided of course that the motion continues 
long enough. 
But the chief difficulty is that the analytical demonstration 
will work either forward or backward in time, as Mr. Burbury 
points out. Assuming that the motion of ihe sy stems, or of 
the liquid in the illustration, extends backward in time wn- 
interruptedly, we shall inevitably come to a prior time. say ¢), 
at which the average 7 in an element is less than at ¢, or at 
which the black and white portions of the ones are better 
mixed than at the later epoch. Prof. Gibbs has not over- 
looked this fact, and has, | think, given the true solution of 
the difficulty, although so briefly that the true import of his 
remarks may easily be overlooked. He says (p. 150) “ It is 
to be observed that if the average index of probability [y] in 
an ensemble may be said in some sense to have a less value 
at one time than at another, it is not necessarily priority in 
time which determines the greater average index. If a dis- 
tribution, which is not one of statistical equilibrium, should 
be given for a time ¢!, and the distribution at an earlier time 
i!" should be dejined as that given by the corresponding phases" , 
if we increase the interval leaving ?! fixed and taking ¢” at 
an earlier and earlier date, the distribution at ¢” will in 
general approach a limiting distribution which is in statistical 
equilibrium. The determining difference in such cases is 
that between a definite distribution at a definite time and the 
* Italics are mine. 
