Variation of Entropy. 13 
limit of a varying distribution when the moment considered 
is carried either forward or backward indefinitely. 
‘“ But while the distinction of prior and subsequent events 
may be immaterial with respect to mathematicai fictions, it 
is quite otherwise with respect to the events of the real world. 
It should not be forgotten, when our ensembles are chosen to 
illustrate the probabilities of events in the real world, that 
while the probabilities of subsequent events may often be 
determined from the probabilities of prior events, it is rarely 
the case that probabilities of prior events can be determined 
from those ot subsequent events, for we are rarely justified 
in excluding the consideration of the antecedent probability 
of the prior events.”’ 
Let us consider first the rotating cylinder of liquid. It is 
quite true, if we imagine the motion to be continued back- 
ward in time from the instant when we have the black sector 
according to the same distribution of angular velocity along 
the radius, that we shail have the black and white portions 
more and more drawn out into thin ribbons the further back 
we go. Butthatthere should have been the nice adjustment 
of distribution of colouring matter and of angular velocity 
oO 
necessary in order that phawe ribbons should, at a given in- 
stant, resolve themselves into the black and white sectors is 
exceedingly improbable. In all reasonable probability such 
a distribution is essentially an initial one, that is one pro- 
duced by outside causes and not by antecedent motion of the 
same type; as a matter of fact we do not separate liquids by 
stirring them. 
In the same manner the thermodynamic states of natural 
bodies which correspond to ensembles not in statistical equi- 
librium are, except in very improbable cases, produced b 
external causes. In Chapter XIIT., in which Prof. Gibbs 
considers the effects of external influences upon an ensemble 
of systems, he shows that an ensemble originally in statistical 
equilibri ium will have that equilibriam disturbed by changes 
in the “ external coordinates.” This is truly an initial state, 
and thereafter the mean value of 7 will decrease* as iis 
goes on; that it would also decrease if we supposed the same 
motions carried back into the past is of no practical im- 
portance since that is not the way in which the existing state 
of things has arisen. An ensemble may unquestionably be 
arranged, as to distribution and motion, so that its 7 shall 
increase, but only up to a certain time, after which it will 
decrease ; that such a preliminary arrangement should 
* It is to be remembered that it is —y which is analogous to entropy. 
