supposed Irreversible Processes. 483 



Pi . . . P,. . . . P». But every P which belongs to an interval for 

 which H differs much from H is for that reason very small. 

 Also for each interval P is less than unity. Therefore if H' is 

 much greater than H , the continued product V 1 . . . P 3l is 

 inappreciably small, aud infinitely smaller than it would be if 

 H were throughout the interval t' nearly equal to H . The 

 existence of our directing condition caused H to diminish, 

 and its continued existence prevents H from again increasing 

 beyond a small amount. The process, aided as we have 

 assumed by Maxwell's demons, is in a defined sense irre- 

 versible. 



22. Let us next assume that no external interference takes 

 place; but the system, once started, is left undisturbed to its 

 own mutual actions. What, under these circumstances, 

 becomes of the chance we considered in Art. 21 ? It there 

 appeared as the continued product of a number of factors, 

 many of which were very small. It is now the single chance 

 of the system having, when H = H , that particular set of 

 coordinates and velocities of which the subsequent increase 

 of H to H' is a necessary consequence. Let 0* denote this 

 particular state. Then O t is itself the necessary consequence 

 of a certain initial state C. And therefore the chance that H 

 shall ascend from H at time t to H/ at time t + t f is the chance 

 that the initial state, formed as we have formed it, shall be C. 

 It may be that C is a very improbable initial state. If that 

 can be proved, then the ascent of H from H to H' is impro- 

 bable in the same degree. I am not aware that it ever has 

 been proved. To say that C is improbable as an initial state 

 because it leads to the subsequent increase of H would be 

 petitio principii. If, however, it can be proved aliunde that 

 C is improbable as an initial state, then we have a clearly 

 defined sense in which one subsequent state is more or less 

 probable than another, viz. as it is the necessary consequence 

 of a more or less probable C. All successive states in the 

 some course are equally probable. 



23. There is indeed another sense in which, I think with 

 less propriety, one state may be called more probable than 

 another. We may prove namely that, P and P' being two 

 possible states of the system, the initial state, formed as we 

 have formed it or in some equivalent way, is more likely to 

 be P than to be P / '. That seems to be the sense in which 

 Boltzmann asserts that the system, as H diminishes, passes 

 from a more to a less probable state. 



It follows from Art. 15 that if any of the factors f¥' -/F 



differ from zero, the state in which — is negative is more 



2 L 2 



