154 EFFECT OF VARIOUS PROCESSES 



when it takes place. Therefore it does not affect the index of 

 probability of phase (77) of any system, or the average value 

 of the index (?/)' at that time. And if these quantities are 

 constant in time before the variation of the external coordi- 

 nates, and after that variation, their constancy hi time is not 

 interrupted by that variation. In fact, in the demonstration 

 of the conservation of probability of phase in Chapter I, the 

 variation of the external coordinates was not excluded. 



But a variation of the external coordinates will in general 

 disturb a previously existing state of statistical equilibrium. 

 For, although it does not affect (at the first instant) the 

 distribution-in-phase, it does affect the condition necessary for 

 equilibrium. This condition, as we have seen in Chapter IV, 

 is that the index of probability of phase shall be a function of 

 phase which is constant in time for moving systems. Now 

 a change in the external coordinates, by changing the forces 

 which act on the systems, will change the nature of the 

 functions of phase which are constant in time. Therefore, 

 the distribution in phase which was one of statistical equi- 

 librium for the old values of the external coordinates, will not 

 be such for the new values. 



Now we have seen, in the last chapter, that when the dis- 

 tribution-in-phase is not one of statistical equilibrium, an 

 ensemble of systems may, and in general will, after a longer or 

 shorter time, come to a state which may be regarded, if very 

 small differences of phase are neglected, as one of statistical 

 equilibrium, and in which consequently the average value of 

 the index (?;) is less than at first. It is evident, therefore, 

 that a variation of the external coordinates, by disturbing a 

 state of statistical equilibrium, may indirectly cause a diminu- 

 tion, (in a certain sense at least,) of the value of rj. 



But if the change in the external coordinates is very small, 

 the change in the distribution necessary for equilibrium will 

 in general be correspondingly small. Hence, the original dis- 

 tribution in phase, since it differs little from one which would 

 be in statistical equilibrium with the new values of the ex- 

 ternal coordinates, may be supposed to have a value of v 



