CONTENTS. xvii 



PAGE 



Theorem VIII. The average index of the whole ensemble com- 

 pared with the average indices of parts of the ensemble . . 135-137 

 Theorem IX. Effect on the average index of making the distribu- 

 tion-in-phase uniform within any limits 137-138 



CHAPTER XII. 



ON THE MOTION OF SYSTEMS AND ENSEMBLES OF SYS- 

 TEMS THROUGH LONG PERIODS OF TIME. 



Under what conditions, and with what limitations, may we assume 

 that a system will return in the course of time to its original 

 phase, at least to any required degree of approximation? . . 139-142 



Tendency in an ensemble of isolated systems toward a state of sta- 

 tistical equilibrium 143-151 



CHAPTER XIII. 



EFFECT OF VARIOUS PROCESSES ON AN ENSEMBLE OF 

 SYSTEMS. 



Variation of the external coordinates can only cause a decrease in 

 the average index of probability 152-154 



This decrease may in general be diminished by diminishing the 

 rapidity of the change in the external coordinates .... 154-157 



The mutual action of two ensembles can only diminish the sum of 

 their average indices of probability 158, 159 



In the mutual action of two ensembles which are canonically dis- 

 tributed, that which has the greater modulus will lose energy . 160 



Repeated action between any ensemble and others which are canon- 

 ically distributed with the same modulus will tend to distribute 

 the first-mentioned ensemble canonically with the same modulus 161 



Process analogous to a Carnot's cycle 162,163 



Analogous processes in thermodynamics 163, 164 



CHAPTER XIV. 



DISCUSSION OF THERMODYNAMIC ANALOGIES. 



The finding in rational mechanics an a priori foundation forthermo- 

 dynamics requires mechanical definitions of temperature and 

 entropy. Conditions which the quantities thus defined must 

 satisfy 165-167 



The modulus of a canonical ensemble (0), and the average index of 

 probability taken negatively (rj), as analogues of temperature 

 and entropy 167-169 



