xiv CONTENTS. 



CHAPTER IV. 



ON THE DISTRIBUTION-IN-PHASE CALLED CANONICAL, IN 

 WHICH THE INDEX OF PROBABILITY IS A LINEAR 



FUNCTION OF THE ENERGY. 



PAGE 



Condition of statistical equilibrium 32 



Other conditions which the coefficient of probability must satisfy . 33 



"""" Canonical distribution Modulus of distribution 34 



^ must be finite 35 



The modulus of the canonical distribution has properties analogous 



to temperature 35-37 



Other distributions have similar properties 37 



Distribution in which the index of probability is a linear function of 



the energy and of the moments of momentum about three axes . 38, 39 

 Case in which the forces are linear functions of the displacements, 



and the index is a. linear function of the separate energies relating 



to the normal types of motion 39-41 



Differential equation relating to average values in a canonical 



ensemble 42-44 



This is identical in form with the fundamental differential equation 



of thermodynamics 44, 45 



CHAPTER V. 



AVERAGE VALUES IN A CANONICAL ENSEMBLE OF SYS- 

 TEMS. 

 Case of v material points. Average value of kinetic energy of a 



single point for a given configuration or for the whole ensemble 



= f 46, 47 



Average value of total kinetic energy for any given configuration 



or for the whole ensemble = % v 47 



System of n degrees of freedom. Average value of kinetic energy, 



for any given configuration or for the whole ensemble = f . 48-50 



Second proof of the same proposition 50-52 



Distribution of canonical ensemble in configuration 52-54 



Ensembles canonically distributed in configuration 55 



Ensembles canonically distributed in velocity 56 



CHAPTER VI. 



EXTENSION1-IN-CONFIGURATION AND EXTENSION-TN- 

 VELOCITY. 



Extension-in-configuration and extension-in-velocity are invari- 

 ants . 57-59 



