xvi CONTENTS. 



CHAPTER IX. 



THE FUNCTION </> AND THE CANONICAL DISTRIBUTION. 



When n > 2, the most probable value of the energy in a canonical 

 ensemble is determined by d(j> j de = 1 / e 100,101 



When n > 2, the average value of d$ j de in a canonical ensemble 

 isl/e 101 



When n is large, the value of < corresponding to d(f>/de=l/Q 

 (<o) js nearly equivalent (except for an additive constant) to 

 the average index of probability taken negatively ( fj) . . 101-104 



Approximate formulae for < + fj when n is large 104-106 



When n is large, the distribution of a canonical ensemble in energy 

 follows approximately the law of errors 105 



This is not peculiar to the canonical distribution 107, 108 



Averages in a canonical ensemble 108-114 



CHAPTER X. 



ON A DISTRIBUTION IN PHASE CALLED MICROCANONI- 

 CAL IN WHICH ALL THE SYSTEMS HAVE THE SAME 

 ENERGY. 



The microcanonical distribution denned as the limiting distribution 

 obtained by various processes 115, 116 



Average values in the microcanonical ensemble of functions of the 

 kinetic and potential energies 117-120 



If two quantities have the same average values in every microcanon- 

 ical ensemble, they have the same average value in every canon- 

 ical ensemble 120 



Average values in the microcanonical ensemble of functions of the 

 energies of parts of the system 121-123 



Average values of functions of the kinetic energy of a part of the 

 system 123, 124 



Average values of the external forces in a microcanonical ensemble. 

 Differential equation relating to these averages, having the form 

 of the fundamental differential equation of thermodynamics . 124-128 



CHAPTER XI. 



MAXIMUM AND MINIMUM PROPERTIES OF VARIOUS DIS- 

 TRIBUTIONS IN PHASE. 



Theorems I- VI. Minimum properties of certain distributions . 129-133 

 Theorem VII. The average index of the whole system compared 

 with the sum of the average indices of the parts 133-135 



