CHAPTER IX. 

 THE FUNCTION < AND THE CANONICAL DISTRIBUTION. 



IN this chapter we shall return to the consideration of the 

 canonical distribution, in order to investigate those properties 

 which are especially related to the function of the energy 

 which we have denoted by </>. 



If we denote by JV, as usual, the total number of systems 

 in the ensemble, 



will represent the number having energies between the limits 

 e and e + de. The expression 



Ne 



(317) 



represents what may be called the density-in-energy. This 

 vanishes for e = GO, for otherwise the necessary equation 



(318) 



could not be fulfilled. For the same reason the density-in- 

 energy will vanish for e = co, if that is a possible value of 

 the energy. Generally, however, the least possible value of 

 the energy will be a finite value, for which, if n > 2, e* will 

 vanish,* and therefore the density-in-energy. Now the density- 

 in-energy is necessarily positive, and since it vanishes for 

 extreme values of the energy if n > 2, it must have a maxi- 

 mum in such cases, in which the energy may be said to have 



* See page 96. 



