CHAPTER VIII. 



ON CERTAIN IMPORTANT FUNCTIONS OF THE 

 ENERGIES OF A SYSTEM. 



IN order to consider more particularly the distribution of a 

 canonical ensemble in energy, and for other purposes, it will 

 be convenient to use the following definitions and notations. 



Let us denote by J^the extension-in-phase below a certain 

 limit of energy which we shall call e. That is, let 



> x . . . dq n , (265) 



the integration being extended (with constant values of the 

 external coordinates) over all phases for which the energy is 

 less than the limit e. We shall suppose that the value of this 

 integral is not infinite, except for an infinite value of the lim- 

 iting energy. This will not exclude any kind of system to 

 which the canonical distribution is applicable. For if 



>i dq n 



taken without limits has a finite value,* the less value repre- 

 sented by 



e 



/... 



u 







taken below a limiting value of 6, and with the e before the 

 integral sign representing that limiting value, will also be 

 finite. Therefore the value of V, which differs only by a 

 constant factor, will also be finite, for finite e. It is a func- 

 tion of e and the external coordinates, a continuous increasing 



* This is a necessary condition of the canonical distribution. See 

 Chapter IV, p. 35. 



