OF AN ENSEMBLE OF SYSTEMS. 45 



only defined by the equation itself, and incompletely defined 

 in that the equation only determines its differential, and the 

 constant of integration is arbitrary. On the other hand, the 

 77 in the statistical equation has been completely defined as 

 the average value in a canonical ensemble of systems of 

 the logarithm of the coefficient of probability of phase. 



We may also compare equation (112) with the thermody- 

 namic equation 



A^ = T ] dTA l da l A z da< i etc., (117) 



where ^r represents the function obtained by subtracting the 

 product of the temperature and entropy from the energy. 



How far, or in what sense, the similarity of these equations 

 constitutes any demonstration of the thermodynamic equa- 

 tions, or accounts for the behavior of material systems, as 

 described in the theorems of thermodynamics, is a question 

 of which we shall postpone the consideration until we have 

 further investigated the properties of an ensemble of systems 

 distributed in phase according to the law which we are con- 

 sidering. The analogies which have been pointed out will at 

 least supply the motive for this investigation, which will 

 naturally commence with the determination of the average 

 values in the ensemble of the most important quantities relating 

 to the systems, and to the distribution of the ensemble with 

 respect to the different values of these quantities. 



