OF AN ENSEMBLE OF SYSTEMS. 35 



It is evident that the canonical distribution is entirely deter- 

 mined by the modulus (considered as a quantity of energy) 

 and the nature of the system considered, since when equation 

 (92) is satisfied the value of the multiple integral (93) is 

 independent of the units and of the coordinates employed, and 

 of the zero chosen for the energy of the system. 



In treating of the canonical distribution, we shall always 

 suppose the multiple integral in equation (92) to have a 

 finite value, as otherwise the coefficient of probability van- 

 ishes, and the law of distribution becomes illusory. This will 

 exclude certain cases, but not such apparently, as will affect 

 the value of our results with respect to their bearing on ther- 

 modynamics. It will exclude, for instance, cases in which the 

 system or parts of it can be distributed in unlimited space 

 (or in a space which has limits, but is still infinite in volume), 

 while the energy remains beneath a finite limit. It also 

 excludes many cases in which the energy can decrease without 

 limit, as when the system contains material points which 

 attract one another inversely as the squares of their distances. 

 Cases of material points attracting each other inversely as the 

 distances would be excluded for some values of , and not 

 for others. The investigation of such points is best left to 

 the particular cases. For the purposes of a general discussion, 

 it is sufficient to call attention to the assumption implicitly 

 involved in the formula (92).* 



The modulus has properties analogous to those of tem- 

 perature in thermodynamics. Let the system A be defined as 

 one of an ensemble of systems of m degrees of freedom 

 distributed in phase with a probability-coefficient 



*% 



e , 



* It will be observed that similar limitations exist in thermodynamics. In 

 order that a mass of gas can be in thermodynamic equilibrium, it is necessary 

 that it be enclosed. There is no thermodynamic equilibrium of a (finite) mass 

 of gas in an infinite space. Again, that two attracting particles should be 

 able to do an infinite amount of work in passing from one configuration 

 (which is regarded as possible) to another, is a notion which, although per- 

 fectly intelligible in a mathematical formula, is quite foreign to our ordinary 

 conceptions of matter. 



