THERMODYNAMIC ANALOGIES. 179 



spending to temperature is a notion but vaguely denned. Now 

 if the state of a system is given by its energy and the external 

 coordinates, it is incompletely denned, although its partial defi- 

 nition is perfectly clear as far as it goes. The ensemble of 

 phases microcanonically distributed, with the given values of 

 the energy and the external coordinates, will represent the im- 

 perfectly defined system better than any other ensemble or 

 single phase. When we approach the subject from this side, 

 our theorems will naturally relate to average values, or most 

 probable values, in such ensembles. 



In this case, the choice between the variables of (485) or of 

 (489) will be determined partly by the relative importance 

 which is attached to average and probable values. It would 

 seem that in general average values are the most important, and 

 that they lend themselves better to analytical transformations. 

 This consideration would give the preference to the system of 

 variables in which log V is the analogue of entropy. Moreover, 

 if we make <f> the analogue of entropy, we are embarrassed by 

 the necessity of making numerous exceptions for systems of 

 one or two degrees of freedom. 



On the other hand, the definition of < may be regarded as a 

 little more simple than that of log F", and if our choice is deter- 

 mined by the simplicity of the definitions of the analogues of 

 entropy and temperature, it would seem that the < system 

 should have the preference. In our definition of these quanti- 

 ties, V was defined first, and e^ derived from V by differen- 

 tiation. This gives the relation of the quantities in the most 

 simple analytical form. Yet so far as the notions are con- 

 cerned, it is perhaps more natural to regard Fas derived from 

 C* by integration. At all events, e* may be defined inde- 

 pendently of F", and its definition niay be regarded as more 

 simple as not requiring the determination of the zero from 

 which V is measured, which sometimes involves questions 

 of a delicate nature. In fact, the quantity e* may exist, 

 when the definition of V becomes illusory for practical pur- 

 poses, as the integral by which it is determined becomes infinite. 



The case is entirely different, when we regard the tempera- 



