172 THERMODYNAMIC ANALOGIES. 



where the second and third members of the equation denote 

 average values in an ensemble in which the compound system 

 is microcanonically distributed in phase. Let us suppose the 

 two original systems to be identical in nature. Then 



The equation in question would require that 



i. e., that we get the same result, whether we take the value 

 of de l /dlog V} determined for the average value of e 1 in the 

 ensemble, or take the average value of de^dlog F" r This 

 will be the case where de^dlog V^ is a linear function of e r 

 Evidently this does not constitute the most general case. 

 Therefore the equation in question cannot be true in general. 

 It is true, however, in some very important particular cases, as 

 when the energy is a quadratic function of the p's and ^'s, or 

 of the p's alone.* When the equation holds, the case is anal- 

 ogous to that of bodies in thermodynamics for which the 

 specific heat for constant volume is constant. 



Another quantity which is closely related to temperature is 

 dcfr/de. It has been shown in Chapter IX that in a canonical 

 ensemble, if n > 2, the average value of d(f>fde is I/, and 

 that the most common value of the energy in the ensemble is 

 that for which d$/de = I/. The first of these properties 

 may be compared with that of de/dlog V, which has been 

 seen to have the average value in a canonical ensemble, 

 without restriction in regard to the number of degrees of 

 freedom. 



With respect to microcanonical ensembles also, dfyjde has 

 a property similar to what has been mentioned with respect to 

 de/d log V. That is, if a system microcanonically distributed 

 in phase consists of two parts with separate energies, and each 



* This last case is important on account of its relation to the theory of 

 gases, although it must in strictness be regarded as a limit of possible cases, 

 rather than as a case which is itself possible. 



