THERMODYNAMIC ANALOGIES. 171 



Since the energies of the parts of a body cannot be supposed 

 to remain absolutely constant, even where this is the case 

 with respect to the whole body, it is evident that if we regard 

 the temperature as a function of the energy, the taking of 

 average or of probable values, or some other statistical process, 

 must be used with reference to the parts, in order to get a 

 perfectly definite value corresponding to the notion of tem- 

 perature. 



It is worthy of notice in this connection that the average 

 value of the kinetic energy, either in a microcanonical en- 

 semble, or in a canonical, divided by one half the number of 

 degrees of freedom, is equal to e~* "FJ or to its average value, 

 and that this is true not only of the whole system which is 

 distributed either microcanonically or canonically, but also 

 of any part, although the corresponding theorem relating to 

 temperature hardly belongs to empirical thermodynamics, since 

 neither the (inner) kinetic energy of a body, nor its number 

 of degrees of freedom is immediately cognizable to our facul- 

 ties, and we meet the gravest difficulties when we endeavor 

 to apply the theorem to the theory of gases, except in the 

 simplest case, that of the gases known as monatomic. 



But the correspondence between &~* V or dejd log V and 

 temperature is imperfect. If two isolated systems have such 

 energies that 



de-L de 2 



d log FI ~~ d log F 2 ' 



and the two systems are regarded as combined to form a third 

 system with energy 



12 = ex + e 2> 

 we shall not have in general 



deiz de l de z 



dlog F 12 ~~ dlog Fi ~ dlog F 2 ' 



as analogy with temperature would require. In fact, we have 

 seen that 



d log F 12 d log Fit M ~~ d log Fj 



