174 THERMODYNAMIC ANALOGIES. 



between de/d log V and temperature would be complete, as has 

 already been remarked. We should have 



de l e^ c?6 2 _ e 2 



n' 9 dlo V~' 



_ = 

 MM rflog F! dlogV 2 ' 



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

 similar equations with % , J ra 2 , -|- w 12 , instead of ^ , w 2 , w 12 , 

 when the energy is a quadratic function of the >'s alone. 



More characteristic of dcf>/de are its properties relating to 

 most probable values of energy. If a system having two parts 

 with separate energies and each with more than two degrees 

 of freedom is microcanonically distributed in phase, the most 

 probable division of energy between the parts, in a system 

 taken at random from the ensemble, satisfies the equation 



^ = ^, (488) 



de l de 2 



which corresponds to the thermodynamic theorem that the 

 distribution of energy between the parts of a system, in case of 

 thermal equilibrium, is such that the temperatures of the parts 

 are equal. 



To prove the theorem, we observe that the fractional part 

 of the whole number of systems which have the energy of one 

 part (ej) between the limits e/ and e/ is expressed by 



r*f. ****>, 



T i 



where the variables are connected by the equation 

 j -|- 2 = constant = e i2 . 



The greatest value of this expression, for a constant infinitesi- 

 mal value of the difference e x " e/, determines a value of e 1 , 

 which we may call its most probable value. This depends on 

 the greatest possible value of fa + fa. Now if n^ > 2, and 

 w 2 > 2, we shall have fa = oo for the least possible value of 



