THERMODYNAMIC ANALOGIES. 173 



with more than two degrees of freedom, the average values in 

 the ensemble of d(f)/de for the two parts are equal to one 

 another and to the value of same expression for the whole. 

 In our usual notations 



"^12 



de 2 L 2 ~" de lz 



if TI X > 2, and n 2 > 2. 



This analogy with temperature has the same incompleteness 

 which was noticed with respect to de/dlog V, viz., if two sys- 

 tems have such energies (ej and e 2 ) that 



and they are combined to form a third system with energy 



*ia = 1 + 2 , 



we shall not have in general 



c?0 12 _ dfa __ d<f> z 

 di 2 de L de z ' 



Thus, if the energy is a quadratic function of the p's and <?'s, 

 we have * 



e 12 la e l + e 2 



where n t , w 2 , w 12 , are the numbers of degrees of freedom of the 

 separate and combined systems. But 



dfa d<f> 2 HI + n% 2 

 de l ~ de 2 " e 1 + e z 



If the energy is a quadratic function of the p's alone, the case 

 would be the same except that we should have J n^ , J w 2 , J w 12 , 

 instead of w x , w 2 , w 12 . In these particular cases, the analogy 



* See foot-note on page 93. We have here made the least value of the 

 energy consistent with the values of the external coordinates zero instead 

 of e a , as is evidently allowable when the external coordinates are supposed 

 invariable. 



