178 THERMODYNAMIC ANALOGIES. 



when 



( - 

 < 



etc. + m-^ a\ da t + etc. = 0. 



This requires % = 0, a z = 0, etc., 



and (-vM =^i, fir] = A', etc. 



\i/^,a \da*J^ a 



This shows that for any given values of E, aj, a 2 , etc. 



( -7 ) , ( -: ) , etc., represent the forces (in the oren- 

 \aai/*,a \da z j^ a 



eralized sense) which the external bodies would have to exert 

 to make these values of a^ , 2 , etc., the most probable under 

 the conditions specified. When the differences of the external 

 forces which are exerted by the different systems are negli- 

 gible, (d/da^)$ tm etc., represent these forces. 



It is certainly in the quantities relating to a canonical 

 ensemble, e, , ??, JL 1? etc., a x , etc., that we find the most 

 complete correspondence with the quantities of the thermody- 

 namic equation (482). Yet the conception itself of the canon- 

 ical ensemble may seem to some artificial, and hardly germane 

 to a natural exposition of the subject; and the quantities 

 de . Tr -, . de de 



S v > a. ete " i. etc " ore ' - 



etc., flj, etc., which are closely related to ensembles of constant 

 energy, and to average and most probable values in such 

 ensembles, and most of which are defined without reference 

 to any ensemble, may appear the most natural analogues of 

 the thermodynamic quantities. 



In regard to the naturalness of seeking analogies with the 

 thermodynamic behavior of bodies in canonical or microca- 

 nonical ensembles of systems, much will depend upon how we 

 approach the subject, especially upon the question whether we 

 regard energy or temperature as an independent variable. 



It is very natural to take energy for an independent variable 

 rather than temperature, because ordinary mechanics furnishes 

 us with a perfectly defined conception of energy, whereas the 

 idea of something relating to a mechanical system and corre- 



