128 THE MICROCANONICAL DISTRIBUTION. 



The result is the same for any value of n. For in the varia- 

 tions considered the kinetic energy will be constantly zero, 

 and the potential energy will have the least value consistent 

 with the external coordinates. The condition of the least 

 possible potential energy may limit the ensemble at each in- 

 stant to a single configuration, or it may not do so ; but in any 

 case the values of A 1 , A v etc. will be the same at each instant 

 for all the systems of the ensemble,* and the equation 



de + A l da^ -f A z da z + etc. = 



will hold for the variations considered. Hence the functions 

 F^ , F% , etc. vanish in any case, and we have the equation 



d V= e*de + e* Z^d^ + e+~Z^dat + etc., (416) 



de + ~A\,da l + Z^Lrfa 2 + etc. 

 or dlogV=;- _ ' 6 - (417) 



or again 



de = e~* V d log V - "27] dot - lj| e da 2 - etc. (418) 



It will be observed that the two last equations have the form 

 of the fundamental differential equations of thermodynamics, 

 er-^V corresponding to temperature and log V to entropy. 

 We have already observed properties of &"*> V suggestive of an 

 analogy with temperature, f The significance of these facts 

 will be discussed in another chapter. 



The two last equations might be written more simply 



de + 37| dct! + Af da z + etc. 



* ' - 7 - j 



er-4 

 de = e~^ d V "37) da^ ~A^\ da 2 etc., 



and still have the form analogous to the thermodynamic 

 equations, but e~^ has nothing like the analogies with tempera- 

 ture which we have observed in e~^ V. 



* This statement, as mentioned before, may have exceptions for particular 

 values of the external coordinates. This will not invalidate the reasoning, 

 which has to do with varying values of the external coordinates. 



t See Chapter IX, page 111 ; also this chapter, page 119. 



