THERMODYNAMIC ANALOGIES. 181 



ensemble will be increased, but not without limit. The anom- 

 alies of the energy of the bath, considered in comparison with 

 its whole energy, diminish indefinitely as the quantity of the 

 bath is increased, and become in a sense negligible, when 

 the quantity of the bath is sufficiently increased. The 

 ensemble of phases of the body, and of the thermometer, 

 approach a standard form as the quantity of the bath is in- 

 definitely increased. This limiting form is easily shown to be 

 what we have described as the canonical distribution. 



Let us write e for the energy of the whole system consisting 

 of the body first mentioned, the bath, and the thermometer 

 (if any), an4 let us first suppose this system to be distributed 

 canonically with the modulus . We have by (205) 



and since e p = = 



de _ n de 

 H~~2de p ' 

 If we write Ae for the anomaly of mean square, we have 



d 

 If we set 



A will represent approximately the increase of which 

 would produce an increase in the average value of the energy 

 equal to its anomaly of mean square. Now these equations 

 give 



(A)* = - 

 n 



which shows that we may diminish A indefinitely by increas- 

 ing the quantity of the bath. 



Now our canonical ensemble consists of an infinity of micro- 

 canonical ensembles, which differ only in consequence of the 

 different values of the energy which is constant in each. If 

 we consider separately the phases of the first body which 



