ON AN ENSEMBLE OF SYSTEMS. 161 



Hence ^" + |'=^' + C (470) 



which may be written 



%'-V^^^ (471) 



This may be compared with the thennodynamic principle that 

 when a body (which need not be in thermal equilibrium) is 

 brought into thermal contact with another of a given tempera- 

 ture, the increase of entropy of the first cannot be less (alge- 

 braically) than the loss_of heat by the second divided by its 

 temperature. Where W is negligible, we may write 



V' + |^' + | . (472) 



Now, by Theorem III of Chapter XI, the quantity 



! . * + | (473) 



has a minimum value when the ensemble to which ^ and e x 

 relate is distributed canonically with the modulus 2 . If the 

 ensemble had originally this distribution, the sign < in (472) 

 would be impossible. In fact, in this case, it would be easy to 

 show that the preceding formulae on which (472) is founded 

 would all have the sign = . But when the two ensembles are 

 not both originally distributed canonically with the same 

 modulus, the formulae indicate that the quantity (473) may 

 be diminished by bringing the ensemble to which e a and y l 

 relate into connection with another which is canonically dis- 

 tributed with modulus 2 , and therefore, that by repeated 

 operations of this kind the ensemble of which the original dis- 

 tribution was entirely arbitrary might be brought approxi- 

 mately into a state of canonical distribution with the modulus 

 <B) 2 . We may compare this with the thermodynamic principle 

 that a body of which the original thermal state may be entirely 

 arbitrary, -may be brought approximately into a state of ther- 

 mal equilibrium with any given temperature by repeated con- 

 nections with other bodies of that temperature. 



11 



