182 THERMODYNAMIC ANALOGIES. 



occur in the canonical ensemble of the whole system, these 

 phases will form a canonical ensemble of the same modulus. 

 This canonical ensemble of phases of the first body will con- 

 sist of parts which belong to the different microcanonical 

 ensembles into which the canonical ensemble of the whole 

 system is divided. 



Let us now imagine that the modulus of the principal ca- 

 nonical ensemble is increased by 2 A (8), and its average energy 

 by 2Ae. The modulus of the canonical ensemble of the 

 phases of the first body considered separately will be increased 

 by 2 A . We may regard the infinity of microcanonical en- 

 sembles into which we have divided the principal canonical 

 ensemble as each having its energy increased by 2Ae. Let 

 us see how the ensembles of phases of the first body con- 

 tained in these microcanonical ensembles are affected. We 

 may assume that they will all be affected in about the same 

 way, as all the differences which come into account may be 

 treated as small. Therefore, the canonical ensemble formed by 

 taking them together will also be affected in the same way. 

 But we know how this is affected. It is by the increase of 

 its modulus by 2 A, a quantity which vanishes when the 

 quantity of the bath is indefinitely increased. 



In the case of an infinite bath, therefore, the increase of the 

 energy of one of the microcanonical ensembles by 2Ae, pro- 

 duces a vanishing effect on the distribution in energy of the 

 phases of the first body which it contains. But 2Ae is more 

 than the average difference of energy between the micro- 

 canonical ensembles. The distribution in energy of these 

 phases is therefore the same in the different microcanonical 

 ensembles, and must therefore be canonical, like that of the 

 ensemble which they form when taken together.* 



* In order to appreciate the above reasoning, it should be understood that 

 the differences of energy which occur in the canonical ensemble of phases of 

 the first body are not here regarded as vanishing quantities. To fix one's 

 ideas, one may imagine that he has the fineness of perception to make these 

 differences seem large. The difference between the part of these phases 

 which belong to one microcanonical ensemble of the whole system and the 

 part which belongs to another would still be imperceptible, when the quan- 

 tity of the bath is sufficiently increased. 



