180 THERMODYNAMIC ANALOGIES. 



ture as an independent variable, and we have to consider a 

 system which is described as having a certain temperature and 

 certain values for the external coordinates. Here also the 

 state of the system is not completely denned, and will be 

 better represented by an ensemble of phases than by any single 

 phase. What is the nature of such an ensemble as will best 

 represent the imperfectly defined state ? 



When we wish to give a body a certain temperature, we 

 place it in a bath of the proper temperature, and when we 

 regard what we call thermal equilibrium as established, we say 

 that the body has the same temperature as the bath. Per- 

 haps we place a second body of standard character, which we 

 call a thermometer, in the bath, and say that the first body, 

 the bath, and the thermometer, have all the same temperature. 



But the body under such circumstances, as well as the 

 bath, and the thermometer, even if they were entirely isolated 

 from external influences (which it is convenient to suppose 

 in a theoretical discussion), would be continually changing in 

 phase, and in energy as well as in other respects, although 

 our means of observation are not fine enough to perceive 

 these variations. 



The series of phases through which the whole system runs 

 in the course of time may not be entirely determined by the 

 energy, but may depend on the initial phase in other respects. 

 In such cases the ensemble obtained by the microcanonical 

 distribution of the whole system, which includes all possible 

 time-ensembles combined in the proportion which seems least 

 arbitrary, will represent better than any one time-ensemble 

 the effect of the bath. Indeed a single time-ensemble, when 

 it is not also a microcanonical ensemble, is too ill-defined a 

 notion to serve the purposes of a general discussion. We 

 will therefore direct our attention, when we suppose the body 

 placed in a bath, to the microcanonical ensemble of phases 

 thus obtained. 



If we now suppose the quantity of the substance forming 

 the bath to be increased, the anomalies of the separate ener- 

 gies of the body and of the thermometer in the microcanonical 



