THERMODYNAMIC ANALOGIES. 183 



As a general theorem, the conclusion may be expressed in 

 the words : If a system of a great number of degrees of 

 freedom is microcanonically distributed in phase, any very 

 small part of it may be regarded as canonically distributed.* 



It would seem, therefore, that a canonical ensemble of 

 phases is what best represents, with the precision necessary 

 for exact mathematical reasoning, the notion of a body with 

 a given temperature, if we conceive of the temperature as the 

 state produced by such processes as we actually use in physics 

 to produce a given temperature. Since the anomalies of the 

 body increase with the quantity of the bath, we can only get 

 rid of all that is arbitrary in the ensemble of phases which is 

 to represent the notion of a body of a given temperature by 

 making the bath infinite, which brings us to the canonical 

 distribution. 



A comparison of temperature and entropy with their ana- 

 logues in statistical mechanics would be incomplete without a 

 consideration of their differences with respect to units and 

 zeros, and the numbers used for their numerical specification. 

 If we apply the notions of statistical mechanics to such bodies 

 as we usually consider in thermodynamics, for which the 

 kinetic energy is of the same order of magnitude as the unit 

 of energy, but the number of degrees of freedom is enormous, 

 the values of B, de/dlogV, and de/d<f> will be of the same 

 order of magnitude as 1/w, and the variable part of ?;, log V, 

 and <j> will be of the same order of magnitude as w.f If these 

 quantities, therefore, represent in any sense the notions of tem- 

 perature and entropy, they will nevertheless not be measured 

 in units of the usual order of magnitude, a fact which must 

 be borne in mind in determining what magnitudes may be 

 regarded as insensible to human observation. 



Now nothing prevents our supposing energy and time in 

 our statistical formulae to be measured in such units as may 



* It is- assumed and without this assumption the theorem would have 

 no distinct meaning that the part of the ensemble considered may be 

 regarded as having separate energy. 



t See equations (124), (288), (289), and (314) ; also page 106. 



