﻿the Partition of Energy. 835 



this way because the function 2=,S s t. + &E also has the 

 increasing property, where b is ;tny universal constant. 

 Now when we set out to define the absolute temperature 

 scale, we must start with the general function 2 which has 

 the increasing property, for we have as yet no right to 

 choose any particular value for b. If we attempt to define T 

 by the relation B§/dE= l/T, we find 



which can never determine absolutely the relation of T to 3- 

 so long as b is undetermined. 



This impasse is one aspect of the fact that in thermo- 

 dynamics the absolute temperature and the entropy are 

 introduced in the same chain of argument — the absolute 

 temperature as integrating factor and the entropy {is the 

 resulting integral. Thus — and this is a point that has 

 been overlooked by some writers — it is impossible to identity 

 the entropy by using assemblies in which temperature is 

 the only variable, for any function of the temperature 

 is then a possible integrating factor. There is only one way 

 of making the identification, and that is to evaluate dQ, the 

 element of heat, for an assembly of more than one variable 

 from our statistical principles, and to show that a certain 

 unique * function of the temperature 3 is an integrating 

 factor for it. The use of functions with the increasing- 

 property can apparently never lead to precise results without 

 this appeal to dQ. We shall therefore abandon the whole of 

 the development of the preceding sections (4-7), including 

 the Boltzmann Hypothesis, and shall establish from first 

 principles that in fact the quantity dQ has a unique inte- 

 grating factor depending only on S, and that this does lead 

 to Sst. for the entropy. 



§ 7. The Entropy from first principles. 



By the definition of dQ, f, we have 



dQ =dE+2Xtf.r, ..... (7-1) 



where E is the energy of the assembly, the .?;'s are certain 

 parameters defining the external fields, and the X's the 

 associated forces. Let us suppose a generalized assembly 



* Of course, an arbitrary constant multiplier excepted. 



t The u heat " dQ taken in in any small change is defined in thermo- 

 dynamics to "be the increase in internal energy plus the external work 

 done by the assembly. See e. g. Born, loc. cit. 



