THE yiiW STATISTIC A I. MI.CIIANICS 375 



which by the way is one of four equations collectively known as Maxwell's 

 relations — a memorial of Maxwell's creative work in early thermodynamics, 

 as the name of the distribution-in-energy law is of his work in early statistics. 

 Substituting from (35) and (39) into (36), we find: 



iLS = (dP/dT)vdV + {II,/T)dT (40) 



a usable and a useful expression for the entropy S — usable, that is to say, 

 to anyone who knows the heat-capacity H^ and the derivative (dP/dT)y, 

 as functions of volume and temperature, for the substance in question. 

 Now take it on faith that there exists a gas having the following qualities: 

 first, its pressure and volume and absolute temperature are linked to- 

 gether by the equation, 



PV = LT (41) 



L being a constant; whence follows, 



{dP/dT)v = L/V (42) 



second, if the attempt is made to express its energy U as function of T 

 and any one of the remaining variables (to wit, P or V or S), then the latter 

 variable drops right out of the picture, leaving t/ as a function of T alone. 

 I suppose this seems a needlessly longwinded way of saying that U does not 

 depend on P or V, but it is necessary to provide for the fact that U may be 

 expressed as (say) a function of S and V, whereupon it will be found that 

 neither variable drops out of the picture. This is one of the features that 

 make the science of thermodynamics very like a maze. 



third, the heat-capacity //,. is independent of all the variables. 



With these stipulations, (36) becomes: 



dS = {L/V)dV -f {HjT)dT (43) 



integrating which, we readily find that for the peculiar kind of gas presented 

 just above as an article of faith, the entropy is given by the formula: 



S ^ L\nV + H^nT + C (44) 



The symbol C stands for one of the most useless things in the world: an 

 arbitrary additive constant of integration. The only purpose normalh' 

 served by such a constant is, to prevent people from thinking that the 

 equation is right if the constant is left off. Its presence means that the 

 absolute value of S is undeterminable, is beyond the reach of experiment to 

 determine. Nevertheless this constant is one of the principal themes of 

 statistical theory; and we shall see that in defiance of what I have just 

 said, and no part of which I retract, it does make sense to assign a particular 

 \-alue to this constant, and remarkably good sense at that. 



