374 BELL SYSTEM TECHNICAL JOURNAL 



any two of the five may be taken as these two, the remainder becoming the 

 dependent variables. 



From (31) we deduce, to begin with, 



T = (dU/dS)v (32) 



an equation which shows that if ever someone sets up a theory in which 

 entropy is expressed as a function of energy and vice versa, it is per se a 

 theory of absolute temperature. This, however, will find its due place 

 later. What is of instant importance is a second deduction, 



(dS/dT)v = T~\dU/dT)y {2,3,) 



for making use of which we take note of the fact (not explicitly stated till 

 now) that when the volume does not change no mechanical work is done 

 upon or by the substance, and therefore all of the change in energy is that 

 brought about by the inflow or outflow of heat. This fact is expressed in 

 another equation, 



(dU/dT)v = H, (34) 



Hv standing for the amount of heat that must be fed into the substance to 

 raise its temperature, at constant volume, by one degree — the "heat- 

 capacity at constant volume," as some would call it. Combining the two, 



(dS/dT)v = H,/T (35) 



Envisage now the entropy ^S* as a function of volume and temperature, 

 and view the equation: 



dS = idS/dV)r dV + {dS/dT)y dT (36) 



An equivalent for the coefficient of dT has been provided, and now it is 

 needful to find one for the coefficient of dV. To do this we use the function 

 {U — TS), to be denoted by A, which by aid of (31) is seen to have the 

 following differential: 



dA = -PdV - SdT (37) 



Out of this one draws the following two deductions, 



{dA/dV)T = -P, {dA/dT)v = -S (38) 



Differentiating both sides of each of these equations, the former with respect 

 to T while holding V constant, the latter with respect to V while holding T 

 constant, one gets two expressions for what is one and the same quantity, 

 to wit, the second derivative d'A/dTdV. Equating these two expressions, 

 and saying goodbye to A which has fulfilled its purpose, one has, 



{dS/dV)T = {dP/dT)y (39) 



