﻿751 Prof. J. P. Kuenen on the 



the relation into the desired shape. These last transformations 

 usually involve long and, as will appear, unnecessary calcu- 

 lation. The simplification in question consists in changing 

 the variables in the original equation *. 



The simplest case occurs when there is only one force to be 

 considered. As an example I shall take a system under 

 uniform pressure p. The fundamental equation is 



Tds = du+pdv; 



or bringing the complete differential du to one side, 



du = Tds—pdc (3) 



Bv the property of complete differentials we have at once 



(4) 



BT _ d/> 



one of the so-called " Maxwell's thermodynamical relations. " 

 The remaining three may be deduced from this one by 

 changing variables, but much more simply by transform- 

 ing the differential equation itself. If we subtract from 

 both sides of equation (3) the differentials d(Ts) or d(—pv) 

 or the sum of both, it assumes one of the following forms : 



d(u — Ts)= — sdT — pdv, 



d(u+pv)= Tds + vdp, 



d(u — Ts-\-pv)= —sdT-\-vdj>. 



As the first side of each of these equations is again a 

 complete differential, the remaining three relations follow 

 immediately. 



Incidentally this method makes it clear why there are 

 four, and only four, relations of the simple form (4). At 

 the same time it brings out very clearly why Gibbs intro- 

 duced the three new thermodynamical functions yjr = u — Ts, 

 ^/ — ti-\-pv, and £ = w — Ts+pv, and no more. 



The same method may be applied when there is more 

 than one force to be considered : I will at once take the 

 general case where there are n general coordinates a, ft, j ... 

 with corresponding forces A, B, C, the direction of the forces 



* This method, though seldom applied, cannot be considered as new: 

 it is virtually contained in Gibbs's treatise, and after writing this note I 

 find that it is used in substantially the same form as here by G. H. Bryan, 

 Encyklopadie der Math. Wiss. v. p. 1 (Teubner, 1903) ; but as it appears 

 little known it may be useful to draw attention to it here. 



