14 Prof. Clausius on the Determination of the 



relation subsists, which results from (9) if we therein place 

 T=2/: 



E', T =TE* T -g- (26) 



Differentiating this equation with respect to T, we have 

 dEVr^^T d (dw\ 



Now bearing in mind that 



.„ d (dw\ d ( dw\ 



the last equation becomes 



dW xT _ T ^E^ T d /dw\ ^) 



dT ~ dT dxXdT/ ' V ; 



By aid of this equation we can immediately refer either of the 

 equations (24) and (25) to the other. 



By integration of the complete differential equations (22) and 

 (23), each of the two magnitudes S and U can be determined 

 except as to a constant that still remains unknown. * 



Of course any other variable y might be substituted for the 

 variable T in these complete differential equations, if it appeared 

 appropriate for special purposes to make the substitution, as 

 might be done without any difficulty if T is supposed known as 

 a function of x and y } and therefore does not require to be further 

 dwelt upon. 



§8. 



All the foregoing equations are developed in such a way that 

 no limiting conditions are set up in relation to the external 

 forces which act upon the body, and to which the external work 

 has reference. We will now consider a particular case, which 

 is of specially frequent occurrence, rather more closely. I refer 

 to the case where the only extraneous force which acts either to 

 hinder or promote the change of condition in the body, and so 

 occasions a positive or negative expenditure of work, is a pres- 

 sure uniformly distributed over the whole surface of the body, and 

 everywhere directed perpendicularly to the surface of the body. 



In this case external work will be performed only in the case 

 of changes of volume, and the expression for it is very simple. 

 Thus if p is the pressure referred to a unit of surface, the exter- 

 nal work (Arbeit) expressed in mechanical units, accompanying 

 a change of volume represented by dv, is dW=pdv, and hence 



