92 Prof. Clausius on the Internal Work 



If we suppose this expression integrated, starting with any 

 initial condition in which Z has the value Z , we get 



Z=Z +Ap^ (3) 



The magnitude Z is thus determined, with the exception of a 

 constant dependent upon the initial condition that is chosen. 



If the temperature of the body is not the same at every part, 

 we can regard it as divided into any number we choose of separate 

 parts, and let the elements dZ and dh in equation (2) refer to 

 any one of them, and at once substitute for T the value of the 

 absolute temperature of that part. If we then unite by summa- 

 tion the infinitely small changes of disgregation of the separate 

 parts, or by integration, if there is an infinite number of them, 

 we obtain the similarly infinitely small change of disgregation 

 of the entire body, and from this we can obtain, likewise by inte- 

 gration, any desired finite change of disgregation. 



We will now return to equation (1), and by help of equation 

 (2) we will eliminate from it the element of work dh. Thus we 

 set 



d£l + dR+TdZ=zO; (4) 



or, dividing by T, 



^H +rfZ = ( 5 ) 



If we suppose this equation integrated for a finite change of con- 

 dition, we have 



f^ + £z = (II.) 



Supposing the body not to be of uniform temperature through- 

 out, we may imagine it broken up again into separate parts, and 

 can make the elements dQ, dJI, and dZ in equation (5) refer in 

 the first instance to one part, and for T we can put the absolute 

 temperature of this part. The symbols of integration in (II.) 

 are then to be understood as embracing the alterations of all the 

 parts. We must here remark that cases in which one conti- 

 nuous body is of different temperatures at different parts, so that 

 a passage of heat immediately takes place by conduction from the 

 warmer to the colder parts, must be for the present disregarded, 

 because such a passage of heat is not reversible, and we have 

 provisionally confined ourselves to the consideration of reversible 

 alterations. 



Equation (II.) is the mathematical expression of the above 

 theorem for which we have been seeking, for all reversible altera- 

 tions of condition of a body ; and it is clearly evident that it also 



