202 Prof. Clausius on the Internal Work 



fices for equation (III.) ; for if II is a function of the temperature 



only, the differential expression -=j- takes the form f(T)dT, in 



which /(T) is ohviously a real function which can have but one 

 meaning, and the integral of this expression must plainly be 

 equal to nothing if the initial and final values of T are the same. 



The necessity of this theorem may be demonstrated thus. 



In order to be able to refer the alterations of condition to 

 alterations of certain magnitudes, we will assume that the manner 

 in which the body changes its condition is not altogether arbi- 

 trary, but is subject to such conditions that the condition of the 

 body is determined by its temperature, and by any second mag- 

 nitude which is independent of the temperature. This second 

 magnitude must plainly be connected with the arrangement of 

 the constituent particles : we may, for example, consider the 

 disgregation of the body as such a magnitude ; it may, however, 

 be any other magnitude dependent on the arrangement of the 

 constituent particles. A case which often occurs, and one which 

 has been frequently discussed, is that in which the volume of the 

 body is the second magnitude, which can be altered indepen- 

 dently of the temperature, and which, together with the tempe- 

 rature, determines the condition of the body. We will take X 

 as a general expression for the second magnitude, so that the 

 two magnitudes T and X together determine the condition of 

 the body. 



Since, however, the quantity of heat, H, present in the body 

 is a magnitude which in any case is completely determined by the 

 condition of the body at any instant, it must here, where the con- 

 dition of the body is determined by the magnitudes T and X, be 

 a function of these two magnitudes. Accordingly, we may write 

 the differential dH in the following form, 



dH = Md~T + NdX, (17) 



where M and N are functions of T and X, which must satisfy 

 the well-known equation of condition to which the differential 

 coefficients of a function of two independent variables are subject; 

 that is, the equation 



d\~dT (18) 



JdJi 

 Tp- is to become equal to nothing each 



time that the magnitudes T and X return to the same values as 



JTJ 



they had at the beginning, -^ must also be the complete dif- 

 ferential of a function of T and X. And since we may write, as 



