30 Prof. Clausius on the Determination of 



first equation with respect to v, and the second with respect to 

 T; we then obtain 



T d^ d*J 



A' dTdv ~ dTdv 



1_ dZ T d?Z_ d 2 J dp^ 

 A ' do + A ' dTdv ~ dTdv + dT 



By subtracting the first of these equations from the second, 

 and multiplying the remainder by A, we get the expression 

 sought, namely, 



~~dv ~ K dT [6) 



If we combine this expression with the expression for -7= 



which results from the first of the equations (2), we can form 

 the complete differential equation which follows, 



#i£-g*f+A** (4) 



In order to integrate this equation, we will take as our start- 

 ing-point a condition in which the temperature and volume are 

 T and v , and will denote the corresponding value of Z by Z . 

 Let us now suppose that, in the first place, the temperature 

 varies from T to any other value T, the volume remaining 

 unchanged at v — and that, in the second place, the volume varies 

 from v to v, while the temperature remains constant at T ; we 

 shall then, by following in our integration the order of changes 

 of state here denoted, obtain the equation 



^W^SL/^^ 



(5) 



In my paper I have compared the quantity Z, determined in 

 the manner that has been explained, with a quantity which Pro- 

 fessor Rankine has denoted by F, and which is defined by the 

 equation 



*-j%* ...••... (6) 



where the integration ought to be taken from a given initial 

 volume to the actual volume, the temperature being supposed 

 to remain constant. I have stated that this quantity F is not 



identical with the quantity — Z, but that it differs from it in 



general by a function of T. It is easy to see that the function 



