2 Prof. Clausius on the Determination of the 



contained a principle applicable to reversible processes in which 

 the initial and final conditions are not identical, a principle which 

 not only have I tacitly regarded as implied therein, but have 

 several times explicitly stated. 



If, indeed, the integral I 7^- becomes equal to nought every 



time that the body, after having passed through various changes, 



comes back to its initial condition, the expression -™- must be 



the complete differential of a magnitude which is determined by 

 the existing condition of the body at any given time, without 

 reference to the way in which it came into this condition — that 

 is to say, to the intermediate conditions which it has succes- 

 sively assumed between its initial and its actual conditions. 

 Accordingly, in the case of the condition of the body being 

 determined by any two independent variables, the expression 



7p- must be the complete differential of a function of these vari- 

 ables; hence, after it has been obtained as a differential expres- 

 sion having reference to these variables, it must satisfy the know T n 

 conditions of integrability. 



The difference between differential expressions which satisfy 

 these conditions, and such as do not satisfy them, is so often 

 mentioned in my papers, that in the collection of my memoirs I 

 have actually made it the subject of a special mathematical in- 

 troduction. With regard to the expression 7^-, it is especially 



stated that it is a complete differential, and accordingly fulfils 

 the conditions of integrability, and among others that stated 

 in my paper " On a Modified Form of the Second Fundamental 

 Theorem of the Mechanical Theory of Heat"*. 



This being once established, it was self-evident that the inte- 



gral \-f¥r must be susceptible of being expressed as a function 



of two variables by aid of the proper magnitudes, which in the 

 calculation were supposed to be known. Such a calculation has 

 now been worked out by Bauschinger, the equation which ex- 

 presses the first fundamental theorem of the mechanical theory 

 of heat being also applied, and special symbols introduced for 



the differential coefficients -^ (if the volume v is assumed to be 



dQ 



constant) and -,- (if the temperature T is assumed as constant). 



* Poggendorff' sAnnalen, vol. xciii. p. 502; Clausius's Abhandlungen- 

 sammlung, part 1, p. 150. [Phil. Mag. S. 4. vol. xii. p. 95.] 



