Fundamental Theorem in the Mechanical Theory of Heat » 85 

 we easily see that it is divisible into the two equations 



dt" dt' 

 and 



dv dv ' 



Let the first of these be differentiated according to v and the 

 second according to t. In doing so we may apply to U the well- 

 known theorem, that when a function of two independent varia- 

 bles is successively differentiated according to both, the order in 

 which this is done does not affect the result. This theorem, 

 however, does not apply to the magnitude Q, and we must use 

 symbols which will show the order of differentiation. This is 

 done in the following equations j — 



d /dQ\ _ d^V 

 dvKdtJ'^dt.dv' 

 d/d(i\d^V ^ dp 

 dt\dv / dtdv ' dt ' 

 By subtraction, we have 



d/dQ\ dfdQ\_ dp 

 Jt\'dh/''Jv\dt)''^'dt' • • • • W) 



an equation which no longer contains U. 



The equations (2) and (3) can be still further specialized by 

 applying them to particular classes of bodies. In my former 

 memoir I have shown these special applications in two of the 

 most important cases, viz. permanent gases and vapours at a 

 maximum density. On this account I will not here pursue the 

 subject further, but pass on to the consideration of the second 

 fundamental theorem in the mechanical theory of heat. 



Theorem of the equivalence of transformations. 



Carnot's theorem, when brought into agreement with the first 

 fundamental theorem^ expresses a relation between two kinds of 

 transformations, the transformation of heat into work, and the 

 passage of heat from a warmer to a colder body, which may be 

 regarded as the transformation of heat at a higher into heat at a 

 lower temperature. In its original form it may be enunciated 

 in some such manner as the following : — In all cases where a • 

 quantity of heat is converted into work, and where the body effect- 

 ing this transformation ultimately returns to its original condition, 

 another quantity of heat must necessarily be transferred from a 

 warmer to a colder body ; and the magnitude of the last quantity 

 of heat, in relation to the first, depends only upon the temperatures 



