Fundamental Theorem in the Mechanical Theory of Heat, 98 



it is evident that the passage of the quantity of heat Q, from the 

 temperature t^ to the temperature /g? lias the same equivalence- 

 value as a double transformation of the first kind, that is to say, 

 the transformation of the quantity Q from heat, at the tempera- 

 ture /j into work, and from work into heat at the temperature t^, 

 A discussion of the question how far this external agreement is 

 based upon the nature of the process itself would be out of 

 place here ; but at all events, in the mathematical determination 

 of the equivalence-value, every transmission of heat, no matter 

 how effected, can be considered as such a combination of two 

 opposite transformations of the first kind. 



By means of this rule, it will be easy to find a mathematical 

 expression for the total value of all the transformations of both 

 kinds, which are included in any circular process, however com- 

 plicated. For instead of examining what part of a given 

 quantity of heat received by a reservoir of heat, during the cir- 

 cular process, has arisen from work, and whence the other part 

 has come, every such quantity received may be brought into 

 calculation as if it had been generated by work, and every 

 quantity lost by a reservoir of heat, as if it had been converted 

 into work. Let us assume that the several bodies Kj, Kg, K3, &c., 

 serving as reservoirs of heat at the temperatures /j, t^, /g, &c., 

 have received during the process the quantities of heat Q^, Qg, Q3, 

 &c., whereby the loss of a quantity of heat will be counted as 

 the gain of a negative quantity of heat ; then the total value N 

 of all the transformations will be 



N=| + ^ + | + &c..=X§ (10) 



It is here assumed that the temperatures of the bodies Kj, Kg, Kg, 

 &c. are constant, or at least so nearly constant, that their varia- 

 tions may be neglected. When one of the bodies, however, 

 either by the reception of the quantity of heat Q itself, or 

 through some other cause, changes its temperature during the 

 process so considerably, that the variation demands considera- 

 tion, then for each element of heat dQ we must employ that 

 temperature which the body possessed at the time it received it, 

 whereby an integration will be necessary. For the sake of 

 generality, let us assume that this is the case with all the bodies ; 

 then the foregoing equation will assume the form 



N = 



•^Q 



:f^ (") 



wherein the integral extends over all the quantities of heat 

 received by the several bodies. 



If the process is ?'eversible, then, however complicated it ma; 

 be, we can prove, as in the simple process before considere 



i 



