92i M. R. Clausius on a modified Form of the second 



difference, because the temperature of the generated heat is 

 arbitrary, and may therefore have the same value as the tem- 

 perature of one of the two bodies, in which case a third body 

 would be superfluous. Consequently, for the two quantities of 

 heat Q and Q'— Q, an equation of the same form as (6) must 

 hold, t. e. 



(Q'-Q)/[0+QF(^,/')=o. 



Eliminating the magnitude Q' by means of (7), and dividing by 

 Q, this equation becomes 



Y(t,l<)=f(t')-f{l), (8) 



80 that the temperatures t and ^' being arbitrary, the function 

 of two temperatures which applies to the second kind of trans- 

 formation is reduced, in a general manner, to the function of 

 one temperature which applies to the fii'st kind. 



For brevity, we will introduce a simpler symbol for the last 

 function, or rather for its reciprocal, inasmuch as the latter will 

 afterwards be shown to be the more convenient of the two. Let 

 us therefore make 



m=\> (9) 



so that T is now the unknown function of the temperature 

 involved in the equivalence- values. Further, Tj, Tg, &c. shall 

 represent particular values of this function, corresponding to the 

 temperatures ^j, t^ &c. 



According to this, the second fundamental theorem in the 

 mechanical theory of heat, which in this form might appro- 

 priately be called the theorem of the equivalence of transform- 

 ations, may be thus enunciated : 



If two transformations which, without necessitating any other 

 permanent change, can mutually replace one another, be called 

 equivalent, then the generation of the quantity of heat Q of the 

 temperature t from work, has the equivalence-value 



and the passage of the quantity of heat Q from the temperature 

 t^ to the temperature i^ has the value 



^(%~t)' 



wherein T is a function of the temperature, independent of the 

 nature of the process by which the transformation is effected. 

 If, to the last expression, we give the form 



Q Q 



