20 Prof. ThouisoQ on the Dynamical Theory of Heat. 



engine which fulfills Caruot's condition of complete reversibility. 

 Hence, by Prop. II., it must produce the same amount of work 

 for the same quantity of heat absorbed in the first opei-ation, as 

 any other substance similarly operated upon through the same 



I'ange of temperatures. But Jj-r.dv is obviously the whole work 



done in the complete cycle, and (by the definition of M in § 20) 

 Mdv is the quantity of heat absorbed in the first operation. 

 Hence the value of 



^T dv ^ 



dt^'^'' dt 



Mrfi; ' ''^' M ''' 

 must be the same for all substances, with the same values of / 

 and T ; or, since t is not involved except as a factor, we must have 

 dp 



1"=^^ (^) 



where j«, depends only on t ; from M'hich we conclude the pro- 

 position which was to be proved. 



dp 



22. The very remarkable theorem that -=-? niust be the same 



for all substances at the same temperature, was first given 

 (although not iu precisely the same terms) by Carnot, and de- 

 monstrated by him, according to the principles he adopted. We 

 have now seen that its truth may be satisfactorily established 

 without adopting the false part of his principles. Hence all Car- 

 not's conclusions, and all conclusions derived by others from his 

 theory, which depend merely on equation (3), require no modifi- 

 cation when the dynamical theory is adopted. Thus, all the 

 conclusions contained in Sections I., II., and III., of the Ap- 

 pendix to my Account of Carnot's Theory, and in the paper im- 

 mediately following it in the Transactions, entitled "Theoretical 

 Considerations on the Effect of Pressure in Lowering the Freezing 

 Point of Water," by my elder brother, still hold. Also, we see 

 that Carnot's expression for the mechanical effect derivable from 

 a given quantity of heat by means of a perfect engine in which 

 the range of temperatures is infinitely small, expresses truly the 

 greatest effect which can possibly be obtained in the circum- 

 stances ; although it is in reality only an infinitely small fraction 

 of the whole mechanical equivalent of the heat supplied; the 

 remamder being irrecoverably lost to man, and therefore "wasted/' 

 although not annihilated. 



23. On the other hand, the expression for the mechanical 

 effect obtainable from a given quantity of heat entering an engine 



