250 Mr. W. J. M. Rankine on the Mechanical Action of tieat. 



heat is received and emitted by the working substance; which 

 function, for each pair of temperatures, is the same for all sub- 

 stances in nature. 



This la\\' is laid down by M. Clausius, as it originally had 

 been by Carnot, as an independent axiom; and I had at first 

 doubts as to the soundness of the reasoning by which he main- 

 tained it. Having stated those doubts to Professor Thomson, I 

 am indebted to him for having induced me to investigate the 

 subject thoroughly ; for although I have not yet seen his paper, 

 nor become acquainted with the method by which he proves 

 Carnot^s law, I have received from him a statement of some of 

 his more important results. 



(42.) I have now come to the conclusions, — 



First. That Cai'not's latu is not an independent principle in the 

 theory of heat, but is deducible, as a consequence, from the equa- 

 tions of the mutual conversion of heat and expansive poiver, as 

 given in the first Section of this paper. 



Secondly. That the function of the temperatures of reception 

 and emission, which expresses the maximum ratio of the heat con- 

 verted into poiver to the total heat received by the ivorking body, is 

 the ratio of the difference of those temperatures to the absolute 

 temperature of reception diminished by the constant, ivhich I have 

 called K = CnfMb, and which must, as I have shown in the Intro- 

 duction, be the same for all substances, in order that molecular 

 equilibrium may be possible. 



(43.) Let abscisste, parallel to OX in the diagram,Pl.III. fig.2, 

 denote the volumes successively assumed by the working body ; 

 and ordiuates, parallel to OY, the corresponding pressures. Let 

 Tj be the constant absolute temperature at which the reception 

 of heat by the body takes place ; Tq the constant absolute tem- 

 perature at which the emission of heat takes place. Let AB be 

 a curve such that its ordinates denote the pressures, at the tem- 

 perature of reception t,, corresponding to the volumes denoted 

 by abscissfe. Let DC be a similar curve for the temperature of 

 emission Tq. Let AD and BC be two curves, expressing by 

 their coordinates how the pressure and volume must vary in 

 order that the body may change its temperature without receiving 

 or emitting heat ; the former corresponding to the most con- 

 densed, and the latter to the most expanded state of the body 

 during the working of the machine. 



The quantity of heat received or emitted during an operation 

 on the body, involving indefinitely small variations of volume 

 and temperature, is expressed by adding to equation (6) of Sec- 

 tion IV. the heat due to change of temperature only in virtue 

 of the real specific heat. We thus obtain the differential equa- 

 tion 



