206 _ MR W. J. M. RANKINE ON THE ECONOMY OF 
the method by which he proves Carno1’s law, I have received from him a state- 
ment of some of his more important results. 
(42.) I have now come to the conclusions,—First: That Carnov’s Lam is not 
an independent principle in the theory of heat; but is deducible, as a consequence, 
Jrom the equations of the mutual conversion of heat and expansive power, 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 converted into power to the total heat 
received by the working body, is the ratio of the difference of those temperatures, to 
the absolute temperature of reception diminished by the constant, which I have 
called x=Cnp.b, and which must, as I have shewn in the Introduction, be the 
same for all substances, in order that molecular equilibrium may be possible. 
(43.) Let abscissee. parallel to OX in the diagram, Plate VIII. fig. 2, denote 
the volumes successively assumed by the working body, and ordinates, parallel 
to OY, the corresponding pressures. Let 7, be the constant absolute temperature 
at which the reception of heat by the body takes place: 7,, the constant absolute 
temperature at which the emission of heat takes place. Let AB be a curve such 
that its ordinates denote the pressures, at the temperature of reception +,, cor- 
responding to the volumes denoted by abscissee. Let DC be a similar curve for 
the temperature of emission 7,. Let AD and BC be two curves, expressing by 
their co-ordinates 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 condensed 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 Section Fourth the heat due to change of tempera- 
ture only, in virtue of the real specific heat. We thus obtain the differential 
6q—8Q=-2-8 [ov (E-Ty)- 87 - | 
—kdér 
In which the negative sign denotes absorption, and the positive emission. 
equation 
dU aU ; 2 ; 
If we now put for 7+ oo their values according to Equation (11.), we find 
8y-8Q=-(r-0 . OV 
dP 
-{se gn (t-S)+o-oF sav \ dr - (63) 
The first term represents the variation of heat due to variation of volume 
only; the second, that due to variation of temperature. Let us now apply this 

