M. CLAPEYKON ON THE MOTIVE POWER OF HEAT. 357 



the body B, and continue to compress the gas until it is again reduced 

 to the volume me. The pressure will then again be equal to ae, as we 

 have shown in the preceding paragraph ; and in the same manner also 

 it will be proved, that the quadrilatei'al figure abed will be the mea- 

 sure of the quantity of action produced by the transmission to the body 

 B, of the heat derived from the body A, during the expansion of the 

 gas. 



Now it is easy to sliow that this quadrilateral figure is a parallelo- 

 gram ; this results from the infinitely small values assigned to the va- 

 riations of the volume and pressure : let us conceive that perpendicu- 

 lars are erected upon each point of the plane upon which the quadri- 

 lateral figure abed IS, traced, and that on each of them, commencing 

 at their foot, are described two quantities T and Q, the first equal to 

 the temperature, the second to the absolute quantity of heat possessed 

 by the gas, when the volume and the pressure have the value assigned 

 to them by the absciss v and the ordinate p which correspond to 

 each point. 



The lines ab and ed belong to the pi'ojections of tAvo curves of 

 equality of temperature, passing through two points infinitely near, 

 taken upon the surface of temperatures ; ab and cd are therefore 

 parallel: ad and be Avill be also projections from two curves, for which 

 Q = const., and which would also pass through two points infinitely 

 near, taken upon the surface Q =f(pv); these two elements are there- 

 fore also parallel. The quadrilateral figure abed is therefore a paral- 

 lelogram, and it is easy to see that its area may be obtained by multi- 

 plying the variation of the volume during the contact of the gas with 

 the body A or the body B, that is to say, cff, or its equal //«, by bn, the 

 difference of the pressures supported during these two operations, and 

 corresponding to the same value of the volume v. Now, eff, or f/t, 

 being the differentials of the volume, are equal to dv; bn will be ob- 

 tained by differentiating the equation jo w = R (267 + v), supposing v 



constant ; we shall then have b?i ^= dp = R — The expression of 



V ' 



the quantity of action developed will therefore be R , 



It remains to determine the quantity of heat necessary to produce 

 this effect : it is equal to that which the gas has derived from the body 

 A, whilst its volume has increased by d v, at the same time preserving 

 the same temperature t. Now Q being the absolute quantity of heat 

 possessed by the gas, ought to be a certain function of j9 and of v, con- 

 sidered as independent variables ; the quantity of heat absorbed by the 

 gas will therefore be 



dQ=^dv + ^^dp; 

 dv dp 



