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M: CLAPEYRON ON THE MOTIVE POWER OF HEAT. 857 
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 quadrilateral figure a bcd 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 show 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 abcd 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 projections of two curves of 
equality of temperature, passing through two points infinitely near, 
taken upon the surface of temperatures; ab and ed are therefore 
parallel: ad and be will 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, eg, or its equal fh, by bn, the 
difference of the pressures supported during these two operations, and 
corresponding to the same value of the volume v. Now, eg, or fh, 
being the differentials of the volume, are equal to dv; 6m will be ob- 
tained by differentiating the equation pv = R (267 + v), supposing v 
constant ; we shall then have bx = dp = R es The expression of 
didv 
eat 
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 dv, at the same time preserving 
the same temperature ¢ Now Q being the absolute quantity of heat 
possessed by the gas, ought to be a certain function of p and of v, con- 
sidered as independent variables; the quantity of heat absorbed by the 
gas will therefore be 
dQ= dy +7 dp; 
