M. CLAPEYRON ON THE MOTIVE POWEK OF HEAT. 359 



whence 



dQ dQ f.^ 



v-p— p-:^ = CR, 

 a V dp 



and consequently 



Rdt _ dt 



dU dQ. C- 



V — p — - 



dv dp 



The function C by which the logarithm of the pressure in the 

 value of Q is multiplied is, as we see, of great importance ; it is inde- 

 pendent of the nature of the gases, and is a function of the temperature 

 alone ; it is essentially positive, and serves as a measure of the maximum 

 quantity of action developed by the heat. 



We have seen that of the four quantities Q, t, p, and v, two being 

 known, the other two follow from them ; they ought therefore to be 

 united together by two equations ; one of them, 



/) f = R (267 + t), 

 results from the combined laws of Mariotte and Gay-Lussac. The 

 equation 



Q = R(B-Clogjo), 



deduced from our theorj', is the second. However, the numerical de- 

 termination of the alterations produced in the gases, when the volume 

 and the pressure are varied in an arbitrary manner, requires a know- 

 ledge of the functions B and C. 



We shall see upon another occasion that a value approaching to the 

 function C may be obtained through a considerable extent of the ther- 

 mometrical scale ; besides, being determined for one gas it will be de- 

 termined for all. As to the function B, it may vary in different gases; 

 however, it is probable that it is the same for all the simple gases: that 

 they ail have the same capacity for heat, is at least the apparent result 

 of the indications of experiment. 



Let us return to the equation 



Q = R (B - C \ogp). 



We will compress a gas occupying the volume v, under the pressure 

 p, until the volume becomes v', and allow it to cool till the tempera- 

 ture sinks to the same point. Let jo' be the new value of the pressure; 

 let Q' be the new val ue of Q ; we shall have 



Q - Q' = R C log -P^ = R C log 4- 

 p v' 



The function C being the same for all the gases, it is evident that 

 equal volumes of all the elastic jluids, taken at the same temperature 

 arid under the same pressure, being compressed or expanded by the same 

 fraction of their volume, disengage or absorb the same absolute quantity 

 of heat. This law M. Dulong has deduced from direct experiment. 



