358 M. CLAPEYRON ON THE MOTIVE POWER OF HEAT. 
but the temperature remaining constant during the variation of the vo- 
lume, we have 
vdp + pdv=0,whenced p = — £ dv, 
and consequently 
dQ= Ge - -. oa) t 
If we divide the effect produced by this value of dQ, we shall 
have 
Rat 
qQ_ dQ 
rer: Pap 
for the expression of the maximum effect which can be developed by 
the passage of a quantity of heat equal to unity, from a body main- 
tained at the temperature ¢ to a body maintained at the temperature 
t— dt. 
We have shown that this quantity of action developed is indepen- 
dent of the agent which has served to transmit the heat; it is there- 
fore the same for all the gases, and is equally independent of the pon- 
derable quantity of the body employed: but there is nothing that 
proves it to be independent of the temperature; v s = = ought 
therefore to be equal to an unknown function of ¢, which is “es same 
for all the gases. 
Now by the equation py = R (267 +t), ¢is itself the function of 
the product p v; the partial differential equation is therefore 
»@Q2 dQ _ 
——p-—=F 
Se. a fudnr oon oak 
Having for its integral 
Q=f(p.v) —F (p.v) log [ (hyp)p]. 
No change is effected in the generality of this formula by substitut- 
ing for these two arbitrary functions of the product p v, the functions 
B and C of the temperature, multiplied by the coefficient R ; we shall 
thus have 
Q=R (B — Clog p). 
That this value of Q satisfies all the conditions to which it is sub- 
ject may be easily verified; in fact we have 
40 _p (2BP _ Iq 4C 
dv di R PTR 
dQ dB vw ie aC v _ 1 
dp dt R ce tdi Pp)’ 
