Page:Scientific Memoirs, Vol. 1 (1837).djvu/370

358 but the temperature remaining constant during the variation of the volume, we have and consequently

If we divide the effect produced by this value of $$dQ$$, we shall have 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 maintained at the temperature $$t$$ to a body maintained at the temperature $$t - dt$$.

We have shown that this quantity of action developed is independent of the agent which has served to transmit the heat; it is therefore the same for all the gases, and is equally independent of the ponderable quantity of the body employed: but there is nothing that proves it to be independent of the temperature; $$v\frac - p \frac$$ ought therefore to be equal to an unknown function of $$t$$, which is the same for all the gases.

Now by the equation $$pv = R (267 + t)$$, $$t$$ is itself the function of the product $$pv$$; the partial differential equation is therefore having for its integral

No change is effected in the generality of this formula by substituting for these two arbitrary functions of the product $$pv$$, the functions $$B$$ and $$C$$ of the temperature, multiplied by the coefficient $$R$$; we shall thus have

That this value of $$Q$$ satisfies all the conditions to which it is subject may be easily verified; in fact we have