Page:Scientific Memoirs, Vol. 1 (1837).djvu/377

body by a quantity $b\,c$, in such a manner that all the heat developed by the diminution of volume may be absorbed by the body $B$, and the temperature remain equal to its primitive value $T$. The volume $V$ also again becoming the same as it was at the commencement of the operation, it is certain that the pressure will return to its primitive value $b\,d$, as will also the absolute quantity of heat $Q$.

If we now connect the four points $f$, $g$, $e$, $d$ by right lines we shall form a quadrilateral figure, the area of which will measure the quantity of action developed during the operation described. Now it is easy to see that $f\,g$ and $d\,c$ are two elements infinitely near, described upon two curves infinitely near, the equations of which will be $T+d\,T=\mathrm {const.}$, and $T=\mathrm {const.}$ They ought therefore to be considered as parallel; the two ordinates which terminate the quadrilateral figure in the other direction being also parallel, the figure is parallelogrammical, and measures $b\,c\times d\,f$.

Now $f\,d$ is nothing but the increase experienced by the pressure $p$, the volume $v$ remaining constant, and $T$ becoming $T+d\,T$. We have therefore

$d\,f={\frac {d\,p}{d\,T}}d\,T,$

whence

$f\,d={\frac {1}{\frac {d\,T}{d\,p}}}d\,T.$

And $b\,c$ being the increase of volume $d\,v$

$f\,d\times b\,c={\frac {d\,v.\,d\,T}{\frac {d\,T}{d\,p}}}.$

It only remains to determine the heat consumed in the production of this quantity of mechanical action.

We have first raised the temperature of the body subjected to experiment by the quantity $d\,T$ without changing its primitive volume $v$; afterwards, when it had become $v+d\,v$, we have lowered its temperature by the same quantity $d\,T$ without varying its primitive volume $v+d\,v$. Now it may easily be seen that this double operation can be effected without loss of heat; let us suppose that $n$ being a number indefinitely great, the interval of temperature $d\,T$ be divided into a number $n$ of new intervals ${\frac {d\,T}{n}}$, and that we have $n+1$ sources of heat maintained at the temperatures $T$, $T+{\frac {d\,T}{n}}$, $T+{\frac {2d\,T}{n}}$,........ $T+{\frac {(n-1)d\,T}{n}}$ and $T+d\,T$.

To raise the temperature of the body upon which we are operating

2 c 2