Page:Scientific Memoirs, Vol. 1 (1837).djvu/378

from $T$ to $T+d\,T$, we bring it successively into contact with the second, the third, and the $(n+1)$th of these sources, until it has acquired the temperature of each of them. When, on the contrary, the volume $v$ of the body being increased by $d\,v$, we wish to give it the temperature $T$, we bring it successively into contact with the $n$th, the $(n-1)$th, and the first of these sources, until it has acquired the temperature of each of them. We then return to these sources the heat that has been borrowed from them in the first part of the operation; for it is not necessary to attend to the differences of an order of inferior magnitude, arising from changes that may have been produced in the specific caloric of the body, in consequence of the variations of $v$ and $Q$.

Nothing therefore will have been lost or gained by any of these sources, excepting always the source of which the temperature is $T+d\,T$, which will have lost the heat necessary to elevate the temperature of the body upon which we are operating from $T+{\frac {(n-1)\;d\,T}{n}}$ to $T+d\,T$, and the source maintained at the temperature $T$, which will have acquired the heat necessary to reduce the temperature of the same body from $T+{\frac {d\,T}{n}}$ to $T$. If we suppose $n$ to be infinitely great, these quantities of heat may be neglected.

We see, therefore, that when the body in question, (its temperature being thus reduced to $T$,) is brought into contact with the source of heat $B$, the heat communicated to it from the source $A$ will be all it has gained from the commencement of the operation. In consequence of the reduction of its volume in contact with the body $B$, it will be found at its original volume and temperature; the quantities $Q$ and $P$ will therefore have re-assumed their primitive value; it is therefore certain that all the heat borrowed from the source $A$, and nothing but that heat, will have been given to the body $B$.

Whence it results that the effect produced,

${\frac {d\,v\;d\,T}{\frac {d\,T}{d\,p}}}$

is owing to the transmission of the heat absorbed by the body subjected to the experiment during its contact with the source $A$, and which has afterwards flowed into the source $B$.

The temperature having remained constant during the contact with the source $A$, it follows that the variations $d\,p$ and $d\,v$ of the pressure and the volume are connected by the relation

${\frac {d\,T}{d\,p}}d\,p+{\frac {d\,T}{d\,v}}d\,v=0$

These variations $d\,p$ and $d\,v$ occasion a variation in the absolute quantity of heat $Q$, the expression of which is