Page:Reflections on the Motive Power of Heat.djvu/272

and from equations (3), (4),

$eF'tdt=N\log vdt;$

or, dividing by F'tdt,

$e={\frac {N}{F't}}\log v=T\log v.$

Calling T the fraction ${\frac {N}{F't}}$ which is a function of t only, the equation

$e=T\log v$

is the analytical expression of the law stated pp. 80, 81. It is common to all gases, since the laws of which we have made use are common to all.

If we call s the quantity of heat necessary to change the air that we have employed from the volume 1 and from the temperature zero to the volume v and to the temperature t, the difference between s and e will be the quantity of heat required to bring the air at the volume 1 from zero to t. This quantity depends on t alone; we will call it U. It will be any function whatever of t. We shall have

$s=e+U=T\log v+U.$

If we differentiate this equation with relation to t alone, and if we represent it by T' and U', the differential coefficients of T and U, we shall get

${\frac {ds}{dt}}=T'\log v+U';$

(5)