82.

The difference of the two expressions [eqn:(93)] (93) and [eqn:(92)] (92) is equal to the whole change [eqn:(90)] (90), hence $-\frac{8\pi }{3}F\frac{\nu v}{c}\frac{\partial 𝖪}{\partial \nu }\delta t=\delta (V𝗎),$ or, according to [eqn:(24)] (24), $-\frac{1}{3}F\nu v\frac{\partial 𝗎}{\partial \nu }\delta t=\delta (V𝗎),$ or, finally, since $Fv\delta t$ is equal to the decrease of the volume $V$, $\frac{1}{3}\nu \frac{\partial 𝗎}{\partial \nu }\delta V=\delta (V𝗎)=𝗎\delta V+V\delta 𝗎,\mathrm{\backslash Label}[eqn](94)\text{upshape (94)}$ whence it follows that $\delta 𝗎=(\frac{\nu }{3}\frac{\partial 𝗎}{\partial \nu }-𝗎)\frac{\delta V}{V}.\mathrm{\backslash Label}[eqn](95)\text{upshape (95)}$ This equation gives the change of the energy density of any definite frequency $\nu$, which occurs on an infinitely slow adiabatic compression of the radiation. It holds, moreover, not only for black radiation, but also for radiation originally of a perfectly arbitrary distribution of energy, as is shown by the method of derivation.

Since the changes taking place in the state of the radiation in the time $\delta t$ are proportional to the infinitely small velocity $v$ and are reversed on changing the sign of the latter, this equation holds for any sign of $\delta V$; hence the process is reversible.