79.

With the aid of the theorems established we are now in a position to calculate the change of the density of radiation for

every frequency for the case of infinitely slow adiabatic compression of the perfectly evacuated hollow cylinder, which is filled with uniform radiation. For this purpose we consider the radiation at the time $t$ in a definite infinitely small interval of frequencies, from $\nu$ to $\nu +d\nu$, and inquire into the change which the total energy of radiation contained in this definite constant interval suffers in the time $\delta t$.

At the time $t$ this radiant energy is, according to [sect:23.] Sec. 23, $V𝗎d\nu$; at the time $t+\delta t$ it is $(V𝗎+\delta (V𝗎))d\nu$, hence the change to be calculated is $\delta (V𝗎)d\nu .\mathrm{\backslash Label}[eqn](90)\text{upshape (90)}$ In this the density of monochromatic radiation $𝗎$ is to be regarded as a function of the mutually independent variables $\nu$ and $t$, the differentials of which are distinguished by the symbols $d$ and $\delta$.

The change of the energy of monochromatic radiation is produced only by the reflection from the moving reflector, that is to say, firstly by certain rays, which at the time $t$ belong to the interval $(\nu ,d\nu )$, leaving this interval on account of the change in color suffered by reflection, and secondly by certain rays, which at the time $t$ do not belong to the interval $(\nu ,d\nu )$, coming into this interval on account of the change in color suffered on reflection. Let us calculate these influences in order. The calculation is greatly simplified by taking the width of this interval $d\nu$ so small that $d\nu \text{ is small compared with }\frac{v}{c}\nu ,\mathrm{\backslash Label}[eqn](91)\text{upshape (91)}$ a condition which can always be satisfied, since $d\nu$ and $v$ are mutually independent.