84a.

The characteristic feature of this new distribution of energy may be stated as follows: If we denote all quantities referring to the new state by the addition of an accent, we have the following equation in addition to [eqn:(99)]

(99) ${𝗎}^{\prime }={\nu }^{\prime 3}\varphi ({\nu }^{\prime 3}{V}^{\prime }).$

Therefore, if we put

${\nu }^{\prime 3}{V}^{\prime }={\nu }^{3}V,\mathrm{\backslash Label}[eqn](99a)\text{upshape (99a)}$

we shall also have

$\frac{{𝗎}^{\prime }}{{\nu }^{\prime 3}}=\frac{𝗎}{{\nu }^3}\text{and}{𝗎}^{\prime }{V}^{\prime }=𝗎V,\mathrm{\backslash Label}[eqn](99b)\text{upshape (99b)}$

i.e., if we coordinate with every frequency $\nu$ in the original state that frequency ${\nu }^{\prime }$ which is to $\nu$ in the inverse ratio of the cube roots of the respective volumes, the corresponding energy densities ${𝗎}^{\prime }$ and $𝗎$ will be in the inverse ratio of the volumes.

The meaning of these relations will be more clearly seen, if we write $\frac{V^{\prime }}{{\lambda }^{\prime 3}}=\frac{V}{{\lambda }^3}.$ This is the number of the cubes of the wave lengths, which correspond to the frequency $\nu$ and are contained in the volume of the radiation. Moreover $𝗎d\nu V=𝖴d\nu$ denotes the radiant energy lying between the frequencies $\nu$ and $\nu +d\nu$, which is contained in the volume $V$. Now since, according to [eqn:(99a)] (99a), $\sqrt[3]{V^{\prime }}d{\nu }^{\prime }=\sqrt[3]{V}d\nu \text{or}\frac{d{\nu }^{\prime }}{{\nu }^{\prime }}=\frac{d\nu }{\nu }\mathrm{\backslash Label}[eqn](99c)\text{upshape (99c)}$ we have, taking account of [eqn:(99b)] (99b), ${𝖴}^{\prime }\frac{d{\nu }^{\prime }}{{\nu }^{\prime }}=𝖴\frac{d\nu }{\nu }.$ These results may be summarized thus: On an infinitely slow reversible adiabatic change in volume of radiation contained in a cavity and uniform in all directions, the frequencies change in such a way that the number of cubes of wave lengths of every frequency contained in the total volume remains unchanged, and the radiant energy of every infinitely small spectral interval changes in proportion to the frequency. These laws hold for any original distribution of energy whatever; hence, e.g., an originally monochromatic radiation remains monochromatic during the process described, its color changing in the way stated.