92.

If the analytical expression of the function $𝗌$ were known, the law of energy distribution in the normal spectrum could immediately be deduced from it; for the normal spectral distribution of energy or that of black radiation is distinguished from all others by the fact that it has the maximum of the entropy of radiation $S$.

Suppose then we take $𝗌$ to be a known function of $\nu$ and $𝗎$. Then as a condition for black radiation we have $\delta S=0,\mathrm{\backslash Label}[eqn](114)\text{upshape (114)}$ for any variations of energy distribution, which are possible with a constant total volume $V$ and constant total energy of radiation $U$. Let the variation of energy distribution be characterized by making an infinitely small change $\delta 𝗎$ in the energy $𝗎$ of every separate definite frequency $\nu$. Then we have as fixed conditions $\delta V=0\text{and}{\int }_0^{\infty }\delta 𝗎d\nu =0.\mathrm{\backslash Label}[eqn](115)\text{upshape (115)}$ The changes $d$ and $\delta$ are of course quite independent of each other.

Now since $\delta V=0$, we have from [eqn:(114)] (114) and [eqn:(113)] (113) ${\int }_0^{\infty }\delta 𝗌d\nu =0,$ or, since $\nu$ remains unvaried ${\int }_0^{\infty }\frac{\partial 𝗌}{\partial 𝗎}\delta 𝗎d\nu =0,$ and, by allowing for [eqn:(115)] (115), the validity of this equation for all values of $\delta 𝗎$ whatever requires that $\frac{\partial 𝗌}{\partial 𝗎}=\text{const}.\mathrm{\backslash Label}[eqn](116)\text{upshape (116)}$ for all different frequencies. This equation states the law of energy distribution in the case of black radiation.