110.

Let us now perform the spectral resolution of the last two equations. To begin with we have from [eqn:(22)] (22): $u={\int }_0^{\infty }{𝗎}_{\nu }d\nu =\frac{3}{8\pi }{\sum }_1^{\infty }{C}_n^2.\mathrm{\backslash Label}[eqn](154)\text{upshape (154)}$ On the right side of the equation the sum $\sum$ consists of separate

terms, every one of which corresponds to a separate ordinal number $n$ and to a simple periodic partial vibration. Strictly speaking this sum does not represent a continuous sequence of frequencies $\nu$, since $n$ is an integral number. But $n$ is, according to [eqn:(150)] (150), so enormously large for all frequencies which need be considered that the frequencies $\nu$ corresponding to the successive values of $n$ lie very close together. Hence the interval $d\nu$, though infinitesimal compared with $\nu$, still contains a large number of partial vibrations, say $n^{\prime }$, where $d\nu =\frac{n^{\prime }}{𝖳}.\mathrm{\backslash Label}[eqn](155)\text{upshape (155)}$ If now in [eqn:(154)] (154) we equate, instead of the total energy densities, the energy densities corresponding to the interval $d\nu$ only, which are independent of those of the other spectral regions, we obtain ${𝗎}_{\nu }d\nu =\frac{3}{8\pi }{\sum }_n^{n+n^{\prime }}{C}_n^2,$ or, according to [eqn:(155)] (155), ${𝗎}_{\nu }=\frac{3𝖳}{8\pi }·\frac{1}{n^{\prime }}{\sum }_n^{n+n^{\prime }}{C}_n^2=\frac{3𝖳}{8\pi }·\overline{C_n^2\mathrm{\backslash Strut}},\mathrm{\backslash Label}[eqn](156)\text{upshape (156)}$ where we denote by $\overline{C_n^2\mathrm{\backslash Strut}}$ the average value of $C_n^2$ in the interval from $n$ to $n+n^{\prime }$. The existence of such an average value, the magnitude of which is independent of $n$, provided $n^{\prime }$ be taken small compared with $n$, is, of course, not self-evident at the outset, but is due to a special property of the function ${⋿}_z$ which is peculiar to stationary heat radiation. On the other hand, since many terms contribute to the mean value, nothing can be said either about the magnitude of a separate term $C_n^2$, or about the connection of two consecutive terms, but they are to be regarded as perfectly independent of each other.

In a very similar manner, by making use of [eqn:(24)] (24), we find for the specific intensity of a monochromatic plane polarized ray, travelling in any direction whatever, ${𝖪}_{\nu }=\frac{3c𝖳}{64{\pi }^2}\overline{C_n^2\mathrm{\backslash Strut}}.\mathrm{\backslash Label}[eqn](157)\text{upshape (157)}$

From this it is apparent, among other things, that, according to the electromagnetic theory of radiation, a monochromatic light or heat ray is represented, not by a simple periodic wave, but by a superposition of a large number of simple periodic waves, the mean value of which constitutes the intensity of the ray. In accord with this is the fact, known from optics, that two rays of the same color and intensity but of different origin never interfere with each other, as they would, of necessity, if every ray were a simple periodic one.

Finally we shall also perform the spectral resolution of the mean value of ${⋿}_z^2$, by writing ${⋿}_z^2=J={\int }_0^{\infty }{𝖩}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](158)\text{upshape (158)}$

Then by comparison with [eqn:(151)] (151), [eqn:(154)] (154), and [eqn:(156)] (156) we find ${𝖩}_{\nu }=\frac{4\pi }{3}{𝗎}_{\nu }=\frac{𝖳}{2}\overline{C_n^2\mathrm{\backslash Strut}}.\mathrm{\backslash Label}[eqn](159)\text{upshape (159)}$ According to [eqn:(157)] (157), ${𝖩}_{\nu }$ is related to ${𝖪}_{\nu }$, the specific intensity of radiation of a plane polarized ray, as follows: ${𝖪}_{\nu }=\frac{3c}{32{\pi }^2}{𝖩}_{\nu }.\mathrm{\backslash Label}[eqn](160)\text{upshape (160)}$