109.

Though we have no detailed special information about the function ${⋿}_z$, nevertheless its relation to the radiation of heat affords some important information as to a few of its general properties.

Firstly, for the space density of radiation in a vacuum we have, according to Maxwell's theory,

$u=\frac{1}{8\pi }(\overline{{⋿}_x^2\mathrm{\backslash Strut}}+\overline{{⋿}_y^2\mathrm{\backslash Strut}}+\overline{{⋿}_z^2\mathrm{\backslash Strut}}+\overline{{𝖧}_x^2\mathrm{\backslash Strut}}+\overline{{𝖧}_y^2\mathrm{\backslash Strut}}+\overline{{𝖧}_z^2\mathrm{\backslash Strut}}).$

Now the radiation is uniform in all directions and in the stationary state, hence the six mean values named are all equal to one another, and it follows that

$u=\frac{3}{4\pi }\overline{{⋿}_z^2\mathrm{\backslash Strut}}.\mathrm{\backslash Label}[eqn](151)\text{upshape (151)}$

Let us substitute in this equation the value of ${⋿}_z$ as given by [eqn:(149)] (149). Squaring the latter and integrating term by term through a time interval, from $0$ to $t$, assumed large in comparison with all periods of vibration ${\displaystyle \frac{1}{\nu }}$ but otherwise arbitrary, and then dividing by $t$, we obtain, since the radiation is perfectly stationary,

$u=\frac{3}{8\pi }\sum C_n^2.\mathrm{\backslash Label}[eqn](152)\text{upshape (152)}$

From this relation we may at once draw an important conclusion as to the nature of ${⋿}_z$ as a function of time.

Namely, since the Fourier's series [eqn:(149)] (149) consists, as we have seen, of a great many terms, the squares, $C_n^2$, of the separate amplitudes of vibration the sum of which gives the space density of radiation, must have exceedingly small values.

Moreover in the integral of the square of the Fourier's series the terms which depend on the time $t$ and contain the products of any two different amplitudes all cancel; hence the amplitudes $C_n$ and the phase-constants ${\theta }_n$ must vary from one ordinal number to another in a quite irregular manner.

We may express this fact by saying that the separate partial vibrations of the series are very small and in a "chaotic" Compare footnote to [page:116]page page:116 (Tr.). state.

For the specific intensity of the radiation travelling in any direction whatever we obtain from [eqn:(21)] (21)

$K=\frac{cu}{4\pi }=\frac{3c}{32{\pi }^2}\sum C_n^2.\mathrm{\backslash Label}[eqn](153)\text{upshape (153)}$