179.

Let us suppose that a large number $N$ of similar oscillators with parallel axes, acting quite independently of one another, are distributed irregularly in a volume-element of the field of radiation, the dimensions of which are so small that within it the intensities of radiation $𝖪$ do not vary appreciably. We shall investigate the mutual action between the oscillators and the radiation which is propagated freely in space.

As before, the state of the field of radiation may be given by the magnitude and the azimuth of vibration $\psi$ of the principal intensities ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$ of the pencils which strike the system of oscillators, where ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$ depend in an arbitrary way on the direction angles $\theta$ and $\varphi$. On the other hand, let the state of the system of oscillators be given by the densities of distribution $w_1$, $w_2$, $w_3,\dots$ [eqn:(166)] (166), with which the oscillators are distributed among the different region elements, $w_1$, $w_2$, $w_3,\dots$ being any proper fractions whose sum is $1$. Herein, as always, the $n$th region element is supposed to contain the oscillators with energies between $(n-1)h\nu$ and $nh\nu$.

The energy absorbed by the system in the time $dt$ within the conical element $d\Omega$ is, according to [eqn:(321)] (321), $\frac{\pi Ndt}{cL}{\sin }^2\theta (𝖪{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi )d\Omega .\mathrm{\backslash Label}[eqn](322)\text{upshape (322)}$ Let us now calculate also the energy emitted within the same conical element.