29.

We shall now consider from a different point of view the radiation phenomena in the interior of a very extended homogeneous isotropic medium which is in thermodynamic equilibrium.

That is to say, we shall confine our attention, not to a definite volume-element, but to a definite pencil, and in fact to an elementary pencil ([sect:21.] Sec. 21).

Let this pencil be specified by the infinitely small focal plane $d\sigma$ at the point $O$ ([fig:2]Fig. 2), perpendicular to the axis of the pencil, and by the solid angle $d\Omega$, and let the radiation take place toward the focal plane in the direction of the arrow. We shall consider exclusively rays which belong to this pencil.

The energy of monochromatic plane polarized radiation of the pencil considered passing in unit time through $d\sigma$ is represented, according to [eqn:(11)] (11), since in this case $dt=1$, $\theta =0$, by $d\sigma d\Omega {𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](30)\text{upshape (30)}$

The same value holds for any other cross-section of the pencil. For first, ${𝖪}_{\nu }d\nu$ has everywhere the same magnitude ([sect:25.] Sec. 25), and second, the product of any right section of the pencil and the solid angle at which the focal plane $d\sigma$ is seen from this section has the constant value $d\sigma d\Omega$, since the magnitude of the cross-section increases with the distance from the vertex $O$ of the pencil in the proportion in which the solid angle decreases.

Hence the radiation inside of the pencil takes place just as if the medium were perfectly diathermanous.

On the other hand, the radiation is continuously modified along its path by the effect of emission, absorption, and scattering. We shall consider the magnitude of these effects separately.