15.

Let us now consider quite generally the pencil, which is propagated from a point of the element $d\sigma$ as vertex in all directions of space and on both sides of $d\sigma$. A certain direction may be specified by the angle $\theta$ (between $0$ and $\pi$), as already used, and by an azimuth $\varphi$ (between $0$ and $2\pi$). The intensity in this direction is the energy propagated in an infinitely thin cone limited by $\theta$ and $\theta +d\theta$ and $\varphi$ and $\varphi +d\varphi$. The solid angle of this cone is $d\Omega =\sin \theta ·d\theta ·d\varphi .\mathrm{\backslash Label}[eqn](5)\text{upshape (5)}$ Thus the energy radiated in time $dt$ through the element of area $d\sigma$ in the direction of the cone $d\Omega$ is: $dtd\sigma \cos \theta d\Omega K=K\sin \theta \cos \theta d\theta d\varphi d\sigma dt.\mathrm{\backslash Label}[eqn](6)\text{upshape (6)}$

The finite quantity $K$ we shall term the "specific intensity" or the "brightness," $d\Omega$ the "solid angle" of the pencil emanating from a point of the element $d\sigma$ in the direction $(\theta ,\varphi )$. $K$ is a positive function of position, time, and the angles $\theta$ and $\varphi$. In general the specific intensities of radiation in different directions are entirely independent of one another. For example, on substituting $\pi -\theta$ for $\theta$ and $\pi +\varphi$ for $\varphi$ in the function $K$, we obtain the specific intensity of radiation in the diametrically opposite direction, a quantity which in general is quite different from the preceding one.

For the total radiation through the element of area $d\sigma$ toward one side, say the one on which $\theta$ is an acute angle, we get, by integrating with respect to $\varphi$ from $0$ to $2\pi$ and with respect to $\theta$ from $0$ to ${\displaystyle \frac{\pi }{2}}$ ${\int }_0^{2\pi }d\varphi {\int }_0^{\frac{\pi }{2}}d\theta K\sin \theta \cos \theta d\sigma dt.$

Should the radiation be uniform in all directions and hence $K$ be a constant, the total radiation on one side will be $\pi Kd\sigma dt.\mathrm{\backslash Label}[eqn](7)\text{upshape (7)}$