98.

In a space filled with any radiation whatever the entropy radiated in the time $dt$ through an element of area $d\sigma$ in the direction of the conical element $d\Omega$ is given by an expression of the form $dtd\sigma \cos \theta d\Omega L=L\sin \theta \cos \theta d\theta d\varphi d\sigma dt.\mathrm{\backslash Label}[eqn](127)\text{upshape (127)}$

The positive quantity $L$ we shall call the "specific intensity of entropy radiation" at the position of the element of area $d\sigma$ in the direction of the solid angle $d\Omega$. $L$ is, in general, a function of position, time, and direction.

The total radiation of entropy through the element of area $d\sigma$ toward one side, say the one where $\theta$ is an acute angle, is obtained by integration with respect to $\varphi$ from $0$ to $2\pi$ and with respect to $\theta$ from $0$ to ${\displaystyle \frac{\pi }{2}}$. It is $d\sigma dt{\int }_0^{2\pi }d\varphi {\int }_0^{\frac{\pi }{2}}d\theta L\sin \theta \cos \theta .$ When the radiation is uniform in all directions, and hence $L$ constant, the entropy radiation through $d\sigma$ toward one side is $\pi Ld\sigma dt.\mathrm{\backslash Label}[eqn](128)\text{upshape (128)}$

The specific intensity $L$ of the entropy radiation in every direction consists further of the intensities of the separate rays belonging to the different regions of the spectrum, which are propagated independently of one another. Finally for a ray of definite color and intensity the nature of its polarization is characteristic. When a monochromatic ray of frequency $\nu$ consists of two mutually independent "Independent" in the sense of "non-coherent." If, e.g., a ray with the principal intensities $𝖪$ and ${𝖪}^{\prime }$ is elliptically polarized, its entropy is not equal to $𝖫+{𝖫}^{\prime }$, but equal to the entropy of a plane polarized ray of intensity $𝖪+{𝖪}^{\prime }$. For an elliptically polarized ray may be transformed at once into a plane polarized one, e.g., by total reflection. For the entropy of a ray with coherent components see below [sect:104.] Sec. 104, et seq. components, polarized at right angles to each other, with the principal intensities of energy radiation ([sect:17.] Sec. 17) ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$, the specific intensity of entropy radiation is of the form $L={\int }_0^{\infty }d\nu ({𝖫}_{\nu }+{𝖫}_{\nu }^{\prime }).\mathrm{\backslash Label}[eqn](129)\text{upshape (129)}$

The positive quantities $L_{\nu }$ and $L_{\nu }^{\prime }$ in this expression, the principal intensities of entropy radiation of frequency $\nu$, are determined by the values of ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$. By substitution in [eqn:(127)] (127), this gives for the entropy which is radiated in the time

$dt$ through the element of area $d\sigma$ in the direction of the conical element $d\Omega$ the expression $dtd\sigma \cos \theta d\Omega {\int }_0^{\infty }d\nu ({𝖫}_{\nu }+{𝖫}_{\nu }^{\prime }),$ and, for monochromatic plane polarized radiation, $dtd\sigma \cos \theta d\Omega {𝖫}_{\nu }d\nu ={𝖫}_{\nu }d\nu \sin \theta \cos \theta d\theta d\varphi d\sigma dt.\mathrm{\backslash Label}[eqn](130)\text{upshape (130)}$ For unpolarized rays ${𝖫}_{\nu }={𝖫}_{\nu }^{\prime }$ and [eqn:(129)] (129) becomes $L=2{\int }_0^{\infty }{𝖫}_{\nu }d\nu .$ For radiation which is uniform in all directions the total entropy radiation toward one side is, according to [eqn:(128)] (128), $2\pi d\sigma dt{\int }_0^{\infty }{𝖫}_{\nu }d\nu .$