185.

We now turn to the discussion of the second principle, the principle of the increase of entropy, and follow closely the above discussion regarding the energy. When there is no oscillator in the field, every one of the elementary pencils, infinite in number,

retains during rectilinear propagation both its specific intensity and its entropy without change, even when reflected at the surface, assumed as plane and reflecting, which bounds the field. The system of oscillators, however, produces a change in the incident pencils and hence also a change in the entropy of the radiation propagated in the field. For the calculation of this change we need to investigate only those monochromatic rays which lie close to the natural frequency $\nu$ of the oscillators, since the rest are not altered at all by the system.

The system of oscillators is struck in the direction $(\theta ,\varphi )$ within the conical element $d\Omega$ converging toward the system by a pencil polarized in some arbitrary way, the spectral intensity of which is given by the sum of the two principal intensities $𝖪$ and ${𝖪}^{\prime }$ with the azimuth of vibration $\psi$ and ${\displaystyle \frac{\pi }{2}}+\psi$ respectively, which are assumed to be non-coherent. According to [eqn:(141)] (141) and [sect:182.] Sec. 182 this pencil conveys the entropy $q\Delta \nu [𝖫(𝖪)+𝖫({𝖪}^{\prime })]d\Omega dt\mathrm{\backslash Label}[eqn](345)\text{upshape (345)}$ to the system of oscillators in the time $dt$, where the function $𝖫(𝖪)$ is given by [eqn:(278)] (278). Hence this amount of entropy is taken from the field of radiation on the side of the rays arriving within $d\Omega$. In compensation a pencil starts from the system on the other side in the same direction $(\theta ,\varphi )$ within $d\Omega$ having the components ${𝖪}^{″}$ and ${𝖪}^{‴}$ with the azimuth of vibration ${\displaystyle \frac{\pi }{2}}$ and $0$ respectively, but its entropy radiation is not represented by $𝖫({𝖪}^{″})+𝖫({𝖪}^{‴})$, since ${𝖪}^{″}$ and ${𝖪}^{‴}$ are not non-coherent, but by $𝖫({𝖪}_0)+𝖫({𝖪}_0^{\prime })\mathrm{\backslash Label}[eqn](346)\text{upshape (346)}$ where ${𝖪}_0$ and ${𝖪}_0^{\prime }$ represent the principal intensities of the pencil.

For the calculation of ${𝖪}_0$ and ${𝖪}_0^{\prime }$ we make use of the fact that, according to [eqn:(330)] (330) and [eqn:(335)] (335), the radiation ${𝖪}^{″}$ and ${𝖪}^{‴}$, of which the component ${𝖪}^{‴}$ vibrates in the azimuth $0$, consists of the following three components, non-coherent with one another:

${𝖪}_1=𝖪{\sin }^2\psi +𝖪{\cos }^2\psi (1-\beta {\sin }^2\theta )=𝖪(1-\beta {\sin }^2\theta {\cos }^2\psi )$ with the azimuth of vibration ${\mathrm{tg}}^2{\psi }_1={\displaystyle \frac{{\mathrm{tg}}^2\psi }{1-\beta {\sin }^2\theta }}$,

${𝖪}_2={𝖪}^{\prime }{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi (1-\beta {\sin }^2\theta )={𝖪}^{\prime }(1-\beta {\sin }^2\theta {\sin }^2\psi )$ with the azimuth of vibration ${\mathrm{tg}}^2{\psi }_2={\displaystyle \frac{{\cot }^2\psi }{1-\beta {\sin }^2\theta }}$, and, ${𝖪}_3=\beta {\sin }^2\theta {𝖪}_e$ with the azimuth of vibration $\mathrm{tg}{\psi }_3=0$.

According to [eqn:(147)] (147) these values give the principal intensities ${𝖪}_0$ and ${𝖪}_0^{\prime }$ required and hence the entropy radiation [eqn:(346)] (346). Thereby the amount of entropy $q\Delta \nu [𝖫({𝖪}_0)+𝖫({𝖪}_0^{\prime })]d\Omega dt\mathrm{\backslash Label}[eqn](347)\text{upshape (347)}$ is added to the field of radiation in the time $dt$. All told, the entropy change of the field of radiation in the time $dt$, as given by subtraction of the expression [eqn:(345)] (345) from [eqn:(347)] (347) and integration with respect to $d\Omega$, is $dt\Delta \nu \int qd\Omega [𝖫({𝖪}_0)+𝖫({𝖪}_0^{\prime })-𝖫(𝖪)-𝖫({𝖪}^{\prime })].\mathrm{\backslash Label}[eqn](348)\text{upshape (348)}$