186.

The principle of increase of entropy requires that the sum of the entropy change [eqn:(348)] (348) of the field of radiation and the entropy change [eqn:(349)] (349) of the system of oscillators be always positive, or zero in the limiting case. That this condition is in fact satisfied we shall prove only for the special case when all rays falling on the oscillators are unpolarized, i.e., when ${𝖪}^{\prime }=𝖪$.

In this case we have from [eqn:(147)] (147) and [sect:185.] Sec. 185. $\left.

$$\begin{gathered} \&{𝖪}_0 \\ \&{𝖪}_0^{\prime } \end{gathered}$$

\right} = \tfrac{1}{2} {2\mathsf{K} + \beta \sin^{2} \theta(\mathsf{K}{e} - \mathsf{K}) ± \beta \sin^{2} \theta(\mathsf{K})},$ and hence ${𝖪}_0=𝖪+\beta {\sin }^2\theta ({𝖪}_e-𝖪),{𝖪}_0^{\prime }=𝖪.$ The entropy change [eqn:(348)] (348) of the field of radiation becomes $dt\Delta \nu \int qd\Omega \{𝖫({𝖪}_o)-𝖫(𝖪)\}=dt\Delta \nu \int qd\Omega \beta {\sin }^2\theta ({𝖪}_e-𝖪)\frac{d𝖫(𝖪)}{d𝖪}$ or, by [eqn:(338)] (338) and [eqn:(278)] (278), $=\frac{\pi kNdt}{hc\nu L}\int d\Omega {\sin }^2\theta ({𝖪}_e-𝖪)\log (1+\frac{h{\nu }^3}{c^2𝖪}).$} - \mathsf{K

On adding to this the entropy change [eqn:(349)] (349) of the system of oscillators and taking account of [eqn:(320)] (320), the total increase in entropy in the time $dt$ is found to be equal to the expression $\Squeeze[0.95]{$\displaystyle\frac{\pi kN\, dt}{ch\nu L} \int d\Omega \sin^{2} \theta \biggl{ \mathsf{K} \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n} + (\mathsf{K}_{e} - \mathsf{K}) \log\biggl(1 + \frac{h\nu^{3}}{c^{2}\mathsf{K}}\biggr) \biggr}$\}$ where $\zeta =1-\eta .\mathrm{\backslash Label}[eqn](350)\text{upshape (350)}$

We now must prove that the expression

$$\begin{gathered} F=\int d\Omega {\sin }^2\theta \backslash \{𝖪{\sum }_1^{\infty }(w_n-\zeta w_{n-1})\log w_n \\ +({𝖪}_e-𝖪)\log (1+\frac{h{\nu }^3}{c^2𝖪})\backslash \}\mathrm{\backslash Label}[eqn](351)\text{upshape (351)} \end{gathered}$$

is always positive and for that purpose we set down once more the meaning of the quantities involved. $𝖪$ is an arbitrary positive function of the polar angles $\theta$ and $\varphi$. The positive proper fraction $\zeta$ is according to [eqn:(350)] (350), [eqn:(265)] (265), and [eqn:(320)] (320) given by $\frac{\zeta }{1-\zeta }=\frac{3c^2}{8\pi h{\nu }^3}\int 𝖪{\sin }^2\theta d\Omega .\mathrm{\backslash Label}[eqn](352)\text{upshape (352)}$ The quantities $w_1$, $w_2$, $w_3,\dots$ are any positive proper

fractions whatever which, according to [eqn:(167)] (167), satisfy the condition ${\sum }_1^{\infty }{w}_n=1\mathrm{\backslash Label}[eqn](353)\text{upshape (353)}$ while, according to [eqn:(342)] (342), $w_0=\frac{1-\zeta }{\zeta }.\mathrm{\backslash Label}[eqn](354)\text{upshape (354)}$ Finally we have from [eqn:(336)] (336) ${𝖪}_e=\frac{h{\nu }^3\zeta }{c^2}{\sum }_1^{\infty }n{w}_n.\mathrm{\backslash Label}[eqn](355)\text{upshape (355)}$