189.

In order to show finally that the value [eqn:(362)] (362) of $\Phi$ is really the minimum value, we form from [eqn:(357)] (357) the second variation ${\delta }^2\Phi ={\sum }_1^{\infty }(\delta w_n-\zeta \delta w_{n-1})\frac{\delta w_n}{w_n}-\frac{\zeta \delta w_{n-1}}{w_n}\delta w_n+\frac{\zeta w_{n-1}}{w_n^2}\delta w_n^2,$ where all terms containing the second variation ${\delta }^2{w}_n$ have been omitted since their coefficients are, by [eqn:(360)] (360), independent of $n$ and since ${\sum }_1^{\infty }{\delta }^2{w}_n=0.$

This gives, taking account of [eqn:(361)] (361), ${\delta }^2\Phi ={\sum }_1^{\infty }\frac{2\delta w_n^2}{(1-\zeta ){\zeta }^{n-1}}-\frac{2\zeta \delta w_{n-1}\delta w_n}{(1-\zeta ){\zeta }^{n-1}}$ or ${\delta }^2\Phi =\frac{2\zeta }{1-\zeta }{\sum }_1^{\infty }\frac{\delta w_n^2}{{\zeta }^n}-\frac{\delta w_{n-1}\delta w_n}{{\zeta }^{n-1}}.$ That the sum which occurs here, namely, $\frac{\delta w_1^2}{\zeta }-\frac{\delta w_1\delta w_2}{\zeta }+\frac{\delta w_2^2}{{\zeta }^2}-\frac{\delta w_2\delta w_3}{{\zeta }^2}+\frac{\delta w_3^2}{{\zeta }^3}-\frac{\delta w_3\delta w_4}{{\zeta }^3}+\dots \mathrm{\backslash Label}[eqn](363)\text{upshape (363)}$ is essentially positive may be seen by resolving it into a sum of squares. For this purpose we write it in the form ${\sum }_1^{\infty }\frac{1-{\alpha }_n}{{\zeta }^n}\delta w_n^2-\frac{\delta w_n\delta w_{n+1}}{{\zeta }^n}+\frac{{\alpha }_{n+1}}{{\zeta }^{n+1}}\delta w_{n+1}^2,$ which is identical with [eqn:(363)] (363) provided ${\alpha }_1=0$. Now the $\alpha$'s may be so determined that every term of the last sum is a perfect square, i.e., that $4·\frac{1-{\alpha }_n}{{\zeta }^n}·\frac{{\alpha }_{n+1}}{{\zeta }^{n+1}}={\left(\frac{1}{{\zeta }^n}\right)}^2$ or ${\alpha }_{n+1}=\frac{\zeta }{4(1-{\alpha }_n)}.\mathrm{\backslash Label}[eqn](364)\text{upshape (364)}$ By means of this formula the $\alpha$'s may be readily calculated. The first values are: ${\alpha }_1=0,{\alpha }_2=\frac{\zeta }{4},{\alpha }_3=\frac{\zeta }{4-\zeta },\dots .$

plus 0.75em minus 0.25em Continuing the procedure ${\alpha }_n$ remains always positive and less than ${\alpha }^{\prime }=\frac{1}{2}(1-\sqrt{1-\zeta })$. To prove the correctness of this statement we show that, if it holds for ${\alpha }_n$, it holds also for ${\alpha }_{n+1}$. We assume, therefore, that ${\alpha }_n$ is positive and $<{\alpha }^{\prime }$. Then from [eqn:(364)] (364) ${\alpha }_{n+1}$ is positive and $<{\displaystyle \frac{\zeta }{4(1-{\alpha }^{\prime })}}$. But ${\displaystyle \frac{\zeta }{4(1-{\alpha }^{\prime })}}={\alpha }^{\prime }$. Hence ${\alpha }_{n+1}<{\alpha }^{\prime }$. Now, since the assumption made does actually hold for $n=1$, it holds in general. The sum [eqn:(363)] (363) is thus essentially positive and hence the value [eqn:(362)] (362) of $\Phi$ really is a minimum, so that the increase of entropy is proven generally.

The limiting case [eqn:(361)] (361), in which the increase of entropy vanishes, corresponds, of course, to the case of thermodynamic equilibrium between radiation and oscillators, as may also be seen directly by comparison of [eqn:(361)] (361) with [eqn:(271)] (271), [eqn:(265)] (265), and [eqn:(360)] (360).

[190.]190. Conclusion.–-The theory of irreversible radiation processes here developed explains how, with an arbitrarily assumed initial state, a stationary state is, in the course of time, established in a cavity through which radiation passes and which contains oscillators of all kinds of natural vibrations, by the intensities and polarizations of all rays equalizing one another as regards magnitude and direction. But the theory is still incomplete in an important respect. For it deals only with the mutual actions of rays and vibrations of oscillators of the same period. For a definite frequency the increase of entropy in every time element until the maximum value is attained, as demanded by the second principle of thermodynamics, has been proven directly. But, for all frequencies taken together, the maximum thus attained does not yet represent the absolute maximum of the entropy of the system and the corresponding state of radiation does not, in general, represent the absolutely stable equilibrium (compare [sect:27.] Sec. 27). For the theory gives no information as to the way in which the intensities of radiation corresponding to different frequencies equalize one another, that is to say, how from any arbitrary initial spectral distribution of energy the normal energy distribution corresponding to black radiation is, in the course of time, developed. For the oscillators on which the consideration was based influence only the intensities of rays which correspond