187.

To give the proof required we shall show that the least value which the function $F$ can assume is positive or zero. For this purpose we consider first that positive function, $𝖪$, of $\theta$ and $\varphi$, which, with fixed values of $\zeta$, $w_1$, $w_2$, $w_3,\dots$ and ${𝖪}_e$, will make $F$ a minimum. The necessary condition for this is $\delta F=0$, where according to [eqn:(352)] (352) $\int \delta 𝖪{\sin }^2\theta d\Omega =0.$ This gives, by considering that the quantities $w$ and $\zeta$ do not depend on $\theta$ and $\varphi$, as a necessary condition for the minimum, $\delta F=0=\int d\Omega {\sin }^2\theta \delta 𝖪\{-\log (1+\frac{h{\nu }^3}{c^2𝖪})-\frac{{𝖪}_e-𝖪}{{\displaystyle \frac{c^2𝖪}{h{\nu }^3}}+1}·\frac{1}{𝖪}\}$ and it follows, therefore, that the quantity in brackets, and hence also $𝖪$ itself is independent of $\theta$ and $\varphi$. That in this case $F$ really has a minimum value is readily seen by forming the second variation ${\delta }^{2}F=\int d\Omega {\sin }^2\theta \delta 𝖪\delta \{-\log (1+\frac{h{\nu }^3}{c^2𝖪})-\frac{{𝖪}_e-𝖪}{{\displaystyle \frac{c^2𝖪}{h{\nu }^3}}+1}·\frac{1}{𝖪}\}$ which may by direct computation be seen to be positive under all circumstances.

In order to form the minimum value of $F$ we calculate the value of $𝖪$, which, from [eqn:(352)] (352), is independent of $\theta$ and $\varphi$. Then it follows, by taking account of [eqn:(319a)] (319a), that $𝖪=\frac{h{\nu }^3}{c^2}\frac{\zeta }{1-\zeta }$

and, by also substituting ${𝖪}_e$ from [eqn:(355)] (355),

$F=\frac{8\pi h{\nu }^3}{3c^2}\frac{\zeta }{1-\zeta }{\sum }_1^{\infty }(w_n-\zeta w_{n-1})\log w_n-[(1-\zeta )n-1]w_n\log \zeta .$