153.

The probability constant $\eta$ contained in the formulæ for the stationary state is determined by the law of emission enunciated in [sect:147.] Sec. 147. According to this, the ratio of the probability that no emission takes place to the probability that emission does take place is proportional to the intensity $𝖨$ of the vibration exciting the oscillator, and hence $\frac{1-\eta }{\eta }=p𝖨\mathrm{\backslash Label}[eqn](265)\text{upshape (265)}$ where the constant of proportionality is to be determined in such a way that for very large energies of vibration the familiar formulæ of classical dynamics shall hold.

Now, according to [eqn:(264)] (264), $\eta$ becomes small for large values of $\overline{U\mathrm{\backslash Strut}}$ and for this special case the equations [eqn:(264)] (264) and [eqn:(265)] (265) give $\overline{U\mathrm{\backslash Strut}}=ph\nu 𝖨,$ and the energy emitted or absorbed respectively in the time $dt$ by all $N$ oscillators becomes, according to [eqn:(250)] (250), $\frac{N𝖨}{4L}dt=\frac{N\overline{U\mathrm{\backslash Strut}}}{4Lph\nu }dt.\mathrm{\backslash Label}[eqn](266)\text{upshape (266)}$

On the other hand, H. Hertz has already calculated from Maxwell's theory the energy emitted by a linear oscillator vibrating periodically. For the energy emitted in the time of one-half of one vibration he gives the expressionH. Hertz, Wied. Ann. 36, p. 12, 1889. $\frac{{\pi }^4{E}^2{l}^2}{3{\lambda }^3}$ where $\lambda$ denotes half the wave length, and the product $El$ (the $C$ of our notation) denotes the amplitude of the moment $f$ ([sect:135.] Sec. 135) of the vibrations. This gives for the energy emitted in the time of a whole vibration $\frac{16{\pi }^4{C}^2}{3{\lambda }^3}$ where $\lambda$ denotes the whole wave length, and for the energy emitted by $N$ similar oscillators in the time $dt$ $N\frac{16{\pi }^4{C}^2{\nu }^4}{3c^3}dt$ since $\lambda ={\displaystyle \frac{c}{\nu }}$. On introducing into this expression the energy $U$ of an oscillator from [eqn:(205)] (205), [eqn:(207)] (207), and [eqn:(208)] (208), namely $U=2{\pi }^2{\nu }^{2}L{C}^2,$ we have for the energy emitted by the system of oscillators $N\frac{8{\pi }^2{\nu }^{2}U}{3c^{3}L}dt\mathrm{\backslash Label}[eqn](267)\text{upshape (267)}$

and by equating the expressions [eqn:(266)] (266) and [eqn:(267)] (267) we find for the factor of proportionality $p$ $p=\frac{3c^3}{32{\pi }^{2}h{\nu }^3}.\mathrm{\backslash Label}[eqn](268)\text{upshape (268)}$