112.

The elementary processes of radiation exhibit regularities only when the vibrations are restricted to a narrow spectral region, that is to say in the case of spectroscopically resolved light, and especially in the case of the natural spectral lines. If, e.g., the amplitudes $C_n$ of the Fourier's series [eqn:(149)] (149) differ from zero only between the ordinal numbers $n=n_0$ and $n=n_1$, where ${\displaystyle \frac{n_1-n_0}{n_0}}$ is small, we may write ${⋿}_z=C_0\cos (\frac{2\pi n_{0}t}{𝖳}-{\theta }_0),\mathrm{\backslash Label}[eqn](161)\text{upshape (161)}$ where

$$\begin{gathered} C_0\cos {\theta }_0\&={\sum }_{n_0}^{n_1}{C}_n\cos (\frac{2\pi (n-n_0)t}{𝖳}-{\theta }_n) \\ {C}_0\sin {\theta }_0\&=-{\sum }_{n_0}^{n_1}{C}_n\sin (\frac{2\pi (n-n_0)t}{𝖳}-{\theta }_n) \end{gathered}$$

and ${⋿}_z$ may be regarded as a single approximately periodic vibration of frequency ${\nu }_0={\displaystyle \frac{n_0}{𝖳}}$ with an amplitude $C_0$ and a phase-constant ${\theta }_0$ which vary slowly and irregularly.

The smaller the spectral region, and accordingly the smaller ${\displaystyle \frac{n_1-n_0}{n_0}}$, the slower are the fluctuations ("Schwankungen") of $C_0$ and ${\theta }_0$, and the more regular is the resulting vibration and also the larger is the difference of path for which radiation can interfere with itself.

If a spectral line were absolutely sharp, the radiation would have the property of being capable of interfering with itself for differences of path of any size whatever. This case, however, according to [sect:18.] Sec. 18, is an ideal abstraction, never occurring in reality.

IIIEntropy and Probability

[Fundamental Definitions and Laws] IFundamental Definitions and Laws. Hypothesis of Quanta