17.

The specific intensity $K$ of the whole energy radiated in a certain direction may be further divided into the intensities of the separate rays belonging to the different regions of the spectrum which travel independently of one another. Hence we consider the intensity of radiation within a certain range of frequencies, say from $\nu$ to ${\nu }^{\prime }$. If the interval ${\nu }^{\prime }-\nu$ be taken sufficiently small and be denoted by $d\nu$, the intensity of radiation within the interval is proportional to $d\nu$. Such radiation is called homogeneous or monochromatic.

A last characteristic property of a ray of definite direction, intensity, and color is its state of polarization. If we break up a ray, which is in any state of polarization whatsoever and which travels in a definite direction and has a definite frequency $\nu$, into two plane polarized components, the sum of the intensities of the components will be just equal to the intensity of the ray as a whole, independently of the direction of the two planes, provided the two planes of polarization, which otherwise may be taken at random, are at right angles to each other. If their position be denoted by the azimuth $\psi$ of one of the planes of vibration (plane of the electric vector), then the two components of the intensity may be written in the form ${2} &\mathsf{K}{\nu} \cos^{2} \psi &&+ \mathsf{K}}' \sin^{2} \psi \ \LeftText{and} &\mathsf{K{\nu} \sin^{2} \psi &&+ \mathsf{K}' \cos^2 \psi.

\Label[eqn]{(8)}\tag*{\upshape (8)}$ Herein $𝖪$ is independent of $\psi$. These expressions we shall call

the "components of the specific intensity of radiation of frequency $\nu$." The sum is independent of $\psi$ and is always equal to the intensity of the whole ray ${𝖪}_{\nu }+{𝖪}_{\nu }^{\prime }$. At the same time ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$ represent respectively the largest and smallest values which either of the components may have, namely, when $\psi =0$ and $\psi ={\displaystyle \frac{\pi }{2}}$. Hence we call these values the "principal values of the intensities," or the "principal intensities," and the corresponding planes of vibration we call the "principal planes of vibration" of the ray. Of course both, in general, vary with the time. Thus we may write generally $𝖪={\int }_0^{\infty }d\nu ({𝖪}_{\nu }+{𝖪}_{\nu }^{\prime })\mathrm{\backslash Label}[eqn](9)\text{upshape (9)}$ where the positive quantities ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$, the two principal values of the specific intensity of the radiation (brightness) of frequency $\nu$, depend not only on $\nu$ but also on their position, the time, and on the angles $\theta$ and $\varphi$. By substitution in [eqn:(6)] (6) the energy radiated in the time $dt$ through the element of area $d\sigma$ in the direction of the conical element $d\Omega$ assumes the value $dtd\sigma \cos \theta d\Omega {\int }_0^{\infty }d\nu ({𝖪}_{\nu }+{𝖪}_{\nu }^{\prime })\mathrm{\backslash Label}[eqn](10)\text{upshape (10)}$ and for monochromatic plane polarized radiation of brightness ${𝖪}_{\nu }$: $dtd\sigma \cos \theta d\Omega {𝖪}_{\nu }d\nu =dtd\sigma \sin \theta \cos \theta d\theta d\varphi {𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](11)\text{upshape (11)}$ For unpolarized rays ${𝖪}_{\nu }={𝖪}_{\nu }^{\prime }$, and hence $K=2{\int }_0^{\infty }d\nu {𝖪}_{\nu },\mathrm{\backslash Label}[eqn](12)\text{upshape (12)}$ and the energy of a monochromatic ray of frequency $\nu$ will be: $2dtd\sigma \cos \theta d\Omega {𝖪}_{\nu }d\nu =2dtd\sigma \sin \theta \cos \theta d\theta d\varphi {𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](13)\text{upshape (13)}$ When, moreover, the radiation is uniformly distributed in all directions, the total radiation through $d\sigma$ toward one side may be found from [eqn:(7)] (7) and [eqn:(12)] (12); it is $2\pi d\sigma dt{\int }_0^{\infty }{𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](14)\text{upshape (14)}$