181.

Let us now, as a preparation for the following deductions, consider more closely the properties of the different pencils passing the system of oscillators. From all directions rays strike the volume-element that contains the oscillators; if we again consider those which come toward it in the direction $(\theta ,\varphi )$ within the conical element $d\Omega$, the vertex of which lies in the volume-element, we may in the first place think of them as being resolved into their monochromatic constituents, and then we need consider further only that one of these constituents which corresponds to the frequency $\nu$ of the oscillators; for all other rays simply pass the oscillators without influencing them or being influenced by them. The specific intensity of a monochromatic ray of frequency $\nu$ is $𝖪+{𝖪}^{\prime }$ where $𝖪$ and ${𝖪}^{\prime }$ represent the principal intensities which we assume as non-coherent. This ray is now resolved into two components according to the directions of its principal planes of vibration ([sect:176.] Sec. 176).

The first component, $𝖪{\sin }^2\psi +{𝖪}^{\prime }{\cos }^2\psi ,$ passes by the oscillators and emerges on the other side with no change whatever. Hence it gives a plane polarized ray, which starts from the system of oscillators in the direction $(\theta ,\varphi )$ within the solid angle $d\Omega$ and whose vibrations are perpendicular to the axis of the oscillators and whose intensity is $𝖪{\sin }^2\psi +{𝖪}^{\prime }{\cos }^2\psi =K^{″}.\mathrm{\backslash Label}[eqn](330)\text{upshape (330)}$

The second component, $𝖪{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi ,$ polarized at right angles to the first consists again, according to [sect:176.] Sec. 176, of two parts {2} &(\mathsf{K} \cos^{2} \psi &&+ \mathsf{K}' \sin^{2} \psi) \cos^{2} \theta \Label[eqn]{(331)}\tag{\upshape (331)} \ \LeftText{and} &(\mathsf{K} \cos^{2} \psi &&+ \mathsf{K}' \sin^{2} \psi) \sin^{2} \theta, \Label[eqn]{(332)}\tag

of which the first passes by the system without any change, since its direction of vibration is at right angles to the axes of the oscillators, while the second is weakened by absorption, say by the small fraction $\beta$. Hence on emergence this component has only the intensity $(1-\beta )(𝖪{\cos }^2\psi +K^{\prime }{\sin }^2\psi ){\sin }^2\theta .\mathrm{\backslash Label}[eqn](333)\text{upshape (333)}$ It is, however, strengthened by the radiation emitted by the system of oscillators [eqn:(329)] (329), which has the value ${\beta }^{\prime }(1-\eta )\sum n{w}_n{\sin }^2\theta ,\mathrm{\backslash Label}[eqn](334)\text{upshape (334)}$ where ${\beta }^{\prime }$ denotes a certain other constant, which depends only on the nature of the system and whose value is obtained at once from the condition that, in the state of thermodynamic equilibrium, the loss is just compensated by the gain.

For this purpose we make use of the relations [eqn:(325)] (325) and [eqn:(327)] (327) corresponding to the stationary state, and thus find that the sum of the expressions [eqn:(333)] (333) and [eqn:(334)] (334) becomes just equal to [eqn:(332)] (332); and thus for the constant ${\beta }^{\prime }$ the following value is found: ${\beta }^{\prime }=\beta \frac{3c}{32{\pi }^{2}p}=\beta \frac{h{\nu }^3}{c^2}.$ Then by addition of [eqn:(331)] (331), [eqn:(333)] (333) and [eqn:(334)] (334) the total specific intensity of the radiation which emanates from the system of oscillators within the conical element $d\Omega$, and whose plane of vibration is parallel to the axes of the oscillators, is found to be ${𝖪}^{‴}=𝖪{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi +\beta {\sin }^2\theta ({𝖪}_e-(𝖪{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi ))\mathrm{\backslash Label}[eqn](335)\text{upshape (335)}$ where for the sake of brevity the term referring to the emission is written $\frac{h{\nu }^3}{c^2}(1-\eta )\sum n{w}_n={𝖪}_e.\mathrm{\backslash Label}[eqn](336)\text{upshape (336)}$

Thus we finally have a ray starting from the system of oscillators in the direction ( θ , ϕ ) within the conical element d Ω and consisting of two components 𝖪 ″ and 𝖪 ‴ polarized perpendicularly to each other,