183.

It is now easy to state the relation of the two principles of thermodynamics to the irreversible processes here considered. Let us consider first the conservation of energy. If there is no oscillator in the field, every one of the elementary pencils, infinite in number, retains, during its rectilinear propagation, both its specific intensity $𝖪$ and its energy without change, even though it be reflected at the surface, assumed as plane and reflecting, which bounds the field ([sect:166.] Sec. 166). The system of oscillators, on the other hand, produces a change in the incident pencils and hence also a change in the energy of the radiation propagated in the field. To calculate this we need consider only those monochromatic rays which lie close to the natural frequency $\nu$ of the oscillators, since the rest are not altered at all by the system.

The system is struck in the direction $(\theta ,\varphi )$ within the conical element $d\Omega$ which converges toward the system of oscillators by a pencil polarized in some arbitrary way, the intensity of which is given by the sum of the two principal intensities $𝖪$ and $𝖪$'. This pencil, according to [sect:182.] Sec. 182, conveys the energy $q\Delta \nu (𝖪+{𝖪}^{\prime })d\Omega dt$ to the system in the time $dt$; hence this energy is taken from the field of radiation on the side of the rays arriving within $d\Omega$. As a compensation there emerges from the system on the other side in the same direction $(\theta ,\varphi )$ a pencil polarized in some definite way, the intensity of which is given by the sum of the two components ${𝖪}^{″}$ and ${𝖪}^{‴}$. By it an amount of energy $q\Delta \nu ({𝖪}^{″}+{𝖪}^{‴})d\Omega dt,$ is added to the field of radiation. Hence, all told, the change in energy of the field of radiation in the time $dt$ is obtained by subtracting

the first expression from the second and by integrating with respect to $d\Omega$. Thus we get $dt\Delta \nu \int ({𝖪}^{″}+{𝖪}^{‴}-𝖪-{𝖪}^{\prime })qd\Omega ,$ or by taking account of [eqn:(330)] (330), [eqn:(335)] (335), and [eqn:(338)] (338) $\frac{\pi Ndt}{cL}\int d\Omega {\sin }^2\theta ({𝖪}_e-(𝖪{\cos }^2\psi +{𝖪}^{\prime }{\sin }^2\psi )).\mathrm{\backslash Label}[eqn](339)\text{upshape (339)}$