42.

We shall now consider a system in a state of thermodynamic equilibrium, contained within an enclosure impermeable to heat and consisting of $n$ emitting and absorbing adjacent bodies of any size and shape whatever. As in [sect:36.] Sec. 36, we again confine our attention to a monochromatic plane polarized pencil which proceeds from an element $d\sigma$ of the bounding surface of the two media in the direction toward the first medium ([fig:3]Fig. 3, feathered arrow) within the conical element $d\Omega$. Then, as in [eqn:(34)] (34), the energy supplied by the pencil in unit time is $d\sigma \cos \theta d\Omega {𝖪}_{\nu }d\nu =I.\mathrm{\backslash Label}[eqn](43)\text{upshape (43)}$

This energy of radiation $I$ consists of a part coming from the first medium by regular or diffuse reflection at the bounding surface and of a second part coming through the bounding surface from the second medium.

We shall, however, not stop at this mode of division, but shall further subdivide $I$ according to that one of the $n$ media from which the separate parts of the radiation $I$ have been emitted.

This point of view is distinctly different from the preceding, since, e.g., the rays transmitted from the second medium through the bounding surface into the pencil considered have not necessarily been emitted in the second medium, but may, according to circumstances, have traversed a long and very complicated path through different media and may have undergone therein the effect of refraction, reflection, scattering, and partial absorption any number of times.

Similarly the rays of the pencil, which coming from the first medium are reflected at $d\sigma$, were not necessarily all emitted in the first medium. It may even happen that a ray emitted from a certain medium, after passing on its way through other media, returns to the original one and is there either absorbed or emerges from this medium a second time.

We shall now, considering all these possibilities, denote that part of $I$ which has been emitted by volume-elements of the first medium by $I_1$ no matter what paths the different constituents have pursued, that which has been emitted by volume-elements of the second medium by $I_2$, etc. Then since every part of $I$ must have been emitted by an element of some body, the following equation must hold, $I=I_1+I_2+I_3+\dots I_n.\mathrm{\backslash Label}[eqn](44)\text{upshape (44)}$