20.

When the principal intensities ${𝖪}_{\nu }$ and ${𝖪}_{\nu }^{\prime }$ of all monochromatic rays are given at all points of the medium and for all directions, the state of radiation is known in all respects and all

questions regarding it may be answered. We shall show this by one or two applications to special cases. Let us first find the amount of energy which is radiated through any element of area $d\sigma$ toward any other element $d{\sigma }^{\prime }$. The distance $r$ between the two elements may be thought of as large compared with the linear dimensions of the elements $d\sigma$ and $d{\sigma }^{\prime }$ but still so small that no appreciable amount of radiation is absorbed or scattered along it. This condition is, of course, superfluous for diathermanous media.

From any definite point of $d\sigma$ rays pass to all points of $d{\sigma }^{\prime }$. These rays form a cone whose vertex lies in $d\sigma$ and whose solid angle is $d\Omega =\frac{d{\sigma }^{\prime }\cos ({𝗇}^{\prime },r)}{r^2}$

where ${𝗇}^{\prime }$ denotes the normal of $d{\sigma }^{\prime }$ and the angle $({𝗇}^{\prime },r)$ is to be taken as an acute angle. This value of $d\Omega$ is, neglecting small quantities of higher order, independent of the particular position of the vertex of the cone on $d\sigma$.

If we further denote the normal to $d\sigma$ by $𝗇$ the angle $\theta$ of [eqn:(14)] (14) will be the angle $(𝗇,r)$ and hence from expression [eqn:(6)] (6) the energy of radiation required is found to be: $K·\frac{d\sigma d{\sigma }^{\prime }\cos (𝗇,r)\cos ({𝗇}^{\prime },r)}{r^2}·dt.\mathrm{\backslash Label}[eqn](17)\text{upshape (17)}$ For monochromatic plane polarized radiation of frequency $\nu$ the energy will be, according to equation [eqn:(11)] (11), ${𝖪}_{\nu }d\nu ·\frac{d\sigma d{\sigma }^{\prime }\cos (𝗇,r)\cos ({𝗇}^{\prime },r)}{r^2}·dt.\mathrm{\backslash Label}[eqn](18)\text{upshape (18)}$

The relative size of the two elements $d\sigma$ and $d{\sigma }^{\prime }$ may have any value whatever. They may be assumed to be of the same or of a different order of magnitude, provided the condition remains satisfied that $r$ is large compared with the linear dimensions of each of them. If we choose $d\sigma$ small compared with $d{\sigma }^{\prime }$, the rays diverge from $d\sigma$ to $d{\sigma }^{\prime }$, whereas they converge from $d\sigma$ to $d{\sigma }^{\prime }$ if we choose $d\sigma$ large compared with $d{\sigma }^{\prime }$.