6. The Coefficient of Emission

Summing up everything said so far, we may equate the total energy in a range of frequency from $\nu$ to $\nu +d\nu$ emitted in the time $dt$ in the direction of the conical element $d\Omega$ by a volume-element $d\tau$ to $dt·d\tau ·d\Omega ·d\nu ·2{ϵ}_{\nu }.\mathrm{\backslash Label}[eqn](1)\text{upshape (1)}$ The finite quantity ${ϵ}_{\nu }$ is called the coefficient of emission of the medium for the frequency $\nu$. It is a positive function of $\nu$ and refers to a plane polarized ray of definite color and direction. The total emission of the volume-element $d\tau$ may be obtained from this by integrating over all directions and all frequencies. Since ${ϵ}_{\nu }$ is independent of the direction, and since the integral over all conical elements $d\Omega$ is $4\pi$, we get: $dt·d\tau ·8\pi {\int }_0^{\infty }{ϵ}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](2)\text{upshape (2)}$