19. The Intensity of Radiation

From equation [eqn:(9)] (9) it is seen that the quantity ${𝖪}_{\nu }$, the intensity of radiation of frequency $\nu$, and the quantity $K$, the intensity of radiation of the whole spectrum, are of different dimensions. Further it is to be noticed that, on subdividing the spectrum according to wave lengths $\lambda$, instead of frequencies $\nu$, the intensity of radiation $E_{\lambda }$ of the wave lengths $\lambda$ corresponding to the frequency $\nu$ is not obtained simply by replacing $\nu$ in the expression for ${𝖪}_{\nu }$ by the corresponding value of $\lambda$ deduced from $\nu =\frac{q}{\lambda }\mathrm{\backslash Label}[eqn](15)\text{upshape (15)}$ where $q$ is the velocity of propagation. For if $d\lambda$ and $d\nu$ refer to the same interval of the spectrum, we have, not $E_{\lambda }={𝖪}_{\nu }$, but $E_{\lambda }d\lambda ={𝖪}_{\nu }d\nu$. By differentiating [eqn:(15)] (15) and paying attention to the signs of corresponding values of $d\lambda$ and $d\nu$ the equation $d\nu =\frac{qd\lambda }{{\lambda }^2}$ is obtained. Hence we get by substitution: $E_{\lambda }=\frac{q{𝖪}_{\nu }}{{\lambda }^2}.\mathrm{\backslash Label}[eqn](16)\text{upshape (16)}$ This relation shows among other things that in a certain spectrum the maxima of $E_{\lambda }$ and ${𝖪}_{\nu }$ lie at different points of the spectrum.