89.

Since $E_{\lambda }$ vanishes for $\lambda =0$ as well as for $\lambda =\infty$, $E_{\lambda }$ must have a maximum with respect to $\lambda$, which is found from the equation $\frac{d{E}_{\lambda }}{d\lambda }=0=-\frac{5}{{\lambda }^6}F\left(\frac{\lambda T}{c}\right)+\frac{1}{{\lambda }^5}\frac{T}{c}\dot{F}\left(\frac{\lambda T}{c}\right)$ where $\dot{F}$ denotes the differential coefficient of $F$ with respect to its argument. Or $\frac{\lambda T}{c}\dot{F}\left(\frac{\lambda T}{c}\right)-5F\left(\frac{\lambda T}{c}\right)=0.\mathrm{\backslash Label}[eqn](108)\text{upshape (108)}$ This equation furnishes a definite value for the argument ${\displaystyle \frac{\lambda T}{c}}$, so

that for the wave length ${\lambda }_m$ corresponding to the maximum of the radiation intensity $E_{\lambda }$ the relation holds ${\lambda }_{m}T=b.\mathrm{\backslash Label}[eqn](109)\text{upshape (109)}$ With increasing temperature the maximum of radiation is therefore displaced in the direction of the shorter wave lengths.

The numerical value of the constant $b$ as determined by Lummer and PringsheimO. Lummer und E. Pringsheim, l. c. is $b=0.294\text{cm}\text{degree}.\mathrm{\backslash Label}[eqn](110)\text{upshape (110)}$

PaschenF. Paschen, Annal. d. Physik 6, p. 657, 1901. has found a slightly smaller value, about $0.292$.

We may emphasize again at this point that, according to [sect:19.] Sec. 19, the maximum of $E_{\lambda }$ does not by any means occur at the same point in the spectrum as the maximum of ${𝖪}_{\nu }$ and that hence the significance of the constant $b$ is essentially dependent on the fact that the intensity of monochromatic radiation is referred to wave lengths, not to frequencies.