161.

For that wave length ${\lambda }_m$ the maximum of the intensity of radiation corresponds in the spectrum of black radiation, we find from [eqn:(276)] (276) ${\left(\frac{d{E}_{\lambda }}{d\lambda }\right)}_{\lambda ={\lambda }_m}=0.$ On performing the differentiation and putting as an abbreviation $\frac{ch}{k{\lambda }_{m}T}=\beta ,$ we get $e^{-\beta }+\frac{\beta }{5}-1=0.$ The root of this transcendental equation is $\beta =4.9651,\mathrm{\backslash Label}[eqn](289)\text{upshape (289)}$ and accordingly ${\lambda }_{m}T={\displaystyle \frac{ch}{\beta k}}$, and this is a constant, as demanded by Wien's displacement law. By comparison with [eqn:(109)] (109) we find the meaning of the constant $b$, namely, $b=\frac{ch}{\beta k},\mathrm{\backslash Label}[eqn](290)\text{upshape (290)}$ and, from [eqn:(277)] (277), $b=\frac{c_2}{\beta }=\frac{1.436}{4.9651}=0.289\text{cm}\text{degree},\mathrm{\backslash Label}[eqn](291)\text{upshape (291)}$ while Lummer and Pringsheim found by measurements $0.294$ and Paschen $0.292.$