160.

For the total space density, $u$, of black radiation at any temperature $T$ we obtain, from [eqn:(275)] (275), $u={\int }_0^{\infty }𝗎d\nu =\frac{8\pi h}{c^3}{\int }_0^{\infty }\frac{{\nu }^{3}d\nu }{e^{\frac{h\nu }{kT}}-1}$ or $u=\frac{8\pi h}{c^3}{\int }_0^{\infty }(e^{-\frac{h\nu }{kT}}+e^{-\frac{2h\nu }{kT}}+e^{-\frac{3h\nu }{kT}}+\dots ){\nu }^{3}d\nu$ and, integrating term by term, $u=\frac{48\pi h}{c^3}{\left(\frac{kT}{h}\right)}^4\alpha \mathrm{\backslash Label}[eqn](286)\text{upshape (286)}$ where $\alpha$ is an abbreviation for $\alpha =1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\dots =1.0823.\mathrm{\backslash Label}[eqn](287)\text{upshape (287)}$

This relation expresses the Stefan-Boltzmann law [eqn:(75)] (75) and it also tells us that the constant of this law is given by $a=\frac{48\pi \alpha k^4}{c^3{h}^3}.\mathrm{\backslash Label}[eqn](288)\text{upshape (288)}$