... form

The reason is that in Eq. (6) $\partial n/\partial t$ is linearly dependent on the second derivative $\partial^{2}n/\partial x^{2}$, i.e. the coefficient of $\partial^{2}n/\partial x^{2}$ in Eq. (6) does not contain n. Therefore it is seen from Eq. (15) that only the form $j(x)\sim \left [\partial n/\partial x+f(n)\right ]$or $j(x)=g(x)\left [\partial n/\partial x+f(n)\right ]$ is possible and the proportional coefficient g(x) has to be independent of n. Furthermore, it can be deduced that the function f(n) must be taken as f(n)=n(n+1) due to the fact that when the photon gas reaches thermal equilibrium we have j(x)=0, hence $\partial n/\partial x=-f(n)$. On the other hand, in this case the distribution function is Planckian, $n=({\rm e}^{x}-1)^{-1}$, thus $\partial n/\partial x=-n(1+n)$. Therefore we get f(n)=n(n+1), and Eq. (16) is obtained where g(x) is an undetermined coefficient.

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...$I(\nu,T)\equiv I(x,T)\propto x^{3}n(x,T)$

The energy density of the radiation field is given by $u_{\nu}{\rm d}\nu=\frac{8\pi \nu^{2}}{c^{3}}n(\nu,t) h\nu {\rm d}\nu \propto nx^{3}$, therefore the spectral intensity is expressed by $I_{\nu}=\frac{c}{4\pi}u_{\nu}\propto nx^{3}$, or $n(x,t)\propto x^{-3}I_{\nu}$.

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