84.

We have already emphasized ([sect:79.] Sec. 79) that $𝗎$ must be regarded as a function of two independent variables, of which we have taken as the first the frequency $\nu$ and as the second the time $t$. Since, now, in equation [eqn:(95)] (95) the time $t$ does not explicitly appear, it is more appropriate to introduce the volume $V$, which depends only on $t$, as the second variable instead of $t$ itself. Then equation [eqn:(95)] (95) may be written as a partial differential equation as follows: $V\frac{\partial 𝗎}{\partial V}=\frac{\nu }{3}\frac{\partial 𝗎}{\partial \nu }-𝗎.\mathrm{\backslash Label}[eqn](97)\text{upshape (97)}$ From this equation, if, for a definite value of $V$, $𝗎$ is known as a function of $\nu$, it may be calculated for all other values of $V$ as a function of $\nu$. The general integral of this differential equation, as may be readily seen by substitution, is $𝗎=\frac{1}{V}\varphi ({\nu }^{3}V),\mathrm{\backslash Label}[eqn](98)\text{upshape (98)}$ where $\varphi$ denotes an arbitrary function of the single argument ${\nu }^{3}V$. Instead of this we may, on substituting ${\nu }^{3}V\varphi ({\nu }^{3}V)$ for $\varphi ({\nu }^{3}V)$, write $𝗎={\nu }^3\varphi ({\nu }^{3}V).\mathrm{\backslash Label}[eqn](99)\text{upshape (99)}$ Either of the last two equations is the general expression of Wien's displacement law.

If for a definitely given volume $V$ the spectral distribution of energy is known (i.e., $𝗎$ as a function of $\nu$), it is possible to deduce therefrom the dependence of the function $\varphi$ on its argument, and thence the distribution of energy for any other volume $V^{\prime }$, into which the radiation filling the hollow cylinder may be brought by a reversible adiabatic process.