93.

The constant of equation [eqn:(116)] (116) bears a simple relation to the temperature of black radiation. For if the black radiation, by conduction into it of a certain amount of heat at constant volume $V$, undergoes an infinitely small change in energy $\delta U$, then, according to [eqn:(73)] (73), its change in entropy is $\delta S=\frac{\delta U}{T}.$ However, from [eqn:(113)] (113) and [eqn:(116)] (116), $\delta S=V{\int }_0^{\infty }\frac{\partial 𝗌}{\partial 𝗎}\delta 𝗎d\nu =\frac{\partial 𝗌}{\partial 𝗎}V{\int }_0^{\infty }\delta 𝗎d\nu =\frac{\partial 𝗌}{\partial 𝗎}\delta U$ hence $\frac{\partial 𝗌}{\partial 𝗎}=\frac{1}{T}\mathrm{\backslash Label}[eqn](117)\text{upshape (117)}$ and the above quantity, which was found to be the same for all frequencies in the case of black radiation, is shown to be the reciprocal of the temperature of black radiation.

Through this law the concept of temperature gains significance also for radiation of a quite arbitrary distribution of energy.

For since $𝗌$ depends only on $𝗎$ and $\nu$, monochromatic radiation, which is uniform in all directions and has a definite energy density $𝗎$, has also a definite temperature given by [eqn:(117)] (117), and, among all conceivable distributions of energy, the normal one is characterized by the fact that the radiations of all frequencies have the same temperature.

Any change in the energy distribution consists of a passage of energy from one monochromatic radiation into another, and, if the temperature of the first radiation is higher, the energy transformation causes an increase of the total entropy and is hence possible in nature without compensation; on the other hand, if the temperature of the second radiation is higher, the total entropy decreases and therefore the change is impossible in nature, unless compensation occurs simultaneously, just as is the case with the transfer of heat between two bodies of different temperatures.