63.

We shall now deduce from the last equation a consequence which is based on the fact that the state of the system considered, and therefore also its entropy, is determined by the values of two independent variables. As the first variable we shall take $V$, as the second either $T$, $u$, or $p$ may be chosen. Of these three quantities any two are determined by the third together with $V$. We shall take the volume $V$ and the temperature $T$ as independent variables. Then by substituting from [eqn:(66)] (66) and [eqn:(70)] (70) in [eqn:(73)] (73) we have $dS=\frac{V}{T}\frac{du}{dT}dT+\frac{4u}{3T}dV.\mathrm{\backslash Label}[eqn](74)\text{upshape (74)}$ From this we obtain ${\left(\frac{\partial S}{\partial T}\right)}_V=\frac{V}{T}\frac{du}{dT}{\left(\frac{\partial S}{\partial V}\right)}_T=\frac{4u}{3T}.$

On partial differentiation of these equations, the first with respect to $V$, the second with respect to $T$, we find $\frac{{\partial }^{2}S}{\partial T\partial V}=\frac{1}{T}\frac{du}{dT}=\frac{4}{3T}\frac{du}{dT}-\frac{4u}{3T^2}$ or $\frac{du}{dT}=\frac{4u}{T}$ and on integration $u=a{T}^4\mathrm{\backslash Label}[eqn](75)\text{upshape (75)}$ and from [eqn:(21)] (21) for the specific intensity of black radiation $K=\frac{c}{4\pi }·u=\frac{ac}{4\pi }{T}^4.\mathrm{\backslash Label}[eqn](76)\text{upshape (76)}$ Moreover for the pressure of black radiation $p=\frac{a}{3}{T}^4,\mathrm{\backslash Label}[eqn](77)\text{upshape (77)}$ and for the total radiant energy $U=a{T}^4·V.\mathrm{\backslash Label}[eqn](78)\text{upshape (78)}$ This law, which states that the volume density and the specific intensity of black radiation are proportional to the fourth power of the absolute temperature, was first established by J. StefanJ. Stefan, Wien. Berichte, 79, p. 391, 1879. on a basis of rather rough measurements. It was later deduced by L. BoltzmannL. Boltzmann, Wied. Annalen, 22, p. 291, 1884. on a thermodynamic basis from Maxwell's radiation pressure and has been more recently confirmed by O. Lummer and E. PringsheimO. Lummer und E. Pringsheim, Wied. Annalen, 63, p. 395, 1897. Annalen d. Physik 3, p. 159, 1900. by exact measurements between $100°$ and $1300°\mathrm{C}.$, the temperature being defined by the gas thermometer. In ranges of temperature and for requirements of precision for which the readings of the different gas thermometers no longer agree sufficiently or cannot be obtained at all, the Stefan-Boltzmann law of radiation can be used for an absolute definition of temperature independent of all substances.