Page:Zur Dynamik bewegter Systeme.djvu/22

or because:

$R'_{0}={\frac {cV}{\sqrt {c^{2}-q^{2}}}}p'_{0}+{\frac {cT}{\sqrt {c^{2}-q^{2}}}}S'_{0}-H'_{0}$

$R={\frac {c}{\sqrt {c^{2}-q^{2}}}}R'_{0}$

(45)

By introduction of thermal function R, the momentum Gsimply writes by (41):

$G={\frac {q}{c^{2}}}R={\frac {q}{c{\sqrt {c^{2}-q^{2}}}}}R'_{0}$.

(46)

§ 14.

The special relations which are contained in the above equations can all be summarized in a single differential equation, which is completely general for the function H of the three variables q, V, T. Namely, if we substitute in equation (46) for G the expression ${\frac {\partial H}{\partial q}}$, and for R the value E + pV, it follows with respect to (10) the equation:

$T{\frac {\partial H}{\partial T}}+V{\frac {\partial H}{\partial V}}-{\frac {c^{2}-q^{2}}{q}}{\frac {\partial H}{\partial q}}-H=0$.

(47)

This differential equation represents the general expression for the application of the principle of relativity on the kinetic potential. Its general integral is expressed by (38), of which one can easily convince himself. Thereafter, the kinetic potential H is a homogeneous function of first degree of the three variables T, V, and ${\sqrt {c^{2}-q^{2}}}$.

§ 15.

Let us now at first give a special application to the black cavity radiation. Hereafter, all laws of motion of cavity radiation are resulting directly from the simple known thermodynamic formulas for static cavity radiation. Namely, for which the Stefan-Boltzmann law is given:

$E_{0}=aT^{4}V$.

Furthermore, Maxwell's radiation pressure is given by:

$p_{0}={\frac {1}{3}}aT^{4}$,

and the entropy of stationary radiation:

$S_{0}=\int {\frac {dE_{0}+p_{0}dV}{T}}={\frac {4}{3}}aT^{3}V$.