Page:Zur Thermodynamik bewegter Systeme.djvu/12

where $\omega _{R}$ is the relative velocity ($\omega _{A}$ and $\omega _{R}$ are in general functions of $\phi$ or $\varphi$).

Thus

${\mathfrak {G}}={\frac {2\pi v}{c^{2}}}q\int _{0}^{\pi }u(\phi )\sin \phi \ d\phi +{\frac {2\pi v}{c^{2}}}\int _{0}^{\pi }u(\phi )\sin \phi \cos \phi \omega _{R}\ d\phi .$

The first summand is equal to ${\tfrac {1}{c^{2}}}qU$; the second one gives the surplus of the energy emanating from the basis surface over the inflowing energy, it is thus connected with the pressure work $pq$, by which we come to equation (10) again.

6. The change of volume.

A resting system in mechanical and thermal equilibrium is given, i.e. in which all bodies have the same pressure and the same temperature. If this system is adiabatically set into motion (each body adiabatically for itself), then pressure and temperature of every single body is changing, namely in different measure for every single body as we must assume from the outset. Thus the equilibrium is disturbed; if it is restored again, then the individual bodies must change their volumes. If these volume changes are different for different bodies, then they are principally observable. However, if the mechanical and thermal equilibrium is restored again by the change of dimensions of all bodies in the same way, then an influence of the common translatory motion is not observable.

This is indeed the case; first it can be shown that the pressure of a body remain unchanged when $\beta$ is adiabatically changed by $d\beta$ and $v$ by $-v{\tfrac {\beta d\beta }{1-\beta ^{2}}}$ at the same time. Thus it must be

$dp={\frac {\partial p}{\partial U_{0}}}dU_{0}+{\frac {\partial p}{\partial v}}dv+{\frac {\partial p}{\partial \beta }}d\beta =0$

(11)