Page:Zur Thermodynamik bewegter Systeme.djvu/11

or, since we denote $uv$ with $U$

${\mathfrak {G}}={\frac {1}{c^{2}}}(pv+U)q.$[1]

(10)

Thus the types of inner energy don't matter, as long as they are only of electromagnetic nature (we imagine that they are composed of radiating energy and energy of arbitrarily moving electrons). Also the relative velocity and the energy flow don't matter as well; the individual energy types can of course flow with different velocities.

Of course, one comes to the same result when the individual energy flows are taken into account. For instance, let $u(\phi )\sin \phi \ d\phi$ be the density of a specific energy type, moving in a relative direction that encloses an angle between $\phi$ and $\phi +d\phi$ with the direction of motion. Then the total energy of this type is

$U=2\pi \ v\int _{0}^{\pi }\ u(\phi )\ \sin \phi \ d\phi .$

We obtain the momentum when we multiply the absolute flow, i.e. $u(\phi )\sin \phi \ d\phi \cdot \omega _{A}$ (where $\omega _{A}$ is the flow-velocity) with $\cos \varphi$, where $\varphi$ is the angle between the absolute flow direction and the direction of motion. Thus:

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

However, now it is^[2]

$\omega _{A}\cos \varphi =q+\omega _{R}\cos \phi$,

1. The method given here is based on the consideration already given by me in an earlier paper (these proceedings, CXIII., p. 1039, 1904). Equation (10) was already derived by Planck. The method of Planck, however, has nothing at all to do with the one used here.
2. For instance, compare F. Hasenöhrl, Ann. d. Phys., 15, p. 347, 1904 (however, only radiating energy is considered there. We now have to replace the quantities denoted as $c$ and $c'$ by $\omega _{A}$ and $\omega _{R}$. Since the relations are purely geometrical, this replacement is allowed without further ado.