Page:Zur Thermodynamik bewegter Systeme.djvu/11

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

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:

$\mathfrak{G}=\frac{2\pi v}{c^{2}}\int_{0}^{\pi}u(\phi)\cdot\omega_{A}\cdot\cos\varphi\cdot\sin\phi\ d\phi.$|undefined

However, now it is

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