Page:Zur Thermodynamik bewegter Systeme.djvu/9

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$H$ plays the role of the inner energy in a system moving with constant velocity.

Let the entropy of the resting system be $S_{0}(U_{0}, v)$, that of the moving one can be expressed by $S(H, v)$. The relations hold

$\left(\frac{\partial S_{0}}{\partial U_{0}}\right)_{v}=\frac{1}{T_{0}},\quad \left(\frac{\partial S_{0}}{\partial v}\right)_{U_{0}}=\frac{p_{0}}{T_{0}};$

but also

$\left(\frac{\partial S}{\partial H}\right)_{v}=\frac{1}{T},\quad \left(\frac{\partial S}{\partial v}\right)_{H}=\frac{p}{T};$

since it is indeed (at constant $\beta$)

$dS=\frac{1}{T}(dH+pdv)$.

Since the system was adiabatically brought from the state of rest to that of motion, the entropy has the same value in both cases, thus:

$S_{0}(U_{0}, v) = S(H, v)\,$

therefore also

$\frac{\partial S_{0}}{\partial U_{0}}=\frac{\partial S}{\partial U_{0}}=\frac{\partial S}{\partial H}\frac{\partial H}{\partial U_{0}},$

$\left(\frac{\partial S_{0}}{\partial v}\right)_{U_{0}}=\left(\frac{\partial S}{\partial v}\right)_{U_{0}}=\frac{\partial S}{\partial H}\left(\frac{\partial H}{\partial v}\right)_{U_{0}}+\left(\frac{\partial S}{\partial v}\right)_{H}.$

From that, also equations (6) and (8) are immediately given.

5. Momentum.

Thus far we have presupposed the existence of momentum, without making a special assumption concerning its value. Now we want to assume in agreement with the theory of Lorentz and Abraham, that momentum is equal to the space integral of the (absolute)