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)


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)