# Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/2

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then (15) becomes

$-\beta\varkappa\frac{\partial H}{\partial\varkappa}+\beta H-\beta v\left(\frac{\partial H}{\partial v}\right)_{S_{0}}=0,$

or when $\beta$ is different from zero:

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

It follows from this equation, that $H/\varkappa$ must be a function of $v/\varkappa$, which of course must also depend on $S_0$. Furthermore, $H$ must be identical with $U_0$ for $\beta = 0,\ \varkappa = 1$. We satisfy these requirements when we put

$\frac{H}{\varkappa}=F\left(S_{0},\frac{v}{\varkappa}\right)$.

$F\left(S_{0},\tfrac{v}{\varkappa}\right)$ is evidently the energy amount of the resting system, when it is adiabatically expanded from $v$ to $v/\varkappa$; if we denote this energy value with $U'_{0}$, then

$H=\sqrt{1-\beta^{2}}\cdot U'_{0}.$

Now, if we assume in accordance with Lorentz's hypothesis, that the velocity change is accompanied with a volume change proportional to $\sqrt{1-\beta^{2}}$, then $U'_{0}$ is the energy of the resting body; then we remove the prime and thus put:

 $H=\sqrt{1-\beta^{2}}\cdot U{}_{0}.$ (16)

8. Summary of results.

With the aid of equations (2) and (10), momentum and total energy ($U$) can be expressed by the state variables of the resting system. (We have to consider here, that equations (1), (3), (4) and (5) may not be applied now; they only hold for velocity changes at constant volume.)