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

 then (15) becomes

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

or when $$\beta$$ is different from zero:

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

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

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

Now, if we assume in accordance with '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:

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.)