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

then (15) becomes

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

or when ${\displaystyle \beta }$ is different from zero:

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

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

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

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

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

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

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

8. Summary of results.

With the aid of equations (2) and (10), momentum and total energy (${\displaystyle U}$) 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.)