# Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/3

We obtain

${\displaystyle U=H+\beta \phi ={\sqrt {1-\beta ^{2}}}\,U_{0}+\beta ^{2}(U+pv),}$

from which it is given under consideration of the first equation (14):

 ${\displaystyle U={\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+\beta ^{2}p_{0}v_{0}).}$ (17)

Finally, the momentum is according to (10):

 ${\displaystyle {\mathfrak {G}}={\frac {\beta }{c}}(pv+U)={\frac {\beta }{c{\sqrt {1-\beta ^{2}}}}}(U_{0}+p_{0}v_{0}).}$ (18)

If we summarize everything, we come to the result:

If a body whose state at rest is given by the variables ${\displaystyle v_{0},U_{0},p_{0},T_{0},S_{0}}$, is adiabatically brought to velocity ${\displaystyle \beta c}$, then the state variables assume the value:

 ${\displaystyle v=v_{0}{\sqrt {1-\beta ^{2}}}}$ (14)
 ${\displaystyle p=p_{0}\,}$
 ${\displaystyle T=T_{0}{\sqrt {1-\beta ^{2}}}}$ (14)
 ${\displaystyle U={\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+\beta ^{2}p_{0}v_{0})}$ (17)
 ${\displaystyle H={\sqrt {1-\beta ^{2}}}\cdot U{}_{0}}$ (16)
 ${\displaystyle S=S_{0}\,}$
 ${\displaystyle {\mathfrak {G}}={\frac {\beta }{c}}{\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+p_{0}v_{0}).}$ (18)

These equations are in agreement with the results of the paper of Planck[1]. Besides thermodynamics, Planck used the relativity principle, while stating equation (10) for momentum is essential in our work.

1. Berliner Berichte, 1907, p. 542.