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

We obtain

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

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

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

Finally, the momentum is according to (10):

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

If we summarize everything, we come to the result:

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

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