# Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/1

Continuation

(Presented in the session of February 6, 1908.)[1]

7. Calculation of quantity H.

In order to express ${\displaystyle H}$ by the variables ${\displaystyle U_{0},v,\beta }$, we insert the value for ${\displaystyle p}$ from (6) into (13) and obtain

 ${\displaystyle (1-\beta ^{2}){\frac {\partial H}{\partial \beta }}+\beta H+\beta v\left(p_{0}{\frac {\partial H}{\partial U_{0}}}-{\frac {\partial H}{\partial v}}\right)=0.}$ (15)

This partial differential equation assumes a simpler form, when the quantities ${\displaystyle \beta ,v,S_{0}}$ are chosen instead of ${\displaystyle \beta ,v,U_{0}}$ as independent variables. Namely, ${\displaystyle S_{0}}$ shall be the value of entropy again, when the system is adiabatically brought to rest; of course ${\displaystyle S=S_{0}}$. Thus we think of ${\displaystyle U_{0}}$ as being expressed by entropy and volume; if for example

${\displaystyle U_{0}=F(S_{0},v)\,}$.

Then it is:

${\displaystyle {\frac {\partial }{\partial v}}-p_{0}{\frac {\partial }{\partial U_{0}}}=\left({\frac {\partial }{\partial v}}\right)_{S_{0}}}$,

because according to (7), ${\displaystyle U_{0}}$ is changed by ${\displaystyle -p_{0}dv}$ at adiabatic volume change. If we furthermore introduce the variable

${\displaystyle \varkappa ={\sqrt {1-\beta ^{2}}}}$

instead of ${\displaystyle \beta }$,

1. Compare these proceedings, CXVI, p. 1391 (1907).