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

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

(15)

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

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

Then it is:

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

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

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

instead of $\beta$,

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