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

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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).