Page:Zur Thermodynamik bewegter Systeme.djvu/14

 Namely, it is according to (8):

However, it is now

$T_{0}\frac{\partial p}{\partial U_{0}}=T_{0}\left(\frac{\partial p_{0}}{\partial U_{0}}\frac{\partial H}{\partial U_{0}}+p_{0}\frac{\partial^{2}H}{\partial U_{0}^{2}}-\frac{\partial^{2}H}{\partial U_{0}\partial v}\right).$|undefined

If we insert herein

$T_{0}\frac{\partial p_{0}}{\partial U_{0}}=p_{0}\frac{\partial T_{0}}{\partial U_{0}}-\frac{\partial T_{0}}{\partial v},$|undefined

a relation known as following from the thermodynamics of resting bodies, then it becomes:

Thus

$\frac{\partial T}{\partial\beta}=-\frac{\beta}{1-\beta^{2}}\cdot T-\frac{\beta}{1-\beta^{2}}v\left(p_{0}\frac{\partial T}{\partial U_{0}}-\frac{\partial T}{\partial v}\right)$|undefined

and

$dT=-\frac{\beta d\beta}{1-\beta^{2}}T.$|undefined

The change of temperature is thus the same for all bodies.

This expression as well as the expression for $$dv$$ (12) can be immediately integrated. Then we obtain: