Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/2

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then (15) becomes

-\beta\varkappa\frac{\partial H}{\partial\varkappa}+\beta H-\beta v\left(\frac{\partial H}{\partial v}\right)_{S_{0}}=0,

or when \beta is different from zero:

\varkappa\frac{\partial}{\partial\varkappa}\left(\frac{H}{\varkappa}\right)+v\left(\frac{\partial}{\partial v}\right)_{S_{0}}\left(\frac{H}{\varkappa}\right)=0.

It follows from this equation, that H/\varkappa must be a function of v/\varkappa, which of course must also depend on S_0. Furthermore, H must be identical with U_0 for \beta = 0,\ \varkappa = 1. We satisfy these requirements when we put

\frac{H}{\varkappa}=F\left(S_{0},\frac{v}{\varkappa}\right).

F\left(S_{0},\tfrac{v}{\varkappa}\right) is evidently the energy amount of the resting system, when it is adiabatically expanded from v to v/\varkappa; if we denote this energy value with U'_{0}, then

H=\sqrt{1-\beta^{2}}\cdot U'_{0}.

Now, if we assume in accordance with Lorentz's hypothesis, that the velocity change is accompanied with a volume change proportional to \sqrt{1-\beta^{2}}, then U'_{0} is the energy of the resting body; then we remove the prime and thus put:

H=\sqrt{1-\beta^{2}}\cdot U{}_{0}. (16)


8. Summary of results.

With the aid of equations (2) and (10), momentum and total energy (U) can be expressed by the state variables of the resting system. (We have to consider here, that equations (1), (3), (4) and (5) may not be applied now; they only hold for velocity changes at constant volume.)