Page:Zur Dynamik bewegter Systeme.djvu/12

way (reversible and adiabatic) at the speed of ${\dot {x}}=v,\ {\dot {y}}=0,\ {\dot {z}}=0$, so that the final volume V₂ is in relation to the initial volume V₁ by the relation:

$V_{2}=V_{1}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$,

(16)

then the final state 2 for the primed system is identical in all respects to the initial state 1 for the unprimed system.

The correctness of this proposition follows from the consideration that the condition of the body is defined by five independent variables, for which we can choose the volume and entropy in addition to the three velocity components. Now, under those conditions ${\dot {x}}'_{2},{\dot {y}}'_{2}$ and ${\dot {z}}'_{2}=0$ are the 3 velocity components of the body in the final state for the primed system, furthermore by (15) the entropy S'₂ = S₂ = S₁, finally the volume by (14):

$V'_{2}=V_{2}{\frac {c^{2}+v{\dot {x}}'_{2}}{c{\sqrt {c^{2}-v^{2}}}}}=V_{2}{\frac {c}{\sqrt {c^{2}-v^{2}}}}=V_{1}$,

so every 5 condition-variables in the final state 2 for the primed system have the same value as in the initial state 1 for the unprimed system, thus the above theorem is proved.

§ 6.

Now we think of any number of different bodies separated from each other, which initially are at rest for the unprimed system and which all have the same temperature T₁ and are subjected to the same pressure p₁. Each of these bodies will somehow be brought to the speed v in a reversible and adiabatic way, and its final volume will be regulated according to relation (16). Then finally, all bodies have in turn a common temperature T₂ and a common pressure p₂. Because for the primed system every body is finally in the same condition as initially for the unprimed frame, thus for the primed system the final temperatures and the final pressure are all equal. However, the same is true for the unprimed frame; for two bodies, having the same temperature and same pressure for one reference system, i.e., they are in thermal and mechanical equilibrium with each other, have the same property also in every other frame of reference.

Thus we can state the following theorem: Different types of bodies of same temperature and same pressure, which are somehow brought from velocity 0 to velocity v (separately and in a reversible and adiabatic way)