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 in specifiable amount. We therefore don't wish to make here such a decomposition, but instead we want to rely on the principle of relativity and develop its consequences for the considered case.

§ 3.

The principle of relativity says that instead of the previously used reference frame (x, y, z, t) we can use with exactly the same justification also the following reference frame:

$$x'=\frac{c(x-vt)}{\sqrt{c^{2}-v^{2}}},\quad y'=y,\quad z'=z,\quad t'=\frac{c^{2}t-vx}{c\sqrt{c^{2}-v^{2}}}$$

for the basic equations of mechanics, electrodynamics and thermodynamics, and therefore describe them as "at rest". We want to denote in the following all quantities measured in the new reference frame by a prime, and denote accordingly the two reference systems as "primed" and "unprimed". Then the content of the principle of relativity can also be expressed in this way: All the equations between primed, unprimed or both quantities remain true, if we replace the primed quantities by the unprimed quantities of the same name, and simultaneously replace the unprimed by the primed quantities. And we have to set c' = c and v' = -v.

This general theorem, which is of course valid for the defining equations (from above) of the primed coordinates, provides for any relation derived, a reciprocal relation that is often useful for verification.

§ 4. Now, our next task is to establish the relation between each of the previously used quantities and the primed quantities of the same name. It will be shown that this may be done in a completely unambiguous way, so that we finally, for example, can calculate from the energy of a body at rest in one reference frame, the energy of the same body in the other reference frame, for which it possesses a certain finite speed.

First, for the primed velocity components ($$\dot{x}'=\frac{dx'}{dt'}$$, etc.) it is found in a purely mathematical way: