§ 5. **Determination of momentum density and energy density.**

The various theories of electrodynamics of moving bodies, are differing by the relations assumed between the four vectors ${\mathfrak {E'H'DB}}$ arising in the main equations. However, before we pass to the discussion of special theories, we want to pursue the general developments; there, only a quite general presupposition shall be made about the form of these relations: The vectors ${\mathfrak {E'H'DB}}$ shall be connected by equations, which namely contain the velocity vector ${\mathfrak {w}}$ itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

$\mathrm {div} {\mathfrak {S}}'=c\{{\mathfrak {H}}'\mathrm {curl} {\mathfrak {E}}'-{\mathfrak {E}}'\mathrm {curl} {\mathfrak {H}}'\}$

this becomes with respect to the two first main-equations:

${\mathfrak {JE}}'+\mathrm {div} {\mathfrak {S}}'=-{\mathfrak {E}}'{\frac {\partial '{\mathfrak {D}}}{\partial t}}-{\mathfrak {H}}'{\frac {\partial '{\mathfrak {B}}}{\partial t}}$

From main equation (III) and relation (14) it follows:

(14a) |
${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}-P'={\mathfrak {E}}'{\frac {\partial '{\mathfrak {D}}}{\partial t}}+{\mathfrak {H}}'{\frac {\partial '{\mathfrak {B}}}{\partial t}}$ |

a condition, which one can also be written in accordance with (4):

(14b) |
${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}-P'={\mathfrak {E}}'{\frac {\delta {\mathfrak {D}}}{\delta t}}+{\mathfrak {H}}'{\frac {\delta {\mathfrak {B}}}{\delta t}}-{\mathfrak {E}}'({\mathfrak {D}}\nabla ){\mathfrak {w}}-{\mathfrak {H}}'({\mathfrak {B}}\nabla ){\mathfrak {w}}$ |

and which finally, by using main equation (V), passes into:

(15) |
${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}={\mathfrak {E}}'{\frac {\delta {\mathfrak {D}}}{\delta t}}+{\mathfrak {H}}'{\frac {\delta {\mathfrak {B}}}{\delta t}}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\mathrm {div} {\mathfrak {w}}$ |

This relation serves to determine the densities of energy and momentum in their dependence from the electromagnetic vectors.

They read with respect to (2a):

(15a) |
${\dot {\psi }}={\mathfrak {w{\dot {g}}}}+(\psi -{\mathfrak {wg}})\mathrm {div} {\mathfrak {w}}={\mathfrak {E'{\dot {D}}+H'{\dot {B}}}}+{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\mathrm {div} {\mathfrak {w}}$ |

Since the way of time differentiation now employed, is satisfying the calculation rules, it follows when it is set for brevity's sake:

(16) |
$\psi -{\mathfrak {wg}}=\varphi$ |

(17) |
${\dot {\varphi }}+{\mathfrak {g{\dot {w}}}}-{\mathfrak {E'{\dot {D}}-H'{\dot {B}}}}+\left\{\varphi -{\frac {1}{2}}{\mathfrak {E'D}}-{\frac {1}{2}}{\mathfrak {H'B}}\right\}\mathrm {div} {\mathfrak {w}}=0$ |

As mentioned in the beginning of the paragraph, the relations which connect ${\mathfrak {D,B}}$ with ${\mathfrak {E'H'}}$ shall contain the velocity vector ${\mathfrak {w}}$, though not its derivative with respect to time and space. The same is to be demanded from the expressions,