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We refuse to express the thus stated law by words. By elevating it to a general fundamental-law, we have completed the system of equations of motion (I)—(V), since the electric force, in connection with possible other forces, determines the motion of ions.

Concerning the latter, we still want to introduce the assumption, that the ions never rotate.^[1]

The conservation of energy.

§ 13. To justify our hypotheses, it is necessary to show its agreement with the energy law. We consider an arbitrary system of ponderable bodies that contain ions, around which only the aether exists up to an infinite distance, and around it we put an arbitrarily closed surface $\sigma$. During an element of time $dt$, the work that affects ponderable matter and which stems from ${\mathfrak {E}}$, is

$4\pi V^{2}d\ t\int \rho \left({\mathfrak {d}}_{x}{\mathfrak {v}}_{x}+{\mathfrak {d}}_{y}{\mathfrak {v}}_{y}+{\mathfrak {d}}_{z}{\mathfrak {v}}_{z}\right)d\ \tau {,}$

where it is to be noted, that no work is done by the forces (which are derived from ${\mathfrak {E}}_{2}$), because they are always perpendicular to the direction of motion. Furthermore, if dA is the work of all other forces acting on matter, and L is the ordinary mechanical energy of that matter, then

$d\ A=d\ L-4\pi V^{2}d\ t\int \rho \left({\mathfrak {d}}_{x}{\mathfrak {v}}_{x}+{\mathfrak {d}}_{y}{\mathfrak {v}}_{y}+{\mathfrak {d}}_{z}{\mathfrak {v}}_{z}\right)d\ \tau .$

(9)

The integral is related to the space filled with ponderable matter; but we can also extend it over the entire space enclosed by $\sigma$. All other space integrals in this § are to be understood in the latter sense.

1. In an earlier published derivation of the equations of motion (La théorie électromagnétique de Maxwell et son application aux corps mouvants), I have discussed the necessary conditions.