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In equations (11), L + U thus means the whole energy within surface $\sigma$, and therefore the view is near at hand, that a quantity of energy

$V^{2}d\ t\int \ [{\mathfrak {d.H}}]_{n}d\ \sigma$

has traveled through the surface into the outside. It is most simple, if we put for the "energy flow" related to unit time and area

$V^{2}[{\mathfrak {d.H}}]_{n}.$

(13)

By that we come to the known theorem formulated by Poynting. Here, we don't discuss the subtle, related question concerning the localization of the energy. We can restrict ourselves with the fact, that the entire energy located in an arbitrary space — the "electric" portion calculated by formula (12) — always varies, as if the energy would travel according to the way determined by (13).

Tensions in the aether.

§ 15. The forces determined by our formula (V), not only require the motion of ions in ponderable bodies, but also in some circumstances can unify themselves to an action, that tends to set the body into motion. In this way all "ponderomotive" forces emerge, as for example the ordinary electrostatic and electrodynamic ones, as well as the pressure that is exerted by light rays on a body.

We want to consider the body as rigid, and calculate (by simple addition) all the forces that were exerted by the aether in the direction of the x-axis, i.e. the total force $\Xi$ in this direction. The investigation should be based on the things said at the beginning of § 13.

We immediately obtain

${\begin{array}{cl}\Xi &=4\pi V^{2}\int {\mathfrak {d}}_{x}\rho \ d\ \tau +\int \rho [{\mathfrak {v.H}}]_{x}\ d\ \tau =\\&=4\pi V^{2}\int {\mathfrak {d}}_{x}\rho \ d\ \tau +\int \rho \left({\mathfrak {v}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {v}}_{z}{\mathfrak {H}}_{y}\right)\ d\ \tau {,}\end{array}}$