Page:Elektrische und Optische Erscheinungen (Lorentz) 026.jpg

we have

$\Xi =2\pi V^{2}\int \left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)\ d\ \sigma +{\frac {1}{8\pi }}\int \left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right)\ d\ \sigma +$ $+{\frac {d}{d\ t}}\int \left({\mathfrak {H}}_{y}{\mathfrak {d}}_{z}-{\mathfrak {H}}_{z}{\mathfrak {d}}_{y}\right)\ d\ \tau {,}$

(15)

Two similar equations serve for the determination of the other components $\mathrm {H}$ and $\mathrm {Z}$ of the ponderomotive action.

Besides it is to be noticed, that $\Xi$, $\mathrm {H}$ and $\mathrm {Z}$ must vanish, as soon space $\tau$ doesn't contain ponderable matter. Then it would be

$2\pi V^{2}\int \left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)\ d\ \sigma +{\frac {1}{8\pi }}\int \left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right)\ d\ \sigma =$ $=-{\frac {d}{d\ t}}\int \left({\mathfrak {H}}_{y}{\mathfrak {d}}_{z}-{\mathfrak {H}}_{z}{\mathfrak {d}}_{y}\right)\ d\ \tau {\text{, u. s. w.}}$

(16)

§ 16. In some cases the space integral the remained in (15), will become independent of t, and if the last member vanishes, namely as soon as we have to deal with a stationary state, may it be with an electric charge, or may it be with a system of constant currents. Then, at least concerning the resultant force, the ponderomotive action can be calculated by integration over an arbitrary surface that encloses the body, and it is near at hand, to view them in a way, so that we (like Maxwell did) attribute to the aether a certain state of tension, and consider the tensions as the cause of the ponderomotive actions.^[1] If we as usual understand by $\left(X_{n},Y_{n},Z_{n}\right)$ the force related to unit area, that the aether exerts at the side (given by n) of an element $d\sigma$ upon the opposite aether, then by (15) we would have to put

$X_{n}=2\pi V^{2}\left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)+{\frac {1}{8\pi }}\left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right){\text{, etc.}}$

(17)

From that, it is easy to derive the values of $X_{x}$, $X_{y}$, $X_{z}$, $Y_{x}$;

1. Also with respect to the resultant force couple, the ponderomotive action on a rigid body is equivalent to the system of tensions (17) on an arbitrary surface $\sigma$ that encloses the body. If we also want to consider the ponderomotive actions on flexible or fluid bodies, then we would have to come back to volume elements. But this would lead too far.