Page:Elektrische und Optische Erscheinungen (Lorentz) 043.jpg

since the action of $K$, ${\overline {\mathfrak {S}}}$, $A$ and $B$ on ${\overline {\mathfrak {S}}}$, $A$ and $B$, has to be considered each time. It is now

$(K,{\overline {\mathfrak {S}}})+(B,{\overline {\mathfrak {S}}})=0,\ (K,A)+(B,A)=0{,}$

so that by the aforementioned expression it only remains

$(K,B)+(B,B)+(A,{\overline {\mathfrak {S}}})+({\overline {\mathfrak {S}}},{\overline {\mathfrak {S}}}).$

(30)

Those forces represented by the first two members would also exist, when ${\overline {\mathfrak {S}}}=0$, and the last two members are independent of the charged body K. An action of K exerted on the conductor as such, doesn't exist.

Besides, in each of the four members (30), the part that depends on ${\mathfrak {p}}$ is of second order. We already know this from $(K,B)+(B,B)$, since this represents an electrostatic effect. $(A,{\overline {\mathfrak {S}}})$ and $({\overline {\mathfrak {S}}},{\overline {\mathfrak {S}}})$, however, represent forces acting on a current, in which the mean electric density is zero. As it can be seen from (V_a), such forces are determined by the value of ${\mathfrak {H}}$, which belongs to the acting system. Inasmuch as ${\mathfrak {H}}$ (that belongs to ${\overline {\mathfrak {S}}}$) depends on ${\mathfrak {p}}$, it is of second order (§ 25), and the compensation charge A only produces by its velocity ${\mathfrak {p}}$ a magnetic force of second order, since its density already contains the factor ${\mathfrak {p}}/V$.

Electrodynamic actions.

§ 27. The question as to how these effects are influenced by earth's motion, can now easily be answered. If we denote the currents in two conductors by ${\overline {\mathfrak {S}}}$ and ${\overline {{\mathfrak {S}}'}}$, and the corresponding compensation charges by A and $A'$, then the action exerted on the second conductor is

$({\overline {\mathfrak {S}}},{\overline {{\mathfrak {S}}'}})+(A,{\overline {{\mathfrak {S}}'}})+({\overline {\mathfrak {S}}},A')+(A,A'){,}$

in which the last two terms are mutually canceled. That $(A,{\overline {{\mathfrak {S}}'}})$ and the ${\mathfrak {p}}$-dependent part $({\overline {\mathfrak {S}}},{\overline {{\mathfrak {S}}'}})$ are of order ${\mathfrak {p}}^{2}/V^{2}$, follows from considerations such as those communicated above.