Page:Elektrische und Optische Erscheinungen (Lorentz) 084.jpg

three equations are summarized, namely in the first of them on the left side, the expression

${\frac {\partial {\mathfrak {H}}'_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}'_{y}}{\partial z}}$.

is stated. For that, we can write with respect to (35)

$[Rot'\ {\mathfrak {H}}']_{x}-{\frac {1}{V^{2}}}\left\{{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {H}}'_{z}}{\partial t'}}-{\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {H}}'_{y}}{\partial t'}}\right\}$,

and thus for the equation itself

$[Rot'\ {\mathfrak {H}}']_{x}=4\pi {\frac {\partial {\mathfrak {D}}_{x}}{dt'}}+{\frac {1}{V^{2}}}{\frac {\partial }{dt'}}\left\{{\mathfrak {p}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {p}}_{z}{\mathfrak {H}}_{y}\right\}=4\pi {\frac {\partial {\mathfrak {D}}'_{x}}{dt'}}$

The two other equations admit of a similar transformation, and therefore we have

$Rot'\ {\mathfrak {H}}'=4\pi {\frac {\partial {\mathfrak {D}}'}{\partial t'}}$

($III_{d})$

Furthermore, as regards the first of equations $IV_{c}$), this one goes over, since

${\frac {\partial {\mathfrak {E}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}=[Rot'\ {\mathfrak {E}}]_{x}-{\frac {1}{V^{2}}}\left\{{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {E}}_{z}}{\partial t'}}-{\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {E}}_{y}}{\partial t'}}\right\}$

into

$[Rot'\ {\mathfrak {E}}]_{x}=-{\frac {\partial {\mathfrak {H}}_{x}}{\partial t'}}+{\frac {1}{V^{2}}}{\frac {\partial }{\partial t'}}\left\{{\mathfrak {p}}_{y}{\mathfrak {E}}_{z}+{\mathfrak {p}}_{z}{\mathfrak {E}}_{y}\right\}=-{\frac {\partial {\mathfrak {H}}'_{x}}{\partial t'}}$,

so that ($IV_{c}$) is equivalent with

$Rot'\ {\mathfrak {E}}=-{\frac {\partial {\mathfrak {H}}'}{\partial t'}}$

$(IV_{d})$

Eventually it follows from

$\varkappa _{1}{\mathfrak {E}}_{x}=4\pi V^{2}{\mathfrak {D}}'_{x},\ \varkappa _{2}{\mathfrak {E}}_{y}=4\pi V^{2}{\mathfrak {D}}'_{y},\ \varkappa _{3}{\mathfrak {E}}_{z}=4\pi V^{2}{\mathfrak {D}}'_{z}$

$(V_{d})$

§ 58. To introduce the new variables also into the limiting conditions, we consider the perpendicular n for the considered point, and also two directions h and k that are perpendicular to one another and to n. There, the direction n shall correspond to a rotation by a right angle from h to k. Consequently it follows from (IX) (§ 56)

$4\pi V^{2}{\mathfrak {D}}'_{n}=4\pi V^{2}{\mathfrak {D}}_{n}+[{\mathfrak {p.H}}]_{n}=4\pi V^{2}{\mathfrak {D}}_{n}+[{\mathfrak {p.H}}']_{n}=$ $=4\pi V^{2}{\mathfrak {D}}{}_{n}+{\mathfrak {p}}_{h}{\mathfrak {H}}'_{k}-{\mathfrak {p}}_{k}{\mathfrak {H}}'_{h}$.

Now, since ${\mathfrak {D}}_{n,}\ {\mathfrak {H}}'_{k}$ and ${\mathfrak {H}}'_{h}$ are steady, then this must also be so for ${\mathfrak {D}}'_{n}$.