Page:Elektrische und Optische Erscheinungen (Lorentz) 075.jpg

to the same coordinate system have the same properties, i.e. that the properties in A and $A'$ can be expressed by the same equations, without change of a constant or sign. In this case we call E a plane of symmetry. The bodies which we now consider, and to which we will restrict ourselves preliminarily, are those, for which three mutually perpendicular symmetry planes of this kind exist.

We give the coordinate planes the direction with of the symmetry-planes, and consider at first the mirror image with respect to the yz-plane. When passing to this image, ${\overline {\mathfrak {E}}}_{x},{\mathfrak {M}}_{x}$ and ${\mathfrak {p}}_{x}$ change their sign, while the other components of ${\mathfrak {{\overline {E}},M}}$ and ${\mathfrak {p}}$ remain completely unchanged. This is only possible, when (after $({\mathfrak {{\dot {M}},p}})_{x}$, etc. are represented as functions of ${\dot {\mathfrak {M}}}_{x},\ {\dot {\mathfrak {M}}}_{y},\ {\dot {\mathfrak {M}}}_{z},\ {\mathfrak {p}}_{x},{\mathfrak {p}}_{y},{\mathfrak {p}}_{z}$) the index x appears in every term of the first formula once, or in every term of the second and third either not at all, or two times. To a similar conclusion we come also with respect to indices y and z. If we additionally consider the mirror images with respect to the zx- and the xy-plane, then we find, that not a single term as $({\mathfrak {{\dot {M}},p}})_{x}$ is applicable, and that from the nine coefficients $\sigma$, only $\sigma _{1.1}$, $\sigma _{2.2}$ and $\sigma _{3.3}$ can be different from zero.

Thus we obtain

${\overline {\mathfrak {E}}}_{x}=\sigma _{1.1}{\mathfrak {M}}_{x},\ {\overline {\mathfrak {E}}}_{y}=\sigma _{2.2}{\mathfrak {M}}_{y},\ {\overline {\mathfrak {E}}}_{z}=\sigma _{3.3}{\mathfrak {M}}_{z}$,

(64)

or

${\frac {4\pi V^{2}}{\sigma _{1.1}}}{\overline {\mathfrak {E}}}_{x}=4\pi V^{2}{\mathfrak {M}}_{x}$, u.s.w.

If we add these formulas to the three summarized in (53), and put

$1+{\frac {4\pi V^{2}}{\sigma _{1.1}}}=\varkappa _{1},\ 1+{\frac {4\pi V^{2}}{\sigma _{2.2}}}=\varkappa _{2},\ 1+{\frac {4\pi V^{2}}{\sigma _{3.3}}}=\varkappa _{3},$

then

$\varkappa _{1}{\overline {\mathfrak {E}}}_{x}=4\pi V^{2}{\mathfrak {D}}_{x}+[{\mathfrak {p.H}}]_{x}$

where, for a certain type of light, $\varkappa _{1},\varkappa _{1}$ and $\varkappa _{3}$ are constants.