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Summary of the equations.

§ 52. Neglecting the primes over the letters — since was continue to only speak about averages — we summarize the equations of motion now in the following way.

In the interior of any body it is given

$Div\ {\mathfrak {D}}=0$,

$(I_{c})$

$Div\ {\mathfrak {H}}=0$,

$(II_{c})$

$Div\ {\mathfrak {H}}'=4\pi {\dot {\mathfrak {D}}}$,

$(III_{c})$

$Div\ {\mathfrak {E}}=-{\dot {\mathfrak {H}}}$,

$(IV_{c})$

$\left.{\begin{array}{c}\varkappa _{1}{\mathfrak {E}}_{x}=4\pi V^{2}{\mathfrak {D}}_{x}+[{\mathfrak {p.H}}]_{x},\ \varkappa _{2}{\mathfrak {E}}_{y}=4\pi V^{2}{\mathfrak {D}}_{y}+[{\mathfrak {p.H}}]_{y}\\\varkappa _{3}{\mathfrak {E}}_{z}=4\pi V^{2}{\mathfrak {D}}_{z}+[{\mathfrak {p.H}}]_{z}\end{array}}\right\}$

$(V_{c})$

and

${\mathfrak {H}}'={\mathfrak {H}}-{\frac {1}{V^{2}}}[{\mathfrak {p.E}}]$,

$(VI_{c})$

since, neglecting magnitudes of second order, we may replace, by the relation (53), $4\pi {\overline {\mathfrak {d}}}$ by ${\mathfrak {E}}/V^{2}$ in equation (54).

At the borderline the conditions apply

${\mathfrak {D}}_{n(1)}={\mathfrak {D}}_{n(2)},\ {\mathfrak {H}}_{n(1)}={\mathfrak {H}}_{n(2)},\ {\mathfrak {E}}_{h(1)}={\mathfrak {E}}_{h(2)},\ {\mathfrak {H}}'_{h(1)}={\mathfrak {H}}'_{h(2)}$,

$(VIII_{c})$

If there is no translation, then ${\mathfrak {H}}'$ falls into ${\mathfrak {H}}$; then the equations ($III_{c}$) and ($V_{c}$) go over into

$Rot\ {\mathfrak {H}}=4\pi {\dot {\mathfrak {D}}}$,

$(III'_{c})$

$\varkappa _{1}{\mathfrak {E}}_{x}=4\pi V^{2}{\mathfrak {D}}_{x}$, etc.

$(V'_{c})$

and the last of the limiting conditions ($VIII_{c}$) into

${\mathfrak {H}}{}_{h(1)}={\mathfrak {H}}{}_{h(2)}$.

Thus for this case, the known equations of motion and limiting conditions of the electromagnetic theory of light are given. From formulas $(I_{c}),(II_{c}),(III'_{c}),(IV_{c})$ and $(V'_{c})$ we derive (when $\varkappa _{1},\varkappa _{2},\varkappa _{3}$ are different from each other) the laws of light motion in crystals of two axes, while the assumption $\varkappa _{1}=\varkappa _{2}=\varkappa _{3}$ leads back to isotropic bodies. Besides, since $\varkappa _{1},\varkappa _{2}$ and $\varkappa _{3}$ depend on the