Page:Elektrische und Optische Erscheinungen (Lorentz) 065.jpg

the surfaces — introduce the coordinates of the point P that lies in $\Sigma$. Thus it is allowed to say, that ${\mathfrak {D}}_{n(1)}$ and ${\mathfrak {D}}_{n(2)}$ are the values at the limiting-surface and that the previous formula expresses the continuity of ${\mathfrak {D}}_{n}$.

Similar formulas as equations (I_c) and (51) are emerging from ($II_{b}$); namely for the interior of a body

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

and for the limiting-surface

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

§ 44. From fundamental equation ($III_{b}$) we derive

$Rot\ {\overline {{\mathfrak {H}}'}}=4\pi {\overline {\rho {\mathfrak {v}}}}+4\pi {\dot {\overline {\mathfrak {d}}}}$,

or, be means of the definition

${\overline {\rho {\mathfrak {v}}}}={\frac {1}{l}}\Sigma e{\mathfrak {v}}={\frac {1}{l}}\Sigma e{\dot {\mathfrak {q}}}={\dot {\mathfrak {M}}}$, $Rot\ {\overline {{\mathfrak {H}}'}}=4\pi {\dot {\mathfrak {D}}}$.

This derivation is true for the interior of a body. To arrive at the limiting condition, we note at first, that (§ 4, h) (by the equation ($III_{b}$) for an arbitrary surface $\sigma$, with the borderline s)

$\int {\mathfrak {H}}_{s}^{'}ds=4\pi \int \left(\rho {\mathfrak {v}}_{n}+{\dot {\mathfrak {d}}}_{n}\right)d\sigma$

and thus also

$\int {\mathfrak {H}}_{s}^{'}ds=4\pi \int \left({\overline {\rho {\mathfrak {v}}_{n}}}+{\dot {\overline {{\mathfrak {d}}_{n}}}}\right)d\sigma =4\pi \int {\dot {\mathfrak {D}}}_{n}d\sigma$

(52)

Now we lay through the point p (Fig. 1 and 2) a plane, that contains the perpendicular of the borderline and the arbitrary direction h tangential to $\Sigma$, and choose as surface $\sigma$ the part of this plane, that lies between $\sigma _{1}$ and $\sigma _{2}$ and which is limited by two lines parallel to that perpendicular. If the length of this layer in the direction h is of order l (§ 39), then we may neglect all magnitudes of order a and we obtain from (52)

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

where the indices 1 and 2 have the same meaning as above. For the two components of ${\overline {{\mathfrak {H}}^{'}}}$ we may take at this place the values in point P again, and thus the equations says, that the tangential components of vector ${\overline {{\mathfrak {H}}^{'}}}$ were steady.