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 (Ic) and (51) are emerging from ($$II_b$$); namely for the interior of a body

and for the limiting-surface

§ 44. From fundamental equation ($$III_b$$) we derive

or, be means of the definition

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)

and thus also

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)

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.