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 We now want to define a new vector $$\mathfrak{D}$$ by the equation

$$\mathfrak{D}=\overline{\mathfrak{d}}+\mathfrak{M}$$,

and call it the dielectric polarization.

This vector, that goes over for the free aether (where $$\mathfrak{M}=0$$) into $$\mathfrak{d}$$, is exactly that, what Maxwell calls "dielectric displacement". Its basic property is according to the above, that for any closed surface

and also in the interior of any body

§ 43. Formula (50) leads to an important limiting-condition, if we apply it to a surface, that lies partly in the first, and partly in the second body. Around a certain point P of the limiting-surface $$\Sigma$$(Fig. 1 and 2) we shall lay a cylinder-surface C that is parallel to the perpendicular in P, and choose for the mentioned area the surface of the space that is cut from layer ($$\sigma_1$$, $$\sigma_2$$). If now the dimensions of the parts limited in $$\sigma_1$$ and $$\sigma_2$$ are of order l (§ 39), then we may consider the parts as elements that are equal, parallel and plane, and as they are much greater than the part of C that lies between $$\sigma_1$$ and $$\sigma_2$$, we can omit the integral taken over the latter

$$\int\mathfrak{D}_{n}d\sigma$$

Thus we find, if we mutually distinguish the values that are valid in $$\sigma_1$$ and $$\sigma_2$$ by the indices 1 and 2, and draw either at $$\sigma_1$$ as well as at $$\sigma_2$$ the perpendicular n from the first to the second body,

In relation to this, we have to notice one thing. In any medium, $$\mathfrak{D}_{x},\mathfrak{D}_{y},\mathfrak{D}_{z}$$ can be represented as slowly (§ 39) varying functions of coordinates, and we would have to substitute in these functions the coordinates of a point of $$\sigma_1$$ or $$\sigma_2$$, to obtain $$\mathfrak{D}_{n(1)}$$ and $$\mathfrak{D}_{n(2)}$$. Instead of this we can without noticeable error — due to the small distance of