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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

$\int {\mathfrak {D}}_{n}{\mathfrak {d}}\sigma =0$

(50)

and also in the interior of any body

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

($I_{c}$)

§ 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,

${\mathfrak {D}}_{n(1)}={\mathfrak {D}}_{n(2)}$

(51)

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