Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/164

would, if these surfaces were each rigid, act on the outer surface with a resultant equal to that of the electrical forces on the outer system $E_{1}$, and on the inner surface with a resultant equal to that of the electrical forces on the inner system.

Let us now consider the space between the surfaces, and let us suppose that at every point of this space there is a tension in the direction of $R$ and equal to ${\frac {1}{8\pi }}R^{2}$ per unit of area. This tension will act on the two surfaces in the same way as the pressures on the other side of the surfaces, and will therefore account for the action between $E_{1}$ and $E_{2}$, so far as it depends on the internal force in the space between $S_{1}$ and $S_{2}$.

Let us next investigate the equilibrium of a portion of the shell bounded by these surfaces and separated from the rest by a surface everywhere perpendicular to the equipotential surfaces. We may suppose this surface generated by describing any closed curve on $S_{1}$, and drawing from every point of this curve lines of force till they meet $S_{2}$.

The figure we have to consider is therefore bounded by the two equipotential surfaces $S_{1}$ and $S_{2}$, and by a surface through which there is no induction, which we may call $S_{0}$.

Let us first suppose that the area of the closed curve on $S_{1}$ is very small, call it $dS_{1}$ and that $C_{2}=C_{1}+dV$.

The portion of space thus bounded may be regarded as an element of volume. If $\nu$ is the normal to the equipotential surface, and $dS$ the element of that surface, then the volume of this element is ultimately $dS\ d\nu$.

The induction through $dS_{1}$ is $R\ dS_{1}$, and since there is no induction through $S_{0}$, and no free electricity within the space considered, the induction through the opposite surface $dS_{2}$ will be equal and opposite, considered with reference to the space within the closed surface.

There will therefore be a quantity of electricity

$e_{1}=-{\frac {1}{4\pi }}R_{1}dS_{1}$

on the first equipotential surface, and a quantity

$e_{2}={\frac {1}{4\pi }}R_{2}dS_{2}$

on the second equipotential surface, with the condition

$e_{1}+e_{2}=0\,$