Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/162

Now $R$ is the resultant due to the combined action of the external system $E_{1}$ and the electrification of the surface $S$. Hence the effect of the pressure $p$ on each element of the inside of the surface considered as a rigid body is equivalent to this combined action.

But the actions of the different parts of the surface on each other form a system in equilibrium, therefore the effect of the pressure $p$ on the rigid shell is equivalent in all respects to the electric attraction of $E_{1}$ on the shell, and this, as we have before shewn, is equivalent to the electric attraction of $E_{1}$ on $E_{2}$ considered as a rigid system.

If we had supposed the pressure $p$ to act on the outside of the shell, the resultant effect would have been equal and opposite, that is, it would have been statically equivalent to the action of the internal system $E_{2}$ on the external system $E_{1}$.

Let us now take the case of two electrified systems $E_{1}$ and $E_{2}$, such that two equipotential surfaces $V=C_{1}$ and $V=C_{2}$, which we shall call $S_{1}$ and $S_{1}$ respectively, can be described so that $E_{1}$ is exterior to $S_{1}$, and $S_{1}$ surrounds $S_{2}$, and $E_{2}$ lies within $S_{2}$.

Then if $R_{1}$ and $R_{2}$ represent the resultant force at any point of $S_{1}$ and $S_{2}$ respectively, and if we make

$p_{1}={\frac {1}{8\pi }}R_{1}^{2}$ and $p_{2}={\frac {1}{8\pi }}R_{2}^{2}$

the mechanical action between $E_{1}$ and $E_{2}$ is equivalent to that between the shells $S_{1}$ and $S_{2}$, supposing every point of $S_{1}$ pressed inwards, that is, towards $S_{2}$ with a pressure $p_{1}$, and every point of $S_{2}$ pressed outwards, that is, towards $S_{1}$ with a pressure $p_{2}$.

105.] According to the theory of action at a distance the action between $E_{1}$ and $E_{2}$ is really made up of a system of forces acting in straight lines between the electricity in $E_{1}$ and that in $E_{2}$, and the actual mechanical effect is in complete accordance with this theory.

There is, however, another point of view from which we may examine the action between $E_{1}$ and $E_{2}$. When we see one body acting on another at a distance, before we assume that the one acts directly on the other we generally inquire whether there is any material connexion between the two bodies, and if we find strings, or rods, or framework of any kind, capable of accounting for the observed action between the bodies, we prefer to explain the action by means of the intermediate connexions, rather than admit the notion of direct action at a distance.

Thus when two particles are connected by a straight or curved rod, the action between the particles is always along the line joining them, but we account for this action by means of a system of