Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/167

 the medium which is consistent with the phenomena so far as we have examined them. We have found that in order to account for the electric attraction between distant bodies without admitting direct action, we must assume the existence of a tension $$p$$ at every point of the medium in the direction of the resultant force $$R$$ at that point. In order to account for the equilibrium of the medium itself we must further suppose that in every direction perpendicular to $$R$$ there is a pressure $$p$$.

By establishing the necessity of assuming these internal forces in the theory of an electric medium, we have advanced a step in that theory which will not be lost though we should fail in accounting for these internal forces, or in explaining the mechanism by which they can be maintained in air, glass, and other dielectric media.

We have seen that the internal stresses in solid bodies can be ascertained with precision, though the theories which account for these stresses by means of molecular forces may still be doubtful. In the same way we may estimate these internal electrical forces before we are able to account for them.

In order, however, that it may not appear as if we had no explanation of these internal forces, we shall shew that on the ordinary theory they must exist in a shell bounded by two equipotential surfaces, and that the attractions and repulsions of the electricity on the surfaces of the shell are sufficient to account for them.

Let the first surface $$S_1$$ be electrified so that the surface-density is

and the second surface $$S_2$$ so that the surface-density is

then, if we suppose that the value of $$V$$ is $$C_1$$ at every point within $$S_1$$ and $$C_2$$ at every point outside of $$S_2$$, the value of $$V$$ between these surfaces remaining as before, the characteristic equation of $$V$$ will be satisfied everywhere, and $$V$$ is therefore the true value of the potential.

We have already shewn that the outer and inner surfaces of the shell will be pulled towards each other with a force the value of which referred to unit of surface is $$p$$, or in other words, there is a tension $$p$$ in the substance of the shell in the direction of the lines of force.