Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/168

 If we now conceive the shell divided into two segments by a surface of no induction, the two parts will experience electrical forces the resultants of which will tend to separate the parts with a force equivalent to the resultant force due to a pressure $$p$$ acting on every part of the surface of no induction which divides them.

This illustration is to be taken merely as an explanation of what is meant by the tension and pressure, not as a physical theory to account for them.

108.] We have next to consider whether these internal forces are capable of accounting for the observed electrical forces in every case, as well as in the case where a closed equipotential surface can be drawn surrounding one of the electrified systems.

The statical theory of internal forces has been investigated by writers on the theory of elasticity. At present we shall require only to investigate the effect of an oblique tension or pressure on an element of surface.

Let $$p$$ be the value of a tension referred to unit of a surface to which it is normal, and let there be no tension or pressure in any direction normal to $$p$$. Let the direction-cosines of p be l, m, n. Let dy dz be an element of surface normal to the axis of x, and let the effect of the internal force be to urge the parts on the positive side of this element with a force whose components are

From every point of the boundary of the element dy dz let lines be drawn parallel to the direction of the tension $$p$$, forming a prism whose axis is in the line of tension, and let this prism be cut by a plane normal to its axis.

The area of this section will be l dy dx, and the whole tension upon it will be pl dy dz and since there is no action on the sides of the prism, which are normal to p, the force on the base dy dz must be equivalent to the force pl dy dx acting in the direction (l, m, n). Hence the component in the direction of x,

If we now combine with this tension two tensions $$p'$$ and $$p$$ in directions $$(l', m', n')$$ and $$(l, m, n)$$ respectively, we shall have