Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/170

 tangential forces urging them in the direction of $$x$$, we find the force on the first face $$-p_{yx}dz\ dx$$, and that on the second

Similarly for the faces dx dy, we find that a force $$-p_{zx}dx\ dy$$ acts on the first face, and

on the second in the direction of $$x$$.

If $$\xi\ dx\ dy\ dz$$ denotes the total effect of all these internal forces acting parallel to the axis of $$x$$ on the six faces of the element, we find

or, denoting by $$\xi$$ the internal force, referred to unit of volume, and resolved parallel to the axis of $$x$$,

with similar expressions for $$\eta$$ and $$\zeta$$, the component forces in the other directions.

Differentiating the values of $$p_{xx},\ p_{yz}$$, and $$p_{zx}$$ given in equations (6), we find

But by Art. 77

Thus, the resultant of the tensions and pressures which we have supposed to act upon the surface of the element is a force whose components are the same as those of the force, which, in the ordinary theory, is ascribed to the action of electrified bodies on the electricity within the element.

If, therefore, we admit that there is a medium in which there is maintained at every point a tension $$p$$ in the direction of the