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

$p_{yx}dz\ dx+\left({\frac {d}{dy}}p_{yz}\right)dz\ dx\ dy$

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

$p_{zx}dx\ dy+\left({\frac {d}{dz}}p_{zx}\right)dx\ dy\ dz$

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

$\xi \ dx\ dy\ dz=\left({\frac {d}{dx}}p_{xx}+{\frac {d}{dy}}p_{yx}+{\frac {d}{dz}}p_{zx}\right)dx\ dy\ dz;$

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

$\xi ={\frac {d}{dx}}p_{xx}+{\frac {d}{dy}}p_{yz}+{\frac {d}{dz}}p_{zx},$

(7)

with similar expressions for $\eta$ and $\zeta$, the component forces in the other directions^[1].

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

$\xi ={\frac {1}{4\pi }}X\left({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}\right).$

(8)

But by Art. 77

$\left({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}\right)=4\pi \rho .$

(9)

${\begin{array}{lll}\mathrm {Hence} &&\xi =\rho X.\\\mathrm {Similarly} &&\eta =\rho Y,\\&&\zeta =\rho Z.\end{array}}$

(10)

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

1. This investigation may be compared with that of the ‚equation of continuity in hydrodynamics‘, and with others in which the effect on an element of volume is deduced from the values of certain quantities at its bounding surface.