Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/169

${\begin{array}{llll}p_{xx}&=pl^{2}&+p'l'^{2}&+p''l''^{2}\\p_{xy}&=plm&+p'l'm'&+p''l''m''\\p_{xz}&=pln&+p'l'n'&+p''l''n''\end{array}}$

(2)

In the case of the electrical tension and pressure the pressures are numerically equal to the tension at every point, and are in directions at right angles to the tension and to each other. Hence, putting

$p'=p''=-p,\,$

(3)

$l^{2}+l'^{2}+l''^{2}=1,\ lm+l'm'+l''m''=0,\ ln+l'n'+l''n''=0,$

(4)

we find

${\begin{array}{l}p_{xx}=\left(2l^{2}-1\right)p,\\p_{xy}=2lmp,\\p_{xz}=2lnp.\end{array}}$

(5)

for the action of the combined tension and pressures.

Also, since $p={\frac {1}{8\pi }}R^{2}$, where $R$ denotes the resultant force, and since $Rl=X,\ Rm=Y,\ Rn=Z,$,

${\begin{array}{ll}p_{xx}&={\frac {1}{8\pi }}\left(X^{2}-Y^{2}-Z^{2}\right),\\\\p_{xy}&={\frac {1}{8\pi }}2XY=p_{yx},\\\\p_{xz}&={\frac {1}{8\pi }}2XZ=p_{zx};\end{array}}$

(6)

where X, Y, Z are the components of $R$, the resultant electromotive force.

The expressions for the component internal forces on surfaces normal to $y$ and $z$ may be written down from symmetry.

To determine the conditions of equilibrium of the element dx dy dz.

This element is bounded by the six planes perpendicular to the axes of coordinates passing through the points (x, y, z) and ($x+dx,\ y+dy,\ z+dz$).

The force in the direction of $x$ which acts on the first face dy dz is $-p_{xx}dy\ dz$, tending to draw the element towards the negative side. On the second face dy dz, for which $x$ has the value $x+dx$, the tension $p_{xx}$ has the value

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

and this tension tends to draw the element in the positive direction. If we next consider the two faces dz dx with respect to the