Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/286

Parallel to $x$, ⁠ $\left(P_{xx}+{\frac {1}{2}}{\frac {dP_{xx}}{dx}}\,dx\right)\,dy\,dz=X_{+x}$, $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ Parallel to $y$, $\left(P_{xy}+{\frac {1}{2}}{\frac {dP_{xy}}{dx}}\,dx\right)\,dy\,dz=Y_{+x}$, Parallel to $z$, $\left(P_{xz}+{\frac {1}{2}}{\frac {dP_{xz}}{dx}}\,dx\right)\,dy\,dz=Z_{+x}$.

(12)

The forces acting on the opposite side, $-X_{-x}$, $-Y_{-x}$ and $-Z_{-x}$, may be found from these by changing the sign of $dx$. We may express in the same way the systems of three forces acting on each of the other faces of the element, the direction of the force being indicated by the capital letter, and the face on which it acts by the suffix.

If $X\,dx\,dy\,dz$ is the whole force parallel to $x$ acting on the element,

whence

$X\,dx\,dy\,dz$	${}=X_{+x}+X_{+y}+X_{+z}+X_{-x}+X_{-y}+X_{-z}$,
	${}=\left({\frac {dP_{xx}}{dx}}+{\frac {dP_{yx}}{dx}}+{\frac {dP_{zx}}{dx}}\right)\,dx\,dy\,dz$,
$X$	${}={\frac {d}{dx}}P_{xx}+{\frac {d}{dy}}P_{yx}+{\frac {d}{dz}}P_{zx}$.

(13)

If $L\,dx\,dy\,dz$ is the moment of the forces about the axis of $x$ tending to turn the element from $y$ to $z$,

whence

$L\,dx\,dy\,dz$	${}={\frac {1}{2}}\,dy(Z_{+y}-Z_{-y})-{\frac {1}{2}}\,dz(Y_{+z}-Y_{-z})$,
	${}=(P_{yz}-P_{zy})\,dx\,dy\,dz$,
$L$	${}=P_{yz}-P_{zy}$.

(14)

Comparing the values of $X$ and $L$ given by equations (9) and (11) with those given by (13) and (14), we find that, if we make

$P_{xx}={\frac {1}{4\pi }}\left(a\alpha -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$,		$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
$P_{yy}={\frac {1}{4\pi }}\left(b\beta -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$,
$P_{zz}={\frac {1}{4\pi }}\left(c\gamma -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$.
$P_{yz}={\frac {1}{4\pi }}b\gamma$,	$P_{zy}={\frac {1}{4\pi }}c\beta$,
$P_{zx}={\frac {1}{4\pi }}c\alpha$,	$P_{xz}={\frac {1}{4\pi }}a\gamma$,
$P_{xy}={\frac {1}{4\pi }}a\beta$,	$P_{yz}={\frac {1}{4\pi }}b\alpha$,

(15)

the force arising from a system of stress of which these are the components will be statically equivalent, in its effects on each