Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/186

The sign of this expression is reversed if we reverse the direction in which we measure $\beta ^{\prime }$. It must therefore represent either a force in the direction of $\beta ^{\prime }$, or a couple in the plane of $\alpha$ and $\beta ^{\prime }$. As we are not investigating couples, we shall take it as a force acting on $\alpha$ in the direction of $\beta ^{\prime }$.

There is of course an equal force acting on $\beta ^{\prime }$ in the opposite direction.

We have for the same reason a force

acting on $\alpha$ in the direction of $\gamma ^{\prime }$, and a force

acting on $\beta$ in the opposite direction.

514.] Collecting our results, we find that the action on $ds$ is compounded of the following forces,

and

$X=(A\alpha \alpha ^{\prime }+B\beta \beta ^{\prime })ii^{\prime }$ in the direction of $r$, $Y=C(\alpha \beta ^{\prime }-\alpha ^{\prime }\beta )ii^{\prime }$ in the direction of $\beta$, $Z=C\alpha \gamma ^{\prime }ii^{\prime }$ in the direction of $\gamma ^{\prime }$.

(9)

Let us suppose that this action on $ds$ is the resultant of three forces, $Rii^{\prime }\,ds\,ds^{\prime }$ acting in the direction of $r$, $Sii^{\prime }\,ds\,ds^{\prime }$ acting in the direction of $ds$, and $S^{\prime }ii^{\prime }\,ds\,ds^{\prime }$ acting in the direction of $ds^{\prime }$, then in terms of $\theta$, $\theta ^{\prime }$ , and $\eta$,

$R=A\cos \theta \cos \theta ^{\prime }+B\sin \theta \sin \theta ^{\prime }\cos \eta$,
$S=-C\cos \theta ^{\prime }$,	$S^{\prime }=C\cos \theta$.

(10)

In terms of the differential coefficients of $r$

$R=A{\frac {dr}{ds}}{dr}{ds^{\prime }}-Br{\frac {d^{2}r}{ds\,ds^{\prime }}}$,
$S=+C{\frac {dr}{ds^{\prime }}}$,	$S^{\prime }=-C{\frac {dr}{ds}}$,

(11)

In terms of $l$, $m$, $n$, and $l^{\prime }$, $m^{\prime }$, $n^{\prime }$,

$R=-(A+B){\frac {1}{r^{2}}}(l\xi +m\eta +n\zeta )(l^{\prime }\xi +m^{\prime }\eta +n^{\prime }\zeta )+B(ll^{\prime }+mm^{\prime }+nn^{\prime })$,
$S=C{\frac {1}{r}}(l^{\prime }\xi +m^{\prime }n+n^{\prime }\zeta )$,	$S^{\prime }=C{\frac {1}{r}}(l\xi +m\eta +n\zeta )$,

(12)

where $\xi$, $\eta$, $\zeta$ are written for $x^{\prime }-x$, $y^{\prime }-y$, and $z^{\prime }-z$ respectively.

515.] We have next to calculate the force with which the finite current $s^{\prime }$ acts on the finite current $s$. The current $s$ extends from $A$ where $s=0$, to $P$, where it has the value $s$. The current $s^{\prime }$ extends from $A^{\prime }$, where $s^{\prime }=0$, to $P^{\prime }$, where it has the value $s^{\prime }$.