Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/187

The coordinates of points on either current are functions of $s$ or of $s^{\prime }$.

If $F$ is any function of the position of a point, then we shall use the subscript ${}_{(s,0)}$ to denote the excess of its value at $P$ over that at $A$, thus

Such functions necessarily disappear when the circuit is closed.

Let the components of the total force with which $A^{\prime }P^{\prime }$ acts on $AA$ be $ii^{\prime }X$, $ii^{\prime }Y$, and $ii^{\prime }Z$. Then the component parallel to $X$ of the force with which $ds^{\prime }$ acts on $ds$ will be $ii^{\prime }{\frac {d^{2}X}{ds\,ds^{\prime }}}ds\,ds^{\prime }$.

Substituting the values of $R$, $S$, and $S^{\prime }$ from (12), remembering that

and arranging the terms with respect to $l$, $m$, $n$, we find

${\frac {d^{2}X}{ds\,ds^{\prime }}}$	${}=l\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi ^{2}+C{\frac {dr}{ds^{\prime }}}+(B+C){\frac {l^{\prime }\xi }{r}}\right\}$,
	${}+m\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi \eta +C{\frac {l^{\prime }\eta }{r}}+B{\frac {m^{\prime }\xi }{r}}\right\}$,
	${}+n\left\{-(A+B){\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}\xi \zeta +C{\frac {l^{\prime }\zeta }{r}}+B{\frac {n^{\prime }\xi }{r}}\right\}$.

(15)

Since $A$, $B$, and $C$ are functions of $r$, we may write

the integration being taken between $r$ and $\infty$ because $A$, $B$, $C$ vanish when $r=\infty$.

516.] Now we know, by Ampère's third case of equilibrium, that when $s^{\prime }$ is a closed circuit, the force acting on $ds$ is perpendicular to the direction of $ds$, or, in other words, the component of the force in the direction of $ds$ itself is zero. Let us therefore assume the direction of the axis of $x$ so as to be parallel to $ds$ by making $l=1$, $m=0$, $n=0$. Equation (15) then becomes

To find ${\frac {dX}{ds}}$, the force on $ds$ referred to unit of length, we must