Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/188

integrate this expression with respect to $s^{\prime }$. Integrating the first term by parts, we find

When $s^{\prime }$ is a closed circuit this expression must be zero. The first term will disappear of itself. The second term, however, will not in general disappear in the case of a closed circuit unless the quantity under the sign of integration is always zero. Hence, to satisfy Ampère's condition,

517.] We can now eliminate $P$, and find the general value of ${\frac {dX}{ds}}$,

${\frac {dX}{ds}}=\left\{{\frac {B+C}{2}}{\frac {\xi }{r}}(l\xi +m\eta n\zeta )+Q\right\}_{(s^{\prime },0)}$
${}+m\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {m^{\prime }\xi -l^{\prime }\eta }{r}}\,ds^{\prime }-n\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {l^{\prime }\zeta -n^{\prime }\xi }{r}}\,ds^{\prime }$.

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When $s^{\prime }$ is a closed circuit the first term of this expression vanishes, and if we make

$\alpha ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {n^{\prime }\eta -m^{\prime }\zeta }{r}}\;ds^{\prime }$, $\beta ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {l^{\prime }\zeta -n^{\prime }\xi }{r}}\,ds^{\prime }$, $\gamma ^{\prime }=\int _{0}^{s^{\prime }}{\frac {B-C}{2}}{\frac {m^{\prime }\xi -l^{\prime }\eta }{r}}\,ds^{\prime }$,

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where the integration is extended round the closed circuit $s^{\prime }$, we may write

Similarly

${\frac {dX}{ds}}=m\gamma ^{\prime }-n\beta ^{\prime }$. ${\frac {dY}{ds}}=n\alpha ^{\prime }-l\gamma ^{\prime }$, ${\frac {dZ}{ds}}=l\beta ^{\prime }-m\alpha ^{\prime }$.

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The quantities $\alpha ^{\prime }$, $\beta ^{\prime }$, $\gamma ^{\prime }$ are sometimes called the determinants of the circuit $s^{\prime }$ referred to the point $P$. Their resultant is called by Ampère the directrix of the electrodynamic action.

It is evident from the equation, that the force whose components are ${\frac {dX}{ds}}$, ${\frac {dY}{ds}}$, and ${\frac {dZ}{ds}}$ is perpendicular both to $ds$ and to this directrix, and is represented numerically by the area of the parallelogram whose sides are $ds$ and the directrix.