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

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

Rh

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, Rh

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

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

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

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.