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

516.] 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 Rh

Such functions necessarily disappear when the circuit is closed.

Let the components of the total force with which $$A^\prime P^\prime$$ acts on $$A A$$ 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 $$ii^\prime\frac{d^2X}{ds\, ds^\prime} ds\, ds^\prime$$.

Hence

Substituting the values of $$R$$, $$S$$, and $$S^\prime$$ from (12), remembering that Rhand arranging the terms with respect to $$l$$, $$m$$, $$n$$, we find

Since $$A$$, $$B$$, and $$C$$ are functions of $$r$$, we may write Rhthe integration being taken between $$r$$ and $$\infty$$ because $$A$$, $$B$$, $$C$$ vanish when $$r = \infty$$.

Hence

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 Rh

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