Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/192

525.] We may now express the components of the force on $ds$ arising from the action of $ds^{\prime }$ in the most general form consistent with experimental facts.

The force on $ds$ is compounded of an attraction

	$R$	${}={\frac {1}{r^{2}}}\left({\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-2r{\frac {d^{2}r}{ds\,ds^{\prime }}}\right)ii^{\prime }\,ds\,ds^{\prime }+r{\frac {d^{2}Q}{ds\,ds^{\prime }}}ii^{\prime }\,ds\,ds^{\prime }$
in the direction of $r$,
	$S$	${}=-{\frac {dQ}{ds^{\prime }}}ii^{\prime }\,ds\,ds^{\prime }$ in the direction of $ds$,
and		$S^{\prime }={\frac {dQ}{ds}}ii^{\prime }\,ds\,ds^{\prime }$ in the direction of $ds^{\prime }$,

(38)

where $Q=\int _{r}^{\infty }C\,dr$, and since $C$ is an unknown function of $r$, we know only that $Q$ is some function of $r$.

526.] The quantity $Q$ cannot be determined, without assumptions of some kind, from experiments in which the active current forms a closed circuit. If we suppose with Ampère that the action between the elements $ds$ and $ds^{\prime }$ is in the line joining them, then $S$ and $S^{\prime }$ must disappear, and $Q$ must be constant, or zero. The force is then reduced to an attraction whose value is

Ampère, who made this investigation long before the magnetic system of units had been established, uses a formula having a numerical value half of this, namely

Here the strength of the current is measured in what is called electrodynamic measure. If $i$, $i^{\prime }$ are the strength of the currents in electromagnetic measure, and $j$, $j^{\prime }$ the same in electrodynamic measure, then it is plain that

Hence the unit current adopted in electromagnetic measure is greater than that adopted in electrodynamic measure in the ratio of ${\sqrt {2}}$ to 1.

The only title of the electrodynamic unit to consideration is that it was originally adopted by Ampère, the discoverer of the law of action between currents. The continual recurrence of ${\sqrt {2}}$ in calculations founded on it is inconvenient, and the electromagnetic system has the great advantage of coinciding numerically