Page:Philosophical magazine 23 series 4.djvu/37

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applied to Statical Electricity. 21

or

Fdr=e_{1}{\frac {d^{2}\Psi _{2}}{dr}}dr,

where F is the resistance and $dr$ the motion.

If the body $e_{2}$ be small, then if $r$ is the distance from $e_{2}$ , equation (123) gives

\Psi _{2}=E^{2}{\frac {e_{2}}{r}};

whence

F=-E^{2}{\frac {e_{1}e_{2}}{r^{2}}};

(127)

or the force is a repulsion varying inversely as the square of the distance.

Now let $\eta _{1}$ and $\eta _{2}$ be the same quantities of electricity measured statically, then we know by definition of electrical quantity

F=-{\frac {\eta _{1}\eta _{2}}{r^{2}}};

(128)

and this will be satisfied provided

\eta _{1}=Ee_{1}\ \mathrm {and} \ \eta _{2}=Ee_{2}

(129)

so that the quantity E previously determined in Prop. XIII. is the number by which the electrodynamic measure of any quantity of electricity must be multiplied to obtain its electrostatic measure.

That electric current which, circulating round a ring whose area is unity, produces the same effect on a distant magnet as a magnet would produce whose strength is unity and length unity placed perpendicularly to the plane of the ring, is a unit current; and E units of electricity, measured statically, traverse the section of this current in one second, — these units being such that any two of them, placed at unit of distance, repel each other with unit of force.

We may suppose either that E units of positive electricity move in the positive direction through the wire, or that E units of negative electricity move in the negative direction, or, thirdly, that ${\tfrac {1}{2}}E$ units of positive electricity move in the positive direction, while ${\tfrac {1}{2}}E$ units of negative electricity move in the negative direction at the same time.