Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/246

electric currents. Fechner supposed every current to consist in a streaming of electric charges, the vitreous charges travelling in one direction, and the resinous charges, equal to them in magnitude and number, travelling in the opposite direction with equal velocity. He further supposed that like charges attract each other when they are moving parallel to the same direction, while unlike charges attract when they are moving in opposite directions. On these assumptions he succeeded in bringing Faraday's induction effects into connexion with Ampère's laws of electrodynamics.

In 1846 Weber,^[1] adopting the same assumptions as Fechner, analysed the phenomena in the following way:—

The formula of Ampère for the ponderomotive force between two elements ds, ds′ of currents i, i′, may be written

$F=ii^{\prime }\ ds\ ds^{\prime }\ \left({\frac {2}{r}}{\frac {d^{2}r}{ds\ ds^{\prime }}}-{\frac {1}{r^{3}}}{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}\right)$.

Suppose now that λ units of vitreous electricity are contained in unit length of the wire s, and are moving with velocity u; and that an equal quantity of resinous electricity is moving with velocity u in the opposite direction; so that

$i=2\lambda u$.

Let λ, u′, denote the corresponding quantities for the other current; and let the suffix ₁, be taken to refer to the action between the positive charges in the two wires, the suffix ₂, to the action between the positive charge in s and the negative charge in s′, the suffix ₃, to the action between the negative charge in s and the positive charge in s′, and the suffix ₄ to the action between the negative charges in the two wires. Then we have

$\left({\frac {dr}{dt}}\right)_{1}=u{\frac {dr}{ds}}+u^{\prime }{\frac {dr}{ds^{\prime }}}$,

1. Elektrodynamische Maasbestimmungen, Leipzig Abhandl., 1846: Ann. d. Phys. lxxiii (1848), p. 193: English translation in Taylor's Scientific Memoirs, v (1862), p. 489.