Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/273

gases, as if the current had passed through the water in both directions. After this F. Savary^[1] had noticed that the discharge of a Leyden jar magnetizes needles in alternating layers, and had conjectured that "the electric motion during the discharge consists of a series of oscillations." A similar remark was made in connexion with a similar observation by Joseph Henry (b. 1799, d. 1878), of Washington, in 1842.^[2]

"The phenomena," he wrote, "require us to admit the existence of a principal discharge in one direction, and then several reflex actions backward and forward, each more feeble than the preceding, until equilibrium is restored." Helmholtz had repeated the same suggestion in his essay on the conservation of energy: and in 1853 W. Thomson^[3] verified it, by investigating the mathematical theory of the discharge, as follows:—

Let C denote the capacity of the jar, i.e., the measure of the charge when there is unit difference of potential between the coatings; let R denote the ohmic resistance of the discharging circuit, and L its coefficient of self-induction. Then if at any instant t the charge of the condenser be Q, and the current in the wire be i, we have i = dQ/dt; while Ohm's law, modified by taking self-induction into account, gives the equation

$Ri+L{\frac {di}{dt}}=-{\frac {Q}{C}}$.

Eliminating i, we have

$L{\frac {d^{2}Q}{dt^{2}}}+R{\frac {dQ}{dt}}+{\frac {1}{C}}Q=0$,

an equation which shows that when R²C < 4L, the subsidence of Q to zero is effected by oscillations of period

$2\pi \left({\frac {1}{LC}}-{\frac {R^{2}}{4L^{2}}}\right)^{-{\tfrac {1}{2}}}$

1. Annales de Chimie, xxxiv (1827), p. 5.
2. Proc. Am. Phil. Soc. ii (1842), p. 193.
3. Phil. Mag. (4) v (1853), p. 400; Kelvin's Math, and Phys. Papers i, P. 540.