Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/277

this shows that any harmonic disturbance, and therefore any disturbance whatever, is propagated along the wire with velocity (CL)^-1/2. The difference between propagation in an aerial wire and propagation in an oceanic cable is, as Thomson remarked, similar to the difference between the propagation of an impulsive pressure through a long column of fluid in a tube when the tube is rigid (case of the aerial wire) and when it is elastic, so as to be capable of local distension (case of the cable, the distension corresponding to the effect of capacity): in the former case, as is well known, the impulse is propagated with a definite velocity, namely, the velocity of sound in the fluid.

The work of Thomson on signalling along cables was followed in 1857 by a celebrated investigation^[1] of Kirchhoff's, on the propagation of electric disturbance along an aerial wire of circular cross-section.

Kirchhoff assumed that the electric charge is practically all resident on the surface of the wire, and that the current is uniformly distributed over its cross-section; his idea of the current was the same as that of Fechner and Weber, namely, that it consists of equal streams of vitreous and resinous electricity flowing in opposite directions. Denoting the electric potential by V, the charge per unit length of wire by e, the length of the wire by l, and the radius of its cross-section by α, he showed that V is determined approximately by the equation^[2]

$V=2e\mathrm {log} (l/\alpha )$.

1. Ann. d. Phys. c (1857), pp. 193, 251: Kirchhoff's Ges. Abhandl., p. 131; Phil. Mag. xiii (1857), p. 393.
2. His method of obtaining this equation was to calculate separately the effects of (1) the portion of the wire within a distance ε on either side of the point considered, where ε denotes a length small compared with l, but large compared with α, and (2) the rest of the wire. He thus obtained the equation

   $V=2e\mathrm {log} {\frac {2e}{\alpha }}+\int {\frac {e^{\prime }ds^{\prime }}{r}}$,

   where the integration is to be taken over all the length of the wire except the portion 2ε: the equation given in the text was then derived by an approximation, which, however, is open to some objection.

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