Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/275

gutta-percha, "may be assimilated exactly to an immense Leyden battery; the glass of the jars represents the gutta-percha; the internal coating is the surface of the copper wire," while the outer coating corresponds to the sea-water. It follows that in all calculations relating to the propagation of electric disturbances along submarine cables, the electrostatic capacity of the cable must be taken into account.

The theory of signalling by cable originated in a correspondence between Stokes and Thomson in 1854. In the case of long submarine lines, the speed of signalling is so much limited by the electrostatic factor that electro-magnetic induction has no sensible effect; and it was accordingly neglected in the investigation. In view of other applications of the analysis, however, we shall suppose that the cable has a self-induction L per unit length, and that R denotes the ohmic resistance, and C the capacity per unit length, V the electric potential at a distance x from one terminal, and i the current at this place. Ohm's law, as modified for inductance, is expressed by the equation

$-{\frac {\partial V}{\partial x}}=L{\frac {\partial i}{\partial t}}+Ri$;

moreover, since the rate of accumulation of charge in unit length at x is - ∂i/∂x, and since this increases the potential at the rate - (1/C)∂i/∂x, we have

$C{\frac {\partial V}{\partial t}}=-{\frac {\partial i}{\partial x}}$.

Eliminating i between these two equations, we have

${\frac {1}{C}}{\frac {\partial ^{2}V}{\partial x^{2}}}=L{\frac {\partial ^{2}V}{\partial t^{2}}}+R{\frac {\partial V}{\partial t}}$,

which is known as the equation of telegraphy.^[1]

Thomson, in one of his letters^[2] to Stokes in 1854, obtained this equation in the form which applies to Atlantic cables, i.e., with the term in L neglected. In this form it is

1. We have neglected leakage, which is beside our present purpose.
2. Proc. Hoy. Soc., May, 1855: Kelvin's Math. and Phys. Papers, ii, p. 61.