Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/278

The next factor to be considered is the mutual induction of the current-elements in different parts of the wire. Assuming with Weber that the electromotive force induced in an clement ds due to another element ds′ carrying a current i′ is derivable from a vector-potential

$i^{\prime }{\frac {(\mathbf {r.ds^{\prime }} )\mathbf {.r} }{r^{3}}}$,

Kirchhoff found for the vector-potential due to the entire wire the approximate value

$w=2i\log(l/\alpha )$,

where i denotes the strength of the current;^[1] the vector-potential being directed parallel to the wire. Ohm's law then gives the equation

$i=-\pi k\alpha ^{2}\left({\frac {\partial V}{\partial x}}+{\frac {1}{c^{2}}}{\frac {\partial w}{\partial t}}\right)$,

where k denotes the specific conductivity of the material of which the wire is composed; and finally the principle of conservation of electricity gives the equation

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

Denoting log (l/α) by γ, and eliminating e, i, w from these four equations, we have

${\frac {\partial ^{2}V}{\partial x^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}+{\frac {1}{2\gamma \kappa \pi \alpha ^{2}}}{\frac {\partial V}{\partial t}}$,

which is, as might have been expected, the equation of telegraphy. When the term in a ∂V/∂t is ignored, as we have seen is in certain cases permissible, the equation becomes

${\frac {\partial ^{2}V}{\partial x^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}$,

1. This expression was derived in a similar way to that for V, by an intermediate formula

   $w=2i\log {\frac {2\epsilon }{\alpha }}+\int {\frac {i^{\prime }ds^{\prime }}{r}}\cos \theta \cos \theta ^{\prime }$,

   where θ und θ′ denote respectively the angles made with r by ds and ds′.