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&prime; carrying a current i&prime; is derivable from a vector-potential

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

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

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

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

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