Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/243

in the earlier part^[1] of the memoir, he calculated the induced currents in various particular cases.

But having arrived at the formulae in this way, Neumann noticed^[2] a peculiarity in them which suggested a totally different method of treating the subject. In fact, on examining the expression for the current induced in a circuit which is in motion in the field due to a magnet, it appeared that this induced current depends only on the alteration caused by the motion in the value of a certain function; and, moreover, that this function is no other than the potential of the ponderomotive forces which, according to Ampère's theory, act between the circuit, supposed traversed by unit current, and the magnet.

Accordingly, Neumann now proposed to reconstruct his theory by taking this potential function as the foundation.

The nature of Neumann's potential, and its connexion with Faraday's theory, will be understood from the following considerations:—

The potential energy of a magnetic molecule M in a field of magnetic intensity B is (B.M); and therefore the potential energy of a current i flowing in a circuit's in this field is

$i\iint _{S'}(\mathbf {B.} d\mathbf {S} )$,

where S denotes a diaphragm bounded by the circuit s; as is seen at once on replacing the circuit by its equivalent magnetic shell S. If the field B be produced by a current i′ flowing in a circuit s′, we have, by the formula of Biot and Savart,

${\begin{matrix}\mathbf {B} &=&i^{\prime }\int _{s^{\prime }}{\frac {[\mathbf {ds^{\prime }.r} ]}{r^{3}}}\\\ &=&i^{\prime }\int _{s^{\prime }}\mathrm {curl} \ {\frac {\mathbf {ds^{\prime }} }{r}}.\end{matrix}}$

1. §§ 1-8. It may be remarked that Neumann, in making use of Ohm's law, was (like everyone else at this time) unaware of the identity of electroscopic force with electrostatic potential,
2. § 9.