Ampère's reasoning rests on the assumption that the magnetic field produced by a current is in all respects of the same nature as that produced by a magnet; in other words, that only one kind of magnetic force exists. This principle of the "unity of magnetic force" Hertz now proposed to supplement by asserting that the electric force generated by a changing magnetic field is identical in nature with the electric force due to electrostatic charges; this second principle he called the "unity of electric force." Suppose, then, that a system of electric currents ι exists in otherwise empty space. According to the older theory, these currents give rise to a vector-potential a1, equal to Pot ι;[1] and the magnetic force H1, is the curl of a1: while the electric force E1, at any point in the field, produced by the variation of the currents, is .
It is now assumed that the electric force so produced is indistinguishable from the electric force which would be set up by electrostatic charges, and therefore that the system of varying currents exerts ponderomotive forces on electrostatic charges; the principle of action and reaction then requires that electrostatic charges should exert ponderomotive forces on a system of varying currents, and consequently (again appealing to the principle of the unity of electric force) that two systems of varying currents should exert on each other ponderomotive forces due to the variations.
But just as Helmholtz,[2] by aid of the principle of conservation of energy, deduced the existence of an electromotive force of induction from the existence of the ponderomotive forces between electric currents (i.e. variable electric systems), so from the existence of ponderomotive forces between variable systems of currents (i.e. variable magnetic systems) we may infer that variations in the rate of change of a variable magnetic system give rise to induced magnetic forces in the surrounding space. The analytical formulae which determine these forces
