1. The Case of Closed Currents.—In the case of the mutual action of two closed currents, experiment revealed to Ampère remarkably simple laws.
I recall rapidly here those which will be useful to us in the sequel:
1º If the intensity of the currents is kept constant, and if the two circuits, after having undergone any deformations and displacements whatsoever, return finally to their initial positions, the total work of the electrodynamic actions will be null.
In other words, there is an electrodynamic potential of the two circuits, proportional to the product of the intensities, and depending on the form and relative position of the circuits; the work of the electrodynamic actions is equal to the variation of this potential.
2º The action of a closed solenoid is null.
3º The action of a circuit C on another voltaic circuit C´ depends only on the 'magnetic field' developed by this circuit. At each point in space we can in fact define in magnitude and direction a certain force called magnetic force, which enjoys the following properties:
(a) The force exercised by C on a magnetic pole is applied to that pole and is equal to the magnetic force multiplied by the magnetic mass of that pole;
(b) A very short magnetic needle tends to take the direction of the magnetic force, and the couple to which it tends to reduce is proportional to the magnetic force, the magnetic moment of the needle and the sine of the dip of the needle;
(c) If the circuit C is displaced, the work of the electrodynamic action exercised by C on C´ will be equal to the increment of the 'flow of magnetic force' which passes through the circuit.
2. Action of a Closed Current on a Portion of Current.—Ampère not having been able to produce an open current, properly so called, had only one way of studying the action of a closed current on a portion of current.