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. This was by operating on a circuit _C_ composed of two parts, the one fixed, the other movable. The movable part was, for instance, a movable wire [alpha][beta] whose extremities [alpha] and [beta] could slide along a fixed wire. In one of the positions of the movable wire, the end [alpha] rested on the _A_ of the fixed wire and the extremity [beta] on the point _B_ of the fixed wire. The current circulated from [alpha] to [beta], that is to say, from _A_ to _B_ along the movable wire, and then it returned from _B_ to _A_ along the fixed wire. _This current was therefore closed._ In a second position, the movable wire having slipped, the extremity [alpha] rested on another point _A'_ of the fixed wire, and the extremity [beta] on another point _B'_ of the fixed wire. The current circulated then from [alpha] to [beta], that is to say from _A'_ to _B'_ along the movable wire, and it afterwards returned from _B'_ to _B_, then from _B_ to _A_, then finally from _A_ to _A'_, always following the fixed wire. The current was therefore also closed.