In the first place, there is no electrodynamic potential; nor was there any, as we have seen, in the case of a closed current acting on an open current.
Next there is, properly speaking, no magnetic force.
And, in fact, we have given above three different definitions of this force:
2º By the director couple which orientates the magnetic needle;
3º By the action on an element of current.
But in the case which now occupies us, not only these three definitions are no longer in harmony, but each has lost its meaning, and in fact:
1º A magnetic pole is no longer acted upon simply by a single force applied to this pole. We have seen in fact that the force due to the action of an element of current on a pole is not applied to the pole, but to the element; it may moreover be replaced by a force applied to the pole and by a couple;
2º The couple which acts on the magnetic needle is no longer a simple director couple, for its moment with respect to the axis of the needle is not null. It breaks up into a director couple, properly so called, and a supplementary couple which tends to produce the continuous rotation of which we have above spoken;
3º Finally the force acting on an element of current is not normal to this element.
In other words, _the unity of the magnetic force has disappeared_.
Let us see in what this unity consists. Two systems which exercise the same action on a magnetic pole will exert also the same action on an indefinitely small magnetic needle, or on an element of current placed at the same point of space as this pole.
Well, this is true if these two systems contain only closed currents; this would no longer be true if these two systems contained open currents.
It suffices to remark, for instance, that, if a magnetic pole is placed at _A_ and an element at _B_, the direction of the element being along the prolongation of the sect _AB_, this element which will exercise no action on this pole will, on the other hand, exercise an action either on a magnetic needle placed at the point _A_, or on an element of current placed at the point _A_.
5. _Induction._--We know that the discovery of electrodynamic induction soon followed the immortal work of Ampère.
As long as it is only a question of closed currents there is no difficulty, and Helmholtz has even remarked that the principle of the conservation of energy is sufficient for deducing the laws of induction from the electrodynamic laws of Ampère. But always on one condition, as Bertrand has well shown; that we make besides a certain number of hypotheses.