gether null. We may also adjust this double current to the apparatus of Fig. 89., giving it the form of Fig. 98.

Fig. 98.

Second Case of Equilibrium.

The action exercised by a rectilinear conductor upon a movable current is exactly equal to that which is exercised upon the same current by a conductor bent and turned in any manner, but comprised within the same limits, provided that the currents which traverse the two finite conductors are the same, or have the same intensity.

The accuracy of this principle is verified by means of the apparatus of Fig. 89., to which the astatic vertical conductor is suspended. For the fixed conductor we place a system of two wires, one of which is rectilinear and the other twisted into the form of a flame, a zigzag, or in any other manner (Fig. 99.). They are so arranged that the vertical branch of the movable astatic conductor is situated between them, and the current that traverses it successively has the same direction in each of them; but at the same time that this direction, which is common to them, is contrary to that of the current in the movable branch. We then see that the latter is equally repelled by the two fixed conductors, and is maintained between them exactly in the middle. It is important that the sinuosities of the twisted conductor, designated by Ampère under the name of sinuous, should not be too great comparatively to the distance of the conductors from the movable current. They may, however, providing that this distance

be sufficiently great, be situated, one of them in a plane different from that of the other. This case of equilibrium served

M. Ampère for showing that we may apply to currents the law of the decomposition and recomposition of ordinary forces, or the law of the parallelogram of forces; which could not be inferred à priori, in consequence of the very special nature of the forces which emanate from electric currents, and which are not similar to the ordinary forces of mechanics.

A closed circuit of any form whatever cannot set in motion any portion of a current forming an arc of a circle of which the centre is on a fixed axis, around which it may freely turn, an axis that is perpendicular to the plane of the circle to which the axis belongs. In this delicate experiment, it is necessary that the arc of the circle may move alone, and without the conductors by which it is placed in the circuit. For this purpose we employ two canals filled with a quantity of mercury, so that the level of the liquid rises by capillarity above the sides of the canals. The conductor, in the form of an arc of a circle, fixed by its middle to the extremity of a horizontal stem coming as it were from the axis, situated

at the centre of the circle of which the arc forms a part, rests delicately by two of its points upon the surface of the mercury of each canal, so as to be simply in contact with it (Fig. 100.).

Fig. 100.

The positive pole of the pile communicates with the mercury of one of the canals, and the negative with the mercury of the other, so that this conductor of an arc of a circle serves to close the circuit, and is itself the only movable part. A wire is presented to it at a certain distance, which is bent into the form of a circle, an ellipsis, or a rectangle; in a word, forming a polygon or a closed curve, and moreover traversed by a current. In whatever manner this fixed conductor is placed in relation to the movable one, no action is manifested.

Fourth Case of Equilibrium.

We take three circular conductors, situated in the same plane, an horizontal plane for example, each movable around an axis, situated beyond their circumference, and to which each of them is connected by a horizontal branch, soldered to one point of this circumference. If these three circular conductors are situated so that their centres are on the same right line; if, moreover, the distances of these centres are respectively proportional to the radii of the circles; that is to say,

if the relation of the radius of the first circle to the radius of the second, and that of the radius of the second to the radius of the third, are to each other as the distance of the first centre from the second, and as the distance of the second centre from the third, the intermediate movable conductor will be in equilibrium between the two extreme fixed conductors, when they are all three traversed by an electric current, moving in the same direction in all, and having the same intensity in each. The mere inspection of the figure (Fig. 101.) is sufficient to explain the manner in which the

Fig. 101.

three circular conductors are traversed successively by the same current, the middle conductor being movable between the two extreme ones, which are fixed.

By means of these four cases of equilibrium, Ampère succeeded not only in determining the form of the mathematical expression of the force that two elements of voltaic currents exercise upon each other, but in finding the value of the constant quantities that enter into this expression, and particularly in deducing from it that the force itself is in inverse ratio to the square of the distance between the two elements. We may remember that the experiment of Biot and Savart

had led to the same law for an element of a current upon an element of a magnet, and which establishes a further analogy between an element of a magnet and an element of an electric current.*

Once having arrived at the mathematical expression of the action of two elements of a current, Ampère deduced from it the action of the assemblage of several elementary currents, either upon a similar assemblage, or upon a finite or indefinite current. The results of the calculation were constantly found to agree with those that had been furnished by experiment. But the most important case is that which includes the mutual action of two solenoids. Ampère designated by this name a system of very small closed currents, having their centres equally distributed on a right line or a curve, which he named direction of the solenoïd. Magnetisation, by impressing fixed directions upon the electric currents by which the molecules of bodies are enveloped, produces solenoïds; in such sort that a solenoïd is the magnetic skeleton of magnetised substances, the magnets being an assemblage of closed currents.

Ampère had demonstrated that the action of a solenoïd depends only on the position of its extremities, and in no degree upon the form of its axis; but he did not succeed in deducing from the calculation applied to the currents of solenoïds all the same consequences as he had deduced from this same application made to the molecular currents, which, according to him, constitute magnets,-results that are perfectly in accordance with the properties of magnets. M. Savary has filled up this blank: he set out from the principle that, if Ampère's formula is true when it is applied to molecular currents, it ought to be equally so when it is applied to the circular currents of solenoïds, and that calculation ought also to give results identical with those of experiment. This experienced philosopher discovered that this, in fact, was the case; and he further showed that solenoids, or electrodynamic cylinders of a very small diameter, act, at distances

* See the final note F, for the calculation relating to the mutual action of electric currents.