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through its centre, and to oscillate about a certain position of equilibrium, if the small oscillations by which it is brought to this position are isochronous, we infer that the force which causes it to oscillate, is, in all the successive positions of the oscillating body, exactly or very nearly, proportional to its angular distance Mech. from the final direction in which it settles; hence results the isochronism of its motions, since it is constantly drawn towards its point of rest by a force sensibly proportional to the angle which remains to be described before it arrives there. The motion of the solid body in these small arcs may then be rigorously compared to that of a simple pendulum oscillating about the same position of equilibrium by the force of gravity. Now we know that the oscillations of such a pendulum, supposed to be of a constant length, vary in duration according to the intensity of the gravity which acts upon it, and that this intensity is reciprocally proportional to the squares of the times employed by the pendulum in making the same number of oscillations, the arcs Mech. being very small. Therefore, if we compare in this way, the squares of the times for different distances of the uniting wire from the needle, supposing always the condition of isochronism to be fulfilled, we shall obtain the ratios of the component forces exerted in these different cases by the uniting wire, parallel to the direction of equilibrium about which the needle oscillates. These ratios, as well as the possibility of the equilibrium itself in the observed position, will therefore be so many conditions to which the total force proceeding from the wire must answer; and consequently we may make use of them in order to discover the absolute law to which this force must be subjected, in order that the conditions in question may take place.

But in order to arrive in this way at the nature of the force itself, we must, in the first place, analyse the mode of application by which it is capable of acting upon the needle and of determining its motions. I have said we were careful to make use of a very short needle. We magnetized it, moreover, in such a way that it was free from consecutive points. The quantities of austral and boreal magnetism which were free, might therefore be considered as sensibly concentrated in two points or poles, situated near the extremities of the needle, at equal distances from these extremities, which distances, the needle being a very fine cylindrical wire, must be equal to the sixth of its whole

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the

length. Now these two poles being of an opposite nature, influence of the uniting wire, whatever it be, must be opposite at these points; that is, if the equal quantities of austral and boreal magnetisms which belong to them, were situated in the same point, and the wire were to act upon one of them according to a certain direction, it must act upon the other in an absolutely opposite direction with an equal force. This opposition may be inferred from the fact, that the uniting wire does not cause unmagnetic needles that are presented to it to move, until it has separated their natural magnetisms; and it actually has no force when presented to needles of tempered steel, or other very hard compounds in which the separation of the natural magnetisms has not been previously effected, and is beyond the power thus exerted. Indeed, such a want of power would not exist, if the particles of the combined magnetisms were merely attracted in directions different but not exactly opposite to each other; for then the resultant of the different efforts exerted by the wire upon the exterior bodies in the combined state of their magnetisms, would not become zero, and consequently, needles formed of magnetic metals would be put in motion by the influence of the uniting wire without being magnetized even transiently, which is contrary to fact.

The state of indifference of these needles, so long as their natural magnetisms are not separated, proves also the equality of the actions exerted by the uniting wire upon equal quantities of the two magnetisms; for, without this equality, needles not actually magnetic, but formed of magnetic substances, would acquire in the presence of the wire an absolute motion of translation in space.

259. These principles being admited, when our needle becomes fixed in a position of equilibrium determined by the influence of the uniting wire, let us draw through its magnetic axis a horizontal plane which shall consequently be perpendicular to the wire. This plane will contain all the forces by which the equilibrium is determined; for we shall have, in the first place, the two poles of the needle in which reside the two quantities of free magnetism subjected to the action of the wire; and we shall have, moreover, the resultant of this action upon each pole, whatever it may be; for, since the wire may be considered as indefinite in the effects which it produces, being the same at whatever part of its length the needle is presented, the two parts of the

wire situated on opposite sides of the horizontal plane drawn through the centre of the needle, will necessarily exert upon it equal forces; whence it follows that their common resultant for each pole must be directed according to this plane. Let us represent the results of this action in figure 135, where AB is the magnetic axis of the needle, A and B its two poles, C its centre, and F the projection of the wire upon the horizontal plane or its intersection with this plane. Now whatever be the nature of the action exerted by the wire upon the pole B, this action will have a certain direction in the plane FBA. Let us suppose that this is BD, so that the quantity of ire boreal magnetism, situated in B, is urged according to BD in virtue of this force. It will be necessary to admit that if an equal quantity of austral magnetism occupied the same point B, it would be acted upon by an equal force and in an opposite direction, that is, according to BD', the continuation of BD. Hence result immediately the direction and intensity of the action exerted by the wire upon the south pole A of the needle, for this pole contains a quantity of free magnetism equal to B; and moreover it is distant from the wire by a quantity FA, equal to FB, in the position of equilibrium in which the needle settles. Now it is easy to prove that these actions exerted by the wire are the same at the same distance from its centre, whatever be the point of its circumference before which the needle is placed; for if it be made to turn upon itself without changing its longitudinal direction or its distance from the needle, the oscillating motion of the latter and its relative position of equilibrium suffer no alteration. Consequently, to obtain the action exerted at A upon any particle of magnetism whatever, whether austral or boreal, we have only to employ the same construction there as at the point B, that is, to draw a line DAD', making with FA, equal to FB, an angle FAD equal to the angle FBD; and taking upon this line from the point A, two opposite portions AD AD', equal to each other; then the first will represent the force exerted upon a quantity B of boreal magnetism s-tuare in A; and the second will represent the force that would be exerted upon an equal quantity of austral magnetism situated at the same point. This case is precisely that of the needle in its position of equilibrium. Thus its mass will be really acted upon by two resultants which will have with each other the preceding relations; that is, one of them, BD, being E. & M.

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applied at the pole B, will tend to move it from B towards D; and the other, AD', equal to the preceding, but applied at the pole A, will tend to move it from A towards D', with an equal intensity. Now it is evident from the most familiar principles of statics, that the two forces BD, AD', being equal, and applied to the two arms of the lever CA, CB, of equal lengths, and tending to turn the needle in opposite directions, they cannot preserve it in a state of equilibrium in the position BCA, perpendicular to CF, unless their directions BD, AD', are equally inclined to the needle; that is, unless the angles DBC, D'AC, are equal, which requires that DBF should be equal to D'AF. But D'AF is equal to D'BF by construction, since the system of lines FA and DAD', is simply the system of lines FB and DBD', transferred from B to A. Consequently, in this state of equilibrium which actually takes place, the two angles DBF, D'BF, are equal to each other. Whence it follows that they are both right angles, because they are adjacent and upon the same straight line. Thus, when an indefinite uniting wire, traversed by the voltaic current, acts upon an element of austral or boreal magnetism, situated at a certain distance FA or FB from its centre, the resultant of the actions which it exerts is perpendicular to the shortest distance from the element to the wire.

260. It is unnecessary to examine here how the different laminae of the wire contribute to form this resultant, or how the infinitely small particles of each lamina are capable of giving resultants transverse to the wire. These particulars necessarily depend on the nature of the forces which the electric current developes in the several integrant particles of which the mass of the wire is composed. A knowledge of these particular forces would doubtless be very useful, and consequently very desirable; but it is by no means necessary in order to establish the reality or the direction of their resultant, which, as we have just seen, are rigorously determined by the simple obervation of the compound results.

261. It is necessary now to fix the absolute direction according to which this resultant is exerted, when it acts on each kind of magnetism, that is, whether it tends to draw it to the right or to the left of the lines FA and FB. It is necessary, moreover, to determine the law according to which it varies at different distances. This has been done both by M. Savart and myself,

by means of two series of experiments which I shall now make known.

The first series was performed by means of the apparatus above described. It is only necessary to render the support moveable which carries the uniting wire, so that we may at pleasure present it successively to the needle at different known distances. We obtain this double object by applying to the foot of this support a horizontal division along which it may be moved, and which has a fixed direction towards the suspension wire of the oscillating needle. Then if we measure directly the horizontal distance of this wire or of the centre of the needle from the uniting wire in any single position of the latter, it is evident that all the other distances will be obtained, by adding to this, or subtracting from it, the distances which the support of the uniting wire may have moved along its horizontal division. Moreover, in order to exhibit the action of the wire alone, without the intervention of any other foreign force, we neutralize the action of the terrestrial magnetism upon the needle, by means of a strong magnet properly placed at a great distance, as we have explained above. These arrangements being made, we place the uniting wire successively at different distances, but so great that the times of oscillation of the needle by its influence may be always sensibly isochronous; and this we determine by experiment, counting with all possible care the number of seconds and half seconds employed by the needle in making a certain constant number of oscillations, for example ten, at each successive distance. Where more exactness is required we take a larger number. Then, since the isochronism of the oscillations allows us to consider the motion as produced by a force parallel to the direction of equilibrium in which the needle stops, it follows that the needle is precisely in the case of a pendulum, made to oscillate about a perpendicular in different latitudes, under the influence of different gravities; and therefore, with respect to the needle, as well as the pendulum, the intensities of the component forces, thus directed, are inversely as the squares of the times of the oscillations. If, therefore, we represent by F the particular imaginary force which would thus cause the needle to perform precisely ten oscillations in a second; and if, in another position of the uniting wire, the same number of oscillations takes place in a different number of seconds expressed by N, the