placed each at one of its extremities. The second method is that of oscillations; it consists in making a magnetised needle oscillate, and in deducing the intensity of the action to which it has been subjected from the comparison between the number of oscillations executed by the needle under the influence of this action, and the number it makes when withdrawn from it.

In both methods there is one element which must be taken into the account, and which does not exist in the measuring of electric forces; it is the directive force of the earth. Thus while, in reference to electricity, the torsion of the wire alone exerts an influence upon the resistance which the movable electrised body opposes to the action of another electrised body; it is not the same when a magnetised needle is in question, for then the force, with which this needle tends to obey the directive action of the earth, is added to the torsion, to oppose the action of an exterior attractive or repulsive force. Thus again, whilst the oscillations of the electrised body are made as if it were not electrised when there is no other body present, those of a steel needle are entirely different, according as this needle is magnetised or not; because, in the former case, the oscillations are influenced by terrestrial magnetism, and, in the latter, this influence does not exist. Let us see how, in each of the two methods, the influence may be appreciated.

In order to obtain the value when the torsion balance is made use of, we must begin by suspending an unmagnetised needle to the thread of the balance, and then turn the piece by which this thread is suspended until the 0° of torsion is in the plane of the magnetic meridian. The needle which had been suspended to the wire is then to be magnetised and to be replaced exactly in the same manner. It follows from this that when, in obedience to the directive action of the earth, the magnetised needle places itself in the magnetic meridian, the torsion of the thread of the balance is found to be null. The thread is now to be twisted by turning the upper piece, either in one direction or the other, so as to bring the north pole of the needle to the west or to the east: the result is the

same in both cases. We may be sure that so long as the angle of deviation which the needle is made to describe does not exceed 20°, it is proportional to the angle of torsion; that is to say, we may be sure that, after having twisted the wire to a certain angle, 35° for example, to make the needle deviate 1° to the east or the west, we must twist it to a double angle, viz., 70°, to produce a deviation of 2° in the same direction; to a triple angle, viz. 105°, for a deviation of 3o, and so on. The directive force of the earth, which tends to bring the needle back into the magnetic meridian, is therefore represented in each case by the angles of torsion, which maintain it at greater or less distances from this meridian; and, as the angles of torsion are proportional to the angular distances, the force itself is proportional to them. It is not rigorously to the angles, but to the sines of the angles of deviation, that the angle of torsion, and consequently the directive force is proportional, as is proved both by the observation made with more considerable deviations, and a simple calculation based upon the consideration of the forces by which the needle is solicited. But this same observation shows, as well as experience does, that when the angle does not exceed 20°, we may, without sensible error, take the angle instead of its sine; for the relation of the angle to the sine, up to 20°, does not exceed that of 1 to 1.02.

It only remains now, when we wish to measure magnetic forces with the torsion balance, to determine for the magnetised needle which is suspended to the thread, and which is termed the proof-needle, the angle of torsion necessary to make it deviate one degree from its natural position. This angle may vary with the torsion thread employed, with the needle that is suspended to it, and with the intensity of the terrestrial magnetism at the place of experiment. Coulomb found that at Paris, and with the needle he made use of, the angle was 35°. When, therefore, any force causes the needle to deviate from the 0° of torsion, and from the magnetic meridian with which it was made to coincide, as we have said, there are two forces that tend to bring it back, and the sum of which is equal, when equilibrium is established, to that which

tends to remove it: these two forces are, the one the torsion represented by the angle of torsion, the other the directive force of the earth. But, in order to add this to the other, we must estimate its value in torsion. Now this is an easy matter when once we know that each degree of deviation corresponds to 35° of torsion. Thus, the force that will maintain the needle at 10° of distance from 0°, will be first the torsion of 10° plus ten times 35°, namely, in all, a force equivalent to 360° of torsion.

In order to appreciate the influence of the directive force in the second method, we must employ the formula of the pendulum; the magnetised needle oscillates, in fact, like a pendulum, only gravity is here replaced by the equally attractive action exercised upon one of the poles of the needle by terrestrial magnetism. It follows that the intensity of the force is in the ratio of the square of the number of oscillations which take place in the same time. This conclusion supposes that the oscillations are made by a dipping needle placed in the direction of the force by which it is actuated, that is to say, in the magnetic meridian, and oscillating in this plane, exactly as the pendulum is placed in the direction of the force of gravity, and oscillates in a vertical plane. However, it is demonstrated that the same formula or law may be applied, without sensible error, to the case in which the needle is a declination needle oscillating in a horizontal plane. When, therefore, we wish to employ the method of oscillations, we must commence by carefully determining for the needle that we employ, and which is still the proof-needle, the number of oscillations it makes in a given time, by taking the precaution of removing from it every magnetised body, or body susceptible of being magnetised, such as iron, so that it may not be actuated by any other force than by that of terrestrial magnetism. It is also understood that experiments, in order to be comparable, should be made, not only with the same magnetised needle, but in the same place*, so that the intensity of the terrestrial magnetism may be constant.

*For the mathematic developments relating to the two methods, see note C.

Law of magnetic Attractions and Repulsions.

We shall begin by applying the methods that we have now unfolded to the determination of the law that magnetic attractions and repulsions obey according to distance.

For employing the former method, that which depends upon the use of the torsion balance, Coulomb had suspended to the torsion thread the same long steel needle in which the directive force of the terrestrial globe was represented by 35° of torsion for 1° of deviation from the magnetic meridian. A second magnetised needle, similar to the first, was placed vertically in the magnetic meridian, so as to act by its north pole upon the north pole of the proof-needle. The disposition of the needle was such, that the two points acting immediately upon each other, or which would have been the line of intersection of the two needles had they been in contact, were an inch from the extremities of each. These points were those of the maximum of the repulsive forces. The movable needle was driven immediately to a distance of 24° from the magnetic meridian, which gave, in order to produce equilibrium to the repulsive force at this distance of 24°, a force of torsion of 24° plus the directive force of the earth, equivalent to 24° x 35° of torsion; in all, 864°. The movable needle was then brought up by turning the upper piece, and it was found that, to bring it back to a distance of 17°, it would have been necessary to make the piece describe three circumferences, or to twist the thread 1080° at the upper The total force was therefore composed, 1st, of the 17° of torsion that the needle was distant from the 0° of torsion, its starting point; 2ndly, of the 1080° of torsion necessary to maintain it at the distance of 17°; 3rdly, of the 17° multiplied by 35, namely, 595° of torsion, which would represent the effect of the directive force of the earth. This makes in all 1692° of torsion to make equilibrium to the repulsive force at the distance of 17°. In order to make the needle attain to a distance of 12°, it would be necessary to turn the upper piece 5 circumferences, namely 2880°, which gives a total torsion of 12° + 2880° + 12° x 35 = 3312°.

part.

Thus the forces of torsion that respectively produce equilibrium with the repulsive forces are, at the distance of

The results are closely approximated to those which would be given by the law of-Inversely to the square of the distances for the intensity of the repulsive force. In fact, setting out from the force 3312°, the other, according to this law, would be

1650 and 828, instead of 1692 and 864, which are given by experiment. These differences are very slight if we consider that an error of a single degree on the observed position of the movable needle makes one of 35° for the force, since the directive force is 35° for each degree of deviation from the magnetic meridian. Besides, we shall remark that the mutual action of the two magnetised needles, not being all concentrated in two single points situated upon each of them, the variation in the distance establishes a variation in the relative position of the acting points, and that, in proportion as the distance increases, there are more points that may act mutually upon each other. Thus we find the force a little greater when the distance increases, which it ought not to be according to the law. We should operate in a similar manner, in order to demonstrate that the law exists equally for attractions; it would merely be necessary to place, opposed to each other, the poles of the two needles having contrary names, first having taken the precaution to place the movable needle, by means of torsion, at a considerable distance from the fixed needle.

We must now see how Coulomb employed the second method. The proof-needle was a steel wire weighing 57.89 grs. Troy, strongly magnetised, and suspended to a silk thread without torsion. This needle made fifteen oscillations per minute under the influence of the terrestrial magnetism. Coulomb then placed vertically in the plane of the magnetic