tion of equilibrium of such a magnet depends upon the relation of the Earth's magnetic force to the deflecting force; and if the deflecting force be constant, the changes of position are connected with the changes of the magnetic force by a simple relation. This principle was first applied to the determination of the diurnal variations of the horizontal magnetic intensity by Professor Christie. A pair of bar magnets were placed in the magnetic meridian passing through the centre of the suspended magnet, one on either side, and with their north ends turned to the south. The suspended magnet was thereby deflected from the magnetic meridian; the magnitude of the deflection depending upon the ratio of the two equilibrating forces.

The torsion of a suspending wire may be employed with advantage, as the deflecting force, instead of the attraction and repulsion of fixed magnets. But the best mode of producing the desired deflection is to suspend the magnet by two equidistant threads, or wires, by the rotation of the upper extremities of which, round their middle point, the magnet may be forced to take up a position inclined at any required angle to the magnetic meridian. This arrangement, which is denominated the bifilar suspension, is due to Sir W. Snow Harris, who substituted it for that of the ordinary balance of torsion in his electrical researches. It was first applied by Gauss to the measurement of the changes of the magnetic force; and the instrument which he devised for that purpose has thence been called the bifilar magnetometer.

(120) The bifilar magnetometer is a magnet bar, suspended horizontally by two equidistant wires, and maintained by the rotation of their upper extremities in a position at right angles to the magnetic meridian.

It is manifest that the weight of the magnet, so suspended, tends to bring it into the position in which the two wires are

in the same plane throughout. The Earth's magnetic force, on the other hand, draws the bar towards the magnetic meridian; and it will consequently rest in the position in which the moments of these opposing forces are equal. Let us in the first place consider the directive force due to the resolved part of the weight.

The line connecting the upper extremities of the wires. being turned through any angle, the wires are no longer vertical. The weight is consequently raised; and its tendency to descend will engender a directive force, by which the line joining the lower extremities of the wires will be urged towards the vertical plane passing through the upper. Let half the weight be conceived to be applied to each of the former points; and let each half be resolved into two forces, one in the direction of the wire itself, and the other horizontal. The former of these is destroyed by the reaction of the fixed point; the latter is equal to

W tan i;

W being the weight of the magnet and its appendages, and i the inclination of the wires to the vertical. The effective part of this force is Wtan i cosv, v being the angle formed by the vertical planes passing through the bearing points above and below. Hence the total moment of the directive force due to the resolved part of the weight is

Wa tan i cosv,

a being half the interval of the wires. But we have

7 being the length of the wires. And / being always great in comparison with a, the angle i is always small, and we may

substitute its sine for its tangent. Hence the total moment of the force due to the weight is

a2
W sin v.
7

(121) The directive force arising from the resolved part of the weight being found, it is easy to form the equation of equilibrium. For this force is opposed by that which the Earth's magnetism exerts upon the bar, i. e. by the force mX, in which X denotes the horizontal component of the Earth's magnetic force, and m the moment of free magnetism of the bar. And the moment of this force to turn the bar is

mX sin u,

u being the angle which the magnetic axis of the bar makes with the magnetic meridian. The equation of equilibrium, therefore, is

mX sin u = Wa sin v.
si

and when the bar is brought into the position perpendicular to the magnetic meridian, in which position the Earth's magnetic force acts with the greatest mechanical advantage, we have

(122) As all the quantities involved in the second member of the preceding equation may be known by direct measurement, we can deduce in this way the value of the product of the Earth's magnetic force into the moment of free magnetism of the bar, which is usually obtained by experiments of vibration, and so employ the instrument in measures of absolute force. The experimental difficulty in such a process would consist in the determination of the quantity a, which should be known to a very small fractional part of its actual value.

This difficulty has been in some degree overcome by the measuring apparatus connected with the suspension, which serves to determine the interval of the wires, at their upper extremities, to the zoooth of an inch. But for many reasons this method is practically inferior to the ordinary process. One of the chief causes of this inferiority is, that the elasticity of the suspending wires-a force not easily valued-conspires with the directive force arising from the weight in determining the position of equilibrium; and if, to remedy this, parallel fibres of silk be adopted for the suspension, we should be involved in the uncertainty of the position of the resultant of these parallel strains, in determining the interval of the suspending threads.

(123) The chief use of the instrument is to observe the variations of the force. The magnet, it has been shown, is acted on by two forces, and rests in the position in which their moments are equal. But one of these forces being variable, the position of equilibrium must vary likewise; and the variations of angle are connected with the variations of force. Differentiating the equation of Art. (121) with respect to X, m, and v, and dividing by the equation itself, we have

the angle Av being expressed in parts of radius.

The angle Av is observed by means of a collimator and fixed telescope, in the same manner as the changes of declination. Let n denote the actual reading of the scale of the collimator, and no the reading at some other epoch; then the corresponding variation of the angle is

A v = (n − no) a.

Again, if t and to denote the temperatures, in degrees of

Fahrenheit, at the two epochs; and 7 the relative change of the magnetic moment corresponding to one degree;

in which we have made, for abridgment, k = a cot v.

(124) The scale-coefficient of the bifilar magnetometer, k, may also be determined by observing the changes of position of the suspended magnet produced by a fixed magnet, whose force conspires with, or is opposed to, the force of the Earth. For, when the distance of this latter magnet is considerable in proportion to its length, the effect of the added force is the same as that produced by a small increase (or diminution) of the Earth's magnetic force; so that if the effect produced by a given added force be observed, in scale divisions of the instrument, the scale coefficient itself will be determined.

To apply this method, let a small magnet be placed in the same horizontal plane with the bifilar magnet, and with its axis in the right line passing through the centre of the latter, and perpendicular to its axis; and let n denote the number of scale divisions corresponding to the angle of deflection. Let the deflecting magnet be then transferred into a similar position with respect to the unifilar magnet, and be placed at the same distance from it; and let n' be the number of scale divisions in the produced deflection. Then, the distances being considerable in proportion to the length of the deflecting magnet, the scale-coefficient of the bifilar magnet will be expressed, in terms of that of the unifilar magnet, by the