of force into an exact method for determining the intensity of a magnetic field at any point, in accordance with the following rule:

Of the indefinitely numerous lines of force which we may consider actually to exist, we shall suppose that at each place there are only as many per square centimetre as there are absolute units in the strength of the field; or, in other words, where each square centimetre is intersected perpendicularly by one line of force, we say that the field has unit strength, while if lines of force cut perpendicularly through the unit of area the strength of field is 5 cm. gr. sec.-'.

That this rule may be applicable, we must have some definite way of mapping out the lines of force. We shall describe in the next paragraph an ideal construction for the simplest kind of magnetic field, namely that of a point-pole, reserving till later the discussion of more complicated cases.

To represent the earth's gravitational field in these latitudes. in accordance with the above convention, we must draw through a horizontal area of 1 sq. cm., 981 vertical lines.

72. Number of lines of force proceeding from a pole of given strength.-From a pole of a thin bar magnet there are in reality an infinite number of lines of force, distributed impartially in all directions, and each following a rectilinear course. But, in accordance with the rule given in the last paragraph, we shall find it best to represent the field by means of a limited number of lines. We shall calculate how many lines of force must be supposed to issue from a pole whose strength is m absolute units.

In accordance with the law of COULOMB, this pole exerts upon a pole of strength + 1, placed at the distance r, the force m/r2. That is, at the distance r the field due to the pole m has the strength

This value of the field-intensity obtains over the surface of a sphere of radius r, that is, over an area 4πr2. Now at every place where the strength of field is , we have to think of 5 lines of force cutting through each square

centimetre of (perpendicular) surface. The spherical surface with which we are now concerned is thus intersected by x 4πr2 or 4πm lines of force. Since we have supposed that there are no magnetic poles present, besides that which we specially consider, no lines of force can originate or disappear on the way from the pole to the spherical surface; so that we must say:

From a pole whose strength is m cm.3 gr.1 sec. ̄1, the number of lines emerging is 4πm.

Since 3.14159

=

it follows that roughly about 12 times as many lines of force diverge from a pole as there are absolute units in the strength of the pole. If the pole is negative the lines will converge to it, and there terminate. And conversely if we have plotted the lines of force throughout the field in accordance with this rule, we can calculate at once the strength of the pole from the number of lines which pass, in one direction or the other, through any given area of the diagram; that is, from the capacity of the source or sink, as Maxwell terms it.

73. Deflection of a declination needle by a bar magnet.-If we bring near to a declination needle a bar magnet of moment M, in the first or second of the positions referred to in § 38, a deflection of the needle will be produced. The deflecting couple we may consider to arise from the action of the two pairs of magnetic poles upon one another, the resisting couple being due to the horizontal component H of the earth's magnetism. The mutual action of the poles may be calculated from COULOMB's law. The needle, being deflected, rests in such a position that all the forces acting upon it are in equilibrium. If the lengths of the needle and the bar magnet are small compared with the distance r between their centres, the deflection is approximately proportional to the inverse cube of the distance r. Supposing the needle to be deflected through an angle 6, and equating to zero the resultant of all the moments acting upon it, we obtain

3tan o

In the first position: M/H =
In the second position: M/H = 73 tan

These relations may be verified by means of a device of WILHELM WEBER. A straight groove is cut in a length of wood,

and is graduated. At the middle of the length of the groove is placed a compass needle, and on setting the apparatus so that the groove is either perpendicular to, or lying in, the magnetic meridian plane, the deflections corresponding to the first and second positions may be observed. The moment of the declination needle disappears from the calculation.

B.-Dynamical methods of measuring magnetic magnitudes; method of oscillations

Magnetic magnitudes may be measured not only by observing deflections when equilibrium has obtained, but by studying the oscillations which result when a body maintained in equilibrium by magnetic forces has been slightly disturbed.

74. Couples acting upon a magnetic needle in a uniform field. In a uniform field of strength 5, a magnetic needle free to turn about an axis perpendicular to the lines of force always sets itself in the direction of those lines. If m is numerically the strength of pole, then on each pole we have the force m dynes. If the needle is turned into a position at right angles to the direction of the lines of force, and if a is the distance between its poles, we have the force Hm acting at one pole on an arm of length a/2 and giving rise to a restoring moment Hm a/2, while an equal moment tending in the same direction of rotation arises from the force acting on the other pole. Thus altogether the moment tending to bring back the needle to its position of equilibrium is 2 5 m a/2=5 ma. Here ma M, the magnetic moment of the needle (§ 67). If the needle makes an angle

=

with the direction of the lines of force, there is only a component of the force m on each pole which contributes to the restoring moment, that component, namely, which is perpendicular to the length of the needle, while the equal and opposite components of force along the needle itself are in equilibrium with one another. At each end of the needle, then, there is a perpendicular component of force m sin; so that the moment tending to restore the needle to its position of rest is

SM sin &

This expression is similar to that which gives the restoring moment in the case of an ordinary pendulum which has been disturbed from its vertical position of rest. Here in place of we have the intensity of the earth's gravitational field at the place of observation, that is, the acceleration g; in place of the magnetic moment M we have the mass of the pendulum multiplied by the distance of its centre of gravity from the axis of suspension. A magnetic needle, when disturbed from its position of rest in a uniform field, must therefore execute oscillations exactly resembling those of a pendulum.

75. Comparison of field-intensities by the method of oscillations. One of the laws of the pendulum tells us that the square of the number of oscillations per unit of time is proportional to the intensity of the forces which act on the pendulum. If the magnetic moment M of the vibrating needle is not subject to variation, we may use the method of oscillations to compare the intensities at different parts of the same field.

By the number of vibrations we are to understand the number of whole vibrations, that is of transits through the position of equilibrium in a determinate sense. As an example, we will apply the method of oscillations to prove COULOMB's law of the inverse square of the distance.

Experiment 38.-A strongly magnetised steel bar, over a metre in length, is fixed in a vertical position, the direction of the magnetic meridian being marked out on the ground. In a horizontal plane, about or of the length of the bar below its upper end, a short magnetic needle suspended from a cocoon fibre is allowed to oscillate; the position chosen for the needle being magnetic north or south of the bar magnet according as this latter has its north-seeking or its south-seeking pole uppermost. The square of the number of oscillations executed by the needle in unit time is determined, as well as the distance r from the axis of the bar to the centre of the needle. Finally the bar magnet is removed, and the needle allowed to oscillate freely in the earth's magnetic field alone, the square of the number of oscillations per second in this case being a measure of the horizontal component of the field, while in the other observations this horizontal com

ponent was superadded to the field of the bar magnet. Accordingly we are to subtract the squared vibration frequency obtained in the last case from the corresponding values previously observed, and the remainders will be proportional to the values of magnetic force due to the bar, that is inversely proportional to the square of the distance r.

76. Quantities which determine the period of oscillation of a needle. The nature of an ordinary pendulum, so far as its oscillations are concerned, is completely determined when we know two quantities. One of these is the product M obtained on multiplying together the mass of the pendulum and the distance of its centre of gravity below the axis of suspension. The other (I) is the so-called moment of inertia, and depends on the nature and distribution of the matter composing the pendulum. It is equal to the sum of all the products obtained when every element of mass is multiplied by the square of its distance from the axis of suspension.

The expression for I involves measurements of mass, as well as those geometrical measurements which serve to determine the distances of the component particles from the axis of rotation. The moment of inertia of a rectangular prismatic bar of length and breadth b, about an axis through its centre at right angles to the plane of land b, is

12+62
12

P. cm.2 gr.;

where P is the mass in grams, which we suppose homogeneously distributed. For a cylindrical bar of length l and radius p, we have

with respect to an axis through the centre of the bar perpendicular to its length.

The period of oscillation of a suspended magnet also depends upon its moment of inertia; we shall show experimentally that the ratio of the quantity M to the moment of inertia I determines the period in question.