" corresponding to one 2 and -μ, with distance

με

μ &c. near the ends

have at the two ends the poles μ, and another and separated by a length 7,; then Z between, and so on. The poles μ and will be the strongest; in fact, the strength of the pairs of poles ought, in a good magnet, to fall off very rapidly as we leave the ends and approach the centre.

So we have the moments 1, μal, &c. And the magnetic moment of the whole needle will be the sum of these, or will be m = m1 + m2 + m3 + &c.

where the first terms, m1 + m2 + &c. are the largest.

The complexity of this result is due to the difference between our ideal magnetisation and the imperfect manner in which we must needs perform it. The magnetisation is not effected uniformly all over and all through the bar; and hence there is not the complete internal neutralisation, with evident magnetism only at the ends, which would result from uniform magnetisation.

To sum up all these terms, 、 + m2 + &c., about each of which too little would be accurately known, were impracticable.

We have, however, other methods of finding m. We may (e.g.) suspend the needle in a uniform field of known strength H, and measure by mechanical means the moment (in terms of unit couple or unit moment) required to keep the needle perpendicular to the lines of force. Let this moment be found to be S.

In fact, we measure the magnetic moment m as a whole by observing the action of a known field upon it.

§ 11. Magnetic Curves.-Consider a bar magnet N S, and a point P in its field, and suppose a + unit pole to be placed there. This will be urged from N in the line N P; urged to S (by a weaker force if it be nearer to N than to S) in the line S P.

If we draw the parallelogram of forces we shall get a resultant force PR lying (in the case given in figure) nearer N P than S P in direction.

As the pole P moves, the component and resultant forces change continuously. And if P is always allowed to follow the

с

direction of the force urging it, then it will trace out a curve of force, called, in the subject of magnetism, a magnetic curve. The curve is such that at any point in it the tangent gives the direction of the force at this point in the field. Now if we put

n

R

FIG. i.

into the field a needle so small that we can consider its

two ends to be practically in the same part of the field, then the needle will be acted upon by a couple, and it will turn until it lies in a straight line with the lines of force. In no other position but this could there be equilibrium for the needle.

Hence, a 'small needle' will always point out the direction of the lines of force at the place it occupies.

Now iron filings do, by induction, become small magnetic needles when they are placed in a magnetic field. Hence, if we

place a magnet under a sheet of glass and scatter over the glass iron filings, tapping the glass to allow of arrangement, we shall find traced out for us many of these lines of force that lie in the plane of the glass.

The accompanying figures show us the general aspect of the curves thus obtained.

In fig. ii. we have respectively the cases of a bar-magnet; a horseshoe-magnet; two bar-magnets with unlike poles opposed ; and the same with similar poles opposed.

§ 12. Magnetic Induction takes place along the Lines of Force. It is along these lines of force that we must consider our molecules, regarded as small magnetic needles, urged to direct themselves.

Hence, if these molecules were perfectly free to move, any mass of iron would by the rearrangement of its molecules become magnetised along the lines of force of the inducing field.

But as the molecules are subject internally to mechanical constraint as well as to each other's action, the resulting magnetisation is determined by the combined influences, internal and external.

If the bar or needle lies so that its greatest length is along the lines of force, and if the form of the bar is symmetrical about this

offn

axis, then the magnetisation should be-whether weak or strongalong the lines of force.

But if the bar be unsymmetrically situated with respect to the lines of force, the resulting magnetisation may be quite unsymmetrical and be oblique to the lines of force.

CHAPTER III.

MAGNETIC MEASUREMENTS.

THE EARTH'S MAGNETISM.

§ 1. Coulomb's Torsion Balance.-In considering the subject of magnetic measurements we shall first describe the torsion balance. This instrument deserves notice as the earliest by which exact magnetic and electrostatic measurements were obtained, and

some study of it will be instructive. In practice it has, however, been now superseded by other instruments.

The figure represents one form of the instrument. A rectangular or cylindrical glass case is provided, either with a graduated scale round the sides, as here shown, or, what is better, with a plane mirror at the bottom, on which is marked a circle graduated in degrees.

In what follows we shall assume that the latter method of graduation has been adopted, and that the centre Above the mirror is sus

pended (when the instrument is used in magnetism) a magnetic needle ab, in such a way that its axis of suspension is immediately above the centre O of the graduated circle.

By looking down from above we can tell over what degree the needle is lying. By moving the eye until the needle and its image coincide, we avoid errors due to the fact that the apparent position of the needle over the scale varies according to our point of view.

The needle is suspended by a fine wire or thread of glass, so hung that its prolongation would pass through O, the centre of the graduated circle.

This thread is fixed at its upper end to a brass piece ed (called a torsion head) that caps the glass tube shown. This piece is so constructed that the thread can be twisted without being displaced laterally, and the angle of twist can be measured.

A sight of the instrument will make clear how this is contrived. Now to twist a wire or thread of glass without displacing it laterally requires a simple couple.

It can be proved experimentally that as long as the thread or wire is not permanently altered by a twist, the couple required to twist it is directly proportional to the angle of twist.

Or if we twist the wire through 10°, 20°, 30°, 170°, &c., we are exerting couples proportional to 10, 20, 30, 170, &c. We cannot measure the couples in absolute C.G.S. units unless we know more about the length, radius, and material of the thread. But we can, without this knowledge, compare couples. If the top of the wire be twisted through ° one way, and the bottom through a the other way, then the total angle of twist will be 0° + ao.

§ 2. Use of Torsion Balance at constant angle.'-We can use the torsion balance to compare the strength of magnetic fields and of magnetic poles; or even to measure them in C.G.S. units if we are acquainted with the 'constants' of the instrument we are using.

270

90°

180

In the accompanying diagram we are supposed to be looking down upon the whole instrument. The small circle represents the graduated circle of the torsion head; and, in the case given, the torsion wire has been twisted either through an angle a°, or through (n × 360° + a°), where n is some whole number.

S 180

FIG. i.

The large circle is the graduated circle on the mirror; and, in the case given, the needle has been deflected through an angle from zero.

The opposite directions of twist of the needle and of the