can turn freely in a hole drilled through the fixed support H. Weights may be hung from the lower end e of the axle A. Through the holes in the upper disc liquid is poured into the drum, mercury being used if the drum is very thick-walled, though otherwise, especially when the drum is of greater size (12 cm. high, 10 cm. in diameter), glycerine will be better.

If the cylindrical element G, filled with fluid (fig. 36), be set in rapid rotation about its axis, its elastic walls become bulged out laterally. At the same time the length of the drum will be diminished, as if its two ends attracted one another. The tendency to contract is so great that a considerable load may be attached to e, and will be lifted up. There is also a pressure exerted on the walls of the vessel, in directions perpendicular to the axis. If we imagine a large number of such rotating cylinders to be closely packed together, they will

mutually hinder one another from undergoing a change of shape, and we shall have a mechanical representation of the tension along the lines of force and the pressure across them.

This model furnishes an instructive example of the way in which a distribution of motion may give rise to apparent actions at a distance, the motions themselves being hidden from our observation. It might be called a gyrostatic model of a tube of force. The kinetic energy of its rotation is at each instant proportional to the angular velocity q (see also below, under D), and

from § 119 we know that the energy per unit volume of a magnetic field is proportional to the square of , so that we may consider the angular velocity q in our mechanical model as the analogue of the field intensity

.

126. Explanation of the mechanical forces due to magnets, in accordance with the hypothesis of permanent motions.-We must conceive of a magnetic field as being completely filled with rotatory motion. That such a distribution of motion is possible without the constituent motions disturbing one another we know from the phenomena of vortices (see also below, under C).

If two fields are superposed upon one another, the tensions along the lines of force, and the pressures across them, increase with the angular velocities. If the body giving rise to one field is fixed, while that giving rise to the other field is movable, the latter body will be displaced by the forces acting upon it.

(J)

(S)

FIG. 37

(a) Behaviour of a north pole in a magnetic field.-Fig. 37 represents a section taken through any given fixed field of force, the positive direction along each line of force being indicated by the arrows attached to it. We must accordingly suppose the lines to be proceeding from a north pole to the left, towards a south pole to the right. Let a north pole n be placed in the field; it behaves as a source of new lines of force, which spread

out from it in all directions indifferently. Eight of these lines are indicated in the figure, that marked v being coincident in direction with the fixed field. In the case of this line the magnetic effects are increased by the superposition of the two fields, so that there is a corresponding increase in the energy of the field and in the tension along the lines of force. In the case of the line h, the field due to the pole is opposed in direction to the original field, so that there is a corresponding diminution of energy, and decrease in the tension. Hence the pole will be urged in the direction of v -that is, in the positive direction of the lines of force.

A north magnetic pole, when placed in a magnetic field, experiences a force in the positive direction of the lines of force.

The effect is increased by the lines of force which pass out in oblique directions from the pole. All lines which are directed forward add something to the tension along the lines of force of the field, while with lines whose general direction is backwards, the opposite is the case (cf. fig. 30). In fig. 37 the direction in which the pole tends to move is indicated by a short thick arrow; if motion in that direction is prevented by some constraint, this latter experiences a thrust, similar to the thrust exerted by a mass placed in the earth's gravitational field and not allowed to fall.

(B) Behaviour of a south pole in a magnetic field.-If a south pole s (fig. 38) be introduced into the same fixed field, the effects observed will be exactly the opposite of the foregoing. For a south pole behaves as a sink towards the lines of force, lines converging towards it and there ending, instead of diverging from it. This is indicated in fig. 38 by the arrows on the eight lines of force which are directed towards the pole, and represent its field. As before we have to distinguish between a region in front of the pole and one behind it, considered in relation to the lines of force of the original field which run from N towards S. The lines proceeding from h towards the pole increase the intensity in their part of the field, giving rise to greater energy per unit volume, and correspondingly greater tension. In the case of v, the opposite takes place, with the result that the pole is urged backwards, towards h.

A south magnetic pole, when placed in a magnetic field, experiences a force against the positive direction of the lines of force.

In fig. 38 the direction in which the pole tends to move is indicated by a short thick arrow.

FIG. 38

127. Kinematic representation of tubes of force.-It might appear difficult, from the infinity of rotatory motions in a magnetic field, to obtain a clear mental picture of the process. If, however, we confine our attention to a single tube of force, all the particles at its bounding surface will be moving in the same direction of rotation. If any such particle, in the course of its rotation about its line of force, passes into the interior of the tube, its place on the surface is immediately filled by a particle of the same kind, moving in the same manner. The tube of force, as we conceive it, is a distribution of motion taking place always in the same sense, the kinematical conception being of a simple kind.

Suitable models of tubes of force may be made from pieces of white rubber tubing (fig. 39), a series of arrows being drawn round at equal intervals to represent the sense of the rotatory motion, and arrows along the length of the tube to indicate the direction of the magnetic field intensity. The two sets of arrows must have their directions related to one another in the manner determined by their convention of § 121. The tubes may be rendered less flexible by threading soft copper wires through them,

(S)

and they may then be bent into any desired form. If we wish to make a tube returning into itself, the two ends may be kept

FIG. 39

together by fitting them over a short cylindrical piece of wood, of suitable diameter.

128. Dynamical models of several important forms of magnets. By means of the model of a tube of force described in the last paragraph, we shall be able to picture to ourselves the rotatory motion assumed to exist in the field of a magnet.

(a) Model of a bar-magnet, fig. 40. The magnet is represented by a wooden bar N S, supported in a horizontal or vertical position upon a stand H, the end N (north-pointing end) being painted red, and the end S (south-pointing end), blue, while J represents the indifferent zone. At each end of the bar six holes are bored, equally spaced out from one another, and into these holes are stuck as many thick copper wires, enclosed by rubber