symmetry as a rotation about an axis. This is important evidence of the axial character of magnetic lines of force,' to which we have already alluded.

We shall find our conclusions verified if we consider what takes place on reflection of the whole field at a mirror plane, passing through the axis of the current conductor, and, consequently, perpendicular to the plane of the latter. The sense of the current, flowing perpendicularly to the mirror, is reversed; while if we suppose something to be simply flowing along the lines of force, its direction of flow is not reversed. On the other hand, a clockwise rotation in a plane perpendicular to the mirror will be changed by reflection into a counter-clockwise rotation. Thus the boundary phenomenon of the current along the conductor possesses the same type of symmetry as the bundle of lines of force passing through it when, and only when, the lines are supposed to be axial in character.

A magnetic field is not reversed in direction by reflection in a plane mirror perpendicular to its lines of force; on the other hand, it is reversed in direction by reflection in a plane passing through its lines of force.

CURIE expresses this result very suggestively by saying that magnetic lines of force possess the symmetry of an axis of rotation, not the symmetry of an arrow. If we make use of the symbols which are ordinarily employed in crystallography, the expression for the symmetry of a magnetic field of force will be

155. The galvanic current as a bundle of lines of force.However far we follow the course of a conductor conveying a current, we find it associated with magnetic lines of force. Hence we conclude that:

Every galvanic current is equivalent to a bundle of lines of force. So far as external magnetic effects are concerned, every circuit in which a current is flowing may be replaced by an equivalent magnetic shell, whose boundary coincides with the path of the current. Thus the current as a whole,

that is to say, the entire circuit round which the current flows, may be regarded as equivalent to a magnet.

AMPÈRE developed this equivalence from the opposite point of view, his aim being to explain all magnetic phenomena in terms of currents. Since a current flowing round a circuit may replace an equivalent magnetic shell, we may suppose a magnet to be built up of currents flowing round minute (molecular) circuits. This conception was successfully developed; it was the first explanation of magnetic phenomena in terms of the galvanic current. We must examine the relations between currents and magnets somewhat closely, as we shall have to make use of the converse transformation.

Although the behaviour of currents and magnetic shells is in many respects identical, yet there are fundamental distinctions which must not be overlooked; and these we now proceed to investigate.

D.-Differences between the fields of magnets and of galvanic

currents

Without entering here into the more elaborate mathematical treatment of the question, we may direct our attention to certain peculiarities of the magnetic fields of currents and magnets, and the differences between them, adopting a mode of investigation which is diagrammatic rather than analytical. The field,' in our sense, will be the region external to any structure which exerts magnetic influence. Although the disposition of the lines of force which thread through a current circuit is in many respects similar to that observed in the case of a magnetic shell, yet there is one way in which the two fields differ fundamentally from one another the field of the current continues undisturbed through the region embraced by the circuit; we can follow the course of the lines of force so as to pass from one side of the circuit to the other without being in any way hindered by the conductor itself (compare fig. 53); on the other hand, in the case of the magnetic shell we are stopped by

the material of the shell, however thin it may be. Thus the difference appertains to the spacial distribution of the field.

156. Simply- and multiply-connected regions.-We shall now consider two magnetic fields; on the one hand, the field or space external to a plate of magnetic material (magnetised transversely) and, on the other hand, the field of a wire bent into a closed circuit, and traversed by a current. In order to make the difference in question as evident as possible, let us think of each field as bounded by an outer envelope which does not intersect the magnetic shell or circuit—for example, the walls, floor and ceiling of the room. If across the region we extend a diaphragm, which reaches as far as the outer envelope in all directions (its bounding line lying entirely upon the envelope), the region, if of ordinary form, will be thus divided into two entirely distinct and separate parts, each of which is completely bounded by a portion of the original envelope and one side of the interposed diaphragm. Such regions are called simply connected' (LISTING, Census räumlicher Gebilde). If two or more diaphragms are needed to effect this partition of the region, the latter is said to be doubly connected' or 'multiply connected.' The magnetic field due to a current in a circuit, in contradistinction to the field of an ordinary magnet, occupies a doubly-connected region. To realise this we must next consider:

(a) The spacial connection of the free field of an ordinary magnet, choosing for the sake of closer similarity a flat magnet or magnetic shell L (fig. 55a). Theoretically, the field extends from the external surface of the magnet L to infinity; practically, we may suppose it limited by some easily specified envelope, such as the walls of the room (in which term we include the floor and ceiling), represented in the figure by A. The region occupied by the field has thus the form of a hollow shell, bounded externally by the envelope A, and internally by the surface of L. A diaphragm SS of arbitrary form is now stretched across this region, the bounding line or lines of the diaphragm lying entirely flecting

on the bounding surface or surfaces of the region. SS extends from A inwards, and must terminate where it intersects the surface of L. In the figure the intersections with

these surfaces are indicated by dotted lines.

FIG. 55b

The diaphragm itself has the form of a flat ring, somewhat as in fig. 55b. It divides the region occupied by the field into two entirely distinct and separate regions R, and R2 (fig. 55a). We have now to consider:

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(b) The region external to a closed circuit (current conductor).

A piece of rubber tube may serve as an illustrative model; its two ends being joined together by fitting them over a short wooden cylinder of suitable diameter.

The field extends from the surface G of the conductor (fig. 56a) to the walls of the room A. But now, however

we may dispose the diaphragm SS, there remains a free communication between the regions R, and R2, through the aperture of the ring (compare figs. 56a and 56b, where different forms of the diaphragm SS and the bounding surfaces A and G are represented in section).

The bounding surfaces are, on the one hand, the surface of the ring G (including, of course, the portion i next the aperture as well as the more exterior portion a), and, on the other hand, the outer envelope A. If we extend a diaphragm from the envelope A to the bounding surface of G, we have once more a ring-shaped diaphragm as indicated in fig. 56c. The outer bounding line of the diaphragm lies on A, the inner boundary upon G. Since the surface S is to be a

simple surface which nowhere intersects itself, it is evidently impossible by means of a single diaphragm of this kind to divide the space external to G into two entirely distinct regions. The partition may, however, be effected by introducing a second diaphragm, as indicated in fig. 56d. Here R1 and R, are entirely separated from one another, each of these two regions being completely enclosed by a portion of the bounding surfaces A and G, and by one side of each of the two diaphragms, S1, S.

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The field due to a current flowing in a circuit occupies a region in which there is a ring-shaped cavity (the space filled by the conductor), and as we have seen such a region is doubly connected.

It is this spacial property of the electro-magnetic field which renders it possible for the lines of force to bend round so as to form closed curves embracing the conductor, and lying entirely in the field, where no sources or sinks exist; neither diverging from poles (divergence = 0) nor converging to them (convergence = 0). In § 169 we shall have occasion to speak of other peculiarities of the electro-magnetic field arising from these spacial properties.