force is seen foreshortened in the figure, and when viewed from above follows the counter-clockwise direction of rotation. Thus,

the observer looking from the point of view of the figure sees the wire pass from left to right between him and the tubes, and from right to left behind the tubes.

C.-Conception of a current

From the consideration of the circuital axis of force as a centric system of magnetic lines of force, we are led to the conception of a current. We have to show how our mode of representing the phenomena leads to the conception of a progressive motion along the axis of force itself, that is, to a motion in the same sense as the elementary motions within the conductor. The most ordinary and familiar conception is of something moving along or even within the conductor, that is, in our example, along or within the wire forming the circuit. We say that there is a 'current' circulating in the wire. In order to have this, the circulation in the conductor, related in a definite manner to some determinate cyclic direction, we may make use of a transformation which is constantly employed in physics, and may appropriately be called AMPÈRE's transformation, having been first suggested by AMPÈRE. We shall consider it in relation to the model, fig. 53.

152. Ampère's transformation.-When a surface is bounded

by a line which nowhere intersects itself, and is completely filled with rotational motions all in the same sense (fig. 54), it may appear to a distant observer as if the motion, instead

FIG. 54

of being extended over the surface, were entirely confined to the boundary. For in the interior each element of surface filled with rotational motion is surrounded by elements in which the motion is similar. Hence it follows that in each portion of the bounding line of such an element motions in opposite senses are taking place; compare, for example, the adjacent surface-elements ƒ, and ƒ in the figure. The motions are of equal intensity; that is to say, if equal velocities correspond throughout to equal surface-elements, there will be a mutual destruction of the external influences, due to the contrary motions along a line separating any two such elements. It is only the outermost surface-elements which are effective; for along those portions of their bounding lines which form part of the boundary UU of the whole surface no compensation takes place, since on the further side of this boundary (indicated by a heavy line in the figure) there are no adjacent elements. The compensation is complete when the circuits within which the rotatory motions take place are very small, as in the present case, where we are dealing with the cross-sections of tubes of force. the whole system of motion, then, the only part which is effective is confined to the boundary of the surface. This causes it to appear not as if any motion took place within the boundary (i.e. over the surface contained by it), but as if something were moving along, or flowing in,' the boundary itself.

Of

This equivalence between a process extended over a bounded surface and a process confined to the boundary of that surface

is a special case of the mathematical proposition known as STOKES'S theorem. This theorem, by means of a general transformation, allows the sum of certain quantities connected with the separate elements of a surface, and satisfying certain conditions, to be expressed as the sum of certain other quantities, corresponding to the separate elements of the line bounding the surface. Thus STOKES's theorem enables us to prove rigorously that twice the sum of all the elements of a surface, each multiplied by a rotational velocity about an axis perpendicular to the element, is equal to the sum of the separate elements of the bounding line, each multiplied by the projection in its own direction of a certain linear velocity. This result is of great importance in connection with those theories which explain the field phenomena of magnets and currents by means of vortex motion. Ampère's transformation is the simplest case of the more general analytical theorem of which we have just made mention.

153. Direction of the current; nomenclature. The cyclic motions which the kinetic theory supposes to exist within the compass of a circuital axis of force are equivalent to a progressive motion along the conductor itself. This motion along the conductor we may call a current, speaking similarly of a current conductor and a current loop. In accordance with AMPÈRE'S transformation, the direction of this current must coincide with that of the elementary motions. adjacent to the conductor; but this is the direction which we have already chosen for the axis of force, namely, from K to Z or from + to, as shown in figs. 49, 52, 53.

We defined the positive sense along the conductor in this way when we gave distinctive names to the terminals.' We now find our convention to be justified, since Ampère's transformation shows that the sense from to along the axis of force corresponds to the direction of motion. along the boundary of a surface enclosed by the conductor. Accordingly, the direction in which a current flows is determined without ambiguity by its magnetic properties. Since we have adopted in Chapter II. a definite convention. as to the positive sense along the lines of force, the relation of a current to its magnetic field is completely determinate, nothing further remaining arbitrary.

Thus the whole system of conceptions is simply and definitely connected. It is only the nomenclature which presents any special difficulty. According to our view there is nothing flowing actually in the wire; there is, indeed, nothing taking place specially in the wire itself; all the processes with which we are concerned take place essentially in the surrounding space, that is to say, in the field. But all such terms as current, direction of the current, conductor, and so on (with which we cannot dispense, since we have no intention of introducing an entirely new nomenclature), are based upon the inadequate mechanical conception of a certain something flowing along the wire.

None of our conceptions have undergone so complete a change as have those relating to the phenomena of currents. There is, accordingly, much to be said for the many attempts made by English physicists to introduce entirely new names along with the new conceptions. Our object, however, is to show how known phenomena are to be interpreted and understood from the new standpoint, and this justifies us in retaining the old nomenclature.

154. The field of force of a circuital current, and its image by reflection. We shall now proceed to give the proof of the axial character of the magnetic field-intensity, to which we have already alluded in § 120. The current flowing along a conducting circuit is only the boundary phenomenon of the system of lines of force which thread through the circuit. The two phenomena are mutually related, so that either implies the other. In accordance with a general law of symmetry of unbounded media, causes and their effects must possess the same type of symmetry.

Let us suppose now that a closed circuit in which a current is flowing is placed in a plane, each side of which behaves as a mirror. Then all points in the space above the circuit will appear by reflection as points in the space below, and, conversely, all points below will by reflection appear above. We may likewise suppose all the processes occurring in the actual field to be reflected so as to give an image of the field, extending on both sides of the reflecting

surface. We have now to examine how the simple motions which we assume to exist in the field are affected by this process of reflection, and hence to draw some conclusion as to their symmetrical properties. The phenomenon of the current itself is evidently unchanged by reflection, because, according to our assumptions, it takes place in the plane of the mirror itself. The corresponding bundle of lines of force, on the other hand, is reversed in direction. If in the actual field the lines of force thread through the circuit from below upwards, in the reflected field they will thread through downwards from above. If, then, they possess the simplest kind of vectorial character, namely, that of a translation or flowing of something along them, a determinate direction. of this flow (from below upwards, or the reverse) cannot be unalterably associated with a determinate cyclic direction of the current; for this latter remains unchanged on reflection, while the direction of flow along the lines is reversed. But such a conclusion is contradicted by experience. For a determinate direction of the current always corresponds to a determinate direction of the lines of force, and, conversely, when the direction of the current is given, the magnetic polarity of the circuit is determinate (§ 151). Thus the vectorial process or arrangement which conditions the existence of the lines of force must be something different from a simple flow or translation along the lines.

Let us now apply the same process of reflection to rotational movements about the lines of force as axes. Rotations in planes parallel to the plane of the mirror are not reversed on reflection, as is immediately evident. Rotations about axes oblique to the mirror plane are transformed by reflection into rotations belonging to the same line of force (compare fig. 53). Motions associated with the lines of force in the actual field, and having the same cyclic direction as the current in the conductor, fulfil the same condition in the reflected field. Thus these rotational motions are affected in the same way by reflection as is the current itself. Whatever the physical constitution of a line of force may be, it is due to something possessing the same type of