of force embrace only the separate turns of the conductor, the outer lines coalesce with those belonging to neighbouring turns. As, in the case of different currents flowing in like directions, these lines of force tend to draw the two conductors together, it follows that there will be a similar electrodynamic action between the neighbouring turns of the solenoid which are parallel to one another, and traversed by currents in the same sense. The whole helix has a tendency to become shorter, the separate turns of wire appearing to attract one another. This effect is shown very strikingly in ROGET's spiral.

Experiment 69.-A long helix is made of thin copper wire, and consists of a single layer of turns wound as closely together as possible. The helix is hung up by one end, the lower end being loaded with a small weight. The end of the copper wire is amalgamated and dips vertically downward into a mercury cup, which is joined to one of a pair of terminals, the other terminal being joined to the upper end of the helix. As soon as the circuit of the current is closed, the turns of the helix become drawn together. Since the uppermost turn is fixed, the lower turns are thus lifted up, so that the amalgamated end of the wire is raised out of the mercury; the lines of force and the pull arising from them cease to exist, the helix falls back under the action of gravity, and the end of the wire becomes once more immersed. Once more the phenomenon of the current is repeated, and the contractile stress acts along the direction. of the field, the same cycle of changes occurring again and again. The end of the spring is thus alternately immersed in the mercury and withdrawn from it, its movements being accompanied by the rhythmical crackling of the vivid sparks which occur on breaking the circuit.

200. Action of different parts of the same current-conductor upon one another. The older theories sought to analyse all observed actions between finite systems into actions between infinitesimal elements of those systems. Thus the forces exerted by current-conductors upon one another were resolved into those arising from so-called current-elements,

to which was ascribed a mutual action according to a definite law, the action between finitely extended conductors being calculated from these elementary actions by addition (integration). If however we confine our attention to the mutual actions of finite systems, such as alone occur in nature, we may apply our doctrines to the elucidation of certain experiments which were designed to show the existence of such actions of current-elements on the remaining portions of a circuit.

Experiment 70 (De la Rives' floating hoop).—Two parallel grooves are cut in a board, at a small distance apart, and are filled with mercury. The grooves must be of such a breadth that in the middle of the reflecting mercury surface there is no appreciable capillary curvature. Currentconductors are connected to these two grooves, which are also placed in conductive connection by means of a hoop of copper wire, this latter having two long horizontal parallel limbs which rest upon the mercury surfaces. The entire hoop is covered with an insulating layer of sealing-wax, except at the extreme ends of the two limbs which are bent downwards, and expose an uncoated copper surface to the mercury. Thus it is through these extremities that the current flows.

When the current circulates, the hoop swims away, in whichever sense the current may be. Thus it would seem as if the portion of the current in the hoop were repelled by the portion in the fixed parts of the circuit.

According to our theory, however, the phenomenon is to be explained by the pressure across the lines of force which are embraced by the path of the current. Every such circuit embraces a number of lines of force determined by its shape and size, by the strength of the current, and by the nature of the surrounding medium, § 155. These lines of force press outwards against the path of the current (compare experiment 54, § 150). If anywhere along this path there is a place where the conductor can easily give way (like the hoop in the present case) the movable portion will be driven outwards by the pressure of the lines of force.

Thus it might appear as if this part of the conductor were repelled by the remainder.

201. Behaviour of crossed currents.-If the two conductors are not parallel, but make any angle with one another, their mutual action may be most simply inferred from a consideration of the lines of force in each of four dihedral regions, into which we may suppose the field divided. The rectilinear conductors in general cross one another obliquely, the shortest line which can be drawn from one to the other being perpendicular to both. Let two planes be drawn, each passing through this common perpendicular and one of the conductors. Thus the field is divided into four dihedral compartments, in each of which the direction of the lines of force is determined by the directions of the currents. Let us fix our attention upon volume elements in outlying parts of the dihedral regions, where it is easier to follow the composition of the lines of force belonging to the two systems. In the two compartments where the currents are flowing both towards or both away from the point of crossing, the disposition of the lines of force will be very similar to that in the biaxal field of parallel same-way directed currents; hence these portions of the conductors will be urged together. On the other hand, the disposition of the lines of force in the other two dihedral regions will be similar to that in the field of parallel oppositely-directed currents, so that the corresponding portions of the conductors will be urged asunder. In other words, the pressure due to the lines of force seeks to widen the dihedral angle of these regions, thus adding to the effect in the two other compartments of the field, where angle tends to become smaller. The total effect is thus a tendency to bring the crossed conductors into parallelism.

These conclusions may be verified by experiments, performed, for example, with Ampère's table. The phenomenon, then, is also seen to be a consequence of the disposition of lines of force, no assumption being needed as to attractions or repulsions between elementary portions of the current-conductors.

U

202. Currents in perpendicular directions; continuous rotation of one current-conductor under the influence of another. Let there be a straight conductor, such as a channel filled with mercury, in connection with which is another straight conductor in a perpendicular direction; for example, a vertical copper wire, whose lower end dips into the mercury, and which is encircled by lines of force due to a current flowing in it. Let us consider the disposition of the lines of force in a plane passing through the two conductors. The annular lines of force due to these two conductors are perpendicular to the plane in question and to one another; in one of the angular compartments into which we may suppose the field divided (as in the last paragraph) the two sets of lines of force intersect the plane in the same direction, and here accordingly they mutually strengthen one another's effect, this being the case in the regions along whose boundaries one current flows towards the point of intersection, and the other current away from it. In the adjoining regions where both currents flow towards the point of intersection, or both away from it, the two sets of lines of force pass through our plane in opposite directions, so that their effects are antagonistic, and the pressure across their direction is correspondingly less. The vertical conductor must therefore experience a force perpendicular to its length, and urging it away from the region first mentioned. If it is free to move parallel to itself, like the wire dipping in the mercury channel, it will be driven along the other conductor, towards that side where the two currents have the same direction relatively to their point of intersection.

This cross-pressure may be applied to produce a continuous rotation of one current-conductor under the electrodynamic influence of another, just as under the influence of the field of a fixed magnet. A copper hoop is capable of rotating upon a fixed pivot, whose axis is vertical, the two limbs of the hoop being bent vertically downwards, and dipping into a circular channel of mercury, from which a current flows through them in all positions, either ascend

ing in both limbs, or descending in both. The circuit is completed through the pivot. Surrounding the vertical limbs of the hoop is another circuit consisting preferably of several turns, and having the plane of its windings horizontal. The mutual action of the two fields produces a continuous rotation of the copper hoop, in the one sense or the other, according to the manner in which the directions of the two currents are related to one another.

The action in the vertical tangent plane which we have considered outweighs any effect upon the more distant, bent portions of the current-path.

203. Magnitude of the electrodynamic action.-At each point of a biaxal or multiaxal magnetic field the resultant effect is to be found by compounding together the effects due to the separate conductors. But the field-intensity due to a current is at each point proportional to the currentstrength (§ 165). In a magnetic field formed by the superposition of two such fields [the field-intensity is the sum of two terms, each proportional to the strength of one of the currents. The stresses at each point are proportional, however, to the square of the corresponding field-intensity, and accordingly contain terms proportional to the product of the two current-strengths. It is to these product terms' in the resultant stresses that the mutual electrodynamic action between the circuits is due]. The mechanical forces exerted upon the current-conductors in the resultant field depend upon the distribution of the lines of force, and especially upon their number; they are proportional to the product of the two current-strengths, or in the case of two portions of the same circuit acting upon one another, to the square of the current-strength.

W. Weber has established the accuracy of this result by a series of very careful experiments with coils conveying currents, one of the coils being bifilarly suspended (bifilar electrodynamometer).