opposite to the mean intensity of the action at a distance of the cut within the space occupied by the ring itself. As, further, we have already seen that the air-gap itself can exert no action, and that this must proceed from the ends, we conclude that such a pair of ends acts at great distances proportionally to the value of the magnetisation. We have thus arrived at a conception which necessitates the closer investigation of the action at a distance of pairs of ends. We are, at the same time, led to a point of view which was formerly the general starting-point, and the one almost exclusively taken into consideration. A noteworthy reaction has, in some quarters, recently set in, which at once considers as inappropriate, antiquated, and useless the ideas of magnetic fluids, of poles, attractions, and Coulomb's law, &c., which were peculiar to that mode of view. We shall proceed most safely in this respect if we take a middle course, and endeavour, as far as possible, to combine and utilise the advantages of both views. § 19. Action at a Distance of a Single End.-Let us imagine the ring cut through, and then stretched out so that it has the form of a long straight bar, and thus its ends are at the greatest distance possible. In the divided ring we assumed that the coiling was uniform, notwithstanding the gap. We shall here make the same assumption, and imagine a uniformly wound closed coil, like that in fig. 6, which does not produce of itself an external magnetic field, but within the windings produces a uniform field in the direction of the feathered arrows. Magnetisation in the same direction is then produced in the ferromagnetic bar, as can be proved by means of secondary coils. closely wound round it. Owing to the action of the ends a magnetic field will now be formed in the surrounding space. We will now investigate this somewhat more closely, by means of an exploring coil as in § 2. If now we investigate the space near one end, N for example, where its own influence decidedly preponderates, we shall find that the field has everywhere a radial direction, and from the end outwards, as shown by the unfeathered arrows. Its numerical value is inversely proportional to the square of the distance from 1 Compare for instance, Prof. Silv. Thompson, Cantor Lectures on the Electromagnet, London, 1890. ACTION AT A DISTANCE OF A SINGLE END 27 the ends as long as this is small compared with the length of the bar. Near the other end the condition is the same, except that the radial field is directed inwards at the end, and is again represented by plain arrows. The latter (S) is usually called the negative, and the former (N) the positive, end.' The positive direction of magnetisation in ferromagnetic substances is always from the negative towards the positive end. The intensity of the field near the ends increases proportionally to the value of magnetisation, as we proved with the divided ring, and is also proportional to the section S of the bar. It is independent of the length, provided this is considerable in comparison with the cross section, as is always supposed to be N FIG. 6 the case. The absolute value of the intensity of the field is therefore given by the equation where r is the distance from the end. Accordingly the power of the ends to exert actions at a distance depends upon the value of the product (IS), which may be called the magnetic strength of the bar.2 With a freely suspended bar the positive end sets nearly north, the negative south. The positive end is by most writers called the north pole, the negative S, the south pole. (See Maxwell, Treatise, vol. 2, § 393.) 2 Sir W. Thomson, Reprint of Papers on Electricity and Magnetism, p. 354, § 454. § 20. General Remarks about the Law of Action between Points. The relation expressed by equation (2) is a somewhat modified conception of what is called Coulomb's law,' to which we shall afterwards return. An essential condition for the validity of that law is, that the end of the bar is regarded as a point, and that, therefore, its dimensions are infinitely small compared with the distance r; the latter, again, is supposed to be small compared with the length of the bar. The experimental demonstration of the relations in question can therefore only be made with very long thin bars. We may here observe that Coulomb's law is by no means a specific magnetic law. It is only a special case of the perfectly general, purely geometrical, law which governs all actions that proceed and are propagated to a distance from points, or rather from centres which may be regarded as points. These actions all diminish as the distance increases, in such a way that their intensity is inversely proportional to 2, and for this simple geometrical reason that the surface of a sphere is proportional to the square of the radius. As the total action, which remains constant, has to be distributed over continually increasing surfaces of concentric spherical shells, the above law of the decrease of its intensity, that is of the action per unit surface, follows at once. Among the known examples of this general relation are the law of gravitation, Coulomb's electrostatic law, the photometric law for the decrease with the distance, of the intensity of light from luminous points. In all these cases the assumption of points as centres is the essential basis for the application of the (1/r2) law of action between points or of inverse squares. A gravitating, an electrified, or a luminous infinitely long straight line, on the contrary, does not act inversely as the square of the distance, but inversely as the distance itself; and here again, for the simple reason that now the surface of a cylinder of given length is proportional to its radius. A further example of this general (1/r) law of the action between lines is Biot and Savart's law of the electromagnetic action of long straight portions of conductors, which has been already discussed in § 5. The action, finally, of a gravitating, electrified, or luminous infinite plane does not depend on the distance; it may, in ATTRACTION OR REPULSION BETWEEN THE ENDS 29 fact, be considered as enclosed by two planes, also infinite, the surface of which is obviously independent of the distance. For the sake of a general expression, we may also, in this case, speak of a (1/*) law of action between planes. § 21. Attraction or Repulsion between the Ends.-The end of a magnet not only produces a field in its vicinity, but it is influenced by an existing extraneous field in such a manner that a mechanical force is exerted on it. This force has either the same or the opposite direction, according as that part of the original field where the end is placed is positive or negative. Its numerical value is equal in absolute measure to the product of the intensity of the field into the magnetic strength (38) of the end (§ 19): 127 If, as a particular case, the field in question radiates from another magnetic end, the resultant action may obviously be explained by saying that according as their signs are the same or opposite there is a repulsion or an attraction, F between the ends, which is the direction of the line joining them, directly proportional to the product of their magnetic strengths, and inversely proportional to the square of the distance. The mathematical formula for this law follows directly from equations (2) and (3), and is as follows: (4). F12 = (I S), (I S) 2 This is the original form of Coulomb's law,' which was discovered by Coulomb by experiments with the torsion balance. An exact confirmation was first obtained by the measurements which Gauss made for this purpose. As the existence of direct mechanical actions at a distance, which a verbal interpretation of this law entails, can scarcely be accepted by modern science, the tendency is to explain these apparent actions at a distance by stresses in the medium which transmits the action. We shall In the notation already mentioned (§ 19, note), it is usual to formulate Coulomb's law as follows. Like (unlike) poles, repel (attract) with a force which is directly proportional to the strength of the poles, and inversely proportional to the square of their distance. 2 Gauss, Intensitas vis Magn. Terrestr. ad Mensuram Absol. Revocata, § 21, Werke, vol. 5, p. 109; 2nd reprint, Göttingen, 1877. discuss these stresses in the gap more in detail in articles §§ 101-110. The introduction of this conception, besides affording a more satisfactory theoretical explanation of the facts, has the advantage of providing a far more suitable basis for most practical problems than Coulomb's law does. The latter, indeed, still gives the simplest representation of the mechanical actions in all cases in which it is a question of the reciprocal influence of a small number of ends of bars, as is still the case in many of the usual experimental methods. § 22. Action at a Distance of a Pair of Ends.—We will now return to the action at a distance of our bar magnet, and consider this at distances which are great compared with the length of the bar; the numerical value and direction of the magnetic field is obtained for all points by the superposition of the components due to the two ends as given from Coulomb's law. Two special cases are to be distinguished, the proof of which need not here be gone into. In the first place, the intensity of the field at a point on the prolongation of the geometrical axis of the bar is directed along the axis, and its value is given by the equation in which L is the length of the bar, D the distance of the given point from the centre of the bar. Secondly, at points in the equatorial plane of the bar, traced by the line EE, in fig. 6, the value is where the direction is still parallel to the axis. We thus see how the length of a bar now enters as a factor into the equations, and accordingly (3 S L) and not (38) is the measure of its power to produce a field at distant points. But SL is the volume of the bar, and therefore the product of the magnetisation into the volume is the determining quantity. In accordance with the usual practice we shall call this the magnetic moment of the bar, and denote it by M; it is analogous to the corresponding quantity for coils, which is also thus designated (§ 6). If the action at a distance is used to measure |