The magnetic moment of the current is the product of its strength by the area of the circuit, or y 4, and the resolved part of this in the direction of the magnetizing force is y 4 cos 0, or, by (5), If there are n such molecules in unit of volume, and if their axes are distributed indifferently in all directions, then the average value of cos2 will be, and the intensity of magnetization of the substance will be nXA2 -.. L (7) (8) The magnetization of the substance is therefore in the opposite direction to the magnetizing force, or, in other words, the substance is diamagnetic. It is also exactly proportional to the magnetizing force, and does not tend to a finite limit, as in the case of ordinary magnetic induction. See Arts. 442, &c. 839.] If the directions of the axes of the molecular channels are arranged, not indifferently in all directions, but with a preponderating number in certain directions, then the sum A2 extended to all the molecules will have different values according to the direction of the line from which is measured, and the distribution of these values in different directions will be similar to the distribution of the values of moments of inertia about axes in different directions through the same point. Such a distribution will explain the magnetic phenomena related to axes in the body, described by Plücker, which Faraday has called Magne-crystallic phenomena. See Art. 435. 840.] Let us now consider what would be the effect, if, instead of the electric current being confined to a certain channel within the molecule, the whole molecule were supposed a perfect conductor. Let us begin with the case of a body the form of which is acyclic, that is to say, which is not in the form of a ring or perforated body, and let us suppose that this body is everywhere surrounded by a thin shell of perfectly conducting matter. We have proved in Art. 654, that a closed sheet of perfectly conducting matter of any form, originally free from currents, be 842.] PERFECTLY CONDUCTING MOLECULES. 423 comes, when exposed to external magnetic force, a current-sheet, the action of which on every point of the interior is such as to make the magnetic force zero. It may assist us in understanding this case if we observe that the distribution of magnetic force in the neighbourhood of such a body is similar to the distribution of velocity in an incompressible fluid in the neighbourhood of an impervious body of the same form. It is obvious that if other conducting shells are placed within the first, since they are not exposed to magnetic force, no currents will be excited in them. Hence, in a solid of perfectly conducting material, the effect of magnetic force is to generate a system of currents which are entirely confined to the surface of the body. 841.] If the conducting body is in the form of a sphere of radius r, its magnetic moment is -3 X, and if a number of such spheres are distributed in a medium, so that in unit of volume the volume of the conducting matter is k, then, by putting μ1=1, and μ0 in equation (17), Art. 314, we find the coefficient of magnetic permeability, Since the mathematical conception of perfectly conducting bodies leads to results exceedingly different from any phenomena which we can observe in ordinary conductors, let us pursue the subject somewhat further. 842.] Returning to the case of the conducting channel in the form of a closed curve of area A, as in Art. 836, we have, for the moment of the electromagnetic force tending to increase the angle 0, This force is positive or negative according as 0 is less or greater than a right angle. Hence the effect of magnetic force on a perfectly conducting channel tends to turn it with its axis at right angles to the line of magnetic force, that is, so that the plane of the channel becomes parallel to the lines of force. An effect of a similar kind may be observed by placing a penny or a copper ring between the poles of an electromagnet. At the instant that the magnet is excited the ring turns its plane towards the axial direction, but this force vanishes as soon as the currents are deadened by the resistance of the copper *. 843.] We have hitherto considered only the case in which the molecular currents are entirely excited by the external magnetic force. Let us next examine the bearing of Weber's theory of the magneto-electric induction of molecular currents on Ampère's theory of ordinary magnetism. According to Ampère and Weber, the molecular currents in magnetic substances are not excited by the external magnetic force, but are already there, and the molecule itself is acted on and deflected by the electromagnetic action of the magnetic force on the conducting circuit in which the current flows. When Ampère devised this hypothesis, the induction of electric currents was not known, and he made no hypothesis to account for the existence, or to determine the strength, of the molecular currents. We are now, however, bound to apply to these currents the same laws that Weber applied to his currents in diamagnetic molecules. We have only to suppose that the primitive value of the current y, when no magnetic force acts, is not zero but yo. The strength of the current when a magnetic force, X, acts on a molecular current of area 4, whose axis is inclined to the line of magnetic force, is and the moment of the couple tending to turn the molecule so as in the investigation in Art. 443, the equation of equilibrium becomes X sin 0-BX2 sin 0 cos 0 D sin (a−0). (17) The resolved part of the magnetic moment of the current in the direction of X is 845.] MODIFIED THEORY OF INDUCED MAGNETISM. 425 844.] These conditions differ from those in Weber's theory of magnetic induction by the terms involving the coefficient B. If BX is small compared with unity, the results will approximate to those of Weber's theory of magnetism. If BX is large compared with unity, the results will approximate to those of Weber's theory of diamagnetism. Now the greater yo, the primitive value of the molecular current, the smaller will B become, and if I is also large, this will also diminish B. Now if the current flows in a ring channel, the value where R is the radius of the mean line of of L depends on log R , the channel, and r that of its section. The smaller therefore the section of the channel compared with its area, the greater will be L, the coefficient of self-induction, and the more nearly will the phenomena agree with Weber's original theory. There will be this difference, however, that as X, the magnetizing force, increases, the temporary magnetic moment will not only reach a maximum, but will afterwards diminish as X increases. If it should ever be experimentally proved that the temporary magnetization of any substance first increases, and then diminishes. as the magnetizing force is continually increased, the evidence of the existence of these molecular currents would, I think, be raised almost to the rank of a demonstration. 845.] If the molecular currents in diamagnetic substances are confined to definite channels, and if the molecules are capable of being deflected like those of magnetic substances, then, as the magnetizing force increases, the diamagnetic polarity will always increase, but, when the force is great, not quite so fast as the magnetizing force. The small absolute value of the diamagnetic coefficient shews, however, that the deflecting force on each molecule must be small compared with that exerted on a magnetic molecule, so that any result due to this deflexion is not likely to be perceptible. If, on the other hand, the molecular currents in diamagnetic bodies are free to flow through the whole substance of the molecules, the diamagnetic polarity will be strictly proportional to the magnetizing force, and its amount will lead to a determination of the whole space occupied by the perfectly conducting masses, and, if we know the number of the molecules, to the determination of the size of each. CHAPTER XXIII. THEORIES OF ACTION AT A DISTANCE. On the Explanation of Ampère's Formula given by Gauss and Weber. 846.] The attraction between the elements ds and ds' of two circuits, carrying electric currents of intensity i and i', is, by Ampère's formula, or ii ds ds' (1) (2) the currents being estimated in electromagnetic units. See Art. 526. The quantities, whose meaning as they appear in these expressions we have now to interpret, are and the most obvious phenomenon in which to seek for an interpretation founded on a direct relation between the currents is the relative velocity of the electricity in the two elements. 847.] Let us therefore consider the relative motion of two particles, moving with constant velocities v and along the elements ds and ds' respectively. The square of the relative velocity of these particles is u2 = v2 - 2 vv′cos € + v22 ; and if we denote by r the distance between the particles, (3) |