M A G N E T I S M 243 vr ; peri- Van Rees, Lament, and Rothlauf favour this last formula ; but none of these experimenters give any proper account of the ends, which must be specially represented in all but those cases where the magnetic moment is zero there. Schaper finds that the results of experiment can be adequately represented by means of end distributions, and a lateral surface distribution following the law A + B^ 3 . See his paper above quoted, p. 242. l Carrying Power of a Magnet. It is obvious that the magnetization of a piece of iron must affect its force of " cohesion. The most familiar case is that of a magnet to which an armature is fitted. If the surfaces of the pole, and armature be carefully ground flat, so as to fit, we may regard the magnet and the armature as continuations of each other. The force of cohesion here is mainly due to the magnetism; and the force required to separate the two is called the " carrying power " of the magnet. To simplify the question, let us consider a cylindrical bar of section w, uniformly magnetized in the direction of its length with intensity I. Suppose the bar cut so that the normal to the plane of section makes an angle 6 with I, and let the surfaces of section be separated infinitely little, then the surface density of Poisson s distribution will be Icos$ on each surface. Assuming that the cohesion is caused solely by the attraction of these surface layers, we get for the carrying power P = 27rlcos0 x Icos$wsec$, i.e., P = 27rI 2 w cos 0. The carrying power is therefore greatest, viz., 27ro>r 2 , when the surface of the pole is perpendicular to the lines of magnetization. A great variety of experiments have been made on thi.s subject by Joule, Dub, Tyndall, Lament, and others, mostly, however, under circumstances that do not admit of the application of the above theory. For an account of what lias been done, the reader should consult Wiede- rnann s Galvanismus, ii. 425 sq. The most recent inves tigations on the subject will be found in the papers of i Rowland, quoted below, p. 255, and in papers by Stefan and Wassmuth in the Monatsberichte der Weiner Akademie for 1880 and 1882. 2 The facts are not so simple as the above theory would indicate ; but Wassmuth finds a modi fied form of it to agree sufficiently well with observation. MATHEMATICAL THEORY OF MAGNETIC INDUCTION. The two fundamental axioms of this theory are the following: 1. The induced magnetism in any element of a body depends merely on the magnitude and direction of the re sultant magnetic force (|)) at the element. 2. The magnetic moment induced by any force |5 is the resultant of the magnetic moments induced separately by any forces of which |p is the resultant. With reference to axiom 1 it is to be remarked that account must be taken of the physical condition of the body as to temperature, and so forth ; but it is implied that no account is to be taken of its magnetic state, except in so far as that affects the resultant magnetic force. In other words, it is asserted that the moment induced by any force does not depend upon any pre-existing magnetic moment in the element, and is the same whatever forces may have acted on the element previously. The full significance of these statements will be better appreciated when we come to consider the exceptions to them in case of strongly mag netic bodies. It should also be noticed that it is supposed 1 Various experimenters have attempted to determine the " indif ference zone" of magnets under different circumstances, i.e. , the line separating the positive and negative parts of the surface distribution. For information as to this and other matters under the present head omitted for want of space, see Wiedemann, Galvanismus, ii. 277, 356, 396, 401 ; and Lamout, Handbuch, 6, 27, 63, 64, 65. 2 Abstracted in Wied. BeiU. 1880 and 1882; see also Von Walten- hofen, Wien. Ber., 1870, and Siemens, Berl. Monatsber., 1881. that the body has reached a state of magnetic equilibrium, and that by whole resultant magnetic force is understood, not only that arising from the given inducing system, includ ing pre-existing magnetism in the body itself, but also that arising from induced magnetism. In the mathematical theory no distinction is drawn between the part of the induced magnetism which dis appears when the inducing force is removed, and that which remains. If anywhere we contemplate what happens after the removal of the force, it is assumed that all the induced magnetism disappears. This important restriction must be borne in mind in applying the results in practice. Axiom 2 enables us to assign at once the law connecting the components of induced magnetization A p B 1? Cj with the components a, /?, y of the resultant force. If r lt g 3 , p 2 be the components parallel to the three coordinate axes of "the induced magnetization caused by a unit resultant force parallel to the axis of x, then, by the axiom, the components of magnetization induced by a force a in the same direction will be ia, q.ja, p.p.; similarly, if p y r 2 , g l be the com ponents due to unit fore? parallel to the ?/ axis, then the components due to /3 will be p 3 /2, r 2 (3, q$ ; and -finally, ^ !?2> Pa r s ke components due to unit force parallel to z axis, the components due to y will be q. 2 y, p^y, r 3 y. Compounding all the^e, according to the axiom, we get General lawofin- du< tiou. Hence the most general expressions for the components of magnetization compatible with our axioms are three linear functions of the components of the resultant force. Here it is necessary to introduce a classification of bodies according to their magnetic properties. If equal, similar, and similarly situated elements cut Homo- from different parts of a body have identical magnetic pro- geneity perties, it is said to be " magnetically homogeneous," if not, f 11 ^ LC. i j )) Jietero- heterogeneous. ^ geneity. If equal and similar elements cut around the same point isotropy in different directions be identical in their magnetic proper- andreolo- ties, the body is said to be magnetically "isotropic "; if tr P v - not, " seolotropic." These are not cross classifications ; for a body (e.g., Iceland spar) may be eeolotropic and yet homogeneous, and it might be heterogeneous and yet isotropic. We must regard the coefficients p, q, r of (71) as belonging to a point of the body ; and we see that in a homogeneous body they will be the same for all points, whereas in a heterogeneous body they will vary from point to point, i.e., they will be functions continuous or discontinuous of the position of the point. In the case of an isotropic body it is obvious a priori Law of that the induced magnetization must be coincident in induc- direction with the resultant force ; the conditions for this I" ! 1 fc are that the coefficients p and q should all vanish, and that r i = r 2 = r s ~ K - The equations (71) thus reduce to A 1 = /ta, P>! = /C)8, C! = KV .... (72). In an a^olotropic body, on the other hand, the coefficients may be all different from zero and from one another ; but, as we shall see, at all events in the ideal case contemplated by the mathematical theory, the conservation of energy reduces the number of independent constants by three; while a proper choice of axes reduces it by three more ; so that the magnetic properties of any element of an aiolotropic body depend virtually on three independent constants. The theory here given is the generalization of Foisson s theory due to Sir William Thomson. It aims at giving the simplest possible exposition of the results of experiment with the fewest assumptions as to the molecular structure
of bodies. We first discuss specially a few of the cases