232 MAGNETISM direction of I is everywhere the same, the whole potential is the component parallel to the magnetic axis of the body of the resultant force at P of a volume distribution p = I throughout its whole extent. This gives at once the expression of (29) for V. In the case of a uniformly magnetized sphere of radius a, the axis "being parallel to the axis of x, and the centre at the origin, we get, if r be the distance of P from the origin, for external points, V = $irl s a:/? 3 ....... (30); in other words, the external action is that of a magnet of infinitely small dimensions, having the same moment and axis, placed at the centre. For internal points V = fcrLe ....... (31); whence it appears that tlio magnetic force inside the sphere is con stant in magnitude and in direction, being opposite to the uniform magnetization, and equal to $vl. The potential of a uniformly magnetized ellipsoid may be simi larly treated. Let the origin be at the centre, and the axes along the principal diameters of the ellipsoid, whose lengths are 2a, 2b, 2c, and let I, m, n be the direction cosines of its magnetic axis. Consider first an external point. Then, 1 if " .__. , M=&c., N = &c. (3C), where a is the positive root of __~ (. * + ~ _ 1 a- + <f> b- + <j> c~ + <f> Ave have and where it must be remembered that L, M, N are functions of x, y, z, inasmuch as a is so. If (x, y, z) be an internal point, X, Y, Z are the components of the force due to a similar and similarly situated ellipsoid through (x, y, z). Let its axes be pa, pb, pc ; we now have where or, writing (34). We thus obtain for V, V = AL + BM7/ + CNs, ..... (35), wlicre L, M, N are now constants, which remain the same so long as the ratios of the axes remain unaltered. The components of the force inside the ellipsoid are a=-AL, 0--BM, 7=-CN . . . (36). The force is therefore uniform ; but its direction does not coincide with that of the magnetization, inless the latter be parallel to one of the principal diameters, and then the force is opposite in direc tion to the magnetization. It will be observed that the force inside similar ellipsoids similarly magnetized to the same intensity is always the same. For an oblate ellipsoid of revolution, in, which S = c = a/Vl - c 2 , For a very flat oblate ellipsoid of revolution L = 4?r, M For a prolate or ovary ellipsoid of revolution, in which r,=/, = c /l-e-, 1 - r" 2c 3 1 -M-2/4 V ^ lot -*- J- T^ ^ log -1 2.C I - c From the fornmhe for an ellipsoid we could easily deduce those for an infinitely long elliptic or circular cylinder ; we have merely to make one of the axes inlinitc. We find in this way, for instance, that the force inside a circular cylinder of infinite length magnetized transversely is -2ir. The reader will find it interesting to examine the values of the magnetic induction m the foregoing cases, and to verify its normal continuity at the surface of the magnet. Lamellar Mayncts form another very important class. In them the components of magnetization are derivable by differentiation 1 Sue Thomson and Tail, Natural Philosoplw, vol. i. J 522. from a function q>(x, y, z), which is sometimes called the "potential Lamellar of magnetization," a so that magneti zation. . . . (37). d$ L= Tz It is obvious at once that the family of surfaces fy(x, y, z} = const, cut the lines of magnetization at right angles ; for, if dx, dy, dz be the projections of the element of any line on the surface, we have by differentiation d(j> 7 dcf> 7 d<}> dx dij dz i. c. , Adx + ~Bdy + Cdz - , which is the analytical expression of the property in question. We Magnetic may therefore suppose a lamellar magnet divided up by these sur- shells. faces of magnetization into an infinite number of infinitely thin normally magnetized shells or lamellae. It can be shown that the product of the intensity of magnetization by the thickness at each point of any such shell is the same ; for, if fy(x, y, z) c and (p(x, y, z) = c + Sc be the equations to the two surfaces bounding the shell, Sis the normal distance between them at any point, we have hence ay d<p dy d$ dz + -7 -. t- j- -j- ov = oc, dy dv dz dv J i.e., I8v = 8c = constant for the same shell, which was to be proved. This product is called the strength of the shell. A shell, which is everywhere normally magnetized, hut whoso strength is not constant, is called a complex shell " ; a magnet made up of such shells is called a "complex lamellar magnet." The con dition to be satisfied by A, B, C in this case is simply that the lines of magnetization must be orthogonal to a family of surfaces, i.e., Adx+Kdy + Cdz must be convertible into a perfect differential by multiplication by a factor ; otherwise that ,/dB dC fdC dA _/rfA A + B - -T- + C I -j dz dy J dx dz J dy dx = 0. The potential at P of a simple magnetic shell of strength i is given by the formula iw . . . (38), where co is the solid angle subtended at the point P. 3 There is a convention here as to sign, viz., that side of the shell is positive towards which the lines of magnetization pass, and the solid angle subtended at points infinitely near that side is positive, while that subtended at points infinitely near the other side is negative. If we cause P to move from the positive side away to infinity, then back from infinity to the negative side, or to move anyhow from infinitely near the positive side to a point infinitely near the nega tive side without cutting through the shell, it will decrease con tinuously by 47ri ; if we pass through the shell from a point infinitely near on the negative side to a point infinitely near on the positive side, there will be a sudden increase of 4-Tri ; tangentially to the shell there is continuity. The potential of a closed shell is evidently zero for any external point, 4?rz for an internal point according as the positive or negative side is innermost. It appears also that the potential of a simple magnetic shell depends merely on its strength and on its boundary, just as that of a magnetic solenoid depends merely on its strength and the position of its ends. A lamellar magnet will in general be made up partly of closed Potentia shells, and partly of shells whose boundaries lie on the surface ; of lamel only the latter of course can influence the potential at external lar mag- points. The general expression for the potential at any point net. ^^ i + -T- dx dy dy V + n M I* where 6 is the angle between D and the outward normal to c?S, and <f> the value of <f> at the point |, TJ, (zero of course if {, TJ, be outside the magnet). The value of V thus found is not discontinuous at the surface as might be supposed, for both the surface integral and 2 To be distinguished of course from the maffnelic potential.
3 Gauss, Allgeincine T/teorie des Enlntaunetismm, 38.