228 MAGNETISM Next suppose the angle 6 fixed, and the magnet free to rotate about a horizontal axis inclined at an angle 90 -0 to OX. The couple tending to diminish the angle <f> is KIl (cost cos 6 sin ^ - sin t cos0). The position of equilibrium is given by the equation tan < = sec 6 taut. The angle at which the axis is depressed below the horizon is therefore least when = 0, and greatest when = 90j its value being i in the former case, and 90 in the latter, as stated above, p. 221. In general, if A , ju , v and A, /i, v be the direction cosines of the direction of the field and of the axis of the magnet respectively, then, resolving the forces acting at N and S, we see at once that the three components of the magnetic couple are A M ) (2). Magnetic moment resolved as a Finite magnet replaced infinite number of in- finitely Intensity oi magnet- IOU These are clearly the same as the components of the couple on a system of three magnets whose axes are parallel to OX, OY, OZ, mid whose magnetic moments are KA, Kyu., KP. Hence, so far as the action of a uniform field is concerned, we may resolve the magnetic moment like a vector, and replace a given magnet by others the resultant of whose moments is the moment of the given magnet. It appears therefore that in a uniform field every magnet behaves as if it were made up of a certain quantity of positive magnetism and an equal quantity of negative magnetism placed at such a distance apart on a line parallel to the magnetic axis that the product of the quantity of magnetism into that distance has a value equal to the magnetic moment of the magnet. It is very im portant to observe that the magnetic moment alone appears in the above formulae for the magnetic action. We can not therefore separately determine from observations in a uniform field either the quantity of positive or negative magnetism in a magnet or the distance between the magnetic centres of mass. Let M be any magnet, and P a point whose distance from j any point of M is infinitely great compared with the linear | dimensions of M. Then, since all the lines drawn from P to diff erenfc points of M are sensibly parallel and equal in length, we may suppose the positive and negative magnetism of M to be collected at their mass centres, i.e., ^ fc foe replaced by an ideal magnet. It is also obvious that, throughout a region around P whose linear dimensions are of the same order as those of M, the field due to M may ba regarded as uniform. Hence we conclude that in cal culating the mutual action of two magnets M and M we may replace each of them by an ideal magnet, provided the distance between them be infinitely great compared with the linear dimensions of either. This condition may be satis fied either by making the distance between the magnets very great if their dimensions be finite, or by making their dimensions infinitely small if the distance between them be finite. The second alternative suggests at once a method for representing the magnetic action of magnetized bodies at finite distances. We may divide up the body into portions whose linear dimensions are infinitely small compared with their distance from any point at which their action is to be considered ; each of these portions is itself a magnet, and may be replaced by an ideal magnet having the same axis and moment. The whole magnetic action is obtained by integrating the action of all the ideal magnets of which the body is thus supposed to be com posed, Let X, /j., v be the direction cosines of the magnetic axis of any element dv of a magnet, and I such that Idv is the ma g ne tic moment of the element, and let IX = A, Lu. = B, Ii/ = C; then I is called the "intensity of magnetization" at the point where the element is taken. I may be regarded as a vector which specifies the magnetization of the body ; in general it varies continuously from point to point ; if it has the same value and direction at every point, the body is said to be uniformly magnetized. A line drawn so that the direction of I at every point of it is tangential to it is called a " line of magnetization." It is clear from what has already been shown that we may if we choose replace the element dv by three ideal magnets whose axes are parallel to the coordinate axes, and whose moments are Kdv, }*>dv, Cdv respectively. If then K be the magnetic moment of the whole magnet, SK the moment of any element Sv, and p, q, r the direction cosines of the axis of the whole magnet, we have K = Z2c, = - Z2 ; and, remem- boring that K = K for every element, Line of ma s n et- 1Z> Resultau magnetic: moment and axis. Sxr x N 1 2(8KA) " K K K We may therefore write, replacing summation by integration, Let SN be an ideal magnet of infinitely small length I, let in be its magnetic moment, and m nl. Let Q be its middle point, and the angle PQN = 0, N being the positive or north-seeking }>ole ; and let QP = D. Then the potential at P due to this magnet is , Potentia of in- finitely small magnet. Expanding and neglecting powers of above the first, we get for the potential TOCOS ,^ D* Hence the potential at P ({, TJ, of an infinitely small magnet Potentia Adv at (x, y, z), having its axis parallel to the axis of x, is of finite A ( - z)/D :i , and similarly for the other two. AVe therefore obtain magnet. for the potential of the whole magnet (5). d d d A ~^. "T M ~y^ ~r V ~v~ i 7 dx dy dz ) L> Taking the second of these expressions and integrating by parts in the usual way, we get where dx dy dz I, m, n being the direction cosines of the outward normal to any ele ment dS of the surface of the magnet, and 6 the angle between the normal and the direction of magnetization at dS. Hence the action of any magnet may be represented by means of a certain volume distribution (p) and a certain surface distribution (cr) of free magnetism. This important proposition is due to Poisson. 1 The fact, in itself obvious, that the sum of all the magnetism of Poisson s distribution must be zero, gives the theorem Poisson s distnln -ff -//IcQ&OdS which admits of course of direct analytical proof. The magnet may also be replaced, so far as its external Gaussian action is concerned, by a distribution wholly on its surface, listl as was shown by Gauss. 2 This will be seen at once if we replace the positive and negative magnetism throughout the body by positive and negative electricity, and suppose the surface of the magnet covered with a conducting layer in connexion with the earth. The surface will thus become charged with a distribution of positive and negative electricity whose total sum is zero, such that the potential of the surface is zero, and hence the potential at every external point zero. The potential of this surface layer 1 Mem. de I Instititt, toin. v., 1821.
2 Jntensitas T7s, 2 (1832), and Allgemeine Lehrsiitzc, 36 (1839).