244 Synthetic solution fo !j a . uniform" 1 field. Reduc- tion in the num- indue- tion co- efficients, more important in practice, and then give a brief account of the general theory with a view to establish some general principles to guide us in the subsequent account of the (often very complex) phenomena observed by experimenters. Homogeneous JEolotropic /Sphere in a Uniform Field of Inductive Force. We suppose that the sphere, to begin with, is not magnetized. If the sphere were uniformly magnet ized, 1 with components A a , B 1? C 1 , then (see above, p. 232) the force inside the sphere due to this magnetization would have for its components This uniform force combined with the given uniform force (a , /? , y ) of the inductive field would result in a uniform force It is obvious therefore that the assumption of uniform magnetization will enable us to satisfy the law of induction. In point of fact, substituting in (71) and transposing, we get three linear equations to determine A 15 B lf C in terms of a , /3 , y G , viz., It is easy, by means of these and formulas given above, to calculate the couple exerted on the inductively magnet ized sphere. If we put a = 0, /3 = Fcos0, y = Fsin0, we can calculate the work done on the sphere in turning through 180 about an axifs perpendicular to the direction of the field. This, by the conservation of energy, ought to vanish, and we thus get the conditions p^ q v p 2 = 2 , p 3 = <? 3 . The equations (74) therefore reduce to Hence, if a, , y be parallel to a radius of the central quadric A 1? B 1( Cj will be normal to the diametral plane of that radius. We have, therefore, by the theory of surfaces of the second degree, the following conclusions. Three 1. The induced magnetization is not in general in the principal direction of the inducing force ; but there are in general magnetic at everv point three directions, called the three principal magnetic axes, mutually at right angles to each other, for which the directions of the induced magnetization and of the inducing force coincide. If the axes of coordinates be parallel to these principal axes, the equations (75) reduce to A J = r 1 o, B^rjiB, C^r^y .... (76). Principal The values of r v r 2 , r* 3 in this case are called the "principal magnetic magnetic inductive susceptibilities." Bodies for which these luctive coefficient are a n positive are called paramagnetic or bilities. ferromagnetic. Bodies for which they are all negative are called diamagnetic. No substance is known for which some are positive and others negative, although this is a mathematically possible case. Since intensity of magnet ization and resultant magnetic force are of the same dimension JJL M T J, r v r 2 , r 3 are pure numbers; for all substances except iron, nickel, and cobalt, they are extremely small. The value of the coefficients r and p for any other axes can be expressed in terms of the three principal susceptibilities by means of simple formulae which we need not stop to deduce. A. physical meaning can be given to r v as follows. Let the body be homogeneous, and let us cut from it a cylindrical piece whose axis is parallel to the principal axis of susceptibility r r Place this cylinder in the direction of 1 Here and in future the suffix denotes components of magnetizing force, &c. , due to given or pre-existent magnetization ; while the suffix 1 denotes those due to induced magnetization. Letters without suffixes denote totals ; e.g., = 00 + 0^ A = A + A 1 , and so on. the lines of force in a uniform field of unit strength, then, provided the cylinder be infinitely thin, and of longitudinal dimensions infinitely great compared with its lateral, the internal force due to the induced magnetization will be zero (see above, p. 229), and it will be magnetized induc tively with a uniform intensity r r Similarly for r 2 , r 3 . The three coefficients if l = 1 + 4*-? ! , w 2 used later on, are called by Thomson the three principal permeabilities of the body at any point. These are of course pure numbers, and they are positive for all known substances. 2. If the susceptibilities for any two principal as3S be equal, then every axis in the plane of these two is a principal axis. 3. If all three principal susceptibilities be equal at any point, then every axis through that point is a principal axis, and the susceptibility for every such axis is the same. The body is therefore isotropic at that point, and the direction of the induced magnetization coincides with the direction of the inductive force for every direction of the latter. Returning to the problem of the teolotropic sphere, let us simplify our equations by taking the coordinate axes parallel to the common directions of the principal axes throughout the homogene ous sphere. We then get for the components of magnetization Princi]> perme; bilitieh,. Using these formuloe, we get, by means of (22), for the compon ents of the couple acting on the sphere (of volume v), Magne ization and couple for sphere unifon field. There is of course no force of translation. As a special case let us suppose- r v r 2 , and r 3 to be in descending order of algebraical magnitude, and suspend the sphere with the axis of r x perpendicular to the lines of force. We may put /3 = Fcos0, y = Fsin#, where 6 is the angle between the axis of y (r 2 ) and the direction of the field, then we have | = ii>F 2 (r. -r.) sin 20/(l + -nr. )(1 + f TIT. ). Hence the sphere tends to turn so as to place the axis of algebraically greatest susceptibility parallel to the lines of force. It will be in equilibrium when either principal axis is parallel to the lines of force ; but in stable equili brium only when the axis of greatest permeability is in that position. It is to be noticed that the couple is pro portional to the square of the strength of the field. There is another way of expressing these results more in accordance with the ideas of Faraday. If N be the surface integral of magnetic induction taken over the meridian section (u) of the sphere perpendicular to the direction of the vector |3 inside, or, as we may call it, the number of lines of force that pass through the sphere, then we have cos 2 + V de 2 4 ir 2 R ^2 + ^-3 + W R being the radius of the sphere. From these formulae we can draw the following conclu sions : 1. The number of lines of force that pass through the sphere is greatest, viz., 3Fww 2 /(i5r 2 + 2), when the axis of greatest permeability is parallel to the direction of the un- disturbed field, and least, viz., 3Fww 3 /(i=r 3 + 2), when the axis of least permeability is in the same position. 2. In any position the number of lines passing through the $2)herical space is greater for a paramagnetic body, and Deduc-
tion of