MAGNETISM 247 with Faraday s -view of the matter. If a, b, c be the components of the magnetic induction parallel to the principal axes of the sphere, then we get = (! +f7rSj)RZ, & = &e., e = &c. Whence if N denote the total induction through the sphere l in the direction of the undisturbed field, we have, co being the area of its meridian section, N = oi(al + bm + en) = R + f TrR^Z 2 + s 2 w 2 + s 3 ?i 2 ). N is thus a maximum or a minimum when s x Z 2 + s 2 m 2 + s 3 ?i 2 is a maximum or a minimum. We have therefore established quite generally Faraday s law that an seolotropic sphere suspended in a uniform field ivith freedom to rotate about any diameter tvill be in stable or unstable equilibrium according as the number of lines of force that pass through it is a maximum or a minimum. A particular case of this theorem has already been proved above (p. 245) for strongly magnetic bodies. We next apply the formula (87) to deduce the force of translation in a heterogeneous field. 1. We see that in a uniform field W is constant so long as there is translation merely without rotation, i.e., there is no tendency in an seolotropic or isotropic sphere to move bodily in a uniform field. 2. If we suppose the sphere isotropic (i.e., s l = s 2 = s s = T), then W = - |rv(a5 + {3%+yl) = - |rvR 2 . Hence the force tending to move the sphere in the direction of ds is j Ci VV 1 (fftl") ,~rtrt /QQ -r = -krv j rwR-r- (o). ds ds ds leduced. j n Q^QJ. W0 rds, the small sphere is subject to a force of sotropic which the scalar potential is ^rvll 2 . If then we draw the isodynamic surfaces R 2 = const., the force on an inductively magnetized isotropic sphere will be everywhere at right angles to these ; that is, the direction of this force at every point will be tangential to the lines of slope of the resultant force, viz., in the direction in which that force varies most rapidly. In the case of paramagnetic bodies, for which T is positive, the spheres will tend to move from places of weaker to places of stronger resultant force ; in the case of diamagnetic bodies, for which T is negative, from places of stronger to places of weaker force. This is the famous law found experimentally by Faraday, and after wards theoretically established by Sir William Thomson. It must be carefully borne in mind that the lines of slope of the field are not necessarily coincident with the lines of force, but may cross them at any degree of obliquity. As strange mistakes have been made in this matter, 2 it may be well to illustrate this statement by a few examples. In the case of an isolated north pole the lines of slope coincide with the lines of force, which are straight lines radiating from the pole. In this case a paramagnetic sphere would approach and a diamagnetic sphere recede from the pole along the lines of force. The lines of force for a rectilinear electric current are circles of which it is the axis ; the lines of slope are straight lines radiating from the current. A paramagnetic sphere would therefore move towards, a diamagnetic sphere away from the current in a direction perpendicular to the lines of force: In the case of an infinitely small magnet, whose lines of force are given by r = c sin 2 0, the lines of slope are given by r = c sin 4 0/ cos 6 and the angle between the line of force and the line of slope at the point (r,Q) is tan " 1 {tan0(l + cos 2 0)/(3 + 5 cos 2 0/} The theory of isodynamics and lines of slope in the case of plane fields of force, i.e., those for which the potential is given by the pmifltirm //- r //7v- -l~ //- V ///)/- fi ia vomn ptaMtr oi-mWlrt TP I" /v t nm ara- ay s vw of ransla- ional orce phere. Isodyna mic sur- aces and lines of slope. equation d 2 V/dx^ + d-V/dy" = 0, is remarkably simple. If { = x + iy, i = x-iy, we know that V = 0() + ^(Tj), <j> and <|/ being functions depending on the particular case. When these are known the isodynamics are given by <t> (W(i) = const. and the lines of slope by 3. Next suppose an seolotropic sphere allowed to move 1 That is, the number of lines of force passing through the sphere (see above, p. 244). 2 See Wiedemann, Galvanismus (ed. 1874), ii. 665 ; Todhunter, Natural Philosophy for Beginners, pt. L 387. without rotation in any direction ds. Let the direction cosines of the field relative to its principal magnetic axes tropic be l,m,n, then these are constant during the displacement; ~~ and, if R be the intensity of the field, a = R/, /8 = Rw, y = Rre ; whence ..71IF -7/TO . . (89). ds Hence, as before, the resultant force of translation on the sphere is along the line of slope, in the direction in which the force increases if the body be wholly paramagnetic, in the opposite direction if it be tvholly diamagnetic. Besides depending on the nature of the field, the force of translation, on account of the factor s^ 2 + s z m 2 + s 3 n 2 , depends on the position of the body relative to the lines of force. Bearing in mind the theory of the radii of an ellipsoid, we have the following proposition : The force of translation on an seolotropic sphere is greatest when its axis of (numerically) greatest magnetic susceptibility is parallel to the lines of force, and least when the axis of (numerically) least susceptibility is in the same position. Or, using permeability instead of susceptibility, The force of translation is greatest for a paramagnetic sphere when its axis of greatest permeability is parallel to the lines of force, for a diamagnetic sphere when the axis of least permeability is parallel to the lines of force, and vice versa. Or, yet again, in the words of Faraday : The force of translation exerted upon a paramagnetic sphere is greatest ivhen it is so placed that the greatest number of lines of force pass through it, whereas in the case of a diamagnetic sphere the force is greatest ivhen it is so placed that the least number of lines of force pass through it, and vice versa. Approximate Theory of the Action on Bodies of Finite Size in a Non-Uniform Field. We have seen that, if the square of the susceptibility be negligible, the effect of the form of the body and the disturbance of the field arising from the induced magnetism may be neglected. In that case we may replace the spheres of the foregoing discussion by cubes, and determine the action on a body of finite size by integrating the action on the elementary cubes of which it is composed. Thus the potential energy will be A/OTtVo + s 2/^o + s 3yoM v > an d the body need not neces sarily be homogeneous. From this expression we can deduce the force under given circumstances. It is quite easy to see, without any mathematical calcula tion, what will happen in a field of force which diminishes in intensity outwards from an axial line. If we suspend an elongated paramagnetic body with its centre in the axis of the field, it will evidently be in stable equilibrium with its longest dimension placed axially ; for if it were slightly displaced every little cube of it would move into a place of weaker force, and would therefore tend to return. If, on the other hand, the body were diamagnetic, it would be in stable equilibrium in an equatorial position ; for any displacement from that position would bring every little cube nearer the axis of the field, i.e., into a place of stronger force, and therefore each such cube would tend to return. General Problem of Magnetic Induction. It will be instructive to consider for a little the theory of induced magnetism in its most general form. We shall suppose the induction to arise from given magnetic force (a n ,j8 ,7ft), arising from pre-existing magnetism (A ,B ,C ) or other wise. Letters with suffix 1 denote components of induced magnet ism, of force arising therefrom, and so on. Letters without suffix denote components of total force, total magnetization, &c. Thus V , Vj, V denote the potentials due to pre-existent, induced, and total magnetism respectively; and we nave V = V + V 1? and the like relation in other cases. . We suppose all the media within the field to have definite permeability ; but there may be aeolotropy and heterogeneity to any extent, and discontinuity along given surfaces. Approxi- I " atc O f finite
size.