234 MAGNETISM fr >m the centre of M to the centre of M , and 12 the angle between the axes, this reduces to KK W= ~-(cos0 12 -3cos0 1 cos0 2 ) (49). Action From this formula we can derive at once by differentiation the between force of translation and the couple about the centre of M , which two in- represent the action of M upon it. An elegant synthesis of this finitely action has been given for the most general case by Tait. 1 It will small be sufficient to confine ourselves here to the case where the magnetic magnets, axes are in one plane. In this case 0] 2 = 0i 2 , and W becomes KK (sin 0! sid 2 - 2cos0 1 cos0 2 )/? 5 . Denoting by X, Y, L the forces of translation parallel to MM and perpendicular to MM (so as to decrease X ) and the couple tending to decrease 2 , we have X = - ~ = dW KK sin 2 - 2 cos 1 cos 2 ) 3KK (sin 0j cos 2 + 2 cos e 1 sin ft. 2 ) . (50). J Force of One most important conclusion follows at once from these for- transla- inuhe, viz., that the translatory forces vary inversely as the fourth tion = power of the distance, whereas the directive couple varies only (dist)- 4 . inversely as the third power. Hence the couple may be quite sen- Couple cc sible at distances for which the force of translation is inappreciably (dist) -3. small. These conclusions apply of course equally to any pair of magnetized bodies, provided the distance between them be suffi ciently great as compared with their linear dimensions. This, applied to the case of the earth, at once explains the phenomena that puzzled Norman and the earlier magnetic philosophers so greatly. The following particular cases are important (fig. 25) : (A) (B) -**---, x> 3KK (D) 1 = - r = 0, L = 0. r = 0, L-=0. _3KK ^A- , L 3KK 2KK KK Deflect- The last two cases are especially important : the position of the deflecting magnet in (C) is described as "end on " (erster Hauptlage), in (D) as "broadside on" (zweiter Hauptlage) ; it will be noticed that the couple in the former case is double that in the latter. magnet "end on" and broad side on." D C n & Closer approxi mation for de flecting couple end on and broad side on. If the terms of the second and third order be taken into account, and the magnet n s be deflected through an angle <|> from its original position by a deflecting magnet (I.) originally end on and (II.) originally broadside on, we get for the couples (51), (52), where T 1 and T are odd functions of the relative position of M and M , but T 2 and T 2 are even. In the case where M and M are symmetrical about three orthogonal planes, and being the centres of symmetry, r l and T vanish, and the writer has obtained for the values of T 2 and T 2 T a - - 6 { K(3A - 4 A 2 + A 8 ) - K (2 A x - A 2 - A 3 ) } T 2 =-f{K(12A 1 -llA 2 -A 3 )-K (3A 1 -4A 2 + A 3 where A 1; A 2 , &c., have the meanings above assigned in (45). 2 Sphere Magnetized in any Manner. This is the most Gauss s interesting of all the cases that fall under the present theory of ssction, both from its being amenable to mathematical J e P es " tvitil inn, cf - treatment and on account of its historical interest. It was lie ti.sm. first discussed in the beautiful memoir, entitled Allgemeine Theorie des Erdmagnetismusf in which Gauss laid the foundation of the rational theory of terrestrial magnetism. The following is a brief account of the theory, which has not been greatly added to since he left it. Let X, Y, Z be the components of the earth s resultant magnetic force at any point on its surface, in the directions of geographical north, geographical west, and vertically upwards respectively. The force is completely known when these are given, since it depends on thres elements only. If H, 8, i have the meanings formerly assigned (p. 220, 221, 227), we have of course H = /X- + Y 2 , tan8 = Y/X, tan< = Z//X 2 + Y 2 . Again, if V be the magnetic potential of the earth, I the latitude, and X the longitude of any point on its surface, then, supposing the earth to be a sphere of radius a, we have Y X = -rr- . Y = at 7 - ; -i , Z - -j , acost a dr r denoting the distance of any point from the centre of the earth. When V is known, there-fore, the force is com pletely determined. If now we suppose all the magnetized matter (or its equivalent say, electric currents) to be within the earth, it follows, from the theory of spherical harmonics, that we can write down a convergent series for its potential at all external points, when the potential at every point of its surface is given. 4 In fact, if the expansion of this surface potential in terms of surface harmonics be Si + Sg^- . . . +S.-+ . . . , we have for all external points V-SJ V r The number of terms of this series that must be retained in order to obtain a sufficiently accurate representation of the phenomena will of course depend on circumstances, and can only be ascertained by trial. S 2 , S- are Quart. Jour. Math., 1860 ; and Quaternions, 414. functions of known form, containing respectively 3, 5, ... 2i+l constants; hence, if terms beyond the i th order may be neglected, the expression for V will contain t 2 +2i arbitrary constants. These must be determined by obser vation, and then the magnetic action at all points on the surface or outside the earth is known irrespective of the internal distribution of the magnetic causes. If we look at the matter from the general point of view Data that V is determined when its surface value is known, we sufficient have the following propositions. ^^ ^" e I. V is determined when the vertical force is known at ea rth s every point of the earth s surface. magnetic For, let the surface value of Z be expanded in a series of ac ^ ODl surface harmonics of which the i th is Z, ; then, equating this to the I th harmonic in the surface value of Z = - cEV/dr derived from (54), we have (i + l)Si = aZ,-, which determines S,-. Thus the proposition is proved. 2 Cf. Riecke, I.e. 3 Res. d. Mag. Vereins, 1838. 4 See Thomson arid Tait, vol. i. chap. 1, App. A and B. 5 The term S of course vanishes, since the sum of the positive and
negative magnetism within the earth is zero.