M A G N E T I S M 235 II. The surface value of V, and hence its general value for external points, is determined if the northward com ponent of the magnetic force be known at every point of the earth s surface. This follows at once from the fact that the difference of the values of V at any two places is the line integral of the magnetic force along any line joining them ; thus, if V be the value of V at the geographical north pole, we have V--a But the constant V does not affect the general value of V; hence the proposition is established. III. The same conclusion follows if the westward horizontal component be known all over the earth s surface and the northward component along any one meridian. In fact, if V be the potential at any place whose latitude is I and longitude A, then V =-af l Ml - af K cos ld + V , J v _ J^o 2 the first integration being performed along the given meridian, the second along the parallel of latitude corresponding to the place. From I., II., and III. we have the remarkable con clusion that, if the vertical component be given all over the earth, or the northward component, or the westward component and the northward along one parallel, then in each case the other two elements are determined. Approxi- Gauss gives another interesting application of the line integral of mation magnetic force. If this integral be taken all round any closed curve by means or polygon, the result is zero. Let us express this for any geodesic of the triangle ABC, at whose vertices the horizontal force has the values line in- > te<*ral of Hj, H.,, H 3 . If the inclinations of H to BC at B and C be a and a , -^ ^ ^ ^ to CA at C and A ft and ft , to AB at A and B 7 and 7 , then, if the arcs BC, CA, AB be not too long, ve may replace the com ponent along BC at every point by the average of its values at B and C, and so on. We thus get If we suppose the values of H at B and C to be known, and tlie values of the declination to be known at all three places, the above equation determines the value of H at A. Calculating in this way from observed values at Gottingen, Milan, and Paris, Gauss found for H at Paris 51696, the observed value being 51804. External It has been supposed hitherto that the magnetic causes and are entirely internal to the earth. The foregoing theory internal enables us to test how far this assumption is correct, earth s ^ we suppose that there are external causes, then the potential magnet c a ^ internal points due to these will be action. n , , , P i . rp / r 2 T 1 + 1, r ll IT . I; 1 a 2 a ) T , Tj, T 2 . . . T; being the different harmonics in the surface value of the part of the potential due to external causes. Suppose now the whole vertical force deduced from observation for all parts of the earth s surface, and expanded in a series of surface harmonics, the i th of which is Zj ; then, since this is the sum of the i ih har monics in the parts due to internal and to external causes, we have Further, suppose the surface value of V determined from observa tions of horizontal force, and let the i th harmonic in it be V,-, then we have From equations (57) and (58) we can determine S 4 - and T;, and thus settle how much is due to external and how much to internal causes. It does not appear from obfcerva- tion that any sensible part of the mean value of V arises from causes external to the earth. Gaussian We have seen already that the aeti >n of any body can be repre- distribu- sented at external points by an ideal layer of positive and negative tion for magnetism. Gauss finds for the surface density of the layer in a sphere, the case of a spherical body like the earth, the expression (V/a - 2Z)/4ir, which may be deduced immediately from the for mulae already given. If we drav/ a series of equipntentiul surfaces correspond- ing to small equidiflferent values ol V, these will cut the Magnetic arth s surface in a series of equipotential lines, which are parallels called the " magnetic parallels." These lines obviously and . ,. i ii- r 11 M. mi i . i < meridians, have the following properties. Ihe horizontal force is every where perpendicular to them, and is at any point inversely proportional to the distance between two consecutive lines there. So that, if these lines were drawn upon a terrestrial globe, their crowding would indicate increase of horizontal force. The lines of horizontal force, or "magnetic meridians," the tangent at every point of which is parallel to the horizontal component, are everywhere orthogonal to th 3 magnetic parallels, and their positive direction is from p irallels of greater potential to parallels of less potential. If, as has been tacitly assumed hitherto in accordance with the results of observation, the potential on the earth s surface have but one maximum and one minimum, then the parallels will be closed curves expanding successively from the maximum point and then closing again round the mini mum point, and the magnetic meridians will all run between these two points. It is clear that at each of these points the equipotential surface and the earth s surface touch ; at the minimum point the line of total resultant force will piss to the earth, at the maximum point from it ; at the former, therefore, the north end of a freely suspended needle will dip vertically downwards, at the latter the south end will do the same. This is the simplest possible case for a magnetized sphere. It is easy to see that, if we define a north pole 1 as a point on the earth s surface at which the horizontal intensity vanishes, and the dip is 90, there might be more than one such point. Consider the series of equipotential surfaces 1, 2, 3, 4, 5, 6 in fig. 26, 2 each of which has two eminences with a depression Fig. 26. between them. The lines a, b, c, d, e, / are the sections of these by the earth s surface. 1 just touches the surface in a ; and, if the potential increase in the order in which the surfaces are numbered, a will be a north pole. The section by 2 is the single oval b. 3 touches the surface in c , which is clearly another north pole, and also meets the surface in a single oval c equipotential with c. The section by 4 is the double oval d, d . The depression on 5 False touches the surface at F, and meets it in a figure of 8, e, e , P oles - on which F is the double point. F is therefore yet another north pole according to our definition ; it differs, however, from an ordinary north pole in one important respect ; for the law that the north end of the compass points from parallel of greater to parallel of less potential shows at once that near F and inside the 8-shaped parallel the south end will point to F, whereas at a neighbouring point outside the north end will point to F. Such a point is called a false north pole, and we see that the existence of two true north poles necessitates the existence of a false north p -le; and in general it may be established 3 that, 1 Of course pole as thus defined has nothing to do with pole in any of the former senses, e.g. , the line joining its N and S poles is not parallel to the earth s magnetic axis. , 2 Gauss, I.e., % 12. Cf. Mascart and Joubert, Lemons sur I Mec incite et sur le Mar/net! xme, torn. i. 436, 1882. 3 See Gauss, All. Theorie des Erdmagnetismus, 12; Maxwell,
vol. i. 113, vol. ii. 468.