On Methods of Representing the Distribution of Magnetic Force over the Earth's Surface Author(s): S. Chapman Source: The Geographical Journal, Vol. 53, No. 3 (Mar., 1919), pp. 166-172 Published by: geographicalj Stable URL: http://www.jstor.org/stable/1779265 Accessed: 25-04-2016 07:38 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Wiley, The Royal Geographical Society (with the Institute of British Geographers) are collaborating with JSTOR to digitize, preserve and extend access to The Geographical Journal This content downloaded from 142.51.1.212 on Mon, 25 Apr 2016 07:38:09 UTC All use subject to http://about.jstor.org/terms ( *66 ) ON METHODS OF REPRESENTING THE DISTRI? BUTION OF MAGNETIC FORCE OVER THE EARTH'S SURFACE S, Chapman, M.A., D.Sc, Fellow of Trinity College, Cam- bridge, and Chief Assistant at the Royal Observatory, Greenwich ? i. 'TT>HE magnetic force at any point of the Earth's surface requires, X like any other force, three elements for its specification. These elements may be the components of the force along any three non-coplanar directions, or the magnitude of the force together with two directions. It is customary, and convenient, to indicate the distribution of the values of any such element by a set of isomagnetic lines, having the property that each line passes through all the points on the sphere at which the element has a given value associated with the line. There are, for instance, the isomagnetic curves of declination, dip, and total or horizontal intensity, ? 2. Two classes of isomagnetic curves may be distinguished, according as to whether the determination of the corresponding element does or does not require a reference to some special direction or diameter (axis) of the sphere. The first class, which may be termed intrinsic isomagnetics, includes the curves of equal total, vertical, or horizontal intensity, and of dip; these involve reference either to no direction at all, or else only to the radial direction associated with the point where the element is measured. It is not possible, however, to give a complete specification of the distribution of force by intrinsic isomagnetics, since there is no natural datum from which to measure the direction of the horizontal com- ponent of the force. This direction is usually indicated by means of the curves of equal declination or " variation," so important in the practice of navigation. These curves, however, are relative isomagnetics, as are also the curves of equal axial inclination drawn by Mr. Reeves, since the directions,' in the one ease of declination, in the other of axial inclination, are measured relatively to a special diameter, and to the meridian curves defined by it. The choice of the axis of rotation for this axis of reference is dictated only by considerations of practical or theoretical advantage, and corresponding sets of curves might be drawn with respect to any other axis of reference. ? 3. One other important intrinsic set of magnetic curves may be noticed, although they are not strictly isomagnetic curves. These are the lines of horizontal (tangential) force, which have the property that the direction of the line passing through any point is the direction of the horizontal com- ponent of force at the point. These lines are, of course, entirely different from the lines of equal declination. They have two foci, one at each mag? netic pole. These are also foci for the lines of equal declination, but the latter have two additional foci at the geographical poles. It seems likely to This content downloaded from 142.51.1.212 on Mon, 25 Apr 2016 07:38:09 UTC All use subject to http://about.jstor.org/terms MAGNETIC FORCE OVER THE EARTH'S SURFACE 167 be a general feature of relative isomagnetics that they are in some ways more complicated than the corresponding intrinsic curves. We may also note that a complete intrinsic specification of the distribution of magnetic force over the surface of the Earth is afforded by the lines of force in conjunc* tion with any two of the intrinsic isomagnetics already mentioned (?2). If, on the other hand, an axis of reference is chosen in order to afford a datum for the measurement of declination, the most natural trio of isomagnetic curves to adopt for a complete magnetic specification is undoubtedly that comprising the curves of equal tofal intensity (intrinsic), and the curves of equal declination and of equal axial inclination. Mr. Reeves' work for the first time affords us a representation of these latter. ? 4. A noteworthy feature of Mr. Reeves' curves is that they are of a simpler and more regular character than are the curves of equal dip.* This constitutes a claim to careful study, in order that the interpretation of the fact may be made clear. Mr. Reeves concludes that it indicates the special suitability of the geographical axis as axis of reference, and as the axis, probably, of the normal or primary magnetizing agency. The geographical axis has, indeed, a strong claim to be the most suitable axis to take, especially since the corresponding isoaxiclinals form the natural comple- ment to the lines of equal declination (? 3). But theoretical considera- tions indicate that the uniformity and regularity of Mr. Reeves' curves are, in the main, not due to any special physical significance of the geographical axis, and that the same features would probably appear if any other axis had been chosen, provided that its poles, like the geographical poles, were not very distant from the magnetic poles. Moreover, as Prof. Schuster has remarked, the situation of the centre of Mr. Reeves' curves (about one- third of the distance from the geographical to the magnetic poles) is in excellent agreement with what theory suggests, on the supposition that the Earth is approximately uniformly magnetized along a direction roughly agreeing with that of the line joining the two magnetic poles. Thus Mr. Reeves' curves must be regarded rather as affording a novel and interest? ing confirmation of the usual view of an approximately uniform, oblique magnetization of the Earth, than as indicating the geographical axis as, magnetically, the normal or fundamental axis. ? 5. In order to elucidate these points, I have followed Prof. Schuster in considering the isoaxiclinals for a uniformly magnetized sphere, corre? sponding to various axes of reference. Imagine a sphere, with centre O, uniformly magnetized along the direction of the magnetic axis MOM', M * This would appear to be true except near the equator. As will be seen later, however, theoretical considerations (? 7) suggest that the isoaxiclinals?as, for brevity, the lines of equal axial inclination may be termed?should break up into two families of curves near the equator, with a curve intersecting itself twice as the transition stage. This feature did not appear on Mr. Reeves' charts as exhibited on February 18, but I think it probable that on careful examination the data will be found to indicate this feature. This content downloaded from 142.51.1.212 on Mon, 25 Apr 2016 07:38:09 UTC All use subject to http://about.jstor.org/terms i68 ON METHODS OF REPRESENTING THE DISTRIBUTION OF and M' being the magnetic poles. By symmetry, all the intrinsic iso? magnetics are circles of latitude, and the lines of horizontal force are meridians. The relative isomagnetics, when referred to the magnetic axis, are also circles of latitude (the isogonic lines, in fact, in this ease become an isogonic surface, for the horizontal magnetic force is every- where &\xzo,te& along the meridians); the axial inclination is the simple sum of the dip and the latitude. Our object is to determine the iso- axiclinals when any other axis of reference, ROR', is chosen, having its poles R, R' distant respectively from M, M' by an angle a. Fig. i. Fig. 2. By the theory of a uniformly magnetized sphere (to be found in any mathematical treatise on magnetism) the magnetic force at any point P on the sphere lies in the plane POM, and its direction makes an angle $ with the magnetic axis, where / x . 3 sin 20 (i) tan w = -????? v ' T 1 + 3 cos 20 0 being the angular distance of P from the magnetic pole M. In the left hand of Fig. i the directions of the tangents to the magnetic force in different magnetic latitudes are indicated. As is clear from the formula, $ increases from o? to i8o? as 0 increases from o? to oo? (i.e. as P moves from the pole to the magnetic equator). Its mean rate of change is hence twice that of 0, as also twice that of dip, which decreases from 90 to o?. When 0 is small, tp is approximately equal to f 0,* so that t[s at first changes at f times the rate of 0, and three times as quickly as the complement of * Thus, writing t// for tan t//, 20 for sin 20, and 1 for cos 20, when 0 is small we have the following approximate relation in place of (1): % . 20 This content downloaded from 142.51.1.212 on Mon, 25 Apr 2016 07:38:09 UTC All use subject to http://about.jstor.org/terms MAGNETIC FORCE OVER THE EARTH'S SURFACE 169 the dip, ijs ? 0 (and therefore also three times as quickly as the dip itself).