Proceeding then from these laws, M. Hansteen pursues the following investigations, based upon them, viz. The line of repose of an infinitely small magnetic needle within the sphere of action of a linear magnet. The dip of the magnetic line of repose towards the surface of a sphere, having in its centre an infinitely small linear magnet. The same, the magnet being eccentric. The situations of dip 0 and dip 90°, with the intermediate lines of dip, in the two cases, first, when the magnet is in the centre of the sphere; second, when it is eccentric. The magnetic intensity, and the isodynamic lines in both the preceding suppositions. The action of a magnet, being in shape a parallelogram, upon a magnetic point in the prolongation of its axis, and in its equator. The action of a cylindrical magnet upon a magnetic point in the prolongation of its axis, and in its equator. Upon these investigations are founded problems contained in the succeeding or sixth chapter, which is entitled "Application of the Theory of Magnets to the Theory of the Dip, Variation, and Force, at any given place of the earth's surface of known geographical position.' Suppose a B, fig. 1., a single magnetic axis in the interior of the earth; its prolongation, till it meets the surface of the earth, forms A B, the magnetic chord, of which A and B are the extremities. It is possible that the centre of the chord y, the mathematical centre of the magnet, and its neutral point (or the point in which the opposite forces are equal, and where there is consequently neither attraction nor repulsion), might be three different points; but they are at present considered to coincide all three in y, the centre of the chord. The circle E BFA is a great circle passing through C, the centre of the earth, and the magnetic chord. Cy is the eccentricity of the chord. The magnetic equator is a great circle passing through the centre of the earth, perpendicular to the magnetic chord, and passing through its centre y its poles are a and b, the extremities of a diameter of the earth parallel to the magnetic chord. If the magnetic axis were not eccentric, and the chord passed through the centre of the earth, its extremities A and B would coincide with a and b, the poles of the magnetic equator. The radius of the earth being unity, the eccentricity Cy is the sine of the arcs A a and B b which measure the distance between the ends of the magnetic chord and the poles of the magnetic equator. EF is a magnetic diameter of the earth passing through y; E is the point on the earth's surface most distant, and F the point least distant, from the centre of the magnetic chord; or the apocentric and pericentric points. Every plane section of the earth passing through the magnetic chord is a magnetic meridian; all of which are small circles except the first, E B FA, which passes through the apocentric and pericentric points. A e B is a magnetic meridian, and e its point of intersection with the magnetic equator. The first meridian passes through the ends of the magnetic chord, the poles of the magnetic equator, and the apocentric and pericentric points. When the chord has no eccentricity there is no first meridian determined by nature. Every plane passing through the axis of the magnetic equator is a magnetic vertical circle: every place has its own, and all are great circles. Were there no eccentricity in the magnetic axis, every magnetic meridian would be a magnetic vertical circle. The magnetic polar colure is a great circle passing through the poles of the earth and those of the magnetic equator. The diametral colure is a great circle passing through the poles of the earth and the apocentric and pericentric points. Thus in fig. 2, in which Pp are the poles of the earth, PNp M is the polar colure, and PFp the diametral colure. A B and a bare as in fig. 1, and the great circle passing through them is the first magnetic meridian. Q FR is the magnetic equator cutting the geographical equator M N in E, the pole of the polar colure; PBp and PAp are geographical meridians passing through the ends of the magnetic chord. If in fig. 1. L be a place on the earth's surface of known geographical position, Ly is its magnetic radius, or a line drawn from the place to the centre of the magnetic chord; Lye is its true magnetic latitude, or the angle formed by its magnetic radius and the magnetic equator; Ly B is its true magnetic polar distance; Eye is its true magnetic longitude, or the angle between the magnetic meridian of L and the first magnetic meridian. B Le A being its magnetic meridian, a perpendicular Cc to the magnetic equator from C gives c the centre of the meridian, Cc its eccentricity, and Le its radius. Le is then the eccentric magnetic latitude measured at c, the eccentric centre. In fig. 3. b LQ is a magnetic vertical circle through L, cutting the magnetic equator in QC; LR is the intersection of bLQ with the magnetic meridian of L; the arc LQ is the apparent magnetic latitude of L intercepted on the vertical circle between L and the magnetic equator, or it is the angle LCQ; EQ is the apparent magnetic longitude, or the arc of the magnetic equator intercepted between E, the apocentric point, and the vertical magnetic circle passing through L; or it is the spheric angle EbL, or the plane angle EC Q. The relations which these several quantities bear to each other, and the deduction, when they are known, of the angles which the horizontal needle will make with the geographical meridian, and the needle freely suspended (the dipping-needle, for example) with the horizon of a place, are shown in the first six problems of this chapter. Problems 7 and 8 contain the deduction of the dip, variation, and force at any given point of the surface of a sphere which has two such magnetic axes, the geographical positions of which are known, as well as the proportion between their absolute forces. The 9th and 10th problems show the method of deducing the proportion between the absolute forces of the axes, when the situation and length of the axes are known, and either the dip or the variation is observed. The expressions by which the values may be found of the several quantities treated of in these problems, collected in one view, are subjoined. In these expressions a = Aa = = Bb the arc between the ends of the magnetic chord and the poles of the magnetic equator. PbF the angle between the first magnetic meridian and the polar colure. ==NER=Pb = the angle between the geographic and magnetic equators. 3 = MP b = the geographical longitude of the north pole of the magnetic equator. LCC = the angle of the magnetic meridian with the μ=LQ= <LCQ= the apparent magnetic latitude. Φ u = = EbL = the true magnetic latitude. v = Lce = the eccentric magnetic latitude. p = = 9 M PL = the geographical longitude reckoned east ward. i = d La' (fig. 1.) = the oblique dip; or the angle of the magnetic line of repose and the tangent to the magnetic meridian. a' LM (fig. 1.). 4 = 2bLP (fig. 3.) the angle of the magnetic vertical R = Lc = the radius of the magnetic meridian. M and M' I = the dip one axis. c = the angle of the forces of the two axes. D = the variation due to the compound action of the two I = the dip Kthe force axes. Formulæ. 1. sin μ = cos e . sin p + sin ɛ . cosp. cos (q 2. cot (v + 8) = cos e . cot (q — ?) — sin ɛ . tan p. cosec (9—3). 3. cot 4 = cot ɛ. cosec (9-). cos p- sin p. cot (q—¿). |