Abstract. Maxwell's multiple pole theory provides the basis for a convenient means of determining the spherical harmonics of geomagnetic fields from directional paleomagnetic data. The relationship between the Maxwell poles and axes and the corresponding spherical harmonics was given by Maxwell. We show that the distribution of virtual geomagnetic poles (VGPs) is symmetrical about the Maxwell axes and converge to their poles. Utilizing this property of VGPs and a complimentary property of the distribution of equatorial virtual poles (EVPs), which are defined as points 90 ø from the VGP on the site meridian, leads to a means of obtaining the Gaussian spherical harmonic coefficients from paleomagnetic data. This VGP method involves the minimization of the horizontal components of the magnetic fields at VGPs and the radial components of the magnetic field at EVPs to yield the best fitting spherical harmonic coefficients. A hybrid variant of the VGP method involving this minimization and fitting the available mean vectors measured at sites has also been developed. Tests of the VGP and hybrid methods on model fields derived from the International Geomagnetic Reference Field (IGRF) 1995 demonstrate that they are effective means of determining the Gauss coefficients for any field requiring no a priori assumptions about the field. The hybrid method has been applied in a preliminary analysis of the time-averaged paleomagnetic data for the past 5 Myr and for the Bruhnes chron. The resulting mean fields were dominantly dipolar, but there were also persistent second order features, suggesting some longterm smaller-scale control over the dynamo process. The hybrid method was also applied to data from the Brunhes-Matuyama reversal. Four models were obtained: (1) a mean field model for reversed polarity, (2) a mean field model for normal polarity, (3) a mean field model for the entire reversal, and (4) a time sequence field model. The results were consistent with the Americas being a persistent site of inward field lines in the nondipole field and suggested that the reversal is initiated by decreases in strength of dipole, quadrupole, and octupole terms. The true dipole path of the transitional field tracks across eastern Asia, but a strong radially inward flux bundle moves over Africa. The paucity of demonstrably reliable data in these reversal records, however, requires that these results be interpreted cautiously.

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An arbitrary point distribution consisting of a finite number of points on a unit sphere may be completely and uniquely represented by an analytic function in the form of a spherical harmonic expansion. The applications of this representation function are illustrated in an analysis of the symmetries in the virtual geomagnetic pole (VGP) distribution of the polarity reversal records of the past 10 million years. We find that the longitudinal confinements in the VGP distribution are (a) persistent only in the equatorially symmetric part (of the non-zonal symmetries) of the VGP distribution and (b) strong along the east coast of the North American continent and weak along the longitudes of East Asia-Australia. We also find that the equatorially symmetric patterns in the VGP distribution appear to extend preferentially into the Pacific Ocean and are relatively depleted in the longitude band associated with Africa.

S U M M A R YThe spatio-temporal variations in the main geomagnetic field are analysed using Maxwell's source representation of the potential function defined by Maxwell's theory of poles. Maxwell sources up to order 4 are obtained for three main geomagnetic field models. The temporal variations in the main geomagnetic field are shown in terms of the variations in the parameters defined in Maxwell's source representation, for example, the Maxwell moments, the Maxwell axes, the intrinsic energy of the Maxwell source, etc. The characteristics of these variations are discussed. Applications of Maxwell's source representation of the potential function to the studies of the earth's magnetic field are suggested.The main geomagnetic field B(rr) is given by B(rr) = −∇V (rr). V (rr) is the potential function of B(rr) and is the solution of Laplace equation ∇ 2 V (rr) = 0. In 1839, Gauss invented the spherical harmonics to represent the potential function V (rr) of the main geomagnetic field (Gauss 1839). Because the main geomagnetic field is generated by the sources within the earth's core with the radius r 0 , its potential function on a sphere of the radius r ≥ r 0 is given by V (rr) = ∞ l=1 V l (rr) where the degree-l potential function V l (rr) is given bywhere Y m l (r) is degree l and order m spherical harmonic. g m l is Gauss coefficient. For convenience, the radius of the core is set to r 0 = 1. Correspondingly, the main geomagnetic field on a sphere isHere, we call eq. (1) Gauss's representation of the potential function. The potential function V l (rr) in eq. (1) is determined by 2l + 1 Gauss coefficients g m l .In Gauss's representation, the potential function is expressed as an expansion of orthonormal basis functions Y m l (r). This provides great computational flexibilities for mapping discrete surface measurements of the geomagnetic field in to spatially continuous maps and for solving various differential equations involving the geomagnetic field [such as inverting for the flows on the top of the core from B(rr) at the core-mantle boundary]. However, it has long been realized that Gauss's representation is a mathematical representation of a potential function (following the method of polynomial expansion of a scalar function). It is not a physical representation because the Gauss coefficients (g m l ) are coordinate-dependent (therefore g m l is in principle not a physical quantity). The spherical harmonic function Y m l (r) in eq. (1) does not have physical meaning. Consequently, there are only few parameters that may be defined from Gauss's representation which are physically meaningful for describing the main geomagnetic field. For instance, a widely used parameter defined from Gauss's representation is the degree-l power spectra of B l (rr) which is given by(2) E l describes the energy characteristics of the main geomagnetic field. In practice, one often wishes that there would be other physical parameters (like E l ) which could be used to describe different aspects of temporal and spatial variations ...