Note on surface‐field analysis

Abstract

In a famour memoir Gauss [see 1 of “References” at end of paper] showed that the surface magnetic field of a sphere could be separated into parts of origin external and internal to the sphere, using the method of spherical harmonic analysis. This analysis also afforded a means of computing the field in adjacent regions of free space from a knowledge of the field over the surface of the sphere.Neumann [2], Schmidt [3], Schuster [4], and others have introduced various improvements in the technique of spherical harmonic analysis and have thereby permitted the achievement of results important in the theory of geomagnetism. However, certain limitations of this valuable method have always been keenly in evidence. In such an analysis an infinite sequence of spherical harmonic terms is usually required to completely represent the field. In cases where the surface‐field is of a type adequately represented by only a small number of spherical harmonic terms, so that those remaining may be neglected, the Gaussian method yields a very convenient and useful analysis. But it is sometimes found that the electric or magnetic surface‐field is complicated and/or of a type expressed adequately only by many spherical harmonic terms. A spherical harmonic analysis may then become so laborious that it becomes futile from a practical point of view, although still possible in theory. This particular limitation of the Gaussian method often arises in geomagnetism, where complicated features appear not only in the Earth's general magnetic field but also in many of its additional surface‐fields varying periodically and aperiodically in time.