On a Local Representation of the Fine Structure in the Earth’s Gravitational Field

This paper introduces a class of nearly harmonic functions useful in analyzing gravity above a limited region of the earth’s surface. These “B functions” constitute a complete orthogonal set of functions over that portion of the surface of a sphere which lies between two latitudes and two longitudes. By an iteration process, the coefficients of a series of B functions can be determined in such a manner as to provide an optimum fit to the difference between the prediction of a global gravity model and gravity actually measured on the earth’s surface, thus bypassing the problem of reducing the data to a reference surface. Representation of local gravity as the sum of a global model in terms of spherical harmonics and a local model in terms of B functions appears to be an effective technique for handling the problem of altitude extension of surface data.

On the circle polynomials of Zernike and related orthogonal sets

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.