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An axially symmetric solution of Laplace's equation in oblate spher oidal coordin ates is found, which may be used as t he gravitaLional po tent ial a bout an oblate planet. This p otential, whi ch makes t he H a milton-J acobi equation for a satellite orbi t separable, has a n expansion in zonal harmoni cs in whi ch t he a mpli t udes of t he zeroth and seco nd harmon ics can be adjusted to agree exactly with t he valu es for a ny axia lly symmetri c pla net and a fourt h harmoni c whi ch t hen agrees approximately wit h t he latest value for t hat of t he ear th . The net result is t here fore a redu ction of t he problem of satellite mot ion to qu ad ratures, wit h use of a potential fi eld t hat is m uch closer to t he empiri cally accepted one for t he earth t ha n any heretofore used as t he star t ing point of a calculation. It may t hus be possible to do t he gravitational t heory of a satellite orbit very acc urately wit hout use of pe rt urbation t heory. T he m et hod can take into acco unt a first harmon ic in t he potent ial, in case observat ions are reduced to a center whi ch does not coin cid e wit h t he center of mass of t he planet.

New method of solution for unretarded satellite orbits