Non-diagonal Terms Research Articles

THE DETERMINATION OF THE EARTH'S GRAVITATIONAL FIELD (Second Paper)

Summary The data used in a recent paper on the same subject have been rediscussed by a more sensitive method for detecting variations of gravity capable of being represented by spherical harmonics of low degrees. In the first place all cross (non-diagonal) terms in the normal equations for the coefficients were neglected; in the second approximation the terms most strongly suggested by the first approximation were determined by the method of least squares. One second harmonic and three third harmonics, besides the terms included in the usual formulae, were found to be determinable. The variation shows no relation to the elevation of the surface and is inconsistent with accurate compensation. A new determination of the constant and the main ellipticity terms is made incidentally. No fourth harmonics are determinable on the data, but a suggestion is offered for the treatment of an independent term in Pp if occasion should arise.

The Self Consistent Field and the Structure of Atoms

The method proposed by Hartree for the solution of problems in atomic structure is examined as to its accuracy as a method of solving Schr\"odinger's equation. A wave function is set up at once from his method, and the matrix of the energy computed with respect to it. The non-diagonal terms are shown to be small, indicating that the function is a good approximation to a real solution. The energy levels are found by perturbation theory from this matrix, and are compared with the term values as found by Hartree. His values should be corrected for three reasons: he has neglected the fact that electron distributions are not really spherical; he has not considered the resonant interactions between electrons; and he has made an approximation which amounts to neglecting the polarization energy. The sizes of these corrections are estimated, and they are found to be of the order of the errors actually present in the numerical cases he has worked out.