On a more precise stokes' series

Summary A relation is established between coefficients of an expansion of the gravitational potential into a series of Legendre's function of the second kind and coefficients of an expansion of gravity anomalies on the surface of the reference ellipsoid into a series of the same functions. This connection can be useful in geodetic computations which take into account the Earth's flattening.     

On the evaluation of space networks

Резюме Описан способ определения координат точек земной поверхности на основе измерения одного базиса, горизонтальных и вертикальных углов (принципиальная возможность решения этой задачи выяснена Молоденским в 1949 г.). Предварительные значения координат и элементов ориентирования местных координатных систем при уравнивании пространственных сетей рекомендовано определять по необходимым измерениям. Приведены дифференциалные формулы геодезических и астрономических азимутов и зенитных расстояний.

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Address: Verchnjaja Pervomajskaja 4b, Moskva E-264, USSR.     

Gravity potential at the Major Vertical Datum as primary geodetic constant

Summary On the basis of the fundamental relations of the Molodensky's Earth's figure theory (1945), admitting the inequality of the gravity potentials at the Major Vertical Datum W₀ and on the surface of the reference level ellipsoid U₀, and taken into account that potential W₀ enters into equations directly, it is recomended, W₀ should be adopted as a primary geodetic constant. Parameters of the best fitting ellipsoid are not needed for the solution of geodetic problems and for the investigation of the Earth's gravity field. The reason for requiring the reference and actual fields be close is only that the boundary-value problem can be solved in the linear approximation. Dedicated to the Memory of M.S. Molodensky Contribution to the I.A.G. Special Commission SC3 Fundamental Constants (SCFC).     

Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth

A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing potential of an ellipsoidal Earth.     

Determination of Stokes' constants respecting zero -frequency tidal term due to the Moon and the Sun

Summary A principle of determining the Earth's gravitational field through the Stokes' constants is presented. The problem is properly posed according to Hadamard. The zero-frequency tidal terms due to the Moon and the Sun are introduced analytically into the absolute terms of equations. The ellipsoidal coordinates are used. The reference field should be so close to the actual one, that the problem can be solved in a linear approximation. The proximity of the fields mentioned does not need to be of the highest possible degree.Contribution to the I.A.G. Special Commission SC3 Fundamental Constants (SCFC)     

To Pizzetti's Theory of the Level Ellipsoid

The topic of the Earth's reference body, which has now been established as Pizzetti's level rotational ellipsoid, is analysed. Such a body is fully determined by four parameters: a, GM, J ₂ and . At present, the largest discrepancy in determining these parameters occurs in the value of a, which may in future be replaced by the gravity potential of the mean sea level W _o, with respect to Brovar's condition.Pizzetti's four parameters of the reference body are determined by solving the Dirichlet boundary value problem. The Dirichlet problem has only a unique solution, which, however, can be expressed in infinitely many ways. It turns out that the most important part in the form of the solution is played by Lamé's conditions, which determine the type of ellipsoidal coordinates.The solutions given by Pizzetti, Molodensky and another variant are considered. The last variant leads to a simple formula for the potential of the reference ellipsoid, but the formulae for Lamé's coefficients are inconvenient. Of course, all the methods lead to identical solutions, but some of them are more convenient for the historical use of logarithms, whereas others are more appropriate for use in computers.     

Sur l'histoire de la notion du potentiel

Summary Huygens undoubtedly used the notion of the potential in his well-known work “Discours de la cause de la pesanteur” (Leyden, 1690), investigating the Earth's figure by means of Newton's famous canals. Maupertuis used this notion in a similar work published in London in 1733. The notion of potential was used by Clairaut in the “Théorie de la figure de la Terre” (Paris, 1743) and by D'Alembert in many articles (1752, 1761, 1768, 1773, 1780). Euler used it also in many of his works since 1736. These examples of application of the potential were not noticed because of peculiarities of terminology.