To Pizzetti's Theory of the Level Ellipsoid

Abstract

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 ω. 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 Lame'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 Lame'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.