Generalized Legendre Expansions for Any Curve Whatever in Space

Abstract

The preceding chapter considered the transport of geographic coordinates and azimuth, of dynamic height and zenith distance between points separated by a finite distance, and belonging to two different equipotential surfaces of the Earth’s gravity field, taking as the line of transport a space geodesic, i.e., a straight line. The expansions obtained represent a generalization of those known in Geodesy as Legendre expansions (or also as formulae of Delambre), and their interest lies in the fact that these expansions extended into space open up a new way of dealing with reductions to the geoid without introducing superfluous hypotheses. They show how it is possible to develop with complete generality the problem of trigonometric levelling, and to deal with the trigonometric transport of the dynamic height and zenith distance which are intimately linked to geographic coordinates and azimuth. The same expansions moreover offer the possibility of determining experimentally the values of the coefficients which appear in them, and which define the characteristics of the field under examination.

Originally published as: Marussi A (1950) Sviluppi di legendre generalizzati per una curva qualunque dello spazio. Rend Accad Naz Lincei 9: Ser 8, fasc 1-2, 80-83