Abstract
The theory of the calculation of the detailed undulation of the geoid from gravity by Stokes's integral is usually given to first order, and there are some approximations which may be important in mountainous places. Some are trivial but the change of undulation with height is not. Discussion of this involves the definition of orthometric height and of the geoid. With suitable definitions the undulations of the geoid are the same as those of equipotential surfaces at external points, so that the latter may be calculated directly without additional correction for changes with height. The free-air correction to gravity is determined by these definitions and conditions; for geodetic purposes the fundamental quantity is not the change of gravity with height but the difference of geopotential. The correction derived in this paper thus differs slightly from those commonly in use and there are consequently some small errors in the figure of the Earth derived with the use of the letter.
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