In section 1 are stated the needs of geodesy which must be met to allow results to be expressed in one unique reference system. At present the great surveys are in disconnected systems and are partially spheroidal and partially geoidal. section 2 recalls a theorem of Stokes, relating geoid with spheroid. Thence are deduced expressions for the deviations of the vertical and the curvature of the geoid. All these are in the form of integrals of the gravity anomalies over the earth. section 3: this ideal earth is a body bounded by a level surface, and to bring the actual earth into its scope, protuberances of the topography above the co-geoid-a surface differing slightly from the geoid-must be annulled. section 4: the relation between gravity at different levels is found in terms of the local mean geoidal curvature, instead of the customary mean of the whole earth. In section 5 a practical observational method of finding this curvature is proposed, depending on reciprocal vertical angles observed at pairs of points. Atmospheric refraction is involved and the mode of dealing with this is discussed. Some general remarks in section 6 terminate the paper.