On the basis of the gravity material available, the author studies two statistical functions: G2 = the rms (root mean square) anomaly in a square with side s, and E2 = the rms deviation of one actual point anomaly from the actual mean anomaly in a square with side s. The function E2, is called the error of representation, for if inside a square there is only one observed anomaly and this anomaly is accepted to represent the mean anomaly of the entire square, a standard mean error E can be used for the estimation of accuracy. On the other hand, if there are no observations inside the square and the mean anomaly of the square is assumed to be zero, G can be used as the standard mean error. For points or for very small squares, E is zero and G has a maximum value G0. For a hemisphere, G is zero and E has a maximum value G0. There is a critical size at about s = 3°, where E = G. When s is greater, it is not advisable to use the observed anomaly at a single station, as the representative of the mean anomaly of the square, because for zero the error to be expected is smaller. The weighted mean of zero and the observed anomaly is recommended. Because the regions without any observations are still large, it is necessary to estimate the size of the smallest squares, where the mean anomaly can be assumed to be independent of the mean anomalies of the adjacent squares. On the basis of the present gravity data, an estimated value of s = 30° seems to be acceptable. Using the functions E and G and the accepted values s = 3° and s = 30°, the precision obtainable for the gravimetric determination of the elevations N of the geoid (Stokes' formula) and of the deflections δ of the vertical (Vening Meinesz' formula) has been estimated. In the most favorable cases (Central Europe and the central parts of the United States) the standard mean error of N is ±10 meters and that of δ ± 0.″85. The former figure is almost entirely due to the great unexplored areas of the Earth'; the latter depends half on these unexplored areas and half on the small gaps within a distance of 50° from the point where δ is computed.

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Many theoretical discussions published by various geodesists during the last decades seem to have a common trend which means no less than the reformation of the very foundations of geodesy. There are three new ideas which have, in fact, no connection with each other except that they all can be expressed in a rather paradoxical form:1. In gravimetric geodesy, the principal problem has been 'the determination of the geoid.' According to the new theory, the geoid is quite unnecessary, and the determination of it is considered to be an unsolvable problem. In geometric geodesy, most problems have been solved by the aid of geodetic lines on the surface of a reference ellipsoid. With modern technique of computation, this method is a detour which can be straightened by a three-dimensional system of coordinates.3. The theory of errors is often considered to be a special domain of geodesy. The theory of adjustments, however, should not be based on the vague concept of errors, but on modern mathematical statistics.

The gravimetric method for the investigation of geoid is based on a theorem of STOKES (1849): If a surface is given, which encloses all attracting masses and at which the potential of the gravity-i.e. the sum of attracting and centrifugal forces-is constant, the gravity at this surface and everywhere outside it is given too.

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