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By Paolo Lanzano (Eds.)

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The fundamental problem of shiftingthe geodetic datum can be formulated as follows. We change the dimensions of the reference ellipsoid (6a # 0, 6f # 0) retaining, however, its rotational axis parallel to Earth’s axis of rotation; and we attach the reference ellipsoid to a different point P , on Earth (68, # 0, 64, # 0,6h, # 0); we then seek to ascertain the coordinates 68,&$, 6h with respect to this new reference ellipsoid for a fixed point P on Earth (6x = 6 y = 6 z = 0). 1. Three consecutive steps must be taken: 1.

18 I Concepts of Physical Geodesy Gravity at the equator is obtainable from Eq. (33) for 8, = 742: where use has been made here also of the relation (b/a)2 = (1 - f)’ = 1 - 2f +f2. Let us now rewrite the Somigliana formula [Eq. (35)] in the form (42) where + y q 2 + . , B = 1 - (b/a)’ = 2f - f have been evaluated by using the already mentioned expansions, and where we have limited ourselves to terms of the second degree in ,f and y. If one now uses the well-known expansion (1 - X>-112 = 1 + iX + &2 + .

The fundamental equation valid for hydrostatic equilibrium is VP = PW, (1) relating the gradient of the pressure to the gradient of the potential; it is obtained from Eulerian equations of motion by imposing the condition that there be no relative motion. The compatibility conditions of Eq. (1) are obtained by equating to each other the mixed second-order partial derivatives of p ; using Cartesian coordinates, they can be written as 8P a* z& = & ,Iz dP a* -dP a* - The above equations show that the normals to the surfaces p = const.

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