By Paolo Lanzano (Eds.)

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**Example text**

This set of five parameters ( a , f ;N ; 5, y) or any equivalent set is called a geodetic datum. Both Helmert’s and Stokes’s formulas deal with the gravity anomaly ; the other case, its vector Ag: in one case we use its components ( 5 , ~ )in magnitude Ag. Both formulas contain these elements linearly. The essential difference is as follows. The Helmert formula makes use of any profile on land, is limited to a bounded area, and is applicable to a reference ellipsoid whose position with respect to Earth’s center of gravity is unknown.

09 Computational Techniques 33 since the last integral is proportional to the coefficients of the spherical harmonic representation of Ag. The expression provides the error in the geoidal height at a given point P(0,q5), which is incurred should one neglect the gravity anomaly beyond a circular cap of radius t,bo centered at P. It requires the knowledge of the spherical harmonic representation of Ag and the functions S,(cos t,b0). Let us briefly discuss the latter quantities. If we introduce new variables z = sin(t,b/2), t = sin(t,bo/2), we have s,,(t)= - 2(2n + 1)J; ~ ( -1 2z2)pfl(1- 2z2)z dz.

9) and (10) will be proportional to Pj and that their proportionality factors will r2-j, depend on the coefficients of P j appearing in the expansions of and ln(r/u), respectively. Within the framework of the third-order approximation, any power of r will be limited to terms in P o , P,, P,, and P6 [see Eq. (13)]; this simply means that we have to take into account only those U j and 5 for which j = 0, 2, 4, 6. 02 Clairaut Equation Expressed up to Third-Order Terms By using the expression for the various powers of r, Eq.