Observatoire Royal de Belgique, Brussels, Belgium
During a campain of 18 years (1976-1994) thousands of positions have permitted improvements of thousands of orbits or determination of new ones; 4500 discoveries or rediscoveries have been obtained; more than 1800 are recognized as ours by the Minor Planet Center (MPC) from which 171 are numbered at the end of 1995. The Universities or Observatories of ESO, Rio de Janeiro, Uppsala, Belgrade, Turin, Valencia, Teheran, Milan, Torun, Tomsk, Moscow, Brussels and Nacional Obs. of Brasil have participated. Theoretical and New Ideas have been developped: Theorem of the Minimum, Einstein Effect solved by simulation, Theorem of Proper Direction(s), use of Fictitious Errors, Test and Fictitious Stars, the Method of the Witness star and its inverse, the Method of the Witness Asteroid to solve the systematical error of catalogue, use of set of instruments with increasing focal lengths (the final accuracy being the one of the larger one if a suffisant number of stars is taken upon the plates of each instrument). The research of the best formula for the ESO-Schmidt is also performed. The transformation plate-sky is created by the coefficients of 2 polynomials expressed in the plate measurements of one point. These polynomials give the 2 rectilinear coordinates X, Y of the corresponding celestial point. By verifying these 2 equations, PPM basis stars provide 2 INDEPENDENT rectangular algebraic systems. Both systems are squared by matrix multiplication. More vertically rectangular (more basis, in some limits) the system, more accurate the celestial coordinate. The accuracy is the Arithmetical Means of the Absolute Values of the Stars Residuals. It is computed in 4 areas, created by 3 concentric circles centered at the plate center, and in the Whole Plate. It is realized for Basis and Test Stars (basis being sometimes unrelevant), separately in alpha and delta because the independence of the 2 systems. For the same reason (independence), elimination of bad stars cannot be based on formulas mixing residuals in alpha and delta; polynomials and basis can be different for X and Y. The basis population depending of various parameters must be here 1.5 to 2 times the number of coefficients. On ESO-Schmidt plates, the best bijection being degree 3 with $y^{2}, x^{2}y$ and $y^{3}$ omitted for $X, x^{2}, x^{3}$ and $xy^{2}$ omitted for Y, 10 basis stars are suffisant. It is the distorsion of 1rst degree formula with 7 coefficients. It is possible, owing to one or more error-simulations, in whatever accuracy formula (in various matters or sciences), to replace the residuals by the true errors.