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Two new campaigns devoted to the observation of the solar limb distortions were made at the Pic-du-Midi Observatory, in September 2000 and September 2001, by means of the scanning heliometer. This apparatus can be used now routinely to accurately determine solar limb profiles (at two wavelengths), at any heliographic latitudes. Each measurement is made within 44 milliseconds (of time) which permits to record a limb profile together with the seeing. Scans are automatically rejected for seeing larger than 1.3 arc sec. Such conditions are essential to perform high-quality observations necessary to obtain the quadrupole term (l = 2) in the polynomial expansion of the radius contour R(φ)∣ρ = constant = R0left(1+∑lclPl(φ) right). Exceptional meteorological conditions in September 2001 (seeing of the order of 18 cm, for a 50 cm clear aperture of the refractor) enabled us to determine c2 and c4 (see Table I) with an accuracy of a few milli-arc-sec. Results indicate a distorted solar shape, the departures from a pure spherical body not exceeding 20 milli-arc-sec. We propose a model to interpret such results (the combination of a nearly uniform rotating core with a prolate solar tachocline and an oblate surface), which is briefly discussed. Our results are confronted to those obtained from space. We conclude that measurements of the quadrupole term from the ground are possible, but of high difficulty and can be obtained only during excellent weather conditions. The hexadecapole term should be only obtained from space. We show that an astrometric satellite would be required, whose mission would be also to accurately determine the solar rotation profiles (both surface and in depth) in order to unambiguously determine the inertia moments of the Sun through the Jn terms. Such values are also briefly discussed.