Based on the measurements performed in the first 14 months of Gaia operation, we have solved the problem of obtaining the systematic differences between the stellar positions and proper motions of the TGAS (Tycho–Gaia Astrometric Solution) and Tycho-2 catalogues. By dividing the common stars from the TGAS and Tycho-2 catalogues into three G-magnitude groups for mean values of 10m⋅5,11m⋅5,and13m⋅0, we have obtained the systematic differences between the stellar equatorial coordinates and proper motions of both catalogues in the form of a decomposition into vector spherical harmonics by taking into account the magnitude equation. The systematic components have been extracted from the individual differences with a probability of 0.977–0.999. The constructed model of systematic differences allows any position measurements performed using Tycho-2 as a reference catalogue to be transformed to the TGAS frame. An important fact is the existence of a magnitude equation in the systematic differences: when passing from bright (G = 10m) to faint (G = 13m) stars, the systematic position differences change within the range from approximately −40 to 15 mas, while the systematic proper motion differences change from −3 to 3 mas yr−1. The orientation and mutual rotation parameters of the Tycho-2 and TGAS frames have also been found to be different for stars of different magnitudes: when passing from bright to faint stars, the rotation angle of the Tycho-2 frame relative to TGAS changes from 3.51 to 5.63 mas, while the angular velocity of rotation changes from 0.35 to 1.22 mas yr−1. Based on the developed method that allows the extent to which the systematic errors in the equatorial propermotions of stars affect the results of a kinematic analysis of the Galactic proper motions to be estimated within the Ogorodnikov–Milne model, we have shown that the slope of the Galactic rotation curve and the Oort parameter C are most sensitive to the transition from the Tycho-2 frame to the TGAS one. Their relative changes after the transformation to the TGAS frame reach 56 and 100%, respectively. At the same time, the changes in the estimates of the Oort parameters A and B as well as the linear velocity of the Sun relative to the Galactic center, the Galactic rotation period, the ratio of the epicyclic frequency to the angular velocity of Galactic rotation, and the mass of the Galaxy within the Galactocentric distance of the Sun are not so large, being 2−10%.
