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We present a method of comparing the Galactic systems realized by two astrometric catalogues. The systematic differences between positions and proper motions are represented by vector spherical harmonics. To extract the signal from the noise, we use a statistical criterion adapted to using HEALPIX data pixelization to determine the significance of all the accessible harmonics. We also use a new analytical method that includes the magnitude equation in the vector spherical harmonics technique. The influence of the magnitude equation on the determination of the mutual orientation and rotation of the PPMXL and UCAC4 Galactic reference frames has been found in the range of J magnitudes from 10.25 to 15.75 mag. The angles of mutual orientation and the rates of mutual rotation of the Galactic frames under consideration depend on magnitude and can reach the level of 10 mas in orientation and 0.7 mas yr−1 for spin. We make a kinematic study of the low degree harmonics in the representation of the systematic differences between the Galactic proper motions. We have found that, averaged over the magnitude range, the biases of the Oort constants due to systematic differences of proper motions between the two catalogues, which are as large as 〈 A〉 = 1.60 ± 0.41 and 〈 B〉 = −1.91 ± 0.32 km s−1 kpc−1, are greater than the standard errors of their evaluation in the systems of these catalogues. The theoretical equations used in this paper are based on real vector harmonics. We present a set of formulae to convert them into the complex function formalism.

Comparison of the Galactic coordinate frames realized by the PPMXL and UCAC4 catalogues