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[en] Presently, there are two most frequently used parameterizations of linear x-y coupled motion used in the accelerator physics. They are the Edwards-Teng and Mais-Ripken parameterizations. The article is devoted to an analysis of close relationship between the two representations, thus adding a clarity to their physical meaning. It also discusses the relationship between the eigen-vectors, the beta-functions, second order moments and the bilinear form representing the particle ellipsoid in the 4D phase space. Then, it considers a further development of Mais-Ripken parameterization where the particle motion is described by 10 parameters: four beta-functions, four alpha-functions and two betatron phase advances. In comparison with Edwards-Teng parameterization the chosen parametrization has an advantage that it works equally well for analysis of coupled betatron motion in circular accelerators and in transfer lines. Considered relationship between second order moments, eigen-vectors and beta-functions can be useful in interpreting tracking results and experimental data. As an example, the developed formalizm is applied to the FNAL electron cooler and Derbenev's vertex-to-plane adapter. In many applications analysis of coupled betatron motion is an important part of the machine design. The development of accelerator technology has stimulated additional interest in the subject in recent years. Initially betatron coupling was an undesired effect and efforts were made to suppress it. However, over recent two decades the betatron coupling has become an intrinsic part of many accelerator proposals (1-4). Although many studies of the coupled motion have been performed over the last 40 years (5-14), in our opinion there is still no representation of coupled betatron motion that would be as elegant as the Courant-Snyder parametrization (15) for the one-dimensional case. Presently, two different basic representations are most frequently used. The first one was proposed by Edwards and Teng (5, 6) and the second one by Mais and Ripken (7, 8). This article follows the steps of the second representation, where we limit the number of independent parameters to ten to parameterize a 4x4 symplectic transfer matrix. They are the four beta-functions, the four alpha-functions and the two betatron phase advances. The beta-functions have similar meaning to the Courant-Snyder parametrization, and the definition of alpha-functions coincides with the standard one in regions with zero longitudinal magnetic field, where they are equal to negative half-derivatives of the beta-functions. The article also reveals a close correspondence between the proposed parametrization and the Edwards-Teng parametrization, thus adding more clarity to their physical meaning. Section 2 is mainly based on references (6), (8) and (16). They describe the equations of motion, the notation and the basics of the theory developed in the 50's and the 60's. Section 3 sets relations between eigen-vectors, emittances and the particle 4D-ellipsoid in the phase space. Sections 4-6 develop the proposed representation and section 7 shows its correspondence to the Edwards-Teng parametrization.