  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 bi-linear 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 parameterization 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 formalism is applied to the FNAL electron
cooler and Derbenev's vertex-to-plane adapter.
