  In this article, we investigate the mathematical part of De Sitter's theory
on the Galilean satellites, and further extend this theory by showing the
existence of some quasi-periodic librating orbits by applications of KAM
theorems. After showing the existence of De Sitter's family of linearly stable
periodic orbits in the Jupiter-Io-Europa-Ganymede model by averaging and
reduction techniques in the Hamiltonian framework, we further discuss the
possible extension of this theory to include a fourth satellite Callisto, and
establish the existence of a set of positive measure of quasi-periodic
librating orbits in both models for almost all choices of masses.
