  An analytical method for solving the problem of the satellite around the
oblated Earth is discussed. Such a method goes back to Dirk Brouwer (1959) and
seems to be still widely used in real applications. We reformulate in a more
concise way the core of the method which is essentially based on a technique of
compositions of power series. In this way we derive some general formulas which
cover, all at once, all the cases treated in the original paper and allow to
compute some terms which were omitted there. We also show some simple
computation we performed to test the formulas.
