  We use the notion of generalized signatures at a singularity of a smooth
curve of symmetric bilinear forms to determine a formula for the computation of
the Maslov index in the case of a real-analytic path having possibly non
transversal intersections. We discuss some applications of the theory, with
special emphasis on the study of the Jacobi equation along a semi-Riemannian
geodesic.
  The research work exposed in this paper originated from a suggestion given by
the third author relating the spectral flow with the partial signatures. He
pointed out several references and formulated the statement that in the
invertible endpoints case, the spectral flow of a real analytic path of
self-adjoint Fredholm operators is given by the sum of the odd partial
signatures at each degeneracy instant. In the present version of the article,
this statement is part of Proposition 2.9; totally, the contribution of the
third author to the theory consists in Definition 2.3, formulas (2.1) and
(2.2), the first statement of Proposition 2.4, parts of Remark 2.5, Definition
2.6, and parts of the statement and parts of the proof of Proposition 2.9.
Apart from this original contribution, the material contained in this paper was
entirely developed and written by the first two authors at the Universita' di
Camerino (Italy), Universidade de Sao Paulo (Brazil).
