  The computation of the index of the Hessian of the action functional in
semi-Riemannian geometry at geodesics with two variable endpoints is reduced to
the case of a fixed final endpoint. Using this observation, we give an
elementary proof of the Morse Index Theorem for Riemannian geodesics with two
variable endpoints, in the spirit of the original Morse's proof. This approach
reduces substantially the effort required in the proofs of the Theorem given in
previous articles on the subject. Exactly the same argument works also in the
case of timelike geodesics between two submanifolds of a Lorentzian manifold.
For the extension to the lightlike Lorentzian case, just minor changes are
required and one obtains easily a proof of the focal index theorems of Beem,
Ehrlich and Kim.
