  We give a new residual intersection decomposition for the refined
intersection products of Fulton-MacPherson. Our formula refines the celebrated
residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The
new decomposition is more likely to be compatible with the canonical
decomposition of the intersection products and each term in the decomposition
thus has simple geometric meaning. Our study is motivated by its applications
to some geometric problems. In particular, we use the decomposition to find the
distribution of limiting linear subspaces in degenerations of hypersurfaces. A
family of identities for characteristic classes of vector bundles is also
obtained as another consequence. This paper will appear in Duke Math. Jour.
