  We establish the formula for multiplication by the class of a special
Schubert variety in the integral cohomology ring of the flag manifold. This
formula also describes the multiplication of a Schubert polynomial by either an
elementary symmetric polynomial or a complete homogeneous symmetric polynomial.
Thus, we generalize the classical Pieri's rule for symmetric
polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our
primary technique is an explicit geometric description of certain intersections
of Schubert varieties. This method allows us to compute additional structure
constants for the cohomology ring, which we express in terms of paths in the
Bruhat order on the symmetric group.
