RULES FOR LOCALIZED OVERLAPPINGS AND INTERSECTIONS OF P-BRANES

We determine an intersection rule for extremal p-branes which are loca lized in their relative transverse coordinates by solving, in a purely bosonic context, the equations of motion of gravity coupled to a dila ton and n-form field strengths. The unique algebraic rule we obtained does not lead to new solutions while it manages to collect, in a syste matic way, most of the solutions (all those compatible with our ansatz ) that have appeared in the literature. We then consider bound states of zero binding energy where a third brane is accomodated in the commo n and overall transverse directions. They are given in terms of non-ha rmonic functions. A different algebraic rule emerges for these last in tersections, being identical to the intersection rule for p-branes whi ch only depend on the overall transverse coordinates. We clarify the o rigin of this coincidence. The whole set of solutions in ten and eleve n dimensional theories is connected by dualities and dimensional reduc tions. They are related to brane configurations recently used to study non-perturbative phenomena in supersymmetric gauge theories.