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.