  We determine an intersection rule for extremal p-branes which are localized
in their relative transverse coordinates by solving, in a purely bosonic
context, the equations of motion of gravity coupled to a dilaton and n-form
field strengths. The unique algebraic rule we obtained does not lead to new
solutions while it manages to collect, in a systematic 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 common and overall transverse directions. They are
given in terms of non-harmonic functions. A different algebraic rule emerges
for these last intersections, being identical to the intersection rule for
p-branes which only depend on the overall transverse coordinates. We clarify
the origin of this coincidence. The whole set of solutions in ten and eleven
dimensional theories is connected by dualities and dimensional reductions. They
are related to brane configurations recently used to study non-perturbative
phenomena in supersymmetric gauge theories.
