  Five-dimensional spacetimes of two orbifold 3-branes are studied, by assuming
that {\em the two 3-branes are spatially homogeneous, isotropic, and
independent of time}, following the so-called "bulk-based" approach. The most
general form of the metric is obtained, and the corresponding field equations
are divided into three groups, one is valid on each of the two 3-branes, and
the third is valid in the bulk. The Einstein tensor on the 3-branes is
expressed in terms of the discontinuities of the first-order derivatives of the
metric coefficients. Thus, once the metric is known in the bulk, the
distribution of the Einstein tensor on the two 3-branes is uniquely determined.
As applications, we consider two different cases, one is in which the bulk is
locally $AdS_{5}$, and the other is where it is vacuum. In some cases, it is
shown that the universe is first decelerating and then accelerating. The global
structure of the bulk as well as the 3-branes is also studied, and found that
in some cases the solutions may represent the collision of two orbifold
3-branes. The applications of the formulas to the studies of the cyclic
universe and the cosmological constant problem are also pointed out.
