  We discuss a general fourth-order theory of gravity on the brane. In general,
the formulation of the junction conditions (except for Euler characteristics
such as Gauss-Bonnet term) leads to the higher powers of the delta function and
requires regularization. We suggest the way to avoid such a problem by imposing
the metric and its first derivative to be regular at the brane, while the
second derivative to have a kink, the third derivative of the metric to have a
step function discontinuity, and no sooner as the fourth derivative of the
metric to give the delta function contribution to the field equations.
Alternatively, we discuss the reduction of the fourth-order gravity to the
second-order theory by introducing an extra tensor field. We formulate the
appropriate junction conditions on the brane. We prove the equivalence of both
theories. In particular, we prove the equivalence of the junction conditions
with different assumptions related to the continuity of the metric along the
brane.
