  We construct exact time-dependent solutions of the supergravity equations of
motion in which two initially non-singular branes, one with positive and the
other with negative tension, move together and annihilate each other in an
all-enveloping spacetime singularity. Among our solutions are the Horava-Witten
solution of heterotic M-theory and a Randall-Sundrum I type solution, both of
which are supersymmetric, i.e. BPS, in the time-independent case. In the
absence of branes our solutions are of Kasner type, and the source of
instability may ascribed to a failure to stabilise some of the modulus fields
of the compactification. It also raises questions about the viability of models
based on some sorts of negative tension brane.
