  We analyze a pair of reflected Brownian motions in a planar domain $D$, for
which the increments of both processes form mirror images of each other when
the processes are not on the boundary. We show that for $D$ in a class of
smooth convex planar domains, the two processes remain ordered forever,
according to a certain partial order.
  This is used to prove that the second eigenvalue is simple for the Laplacian
with Neumann boundary conditions for the same class of domains.
