  The notions of reflection from outside, reflection from inside and signature
of a billiard trajectory within a quadric are introduced. Cayley-type
conditions for periodical trajectories for the billiard in the region bounded
by $k$ quadrics in $R^d$ and for the billiard ordered game within $k$
ellipsoids in $R^d$ are derived. In a limit, the condition describing periodic
trajectories of billiard systems on a quadric in $R^d$ is obtained.
