  The classical Schl\"afli formula, and its ``higher'' analogs given in [SS03],
are relations between the variations of the volumes and ``curvatures'' of faces
of different dimensions of a polyhedra (which can be Euclidean, spherical or
hyperbolic) under a first-order deformation. We describe here analogs of those
formulas which are vector-valued rather than scalar. Some consequences follow,
for instance constraints on where cone singularities can appear when a constant
curvature manifold is deformed among cone-manifolds.
