  A multidimensional cosmological model with scalar and form fields [1-4] is
studied. An exact S-brane solution (either electric or magnetic) in a model
with l scalar fields and one antisymmetric form of rank m > 1 is considered.
This solution is defined on a product manifold containing n Ricci-flat factor
spaces M_1, ..., M_n. In the case when the kinetic term for scalar fields is
positive definite we singled out a special solution governed by the function
cosh. It is shown that this special solution has Kasner-like asymptotics in the
limits \tau \to + 0 and \tau \to + \infty, where \tau is a synchronous time
variable. A relation between two sets of Kasner parameters \alpha_{\infty} and
\alpha_0 is found. This relation, named as ``scattering law'' (SL) formula, is
coinciding with the ``collision law'' (CL) formula obtained previously in [5]
in a context of a billiard description of S-brane solutions near the
singularity. A geometric sense of SL formula is clarified: it is shown that SL
transformation is a map of a ``shadow'' part of the Kasner sphere S^{N-2} (N =
n+l) onto ``illuminated'' part. This map is just a (generalized) inversion with
respect to a point v located outside the Kasner sphere S^{N-2}. The shadow and
illuminated parts of the Kasner sphere are defined with respect to a point-like
source of light located at v. Explicit formulae for SL transformations
corresponding to SM2- and SM5-brane solutions in 11-dimensional supergravity
are presented.
