  Asymptotic representations for large values of the hyperradius are
constructed for the scattering wave function of a system of $ N $ particles
considered as a generalized function of angular variable coordinates. The
coefficients of the asymptotic representations are expressed in terms of the
$N$-particle scattering matrix. The phenomenon of asymptotic filtration is
discovered, which consists in the fact that only scattering processes
contribute to the leading terms of such an asymptotic representation, in which
all particles are free both before and after interaction. The obtained
representations are used to construct the correct asymptotics of the partial
components of the wave function of $N$ particles in the hyperspherical
representation.
