  Let $W$ be a finite Coxeter group and $V$ its reflection representation. The
orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined
as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant
of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in
$\mathcal{M}_W$. It is shown that this determinant is proportional to a product
of linear factors in the flat coordinates of the form $g$ on the stratum. We
also find multiplicities of these factors in terms of Coxeter geometry of the
stratum.
  This result may be interpreted as a generalisation to discriminant strata of
the Coxeter factorisation formula for the Jacobian of the group $W$. As another
interpretation, we find determinant of the operator of multiplication by the
Euler vector field in the natural Frobenius structure on the strata.
