  We give explicit formulas for the residue of the Chinta-Gunnells average
attached to a finite irreducible root system, at the polar divisor
corresponding to a simple short root. The formula describes the residue in
terms of the average attached to the root subsystem orthogonal to the relevant
simple root. As a consequence, we obtain similar formulas for the residues of
quadratic Weyl group multiple Dirichlet series over the rational function field
and over the Gaussian field. The residue formula also allows us to obtain a new
expression for the Chinta-Gunnells average of a finite irreducible root system,
as an average over a maximal parabolic subgroup of a rational function that has
an explicit description reflecting the combinatorics of the root system.
