  A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formula
on an orbifolded 2-sphere is elaborated and extended to higher dimensions using
the known heat-kernel coefficients for manifolds with piecewise-linear
boundaries. Several results are obtained that relate the coefficients, $b_i$,
in the Shephard-Todd polynomial to the geometry of the fundamental domain. For
the 3-sphere we show that $b_4$ is given by the ratio of the volume of the
fundamental tetrahedron to its Schl\"afli reciprocal.
