Let X be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric m). We derive the variational formulas for the determinant of the Laplacian, \(\textrm{det}\,\Delta ^m\) , on X under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of X in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of \(\textrm{det}\,\Delta ^m\) performed by Aurell and Salomonson in the 90-s.
