In some recent work on uniformization [2], I found it necessary to consider a regular branched covering Riemann surface Ȓ of a given Riemann surface Rf, where Rf is an unlimited branched, but not necessarily regular, covering surface of a portion Rz of the extended complex z-plane Z(2-sphere). The branching of Ȓ over Rf had to be chosen so that Ȓ was regular over Rz, since the uniformization of the functions on Rf is then simpler; in particular, the Schwarzian derivative is then a single-valued function of z.