In paper 7) we concerned ourselves with the conformal mapping onto circular-radial slit covering surfaces over the whole plane and its extremal property. In the present paper we shall concern ourselves with the conformal mapping onto circular-radial slit covering surfaces of annular and circular types and their extremal properties (Theorems 1.1 and 2.1). Especially the extremal property with respect to the radial slits is new. The results are stated only for the case of the planar domain of finite
connectivity. The method suggests the possibility of an extension to the case of a domain of infinite connectivity or an open Riemann surface of finite genus. We shall concern ourselves with this problem in the subsequent paper.