In a recent paper Tsuji [1952] has given some theorems which may be considered as a generalization to Fuchsian groups of the classical theorems of Blichfeldt [1914] and Minkowski in the geometry of numbers. In both cases, however, the generalization is restricted to circular domains. The object of this paper is to carry this generalization further. First, by considering more general groups than the Fuchsian groups; second, by considering a kind of domains more general than the circular domains