In this paper analytical expressions for the extreme velocity ratio of a four-bar linkage are derived by the application of vector loop equations. Alternative forms for Freudenstein's theorems are presented. Either geometrically or analytically, the theorems can be proved concisely. For an arbitrary four-bar linkage, Freudenstein's theorem can examine the occurrence of the phases at which the angular velocity ratio of links becomes extreme, but it cannot straightforwardly determine the exact phases to happen. (C) 1998 Elsevier Science Ltd.