The angular accelerations of the crank and lever of a four-bar mechanism have been related herein to the velocity of their rotopole through an equation developed by Koenig. It is shown that the Koenig equation may be transformed in a simple manner into the collineation-axis equation derived by Freudenstein. Other collineation-axis equations are here derived which permit a direct solution for the angular acceleration of the connecting link of a four-bar mechanism. These equations are differentiated with respect to time yielding fairly simple equations for the determination of the third time derivatives of lever and connecting-link motions. The collineation-axis equations are further employed to determine the families of four-bar mechanisms which satisfy the first three derivatives of a Taylor’s series expansion of a desired functional relation between the crank and lever rotations. The fourth time-derivative relation for these mechanisms is then determined using the complex-number vector notation. Finally the solutions are given to example problems of kinematic analysis and synthesis.