The velocity ratio i = d ψ/d ϕ and its derivatives i′ = d 2 ψ/d ϕ 2 and i″ = d 3 ψ/d ϕ 3 of the four-bar linkage (Fig. 1) along with angular velocity ω an of the input link lead to the angular velocity ω ab of the output link, the angular accelleration ge ab and its derivative ϵ dot ab . Equations for computing i, i′ and i″ are known [2, 11, 4, 6, 7, 9, 10]. It is proper to have them in the form i = f(q) = f( i, λ) and i″ = f( i′, i, D) ; q, λ and D are parameters which can be taken from the drawing of a linkage in a fixed position. The equations are suitable for analysis. Synthesis of linkages is possible if q, λ and D are on the left side of the equations. In this way it is very easy to perform the synthesis for a four-bar function generator with four precision-points in an infinite neighborhood. In the equation i″ = f( i′, i, D) the drawing parameter D is the diameter of the so called Carter-Hall-circle [1,5,7]. With constant values of i, i″ it is the geometrical line of all velocity poles P of four-bar linkages (Fig. 1), whose transmission functions have four precision points in common with a desired function. The author has developed special equations for fixed link, dead center and parallel positions of plane four-bar linkages. In these special cases we often get indeterminate terms with the known equations and then have to seek for new drawing parameters instead of λ or D[8]. Similar derivations give the corresponding equations for slider-crank mechanisms too. All equations are compiled with the respective special positions of the four-bar linkages in a table.