We prove the existence of the generalized Samuel number [wbar]f(g) and the equalityfor any AP filtration f = (In) and any filtration g = (Jn) on a ring A. It is shown that [wbar]f(g) ≥ ⊢f(g) for all AP filtrations f and g where f is separated and nonnilpotent. Two real numbers āf(g) and ƀf(g) are introduced. It is shown that āf(g) = ⊢f(g) if ⊢f(g) exists (resp if [wbar]f(g)exists). Several properties of numbers āf(g) and ƀf(g) are given. It follows a generalization of ([1], Theorem 5. 6) and a generalization of ([6], Theorem 2) given by the formulawhere f, g, h are filtrations on a ring A.