In a previous paper [Ⅰ] we proved the theorem, concerning the behaviour of the sumnability (c, r) for negative indices of the Fourier series of a monotonic function with an infinite limit. It is natural to inquire what happens when the indices are positive.Let be the Fourier series of an L-integrable function f(θ), and σ_n~r(θ) the r-th Cesaro mean of the series at the point θ.Define φ(t) = 1/2 [f(θ + t) + f(θ - t)]. We shall be concerned with behaviour at a single point θ, which we may suppose to be θ=0. And then σ_n~r(O) is the r-th Cesaro mean of the Fourier series of the even function φ(t) at the point t=0.Theorem (i) If r≥1 and limφ(t) = + ∞, then (ii) If0