Neutron diffraction measurements on high-purity Dy and Ho crystals indicate no discontinuity in the order parameter at T_N, and no hysteresis in the value of T_N within 0.05 K between consecutive cooling and heating runs. This finding is also reflected by the results of ultrasonic velocity and attenuation measurements, thus corroborating the continuous nature of the phase transition. Integrated neutron diffraction measurements near T_N yield beta =0.38+or-0.02 for Dy and beta =0.39+or-0.02 for Ho in agreement with renormalization group predictions ( beta =0.39) of Mukamel, Krinsky and Bak. Neutron intensities in the paramagnetic region do not adhere to a conventional t₊^{- gamma} dependence over the complete temperature regime. A possible description for the region near T_N with a 2D-planar spin model is indicated. The critical velocity change Delta nu ₃₃ of Ho near T_N is discussed.