From June 11 to September 16, 1974, the NOAA low-energy proton detector on board the ATS 6 satellite observed 71 cases of ultralow-frequency oscillations of proton flux intensities. The oscillation periods varied from 40 s to 6 min, and the events were observed most frequently during moderate geomagnetic conditions. The flux oscillations occurred at various local times, yet almost two thirds of the events were detected in the near-dusk region of the magentosphere. For a majority of the events in this set a substantial phase shift in flux oscillation was detected between different energy channels and/or between two oppositely oriented detector telescopes. The phase shift is mainly due to the finite gyroradius effect of the protons gyrating in the geomagnetic field. By examining this finite gyroradius effect on the perturbed particle distribution function associated with the wave in a nonuniform magnetic field we are able to determine the propagation direction of the wave from particle observations made by a single spacecraft. Although the type of wave and its excitation mechanism can only be conjectured at the present time, it is concluded that the wave propagates in the westward direction with a phase velocity of about 13 km/s. Furthermore, it also has a very small phase velocity ~0.15 km/s propagating toward the earth. If the wave had been traveling 1 hour or so before it was observed near the dusk magnetosphere, it might have originated in the dark magnetosphere in associating with some changes in geophysical conditions. The statistical correlation between the times of the observed wave events and the onsets of the auroral magnetic bays indicates that although they seldom occurred simultaneously, 80% of the waves were observed within 1 hour of the bay onset. Therefore it is concluded that the condition of the magnetosphere after a substorm is favorable for the occurrence of the ring current proton ultralow-frequency waves.