Intense ultralow-frequency waves are commonly observed on GEOS 1 and 2 spacecraft, around FHe+, the helium gyrofrequency. These waves were identified as ion cyclotron wves (ICW's) by Young et al. (1981), who showed their close connection with a sufficient amount of He+ in the magnetospheric plasma. Motivated by these observations, we present a ray tracing study of ULF waves below FH+, the proton gyrofrequency. In the presence of He+ the dispersion relation of ULF waves is split into three branches, and we have studied the ray paths for these three branches. Of particular interest is the ion cyclotron branch, which is left handed above the new crossover frequency Fcr introduced by the presence of He+ ions. This mode is amplified, in the equatorial region, by anisotropic (T1>T∥) energetic protons. It is well guided along field lines, suffers a polarization reversal when F=Fcr locally, and continues to be guided up to the point where F=Fbi, the bi-ion hybrid frequency. Then it is reflected and returns to the equator, where it is amplified again. Thus such waves undergo several bounces through the amplifying region without significant drift, either azimuthal or radial. This mirroring effect, which can only occur when minor ions, such as He+, are present, is believed to be crucial for ICW amplification. We also show that for each such equatorial crossing the parallel wave number is conserved while the perpendicular one progressively increases. As a consequence, ICW's become quasi-electrostatic, which enables them to acquire a small but finite parallel electric field; it is suggested that this electric field can in turn accelerate thermal electrons parallel to B.