We have studied the element abundances and energy spectra of the small "He-poor" impulsive solar energetic-particle (SEP) events, comparing them with other impulsive SEP events with more-normal abundances of He. He-poor events can have abundances as low as He/O ~ 2, while both impulsive and gradual SEP events usually have source abundances of 30 < He/O < 100 with mean values of 50 - 60. He/C ratios are not only low, but often decrease with energy in He-poor events. Abundance enhancement patterns of other elements with atomic numbers 6 < Z < 56, and likely values of their mass-to-charge ratios A/Q, are generally unaltered in He-poor events, as are the probable source-plasma temperatures of 2.5 - 3.2 MK for all impulsive SEP events. One He-poor event is also an example of a rarer C-poor event with C/O = 0.08 +- 0.04, sup-pressed by a factor over 5 from the mean. We discuss suggestions of a possible A/Q threshold during acceleration and of the sluggish ionization of He entering the corona, because of its uniquely high first ionization potential (FIP), but the suppression of He and the decline of He/C with energy is difficult to explain if both He and C are fully ionized with A/Q = 2 as expected at 2.5 - 3.2 MK. Although less dramatic, a possible excess enhancement of Ne in some impulsive SEP events is also considered. Possible causes of the large ~30% spectral and abundance variations in impulsive events are also discussed. However, the physics of the He-poor events remains a mystery.
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