The betatron mechanism was proposed by Brown and Hoyng (1975) as a means of producing the continuous, quasi-periodic electron acceleration which may occur in long-lasting hard X-ray events. In the present work, two pertinent facets of the betatron model are investigated: The possibility that the multiplicity characteristic of complex impulsive bursts is due to the betatron process; and the possibility that some or all of the second-stage emission during two-stage bursts can be attributed to betatron acceleration. To test for the pattern of X-ray spectral behavior predicted by the betatron model, a number of multiply-impulsive events (cf., Karpen et al., 1979) and two-stage bursts (cf., Frost and Dennis, 1971) were selected from the OSO-5 hard X-ray spectrometer data for in-depth analysis. The purely impulsive emissions show no signs of the effects of betatron action, thus eliminating this process as a potential source of impulsive-phase multiplicity. However, the spectr".! characteristics determined during the first few minutes of the second stage are found to be consistem ~v,h the predictions of the betatron model for the majority of the two-stage events studied. The betatron-acceleration mechanism thus is proposed as a common second-stage phenomenon, closely associated with the diverse phenomena at other wavelengths which characterize this phase of emission. The physical significance of the source parameters derived according to the model-fitting procedure are discussed in detail, and the role of the betatron process is evaluated in the broader context of present-day concepts of the second stage.