Theoretical computations (Hoyle and Schwarzschild, 1955; Faulkner, 1966; Iben and Rood, 1970; Demarque and Mengel, 1971a, b) have identified the horizontal-branch stars in globular clusters with the evolution phase in which helium burns within a convective core and hydrogen burns in a shell outside the convective core. Most computations for such double-energy-source models have indicated that the evolution proceeds smoothly on a nuclear time scale during the horizontal-branch phase, leading to small predicted rates of change in the RR Lyrae pulsation period (Iben and Rood, 1970). Sweigart and Demarque (1972) have recently considered the effects of semiconvection on the horizontal-branch evolution of typical Population II stars and have suggested that changes in the composition distribution within the core may occur on a time scale considerably shorter than the nuclear time scale during the phase immediately preceding core-helium exhaustion. It has been found that the composition distribution generated by the growth of a semiconvective zone in the layers surrounding the convective core can become unstable when Yc , the helium abundance within the convective core, decreases below roughly 0.12. The changes in the internal structure caused by this instability result in relatively rapid movement of the models in the HR diagram and consequently produce large predicted rates of change in the RR Lyrae pulsation period. The possibility that RR Lyrae period changes may be associated with the behavior of the semiconvective zone has been previously suggested by Schwarzschild (1970). A similar instability may occur in the late core-hydrogen burning phase for stars around 10 M ⊙. Percy (1970) has noted the coincidence of β Cephei stars with stellar models containing semiconvective zones. It is tempting to suggest that such an instability in the semiconvective zone could also be related to the β Cephei phenomenon.