Numerical calculations have been carried out which cover the first 4 million years of those evolution phases in which helium burning occurs in a shell in a star of 1 solar mass and of a Population II composition. The thermal instability previously found to occur in these helium-shell-burning phases leads to relaxation oscillations-just as has already been shown by other authors for other types of stars. Thirteen consecutive relaxation cycles are covered by the present computations. In each cycle the main helium- shell flash causes a convective xone to stretch outward from the helium shell. In the initial relaxation cycles this convection xone does not reach the hydrogen-containing layers. However, cycle after cycle the main helium-shell flash increases in strength and, in consequence, causes a larger and larger convection xone. Mter about nine cycles the convection xone finally reaches the hydrogen-containing layers and from then on, for a short while in each cycle, hydrogen is mixed into the hot carbon-rich interior. This physical situation with its likely importance for nucleosynthesis we had searched for originally in connection with the helium-core flash, with, however, negative results; while now, in connection with the helium-shell flashes, the results seem to be encouragingly positive.