Both the protons and the helium ions of the terrestrial magnetosheath typically display T⊥>T∥, where ⊥ and ∥ denote directions perpendicular and parallel to the background magnetic field. Observations of the highly compressed magnetosheath show an inverse correlation between these ion temperature anisotropies and the parallel proton &bgr;. Computer simulations have demonstrated that these correlations are due to wave-particle scattering by electromagnetic ion cyclotron anisotropy instabilities. These correlations correspond to linear theory thresholds of the proton cyclotron and the helium cyclotron instabilities. This paper uses linear Vlasov theory and the assumption of a constant maximum growth rate to obtain closed-form expressions for these thresholds as a function of the relative helium density and the parallel proton &bgr; in a parameter model of the magnetosheath. ¿ American Geophysical Union 1994