True triaxial tests have been carried out in two quartz-rich, high porosity, sandstones, Coconino (n = 17.5%) and Bentheim (n = 24%) by maintaining constant but different σ2 and σ3 and raising σ1 until failure occurred (at σ1,peak). For each constant σ3 level, σ2 was varied from test to test between σ2 - σ3 and σ2 = σ1, σ1,peak for a given σ3 increased with σ2, reached a maximum (up to 15% higher than when σ2 = σ3), and then declined so that when σ1 approached σ1, σ1,peak was about equal to its base value when σ2 = σ3. A separate series of tests was carried out using a novel loading path by maintaining constant Lode angle θ (= 0°). This series of tests characterized the dependence of the octahedral shear stress at failure τoct,f on the octahedral normal stress at failure σoct,f when θ = 0°. The latter tests were used to obtain the necessary parameters employed in a three-invariant failure theory proposed by Rudnicki (2013). The theory was then applied to predicting the variation of σ1,peak with σ2 for a given σ3. The prediction reasonably replicated the typical ascending-then-descending σ1,peak vs. σ2 trend observed experimentally in both sandstones, confirming (with some limitations) the applicability of the Rudnicki's (2013) theory.