General inequalities of type (5) for eigen values were established as early as 1912 by H. Weyl  through consideration of integral equations; he used them for studying the asymptotic behaviour of λ n when n→∞. These inequalities seem to have been forgotten, as some of them were independently re-discovered by several authors (see , p. 314 and , pp. 483 484). —Such inequalities are here derived and applied in various ways to special problems (vibrating strings, rods, membranes and plates). Explicit lower bounds for λ1 are obtained in a somewhat similar manner in terms of the Green's function. For Schrödinger's equation (15'), the very simple inequalities (17') are found. Analogous inequalities are established for a class of equilibrium problems. Many of the inequalities thus obtained express convexity properties (cf. the Post-scriptum above and the paper by Pólya and Schiffer quoted there).