2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76514For a membrane in the plane the multiplicity of the $k$-th eigenvalue is known to be not greater than $2k-1$. Here we prove that it is actually not greater than $2k-3$, for $k\ge 3$.AmSTeX, epsf.tex, 17 encapsulated postscript figures, 17 pages. Third version with minor stylistic improvementsAnalysis of PDEsDifferential Geometry35B05, 35P15, 58G25text