Hoffmann-Ostenhof T., Michor P. W., Nadirashvili N. Bounds on the Multiplicity of Eigenvalues for Fixed Membranes (582K, Postscript) ABSTRACT. For 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 $k\ge3$.