It is shown that the stability limit for pressurized horizontal floating zones is related to the eigenvalues of a membrane of the same shape. In particular, a rectangle of lengthaand widthbis stable if&pgr;2(4/a2+1/b2)≥&rgr;g/&ggr;,and a circle of radiusRis stable if(&bgr;11/R)2≥&rgr;g/&ggr;,where &rgr; is the density,gis the acceleration of gravity, &ggr; is the surface tension constant, and &bgr;11=3.8317 is the first root of the Bessel functionJ1(x). This latter case checks with experiments on melted circular zones of tin.