The percentage of normal, red blood cells that are hemolyzed when placed in hypotonic solutions depends on a variety of factors, two important ones being the initial sphericities of the cells and the tonicities to which they are subjected. Other, less well-understood factors that are important in hemolysis are the initial cell volumes, how much free water they contain and the elasticity of the cell membranes. The purpose of this work is to identify the constraints a red cell must satisfy in order to be hemolyzed. Human erythrocyte data is used in a physical model that compares the balance of hydrostatic stresses in sphered cells that are on the verge of hemolysis. For hemolysis to occur we find there is a critical sphericity index that must be exceeded. It depends on tonicity, the initial, fractional water volume in the cells and the maximum fractional area dilation the cell membranes can withstand. Membrane tensile strength and the non-ideal osmotic behavior of hemoglobin are of relatively minor importance. But when they are taken into account, the hemolysis constraint, in the form of a remarkably simple inequality, compares favorably with clinical tests of erythrocyte osmotic fragility.