This paper is concerned with the vibration analysis of spherical shells, closed at one pole and open at the other, by means of the linear classical bending theory of shells. Frequency equations are derived ha terms of Legendre functions with complex indices, and for axisymmetric vibration the natural frequencies and mode shapes are deduced for opening angles ranging from a shallow to a closed spherical shell. It is found that for all opening angles the frequency spectrum consists of two coupled infinite sets of modes that can be labeled as bending (or flexural) and membrane modes. This distinction is made on the basis of the comparison of the strain energies due to bending and stretching of each mode. It is also found that the membrane modes are practically independent of thickness, whereas the bending modes vary with thickness. Previous analyses with the use of membrane theory have shown that one of two infinite sets of modes is spaced within a finite interval of the frequency spectrum. It is shown in this paper that this set of modes is a degenerate case of bending modes, and, if deduced by means of membrane theory, it is applicable only when the thickness of the shell is zero. When the bending theory is employed, then the frequency interval for this set of modes extends to infinity for every value of thickness that is greater than zero.