A method of solving the band problem is developed and applied which makes use of a composite representation of the wave function and of a variational expression for the energy applicable to trial functions with discontinuities in their values and derivatives on a surface within the cell. The composite representation consists of spherical waves belonging to the given potential within the inscribed sphere, and symmetrized combinations of plane waves for the given k outside the inscribed sphere. The potential is assumed spherical within the inscribed sphere but not outside. The calculated energy is variational even when the spherical waves are computed with trial energies differing from the true energy; hence, the process of iterating on the trial energies is strongly convergent. Results are obtained with the flattened Seitz potential for Li and the flattened Prokofiev potential for Na at many points, including points of general k. The Li energies check with high-precision values obtained independently, give a Fermi level at -0.429 eV and a Fermi surface that bulges toward, but does not touch, the zone face in the  direction. Comparisons are made with a number of other recent methods of solving the band problem. © 1963 The American Physical Society.