An exact solution of the wave equation is found for a form of one-dimensional potential energy which may be of use in discussing polyatomic molecular vibrational energies. An example of its use is given in an analysis of the vibration of the nitrogen in the ammonia molecule. The potential energy for this atom has two minima a distance $2{x}_{m}$ apart, separated by a "hill" of height $H$. The values of ${x}_{m}$ and $H$ are not known directly from band spectral data, and are needed for a full analysis of the spectrum. By joining two potential curves of the sort dealt with in the first part of this paper in a symmetric manner, a curve simulating that for the nitrogen atom in ammonia was formed. It was found that for certain values of the constants fixing this curve, the allowed vibrational energies were the same as the experimentally determined values for ammonia. The corresponding value of ${x}_{m}$ was 0.38A, and that of $H$ was \textonequarter{} electron-volt. These values are probably near the correct values of ${x}_{m}$ and $H$ for ammonia.