The equation governing elastic waves propagating along a solid-solid interface is found to have sixteen (16) independent roots on its eight (8) associated Riemann sheets. The range of existence (in terms of material parameters) for the real root corresponding to the propagation of Stoneley waves has long been known. It is found that outside this range there are two types of behavior. If the material of greater density has a velocity slightly greater than that of the material of lesser density, the unattenuated Stoneley waves make a transition to attenuated Interface waves, i.e., they leak energy away from the interface as they propagate along it. If the more dense material has a velocity more than about three times that of the less dense, then the Interface-wave root disappears and energy is propagated along the interface as Rayleigh waves. This Rayleigh-wave propagation is associated with a different root of the fundamental equation. On the other hand, if the material of greater density has a velocity much lower than that of the material of lower density (a case that is difficult to find physically), then no energy will be propagated along the interface at all. This result was unexpected. Some rather interesting behavior of the 16 roots was noted as the physical parameters were varied over a wide range. In addition to the normal collisions between pairs of roots, and between individual roots and branch points (with attendant Riemann sheet jumping), it was found that some roots go through the point at infinity and return with a change in sign. At least one unexpected case of a multiple root was found. Another case was noted in which a pair of complex roots change quadrants in the complex phase-velocity plane, leading to a discontinuity in root type. Finally, it was noted that, in a cyclic variation of the material parameters, it is possible to choose a path such that the roots, when followed individually, will not return to their original values. In fact, as many as five cycles in parameter space can be accomplished before the roots return. All this strange mathematical behavior seems to have no physical significance, but has been presented to increase understanding of the general behavior of the dispersion relations associated with elastic-wave propagation.