Abstract The suggestion, made in , that hyperbolic regions of slowness and wave surfaces might afford the means whereby orientations for which the transit times of waves are invariant is further examined. General principles of time invariance, consistency of slowness and of energy flux are developed. At linear approximation in increments of slowness and of temperature, invariance of wave travel-time is shown to depend on the existence of real intersections of the slowness surface and another associated surface of the same degree. In the case of elastic waves, the two surfaces are, in general, sextic. In hexagonal media, the sextic for the slowness of elastic waves is resolvable into quadratic and quartic factors and this decomposition permits the problem to be reduced to the consideration of two quartic plane curves. The possibility of intersection of these curves is discussed in detail for some zinc-like materials. While the mathematical validity of the original suggestion is vindicated, the normal observed magnitudes of the temperature dependences of the stiffnesses, compared with the values of thermal expansion coefficients, are such that their effect outweighs the counter-balancing contribution from negative curvature in the equations. Some anomaly in temperature dependence appears essential for the existence of the intersections required for temperature invariance; however, the strength of such anomaly may be abated if the elastic stiffness tensor occasions hyperbolic regions in the slowness surface.