The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product.