Local similarity and related methods are used to analyze the heterogeneous ignition mechanism of solid fuels. Analytic solutions are derived for the surface temperature history in order to determine the time and nature of the transition from a weak combustion mode to a strong one. The existence of the various burning modes is attributed to a time-dependent Damkohler number. It is then shown that non-unique solutions in the local similitude transform plane correspond to discontinuities in the physical plane. The ignition phenomenon is readily associated with the discontinuity so that it is not necessary to resort to arbitrary ignition criteria. The solutions are shown to compare favorably with exact numerical solutions. Finally, the existence of discontinuities suggests that ignition problems may be amenable to asymptotic analysis.