Saito, Quintiere and Williams developed an integral equation derived from quasi-steady thermal flame spread theory to describe spread up a thick solid. Allowance was made in this purely thermal model for a varying rate of pyrolysis and the relationship between flarne length and burning rate was linearised. They discussed asymptotic solutions to their equation and derived the requirements for two-dimensional propagation both for a step function pyrolysis rate m" = constant up to a burnout time and an inverse square root variation. Here their integral equation is solved analytically by Laplace Transform and is also applied to pyrolysis which decreases exponentially with time. The three conditions for propagation are compared, all being of the type tig KQ" = function ----- tB where: K is constant Q" is a characteristic heat release rate tig is the ignition time tB a characteristic of the duration of pyrolysis Q", tig and tB are obtained under the rate of heating imposed by the flame In those flames which spread indefinitely a comparison is made with Hasemi, Yoshida and Nohara's (2) asymptotic theory. Some limitations of quasi-steady theory are considered. Ett BRANDFORSK projekt.