A theoretical investigation of ignition of a premixed combustible gas by a line heat source is described. Laminar flow was examined by numerical solution of the nonsimilar boundary layer equations, for an ideal gas with a one, step reaction and neglecting radiation. A deflagration wave is always present in the plume for these assumptions l However, the flame makes a rapid transition away from the source over a relatively small range of reaction parameters. This transition state provides a practical estimate of limiting ignition conditions--assuming that flames far from the source are not observed due to flow disturbances, transition to turbulence, or quenching by surrounding surfaces. Computations of limiting ignition conditions using this criterion indicate that while plume ignition approximates ignition by finite-sized sources, as the source size becomes small, there are interesting differences between the two cases due to effects of distributed heat addition and surface friction when the source size is finite. Streamwise gradients were appreciable in some circumstances; therefore, further analysis of this problem is suggested--relaxing the boundary layer approximations.