The oxidation of hydrogen under well stirred, flowing conditions is the natural prototype of branched-chain reactions in open, gaseous systems. Experimentally, it exhibits the classical forms for the (p-T$_a$) first and second ignition limits in the flow system. The constant reactant supply ensures that stationary states exist at subcritical conditions whereas the supercritical reaction is a repetitive, oscillatory sequence of events. In a linked pair of papers, we investigate isothermal criticality at the second limit in terms of changing nature of the singularities for the mass conservation equations and derive kinetic relationships that explain the oscillatory features. In this paper the origins of oscillatory ignition are traced analytically to chain-branching via H atoms coupled to the changing third-body efficiency in the elementary process $H + O_2 + M \rightarrow HO_2 + M,$ when water formed during ignition is displaced by the inflow of fresh hydrogen and oxygen. Analytical predictions are made of the periods between successive ignitions and of the conditions at which oscillatory reaction is transformed to a stationary state. A composition limit for the existence of oscillatory ignition in the (lean) mixture $H_2 + 150_2 + 14N_2$ is located experimentally and explained in terms of the analytical interpretations presented here.