The stabilization law for transonic flow expresses the experimentally observed freezing of the local Mach number near the airfoil and ahead of the shock at free stream Mach numbers near one. Mathematically this means that near Mach one the flow over an airfoil is a small perturbation of sonic flow. Using the method of matched asymptotic expansions applied to the transonic small perturbation equation the authors show that the perturbation is a singular one and the flow near the airfoil differs only \(O(K^ 3)\) from sonic flow, where K is the transonic similarity parameter. The authors expand in the physical plane for subsonic and in the hodograph plane for supersonic free stream Mach numbers. In both cases inner and outer expansions are defined and substituted into the governing equation resulting in boundary value problems for the corrections. These still depend on undetermined constants which are found in the form of integral relations by applying Germain's conservation laws.