where 0(z/)ela(®~c^ is the stream function of the disturbance in the usual normal mode analysis, U(y) is the basic velocity distribution, R is the Reynolds number, and D = d/d In the study of this equation for large values of c c R, asymptotic methods of approximation ha important role and, in the early work on the subject by Heisenberg (1924) , Tollmien (1929 ,1947 ), and Lin (1945 , 1955 , two different types of asymptotic approximations were obtained by some what heuristic methods. These

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... oximations correspond to the leading terms of what would now be called inner and outer expansions. Most of the existing calculations, however, have been based on the inconsistent but apparently successful procedure of using outer expansions for the solutions of inviscid type and inner expansions for the solutions of viscous type but it is only recently that a detailed study has been made by Eagles (1969) of the limitations of such mixed approximations. Attempts to improve on these older theories have generally been based on either the comparison equation method or the method of matched asymptotic expansions, and it is of some importance to discuss briefly the essential differences between these two approaches. In applying the method of matched asymptotic expansions to the O rr-Sommerfeld equation, as Eagles (1969) has recently done in a very systematic manner, one is primarily concerned with what may be called the central matching problem (cf. Wasow 1968), i.e. the problem of relating the inner and outer expansions so that they represent different asymptotic approximations to the same solutions. Since all but one of the outer expansions are multiple-valued, however, the solutions which they represent must exhibit the Stokes phenomenon and this leads to a consideration of the lateral connexion problem, i.e. the problem of determining the continuation of a given solution (or, more precisely, its outer expansion) on crossing a Stokes line in the complex plane. These two problems are closely related and, although Eagles (1969) did not consider the lateral connexion problem explicitly, it should be emphasized that a complete solution to the central matching problem must necessarily contain the solution to the lateral connexion problem but not conversely. By using the comparison equation method, however, it is possible to determine the Stokes multipliers in an indirect manner that avoids the need for a complete solution of the central matching problem, and this is the approach that will be adopted in the present paper. The comparison equation method has been extensively studied by Wasow (1953 ), Langer (1957 , Lin (1957aLin ( , b, 1958, Lin & Rabenstein (i960) and others. In all of this work the major aims have been to obtain asymptotic approximations to the solutions of the O rr-Sommerfeld equation that are uniformly valid in a bounded domain containing one critical point and to develop an algorithm by which higher approximations can be systematically obtained. Theories of this type are largely based on the idea of generalizing Langer's (1932) well-known theory for second-order differential equations with a simple turning point to higher-order equations of the Orr-Sommerfeld type. This requires the development of a procedure by which the solutions of the Orr-Sommerfeld equation can be represented asymptotically in terms of the solutions of a suitably chosen comparison equation. The success of this method, however, crucially depends upon being able to satisfy two closely related conditions. First, the comparison equation must be sufficiently simple so that its solutions may be considered known, otherwise little would be achieved; and, secondly, the solutions of the comparison equation must have asymptotic properties that are close to those of the Orr-Sommerfeld equation in order to achieve the desired degree of uniformity in the resulting approximations. These conditions severely limit the class of flows for which the general theory has thus far been developed and would appear to exclude, for example, asymmetrical flows with two critical points. Although the present paper is closely related to the work of Lin (1957 a, b, 1958) and Lin & Rabenstein (i960), our work differs from theirs in some important respects. The most significant difference results from our insistence that all asymptotic expansions be 'complete' in the sense of Olver (1961, 1963, 1964). The concept of a complete asymptotic expansion, as developed by Olver in connexion with his theory of error bounds for asymptotic solutions of certain secondorder differential equations, is based on the observation that it is often more important to obtain a first approximation that is valid in the complete sense than to obtain the whole of the descending £o(°) = °> *o(0) = " 2yo5 m = -5 y 0, | /i(o ) 4= o, gm # 0, # 0, J y0 = y(°) = \(UcjU'c). where (2.9) The solutions A1 c (tj; a, ft, e) Consider first the solution associated with the path and define a standard solution