The boundary layer approximation to a given flow problem is not invariant if different coordinate systems are used in the approximation process. However, a correlation theorem (Theorem 1) is given, which states that the boundary layer solution with respect to any given system can be found, by a simple substitution, from that with respect to any other system. On the basis of this theorem, the dependence of the solution on the choice of coordinates is investigated in detail. The skin friction is invariant, but the flow field is not invariant. At large distances from the wall, the flow field given by boundary layer theory depends almost entirely on the choice of coordinates, rather than on the physical problem.
This dependence may be used to obtain a complete matching between the boundary layer solution and the external flow, in the following sense: Theorem 2 states how a coordinate system can be found such that the boundary layer solution with respect to this system is valid as an approximation for the entire flow field. It contains the external flow and the flow due to displacement thickness.
The discussion is restricted to steady, two-dimensional, incompressible flow without separation. These restrictions, however, are not essential for many of the results
