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for (where we have used our solution for and otherwise
which generates only the zero solution when the boundary conditions are applied. The
boundary conditions relevant to the solution of (3.31) are (from (3.28) and (3.29b))

and

because of the form of this latter boundary condition, it is a little more convenient to
find the asymptotic solution for (Of course, from this it is then possible to
deduce both and if these are required; see later.) From (3.31) we obtain directly
that

where J and K are arbitrary functions, and condition (3.32) then requires that
cf. the solution for The boundary condition (3.33) is satisfied if

and then we obtain directly (because we may use

(for where we have written The two-term asymptotic
expansion for is therefore

as for and
It is clear, for and bounded, that the expansion (3.35) breaks down where
—the far-field. (This same property is exhibited by the expansions for
and The assumption that we have and bounded (and correspondingly,
and for the lower surface), for implies that these aerofoils are sharp
at both the leading and trailing edges—which is certainly what is aimed for in their
design and construction. However, if these edges are suitably magnified than it will
become evident that a real aerofoil must have rounded edges on some scale. This in turn
implies that stagnation points exist, where and then the asymptotic
expansion, (3.35), certainly cannot be uniformly valid near to and even
for y = O(1): a boundary-layer-type structure is required near and near