Here we shall show that those solutions of (1.1) whose initial y vector lies on a certain initial manifold, which depends on E and the initial x vector, are very well approximated for small t and E by the corresponding solutions of a boundary layer equation, in which the initial x vector enters as a parameter, associated with (1.1). Moreover, it will also be shown that as E->O+ the stable initial manifolds associated with (1.1) tend to the stable initial manifold associated with the boundary layer equation. Problems of this nature, i.e., problems in which there is a system of differential equations possessing the property that the setting of a parameter equal to zero reduces the order of the system, have been treated under various hypotheses by several authors, e.g., [2; 3; 4; 5; 6 ] and [7 ]. The same hypotheses as in [5 ] will be assumed here. We now state these hypotheses as well as the principal theorem of [5]; this will then serve to introduce the problem and the results of the present paper. Let f=f,(t, x, y, E) be the matrix with ofil/xj in the ith row and jth column and let f,(t) be the matrix f,(t, p(t), q(t), 0). The matrices fy, gx, gy as well as fy(t), gx(t), gy(t) are similarly defined. HI:f, g, fx, fy, gx, gy are continuous in (t, x, y, E) for 0? t< T if I x-p(t)| +|y-q(t)| +E is sufficiently small. H2: There exists a real nonsingular matrix P(t) CC'(O <t ? T) such that