This paper is concerned with a particular method used in the calculation of asymptotic expansions of solutions to problems which contain a small parameter ∈. The physical idea underlying the method is that the parameter introduces new scales (of length and/or time) into the problem, and that the solution should incorporate these (see, for example, Cole (1)). This may be done in various ways: two-timing (two-scaling, multiple-scaling) does it by demanding that the solution is a function of all relevant scales. Consider, for example, a linear oscillator whose frequency is slowly adjusted, governed by

and subject to some initial condition. Account of the slowly varying frequency is taken by the dependence of Ω on ∈t, where 0 < ∈ ≪ 1, and we assume that Ω > 1 for all arguments. The rapid oscillatory behaviour anticipated in the solution is customarily described by a phase function Θ(isin;t)/∈, a generalization of Ωt the case that Ω is constant, while slow changes due to frequency variation occur on a scale ∈t.