Use of the higher order method of multiple scales with reconstitution to determine the periodic steady state response of harmonically excited non-linear oscillators is considered. The well known primary resonance in the Duffing oscillator is considered as a prototype example. Second order mutliple-scales approximations, determined by using two different time co-ordinates, are compared. The role of a commonly used transformation, namely (T=\Omega t), on the higher order approximations is investigated. It is shown that only a finite approximation of this or any other time transformation is used in the reconstitution step. Implications of this finite approximation for the reconstitution procedure are discussed. The predicted extraneous solutions, those due to the higher order approximation and those due to a "zero" or non-uniformity of the truncated transformation, are analyzed.
Sometimes an heuristic procedure is used to obtain an improved first order multiplescales approximation. In this procedure, a frequency expansion is used in the inertia term, but not in the damping term. The effects of continuing this heuristic procedure to higher orders are also discussed.