Recently, Lim et al (1998) conducted an experimental study on the "history" effect of acceleration on the stability of the Taylor-Couette flow. They found that when the inner cylinder is started from rest, and if the acceleration was higher than a critical value of about (s−1), a new flow regime was formed within the Reynolds number range of the wavy vortex flow. In this regime, toroidal vortices in-between the two cylinders showed striking resemblance to the Taylor vortices found at the critical Taylor number. Accordingly, they referred to the new flow as Second Taylor Vortex flow (or STVF). What is fascinating about Lim et al's discovery is that if the acceleration is less than 2.2 (s−1), the vortices are wavy even though the Reynolds number is the same. In this paper, we extended the work of Lim et al, which was restricted to one radius ratio and one aspect ratio only. The primary objectives of this work are to see how changes in the radius ratio and aspect ratio affect the STVF region. The investigation is divided into two parts. In the first part, attention is focused on three different radius ratios (i.e. η = 0.659, 0.8032 and 0.8936) for a fixed aspect ratio Γ of 30. In the second part, the radius ratio is fixed at η = 0.8032 while the aspect ratio is varied systematically from 50 (the value used by Lim et al (1998)) to 20. In both cases, the inner cylinder is subjected to a wide range of acceleration from 0.01(s−1), to 200(s−1). The results, when presented in an acceleration-Reynolds number parametric space, show that STVF flow is not only a function of the Reynolds number and the acceleration as was first reported by Lim et al, but is also a function of the radius ratio, and aspect ratio. Although the exact relationship between then is complex, but in general, with a fixed aspect ratio Γ = 30 STVF regime shrinks with the increase in the radius ratio. As to the effect of the aspect ratio, it is found that for a fixed radius ratio η = 0.8032, STVF regime increases notably with a reduction in the aspect ratio. To the best of our knowledge, these results have not been reported in the literature before. We believe this finding is significant, because it provides an important linkage between the observations of Lim et al and that of Koschmieder for the supercritical Taylor vortex flow.