1 Introduction

One of the main by-product results of MACHO and EROS projects is the discovery of many double-mode Cepheids (DMCs) pulsating in both the fundamental- and first-overtone mode (F/1O), and the first- and second-overtone (1O/2O) modes, in the LMC and SMC (e.g. Alcock et al. 1995; Alcock et al. 1997; Beaulieu et al. 1996). The presence of many 1O/2O DMCs raises immediately the question of the possible existence of Cepheids pulsating purely in the second overtone mode, but up to now none have been found, probably because it is difficult to identify them unambiguously. They should have short periods and should be found preferentially in low metallicity galaxies (see e.g. Alcock et al. 1997). In fact, while many 1O/2O DMCs have been discovered in the Magellanic Clouds, only one of these stars (CO Aur; Mantegazza 1983) was observed in our Galaxy.

The importance of second overtone mode Cepheids relies on their being, among Cepheids, the third possible benchmark for the stellar interior and evolution theory beside fundamental and first overtone mode Cepheids. Recently Antonello & Kanbur (1997) studied the characteristics of these stars predicted by nonlinear pulsation models, and remarked in particular the effects of the resonance P₂/P₆=2 at $P_2 \sim 1$ d (P is the period) between the second and sixth overtone mode. Resonances represent a powerful comparison tool between observations and theoretical model predictions, because they affect the shape of the curves of stars in a specific period range, for example P₀/P₂=2 at $P_0 \sim 10$ d in fundamental mode Cepheids (Simon & Lee 1981) and P₁/P₄=2 at $P_1 \sim 3$ d in first overtone Cepheids (Antonello et al. 1990). The close comparison allows to probe the stellar interior and to put constraints on the stellar physical parameters. Recently, Beaulieu (1998) mentioned some possible second overtone candidates in the SMC, and it would be interesting to observe them accurately in order to confirm their nature.

The Magellanic Cloud variables were extensively studied about thirty years ago by C. Payne-Gaposchkin and S. Gaposchkin. One of us (Antonello 1993) used their results concerning the asymmetry parameter of Cepheid light curves for studying the differences between fundamental and first overtone mode Cepheids, and, in the case of the SMC (Payne-Gaposchkin & Gaposchkin 1966), he noted four stars with short period and unusual asymmetry parameter (that is unusual light curve) and indicated them as possible second overtone mode candidates. In the present note we report about the results of new observations of three of these stars: HV 1777, HV 1779 and HV 1353.