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Aims: The Danish 1.54-meter telescope at the La Silla observatory was usedfor photometric monitoring of selected eccentric eclipsing binaries located inthe Small Magellanic Cloud. The new times of minima were derived for thesesystems, which are needed for accurate determination of the apsidal motion.Moreover, many new times of minima were derived from the photometric databasesOGLE and MACHO. Eighteen early-type eccentric-orbit eclipsing binaries werestudied.  Methods: Their (O-C) diagrams of minima timings were analysed and theparameters of the apsidal motion were obtained. The light curves of theseeighteen binaries were analysed using the program PHOEBE, giving the lightcurve parameters. For several systems the additional third light also wasdetected.  Results: We derived for the first time and significantly improved therelatively short periods of apsidal motion from 19 to 142 years for thesesystems. The relativistic effects are weak, up to 10% of the total apsidalmotion rate. For one system (OGLE-SMC-ECL-0888), the third-body hypothesis wasalso presented, which agrees with high value of the third light for this systemdetected during the light curve solution.

Apsidal motion and light a curve solution for eighteen SMC eccentric eclipsing binaries