A higher-dimensional toroidal universe containing a perfect fluid obeying the equation of state /ital p/=(..gamma../minus/1)/rho/ is considered. A previousanalysis of the underlying equations of motion is extended to arbitrary valuesof ..gamma... Solutions to the latter are studied near their singular points.It is found that in these regimes the curvature and matter terms aregenerically unimportant (except when ..gamma..=2) and that thesuperexponential inflation of the external dimensions is driven by a cross term/similar to/(/ital R/dot ////ital R/)(/ital r/dot ////ital r/). This term could produce a large amountof entropy. This makes the resolution of the flatness and horizon problems inthese models even easier than believed previously.