  The recent observations of type Ia supernovae strongly support that the
universe is accelerating now and decelerated in the recent past. By assuming a
general relation between the quintessence potential and the quintessence
kinetic energy, a general relation is found between the quintessence energy
density and the scale factor. The potential includes both the hyperbolic and
the double exponential potentials. A detailed analysis of the transition from
the deceleration phase to the acceleration phase is then performed. We show
that the current constraints on the transition time, the equation of state and
the energy density of the quintessence field are satisfied in the model.
