  Charge is transported through superconducting SSS single-electron transistors
at finite bias voltages by a combination of coherent Cooper-pair tunneling and
quasiparticle tunneling. At low transport voltages the effect of an ``odd''
quasiparticle in the island leads to a $2e$-periodic dependence of the current
on the gate charge. We evaluate the $I-V$ characteristic in the framework of a
model which accounts for these effects as well as for the influence of the
electromagnetic environment. The good agreement between our model calculation
and experimental results demonstrates the importance of coherent Cooper-pair
tunneling and parity effects.
