  A general expression is given for the change in free energy when a charge
tunnels through a junction in a one-dimensional array of N metallic islands
with arbitrary capacitances and arbitrary background charges. This is used to
obtain expressions for the (average) threshold voltage of the Coulomb blockade
for a few characteristic geometries. We find that including random background
charges has a large effect on the N-dependence of the threshold voltage: In an
array with identical junction capacitances C and gate capacitances C_g, the
threshold voltage, averaged over the background charge, is proportional to N^a,
where {a} crosses over from 1/2 to 1 when N becomes larger than
2.5\sqrt{C/C_g}.
