  On two-dimensional percolation clusters at the percolation threshold, we
study $<\sigma(M_B,r)>$, the average conductance of the backbone, defined by
two points separated by Euclidean distance $r$, of mass $M_B$. We find that
with increasing $M_B$ and for fixed r, $<\sigma(M_B,r)>$ asymptotically {\it
decreases} to a constant, in contrast with the behavior of homogeneous sytems
and non-random fractals (such as the Sierpinski gasket) in which conductance
increases with increasing $M_B$. We explain this behavior by studying the
distribution of shortest paths between the two points on clusters with a given
$M_B$. We also study the dependence of conductance on $M_B$ slightly above the
percolation threshold.
