  Let $G$ be a finite plane multigraph and $G'$ its dual. Each edge $e$ of $G$
is interpreted as a resistor of resistance $R_e$, and the dual edge $e'$ is
assigned the dual resistance $R_{e'}:=1/R_e$. Then the equivalent resistance
$r_e$ over $e$ and the equivalent resistance $r_{e'}$ over $e'$ satisfy
$r_e/R_e+r_{e'}/R_{e'}=1$. We provide a graph theoretic proof of this relation
by expressing the resistances in terms of sums of weights of spanning trees in
$G$ and $G'$ respectively.
