  We consider the zeros of the partition function of the Ising model with
ferromagnetic pair interactions and complex external field. Under the
assumption that the graph with strictly positive interactions is connected, we
vary the interaction (denoted by $t$) at a fixed edge. It is already known that
each zero is monotonic (either increasing or decreasing) in $t$; we prove that
its motion is local: the entire trajectories of any two distinct zeros are
disjoint. If the underlying graph is a complete graph and all interactions take
the same value $t\geq 0$ (i.e., the Curie-Weiss model), we prove that all the
principal zeros (those in $i[0,\pi/2)$) decrease strictly in $t$.
