  Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$
with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show
that it is possible to transform $D$ into an alternating trigonal diagram, so
that all intermediate diagrams remain trigonal, and the number of crossings
never increases.
