  We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap
\{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on
$t = 0$. We consider the case when the fundamental matrix of the principal
symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is
strictly hyperbolic for $t > 0.$ We prove that $P$ is strongly hyperbolic, that
is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order
terms $Q$.
