  We study a class of third order hyperbolic operators $P$ in $G = \{(t, x):0
\leq t \leq T, x \in U \Subset {\mathbb R}^{n}\}$ with triple characteristics
at $\rho = (0, x_0, \xi), \xi \in {\mathbb R}^n \setminus \{0\}$. We consider
the case when the fundamental matrix of the principal symbol of $P$ at $\rho$
has a couple of non-vanishing real eigenvalues. Such operators are called {\it
effectively hyperbolic}. V. Ivrii introduced the conjecture that every
effectively hyperbolic operator is {\it strongly hyperbolic}, that is the
Cauchy problem for $P + Q$ is locally well posed for any lower order terms $Q$.
This conjecture has been solved for operators having at most double
characteristics and for operators with triple characteristics in the case when
the principal symbol admits a factorization. A strongly hyperbolic operator in
$G$ could have triple characteristics in $G$ only for $t = 0$ or for $t = T$.
We prove that the operators in our class are strongly hyperbolic if $T$ is
small enough. Our proof is based on energy estimates with a loss of regularity.
