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In this paper we consider the Cauchy problem for higher order weaklyhyperbolic equations. We assume that the principal symbol depends only on onespace variable and the characteristic roots $\tau_j$ verify the inequality\[\tau_j^2(x) + \tau_k^2(x) \le M \bigl(\tau_j(x)-\tau_k(x)\bigr)^2\] for someconstant $M$ independent of $x$. We prove that if the lower order terms verifya suitable Levi condition, the Cauchy problem is well-posed in$\mathcal{C}$-infinity.

The Cauchy Problem for properly hyperbolic equations in one space variable