Let $P \left( \frac{\partial}{\partial x_{1}}, \dots , \frac{\partial}{\partial x_{n}} \right) u = 0$ be a quasi-elliptic partial differential equation with constant coefficients. Let $f_{j}(x_{1},\dots,x_{n-1})$, $j = 0$, $j = 0,1,\dots,p-1$, be the Dirichlet data for $(x; x \in R^{n},x_{n} \ge 0)$. The object of this paper is to determine the Gevrey class of $u$ in $\{x; x \in R^{n}, 0 \le x_{n} \le a \}$ ($a$ any positive) if $f_{j} \in S^{\beta_{1},\cdots,\beta_{n-1}}_{\gamma_{1},\cdots,\gamma_{n-1}}$.