  Upper bounds of the first non-trivial eigenvalue $\lambda_1$ of the Laplace
operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means
of a new technique, are obtained. Each of the upper bounds of $\lambda_1$
depends on the length of mean curvature vector field, the dimension $n$, the
volume of $M^n$, and of a vector of $\R^{m+1}$. When $M^n$ does not lie
minimally in a hypersphere of $\R^{m+1}$, classical Reilly's inequality
\cite{Re} is improved and new upper bounds are explicitly computed. For
instance, considering a torus of revolution whose generating circle has a
radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of
revolution, we find $\lambda_1 < \frac{4}{3}(\sqrt{2}-1)\approx 0.552284$,
whereas Reilly's upper bound gives $\lambda_1 < 1/\sqrt{2}\approx 0.707106$.
