  General stability criterions of two-dimensional inviscid parallel flow are
obtained analytically for the first time. First, a criterion for stability is
found as $\frac{U''}{U-U_s}>-\mu_1$ everywhere in the flow, where $U_s$ is the
velocity at inflection point, $\mu_1$ is eigenvalue of Poincar\'{e}'s problem.
Second, we also prove a principle that the flow is stable, if and only if all
the disturbances with $c_r=U_s$ are neutrally stable. Finally, following this
principle, a criterion for instability is found as $\frac{U''}{U-U_s}<-\mu_1$
everywhere in the flow. These results extend the former theorems obtained by
Rayleigh, Tollmien and Fj{\o}rtoft and will lead future works to investigate
the mechanism of hydrodynamic instability.
