Abstract. The Arnold stability criterion suggests that a stationary flow of an ideal incompressible fluid is stable if a certain quadratic form is definite. We show that, in three or more dimensions, this quadratic form is never definite. Typically the form is indefinite, and the spectrum of the associated Hermitian operator ranges from $-\infty$ to $\infty$. The exceptional case is where the velocity field is harmonic (solenoidal and irrotational) in which case the quadratic form is identically zero.