Conditions Satisfied by the Expansion and Vorticity of a Viscous Fluid in a Fixed Container*

where W(y, t) is a quadratic form whose coefficients depend upon those elements &„•(<) of B(t) for which i < j, i, j = 1, 2, • • • n. Since these elements can always be made sufficiently small by a transformation with constant coefficients (which will not affect stability properties) the derivative dV/dt will be a negative definite quadratic form. Hence the trivial solution y(t) = 0 of (2) is asymptotically stable, and therefore the trivial solution x(t) = 0 of (1) is asymptotically stable. This establishes our theorem. 0. Perron* was the first to prove directly that the conditions