A boundary-value problem describing the onset of linear instability in a Benard layer, is considered. The solutions of the sixth-order differential equation arising are expressed as Laplace integrals whose integrands involve a function satisfying a second-order equation with six transition points. W.K.B. approximations to this function, valid in regions associated with each transition point, are related by using global phase-integral methods. This allows solutions of the sixth-order problem to be estimated by steepest descents, and leads to an eigenvalue condition. The eigenvalue estimates are confirmed numerically by using the compound matrix method.