QUASIPERIODIC MATHIEU-HILL EQUATION.

A study is mde of a generalized Mathieu-Hill equation u double prime plus left bracket delta plus epsilon (cos 2t plus alpha cos 2( lambda plus epsilon )t) right bracket u equals 0, where delta , epsilon , alpha , lambda are constants, with vertical epsilon vertical very much less than 1 and lambda a rational fraction. The equation has a quasiperiodic coefficient and Floquet theory is inapplicable. It is shown that a multiple scaling technique enables the stability characteristics of the equation to be calculated and the transition curves between stable and unstable regions of the epsilon - delta plane to be compared. The structure of the analysis is such that the small divisor problem is not encountered.