Abstract: By use of the ingenious integral manipulation introduced by Howard(1961), the instability of the basic flow changed with height in three dimensional nonhydrostatic atmosphere has been investigated in the present paper. The essential condition of instability generation and the distribution law of the phase speed on the complex phase speed coodinate plane are explicitely given. Improved Miles theorem and Howard's semicircle theorem, as well as semi-ellipse theorem have been established. The semi-ellipse theorem means that the complex wave velocity for any unstable mode is rest ricted in a semi-ellipse whose majoraxis and centre coincide with the diameter and that of Howard's semicircle respectively, and whose minor axis depends not only on the parameters J0, R0 associated with the Richardson number and wave number k but also on the at mospheric depth parameter k0.