In order to extend the e~N method to general three-dimensional boundary layers, the conservation law of the imaginary parts for the wave parameters with a fixed wave vector is deduced. The compatibility relationship(CR) and the general theory of ray tracing(RT), which have been extensively used in conservative systems, are applied to a general three-dimensional boundary layer belonging to non-conservative systems. Two kinds of e~N methods, i.e., the e~N -CR method and the e~N -RT method, are established.Both the two kinds of methods can be used to predict the evolutions of the spanwise wavenumber and the amplitude of the disturbances in general three-dimensional boundary layers. The reliability of the proposed methods is verified and validated by performing a direct numerical simulation(DNS) in a hypersonic general three-dimensional boundary layer over an aircraft model. The results are also compared with those obtained by other e~N methods, indicating that the proposed methods have great potential applications in improving the transition prediction accuracy in general three-dimensional boundary layers. In order to extend the e ~ N method to general three-dimensional boundary layers, the conservation law of the imaginary parts for the wave parameters with a fixed wave vector is deduced. The compatibility relationship (CR) and the general theory of ray tracing ( RT), which has been extensively used in conservative systems, are applied to a general three-dimensional boundary layer belonging to non-conservative systems. Two kinds of e ~ N methods, ie, the e ~ N -CR method and the e ~ N -RT method, are established. Both the two kinds of methods can be used to predict the evolutions of the spanwise wavenumber and the amplitude of the disturbances in general three-dimensional boundary layers. The reliability of the proposed methods is verified and validated by performing a direct numerical simulation (DNS) in a hypersonic general three-dimensional boundary layer over an aircraft model. indicating results of the proposed methods have gr eat potential applications in improving the transition prediction accuracy in general three-dimensional boundary layers.