The wave aberration theory of non-symmetrical optical systems is useful for understanding the misalignment in symmetrical systems and designing of off-axis mirror systems. A theory about the third-order aberrations for sub-aperture plane symmetric optical system is developed by using the aberration theory for full-aperture axially symmetric spherical systems. This paper proves the nodal aberration theory, namely, the points in the field with zero third-order aberration will deviate from the field center except for spherical aberration. It also reveals that the nodal aberrations arise from the transformation of the aberrations in full-aperture systems. This theory can be efficiently used in non-symmetric optical system designing process.