After an extensive research on the fundamental theory, the theoretical basis for the Logix gearing will be presented in this paper. This includes the theory for gear meshing with high degree of contact. In comparison to previous studies, the theory of this paper is more restricted going beyond third order parameters into the fourth order, the teeth profile had one order of contact higher than that of the Logix gearing. So that further improved the contact strength of tooth profile. A large category of gearings with high order of contact was presented, while the Logix gear is only one of them, or the special example in realization of my theory. In the Logix gearing, the zigzag curvature center curve of media rank does not always lay about the pitch line and does not extend along the pitch line continuously and steadily. It goes forward and returns. So that, in terms of properties of the transverse engagement, the Logix gearing has not much improvement over the traditional Novikov gearing (in which the transverse engagement is temporally). In my gearing the curvature center of media rank extend along the pitch line continuously and steadily. The transverse engagement factor is larger than that of Logix gearing.