The paper pioneers the higher order conjugation theory that elevates the current Camus and Litvin based conjugation theory to a new level by offering the methodology to enable explicitly selecting conjugate pairs to meet the prescribed intrinsic contact characteristics or performance for contact strength. A conjugation theorem is proposed, in which a three-body conjugation system is characterized by the coinciding instant centers and from which the theory is established. Instantaneous invariants are introduced to characterize the instantaneous motion of a three-body conjugation system. The loci of points that generate conjugate pairs with the common relative curvature as well as first and second order stationary relative curvature are presented. These curvature properties determine the contact pattern and therefore the contact strength. An immediate and important application is on gear tooth profile synthesis. The proposed theory offers the design freedom that breaks the traditional constraint of using a specific curve, such as involute. It therefore fills a void in gear tooth profile synthesis and brings light to the question on seeking the strongest tooth profiles. On the other hand, this is the first kinematic synthesis theory that generate both elements of a higher pair simultaneously for a prescribed contact performance. It is a contrast to the conventional Burmester theory by using RR, PR, or RP dyads to form a linkage.
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