This article is concerned with the mathematical proof that distillation boundaries correspond to locally longest residue curves. Perturbation theory and the implicit function theorem are used to provide an analytical relationship between neighboring trajectories while local congruence and a variety of limiting arguments are used to establish the relative distances between neighboring residue curves. The mathematical conditions on which this analysis is based include (1) The assumptions that trajectories are twice continuously differentiable and do not cross. (2) The comparison of neighboring trajectories that converge to the same stable node. (3) The existence and placement of a proper focal point. (4) Local congruence. While all of these assumptions and conditions are important, it is the existence and proper placement of a focal point and local congruence that are the keys to establishing relative distances between neighboring residue curves and, in turn, the main result that distillation boundaries are defined by locally longest trajectories from any unstable node to all reachable stable nodes. Wherever possible, geometric illustrations are used to clarify key concepts. (c) 2007 American Institute of Chemical Engineers.