2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120056We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer--Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms.v2: Introduction expanded and references added. 20 pages, 4 Postscript figuresDifferential Geometry53C40, 37K35 (Primary) 53A40, 53B25 (Secondary)text