Several de nitions, facts, and theorems stated during the recent talks given by Dr. Donnelly relating to the word problem for Coxeter groups are restated in 1 to provide some background information and notation. Three other facts, due to Casselman, are also stated and proved as lemmas in this section. In 2, we state and prove Tits' Theorem for the word problem on Coxeter groups. Finally, we c...