Table of Contents -- Introduction -- 1 Types and Conjugacy Classes -- 2 Matrix Representation of Types -- 3 Positive Reflections -- 4 Basic Equation -- 5 Contributions of Indecomposable Types -- 6 The Invariant -- 7 Lemmas E E' -- 8 Loxodromic Types -- 9 Pseudo-Loxodromic Types -- 10 Classification of Types -- 11 Types of the First Kind -- 12 Types of the Second Kind -- 13 The Main Theorem -- 14 Lengths using both types of Reflections -- References