In the foregoing chapter we encountered unimodular transformations $$ {{c\tau + d}} $$ where a, b, c, d are integers with ad — bc = 1. This chapter studies such transformations in greater detail and also studies functions which, Iike J(τ), are invariant under unimodular transformations. We begin with some remarks concerning the more general transformations $$ {{cz + d}} $$ (1) where a, b, c, d are arbitrary complex numbers.