The purpose of this paper is to present a rational approach to modelling the triple velocity correlations that appear in the transport equations for the Reynolds stresses. All existing models of these correlations have largely been formulated on phenomenological grounds and are defective in one important aspect: they all neglect to allow for the dependence of these correlations on the local gradients of mean velocity. The mathematical necessity for this dependence will be demonstrated in the paper. The present contribution lies in the novel use of group representation theory to determine the most general tensorial form of these correlations in terms of all the secondand third-order tensor quantities which appear in the exact equations that govern their evolution. The requisite representation did not exist in the literature and had therefore to be developed specifically for this purpose. The outcome of this work is a mathematical framework for the construction of algebraic, explicit and rational models for the triple velocity correlations that are theoretically consistent and include all the correct dependencies. Previous models are reviewed, and all are shown to be incomplete subsets of this new representation, even to lowest order.