The paper defines and analyzes a functional form for a one-output, many-factors production function, which is homothetic (or homogeneous), and exhibits CRES; that is, its ES (Allen-Uzawa elasticities of substitution) vary along isoquants and differ as between pairs of factors, but the ES stand in fixed ratios everywhere. Given data on factor prices, quantities, and assuming competitive cost-minimization, the parameters of CRESH are estimable from a system of log-linear equations, each containing at most three independent variables. The CES function, as wall as its limiting forms (the Cobb-Douglas ($\sigma = 1), Leontief ($\sigma = 0$), and linear ($\sigma = \infty$) functions) are special cases of CRESH. The Mukerji CRES function has an identical unit-isoquant surface, but it is not homothetic. Appendix A analyzes the Mukerji function. Appendix B derives the (implicit) CRESH cost function.