THE AGGREGATE PRODUCTION FUNCTION is one of the basic tools in the modern theory of economic growth. In recent years the increased interest in this subject has brought forth two types of improvement in the specification of the aggregate production function. First, the form of the production function has been noticeably extended from the constant elasticity of substitution functions (CES [1]) to variable elasticity of substitution functions, such as the linear elasticity of substitution function [17], the production function which combines CES and Cobb-Douglas functions [9] and [4], the production function in which the elasticity of derived demand is constant (CEDD) [14] and [15], etc. Secondly, the problems of specifying and measuring the types of technical progress that are consistent with a given form of the production function have been elucidated. For instance, in a theoretical paper [16] the present authors have studied extensively the implications of various types of neutral technical progress that result from the invariant relationships between pairs of important economic variables. Subsequently we have estimated production functions and technical progress on the basis of linear and invariant relationships between pairs of these variables [3]. The purpose of the present paper is to extend the scope of the above mentioned papers. We proceed by dropping the two-variable restriction and allowing general multi-variate analysis among subsets of production variables. In particular, alternative specifications of production functions and of technical progress will be obtained by specifying linear relationships between rates of change of such production variables as labor-capital ratio, output-capital ratio, wage rate, interest rate, marginal rate of substitution, and profit share. A general differential equation is derived from the linear combination of all these variables. Its solution yields a new class of production functions which includes the CES, Cobb-Douglas and their combination as special cases and for which technical change is factor augmenting. This new class of production functions may appropriately be called the log-linear production functions. The model is tested against time series data for two digit industries in the United States 1949-65. The question asked is this: what production functions and what types of technical progress are we testing against each other when we