From the beginning there is a close but not complete parallelism between the theory of approximation by polynomials and that of approximation by trigonometric sums. This is true both for the orthogonal developments in Legendre and Fourier series and for approximation by more general linear combinations of the fundamental monomials. From another point of view, the Legendre and Fourier series may be regarded as special cases under a theory of polynomials orthogonal on curves in the plane of a pair of real variables, the curve being a line segment in one case and a circle in the other.2 The notion of orthogonal polynomials on a plane curve may again be varied by replacing the polynomials by trigonometric sums. The purpose of the following pages is to clarify some of the changes in the theory that are brought about by this replacement. A natural procedure would be to pass directly from polynomials in two variables to double trigonometric sums. It is somewhat simpler, however, to deal with expressions which are trigonometric sums in one variable and polynomials in the other. Such an expression, a trigonometric sum in x with coefficients which are polynomials in y, or a polynomial in y with coefficients which are trigonometric sums in x, will be called here simply a mixed sum. The next sections will be concerned mainly with the beginnings of a theory of orthogonal mixed sums on a curve which is itself characterized by the vanishing of a particular mixed sum, the analogue of an algebraic curve in the case of ordinary polynomials. By another change in point of view the problem can be regarded as pertaining to polynomials in three variables on an algebraic curve in space.3 If u = cos x, v = sin x, a mixed sum S(x, y) is a polynomial P(u, v, y), and the curve S(x, y) = 0 corresponds to the curve defined by the simultaneous vanishing of P(u, v, y) and u2 + V21. It will be convenient however to work with the formulation in terms of two variables. The discussion of mixed sums will be followed by a brief indication of some of the novel features which are encountered for the first time when the orthogonal functions are trigonometric with respect to both variables. The double trigo-