1. The main purpose of this paper is to prove some theorems about conjugate Fourier series, and similar orthogonal expansions, of a positive definite function. At the end of the paper there is given a slight generalization of the well-known continuity theorem for characteristic functions. It was proved several years ago by Dugue [1 ] that if X is a measurable positive-definite function then the Fourier series of 4) is uniformly convergent. Dugue's Theorem was actually stated in terms of characteristic functions, but it is equivalent to the present formulation on account of the Bochner-Khintchine Representation Theorem. More precisely one can state: