where w(z) =fII1(z-cai). We shall denote the partial sums of this series by S.(z; F) or S.(z), n =0, 1, 2, * ,and the Cesaro means of order r by Snr'(z; F) or S(r) (Z) , n = 0, 1, 2, Jacobi [81(1) seems to have been the first to study developments of this type. He was interested in the problem of finding a formal expansion for F(z) of the type 0q (z) (co(z)) P, in which the functions qp(z) are polynomials of degree less than X. The sum of the first n terms of this original series of Jacobi is the polynomial of degree at most }Xn -1 which interpolates to the function F(z) in the points aej, each considered to be of multiplicity n. Thus the nth partial sum of Jacobi's original series is identical with the Xnth partial sum of the series (1.1). A change in the order of the points ai naturally changes (1.1), but does not change Jacobi's original series. We shall call (1.1) the Jacobi series for F(z) with respect to the points ai. The series is a generalization of the Taylor series, to which it reduces if the ai all coincide. The present study of the Jacobi series was undertaken at the suggestion of Professor J. L. Walsh. The purpose of the paper is to develop two general methods for studying the Jacobi series on the boundaries of its regions of convergence, and to obtain thereby certain typical results concerning the behavior of the series on these boundaries. The first method, in which the basic idea (?4) is due to Professor Walsh, consists in the study of certain expres-