* Received December 13, 1929. t Reference to this result has been made by G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928), pp. 612-634 (p. 633 footnote 32)), and by Einar Hille, Note on a power series considered by Hardy and Littlewood, Journal London Math. Soc., 4 (1929), pp. 176-182 (pp. 181-182). There is a slip in Hardy and Littlewood's statement of the result, which is corrected by Hille. The condition r = o on p. 182 of Hille's paper is presumably a misprint for y>'O. Actually, however, I had considered the series (1) only for O<y < 1, and I confine myself to this case here, this being the only one which is of interest for the application in view. The same result has been obtained independently and by a different method by Hille in a paper as yet unpublished but kindly placed at my disposal by the author. The present note is written at the request of Professor Rille. 241 18