Mr. Hardy communicated his share of the paper on February llth. 1904, and discovered shortly afterwards that Prof. Bromwich had at an earlier date arrived independently at the results of § §1-5. § 6 and § § 12-17 are due more particularly to Mr. Hardy, and §} 7-11 to Prof. Eromwich. Some of the earlier results (those relating to double series sumim-d by rows or columns) were also obtained by Mr. A. Brown, to whom the subject had. been suggested by Prof. Bromwich for a dissertation. As regards

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... e latter part of the paper, each of the authors had arrived by conjecture at the other's results, but had not worked out formal proofs at the time when it was decided to unite them in one paper. t The theorem is still true if x approaches 1 by any path (in the complex-plane) which does not touch the circle of convergence ; but it is not with extensions of this kind that we shall be concerned now.