On the Asymptotic Formulae of Riemann and of Laplace

Variable, 2nd Ed., Vol. I, pp. 506-512 (1921). Extensions of our Theorem I in the manner of Hobson are easily possible but unnecessary for our present purposes. 9 E. K. Haviland, loc. cit., p. 550. 10 PSV is the point with the coordinates (x, y). Q P,,, is to be taken in the sense of vector addition. Cf., e.g., H. Bohr and B. Jessen, D. Kgl. Danske Vidensk. Selsk. Skrifter, Ser. 8, 12, 332 (1929). 11 Cf. J. Radon, Wiener Sitzungsber., 128, 1093 (1919). 12 Cf. Radon, loc. cit., pp. 1304-1305. 13 The foregoing proof contains the stronger result that (xO, y°) is a continuity point of F* and hence of F, and this is true of every (xO, y°) such that the exceptional lines of 9i(RO Py) are distinct from the singular lines of q2(E). Thus the singular lines of F are at most those whose form is x = Es + ',p, y = 17i + '3%14 Cf. J. Radon, loc. cit., pp. 1338-1339. I have perceived since writing Note 14 to my above-mentioned article that Radon does not need to show that his function F* defined on the everywhere dense set of points possesses property (#) there. However, when such is the case it is possible at once to infer that F = F* on all points for which the latter is defined. 15 Cf., e.g., v. Mises, Wahrscheinlichkeitsrechnung u. ihre Anwendung, p. 219 (1931). This is the so-called independence relation often used in a formal way. We are not aware of a published general proof of our Theorem V even in the one-dimensional case. 16 Cf., e.g., S. Bochner, loc. cit., p. 402.