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Mathematics of Computation

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Journal: Math. Comp. 2 (1946), 13-46
Corrigendum: Math. Comp. 3 (1949), 398.
Corrigendum: Math. Comp. 2 (1946), 95-96.
Corrigendum: Math. Comp. 2 (1946), 63-64.
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    New York, 1942; see MTAC, v. 1, p. 47-48. K. Hayashi, Sieben- und mehrstellige Tafeln der Kreis. und Hyperbelfunktionen und deren Produkte sowie der Gammafunktion, Berlin, Springer, 1926, p. 5-7, 12-50, 52-86. R. A. Davis, Table of Natural Sines and Radians, Oakland, Cal., Marchant Calculating Machine Co., 1941, 8 p., no. MM193; see MTAC, v. 1, p. 94f, 124. H. H. Turner, “On a method of solving spherical triangles, and performing other astronomical computations, by use of a simple table of squares,” R.A.S., Mo. Notices, v. 75, 1915, p. 530-541. On p. 538-541 is a table of 2 vers $\Delta ,\Delta = [0(6’){4\circ }.9;6\operatorname {D} ],[{5\circ }(6’){59\circ }.9;4\operatorname {D} ],[{60\circ }(6’){119\circ }.9;3\operatorname {D} ]$. L. J. Comrie, The Geocentric Direction Cosines of Seismological Observatories, with Introduction by Harold Jeffreys, London, B.A.A.S., 1938. viii, 14 p. H. Jeffreys and K. E. Bullen, Seismological Tables, London, B.A.A.S., 1940, 48 p. + 1 plate. The terms “decibel” and “neper” were adopted about 1929; see W. H. Martin, “Decibel—the name for the transmission unit,” Bell System Technical J., v. 8, 1929, p. 1-2. A decibel, one tenth of a bel, is a unit in communication engineering, acoustics, and allied fields. This unit is defined by the statement that two amounts of power, ${P_1},{P_2}$, differ by one transmission unit when they are in the ratio of ${10^{.1}}$, and any two amounts of power differ by $N$ transmission units when they are in the ratio ${10^{N(.1)}}$ or $N = 10\log ({P_1}/{P_2})$. The unit bel was derived from the name of Alexander Graham Bell (1847-1922). The neper, derived from the name of John Napier or Neper (1550-1617), is also a transmission unit with basic power ratio ${e^2}$, so that here ${e^{2x}} = {P_1}/{P_2}$ or $x = .5\ln ({P_1}/{P_2})$. S. Ramanujan, “On certain arithmetical functions,” Cambridge Phil. So. Trans., v. 22, no. 9, 1916, p. 174; Collected papers of Srinivasa Ramanujan, Cambridge, 1927, p. 153. In copying $\tau (17)$ Gupta omitted the “—” sign. P. Stäckel, “Die Lückenzahlen $r$-ter Stufe und die Darstellung der geraden Zahlen als Summen und Differenzen ungerader Primzahlen,” Heidelberger Akad. d. Wissen., Sitzungsb., math.-natw. Kl., 1917, no. 15, 52 p. N. M. Shah & B. M. Wilson, “On an empirical formula connected with Goldbach’s theorem,” Cambridge Phil. So., Proc., v. 19, 1919, p. 238-244. V. Brun, “Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare,” Archiv for Math. og Naturvid., v. 34, no. 8, 1916, 19 p. On p. 35 the authors refer to an error in D. N. Lehmer, List of Prime Numbers, Washington, 1914; p. 11, col. 13, for 8151, read 8051. This error was already noted in my Guide to Tables in the Theory of Numbers, Washington, 1941, p. 161, and in Scripta Mathematica, v. 4, 1936, p. 198. C. Størmer, “Solution complète en nombres entiers $m,n,x,y,k$ de l’équation $m\operatorname {arctg} (1/x) + n\operatorname {arctg} (1/y) = k(\pi /4)$,” Norsk Videnskabs Akademie, Christiania, Skrifter, math.-naturw. Kl., 1895, no. 11; see also his “Sur l’application de la théorie des nombres entiers complexes à la solution en nombres rationnels ${x_1}{x_2} \cdots {x_n},{c_1}{c_2} \cdots {c_n},k$ de l’équation: ${c_1}\operatorname {arctg} {x_1} + {c_2}\operatorname {arctg} {x_2} + \cdots + {c_n}\operatorname {arctg} {x_n} = k\pi /4$,” Archiv for Math. og Naturvid., v. 19, no. 3, 1896, 96 p. A reference may be given to a third paper by this author, “Solution complète en nombres entiers de l’équation $m\operatorname {arctang} (1/x) + n\operatorname {arctang} (1/y) = k\pi /4$,” So. Math. d. France, Bull., v. 27, 1899, p. 160-170.
  • Wilhelm Ljunggren, Über einige Arcustangensgleichungen die auf interessante unbestimmte Gleichungen führen, Ark. Mat. Astr. Fys. 29A (1943), no. 13, 11 (German). MR 0012090
  • Harry Bateman, Haley’s Methods for Solving Equations, Amer. Math. Monthly 45 (1938), no. 1, 11–17. MR 1524157, DOI
  • H. Buckley, “Some problems of interreflection,” Intern. Congress on Illumination, Proc., 1928, p. 888.
  • Walter R. Hedeman Jr., The cinema integraph in interreflection problems, J. Math. Phys. Mass. Inst. Tech. 20 (1941), 402–417. MR 6248, DOI

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Article copyright: © Copyright 1946 American Mathematical Society