Development of Gravity Pendulums in the 19th Century

Summary

Since the creation of classical mechanics in the 17th century, the pendulum has been a basic instrument for the determination of the intensity of gravity, which is expressed as the acceleration of a freely falling body. Basis of theory is the simple pendulum, whose time of swing under gravity is proportional to the square root of the length divided by the acceleration due to gravity. Since the length of a simple pendulum divided by the square of its time of swing is equal to the length of a pendulum that beats seconds, the intensity of gravity also has been expressed in terms of the length of the seconds pendulum. The reversible compound pendulum has served for the absolute determination of gravity by means of a theory developed by Huygens. Invariable compound pendulums with single axes also have been used to determine relative values of gravity by comparative times of swing.

The history of gravity pendulums begins with the ball or “simple” pendulum of Galileo as an approximation to the ideal simple pendulum. Determinations of the length of the seconds pendulum by French scientists culminated in a historic determination at Paris by Borda and Cassini, from the corrected observations with a long ball pendulum. In the 19th century, Bessel found the length of the seconds pendulum at Königsberg and Berlin by observations with a ball pendulum and by original theoretical considerations. During the century, however, the compound pendulum came to be preferred for absolute and relative determinations.

Capt. Henry Kater, at London, constructed the first convertible compound for an absolute determination of gravity, and then he designed an invariable compound pendulum, examples of which were used for relative determinations at various stations in Europe and elsewhere. Bessel demonstrated theoretically the advantages of a reversible compound pendulum which is symmetrical in form and is hung by interchangeable knives. The firm of A. Repsold and Sons in Hamburg constructed pendulums from the specifications of Bessel for European gravity surveys.

Charles S. Peirce in 1875 received delivery in Hamburg of a Repsold-Bessel pendulum for the U.S. Coast Survey and observed with it in Geneva, Paris, Berlin, and London. Upon an initial stimulation from Baeyer, founder of Die Europäische Gradmessung, Peirce demonstrated by experiment and theory that results previously obtained with the Repsold apparatus required correction, because of the flexure of the stand under oscillations of the pendulum. At the Stuttgart conference of the geodetic association in 1877, Hervé Faye proposed to solve the problem of flexure by swinging two similar pendulums from the same support with equal amplitudes and in opposite phases. Peirce, in 1879, demonstrated theoretically the soundness of the method and presented a design for its application, but the “double pendulum” was rejected at that time. Peirce also designed and had constructed four examples of a new type of invariable, reversible pendulum of cylindrical form which made possible the experimental study of Stokes’ theory of the resistance to motion of a pendulum in a viscous fluid. Commandant Defforges, of France, also designed and used cylindrical reversible pendulums, but of different length so that the effect of flexure was eliminated in the reduction of observations. Maj. Robert von Sterneck, of Austria-Hungary, initiated a new era in gravity research by the invention of an apparatus with a short pendulum for relative determinations of gravity. Stands were then constructed in Europe on which two or four pendulums were hung at the same time. Finally, early in the present century, Vening Meinesz found that the Faye-Peirce method of swinging pendulums hung on a Stückrath four-pendulum stand solved the problem of instability due to the mobility of the soil in Holland.

The 20th century has witnessed increasing activity in the determination of absolute and relative values of gravity. Gravimeters have been perfected and have been widely used for rapid relative determinations, but the compound pendulums remain as indispensable instruments. Mendenhall’s replacement of knives by planes attached to nonreversible pendulums has been used also for reversible ones. The Geodetic Institute at Potsdam is presently applying the Faye-Peirce method to the reversible pendulum.115 Pendulums have been constructed of new materials, such as invar, fused silica, and fused quartz. Minimum pendulums for precise relative determinations have been constructed and used. Reversible pendulums have been made with “I” cross sections for better stiffness. With all these modifications, however, the foundations of the present designs of compound pendulum apparatus were created in the 19th century.

Footnotes

1 The basic historical documents have been collected, with a bibliography of works and memoirs published from 1629 to the end of 1885, in Collection de mémoires relatifs a la physique, publiés par la Société française de Physique [hereinafter referred to as Collection de mémoires]: vol. 4, Mémoires sur le pendule, précédés d’une bibliographie (Paris: Gauthier-Villars, 1889); and vol. 5, Mémoires sur le pendule, part 2 (Paris: Gauthier-Villars, 1891). Important secondary sources are: C. Wolf, “Introduction historique,” pp. 1-42 in vol. 4, above; and George Biddell Airy, “Figure of the Earth,” pp. 165-240 in vol. 5 of Encyclopaedia metropolitana (London, 1845).

2 Galileo Galilei’s principal statements concerning the pendulum occur in his Discourses Concerning Two New Sciences, transl. from Italian and Latin into English by Henry Crew and Alfonso de Salvio (Evanston: Northwestern University Press, 1939), pp. 95-97, 170-172.

3 P. Marin Mersenne, Cogitata physico-mathematica (Paris, 1644), p. 44.

4 Christiaan Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia adaptato demonstrationes geometricae (Paris, 1673), proposition 20.

5 The historical events reported in the present section are from Airy, “Figure of the Earth.”

6 Abbé Jean Picard, La Mesure de la terre (Paris, 1671). John W. Olmsted, “The ‘Application’ of Telescopes to Astronomical Instruments, 1667-1669,” Isis (1949), vol. 40, p. 213.

7 The toise as a unit of length was 6 Paris feet or about 1,949 millimeters.

8 Jean Richer, Observations astronomiques et physiques faites en l’isle de Caïenne (Paris, 1679). John W. Olmsted, “The Expedition of Jean Richer to Cayenne 1672-1673,” Isis (1942), vol. 34, pp. 117-128.

9 The Paris foot was 1.066 English feet, and there were 12 lines to the inch.

10 Christiaan Huygens, “De la cause de la pesanteur,” Divers ouvrages de mathematiques et de physique par MM. de l’Académie Royale des Sciences (Paris, 1693), p. 305.

11 Isaac Newton, Philosophiae naturalis principia mathematica (London, 1687), vol. 3, propositions 18-20.

12 Pierre Bouguer, La figure de la terre, déterminée par les observations de Messieurs Bouguer et de La Condamine, envoyés par ordre du Roy au Pérou, pour observer aux environs de l’equateur (Paris, 1749).

13 P. L. Moreau de Maupertuis, La figure de la terre déterminée par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le Monnier, l’Abbé Outhier et Celsius, faites par ordre du Roy au cercle polaire (Paris, 1738).

14 Paris, 1743.

15 George Gabriel Stokes, “On Attraction and on Clairaut’s Theorem,” Cambridge and Dublin Mathematical Journal (1849), vol. 4, p. 194.

16 See Collection de mémoires, vol. 4, p. B-34, and J. H. Poynting and Sir J. J. Thomson, Properties of Matter (London, 1927), p. 24.

17 Poynting and Thomson, ibid., p. 22.

18 Charles M. de la Condamine, “De la mesure du pendule à Saint Domingue,” Collection de mémoires, vol. 4, pp. 3-16.

19 Père R. J. Boscovich, Opera pertinentia ad Opticam et Astronomiam (Bassani, 1785), vol. 5, no. 3.

20 J. C. Borda and J. D. Cassini de Thury, “Expériences pour connaître la longueur du pendule qui bat les secondes à Paris,” Collection de mémoires, vol. 4, pp. 17-64.

21 F. W. Bessel, “Untersuchungen über die Länge des einfachen Secundenpendels,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, 1826 (Berlin, 1828).

22 Bessel used as a standard of length a toise which had been made by Fortin in Paris and had been compared with the original of the “toise de Peru” by Arago.

23 L. G. du Buat, Principes d’hydraulique (Paris, 1786). See excerpts in Collection de mémoires, pp. B-64 to B-67.

24 Capt. Henry Kater, “An Account of Experiments for Determining the Length of the Pendulum Vibrating Seconds in the Latitude of London,” Philosophical Transactions of the Royal Society of London (1818), vol. 108, p. 33. [Hereinafter abbreviated Phil. Trans.]

25 M. G. de Prony, “Méthode pour déterminer la longueur du pendule simple qui bat les secondes,” Collection de mémoires, vol. 4, pp. 65-76.

26 Collection de mémoires, vol. 4, p. B-74.

27 Phil. Trans. (1819), vol. 109, p. 337.

28 John Herschel, “Notes for a History of the Use of Invariable Pendulums,” The Great Trigonometrical Survey of India (Calcutta, 1879), vol. 5.

29 Capt. Edward Sabine, “An Account of Experiments to Determine the Figure of the Earth,” Phil. Trans. (1828), vol. 118, p. 76.

30 John Goldingham, “Observations for Ascertaining the Length of the Pendulum at Madras in the East Indies,” Phil. Trans. (1822), vol. 112, p. 127.

31 Basil Hall, “Letter to Captain Kater Communicating the Details of Experiments made by him and Mr. Henry Foster with an Invariable Pendulum,” Phil. Trans. (1823), vol. 113, p. 211.

32 See Collection de mémoires, vol. 4, p. B-103.

33 Ibid., p. B-88.

34 Ibid., p. B-94.

35 Francis Baily, “On the Correction of a Pendulum for the Reduction to a Vacuum, Together with Remarks on Some Anomalies Observed in Pendulum Experiments,” Phil. Trans. (1832), vol. 122, pp. 399-492. See also Collection de mémoires, vol. 4, pp. B-105, B-112, B-115, B-116, and B-117.

36 One was of case brass and the other of rolled iron, 68 in. long, 2 in. wide, and 1/2 in. thick. Triangular knife edges 2 in. long were inserted through triangular apertures 19.7 in. from the center towards each end. These pendulums seem not to have survived. There is, however, in the collection of the U.S. National Museum, a similar brass pendulum, 37-5/8 in. long (fig. ¹⁵) stamped with the name of Edward Kübel (1820-96), who maintained an instrument business in Washington, D.C., from about 1849. The history of this instrument is unknown.

37 See Baily’s remarks in the Monthly Notices of the Royal Astronomical Society (1839), vol. 4, pp. 141-143. See also letters mentioned in footnote 38.

38 This document, together with certain manuscript notes on the pendulum experiments and six letters between Wilkes and Baily, is in the U.S. National Archives, Navy Records Gp. 37. These were the source materials for the information presented here on the Expedition. We are indebted to Miss Doris Ann Esch and Mr. Joseph Rudmann of the staff of the U.S. National Museum for calling our attention to this early American pendulum work.

39 G. B. Airy, “Account of Experiments Undertaken in the Harton Colliery, for the Purpose of Determining the Mean Density of the Earth,” Phil. Trans. (1856), vol. 146, p. 297.

40 T. C. Mendenhall, “Measurements of the Force of Gravity at Tokyo, and on the Summit of Fujiyama,” Memoirs of the Science Department, University of Tokyo (1881), no. 5.

41 J. T. Walker, Account of Operations of The Great Trigonometrical Survey of India (Calcutta, 1879), vol. 5, app. no. 2.

42 Bessel, op. cit. (footnote 21), article 31.

43 C. A. F. Peters, Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (Altona, Germany, 1860), Band 2, p. 3. The correction required if the times of swing are not exactly the same is said to have been given also by Bohnenberger.

44 F. W. Bessel, “Construction eines symmetrisch geformten Pendels mit reciproken Axen, von Bessel,” Astronomische Nachrichten (1849), vol. 30, p. 1.

45 E. Plantamour, “Expériences faites à Genève avec le pendule à réversion,” Mémoires de la Société de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, p. 309.

46 Ibid., pp. 309-416.

47 C. Cellérier, “Note sur la Mesure de la Pesanteur par le Pendule,” Mémoires de la Société de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, pp. 197-218.

48 A. Sawitsch, “Les variations de la pesanteur dans les provinces occidentales de l’Empire russe,” Memoirs of the Royal Astronomical Society (1872), vol. 39, p. 19.

49 J. J. Baeyer, Über die Grösse und Figur der Erde (Berlin, 1861).

50 Comptes-rendus de la Conférence Géodésique Internationale réunie à Berlin du 15-22 Octobre 1864 (Neuchâtel, 1865).

51 Ibid., part III, subpart E.

52 Bericht über die Verhandlungen der vom 30 September bis 7 October 1867 zu Berlin abgehaltenen allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1868). See report of fourth session, October 3, 1867.

53 C. Bruhns and Albrecht, “Bestimmung der Länge des Secundenpendels in Bonn, Leiden und Mannheim,” Astronomisch-Geodätische Arbeiten im Jahre 1870 (Leipzig: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1871).

54 Bericht über die Verhandlungen der vom 23 bis 28 September 1874 in Dresden abgehaltenen vierten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1875). See report of second session, September 24, 1874.

55 Carolyn Eisele, “Charles S. Peirce—Nineteenth-Century Man of Science,” Scripta Mathematica (1959), vol 24, p. 305. For the account of the work of Peirce, the authors are greatly indebted to this pioneer paper on Peirce’s work on gravity. It is worth noting that the history of pendulum work in North America goes back to the celebrated Mason and Dixon, who made observations of “the going rate of a clock” at “the forks of the river Brandiwine in Pennsylvania,” in 1766-67. These observations were published in Phil. Trans. (1768), vol. 58, pp. 329-335.

56 The pendulums with conical bobs are described and illustrated in E. D. Preston, “Determinations of Gravity and the Magnetic Elements in Connection with the United States Scientific Expedition to the West Coast of Africa, 1889-90,” Report of the Superintendent of the Coast and Geodetic Survey for 1889-90 (Washington, 1891), app. no. 12.

57 Eisele, op. cit. (footnote 55), p. 311.

58 The record of Peirce’s observations in Europe during 1875-76 is given in C. S. Peirce, “Measurements of Gravity at Initial Stations in America and Europe,” Report of the Superintendent of the Coast Survey for 1875-76 (Washington, 1879), pp. 202-337 and 410-416. Peirce’s report is dated December 13, 1878, by which time the name of the Survey had been changed to U.S. Coast and Geodetic Survey.

59 Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1876).

60 Ibid. See report for fifth session, September 25, 1875.

61 The experiments at the Stevens Institute, Hoboken, were reported by Peirce to the Permanent Commission which met in Hamburg, September 4-8, 1878, and his report was published in the general Bericht for 1878 in the Verhandlungen der vom 4 bis 8 September 1878 in Hamburg Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1879), pp. 116-120. Assistant J. E. Hilgard attended for the U.S. Coast and Geodetic Survey. The experiments are described in detail in C. S. Peirce, “On the Flexure of Pendulum Supports,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), app. no. 14, pp. 359-441.

62 Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1877). See report of third session, October 7, 1876.

63 Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart abgehaltenen fünften allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1878).

64 Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1880).

65 Assistants’ Reports, U.S. Coast and Geodetic Survey, 1879-80. Peirce’s paper was published in the American Journal of Science (1879), vol. 18, p. 112.

66 Comptes-rendus de l’Académie des Sciences (Paris, 1879), vol. 89, p. 462.

67 Verhandlungen der vom 13 bis 16 September 1880 zu München abgehaltenen sechsten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1881).

68 Ibid., app. 2.

69 Ibid., app. 2a.

70 Verhandlungen der vom 11 bis zum 15 September 1882 im Haag Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1883).

71 Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen siebenten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1884). Gen. Cutts attended for the U.S. Coast and Geodetic Survey.

72 Ibid., app. 6. See also, Zeitschrift für Instrumentenkunde (1884), vol. 4, pp. 303 and 379.

73 Op. cit. (footnote 67).

74 Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), p. 26.

75 Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1889-90 (Washington, 1891), app. no. 12.

76 Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82 (Washington, 1883).

77 Transactions of the Cambridge Philosophical Society (1856), vol. 9, part 2, p. 8. Also published in Mathematical and Physical Papers (Cambridge, 1901), vol. 3, p. 1.

78 Peirce’s comparison of theory and experiment is discussed in a report on the Peirce memoir by William Ferrel, dated October 19, 1890, Martinsburg, West Virginia. U.S. Coast and Geodetic Survey, Special Reports, 1887-1891 (MS, National Archives, Washington).

79 The stations at which observations were conducted with the Peirce pendulums are recorded in the reports of the Superintendent of the U.S. Coast and Geodetic Survey from 1881 to 1890.

80 Comptes-rendus de l’Académie des Sciences (Paris, 1880), vol. 90, p. 1401. Hervé Faye’s report, dated June 21, 1880, is in the same Comptes-rendus, p. 1463.

81 Commandant C. Defforges, “Sur l’Intensité absolue de la pesanteur,” Journal de Physique (1888), vol. 17, pp. 239, 347, 455. See also, Defforges, “Observations du pendule,” Mémorial du Dépôt général de la Guerre (Paris, 1894), vol. 15. In the latter work, Defforges described a pendulum “reversible inversable,” which he declared to be truly invariable and therefore appropriate for relative determinations. The knives remained fixed to the pendulums, and the effect of interchanging knives was obtained by interchanging weights within the pendulum tube.

82 Papers by Maj. von Sterneck in Mitteilungen des K. u. K. Militär-geographischen Instituts, Wien, 1882-87; see, in particular, vol. 7 (1887).

83 T. C. Mendenhall, “Determinations of Gravity with the New Half-Second Pendulum...,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1890-91 (Washington, 1892), part 2, pp. 503-564.

84 W. H. Burger, “The Measurement of the Flexure of Pendulum Supports with the Interferometer,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1909-10 (Washington, 1911), app. no. 6.

85 E. J. Brown, A Determination of the Relative Values of Gravity at Potsdam and Washington (Special Publication No. 204, U.S. Coast and Geodetic Survey; Washington, 1936).

86 M. Haid, “Neues Pendelstativ,” Zeitschrift für Instrumentenkunde (July 1896), vol. 16, p. 193.

87 Dr. R. Schumann, “Über eine Methode, das Mitschwingen bei relativen Schweremessungen zu bestimmen,” Zeitschrift für Instrumentenkunde (January 1897), vol. 17, p. 7. The design for the stand is similar to that of Peirce’s of 1879.

88 Dr. R. Schumann, “Über die Verwendung zweier Pendel auf gemeinsamer Unterlage zur Bestimmung der Mitschwingung,” Zeitschrift für Mathematik und Physik (1899), vol. 44, p. 44.

89 P. Furtwängler, “Über die Schwingungen zweier Pendel mit annähernd gleicher Schwingungsdauer auf gemeinsamer Unterlage,” Sitzungsberichte der Königlicher Preussischen Akademie der Wissenschaften zu Berlin (Berlin, 1902) pp. 245-253. Peirce investigated the plan of swinging two pendulums on the same stand (Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81, Washington, 1883, p. 26; also in Charles Sanders Peirce, Collected Papers, 6.273). At a conference on gravity held in Washington during May 1882, Peirce again advanced the method of eliminating flexure by hanging two pendulums on one support and oscillating them in antiphase (“Report of a conference on gravity determinations held in Washington, D.C., in May, 1882,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82, Washington, 1883, app. no. 22, pp. 503-516).

90 F. A. Vening Meinesz, Observations de pendule dans les Pays-Bas (Delft, 1923).

91 A. Berroth, “Schweremessungen mit zwei und vier gleichzeitig auf demselben Stativ schwingenden Pendeln,” Zeitschrift für Geophysik, vol. 1 (1924-25), no. 3, p. 93.

92 “Pendulum Apparatus for Gravity Determinations,” Engineering (1926), vol. 122, pp. 271-272.

93 Malcolm W. Gay, “Relative Gravity Measurements Using Precision Pendulum Equipment,” Geophysics (1940), vol. 5, pp. 176-191.

94 L. G. D. Thompson, “An Improved Bronze Pendulum Apparatus for Relative Gravity Determinations,” [published by] Dominion Observatory (Ottawa, 1959), vol. 21, no. 3, pp. 145-176.

95 W. A. Heiskanen and F. A. Vening Meinesz, The Earth and its Gravity Field (McGraw: New York, 1958).

96 F. Kühnen and P. Furtwängler, Bestimmung der Absoluten Grösze der Schwerkraft zu Potsdam mit Reversionspendeln (Berlin: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1906), new ser., no. 27.

97 Reported by Dr. F. Kühnen to the fifth session, October 9, 1895, of the Eleventh General Conference, Die Internationale Erdmessung, held in Berlin from September 25 to October 12, 1895. A footnote states that Assistant O. H. Tittmann, who represented the United States, subsequently reported Peirce’s prior discovery of the influence of the flexure of the pendulum itself upon the period (Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1883-84, Washington, 1885, app. 16, pp. 483-485).

98 Assistants’ Reports, U.S. Coast and Geodetic Survey, 1883-84 (MS, National Archives, Washington).

99 C. S. Peirce, “Effect of the Flexure of a Pendulum Upon its Period of Oscillation,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1883-84 (Washington, 1885), app. no. 16.

100 F. R. Helmert, Beiträge zur Theorie des Reversionspendels (Potsdam: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1898).

101 J. A. Duerksen, Pendulum Gravity Data in the United States (Special Publication No. 244, U.S. Coast and Geodetic Survey; Washington, 1949).

102 Ibid., p. 2. See also, E. J. Brown, loc. cit. (footnote 85).

103 Paul R. Heyl and Guy S. Cook, “The Value of Gravity at Washington,” Journal of Research, National Bureau of Standards (1936), vol. 17, p. 805.

104 Sir Harold Jeffreys, “The Absolute Value of Gravity,” Monthly Notices of the Royal Astronomical Society, Geophysical Supplement (London, 1949), vol. 5, p. 398.

105 J. S. Clark, “The Acceleration Due to Gravity,” Phil. Trans. (1939), vol. 238, p. 65.

106 Hugh L. Dryden, “A Reexamination of the Potsdam Absolute Determination of Gravity,” Journal of Research, National Bureau of Standards (1942), vol. 29, p. 303; and A. Berroth, “Das Fundamentalsystem der Schwere im Lichte neuer Reversionspendelmessungen,” Bulletin Géodésique (1949), no. 12, pp. 183-204.

107 T. C. Mendenhall, op. cit. (footnote 83), p. 522.

108 A. H. Cook, “Recent Developments in the Absolute Measurement of Gravity,” Bulletin Géodésique (June 1, 1957), no. 44, pp. 34-59.

109 See footnote 89.

110 C. S. Peirce, “On the Deduction of the Ellipticity of the Earth, from Pendulum Experiments,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), app. no. 15, pp. 442-456.

111 Heiskanen and Vening Meinesz, op. cit. (footnote 95), p. 74.

112 Ibid., p. 76.

113 Ibid., p. 309.

114 Ibid., p. 310.

115 K. Reicheneder, “Method of the New Measurements at Potsdam by Means of the Reversible Pendulum,” Bulletin Géodésique (March 1, 1959), no. 51, p.72.

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