DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19th CENTURY (continued 2)
Reading edition
Peirce, in 1883, had discovered that the recesses cut in his pendulums for the insertion of tongues that carried the knives had resulted in the flexure of the pendulum staff.[98] By experiment, he also found an even greater flexure for the Repsold pendulum. In order to eliminate this source of error, Peirce designed a pendulum with knives that extended from each side of the cylindrical staff, and he received authorization from the superintendent of the Coast and Geodetic Survey to arrange for the construction of such pendulums by Gautier in Paris. Peirce, who had made his plans in consultation with Gautier, was called home before the pendulums were completed, and these new instruments remained undelivered.
In a memoir titled “Effect of the flexure of a pendulum upon its period of oscillation,”[99] Peirce determined analytically the effect on the period of a pendulum with a single elastic connection between two rigid parts of the staff. Thus, Peirce discovered experimentally the flexure of the staff and derived for a simplified case the effect on the period. It is not known if he ever found the integrated effect of the continuum of elastic connections in the pendulum. Lorenzoni, in 1896, offered a solution to the problem, and Almansi, in 1899, gave an extended analysis. After the independent discovery of the problem at the Geodetic Institute, Dr. Helmert took up the problem and criticized the theories of Peirce and Lorenzoni. He then presented his own theory of flexure in a comprehensive memoir.[100] In view of the previous neglect of the flexure of the pendulum staff in the reduction of observations, Helmert directed that the Geodetic Institute make a new absolute determination of the intensity of gravity at Potsdam. For this purpose, Kühnen and Furtwängler used the following reversible pendulums which had been constructed by the firm of A. Repsold and Sons in Hamburg:
1. The seconds pendulum of the Geodetic Institute procured in 1869.
2. A seconds pendulum from the Astronomical Observatory, Padua.
3. A heavy, seconds pendulum from the Imperial and Royal Military-Geographical Institute, Vienna.
4. A light, seconds pendulum from the Imperial and Royal Military-Geographical Institute.
5. A 1/2-second, reversible pendulum of the Geodetic Institute procured in 1892.
Work was begun in 1898, and in 1906 Kühnen and Furtwängler published their monumental memoir, “Bestimmung der Absoluten Grösze der Schwerkraft zu Potsdam mit Reversionspendeln.”
The acceleration of gravity in the pendulum room of the Geodetic Institute was determined to be 981.274 ± 0.003 cm/sec2. In view of the exceptionally careful and thorough determination at the Institute, Potsdam was accepted as the world base for the absolute value of the intensity of gravity. The absolute value of gravity at some other station on the Potsdam system was determined from the times of swing of an invariable pendulum at the station and at Potsdam by the relation T12/T22 = g2/g1. Thus, in 1900, Assistant G. R. Putnam of the Coast and Geodetic Survey swung Mendenhall pendulums at the Washington base and at Potsdam, and by transfer from Potsdam determined the intensity of gravity at the Washington base to be 980.112 cm/sec2.[101] In 1933, Lt. E. J. Brown made comparative measurements with improved apparatus and raised the value at the Washington base to 980.118 cm/sec2.[102]
In view of discrepancies between the results of various relative determinations, the Coast and Geodetic Survey in 1928 requested the National Bureau of Standards to make an absolute determination for Washington. Heyl and Cook used reversible pendulums made of fused silica having a period of approximately 1 second. Their result, published in 1936, was interpreted to indicate that the value at Potsdam was too high by 20 parts in 1 million.[103] This estimate was lowered slightly by Sir Harold Jeffreys of Cambridge, England, who recomputed the results of Heyl and Cook by different methods.[104]
Figure 31.—Map showing the distribution of gravity stations throughout the United States as of December 1908.
Figure 32.—Map showing the distribution of gravity stations throughout the United States in 1923.
In 1939, J. S. Clark published the results of a determination of gravity with pendulums of a non-ferrous Y-alloy[105] at the National Physical Laboratory at Teddington, England, and, after recomputation of results by Jeffreys, the value was found to be 12.8 parts in 1 million less than the value obtained by transfer from Potsdam. Dr. Hugh L. Dryden of the National Bureau of Standards, and Dr. A. Berroth of the Geodetic Institute at Potsdam, have recomputed the Potsdam data by different methods of adjustment and concluded that the Potsdam value was too high by about 12 parts in a million.[106] Determination of gravity at Leningrad by Russian scientists likewise has indicated that the 1906 Potsdam value is too high. In the light of present information, it appears justifiable to reduce the Potsdam value of 981.274 by .013 cm/sec2 for purposes of comparison. If the Brown transfer from Potsdam in 1933 was taken as accurate, the value for the Washington base would be 980.105 cm/sec2. In this connection, it is of interest to note that the value given by Charles S. Peirce for the comparable Smithsonian base in Washington, as determined by him from comparative methods in the 1880’s and reported in the Annual Report of the Superintendent of the Coast and Geodetic Survey for the year 1890-1891, was 980.1017 cm/sec2.[107] This value would appear to indicate that Peirce’s pendulums, observations, and methods of reduction of data were not inferior to those of the scientists of the Royal Prussian Geodetic Institute at Potsdam.
Doubts concerning the accuracy of the Potsdam value of gravity have stimulated many new determinations of the intensity of gravity since the end of World War II. In a paper published in June 1957, A. H. Cook, Metrology Division, National Physical Laboratory, Teddington, England, stated:
At present about a dozen new absolute determinations are in progress or are being planned. Heyl and Cook’s reversible pendulum apparatus is in use in Buenos Aires and further reversible pendulum experiments have been made in the All Union Scientific Research Institute of Metrology, Leningrad (V N I I M) and are planned at Potsdam. A method using a very long pendulum was tried out in Russia about 1910 and again more recently and there are plans for similar work in Finland. The first experiment with a freely falling body was that carried out by Volet who photographed a graduated scale falling in an enclosure at low air pressure. Similar experiments have been completed in Leningrad and are in progress at the Physikalisch-Technische Bundesanstalt (Brunswick) and at the National Research Council (Ottawa), and analogous experiments are being prepared at the National Physical Laboratory and at the National Bureau of Standards. Finally, Professor Medi, Director of the Istituto Nazionale di Geofisica (Rome), is attempting to measure the focal length of the paraboloidal surface of a liquid in a rotating dish.[108]
Application of Gravity Surveys
We have noted previously that in the ancient and early modern periods, the earth was presupposed to be spherical in form. Determination of the figure of the earth consisted in the measurement of the radius by the astronomical-geodetic method invented by Eratosthenes. Since the earth was assumed to be spherical, gravity was inferred to be constant over the surface of the earth. This conclusion appeared to be confirmed by the determination of the length of the seconds pendulum at various stations in Europe by Picard and others. The observations of Richer in South America, the theoretical discussions of Newton and Huygens, and the measurements of degrees of latitude in Peru and Sweden demonstrated that the earth is an oblate spheroid.
Figure 33.—Gravity characteristics of the globe. Deductions as to the distribution of matter in the earth can be made from gravity measurements. This globe shows worldwide variations in gravity as they now appear from observations at sea (in submarines) as well as on land. It is based on data from the Institute of Geodesy at Ohio State University.
The theory of gravitation and the theory of central forces led to the result that the intensity of gravity is variable over the surface of the earth. Accordingly, determinations of the intensity of gravity became of value to the geodesist as a means of determining the figure of the earth. Newton, on the basis of the meager data available to him, calculated the ellipticity of the earth to be 1/230 (the ellipticity is defined by (a-b)/a, where a is the equatorial radius and b the polar radius). Observations of the intensity of gravity were made on the historic missions to Peru and Sweden. Bouguer and La Condamine found that at the equator at sea level the seconds pendulum was 1.26 Paris-lines shorter than at Paris. Maupertuis found that in northern Sweden a certain pendulum clock gained 59.1 seconds per day on its rate in Paris. Then Clairaut, from the assumption that the earth is a spheroid of equilibrium, derived a theorem from which the ellipticity of the earth can be derived from values of the intensity of gravity.
Figure 34.—An exhibit of gravity apparatus at the Smithsonian Institution. Suspended on the wall, from left to right, are the invariable pendulums of Mendenhall (1/2-second), Peirce (1873-1874), and Peirce (1881-1882); the double pendulum of Edward Kübel (see fig. 15, p. 319), and the reversible pendulum of Peirce. On the display counter, from left to right, are the vacuum chamber, telescope and flash apparatus for the Mendenhall 1/4-second apparatus. Shown below these are the four pendulums used with the Mendenhall apparatus, the one on the right having a thermometer attached. At bottom, right, is the Gulf apparatus (cover removed) mentioned in the text, shown with one quartz pendulum.
Early in the 19th century a systematic series of observations began to be conducted in order to determine the intensity of gravity at stations all over the world. Kater invariable pendulums, of which 13 examples have been mentioned in the literature, were used in surveys of gravity by Kater, Sabine, Goldingham, and other British pendulum swingers. As has been noted previously, a Kater invariable pendulum was used by Adm. Lütke of Russia on a trip around the world. The French also sent out expeditions to determine values of gravity. After several decades of relative inactivity, Capts. Basevi and Heaviside of the Indian Survey carried out an important series of observations from 1865 to 1873 with Kater invariable pendulums and the Russian Repsold-Bessel pendulums. In 1881-1882 Maj. J. Herschel swung Kater invariable pendulums nos. 4, 6 (1821), and 11 at stations in England and then brought them to the United States in order to make observations which would connect American and English base stations.[109]
The extensive sets of observations of gravity provided the basis of calculations of the ellipticity of the earth. Col. A. R. Clarke in his Geodesy (London, 1880) calculated the ellipticity from the results of gravity surveys to be 1/(292.2 ± 1.5). Of interest is the calculation by Charles S. Peirce, who used only determinations made with Kater invariable pendulums and corrected for elevation, atmospheric effect, and expansion of the pendulum through temperature.[110] He calculated the ellipticity of the earth to be 1/(291.5 ± 0.9).
The 19th century witnessed the culmination of the ellipsoidal era of geodesy, but the rapid accumulation of data made possible a better approximation to the figure of the earth by the geoid. The geoid is defined as the average level of the sea, which is thought of as extended through the continents. The basis of geodetic calculations, however, is an ellipsoid of reference for which a gravity formula expresses the value of normal gravity at a point on the ellipsoid as a function of gravity at sea level at the equator, and of latitude. The general assembly of the International Union of Geodesy and Geophysics, which was founded after World War I to continue the work of Die Internationale Erdmessung, adopted in 1924 an international reference ellipsoid,[111] of which the ellipticity, or flattening, is Hayford’s value 1/297. In 1930, the general assembly adopted a correlated International Gravity Formula of the form γ = γE(1 + β(sin2 φ) + ε(sin2 2φ)) where γ is normal gravity at latitude φ, γE is the value of gravity at sea level at the equator, β is a parameter which is computed on the basis of Clairaut’s theorem from the flattening value of the meridian, and ε is a constant which is derived theoretically. The plumb line is perpendicular to the geoid, and the components of angle between the perpendiculars to geoid and reference ellipsoid are deflections of the vertical. The geoid is above the ellipsoid of reference under mountains and it is below the ellipsoid on the oceans, where the geoid coincides with mean sea level. In physical geodesy, gravimetric data are used for the determination of the geoid and components of deflections of the vertical. For this purpose, one must reduce observed values of gravity to sea level by various reductions, such as free-air, Bouguer, isostatic reductions. If g0 is observed gravity reduced to sea level and γ is normal gravity obtained from the International Gravity Formula, then Δg = g0 - γ is the gravity anomaly.[112]
In 1849, Stokes derived a theorem whereby the distance N of the geoid from the ellipsoid of reference can be obtained from an integration of gravity anomalies over the surface of the earth. Vening Meinesz further derived formulae for the calculation of components of the deflection of the vertical.
Geometrical geodesy, which was based on astronomical-geodetic methods, could give information only concerning the external form of the figure of the earth. The gravimetric methods of physical geodesy, in conjunction with methods such as those of seismology, enable scientists to test hypotheses concerning the internal structure of the earth. Heiskanen and Vening Meinesz summarize the present-day achievements of the gravimetric method of physical geodesy by stating[113] that it alone can give:
1. The flattening of the reference ellipsoid.
2. The undulations N of the geoid.
3. The components of the deflection of the vertical ζ and η at any point, oceans and islands included.
4. The conversion of existing geodetic systems to the same world geodetic system.
5. The reduction of triangulation base lines from the geoid to the reference ellipsoid.
6. The correction of errors in triangulation in mountainous regions due to the effect of the deflections of the vertical.
7. Geophysical applications of gravity measurements, e.g., the isostatic study of the earth’s interior and the exploration of oil fields and ore deposits.
With astronomical observations or with existing triangulations, the gravimetric method can accomplish further results. Heiskanen and Vening Meinesz state:
It is the firm conviction of the authors that the gravimetric method is by far the best of the existing methods for solving the main problems of geodesy, i.e., to determine the shape of the geoid on the continents as well as at sea and to convert the existing geodetic systems to the world geodetic system. It can also give invaluable help in the computation of the reference ellipsoid.[114]
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:
U.S. GOVERNMENT PRINTING OFFICE: 1965 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402—Price 70 cents.
INDEX
Airy, G. B., 319, 324, 332
Albrecht, Karl Theodore, 322, 338
Al-Mamun, seventh calif of Bagdad, 306
Almansi, Emilio, 339
Aristotle, 306
Baeyer, J. J., 321, 322, 324, 338, 346
Baily, Francis, 317
Basevi, James Palladio, 345
Berroth, A., 342
Bessel, Friedrich Wilhelm, 313, 314, 319, 320, 324, 325, 338, 346
Biot, Jean Baptiste, 325, 329
Bohnenberger, Johann Gottlieb Friedrich, 315
Borda, J. C., 311, 312, 315, 325, 329, 346
Boscovitch, Père R. J., 310, 311
Bouguer, Pierre, 307, 309, 327, 343, 345
Brahe, Tycho, 306
Brown, E. J., 334, 339
Browne, Henry, 304, 314
Bruhns, C., 322, 324, 338
Brunner Brothers (Paris), 329
Cassini, Giovanni-Domenico, 306, 307
Cassini, Jacques, 306
Cassini de Thury, J. D., 311, 312, 315, 325, 329, 346
Cellérier, Charles, 320, 321, 325, 326, 329, 336
Clairaut, Alexis Claude, 308, 309, 343, 345
Clark, J. S., 342
Clarke, A. R., 345
Colbert, Jean Baptiste, 306
Cook, A. H., 342
Cook, Guy S., 339, 342
Defforges, C., 314, 329, 346
De Freycinet, Louis Claude de Saulses, 317
De la Hire, Gabriel Philippe, 306
De Prony, M. G., 314
Dryden, Hugh L., 342
Du Buat, L. G., 314
Duperry, Capt. Louis Isidore, 317
Eratosthenes, 306, 308, 342
Eudoxus of Cnidus, 306
Faye, Hervé, 325, 336, 346, 347
Fernel, Jean, 306
Furtwängler, P., 337
Galilei, Galileo, 304, 305, 346
Gauss, C. F., 320
Gautier, P., 339
Godin, Louis, 307
Goldingham, John, 316, 345
Greely, A. W., 329
Gulf Oil and Development Company, 338
Haid, M., 335
Hall, Basil, 316
Heaviside, W. J., 321, 345
Heiskanen, W. A., 338, 345, 346
Helmert, F. R., 338, 339
Helmholtz, Hermann von, 326
Herschel, John, 319, 328, 345
Heyl, Paul R., 339, 342
Hirsch, Adolph, 322, 324
Huygens, Christiaan, 304, 305, 307, 314, 342, 346
Ibañez, Carlos, 325
Jeffreys, Sir Harold, 342
Jones, Thomas, 318
Kater, Henry, 304, 314, 325, 327, 329, 345, 346
Kühnen, F., 338, 339
La Condamine, Charles Marie de, 307, 310, 311, 343
Laplace, Marquis Pierre Simon de, 309, 313, 320
Lorenzoni, Giuseppe, 336, 339
Lütke, Count Feodor Petrovich, 316, 345
Maupertius, P. L. Moreau de, 308, 343
Maxwell, James Clerk, 324
Medi, Enrico, 342
Mendenhall, Thomas Corwin, 319, 331, 332, 334, 347
Mersenne, P. Marin, 305
Newton, Sir Isaac, 303, 307, 308, 342, 343
Norwood, Richard, 306
Oppolzer, Theodor von, 322, 324
Patterson, Carlile Pollock, 325, 326
Peirce, Charles Sanders, 314, 322, 332, 336, 342, 345
Peters, C. A. F., 322, 324
Picard, Abbé Jean, 306, 308, 342
Plantamour, E., 319, 324
Posidonius, 306
Preston, E. D., 328, 329
Putnam, G. R., 339
Pythagoras, 306
Repsold, A., and Sons (Hamburg), 320, 322, 338, 339, 346
Richer, Jean, 307, 342
Sabine, Capt. Edward, 315, 325, 329, 345
Sawitsch, A., 321, 322
Schumacher, H. C., 320
Schumann, R., 335, 336
Snell, Willebrord, 306
Sterneck, Robert von, 331, 332, 335, 338, 346
Stokes, George Gabriel, 324, 328, 329, 345, 346
Ulloa, Antonio de, 308
Vening Meinesz, F. A., 337, 338, 345
Volet, Charles, 342
Wilkes, Charles, 317, 318