oscillations of the pendulum, determinations of the displacement of the
receiver of the Mendenhall apparatus were part of a relative
determination of gravity by members of the Coast and Geodetic Survey.
Initially, a statical method was used, but during 1908-1909 members of
the Survey adapted the Michelson interferometer for the determinations
of flexure during oscillations from the shift of fringes.[84] The first
Mendenhall pendulums were made of bronze, but about 1920 invar was
chosen because of its small coefficient of expansion. About 1930, Lt. E.
J. Brown of the Coast and Geodetic Survey made significant improvements
in the Mendenhall apparatus, and the new form came to be known as the
Brown Pendulum Apparatus.[85]
[Illustration: Figure 27.--THE MICHELSON INTERFEROMETER. The horizontal
component of the force acting on the knife edge through the swinging
pendulum causes the support to move in unison with the pendulum, and
thereby affects the period of the oscillation. This movement is the
so-called flexure of the pendulum support, and must be taken into
account in the most accurate observations.
In 1907, the Michelson interferometer was adapted to this purpose by the
U.S. Coast and Geodetic Survey. As shown here, the interferometer,
resting on a wooden beam, is introduced into the path of a light beam
reflected from a mirror on the vacuum chamber. Movement of that mirror
causes a corresponding movement in the interference fringes in the
interferometer, which can be measured.]
The original Von Sterneck apparatus and that of Mendenhall provided for
the oscillation of one pendulum at a time. After the adoption of the Von
Sterneck pendulum in Europe, there were developed stands on which two or
four pendulums hung at the same time. This procedure provided a
convenient way to observe more than one invariable pendulum at a station
for the purpose of detecting changes in length. Prof. M. Haid of
Karlsruhe in 1896 described a four-pendulum apparatus,[86] and Dr.
Schumann of Potsdam subsequently described a two-pendulum
apparatus.[87]
[Illustration: Figure 28.--APPARATUS WHICH WAS DEVELOPED IN 1929 by the
Gulf Research and Development Company, Harmarville, Pennsylvania. It was
designed to achieve an accuracy within one ten-millionth of the true
value of gravity, and represents the extreme development of pendulum
apparatus for relative gravity measurement. The pendulum was designed so
that the period would be a minimum. The case (the top is missing in this
photograph) is dehumidified and its temperature and electrostatic
condition are controlled. Specially designed pendulum-lifting and
-starting mechanisms are used. The problem of flexure of the case is
overcome by the Faye-Peirce method (see text) in which two dynamically
matched pendulums are swung simultaneously, 180 deg. apart in phase.]
The multiple-pendulum apparatus then provided a method of determining
the flexure of the stand from the action of one pendulum upon a second
pendulum hung on the same stand. This method of determining the
correction for flexure was a development from a "Wippverfahren" invented
at the Geodetic Institute in Potsdam. A dynamometer was used to impart
periodic impulses to the stand, and the effect was observed upon a
pendulum initially at rest. Refinements of this method led to the
development of a method used by Lorenzoni in 1885-1886 to determine the
flexure of the stand by action of an auxiliary pendulum upon the
principal pendulum. Dr. Schumann, in 1899, gave a mathematical theory of
such determinations,[88] and in his paper cited the mathematical methods
of Peirce and Cellerier for the theory of Faye's proposal at Stuttgart
in 1877 to swing two similar pendulums on the same support with equal
amplitudes and in opposite phases.
[Illustration: Figure 29.--THE GULF PENDULUM is about 10.7 inches long,
and has a period of .89 second. It is made of fused quartz which is
resistant to the influence of temperature change and to the earth's
magnetism. Quartz pendulums are subject to the influence of
electrostatic charge, and provision is made to counteract this through
the presence of a radium salt in the case. The bearings are made of
Pyrex glass.]
In 1902, Dr. P. Furtwaengler[89] presented the mathematical theory of
coupled pendulums in a paper in which he referred to Faye's proposal of
1877 and reported that the difficulties predicted upon its application
had been found not to occur. Finally, during the gravity survey of
Holland in the years 1913-1921, in view of instability of supports
caused by the mobility of the soil, F. A. Vening Meinesz adopted Faye's
proposed method of swinging two pendulums on the same support.[90] The
observations were made with the ordinary Stueckrath apparatus, in which
four Von Sterneck pendulums swung two by two in planes perpendicular to
each other. This successful application of the method--which had been
proposed by Faye and had been demonstrated theoretically to be sound by
Peirce, who also published a design for its application--was rapidly
followed for pendulum apparatus for relative determinations by
Potsdam,[91] Cambridge (England),[92] Gulf Oil and Development
Company,[93] and the Dominion Observatory at Ottawa.[94] Heiskanen and
Vening Meinesz state:
The best way to eliminate the effect of flexure is to use two
synchronized pendulums of the same length swinging on the same
apparatus in the same plane and with the same amplitudes but in
opposite phases; it is clear then the flexure is zero.[95]
In view of the fact that the symmetrical reversible pendulum is named
for Bessel, who created the theory and a design for its application by
Repsold, it appears appropriate to call the method of eliminating
flexure by swinging two pendulums on the same support the Faye-Peirce
method. Its successful application was made possible by Maj. von
Sterneck's invention of the short, 1/4-meter pendulum.
[Illustration: Figure 30.--THE ACCUMULATED DATA OF GRAVITY observations
over the earth's surface have indicated that irregularities such as
mountains do not have the effect which would be expected in modifying
gravity, but are somehow compensated for. The most satisfactory solution
to this still unanswered question has been the theory of isostasy,
according to which variations in the density of the material in the
earth's crust produce a kind of hydrostatic equilibrium between its
higher and lower parts, as they "float" on the earth's fluid core. The
metals of different density floating in mercury in this diagram
illustrate isostasy according to the theory of Pratt and Hayford.]
Absolute Value of Gravity at Potsdam
The development of the reversible pendulum in the 19th century
culminated in the absolute determination of the intensity of gravity at
Potsdam by Kuehnen and Furtwaengler of the Royal Prussian Geodetic
Institute, which then became the world base for gravity surveys.[96]
We have previously seen that in 1869 the Geodetic Institute--founded by
Lt. Gen. Baeyer--had acquired a Repsold-Bessel reversible pendulum which
was swung by Dr. Albrecht under the direction of Dr. Bruhns.
Dissatisfaction with this instrument was expressed by Baeyer in 1875 to
Charles S. Peirce, who then, by experiment and mathematical analysis of
the flexure of the stand under oscillations of the pendulum, determined
that previously reported results with the Repsold apparatus required
correction. Dr. F. R. Helmert, who in 1887 succeeded Baeyer as director
of the Institute, secured construction of a building for the Institute
in Potsdam, and under his direction the scientific study of the
intensity of gravity was pursued with vigor. In 1894, it was discovered
in Potsdam that a pendulum constructed of very flexible material yielded
results which differed markedly from those obtained with pendulums of
greater stiffness. Dr. Kuehnen of the Institute discovered that the
departure from expectations was the result of the flexure of the
pendulum staff itself during oscillations.[97]
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, Kuehnen and
Furtwaengler 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 Kuehnen and Furtwaengler published
their monumental memoir, "Bestimmung der Absoluten Groesze 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/sec^{2}. 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
(T_{1})^{2}/(T_{2})^{2} = g_{2}/g_{1}. 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/sec^{2}.[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/sec^{2}.[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]
[Illustration: Figure 31.--MAP SHOWING THE DISTRIBUTION of gravity
stations throughout the United States as of December 1908.]
[Illustration: 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/sec^{2} 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/sec^{2}. 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/sec^{2}.[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.
[Illustration: 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.
[Illustration: 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 Kuebel (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.
Luetke 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
[gamma] = [gamma]_{E}(1 + [beta]sin^{2} [phi] + [epsilon]sin^{2} 2[phi])
where [gamma] is normal gravity at latitude [phi], [gamma]_{E} is the
value of gravity at sea level at the equator, [beta] is a parameter
which is computed on the basis of Clairaut's theorem from the flattening
value of the meridian, and [epsilon] 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 g_{0} is
observed gravity reduced to sea level and [gamma] is normal gravity
obtained from the International Gravity Formula, then
[Delta]g = g_{0} - [gamma]
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 [xi] and
[eta] 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