The Repsold-Bessel reversible pendulum was designed and initially used
to make absolute determinations of gravity not only at initial stations
such as Kew, the observatory in Paris, and the Smithsonian Institution
in Washington, D.C., but also at stations in the field. An invariable
pendulum with a single knife edge, however, is adequate for relative
determinations. As we have seen, such invariable pendulums had been used
by Bouguer and Kater, and after the experiences with the Repsold
apparatus had been recommended again by Baeyer for relative
determinations. But an invariable pendulum is subject to uncontrollable
changes of length. Peirce proposed to detect such changes in an
invariable pendulum in the field by combining the invariable and
reversible principles. He explained his proposal to Faye in a letter
dated July 23, 1880, and he presented it on September 16, 1880, at the
fourth session of the sixth General Conference of _Die Europaeische
Gradmessung_, in Munich.[73]
As recorded in the Proceedings of the Conference, Peirce wrote:
But I obviate it in making my pendulum both invariable and
reversible. Every alteration of the pendulum will be revealed
immediately by the change in the difference of the two periods
of oscillation in the two positions. Once discovered, it will be
taken account of by means of new measures of the distance
between the two supports.
Peirce added that it seemed to him that if the reversible pendulum
perhaps is not the best instrument to determine absolute gravity, it is,
on condition that it be truly invariable, the best to determine relative
gravity. Peirce further stated that he would wish that the pendulum be
formed of a tube of drawn brass with heavy plugs of brass equally drawn.
The cylinder would be terminated by two hemispheres; the knives would be
attached to tongues fixed near the ends of the cylinder.
During the years 1881 and 1882, four invariable, reversible pendulums
were made after the design of Peirce at the office of the U.S. Coast and
Geodetic Survey in Washington, D.C. The report of the superintendent for
the year 1880-1881 states:
A new pattern of the reversible pendulum has been invented,
having its surface as nearly as convenient in the form of an
elongated ellipsoid. Three of these instruments have been
constructed, two having a distance of one meter between the
knife edges and the third a distance of one yard. It is proposed
to swing one of the meter pendulums at a temperature near 32 deg.
F. at the same time that the yard is swung at 60 deg. F., in order
to determine anew the relation between the yard and the meter.[74]
The report for 1881-1882 mentions four of these Peirce pendulums.
A description of the Peirce invariable, reversible pendulums was given
by Assistant E. D. Preston in "Determinations of Gravity and the
Magnetic Elements in Connection with the United States Scientific
Expedition to the West Coast of Africa, 1889-90."[75] The invariable,
reversible pendulum, Peirce no. 4, now preserved in the Smithsonian
Institution's Museum of History and Technology (fig. 34), may be taken
as typical of the meter pendulums: In the same memoir, Preston gives the
diameter of the tube as 63.7 mm., thickness of tube 1.5 mm., weight
10.680 kilograms, and distance between the knives 1.000 meter.
The combination of invariability and reversibility in the Peirce
pendulums was an innovation for relative determinations. Indeed, the
combination was criticized by Maj. J. Herschel, R.E., of the Indian
Survey, at a conference on gravity held in Washington in May 1882 on the
occasion of his visit to the United States for the purpose of
connecting English and American stations by relative determinations with
three Kater invariable pendulums. These three pendulums have been
designated as nos. 4, 6 (1821), and 11.[76]
[Illustration: Figure 20.--SUPPORT FOR THE PEIRCE PENDULUM, 1889. Much
of the work of C. S. Peirce was concerned with the determination of the
error introduced into observations made with the portable apparatus by
the vibration of the stand with the pendulum. He showed that the popular
Bessel-Repsold apparatus was subject to such an error. His own pendulums
were swung from a simple but rugged wooden frame to which a hardened
steel bearing was fixed.]
Another novel characteristic of the Peirce pendulums was the mainly
cylindrical form. Prof. George Gabriel Stokes, in a paper "On the Effect
of the Internal Friction of Fluids on the Motion of Pendulums"[77] that
was read to the Cambridge Philosophical Society on December 9, 1850, had
solved the hydrodynamical equations to obtain the resistance to the
motions of a sphere and a cylinder in a viscous fluid. Peirce had
studied the effect of viscous resistance on the motion of his
Repsold-Bessel pendulum, which was symmetrical in form but not
cylindrical. The mainly cylindrical form of his pendulums (fig. 19)
permitted Peirce to predict from Stokes' theory the effect of viscosity
and to compare the results with experiment. His report of November 20,
1889, in which he presented the comparison of experimental results with
the theory of Stokes, was not published.[78]
Peirce used his pendulums in 1883 to establish a station at the
Smithsonian Institution that was to serve as the base station for the
Coast and Geodetic Survey for some years. Pendulum Peirce no. 1 was
swung at Washington in 1881 and was then taken by the party of
Lieutenant Greely, U.S.A., on an expedition to Lady Franklin Bay where
it was swung in 1882 at Fort Conger, Grinnell Land, Canada. Peirce nos.
2 and 3 were swung by Peirce in 1882 at Washington, D.C.; Hoboken, New
Jersey; Montreal, Canada; and Albany, New York. Assistant Preston took
Peirce no. 3 on a U.S. eclipse expedition to the Caroline Islands in
1883. Peirce in 1885 swung pendulums nos. 2 and 3 at Ann Arbor,
Michigan; Madison, Wisconsin; and Ithaca, New York. Assistant Preston in
1887 swung Peirce nos. 3 and 4 at stations in the Hawaiian Islands, and
in 1890 he swung Peirce nos. 3 and 4 at stations on the west coast of
Africa.[79]
The new pattern of pendulum designed by Peirce was also adopted in
France, after some years of experience with a Repsold-Bessel pendulum.
Peirce in 1875 had swung his Repsold-Bessel pendulum at the observatory
in Paris, where Borda and Cassini, and Biot, had made historic
observations and where Sabine also had determined gravity by comparison
with Kater's value at London. During the spring of 1880, Peirce made
studies of the supports for the pendulums of these earlier
determinations and calculated corrections to those results for
hydrodynamic effects, viscosity, and flexure. On June 14, 1880, Peirce
addressed the Academy of Sciences, Paris, on the value of gravity at
Paris, and compared his results with the corrected results of Borda and
Biot and with the transferred value of Kater.[80]
In the same year the French Geographic Service of the Army acquired a
Repsold-Bessel reversible pendulum of the smaller type, and Defforges
conducted experiments with it.[81] He introduced the method of measuring
flexure from the movement of interference fringes during motion of the
pendulum. He found an appreciable difference between dynamical and
statical coefficients of flexure and concluded that the "correction
formula of Peirce and Cellerier is suited perfectly to practice and
represents exactly the variation of period caused by swaying of the
support, on the condition that one uses the statical coefficient."
Defforges developed a theory for the employment of two similar pendulums
of the same weight, but of different length, and hung by the same
knives. This theory eliminated the flexure of the support and the
curvature of the knives from the reduction of observations.
Pendulums of 1-meter and of 1/2-meter distance between the knife edges
were constructed from Defforges' design by Brunner Brothers in Paris
(fig. 21). These Defforges pendulums were cylindrical in form with
hemispherical ends like the Peirce pendulums, and were hung on knives
that projected from the sides of the pendulum, as in some unfinished
Gautier pendulums designed by Peirce in 1883 in Paris.
[Illustration: Figure 21.--REVERSIBLE PENDULUM APPARATUS of Defforges,
as constructed by Brunner, Paris, about 1887. The clock and telescope
used to observe coincidences are not shown. The telescope shown is part
of an interferometer used to measure flexure of the support. One mirror
of the interferometer is attached to the pendulum support; the other to
the separate masonry pillar at the left.]
[Illustration: Figure 22.--BECAUSE OF THE GREATER SIMPLICITY of its use,
the invariable pendulum superseded the convertible pendulum towards the
end of the 19th century, except at various national base stations (Kew,
Paris, Potsdam, Washington, D.C., etc.). Shown here are, right to left,
a pendulum of the type used by Peirce at the Hoosac Tunnel in 1873-74,
the Mendenhall 1/2-second pendulum of 1890, and the pendulum designed by
Peirce in 1881-1882.]
[Illustration: Figure 23.--THE OVERALL SIZE of portable pendulum
apparatus was greatly reduced with the introduction of this 1/2-second
apparatus in 1887, by the Austrian military officer, Robert von
Sterneck. Used with a vacuum chamber not shown here, the apparatus is
only about 2 feet high. Coincidences are observed by the reflection of a
periodic electric spark in two mirrors, one on the support and the other
on the pendulum itself.]
[Illustration: Figure 24.--THOMAS C. MENDENHALL (1841-1924). Although
largely self-educated, he became the first professor of physics and
mechanics at the Ohio Agricultural and Mechanical College (later Ohio
State University), and was subsequently connected with several other
universities. In 1878, while teaching at the Tokyo Imperial University
in Japan, he made gravity measurements between Tokyo and Fujiyama from
which he calculated the mean density of the earth. While superintendent
of the U.S. Coast and Geodetic Survey, 1889-94, he developed the
pendulum apparatus which bears his name.]