Repsold-Bessel Reversible Pendulum
As we have noted, Bessel made determinations of gravity with a ball (“simple”) pendulum in the period 1825-1827 and in 1835 at Königsberg and Berlin, respectively. In the memoir on his observations at Königsberg, he set forth the theory of the symmetrical compound pendulum with interchangeable knife edges. [42] Bessel demonstrated theoretically that if the pendulum were symmetrical with respect to its geometrical center, if the times of swing about each axis were the same, the effects of buoyancy and of air set in motion would be eliminated. Laplace had already shown that the knife edge must be regarded as a cylinder and not as a mere line of support. Bessel then showed that if the knife edges were equal cylinders, their effects were eliminated by inverting the pendulum; and if the knife edges were not equal cylinders, the difference in their effects was canceled by interchanging the knives and again determining the times of swing in the so-called erect and inverted positions. Bessel further showed that it is unnecessary to make the times of swing exactly equal for the two knife edges.
The simplified discussion for infinitely small oscillations in a vacuum is as follows: If T1 and T2 are the times of swing about the knife edges, and if h1 and h2 are distances of the knife edges from the center of gravity, and if k is the radius of gyration about an axis through the center of gravity, then from the equation of motion of a rigid body oscillating about a fixed axis under gravity T12 = π2(k2 + h12)/gh1, T22 = π2(k2 + h22)/gh2. Then (h1T12 - h2T22)/(h1 - h2) = (π2/g)(h1 + h2) = τ2.
τ is then the time of swing of a simple pendulum of length h1 + h2. If the difference T1 - T2 is sufficiently small, τ = (h1T1 - h2T2)/(h1 - h2). Prior to its publication by Bessel in 1828, the formula for the time of swing of a simple pendulum of length h1 + h2 in terms of T1, T2 had been given by C. F. Gauss in a letter to H. C. Schumacher dated November 28, 1824.[43]
The symmetrical compound pendulum with interchangeable knives, for which Bessel gave a posthumously published design and specifications,[44] has been called a reversible pendulum; it may thereby be distinguished from Kater’s unsymmetrical convertible pendulum. In 1861, the Swiss Geodetic Commission was formed, and in one of its first sessions in 1862 it was decided to add determinations of gravity to the operations connected with the measurement—at different points in Switzerland—of the arc of the meridian traversing central Europe. [45] It was decided further to employ a reversible pendulum of Bessel’s design and to have it constructed by the firm of A. Repsold and Sons, Hamburg. It was also decided to make the first observations with the pendulum in Geneva; accordingly, the Repsold-Bessel pendulum ([fig. 16]) was sent to Prof. E. Plantamour, director of the observatory at Geneva, in the autumn of 1864. [46]
The Swiss reversible pendulum was about 560 mm. in length (distance between the knife edges) and the time of swing was approximately 3/4-second. At the extremities of the stem of the pendulum were movable cylindrical disks, one of which was solid and heavy, the other hollow and light. It was intended by the mechanicians that equality of times of oscillation about the knife edges would be achieved by adjusting the position of a movable disk. The pendulum was hung by a knife edge on a plate supported by a tripod and having an attachment from which a measuring rod could be suspended so that the distance between the knife edges could be measured by a comparator. Plantamour found it impracticable to adjust a disk until the times of swing about each knife edge were equal. His colleague, Charles Cellérier, [47] then showed that if (T1 - T2)/T1 is sufficiently small so that one can neglect its square, one can determine the length of the seconds pendulum from the times of swing about the knife edges by a theory which uses the distances of the center of gravity from the respective knife edges. Thus, a role for the position of the center of gravity in the theory of the reversible pendulum, which had been set forth earlier by Bessel, was discovered independently by Cellérier for the Swiss observers of pendulums.
In 1866, Plantamour published an extensive memoir “Expériences faites à Genève avec le pendule à réversion.” Another memoir, published in 1872, presented further results of determinations of gravity in Switzerland. Plantamour was the first scientist in western Europe to use a Repsold-Bessel reversible pendulum and to work out methods for its employment.
The Russian Imperial Academy of Sciences acquired two Repsold-Bessel pendulums, and observations with them were begun in 1864 by Prof. Sawitsch, University of St. Petersburg, and others. [48] In 1869, the Russian pendulums were loaned to the India Survey in order to enable members of the Survey to supplement observations with the Kater invariable pendulums nos. 4 and 6 (1821). During the transport of the Russian apparatus to India, the knives became rusted and the apparatus had to be reconditioned. Capt. Heaviside of the India Survey observed with both pendulums at Kew Observatory, near London, in the spring of 1874, after which the Russian pendulums were sent to Pulkowa (Russia) and were used for observations there and in the Caucasus.