In 1879, Maj. J. Herschel, R.E., stated:
The years from 1840 to 1865 are a complete blank, if we except Airy’s relative density experiments in 1854. This pause was broken simultaneously in three different ways. Two pendulums of the Kater pattern were sent to India; two after Bessel’s design were set to work in Russia; and at Geneva, Plantamour’s zealous experiments with a pendulum of the same kind mark the commencement of an era of renewed activity on the European continent.[41]
With the statement that Kater invariable pendulums nos. 4 and 6 (1821) were used in India between 1865 and 1873, we now consider the other events mentioned by Herschel.
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]