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 Konigsberg and 

 Berlin, respectively. In the memoir on his observa- 

 tions at Konigsberg, he set forth the theory of the 

 symmetrical compound pendulum with interchange- 

 able 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 oscil- 

 lations in a vacuum is as follows: If 7"i and T_> 

 are the times of swing about the knife edges, and if 

 /ii and h 2 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 



r i i -j- n-i T 2 (^ 2 + ^l 2 ) 



about a fixed axis under gravitv /,= — ! — : -> 



gh i 



/ ' "' -Then -±- j =— (hi+h 2 ) = T~. 



gh 2 hi—h 2 g 



t is then the time of swing of a simple pendulum 

 of length //, h 2 . If the difference 7~i — T> is suffi- 

 ciently small, T=-^-' — ; Prior to its publication 



hi—hi 



by Bessel in 1828, the formula for the time of 



of a simple pendulum of length h, \ h 2 in terms 

 of 7"i, T 2 had been given b) C. I . Gauss in a letter 



to H. C. Schumacher dated November 28, 1824." 

 The symmetrical compound pendulum with inter- 

 changeable knives, for which Bessel gave a post- 

 humously published design and specifications, 44 has 

 been called a reversible pendulum; it may thereby 

 be distinguished from Kater's unsymmetrical con- 

 vertible 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 measure- 

 ment — 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 pendu- 

 lum in Geneva; accordingly, the Repsold-Bessel pen- 

 dulum (fig. 16) was sent to Prof. E. Plantamour, 

 director of the observatory at Geneva, in the autumn 

 of 1864. » 



The Swiss reversible pendulum was about 560 mm. 

 in length (distance between the knife edges) and the 

 time of swing was approximately % 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 Cel- 



lerier, 4 ' then showed that if — ~ — is sufficiently 



*'- Bessel, op. tit. (footnote 21), article 31. 



43 C. A. F. Peters, Briejivechsel zwischen C F. Gauss und II. C. 

 Schumacher (Altona, Germany, 1860), Band 2, p. 3. The 

 correction required if the times of swing are not exactly the 

 same is said to have been given also by Bohnenberger. 



44 F. \V. Bessel, "Construction cines symmetrisch gefountcn 

 Pendels mit reciproken Axen, von Bessel," Asttonomischt 

 Nachrichtcn (1849), vol. 30, p. 1. 



45 E. Plantamour, "Experiences faites a Geneve avec le 

 pendule a r6version," Memoir es de la Sociitt de Physique el d'his- 

 torire nalurelle de Geneve, 1865 (Geneva, 1866), vol. 18, p. 309. 



• [bid., pp. 309-416. 



47 C. Cellerier, "note sur la Mesure de la Pcsanteur par 

 le Pendule," Mi-moires de la Societi de Physique el d'historire 

 nalurelle de Cenive, 1865 (Geneva, 1866), vol. 18, pp. 197-218. 



320 



BULLETIN 240: CONTRIBUTIONS FROM THE MUSEUM OF HISTORY AND TECHNOLOGY 



