Editing Kater's pendulum (section)

===Repsold&ndash;Bessel pendulum===
[[File:Repsold pendulum.png|thumb|80px|Repsold pendulum.]]
Repeatedly timing each period of a Kater pendulum, and adjusting the weights until they were equal, was time-consuming and error-prone. [[Friedrich Bessel]] showed in 1826 that this was unnecessary. As long as the periods measured from each pivot, ''T''<sub>1</sub> and ''T''<sub>2</sub>, are close in value, the period ''T'' of the equivalent simple pendulum can be calculated from them:<ref>[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA15# Poynting & Thompson 1907], p. 15</ref>

:<math>T^2 = \frac{T_1^2 + T_2^2}{2} + \frac{T_1^2 - T_2^2}{2} \left ( \frac {h_1 + h_2}{h_1-h_2} \right ) \, \qquad \qquad \qquad (2)</math>

Here <math>h_1\,</math> and <math>h_2\,</math> are the distances of the two pivots from the pendulum's center of gravity.   The distance between the pivots, <math>h_1 + h_2\,</math>,  can be measured with great accuracy.   <math>h_1\,</math> and <math>h_2\,</math>, and thus their difference <math>h_1 - h_2\,</math>,  cannot be measured with comparable accuracy.  They are found by balancing the pendulum on a knife edge to find its center of gravity, and measuring the distances of each of the pivots from the center of gravity. However, because <math>T_1^2 - T_2^2\,</math> is so much smaller than <math>T_1^2 + T_2^2\,</math>, the second term on the right in the above equation is small compared to the first, so <math>h_1 - h_2\,</math> doesn't have to be determined with high accuracy, and the balancing procedure described above is sufficient to give accurate results.

Therefore, the pendulum doesn't have to be adjustable at all, it can simply be a rod with two pivots. As long as each pivot is close to the [[center of oscillation]] of the other, so the two periods are close, the period ''T'' of the equivalent simple pendulum can be calculated with equation (2), and the gravity can be calculated from ''T'' and ''L'' with (1).

In addition, Bessel showed that if the pendulum was made with a symmetrical shape, but internally weighted on one end, the error caused by effects of air resistance would cancel out. Also, another error caused by the non-zero radius of the pivot knife edges could be made to cancel out by interchanging the knife edges.

Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract to the Swiss Geodetic Commission, developed a symmetric pendulum 56&nbsp;cm long with interchangeable pivot blades, with a period of about {{frac|3|4}} second. The Repsold pendulum was used extensively by the Swiss and Russian Geodetic agencies, and in the [[Great Trigonometric Survey|Survey of India]]. Other widely used pendulums of this design were made by [[Charles Sanders Peirce|Charles Peirce]] and C. Defforges.