As we have noted, Bessel made determinations of gravity with a ball
("simple") pendulum in the period 1825-1827 and in 1835 at Koenigsberg
and Berlin, respectively. In the memoir on his observations at
Koenigsberg, 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 T_{1} and T_{2} are the times of swing about the knife
edges, and if h_{1} 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 about a fixed axis under gravity
(T_{1})^{2} = [pi]^{2}(k^{2} + (h_{1})^{2})/gh_{1},
(T_{2})^{2} = [pi]^{2}(k^{2} + (h_{2})^{2})/gh_{2}.