In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the lowest position is
t = √(l/g) · log tan (1⁄4π + 1⁄4ψ).
(20)
This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the highest position.
The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path is
| d2s | = −g sin ψ, |
| dt2 |
(21)
where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not merely approximately proportional to the arc s, say
s = k sin ψ,
(22)