| = 4√ | l | · F1 (sin 1⁄2α), |
| g |
(18)
in the notation of elliptic integrals. The function F1 (sin β) was tabulated by A. M. Legendre for values of β ranging from 0° to 90°. The following table gives the period, for various amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.
| α/π | τ | α/π | τ |
| .1 | 1.0062 | .6 | 1.2817 |
| .2 | 1.0253 | .7 | 1.4283 |
| .3 | 1.0585 | .8 | 1.6551 |
| .4 | 1.1087 | .9 | 2.0724 |
| .5 | 1.1804 | 1.0 | ∞ |
The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus
| τ = 2π √ | l | · { 1 + | 12 | sin2 1⁄2α + | 12 · 32 | sin4 1⁄2α + ... }. |
| g | 22 | 22 · 42 |
(19)
If α be small, an approximation (usually sufficient) is
τ = 2π √(l/g) · (1 + 1⁄16α2).