In the spherical top then,
| ½ (φ + ψ)= ∫ | G′ + G | dt | , ½ (φ − ψ)= ∫ | G′ − G | dt | ||
| 1 + z | 2A | 1 − z | 2A |
(5)
depending on the two elliptic integrals of the third kind, with pole at z = ±1; and measuring θ from the downward vertical, their elliptic parameters are:—
| v1 = ∫ ∞1 | √ (z3 − z1) dz | = f1K′i, |
| √ (4Z) |
(6)
| v2 = ∫ −1−∞ | √ (z3 − z1) dz | K + (1 − f2) K′i, |
| √ (4Z) |
(7)
| f1K′ = ∫ ∞1 | √ (z3 − z1) dz |
| √ ( −4Z) |
| = sn−1 √ | z3 − z1 | = cn−1 √ | 1 − z3 | = dn−1 √ | 1 − z2 | , |
| 1 − z1 | 1 − z1 | 1 − z1 |