| Ω1 ξ + Ω2 ηN = + | a2 + c2 | ζ2; |
| 4c2 |
(22)
and then
| ( | dζ | ) | 2 | = | 4c4 | (Ω2ξ − Ω12η)2 |
| dt | (a2 + c2) |
| = | 4c4 | [ (ξ2 + η2) (Ω12 + Ω22) − (Ω1ξ + Ω2η)2 ] |
| (a2 + c2)2 |
| = | 4c4 | [ LM − N2 + { | (a2 + c2)2 | − M | a2 | − N | a2 + c2 | } ζ2 |
| (a2 + c2)2 | 2c2 (a2 + c2) | c2 | 2c2 |
| − | (a2 + c2) (9a2 − c2) | ζ4 ] = Z, |
| 16c4 (a2 − c2) |
(23)
where Z is a quadratic in ζ2, so that ζ is an elliptic function of t, except when c = a, or 3a.
Put Ω1 = Ω cos φ, Ω2 = −Ω sin φ,