(18)
By choosing for f a simple rational fraction, such as ½, 1⁄3, ¼, 1⁄5, ... an algebraical case of motion can be constructed (Annals of Mathematics, 1904).
Thus with G′ − GE = 0, we have E = z1 or z2, never z3; f = 0 or 1; and P is at A or B on the focal ellipse; and then
ῶ = −pt, p = G/2A,
(19)
| ψ + pt = tan−1 | n√ (2Z) | , |
| 2p (z − E) |
(20)
sin θ exp (ψ + pt) i = i√ [(−z2 − z3) (z − z1)] + √ [(z3 − z) (z − z2)],
| z1 = | 1 + z2 z3 | , √ | −z2 − z3 | = | G | = | p | = | G′ | , or |
| z2 + z3 | 2 | 2An | n | 2Anz1 |
(21)