| α | dβ | − | dα | β = n | ρ | eῶi, |
| dt | dt | k |
(22)
the vectorial equation in the plane GKH of the herpolhode of H for a spherical top.
When f1 and f2 in (9) are rational fractions, these multiplicative elliptic functions can be replaced by algebraical functions, qualified by factors which are exponential functions of the time t; a series of quasi-algebraical cases of motion can thus be constructed, which become purely algebraical when the exponential factors are cancelled by a suitable arrangement of the constants.
Thus, for example, with f = 0, f′ = 1, f1 = ½, f2 = ½, as in (24) § 9, where P and P′ are at A and B on the focal ellipse, we have for the spherical top
(1 + cos θ) exp (φ + ψ − qt) i
= √ (sec β − cos θ) √ (cos β − cos θ) + i (√ sec β + √ cos β) √ cos θ,
(23)
(1 − cos θ) exp (φ − ψ − q′t) i
= √ (sec β − cos θ) √ (cos β − cos θ) + i (√sec β − √ cos β) √ cos θ,
(24)
q, q′ = n√ (2 sec β) ± n√ (2 cos β);