and thus the motion in space of any point fixed in the body defined by Λ is determined completely by means of α, β, γ, δ; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name of multiplicative elliptic functions; and
αδ = cos2 ½θ, βγ = −sin2 ½θ,
αδ − βγ = 1, αδ + βγ = cos θ,
√ ( −4αβγδ) = sin θ;
(19)
while, for the motion of a point on the axis, putting Λ = 0, or ∞,
λ = β/δ = i tan ½θeψi, or λ = α/γ = −i cot ½θeψi,
(20)
and
αβ = ½i sin θeψi, αγ = ½i sin θeψi,
(21)
giving orthogonal projections on the planes GKH, CHK; and