Page:EB1911 - Volume 12.djvu/800

and then the four vector components OC′, C′K, KH, HI give a resultant vector OI, representing the angular velocity ω, such that

OI/Q′I = ω/R.

(4)

Fig. 12.

The point I is then fixed on the generating line Q′H of the deformable hyperboloid, and the other generator through I will cut the fixed generator OC of the opposite system in a fixed point O′, such that IO′ is of constant length, and may be joined up by a link, which constrains I to move on a sphere.

In the spherical top then,

1/2 (φ + ψ)= $\int$	G′ + G		dt	,   1/2 (φ − ψ)= $\int$	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 = $\int _{1}^{\infty }$ ∞1	√ (z3 − z1) dz	= ƒ1K′i,
	√ (4Z)

(6)

v2 = $\int _{-\infty }^{-1}$	√ (z3 − z1) dz	K + (1 − ƒ2) K′i,
	√ (4Z)

(7)

f1K′ = $\int _{1}^{\infty }$	√ (z3 − z1) dz
	√ ( −4Z)

= sn−1 √	z3 − z1	= cn−1 √	1 − z3	= dn−1 √	1 − z2	,
	1 − z1		1 − z1		1 − z1

(8)

(1 − ƒ2) K′ = $\int _{z_{1}}^{-}1$	√ (z3 − z1) dz
	√ ( −4Z)

= sn−1 √	−1 − z1	= cn−1 √	1 + z2	= dn−1 √	1 + z3	.
	z2 − z1		z2 − z1		z3 − z1

(9)

Then if v′ = K + (1 − ƒ′)K′i is the parameter corresponding to z = D, we find

f = ƒ₂ − ƒ₁, ƒ′ = ƒ₂ + ƒ₁,

(10)

v = v₁ + v₂, v′ = v₁ − v₂.

(11)

The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, α, β, γ, δ, are introduced, defined in terms of Euler’s angles, θ, ψ, φ, by

α = cos 1/2θ exp 1/2 (φ + ψ)i,

(12)

β = i sin 1/2θ exp 1/2 (−φ + ψ)i,

(13)

γ = i sin 1/2θ exp 1/2 (φ − ψ)i,

(14)

δ = cos 1/2θ exp 1/2 (−φ − ψ)i.

(15)

Next Klein takes two functions or co-ordinates λ and Λ, defined by

λ =	x + yi	=	r + z	,
	r − z		x − yi

(16)

and Λ the same function of X, Y, Z, so that λ, Λ play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ.

These new functions are shown to be connected by the bilinear relation

λ =	αΛ + β	,   αδ − βγ = 1,
	γΛ + δ

(17)

in accordance with the annexed scheme of transformation of co-ordinates—

	Ξ	Η	Ζ
ξ	α2	β2	2αβ
η	γ2	δ2	2γδ
ζ	αγ	βδ	αδ + βγ

where

ξ = x + yi,   η = −x + yi,   ζ = −z,
Ξ = X + Yi,   Η = −X + Yi,   Ζ = −Z;

(18)

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 1/2θ,   βγ = −sin² 1/2θ,

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 1/2θe^ψi, or λ = α/γ = −i cot 1/2θe^ψi,

(20)

and

αβ = 1/2i sin θeψi, αγ = 1/2i sin θe^ψi,

(21)

giving orthogonal projections on the planes GKH, CHK; and

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 ƒ₁ and ƒ₂ 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 ƒ = 0, ƒ′ = 1, ƒ₁ = 1/2, ƒ₂ = 1/2, 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);

(25)

and thence α, β, γ, δ can be inferred.

The physical constants of a given symmetrical top have been denoted in § 1 by M, h, A, C, and l, n, T; to specify a given state of general motion we have G, G′ or CR, D, E, or F, which may be called the dynamical constants; or κ, v, w, v₁, v₂, or ƒ, ƒ′, ƒ₁, ƒ₂, the analytical constants; or the geometrical constants, such as α, β, δ, δ′, k of a given articulated hyperboloid.

There is thus a triply infinite series of a state of motion; the choice of a typical state can be made geometrically on the hyperboloid, flattened in the plane of the local ellipse, of which κ is the ratio of the semiaxes α and β, and am(1 − ƒ) K′ is the eccentric angle from the minor axis of the point of contact P of the generator HQ, so that two analytical constants are settled thereby; and the point H may be taken arbitrarily on the tangent line PQ, and HQ′ is then the other tangent of the focal ellipse; in which case θ₃ and θ₂ are the angles between the tangents HQ, HQ′, and between the focal distances HS, HS′, and k² will be HS·HS′, while HQ, HQ′ are δ, δ′.