Page:EB1911 - Volume 12.djvu/799

Rh (48)

The curvature is zero and H passes through a point of inflexion when C′ comes into the horizontal plane through C; will then be stationary and the curve described by C′ will be looped.

In a state of steady motion, z oscillates between two limits z2 and z3 which are close together; so putting z2 = z3 the coefficient of z in Z is

(49)

(50)

(51)

With z2 = z3, = 0, K = ; and the number of beats per second of the axis is

(52)

beating time with a pendulum of length

(53)

The wheel making R/2 revolutions per second,

(54)

from (8) (9) § 3; and the apsidal angle is

(55)

and the height of the equivalent conical pendulum is given by

(56)

if OR drawn at right angles to OK cuts KC′ in R, and RL is drawn horizontal to cut the vertical CO in L; thus if OC2 represents to scale, then OL will represent.

9. The gyroscope motion in fig. 4 comes to a stop when the rim of the wheel touches the ground; and to realize the motion when the axis is inclined at a greater angle with the upward vertical, the stalk is pivoted in fig. 8 in a lug screwed to the axle of a bicycle hub, fastened vertically in a bracket bolted to a beam. The wheel can now be spun by hand, and projected in any manner so as to produce a desired gyroscopic motion, undulating, looped, or with cusps if the stalk of the wheel is dropped from rest.

As the principal part of the motion takes place now in the neighbourhood of the lowest position, it is convenient to measure the angle from the downward vertical, and to change the sign of z and G.

Equation (18) § 8 must be changed to

(1)

(2)

1 > z3 > z > z2 > −1, D, E > z1, (3)

z1 + z2 + z3 = F = D − G′2 / 2A2n2 = E − G2 / 2A2n2, (4)

and expressed by the inverse elliptic function

(5)

z = z2sn2mt + z3cn2mt, 2 = (z3 − z2) / (z3 − z1). (6)

Equation (25) and (29) § 8 is changed to

(7)

(8)

while and  change places in (26).

The Jacobian elliptic parameter of the third elliptic integral in (7) can be given by, where

(9)

where ƒ is a real fraction,

(10)

(11)

with respect to the comodulus ′.

Then, with z = E, and

2Z E = −{ (G′ − GE) / An}2, (12)

if denotes the apsidal angle of, and T the time of a single beat of the axle, up or down,

(13)

in accordance with the theory of the complete elliptic integral of the third kind.

Interpreted geometrically on the deformable hyperboloid, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ = amƒK′, and the eccentric angle of P, measured from the minor axis, is am(1 − ƒ) K′, the eccentricity of the focal ellipse being the comodulus ′.

A point L is taken in QP such that

QL/OA = znƒK′, (14)

QV, QT, QP = OA (zs, zc, zd) ƒK′; (15)

and with

mT = K, m/n = √ (z3 − z1) /2 = OA/k, (16)

(17)

(18)

By choosing for ƒ a simple rational fraction, such as, , , , ... 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; ƒ = 0 or 1; and P is at A or B on the focal ellipse; and then

= −pt, p = G/2A, (19)

(20)

sin exp ( + pt) i = i√ [(−z2 − z3) (z − z1)] + √ [(z3 − z) (z − z2)],

(21)

sin exp ( + pt)i = i√ [(−z1 − z3) (z − z2)] + √ [(z3 − z) (z − z1)],

(22)

Thus z2 = 0 in (22) makes G′ = 0; so that if the stalk is held out horizontally and projected with angular velocity 2p about the vertical axis OC without giving any spin to the wheel, the resulting motion of the stalk is like that of a spherical pendulum, and given by

= i sin √ (sec  cos ) + √ [(sec  + cos ) (cos  − cos )], (23)

if the axis falls in the lowest position to an angle with the downward vertical.

With z3 = 0 in (21) and z2 = −cos, and changing to the upward vertical measurement, the motion is given by

sin e i = e int √  cos  [√ (1 − cos  cos ) + i√ (cos  cos  − cos2 )], (24)

and the axis rises from the horizontal position to a series of cusps; and the mean precessional motion is the same as in steady motion with the same rotation and the axis horizontal.

The special case of ƒ = may be stated here; it is found that

(25)

2 = a2 ( − x2), (26)

(27)

L = (1 − ) + p/n, (28)

so that p = 0 and the motion is made algebraical by taking L = (1 − ).

The stereoscopic diagram of fig. 12 drawn by T. I. Dewar shows these curves for =, , and  (cusps).

10. So far the motion of the axis OC′ of the top has alone been considered; for the specification of any point of the body, Euler’s third angle must be introduced, representing the angular displacement of the wheel with respect to the stalk. This is given by

(1)

(2)

It will simplify the formulas by cancelling a secular term if we make C = A, and the top is then called a spherical top; OH becomes the axis of instantaneous angular velocity, as well as of resultant angular momentum.

When this secular term is restored in the general case, the axis OI of angular velocity is obtained by producing Q′H to I, making

(3)