(14) QL/OA = znfK',
(15) QV, QT, QP = OA(zs, zc, zd)fK';
and with
(16) mT = K, m/n = [root](z3 - z1)/2 = OA/k,
GT G k QH
(17) -- = --- . -- K = -- K,
2A 2An OA OA
QL + QH HL
(18) II = 1/2[pi]f + ------- K = 1/2[pi]f + -- K.
OA OA
By choosing for f a simple rational fraction, such as 1/2, 1/3, 1/4,
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
(19) [~omega] = -pt, p = G/2A,
n[root](2Z)
(20) [psi] + pt = tan^(-1) -----------,
2p(z - E)
(21) sin [theta] exp ([psi] + pt)i
= i[root][(-z2 - z3)(z - z1)] + [root][(z3 - z)(z - z2)],
1 + z2z3 /-z2 - z3 G p G'
z1 = --------, / -------- = --- = -- = -----, or
z2 + z3 \/ 2 2An n 2Anz1
(22) sin [theta] exp ([psi] + pt)i =
i[root][(-z1 - z3)(z - z2)] + [root][(z3 - z)(z - z1)],
1 + z1z3 /-z1 - z3 G p G'
z2 = --------, / -------- = --- = -- = -----.
z1 + z3 \/ 2 2An n 2Anz2
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
/ /2p^2 \
(23) sin [theta] exp ([psi] + pt)i = i / ( ---- cos [theta] )
\/ \ n^2 /
/ / p^2 \
+ / ( sin^2 [theta] - 2 --- cos [theta] ),
\/ \ n^2 /
= i sin [alpha] [root](sec [alpha] cos [theta])
+ [root][(sec [alpha] + cos [theta])(cos [alpha] - cos [theta])],
if the axis falls in the lowest position to an angle [alpha] with the
downward vertical.
With z3 = 0 in (21) and z2 = -cos [beta], and changing to the upward
vertical measurement, the motion is given by
(24) sin [theta] e^([psi]i)
= e^(int) [root]1/2 cos [beta][[root](1 - cos [beta] cos [theta])
+ i[root](cos [beta] cos [theta] - cos^2 [theta])],
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.