(10) KH^2 = OH^2 - OK^2,
(11) OK^2 sin^2 [theta] = CC'^2 = G^2 - 2GG' cos [theta] + G'^2,
(12) A^2 sin^2 [theta] (d[theta]/dt)^2
= 2A^2n^2(F - cos [theta]) sin^2 [theta] - G^2 + 2GG' cos [theta] - G'^2;
and putting cos [theta] = z,
(13) (dz/dt)^2 = 2n^2(F - z) (1 - z^2) - (G^2 - 2GG'z + G'^2)/A^2
= 2n^2(E - z)(1-z^2) - (G' - Gz)^2/A^2
= 2n^2(D - z)(1-z^2) - (G - G'z)^2/A^2
= 2n^2 Z suppose.
Denoting the roots of Z = 0 by z1, z2, z3, we shall have them arranged
in the order
(14) z1 > 1 > z2 > z > z3 > -1.
(15) (dz/dt)^2 = 2n^2(z1 - z)(z2 - z)(z - z3).
_z
/ /
(16) nt = | dz/ \/(2Z),
_/z3
an elliptic integral of the first kind, which with
/z1 - z3 z2 - z3
(17) m = n \ / -------, [kappa]^2 = -------,
\/ 2 z1 - z2
can be expressed, when normalized by the factor [root](z1 - z3)/2, by
the inverse elliptic function in the form
_z
/ [root](z1 - z3)dz
(18) mt = | ---------------------------------
_/z3 [root][4(z1 - z)(z2 - z)(z - z3)]
/ z - z3 /z2 - z /z1 - z
= sn^(-1)\ / ------- = cn^(-1)\ / ------- = dn^(-1)\ / -------.
\/ z2 - z3 \/ z2 - z3 \/ z1 - z3
(19) z - z3 = (z2 - z3)sn^2mt, z2 - z = (z2 - z3)cn^2mt,
z1 - z = (z1 - z3)dn^2mt.
(20) z = z2sn^2mt + z3cn^2mt.
Interpreted dynamically, the axis of the top keeps time with the beats
of a simple pendulum of length
(21) L = l/1/2(z1 - z3),
suspended from a point at a height 1/2(z1 + z3)l above 0, in such a
manner that a point on the pendulum at a distance
(22) 1/2(z1 - z3)l = l^2/L
from the point of suspension moves so as to be always at the same
level as the centre of oscillation of the top.
The polar co-ordinates of H are denoted by [rho], [varpi] in the
horizontal plane through C; and, resolving the velocity of H
perpendicular to CH,
(23) [rho] d[~omega]/dt = An^2 sin [theta] cos KCH.
(24) [rho]^2 d[~omega]/dt = An^2 sin [theta] CK
= An^2 (G' - G cos [theta])
_ _
/ G' - Gz dt / (G' - Gz)/2An dz
(25) [~omega] = 1/2 | ------- -- = | ------------- ----------,
_/ E - z A _/z3 E - z [root](2Z)
an elliptic integral, of the third kind, with pole at z = E; and then
(26) [~omega] - [psi] = KCH = tan^(-1) KH/CH
A sin [theta] d[theta]/dt [root](2Z)
= tan^(-1) ------------------------- = tan^(-1) ------------,
G' - G cos [theta] (G' - Gz)/An
which determines [psi].
Otherwise, from the geometry of fig. 4,
(27) C'K sin [theta] = OC - OC' cos [theta],
(28) A sin^2 [theta] d[psi]/dt = G - G' cos [theta],
_ _ _
/ G - G'z dt / G - G' dt / G + G' dt
(29) [psi] = | ------- -- = 1/2 | ------ -- + 1/2 | ------ --,
_/ 1 - z^2 A _/ 1 - z A _/ 1 + z A
the sum of two elliptic integrals of the third kind, with pole at z =
[+-]1; and the relation in (25) (26) shows the addition of these two
integrals into a single integral, with pole at z = E.
The motion of a sphere, rolling and spinning in the interior of a
spherical bowl, or on the top of a sphere, is found to be of the same
character as the motion of the axis of a spinning top about a fixed
point.
The curve described by H can be identified as a Poinsot herpolhode,
that is, the curve traced out by rolling a quadric surface with centre
fixed at O on the horizontal plane through C; and Darboux has shown
also that a deformable hyperboloid made of the generating lines, with
O and H at opposite ends of a diameter and one generator fixed in OC,
can be moved so as to describe the curve H; the tangent plane of the
hyperboloid at H being normal to the curve of H; and then the other
generator through O will coincide in the movement with OC', the axis
of the top; thus the Poinsot herpolhode curve H is also the trace made
by rolling a line of curvature on an ellipsoid confocal to the
hyperboloid of one sheet, on the plane through C.
Kirchhoff's _Kinetic Analogue_ asserts also that the curve of H is the
projection of a tortuous elastica, and that the spherical curve of C'
is a hodograph of the elastica described with constant velocity.
Writing the equation of the focal ellipse of the Darboux hyperboloid
through H, enlarged to double scale so that O is the centre,
(30) x^2/[alpha]^2 + y^2/[beta]^2 + z^2/O = 1,
with [alpha]^2 + [lambda], [beta]^2 + [lambda], [lambda] denoting the
squares of the semiaxes of a confocal ellipsoid, and [lambda] changed
into [mu] and [nu] for a confocal hyperboloid of one sheet and of two
sheets.
(31) [lambda] > O > [mu] > -[beta]^2 > [nu] > -[alpha]^2,
then in the deformation of the hyperboloid, [lambda] and [nu] remain
constant at H; and utilizing the theorems of solid geometry on
confocal quadrics, the magnitudes may be chosen so that
(32) [alpha]^2 + [lambda] + [beta]^2 + [mu] + [nu] = OH^2 = 1/2k^2(F - z)
= [rho]^2 + OC^2.
(33) [alpha]^2 + [mu] = 1/2k^2(z1 - z) = [rho]^2 - [rho]1^2,
(34) [beta]^2 + [mu] = 1/2k^2(z2 - z) = [rho]^2 - [rho]2^2,
(35) [mu] = 1/2k^2(z3 - z) = [rho]^2 - [rho]3^2,
(36) [rho]1^2 < 0 < [rho]2^2 < [rho]^2 < [rho]3^2,
(37) F = z1 + z2 + z3,
(38) [lambda] - 2[mu] + [nu] = k^2z, [lambda] - [nu] = k^2,
[lambda] - [mu] 1 + z [mu] - [nu] 1 - z
(39) ---------------- = -----, --------------- = -----
[lambda] - [nu] 2 [lambda] - [nu] 2
with z = cos [theta], [theta] denoting the angle between the
generating lines through H; and with OC = [delta], OC' = [delta]', the
length k has been chosen so that in the preceding equations
(40) [delta]/k = G/2An, [delta]'/k = G'/2An;
and [delta], [delta]', k may replace G, G', 2An; then
2Z 1 /d[theta]\^2 4KH^2
(41) ------- = --- ( -------- ) = -----,
1 - z^2 n^2 \ dt / k^2
while from (33-39)
2Z 4([alpha]^2 + [mu])([beta]^2 + [mu])[mu]
(42) ------- = ----------------------------------------,
1 - z^2 k^2([mu] - [lambda])([mu] - [nu])
which verifies that KH is the perpendicular from O on the tangent
plane of the hyperboloid at H, and so proves Darboux's theorem.
Planes through O perpendicular to the generating lines cut off a
constant length HQ = [delta], HQ' = [delta]', so the line of curvature
described by H in the deformation of the hyperboloid, the intersection
of the fixed confocal ellipsoid [lambda] and hyperboloid of two sheets
[nu], rolls on a horizontal plane through C and at the same time on a
plane through C' perpendicular to OC'.
Produce the generating line HQ to meet the principal planes of the
confocal system in V, T, P; these will also be fixed points on the
generator; and putting
(43) (HV, HT, HP,)/HQ = D/(A, B, C,),
then
(44) Ax^2 + By^2 + Cz^2 = D[delta]^2
is a quadric surface with the squares of the semiaxes given by HV.HQ,
HT.HQ, HP.HQ, and with HQ the normal line at H, and so touching the
horizontal plane through C; and the direction cosines of the normal
being
(45) x/HV, y/HT, z/HP,
(46) A^2x^2 + B^2y^2 + C^2z^2 = D^2[delta]^2,
the line of curvature, called the polhode curve by Poinsot, being the
intersection of the quadric surface (44) with the ellipsoid (46).
There is a second surface associated with (44), which rolls on the
plane through C', corresponding to the other generating line HQ'
through H, so that the same line of curvature rolls on two planes at a
constant distance from O, [delta] and [delta]'; and the motion of the
top is made up of the combination. This completes the statement of
Jacobi's theorem (_Werke_, ii. 480) that the motion of a top can be
resolved into two movements of a body under no force.
Conversely, starting with Poinsot's polhode and herpolhode given in
(44) (46), the normal plane is drawn at H, cutting the principal axes
of the rolling quadric in X, Y, Z; and then
(47) [alpha]^2 + [mu] = x.OX, [beta]^2 + [mu] = y.OY, [mu] = z.OZ,
this determines the deformable hyperboloid of which one generator
through H is a normal to the plane through C; and the other generator
is inclined at an angle [theta], the inclination of the axis of the
top, while the normal plane or the parallel plane through O revolves
with angular velocity d[psi]/dt.
The curvature is useful in drawing a curve of H; the diameter of
curvature D is given by
dp^2 1/2k^2sin^3 [theta] 1/2D 1/4k^2
(48) D = ---- = -----------------------------, ---- = ------.
dp [delta] - [delta]' cos[theta] p KM.KN
The curvature is zero and H passes through a point of inflexion when
C' comes into the horizontal plane through C; [psi] 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
GG' (OM cos[theta] + ON)(OM + ON cos [theta])
(49) 2Z1z3 + z^23 = -1 + ------ = -1 + ----------------------------------------,
A^2n^2 OM.ON
OM^2 + ON^2 OM^2 + ON^2
(50) 2z1z3 = ----------- cos [theta], z1 = -----------,
OM.ON 2OM.ON
OM^2 - 2OM.ON cos [theta] + ON^2 MN^2
(51) 2(z1 - z3) = -------------------------------- = -----.
OM.ON OM.ON
With z2 = z3, [kappa] = [omicron], K = 1/2[pi]; and the number of
beats per second of the axis is
m n /z1 - z3 MN n
(52) ---- = ---- \ / ------- = ------------- -----,
[pi] [pi] \/ 2 [root](OM.ON) 2[pi]
beating time with a pendulum of length
l 4OM.ON
(53) L = ------------ = ------ l.
1/2(z1 - z3) MN^2
The wheel making R/2[pi] revolutions per second,
beats/second MN n C MN
(54) ------------------ = ------------- -- = -- . ---,
revolutions/second [root](OM.ON) R A OC'
from (8) (9) S 3; and the apsidal angle is
1/2[pi] A[mu] n ON 2[root](OM.ON) ON
(55) [mu] ------- = -----.--.1/2[pi] = ------------- . -------------- . 1/2[pi] = -- [pi],
m An m [root](OM.ON) MN MN
and the height of the equivalent conical pendulum [lambda] is given by
[lambda] g n^2 OM KC OL
(56) -------- = ------- = ------ = -- = --- = ---,
l l[mu]^2 [mu]^2 ON KC' OC'
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 OC^2 represents l to
scale, then OL will represent [lambda].
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 [theta] from the downward vertical, and to change the sign of z
and G.