Chapter V: Part 5

so that the gyroscope would reverse if it were possible to make F cos
[alpha] > 86400 A/T^2C (Foppl, _Munch. Ber_, 1904).

A gyroscopic pendulum is made by the addition to it of a fly-wheel,
balanced and mounted, as in Gilbert's barogyroscope, in a ring movable
about an axis fixed in the pendulum, in the vertical plane of motion.

As the pendulum falls away to an angle [theta] with the upward
vertical, and the axis of the fly-wheel makes an angle [phi] with the
vertical plane of motion, the three components of angular momentum are

(24) h1 = K cos [phi], h2 = A .[theta] + K sin [phi], h3 = A .[phi],

where h3 is the component about the axis of the ring and K of the
fly-wheel about its axis; and if L, M', N denote the components of the
couple of reaction of the ring, L may be ignored, while N is zero,
with P = 0, Q = [.[theta]], R = 0, so that

(25) M' = h2 = A :[theta] + K .[phi] cos [phi],

(26) 0 = h3 - h1 .[theta] = A :[phi] - K .[theta] cos [phi].

For the motion of the pendulum, including the fly-wheel,

(27) MK^2 :[theta] = gMH sin [theta] - M'
= gMH sin [theta] - A :[theta] - K .[phi] cos [phi].

If [theta] and [phi] remain small,

(28) A :[phi] = K .[theta], A .[phi] = K([theta] - [alpha]),

(29) (MK^2 + A) :[theta] + (K^2/A) ([theta] - [alpha]) - gMH[theta] = 0;

so that the upright position will be stable if K^2 > gMHA, or the
rotation energy of the wheel greater than 1/2A/C times the energy
acquired by the pendulum in falling between the vertical and
horizontal position; and the vibration will synchronize with a simple
pendulum of length

(30) (MK^2 + A)/[(K^2/gA) - MH].

This gyroscopic pendulum may be supposed to represent a ship among
waves, or a carriage on a monorail, and so affords an explanation of
the gyroscopic action essential in the apparatus of Schlick and
Brennan.

General motion of the top.

8. Careful scrutiny shows that the steady motion of a top is not
steady absolutely; it reveals a small nutation superposed, so that a
complete investigation requires a return to the equations of unsteady
motion, and for the small oscillation to consider them in a
penultimate form.

In the general motion of the top the vector OH of resultant angular
momentum is no longer compelled to lie in the vertical plane COC'
(fig. 4), but since the axis Oh of the gravity couple is always
horizontal, H will describe a curve in a fixed horizontal plane
through C. The vector OC' of angular momentum about the axis will be
constant in length, but vary in direction; and OK will be the
component angular momentum in the vertical plane COC', if the planes
through C and C' perpendicular to the lines OC and OC' intersect in
the line KH; and if KH is the component angular momentum perpendicular
to the plane COC', the resultant angular momentum OH has the three
components OC', C'K, KH, represented in Euler's angles by

(1) KH = A d[theta]/dt, C'K = A sin [theta] d[psi]/dt, OC' = G'.

Drawing KM vertical and KN parallel to OC', then

(2) KM = A d[psi]/dt, KN = CR - A cos [theta] d[psi]/dt
= (C - A)R + A d[phi]/dt

so that in the spherical top, with C = A, KN = A d[phi]/dt.

The velocity of H is in the direction KH perpendicular to the plane
COC', and equal to gMh sin [theta] or An^2 sin [theta], so that if a
point in the axis OC' at a distance An^2 from O is projected on the
horizontal plane through C in the point P on CK, the curve described
by P, turned forwards through a right angle, will be the hodograph of
H; this is expressed by

(3) An^2 sin [theta] e^{([psi] + 1/2[pi])i}
= iAn^2 sin [theta] e^{[psi]i} = d/dt ([rho]e^{[pi]i})

where [rho]e[varpi]i is the vector CH; and so the curve described by P
and the motion of the axis of the top is derived from the curve
described by H by a differentiation.

Resolving the velocity of H in the direction CH,

(4) d.CH/dt = An^2 sin [theta] sin KCH = An^2 sin [theta] KH/CH,

(5) d.1/2CH^2/dt = A^2n^2sin [theta] d[theta]/dt.

and integrating

(6) 1/2CH^2 = A^2n^2(E - cos [theta]),

(7) 1/2OH^2 = A^2n^2(F - cos [theta]),

(8) 1/2C'H^2 = A^2n^2(D - cos [theta]),

where D, E, F are constants, connected by

(9) F = E + G^2/2A^2n^2 = D + G'^2/2A^2n^2.

Then

(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.

Equation (18) S 8 must be changed to

_z3
/z3 - z1 / [root](z3 - z1)dz
(1) mt = nt / ------- = | -----------------,
\/ 2 _/z [root](4Z)

(2) Z = (z - F)(1 - z^2) - (G^2- 2GG'z + G'^2) / 2A^2n^2
= (z - D)(1 - z^2) - (G - G'z)^2 / 2A^2n^2
= (z - E)(1 - z^2) - (G'- Gz)^2 / 2A^2n^2
= (z3 - z)(z - z2)(z - z1),

(3) 1 > z3 > z > z2 > -1, D, E > z1,

(4) z1 + z2 + z3 = F = D - G'^2 / 2A^2n^2 = E - G^2/2A^2n^2,

and expressed by the inverse elliptic function

/z3 - z /z - z2 /z - z1
(5) mt = sn^(-1) / ------- = cn^(-1) / ------- = dn^(-1) / -------,
\/ z3 - z2 \/ z3 - z2 \/ z3 - z1

(6) z = z2sn^2mt + z3cn^2mt, [kappa]^2 = (z3 - z2)/(z3 - z1),

Equation (25) and (29) S 8 is changed to
_ _
/ G'- Gz dt / G'-GE dt Gt
(7) [~omega] = 1/2 | ------- -- = 1/2 | ----- -- - --,
_/ z - E A _/ z-E A 2A
_ _ _
/ G'z - G dt / G' + G dt / G' - G dt
(8) [psi] = | ------- -- = 1/2 | ------ -- - 1/2 | ------ --,
_/ 1 - z^2 A _/ 1 - z A _/ 1 + z A

while [psi] and [varpi] change places in (26).

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

_z3 / _z3 _z2
/ [root](z3 - z1) / /
(9) [nu] = | --------------- dz = | + | = K + (1 - f)Ki',
_/E [root](4Z) _/z2 _/E

where f is a real fraction,

_E
/ [root](z3 - z1)
(10) (1 - f)K' = | --------------- dz,
_/z1 [root](-4Z)

_E /
/ [root](z3 - z1)
(11) fK' = | --------------- dz,
_/z1 [root](-4Z)

/ E - z1 / z2 - E / z3 - E
= sn^(-1) / ------- = cn^(-1) / ------- = dn^(-1) / -------,
\/ z2 - z1 \/ z2 - z1 \/ z3 - z1

with respect to the comodulus [kappa]'.

Then, with z = E, and

(12) 2Z{E} = -{(G'- GE)/An}^2,

if II denotes the apsidal angle of [~omega], and T the time of a single
beat of the axle, up or down,

_z3
GT / [root](-2Z_E) dz
(13) II + -- = | ------------- ----------,
2A _/z2 z - E [root](2Z)

= 1/2[pi]f + KznfK',

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 = amfK', and the eccentric angle of P,
measured from the minor axis, is am(1 - f)K', the eccentricity of the
focal ellipse being the comodulus [kappa]'.

A point L is taken in QP such that

(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.

The special case of f = 1/2 may be stated here; it is found that

p /(1 + x)([kappa] - x) /(1 - x)([kappa] + x)
(25) -- exp ([~omega] - pt)i = / -------------------- + i / --------------------,
a \/ 2 \/ 2

(26) [rho]^2 = a^2 ([kappa] - x^2),

(27) 1/2[lambda]^2 sin [theta] exp ([psi] - pt)i

/(1 - x)([kappa] + x)
= (L - 1 + [kappa] - x) / --------------------
\/ 2

/(1 + x)([kappa] - x)
+ i(L - 1 + [kappa] + x) / --------------------
\/ 2

(28) L = 1/2(1 - [kappa]) + [lambda]p/n,

so that p = 0 and the motion is made algebraical by taking L = 1/2(1 -
[kappa]).

The stereoscopic diagram of fig. 12 drawn by T. I. Dewar shows these
curves for [kappa] = 15/17, 3/5, and 1/3 (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 [phi] must be introduced, representing the angular
displacement of the wheel with respect to the stalk. This is given by

d[phi] d[psi]
(1) ------ + cos [theta] ------ = R,
dt dt

d([phi] + [psi]) / C \ G' + G
(2) ---------------- = ( 1 - -- ) R + ------------------,
dt \ A / A(1 + cos [theta])

d([phi] - [psi]) / C \ G' - G
---------------- = ( 1 - -- ) R + ------------------.
dt \ A / A(1 - cos [theta])

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

HI A - C HI A - C
(3) --- = -----, --- = -----,
Q'H C Q'I A

and then the four vector components OC', C'K, KH, HI give a resultant
vector OI, representing the angular velocity [omega], such that

(4) OI/Q'I = [omega]/R.

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,
_ _
/ G' + G dt / G' - G dt
(5) 1/2([phi] + [psi])= | ------ --, 1/2([phi] - [psi]) = | ------ --
_/ 1 + z 2A _/ 1 - z 2A

depending on the two elliptic integrals of the third kind, with pole
at z = [+-]1; and measuring [theta] from the downward vertical, their
elliptic parameters are:--

_[oo]
/ [root](z3 - z1)dz
(6) v1 = | ----------------- = f1K'i,
_/1 [root](4Z)

_-1
/ [root](z3 - z1)dz
(7) v2 = | ----------------- = K + (1 - f2)K'i
_/[oo] [root](4Z)

_[oo]
/ [root](z3 - z1)dz
(8) f1K' = | -----------------
_/1 [root](-4Z)

/z3 - z1 /1 - z3 /1 - z2
= sn^(-1) / ------- = cn^(-1) / ------ = dn^(-1) / --------,
\/ 1 - z1 \/ 1 - z1 \/ 1 - z1

_-1
/ [root](z3 - z1)dz
(9) (1 - f2)K' = | -----------------
_/z1 [root](-4Z)

/-1 - z1 /1 + z2
= sn^(-1) / ------- = cn^(-1) / -------
\/ z2 - z1 \/ z2 - z1

/1 + z3
= dn^{-1} / -------
\/ z3 - z1

Then if v' = K + (1 - f')K'i is the parameter corresponding to z = D,
we find

(10) f = f2 - f1, f' = f2 + f1,

(11) v = v1 + v2, v' = v1 - v2.

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, [alpha],
[beta], [gamma], [delta], are introduced, defined in terms of Euler's
angles, [theta], [psi], [phi], by

(12) [alpha] = cos 1/2[theta] exp 1/2([phi] + [psi]i),

(13) [beta] = i sin 1/2[theta] exp 1/2(-[phi] + [psi])i,

(14) [gamma] = i sin 1/2[theta] exp 1/2([phi] - [psi])i,

(15) [delta] = cos 1/2[theta] exp 1/2(-[phi] - [psi])i.

Next Klein takes two functions or co-ordinates [lambda] and [Lambda],
defined by

x + yi r + z
(16) [lambda] = ------ = ------,
r - z x - yi

and [Lambda] the same function of X, Y, Z, so that [lambda], [Lambda]
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

[alpha][Lambda] + [beta]
(17) [lambda] = --------------------------,
[gamma][Lambda] + [delta]'

[alpha][delta] - [beta][gamma] = 1,

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

| [Xi] | [Eta] | [Zeta]
-------+----------------+-----------+-----------------------------------
[xi] | [alpha]^2 | [beta]^2 | 2[alpha][beta]
-------+----------------+-----------+-----------------------------------
[eta] | [gamma]^2 | [delta]^2 | 2[gamma][delta]
-------+----------------+-----------+-----------------------------------
[zeta] | [alpha][gamma] | [beta][delta] | [alpha][delta] + [beta][gamma]

where

(18) [xi] = x + yi, [eta] = -x + yi, [zeta] = -z,
[Xi] = X + Yi, [Eta] = -X + Yi, [Zeta] = -Z;

and thus the motion in space of any point fixed in the body defined by
[Lambda] is determined completely by means of [alpha], [beta], [gamma],
[delta]; 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

(19) [alpha][delta] = cos^2 1/2[theta], [beta][gamma] = -sin^2 1/2[theta],
[alpha][delta] - [beta][gamma] = 1, [alpha][delta] + [beta][gamma]
= cos [theta],
[root](-4[alpha][beta][gamma][delta]) = sin [theta];

while, for the motion of a point on the axis, putting [Lambda] = 0, or
[infinity],

(20) [lambda] = [beta]/[delta] = i tan 1/2[theta]e^{[psi]i}, or
[lambda] = [alpha]/[gamma] = -i cot 1/2[theta]e^{[psi]i},

and

(21) [alpha][beta] = 1/2i sin [theta]e^([psi]i),
[alpha][gamma] = 1/2i sin [theta]e^([psi]i),

giving orthogonal projections on the planes GKH, CHK; and

d[beta] d[alpha] [rho]
(22) [alpha] ------- - --------[beta] = n ---- e^([~omega]i),
dt dt k

the vectorial equation in the plane GKH of the herpolhode of H for a
spherical top.

When f1 and f2 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 f = 0, f' = 1, f1 = 1/2, f2 = 1/2, as in (24)
S 9, where P and P' are at A and B on the focal ellipse, we have for
the spherical top

(23) (1 + cos [theta]) exp ([phi] + [psi] - qt)i
= [root](sec [beta] - cos [theta]) [root](cos [beta] - cos [theta])
+ i([root]sec [beta] + [root] cos [beta])[root]cos [theta],

(24) (1 - cos [theta]) exp ([phi] - [psi] - q't)i
= [root](sec [beta] - cos [theta])[root](cos [beta] - cos [theta])
+ i([root]sec [beta] - [root]cos [beta]) [root]cos [theta],

(25) q, q' = n[root](2 sec [beta]) [+-] n[root](2 cos [beta]);

and thence [alpha], [beta], [gamma], [delta] can be inferred.

The physical constants of a given symmetrical top have been denoted in
S 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 [kappa], v, w, v1, v2, or f, f', f1, f2, the
analytical constants; or the geometrical constants, such as [alpha],
[beta], [delta], [delta]', 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
[kappa] is the ratio of the semiaxes [alpha] and [beta], and am(1 - f)
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 [theta]3 and [theta]2 are the angles between the tangents HQ,
HQ', and between the focal distances HS, HS', and k^2 will be HS.HS',
while HQ, HQ' are [delta], [delta]'. As H is moved along the tangent
line HQ, a series of states of motion can be determined, and drawn
with accuracy.

11. Equation (5) S 3 with slight modification will serve with the same
notation for the steady rolling motion at a constant inclination
[alpha] to the vertical of a body of revolution, such as a disk, hoop,
wheel, cask, wine-glass, plate, dish, bowl, spinning top, gyrostat, or
bicycle, on a horizontal plane, or a surface of revolution, as a coin
in a conical lamp-shade.

The point O is now the intersection of the axis GC' with the vertical
through the centre B of the horizontal circle described by the centre
of gravity, and through the centre M of the horizontal circle
described by P, the point of contact (fig. 13). Collected into a
particle at G, the body swings round the vertical OB as a conical
pendulum, of height AB or GL equal to g/[mu]^2 = [lambda], and GA
would be the direction of the thread, of tension gM(GA/GL) dynes. The
reaction with the plane at P will be an equal parallel force; and its
moment round G will provide the couple which causes the velocity of
the vector of angular momentum appropriate to the steady motion; and
this moment will be gM.Gm dyne-cm. or ergs, if the reaction at P cuts
GB in m.

Draw GR perpendicular to GK to meet the horizontal AL in R, and draw
RQC'K perpendicular to the axis Gz, and KC perpendicular to LG.

The velocity of the vector GK of angular momentum is [mu] times the
horizontal component, and

(1) horizontal component / A[mu] sin [alpha] = KC/KC',

so that

(2) gM.Gm = A[mu]^2 sin [alpha](KC/KC'),

A KC' g
(3) -- = --- ------------------ Gm = GQ.Gm.
M KC [mu]^2 sin [alpha]

The instantaneous axis of rotation of the case of a gyrostat would be
OP; drawing GI parallel to OP, and KK' parallel to OG, making tan
K'GC' = (A/C) tan IGC'1; then if GK represents the resultant angular
momentum, K'K will represent the part of it due to the rotation of the
fly-wheel. Thus in the figure for the body rolling as a solid, with
the fly-wheel clamped, the points m and Q move to the other side of G.
The gyrostat may be supposed swung round the vertical at the end of a
thread PA' fastened at A' where Pm produced cuts the vertical AB, and
again at the point where it crosses the axis GO. The discussion of the
small oscillation superposed on the state of steady motion requisite
for stability is given in the next paragraph.

General motion of a gyrostat rolling on a plane.

12. In the theoretical discussion of the general motion of a gyrostat
rolling on a horizontal plane the safe and shortest plan apparently is
to write down the most general equations of motion, and afterwards to
introduce any special condition.

Drawing through G the centre of gravity any three rectangular axes Gx,
Gy, Gz, the notation employed is

u, v, w, the components of linear velocity of G;
p, q, r, the components of angular velocity about the axes,
h1, h2, h3, the components of angular momentum;
[theta]1, [theta]2, [theta]3, the components of angular velocity of
the coordinate axes;
x, y, z, the co-ordinates of the point of contact with the
horizontal plane;
X, Y, Z, the components of the reaction of the plane;
[alpha], [beta], [gamma], the direction cosines of the downward
vertical.

The geometrical equations, expressing that the point of contact is at
rest on the plane, are

(1) u - ry + qz = 0,

(2) v - pz + rx = 0,

(3) w - qx + py = 0.

The dynamical equations are

(4) du/dt - [theta]3v + [theta]2w = g[alpha] + X/M,

(5) dv/dt - [theta]1w + [theta]2u = g[beta] + Y/M,

(6) dw/dt - [theta]2u + [theta]1v = g[gamma] + Z/M,

and

(7) dh1/dt - [theta]3h2 + [theta]2h3 = yZ - zY,

(8) dh2/dt - [theta]1h3 + [theta]3h1 = zX - xZ,

(9) dh3/dt - [theta]2h1 + [theta]1h2 = xY - yX.

In the special case of the gyrostat where the surface is of revolution
round Gz, and the body is kinetically symmetrical about Gz, we take Gy
horizontal and Gzx through the point of contact so that y = 0; and
denoting the angle between Gz and the downward vertical by [theta]
(fig. 13)

(10) [alpha] = sin[theta], [beta] = 0, [gamma] = cos[theta].

The components of angular momentum are

(11) h1 = Ap, h2 = Aq, h3 = Cr + K,

where A, C denote the moment of inertia about Gx, Gz, and K is the
angular momentum of a fly-wheel fixed in the interior with its axis
parallel to Gz; K is taken as constant during the motion.

The axis Gz being fixed in the body,

(12) [theta]1 = p, [theta]2 = q = -d[theta]/dt,
[theta]3 = p cot [theta].

With y = 0, (1), (2), (3) reduce to

(13) u = -qz, v = pz - rx, w = qx;

and, denoting the radius of curvature of the meridian curve of the
rolling surface by [rho],

dx d[theta]
(14) -- = [rho] cos[theta] -------- = -q [rho] cos[theta],
dt dt

dz d[theta]
-- = -[rho] sin[theta] -------- = q [rho] sin[theta];
dt dt

so that

du dq
(15) -- = - -- z - q^2[rho] sin[theta],
dt dt

dv dp dr
(16) -- = -- z - -- x + pq[rho] sin[theta] + qr[rho] sin[theta],
dt dt dt

dw dq
(17) -- = -- x - q^2[rho] cos[theta].
dt dt

The dynamical equations (4)...(9) can now be reduced to

X dq
(18) -- = - -- z - p^2z cot[theta] + q^2(x - [rho] sin[theta])
M dt

+ prx cot[theta] - g sin[theta],

Y dp dr
(19) -- = -- z - -- x - pq(x + z cot[theta] - [rho] sin[theta])
M dt dt

+ qrp cos[theta],

Z dq
(20) -- = -- x + q^2(z - [rho] cos[theta]) + p^2z - prx - g cos[theta],
M dt

dp
(21) -zY = A -- - Apq cot[theta] + qh3,
dt

dq
(22) -zX - xZ = A -- + Ap^2 cot [theta] - ph3,
dt

dh3 dr d
(23) xY = --- = C -- = -Cq --------.
dt dt d[theta]

Eliminating Y between (19) and (23),

/C \ dr dp
(24) (-- + x^2) -- - xz -- + pqx(x + z cot[theta] - [rho] sin[theta])
\M / dt dt

- qrx[rho] cos[theta] = 0,

/C \ dr dp
(A) (-- + x^2) -------- - xz -------- - px(x + z cot[theta]
\M / d[theta] d[theta]

- [rho] sin[theta]) + rx[rho] cos[theta] = 0.

Eliminating Y between (19) and (21)

/A \ dp dr A h3
(25) (-- + z^2) -- - xz -- -- -pq cot[theta] + q --
\M / dt dt M M

- pqz(x + z cot[theta] - [rho] sin[theta]) + qrz[rho] cos[theta] = 0,

dr /A \ dp A h3
(B) -xz -------- + (-- + z^2) -------- + -- p cot[theta] - --
d[theta] \M / d[theta] M M

+ pz(x + z cot[theta] - [rho] sin[theta]) + rz[rho] cos[theta] = 0.

In the special case of a gyrostat rolling on the sharp edge of a
circle passing through G, z = 0, [rho] = 0, (A) and (B) reduce to

/ C \ dr / 1 1 \ dh3
(26) p = ( --- + 1 ) -------- = ( ---- + -- ) --------,
\Mx^2 / d[theta] \Mx^2 C / d[theta]

dp h3 d.p sin[theta] h3sin [theta]
(27) -------- + p cot[theta] = --, -------------- = -------------;
d[theta] A d[theta] A

d^2h3 dh3 CMx^2
(28) ---------- + -------- cot[theta] = -----------,
d[theta]^2 d[theta] A(Mx^2 + C)

a differential equation of a hypergeometric series, of the form of
Legendre's zonal harmonic of fractional order n, given by

(29) n(n + 1) = CMx^2/A(Mx^2 + C).

For a sharp point, x = 0, [rho] = 0, and the previous equations are
obtained of a spinning top.

The elimination of X and Z between (18) (20) (22), expressed
symbolically as

(30) (22) - z(18) + x(20) = 0,

gives

/A \ dq h3 /A \
(C) ( -- + x^2 + z^2 ) -- - p -- + ( -- + z^2 ) p^2 cot [theta] + p^2xz
\M / dt M \M /

+ q^2[rho](x cos[theta] - z sin[theta]) - prx(x + z cot[theta])
- g(x cos[theta] - z sin[theta]) = 0,

and this combined with (A) and (B) will lead to an equation the
integral of which is the equation of energy.

13. The equations (A) (B) (C) are intractable in this general form;
but the restricted case may be considered when the axis moves in
steady motion at a constant inclination [alpha] to the vertical; and
the stability is secured if a small nutation of the axis can be
superposed.

It is convenient to put p = [Omega] sin [theta], so that [Omega] is
the angular velocity of the plane Gzx about the vertical; (A) (B) (C)
become

/C \ dr d[Omega]
(A*) ( -- + x^2) -------- - xz sin[theta] --------
\M / d[theta] d[theta]

- [Omega]x(x sin[theta] - 2z cos[theta]
- [rho] sin^2[theta]) + rx[rho] cos[theta] = 0,

dr /A \ d[Omega] h3 /A \
(B*) -xz -------- + ( -- + z^2) sin[theta] -------- - -- + 2[Omega]( -- + z^2) cos [theta]
d[theta] \M / d[theta] M \M /

+ [Omega]z sin[theta](x - [rho] sin[theta]) - rz[rho] cos[theta] = 0,

/A \ dq h3
(C*) ( -- + x^2 + z^2) -- + q^2p(x cos[theta] - z sin[theta]) - [Omega] -- sin [theta]
\M / dt M

/A \
+ [Omega]^2 ( -- + z^2) sin [theta] cos [theta] + [Omega]^2xz sin^2[theta]
\M /

- [Omega]rx(x sin [theta] + z cos [theta]) - g(x cos [theta] - z sin [theta]) = 0.

The steady motion and nutation superposed may be expressed by

(1) [theta] = [alpha] + L, sin [theta] = sin[alpha] + L cos[alpha],
cos[theta] = cos[alpha] - L sin [alpha],
[Omega] = [mu] + N, r = R + Q,

where L, N, Q are small terms, involving a factor e^(nti), to express
the periodic nature of the nutation; and then if a, c denote the mean
value of x, z, at the point of contact

(2) x = a + L[rho] cos[alpha], z = c - L[rho] sin[alpha],

(3) x sin [theta] + z cos [theta] = a sin [alpha] + c cos [alpha]
+ L(a cos [alpha] - c sin [alpha]),

(4) x cos [theta] - z sin [theta] = a cos [alpha] - c sin [alpha]
- L(a sin [alpha] + c cos [alpha] - [rho]).

Substituting these values in (C*) with dq/dt = -d^2[theta]/dt^2 = n^2L,
and ignoring products of the small terms, such as L^2, LN, ...

/A \ /CR + K CQ\
(C**) ( -- + a^2 + c^2) Ln^2 - ([mu] + N)( ------ + -- )(sin[alpha] + L cos[alpha])
\M / \ M M /

/A \
+ ([mu]^2 + 2[mu]N)( -- + c^2 - 2L[rho]c sin[alpha]) (sin [alpha] cos[alpha] + L cos[alpha])
\M /

+ ([mu]^2 + 2[mu]N) [ac - L[rho](a sin [alpha] - c sin[alpha])] (sin^2 [alpha] + L sin 2[alpha])
- ([mu] + N)(R + Q)(a + L[rho]cos[alpha])[a sin [alpha] + c cos[alpha]
+ L(a cos[alpha] - c sin [alpha])] - g(a cos[alpha] - c sin[alpha])
+ gL(a sin[alpha] + c cos[alpha] - [rho]) = 0,

which is equivalent to

CR + K /A \
(5) -[mu] ------ sin[alpha] + [mu]^2 ( -- + c^2) sin [alpha] cos [alpha]
M \M /

+ [mu]^2 ac sin^2[alpha] - [mu]Ra(a sin[alpha] + c cos[alpha])
- g(a cos[alpha] - c sin[alpha]) = 0,

the condition of steady motion; and

(6) DL + EQ + FN = 0,

where

/A \ CK + K
(7) D = ( -- + a^2 + c^2) n^2 - [mu] ------ cos[alpha]
\M / M

- 2[mu]^2[rho]c sin^2[alpha] cos[alpha]

/A \
+ [mu]^2 ( -- + c^2) cos [alpha] - [mu]^2[rho](a sin [alpha]
\M /

- c cos[alpha]) sin^2[alpha] + [mu]^2ac sin 2[alpha]
- [mu]R[rho] cos [alpha](a sin [alpha] + c cos[alpha])
- [mu]Ra(a cos [alpha] - c sin[alpha])
+ g(a sin[alpha] + c cos[alpha] - [rho]),

C
(8) E = -[mu] -- sin [alpha] - [mu]a(a sin [alpha] + c cos [alpha]),
M

CR + K /A \
(9) F = - ------ sin [alpha] + 2[mu] ( -- + c^2) sin [alpha] cos [alpha]
M \M /

+ 2[mu]ac sin^2 [alpha] - Ra(a sin [alpha] + c cos [alpha]).

With the same approximation (A*) and (B*) are equivalent to

/C \ Q N
(A**) ( -- + a^2) -- - ac sin [alpha]--
\M / L L

- [mu]a(a sin [alpha] + 2c cos [alpha] - [rho] sin^2 [alpha])
+ Ra[rho] cos [alpha] = 0,

Q /A \ N CR + K /A \
(B**) -ac -- + ( -- + c^2) sin [alpha]-- - ------ + 2[mu] ( -- + c^2) cos [alpha]
L \M / L M \M /

+ [mu]c sin [alpha](a - [rho] sin [alpha])
- Rc[rho] cos [alpha] = 0.

The elimination of L, Q, N will lead to an equation for the
determination of n^2, and n^2 must be positive for the motion to be
stable.

If b is the radius of the horizontal circle described by G in steady
motion round the centre B,

(10) b = v/[mu] = (cP - aR)/[mu] = c sin [alpha] - aR/[mu],

and drawing GL vertically upward of length [lambda] = g/[mu]^2, the
height of the equivalent conical pendulum, the steady motion condition
may be written

(11) (CR + K)[mu] sin[alpha] - [mu]^2 sin[alpha] cos[alpha]
= -gM(a cos[alpha] - c sin[alpha])
+ M([mu]^2c sin[alpha] - [mu]Ra) (a sin[alpha] + c cos[alpha])
= gM[b[lambda]^(-1) (a sin [alpha] + c cos [alpha])
- a cos [alpha] + c sin [alpha]]
= gM.PT,

LG produced cuts the plane in T.

Interpreted dynamically, the left-hand side of this equation
represents the velocity of the vector of angular momentum about G, so
that the right-hand side represents the moment of the applied force
about G, in this case the reaction of the plane, which is parallel to
GA, and equal to gM.GA/GL; and so the angle AGL must be less than the
angle of friction, or slipping will take place.

Spinning upright, with [alpha] = 0, a = 0, we find F = 0, Q = 0, and

CR + K /A \
(12) - ------ + 2[mu]( -- + c^2) - Rcp = 0,
M \M /

/A \ CR + K /A \
(13) ( -- + c^2) n^2 = [mu] ------ - [mu]^2( -- + c^2) + [mu]R[rho]c - g(c - [rho]),
\M / M \M /

/A \^2 /CK + R \^2 /A \
(14) ( -- + c^2) n^2 = 1/4( ------ + Rc[rho] ) - g( -- + c^2) (c - [rho]),
\M / \ M / \M /

Thus for a top spinning upright on a rounded point, with K = 0, the
stability requires that

(15) R > 2k' [root]{g(c - [rho])}/(k^2 + c[rho]),

where k, k' are the radii of gyration about the axis Gz, and a
perpendicular axis at a distance c from G; this reduces to the
preceding case of S 3 (7) when [rho] = 0.

Generally, with [alpha] = 0, but a [+-] 0, the condition (A) and (B)
becomes

/C \ Q
(16) ( - + a^2) -- = 2[mu]ac - Ra[rho],
\M / L

Q CR + K /A \
-ac -- = ------ + Rc[rho] - 2[mu]( -- + a^2),
L M \M /

so that, eliminating Q/L,
_ _
| /A \ /C \ | /C \ /CR + K\ C
(17) 2 | ( -- + c^2)( -- + a^2) - a^2c^2| [mu] = ( -- + a^2)( ------ ) + -- Rc[rho],
|_ \M / \M / _| \M / \ M / M

the condition when a coin or platter is rolling nearly flat on the
table.

Rolling along in a straight path, with [alpha] = 1/2[pi], c = 0, [mu]
= 0, E = 0; and

(18) N/L = (CR + K)/A,

/A \
(19) D = ( -- + a^2) n^2 + g(a - [rho]),
\M /

CR + K
F = - ------ - Ra^2,
M

/A \
( -- + a^2) n^2 + g(a - [rho])
N D \M /
(20) -- = - -- = ----------------------------,
L F /C \ K
( -- + a^2) R + -
\M / M
_ _
/A \ (CR + K) | /C \ K |
(21) ( -- + a^2) n^2 = -------- | ( -- + a^2) R + -- | - g(a - [rho]).
\M / A |_ \M / M _|

Thus with K = 0, and rolling with velocity V = Ra, stability
requires

V^2 a - [rho] A a - [rho]
(22) --- > ---------------- > 1/2 -- ---------,
2g C / C \ C C
2 -- ( ---- + 1 ) ---- + 1
A \Ma^2 / Ma^2

or the body must have acquired velocity greater than attained by
rolling down a plane through a vertical height 1/2(a - [rho])A/C.

On a sharp edge, with [rho] = 0, a thin uniform disk or a thin ring
requires

(23) V^2/2g > a/6 or a/8.

The gyrostat can hold itself upright on the plane without advance when
R = 0, provided

(24) K^2/AM - g(a - [rho]) is positive.

For the stability of the monorail carriage of S 5 (6), ignoring the
rotary inertia of the wheels by putting C = 0, and replacing K by G'
the theory above would require

G' / G'\
(25) -- ( aV + -- ) > gh.
A \ A /

For further theory and experiments consult Routh, _Advanced Rigid
Dynamics_, chap. v., and Thomson and Tait, _Natural Philosophy_, S
345; also Bourlet, _Traite des bicycles_ (analysed in Appell,
_Mecanique rationnelle_, ii. 297, and Carvallo, _Journal de l'ecole
polytechnique_, 1900); Whipple, _Quarterly Journal of Mathematics_,
vol. xxx., for mathematical theories of the bicycle, and other bodies.

Gyrostatic chain.

14. Lord Kelvin has studied theoretically and experimentally the
vibration of a chain of stretched gyrostats (_Proc. London Math.
Soc._, 1875; J. Perry, _Spinning Tops_, for a diagram). Suppose each
gyrostat to be equivalent dynamically to a fly-wheel of axial length
2a, and that each connecting link is a light cord or steel wire of
length 2l, stretched to a tension T.

Denote by x, y the components of the slight displacement from the
central straight line of the centre of a fly-wheel; and let p, q, 1
denote the direction cosines of the axis of a fly-wheel, and r, s, 1
the direction cosines of a link, distinguishing the different bodies
by a suffix.

Then with the previous notation and to the order of approximation
required,

(1) [theta]1 = -dq/dt, [theta]2 = dp/dt,

(2) h1 = A[theta]1, h2 = A[theta]2, h3 = K,

to be employed in the dynamical equations

(3) dh1/dt - [theta]3 h2 + [theta]2 h3 = L, ...

in which [theta]3 h1 and [theta]3 h2 can be omitted.

For the kth fly-wheel

(4) -A :q_k + K .p_k = Ta(q_k - s_k) + Ta[q_k - s_(k+1)],

(5) A :p_k + K .q_k = -Ta(p_k - r_k) - Ta[p_k - r_(k+1)];

and for the motion of translation

(6) M :x_k = T[r_(k+1) - r_k], M :y_k = T[s(k+1) - s_k];

while the geometrical relations are

(7) x_(k+1) - x_k = a [p_(k+1) + p_k] + 2lr_(k+1),

(8) y_(k+1) - y_k = a [q_(k+1) + q_k] + 2ls_(k+1).

Putting

(9) x + yi = w, p + qi = [~omega], r + si = [sigma],

these three pairs of equations may be replaced by the three equations

(10) A :[~omega]_k - K .[~omega]_ki + 2Ta[~omega]_k
- Ta([sigma]_(k+1) + [sigma]_k) = 0,

(11) M :[~omega]_k - T([sigma]_(k+1) - [sigma]_k) = 0,

(12) w_(k+1) - w_k - a([~omega]_(k+1) +
[~omega]_k - 2l[sigma]_(k+1) = 0.

For a vibration of circular polarization assume a solution

(13) w_k, [~omega]_k, [sigma]_k = (L, P, Q) exp (nt + kc)i,

so that c/n is the time-lag between the vibration of one fly-wheel and
the next; and the wave velocity is

(14) U = 2(a + l)n/c.

Then

(15) P(-An^2 + Kn + 2Ta) - QTa[e^(ci) + 1] = 0,

(16) -LMn^2 - QT[e^(ci) - 1] = 0,

(17) L[e^(ci) - 1] -Pa[e^(ci) + 1] - 2Qle^(ci) = 0,

leading, on elimination of L, P, Q, to

(2 Ta + Kn - An^2) (1 - Mn^2l/T) - Mn^2
(18) cos c = ---------------------------------------,
2Ta + Kn - An^2 + Mna^2

M n^2 2Ta(a + l) + KNl - An^2l
(19) 2 sin^2 1/2c = -- ----------------------------.
T 2Ta + Kn - An^2 + Mn^2a^2

With K = 0, A = 0, this reduces to Lagrange's condition in the
vibration of a string of beads.

Putting

(20) [rho] = M/2(a + l), the mass per unit length of the chain,

(21) [kappa] = K/2(a + l), the gyrostatic angular momentum per unit
length,

(22) [alpha] = A/2(a + l), the transverse moment of inertia per unit
length,

(23) 1/2c = (a + l)n/U,

equation (19) can be written

(24) {sin (a + l)n/U}^2

[rho] Ta + [kappa]nl - [alpha]n^2l
= (a + l)^2n^2 ----- -------------------------------------------------------------,
T Ta + [kappa]n(a + l) - [alpha]n^2(a + l) + [rho]n^2a^2(a + l)

/ (a + l)n \^2
(25) ( --------------- )
\ sin (a + l)n/U /

T T + ([kappa]n - [alpha]n^2) (1 + l/a) + [rho]n^2a(a + l)
= ----- . --------------------------------------------------------.
[rho] T + ([kappa]n - an^2)l/a

In a continuous chain of such gyrostatic links, with a and l
infinitesimal,

T / [kappa]n - [alpha]n^2 \
(26) U^2 = ----- ( 1 + ------------------------------- )
[rho] \ T + ([kappa]n - [alpha]n^2 l/a) /

for the vibration of helical nature like circular polarization.

Changing the sign of n for circular polarization in the opposite
direction

T / [kappa]n + [alpha]n^2 \
(27) U'^2 = ----- ( 1 - ------------------------------- ).
[rho] \ T - ([kappa]n + [alpha]n^2 l/a) /

In this way a mechanical model is obtained of the action of a
magnetized medium on polarized light, [kappa] representing the
equivalent of the magnetic field, while [alpha] may be ignored as
insensible (J. Larmor, _Proc. Lond. Math. Soc._, 1890; _Aether and
Matter_, Appendix E).