[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.
[Illustration: FIG. 13.]
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.