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.
[Illustration: FIG. 12.]
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.