(4) w, [~omega], [sigma], = (L, P, Q) exp nti,
and eliminating L, P, Q, we obtain
(5) (-An^2 + Kn + gMa)(g - n^2b) - gMn^2a^2 = 0,
and the frequency of a vibration in double beats per second is
n/2[pi], where n is a root of this quartic equation.
For upright spinning on a smooth horizontal plane, take b = [oo] and
change the sign of a, then
(6) An^2 - Kn + gMa = 0,
so that the stability requires
(7) K^2 > 4gAMa.
Here A denotes the moment of inertia about a diametral axis through
the centre of gravity; when the point of the fly-wheel is held in a
small smooth cup, b = 0, and the condition becomes
(8) (A + Ma^2)n^2 - Kn + gMa = 0,
requiring for stability, as before in S 3,
(9) K^2 > 4g(A + M^2)Ma.
For upright spinning inside a spherical surface of radius b, the sign
of a must be changed to obtain the condition at the lowest point, as
in the gyroscopic horizon of Fleuriais.
For a gyrostat spinning upright on the summit of a sphere of radius b,
the signs of a and b must be changed in (5), or else the sign of g,
which amounts to the same thing.
Denoting the components of horizontal displacement of the point of the
fly-wheel by [xi], [eta], then
(10) br = [xi], bs = [eta], b[sigma] = [xi] + [eta]i = [lambda]
(suppose),
(11) [omega] = [alpha][~omega] + [lambda].
If the point is forced to take the motion ([xi], [eta], [zeta]) by
components of force X, Y, Z, the equations of motion become
(12) -A :q + K .p = Ya - Zaq,
(13) A :p + K .q = -Xa + Zap,
(14) M :w = X + Yi, M( :[zeta] - g) = Z;
so that
(15) A :[~omega] - K .[~omega]i + gMa[~omega] + Ma :w
= Ma[~omega] :[zeta],
or
(16) (A + Ma^2) :[~omega] - K .[~omega]i + gMa[~omega] + Ma[lambda] =
Ma[~omega] :[zeta].
Thus if the point of the gyrostat is made to take the periodic motion
given by [lambda] = R exp nti, [zeta] = 0, the forced vibration of the
axis is given by [~omega] = P exp nti, where
(17) P{-(A + Ma^2)n^2 + Kn + gMa} - RMn^2a = 0;
and so the effect may be investigated on the Fleuriais gyroscopic
horizon of the motion of the ship.
Suppose the motion [lambda] is due to the suspension of the gyrostat
from a point on the axis of a second gyrostat suspended from a fixed
point.
Distinguishing the second gyrostat by a suffix, then [lambda] =
b[~omega]1, if b denotes the distance between the points of suspension
of the two gyrostats; and the motion of the second gyrostat influenced
by the reaction of the first, is given by
(18) (A1 + M1h1^2) :[~omega]1 - K1 .[~omega]1 i
= -g(M1h1 + Mb)[~omega]1 - b(X + Yi)
= -g(M1h1 + Mb)[~omega]1 - Mb(a :[~omega] + :[lambda]);
so that, in the small vibration,
R / \
(19) -- ( -(A1 + M1h1^2)n^2 + K1n + g(M1h1 + Mb) ) = Mn^2b(aP + R),
b \ /
(20) R { -(A1 + M1h1^2 + Mb^2)n^2 + K1n + g(M1h1 + Mb)} - PMn^2ab^2 = 0.