(1) [.[psi]] sin [theta], .[theta], .[phi] + .[psi] cos [theta];
so that, denoting the components about OX, OY, OZ by P, Q, R,
(2) P = .[theta] cos [phi] + .[psi] sin [theta] sin [phi],
Q = -.[theta] sin [phi] + .[psi] sin [theta] cos [phi],
R = .[phi] + .[psi] cos [theta].
Consider, for instance, the motion of a fly-wheel of preponderance Mh,
and equatoreal moment of inertia A, of which the axis OC is held in a
light ring ZCX at a constant angle [gamma] with OZ, while OZ is held
by another ring zZ, which constrains it to move round the vertical Oz
at a constant inclination [theta] with constant angular velocity [mu],
so that
(3) .[theta] = 0, .[psi] = [mu];
(4) P = [mu] sin [theta] sin [phi],
Q = [mu] sin [theta] cos [phi],
R = .[phi] + [mu] cos [theta].
With CXF a quadrant, the components of angular velocity and momentum
about OF, OY, are
(5) P cos [gamma] - R sin [gamma], Q, and A(P cos [gamma]
- R sin [gamma]), AQ,
so that, denoting the components of angular momentum of the fly-wheel
about OC, OX, OY, OZ by K or G', h1, h2, h3,
(6) h1 = A(P cos [gamma] - R sin [gamma]) cos [gamma] + K sin [gamma],
(7) h2 = AQ,
(8) h3 = -A(P cos [gamma] - R sin [gamma]) sin [gamma]
+ K cos [gamma];
and the dynamical equation
dh3
(9) --- - h1Q + h2P = N,
dt
with K constant, and with preponderance downward
(10) N = gMh cos zY sin [gamma]
= gMh sin [gamma] sin [theta] cos [phi],
reduces to
d^2[phi]
(11) A -------- sin [gamma]
dt^2
+ A[mu]^2 sin [gamma] sin^2 [theta] sin [phi] cos [phi]
+ A[mu]^2 cos [gamma] sin [theta] cos [theta] cos [phi]
- (K[mu] + gMh) sin [theta] cos [phi] = 0.