* We have next to compare the velocity of the invariable axis with
respect to the body, with that of the body itself round one of the
principal axes. Since the invariable axis is fixed in space, its motion
relative to the body must be equal and opposite to that of the portion
of the body through which it passes. Now the angular velocity of a
portion of the body whose direction-cosines are $l$, $m$, $n$, about
the axis of $x$ is
\begin{displaymath} \frac{\omega_1}{1 - l^2} - \frac{l}{1 -
l^2}(l\omega_1 + m\omega_2 + n\omega-3). \end{displaymath}
Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$, in terms
of $l$, $m$, $n$, and taking account of equation (3), this expression
becomes
\begin{displaymath} H\frac{(a^2 - e^2)}{1 - l^2}l. \end{displaymath}
Changing the sign and putting $\displaystyle l = \frac{\omega_1}{a^2H}$
we have the angular velocity of the invariable axis about that of $x$
\begin{displaymath} = \frac{\omega_1}{1 - l^2} \frac{e^2 - a^2}{a^2},
\end{displaymath}
always positive about the axis of greatest moment, negative about that
of least moment, and positive or negative about the mean axis according
to the value of $e^2$. The direction of the motion in every case is
represented by the arrows in the figures. The arrows on the outside of
each figure indicate the direction of rotation of the body.
* If we attend to the curve described by the pole of the invariable
axis on the sphere in fig. 5, we shall see that the areas described by
that point, if projected on the plane of $yz$, are swept out at the
rate
\begin{displaymath} \omega_1 \frac{e^2 - a^2}{a^2}. \end{displaymath}
Now the semi-axes of the projection of the spherical ellipse described
by the pole are
\begin{displaymath} \sqrt{\frac{e^2 - a^2}{b^2 - a^2}}
\hspace{1cm}\textrm{and}\hspace{1cm} \sqrt{\frac{e^2 - a^2}{c^2 -
a^2}}. \end{displaymath}
Dividing the area of this ellipse by the area described during one
revolution of the body, we find the number of revolutions of the body
during the description of the ellipse--
\begin{displaymath} = \frac{a^2}{\sqrt{b^2 - a^2}\sqrt{c^2 - a^2}}.
\end{displaymath}
The projections of the spherical ellipses upon the plane of $yz$ are
all similar ellipses, and described in the same number of revolutions;
and in each ellipse so projected, the area described in any time is
proportional to the number of revolutions of the body about the axis of
$x$, so that if we measure time by revolutions of the body, the motion
of the projection of the pole of the invariable axis is identical with
that of a body acted on by an attractive central force varying directly
as the distance. In the case of the hyperbolas in the plane of the
greatest and least axis, this force must be supposed repulsive. The
dots in the figures 1, 2, 3, are intended to indicate roughly the
progress made by the invariable axis during each revolution of the body
about the axis of $x$, $y$ and $z$ respectively. It must be remembered
that the rotation about these axes varies with their inclination to the
invariable axis, so that the angular velocity diminishes as the
inclination increases, and therefore the areas in the ellipses above
mentioned are not described with uniform velocity in absolute time, but
are less rapidly swept out at the extremities of the major axis than at
those of the minor.
* When two of the axes have equal moments of inertia, or $b = c$, then
the angular velocity $\omega_1$ is constant, and the path of the
invariable axis is circular, the number of revolutions of the body
during one circuit of the invariable axis, being
\begin{displaymath} \frac{a^2}{b^2 - a^2} \end{displaymath}
The motion is in the same direction as that of the rotation, or in the
opposite direction, according as the axis of $x$ is that of greatest or
of least moment of inertia.
* Both in this case, and in that in which the three axes are unequal,
the motion of the invariable axis in the body may be rendered very slow
by diminishing the difference of the moments of inertia. The angular
velocity of the axis of $x$ about the invariable axis in space is
\begin{displaymath} \omega_1\frac{e^2 - a^2l^2}{a^2(1 - l^2)},
\end{displaymath}
which is greater or less than $\omega_1$, as $e^2$ is greater or less
than $a^2$, and, when these quantities are nearly equal, is very nearly
the same as $\omega_1$ itself. This quantity indicates the rate of
revolution of the axle of the top about its mean position, and is very
easily observed.
* The _instantaneous axis_ is not so easily observed. It revolves round
the invariable axis in the same time with the axis of $x$, at a
distance which is very small in the case when $a$, $b$, $c$, are nearly
equal. From its rapid angular motion in space, and its near coincidence
with the invariable axis, there is no advantage in studying its motion
in the top.
* By making the moments of inertia very unequal, and in definite
proportion to each other, and by drawing a few strong lines as
diameters of the disc, the combination of motions will produce an
appearance of epicycloids, which are the result of the continued
intersection of the successive positions of these lines, and the cusps
of the epicycloids lie in the curve in which the instantaneous axis
travels. Some of the figures produced in this way are very pleasing.
In order to illustrate the theory of rotation experimentally, we must
have a body balanced on its centre of gravity, and capable of having
its principal axes and moments of inertia altered in form and position
within certain limits. We must be able to make the axle of the
instrument the greatest, least, or mean principal axis, or to make it
not a principal axis at all, and we must be able to _see_ the position
of the invariable axis of rotation at any time. There must be three
adjustments to regulate the position of the centre of gravity, three
for the magnitudes of the moments of inertia, and three for the
directions of the principal axes, nine independent adjustments, which
may be distributed as we please among the screws of the instrument.