On a Dynamical Top, for exhibiting the phenomena of the motion of a system of invariable form about a fixed point, with some suggestions as to the Earth's motion

\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.