The third figure shows the sections perpendicular to the axis of least moment of inertia. From $e^2 = 110$ to $e^2 = 107$ the sections are ellipses, $e^2 = 107$ gives two parallel straight lines, and beyond these the curves are hyperbolas. Figure: Figure 4 * The fourth and fifth figures show the sections of the series of cones made by a cube and a sphere respectively. The use of these figures is to exhibit the connexion between the different curves described about the three principal axes by the invariable axis during the motion of the body. Figure: Figure 5 * 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}