HKB
opposite forces, produce equal and opposite changes in the angular momentum of those parts. Hence the whole angular momentum of the system is not affected by these actions and re-actions.
When a system of invariable form revolves about an axis, the angular velocity of every part is the same, and the angular momentum about the axis is the product of the angular velocity and the moment of inertia about that axis.
It is only in particular cases, however, that the whole angular momentum can be estimated in this way. In general, the axis of angular momentum differs from the axis of rotation, so that there will be a residual angular momentum about an axis perpendicular to that of rotation, unless that axis has one of three positions, called the principal axes of the body.
By referring everything to these three axes, the theory is greatly simplified. The moment of inertia about one of these axes is greater than that about any other axis through the same point, and that about one of the others is a minimum. These two are at right angles, and the third axis is perpendicular to their plane, and is called the mean axis.
Angular momenta may be compounded like forces or velocities, by the law of the “parallelogram,” and since these three are at right angles to each other, their resultant is
\begin{displaymath} \sqrt{A^2\omega1^2 + B^2\omega2^2 + C^2\omega_3^2} = H \end{displaymath} (1)
and this must be constant, both in magnitude and direction in space, since no external forces act on the body.
We shall call this axis of angular momentum the invariable axis. It is perpendicular to what has been called the invariable plane. Poinsôt calls it the axis of the couple of impulsion. The direction-cosines of this axis in the body are,
\begin{displaymath} \begin{array}{c c c} \displaystyle l = \frac{A\omega1}{H}, ... ...ga2}{H}, & \displaystyle n = \frac{C\omega_3}{H}. \end{array}\end{displaymath}
Since $I$, $m$ and $n$ vary during the motion, we need some additional condition to determine the relation between them. We find this in the property of the vis viva of a system of invariable form in which there is no friction. The vis viva of such a system must be constant. We express this in the equation
\begin{displaymath} A\omega1^2 + B\omega2^2 + C\omega_3^2 = V \end{displaymath} (2)
Substituting the values of $\omega1$, $\omega2$, $\omega_3$ in terms of $l$, $m$, $n$,
\begin{displaymath} \frac{l^2}{A} + \frac{m^2}{B} + \frac{n^2}{C} = \frac{V}{H^2}. \end{displaymath}
Let $1/A = a^2$, $1/B = b^2$, $1/c = c^2$, $V/H^2 = e^2$, and this equation becomes
\begin{displaymath} a^2l^2 + b^2m^2 + c^2n^2 = e^2 \end{displaymath} (3)
and the equation to the cone, described by the invariable axis within the body, is
\begin{displaymath} (a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2 = 0 \end{displaymath} (4)
The intersections of this cone with planes perpendicular to the principal axes are found by putting $x$, $y$, or $z$, constant in this equation. By giving $e$ various values, all the different paths of the pole of the invariable axis, corresponding to different initial circumstances, may be traced.
Figure: Figure 1