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\omega_1^2 + B^2\omega_2^2 + 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\omega_1}{H}, ... ...ga_2}{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\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V \end{displaymath} (2) Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$ 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 * In the figures, I have supposed $a^2 = 100$, $b^2= 107$, and $c^2= 110$. The first figure represents a section of the various cones by a plane perpendicular to the axis of $x$, which is that of greatest moment of inertia. These sections are ellipses having their major axis parallel to the axis of $b$. The value of $e^2$ corresponding to each of these curves is indicated by figures beside the curve. The ellipticity increases with the size of the ellipse, so that the section corresponding to $e^2 = 107$ would be two parallel straight lines (beyond the bounds of the figure), after which the sections would be hyperbolas. Figure: Figure 2 * The second figure represents the sections made by a plane, perpendicular to the _mean_ axis. They are all hyperbolas, except when $e^2 = 107$, when the section is two intersecting straight lines. Figure: Figure 3