196 THE THEORY OF SCREWS. [197
the initial angular velocity may also be resolved into two angular velocities
on the two harmonic axes. The entire motion will, therefore, be found by
compounding the vibrations about the two harmonic axes. Also the instan-
taneous axis will at every instant be found in the plane of the harmonic
axes, and will oscillate to and fro in their plane.
Since conjugate diameters of an ellipse are always projected into con-
jugate diameters of the projected ellipse, it follows that the harmonic axes
must project into two conjugate diameters of a circle on any horizontal
plane. Hence we see that two vertical planes, each containing one of the
harmonic axes, are at right angles to each other.
We have thus obtained a complete solution of the problem of the small
oscillations of a body about a fixed point under the influence of gravity.