* 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. In order to illustrate the theory of rotation experimentally, we must have a body balanced on its centre of gravity, and capable of having its principal axes and moments of inertia altered in form and position within certain limits. We must be able to make the axle of the instrument the greatest, least, or mean principal axis, or to make it not a principal axis at all, and we must be able to _see_ the position of the invariable axis of rotation at any time. There must be three adjustments to regulate the position of the centre of gravity, three for the magnitudes of the moments of inertia, and three for the directions of the principal axes, nine independent adjustments, which may be distributed as we please among the screws of the instrument. Figure: Figure 6 The form of the body of the instrument which I have found most suitable is that of a bell (fig. 6). $C$ is a hollow cone of brass, $R$ is a heavy ring cast in the same piece. Six screws, with heavy heads, $x$, $y$, $z$, $x'$, $y'$, $z'$, work horizontally in the ring, and three similar screws, $l$, $m$, $n$, work vertically through the ring at equal intervals. $AS$ is the axle of the instrument, $SS$ is a brass screw working in the upper part of the cone $C$, and capable of being firmly clamped by means of the nut $c$. $B$ is a cylindrical brass bob, which may be screwed up or down the axis, and fixed in its place by the nut $b$. The lower extremity of the axle is a fine steel point, finished without emery, and afterwards hardened. It runs in a little agate cup set in the top of the pillar $P$. If any emery had been embedded in the steel, the cup would soon be worn out. The upper end of the axle has also a steel point by which it may be kept steady while spinning. When the instrument is in use, a coloured disc is attached to the upper end of the axle. It will be seen that there are eleven adjustments, nine screws in the brass ring, the axle screwing in the cone, and the bob screwing on the axle. The advantage of the last two adjustments is, that by them large alterations can be made, which are not possible by means of the small screws. The first thing to be done with the instrument is, to make the steel point at the end of the axle coincide with the centre of gravity of the whole. This is done roughly by screwing the axle to the right place nearly, and then balancing the instrument on its point, and screwing the bob and the horizontal screws till the instrument will remain balanced in any position in which it is placed.