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.
When this adjustment is carefully made, the rotation of the top has no
tendency to shake the steel point in the agate cup, however irregular
the motion may appear to be.
The next thing to be done, is to make one of the principal axes of the
central ellipsoid coincide with the axle of the top.
To effect this, we must begin by spinning the top gently about its
axle, steadying the upper part with the finger at first. If the axle is
already a principal axis the top will continue to revolve about its
axle when the finger is removed. If it is not, we observe that the top
begins to spin about some other axis, and the axle moves away from the
centre of motion and then back to it again, and so on, alternately
widening its circles and contracting them.
It is impossible to observe this motion successfully, without the aid
of the coloured disc placed near the upper end of the axis. This disc
is divided into sectors, and strongly coloured, so that each sector may
be recognised by its colour when in rapid motion. If the axis about
which the top is really revolving, falls within this disc, its position
may be ascertained by the colour of the spot at the centre of motion.
If the central spot appears red, we know that the invariable axis at
that instant passes through the red part of the disc.
In this way we can trace the motion of the invariable axis in the
revolving body, and we find that the path which it describes upon the
disc may be a circle, an ellipse, an hyperbola, or a straight line,
according to the arrangement of the instrument.
In the case in which the invariable axis coincides at first with the
axle of the top, and returns to it after separating from it for a time,
its true path is a circle or an ellipse having the axle in its
_circumference_. The true principal axis is at the centre of the closed
curve. It must be made to coincide with the axle by adjusting the
vertical screws $l$, $m$, $n$.
Suppose that the colour of the centre of motion, when farthest from the
axle, indicated that the axis of rotation passed through the sector
$L$, then the principal axis must also lie in that sector at half the
distance from the axle.
If this principal axis be that of _greatest_ moment of inertia, we must
_raise_ the screw $l$ in order to bring it nearer the axle $A$. If it
be the axis of least moment we must _lower_ the screw $l$. In this way
we may make the principal axis coincide with the axle. Let us suppose
that the principal axis is that of greatest moment of inertia, and that
we have made it coincide with the axle of the instrument. Let us also
suppose that the moments of inertia about the other axes are equal, and
very little less than that about the axle. Let the top be spun about
the axle and then receive a disturbance which causes it to spin about
some other axis. The instantaneous axis will not remain at rest either
in space or in the body. In space it will describe a right cone,
completing a revolution in somewhat less than the time of revolution of
the top. In the body it will describe another cone of larger angle in a
period which is longer as the difference of axes of the body is
smaller. The invariable axis will be fixed in space, and describe a
cone in the body.
The relation of the different motions may be understood from the
following illustration. Take a hoop and make it revolve about a stick
which remains at rest and touches the inside of the hoop. The section
of the stick represents the path of the instantaneous axis in space,
the hoop that of the same axis in the body, and the axis of the stick
the invariable axis. The point of contact represents the pole of the
instantaneous axis itself, travelling many times round the stick before
it gets once round the hoop. It is easy to see that the direction in
which the hoop moves round the stick, so that if the top be spinning in
the direction $L$, $M$, $N$, the colours will appear in the same order.
By screwing the bob B up the axle, the difference of the axes of
inertia may be diminished, and the time of a complete revolution of the
invariable axis in the body increased. By observing the number of
revolutions of the top in a complete cycle of colours of the invariable
axis, we may determine the ratio of the moments of inertia.
By screwing the bob up farther, we may make the axle the principal axis
of _least_ moment of inertia.
The motion of the instantaneous axis will then be that of the point of
contact of the stick with the _outside_ of the hoop rolling on it. The
order of colours will be $N$, $M$, $L$, if the top be spinning in the
direction $L$, $M$, $N$, and the more the bob is screwed up, the more
rapidly will the colours change, till it ceases to be possible to make
the observations correctly.
In calculating the dimensions of the parts of the instrument, it is
necessary to provide for the exhibition of the instrument with its axle
either the greatest or the least axis of inertia. The dimensions and
weights of the parts of the top which I have found most suitable, are
given in a note at the end of this paper.
Now let us make the axes of inertia in the plane of the ring unequal.
We may do this by screwing the balance screws $x$ and $x^1$ farther
from the axle without altering the centre of gravity.
Let us suppose the bob $B$ screwed up so as to make the axle the axis
of least inertia. Then the mean axis is parallel to $xx^1$, and the
greatest is at right angles to $xx^1$ in the horizontal plane. The path
of the invariable axis on the disc is no longer a circle but an
ellipse, concentric with the disc, and having its major axis parallel
to the mean axis $xx^1$.
The smaller the difference between the moment of inertia about the axle
and about the mean axis, the more eccentric the ellipse will be; and
if, by screwing the bob down, the axle be made the mean axis, the path
of the invariable axis will be no longer a closed curve, but an
hyperbola, so that it will depart altogether from the neighbourhood of
the axle. When the top is in this condition it must be spun gently, for
it is very difficult to manage it when its motion gets more and more
eccentric.
When the bob is screwed still farther down, the axle becomes the axis
of greatest inertia, and $xx^1$ the least. The major axis of the
ellipse described by the invariable axis will now be perpendicular to
$xx^1$, and the farther the bob is screwed down, the eccentricity of
the ellipse will diminish, and the velocity with which it is described
will increase.
I have now described all the phenomena presented by a body revolving
freely on its centre of gravity. If we wish to trace the motion of the
invariable axis by means of the coloured sectors, we must make its
motion very slow compared with that of the top. It is necessary,
therefore, to make the moments of inertia about the principal axes very
nearly equal, and in this case a very small change in the position of
any part of the top will greatly derange the _position_ of the
principal axis. So that when the top is well adjusted, a single turn of
one of the screws of the ring is sufficient to make the axle no longer
a principal axis, and to set the true axis at a considerable
inclination to the axle of the top.
All the adjustments must therefore be most carefully arranged, or we
may have the whole apparatus deranged by some eccentricity of spinning.
The method of making the principal axis coincide with the axle must be
studied and practised, or the first attempt at spinning rapidly may end
in the destruction of the top, if not the table on which it is spun.