On a Dynamical Top_ for exhibiting the phenomena of the motion of a system of invariable form about a fixed point_ with some suggestions as to the Earth_s motion James Clerk Maxwel

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Title: On a Dynamical Top, for exhibiting the phenomena of the motion of a
system of invariable form about a fixed point, with some
suggestions as to the Earth's motion

Author: James Clerk Maxwell

Release date: February 1, 2004 [eBook #5192]
Most recently updated: January 21, 2021

Language: English

Other information and formats: www.gutenberg.org/ebooks/5192

Credits: Gordon Keener

*** START OF THE PROJECT GUTENBERG EBOOK ON A
DYNAMICAL TOP, FOR EXHIBITING THE PHENOMENA OF THE
MOTION OF A SYSTEM OF INVARIABLE FORM ABOUT A FIXED
POINT, WITH SOME SUGGESTIONS AS TO THE EARTH'S MOTION
***

O n a D y n a m i c a l To p ,
for exhibiting the phenomena of the motion of a
system of invariable form about a fixed point, with
some suggestions as to the Earth’s motion

James Clerk Maxwell

[From the Transactions of the Royal Society of Edinburgh, Vol. XXI. Part
IV.]
(Read 20th April, 1857.)

To those who study the progress of exact science, the common spinning-top
is a symbol of the labours and the perplexities of men who had successfully
threaded the mazes of the planetary motions. The mathematicians of the last
age, searching through nature for problems worthy of their analysis, found in
this toy of their youth, ample occupation for their highest mathematical
powers.

No illustration of astronomical precession can be devised more perfect than
that presented by a properly balanced top, but yet the motion of rotation has
intricacies far exceeding those of the theory of precession.

Accordingly, we find Euler and D’Alembert devoting their talent and their
patience to the establishment of the laws of the rotation of solid bodies.
Lagrange has incorporated his own analysis of the problem with his general
treatment of mechanics, and since his time M. Poinsôt has brought the subject
under the power of a more searching analysis than that of the calculus, in
which ideas take the place of symbols, and intelligible propositions supersede
equations.

In the practical department of the subject, we must notice the rotatory
machine of Bohnenberger, and the nautical top of Troughton. In the first of
these instruments we have the model of the Gyroscope, by which Foucault
has been able to render visible the effects of the earth’s rotation. The beautiful
experiments by which Mr J. Elliot has made the ideas of precession so
familiar to us are performed with a top, similar in some respects to
Troughton’s, though not borrowed from his.

The top which I have the honour to spin before the Society, differs from that
of Mr Elliot in having more adjustments, and in being designed to exhibit far
more complicated phenomena.

The arrangement of these adjustments, so as to produce the desired effects,
depends on the mathematical theory of rotation. The method of exhibiting the
motion of the axis of rotation, by means of a coloured disc, is essential to the
success of these adjustments. This optical contrivance for rendering visible
the nature of the rapid motion of the top, and the practical methods of
applying the theory of rotation to such an instrument as the one before us, are
the grounds on which I bring my instrument and experiments before the
Society as my own.

I propose, therefore, in the first place, to give a brief outline of such parts of
the theory of rotation as are necessary for the explanation of the phenomena
of the top.

I shall then describe the instrument with its adjustments, and the effect of
each, the mode of observing of the coloured disc when the top is in motion,
and the use of the top in illustrating the mathematical theory, with the method
of making the different experiments.

Lastly, I shall attempt to explain the nature of a possible variation in the
earth’s axis due to its figure. This variation, if it exists, must cause a periodic
inequality in the latitude of every place on the earth’s surface, going through
its period in about eleven months. The amount of variation must be very
small, but its character gives it importance, and the necessary observations
are already made, and only require reduction.

On the Theory of Rotation.

The theory of the rotation of a rigid system is strictly deduced from the
elementary laws of motion, but the complexity of the motion of the particles
of a body freely rotating renders the subject so intricate, that it has never been
thoroughly understood by any but the most expert mathematicians. Many
who have mastered the lunar theory have come to erroneous conclusions on
this subject; and even Newton has chosen to deduce the disturbance of the
earth’s axis from his theory of the motion of the nodes of a free orbit, rather
than attack the problem of the rotation of a solid body.

The method by which M. Poinsôt has rendered the theory more manageable,
is by the liberal introduction of “appropriate ideas,” chiefly of a geometrical
character, most of which had been rendered familiar to mathematicians by the
writings of Monge, but which then first became illustrations of this branch of
dynamics. If any further progress is to be made in simplifying and arranging
the theory, it must be by the method which Poinsôt has repeatedly pointed out
as the only one which can lead to a true knowledge of the subject,--that of
proceeding from one distinct idea to another instead of trusting to symbols
and equations.

An important contribution to our stock of appropriate ideas and methods has
lately been made by Mr R. B. Hayward, in a paper, “On a Direct Method of
estimating Velocities, Accelerations, and all similar quantities, with respect to
axes, moveable in any manner in Space.” (Trans. Cambridge Phil. Soc Vol. x.
Part I.)

* In this communication I intend to confine myself to that part of the subject
which the top is intended io illustrate, namely, the alteration of the position of
the axis in a body rotating freely about its centre of gravity. I shall, therefore,
deduce the theory as briefly as possible, from two considerations only,--the
permanence of the original angular momentum in direction and magnitude,
and the permanence of the original vis viva.

* The mathematical difficulties of the theory of rotation arise chiefly from the
want of geometrical illustrations and sensible images, by which we might fix
the results of analysis in our minds.

It is easy to understand the motion of a body revolving about a fixed axle.
Every point in the body describes a circle about the axis, and returns to its
original position after each complete revolution. But if the axle itself be in
motion, the paths of the different points of the body will no longer be circular
or re-entrant. Even the velocity of rotation about the axis requires a careful
definition, and the proposition that, in all motion about a fixed point, there is
always one line of particles forming an instantaneous axis, is usually given in
the form of a very repulsive mass of calculation. Most of these difficulties
may be got rid of by devoting a little attention to the mechanics and geometry
of the problem before entering on the discussion of the equations.

Mr Hayward, in his paper already referred to, has made great use of the
mechanical conception of Angular Momentum.

Definition 1 The Angular Momentum of a particle about an axis is measured
by the product of the mass of the particle, its velocity resolved in the normal
plane, and the perpendicular from the axis on the direction of motion.

* The angular momentum of any system about an axis is the algebraical sum
of the angular momenta of its parts.

As the rate of change of the linear momentum of a particle measures the
moving force which acts on it, so the rate of change of angular momentum
measures the moment of that force about an axis.

All actions between the parts of a system, being pairs of equal and 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.

* Let , , be the moments of inertia about the principal axes through

the centre of gravity, taken in order of magnitude, and let be the

angular velocities about them, then the angular momenta will be , ,

and .

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

(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,

Since , and 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

(2)

Substituting the values of , , in terms of , , ,

Let , , , , and this equation becomes

(3)

and the equation to the cone, described by the invariable axis within the body,
is

(4)

The intersections of this cone with planes perpendicular to the principal axes

are found by putting , , or , constant in this equation. By giving

various values, all the different paths of the pole of the invariable axis,
corresponding to different initial circumstances, may be traced.

Figure:

* In the figures, I have supposed , , and . The
first figure represents a section of the various cones by a plane perpendicular
to the axis of , which is that of greatest moment of inertia. These sections
are ellipses having their major axis parallel to the axis of . The value of
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 would be two parallel straight lines (beyond the
bounds of the figure), after which the sections would be hyperbolas.

Figure:

* The second figure represents the sections made by a plane, perpendicular to
the mean axis. They are all hyperbolas, except when , when the
section is two intersecting straight lines.

Figure:

The third figure shows the sections perpendicular to the axis of least moment
of inertia. From to the sections are ellipses,
gives two parallel straight lines, and beyond these the curves are hyperbolas.

Figure:

* The fourth and fifth figures show the sections of the series of cones made
by a cube and a sphere respectively. The use of these figures is to exhibit the
connexion between the different curves described about the three principal
axes by the invariable axis during the motion of the body.

Figure:

* We have next to compare the velocity of the invariable axis with respect to
the body, with that of the body itself round one of the principal axes. Since
the invariable axis is fixed in space, its motion relative to the body must be
equal and opposite to that of the portion of the body through which it passes.
Now the angular velocity of a portion of the body whose direction-cosines
are , , , about the axis of is

Substituting the values of , , , in terms of , , , and taking
account of equation (3), this expression becomes

Changing the sign and putting we have the angular velocity of the
invariable axis about that of

always positive about the axis of greatest moment, negative about that of
least moment, and positive or negative about the mean axis according to the
value of . The direction of the motion in every case is represented by the
arrows in the figures. The arrows on the outside of each figure indicate the
direction of rotation of the body.

* If we attend to the curve described by the pole of the invariable axis on the
sphere in fig. 5, we shall see that the areas described by that point, if

projected on the plane of , are swept out at the rate

Now the semi-axes of the projection of the spherical ellipse described by the
pole are

Dividing the area of this ellipse by the area described during one revolution
of the body, we find the number of revolutions of the body during the
description of the ellipse--

The projections of the spherical ellipses upon the plane of are all similar
ellipses, and described in the same number of revolutions; and in each ellipse
so projected, the area described in any time is proportional to the number of
revolutions of the body about the axis of , so that if we measure time by
revolutions of the body, the motion of the projection of the pole of the
invariable axis is identical with that of a body acted on by an attractive
central force varying directly as the distance. In the case of the hyperbolas in
the plane of the greatest and least axis, this force must be supposed repulsive.
The dots in the figures 1, 2, 3, are intended to indicate roughly the progress
made by the invariable axis during each revolution of the body about the axis

of , and respectively. It must be remembered that the rotation about
these axes varies with their inclination to the invariable axis, so that the
angular velocity diminishes as the inclination increases, and therefore the
areas in the ellipses above mentioned are not described with uniform velocity

in absolute time, but are less rapidly swept out at the extremities of the major
axis than at those of the minor.

* When two of the axes have equal moments of inertia, or , then the

angular velocity is constant, and the path of the invariable axis is circular,
the number of revolutions of the body during one circuit of the invariable
axis, being

The motion is in the same direction as that of the rotation, or in the opposite
direction, according as the axis of is that of greatest or of least moment of
inertia.

* Both in this case, and in that in which the three axes are unequal, the
motion of the invariable axis in the body may be rendered very slow by
diminishing the difference of the moments of inertia. The angular velocity of
the axis of about the invariable axis in space is

which is greater or less than , as is greater or less than , and, when

these quantities are nearly equal, is very nearly the same as itself. This
quantity indicates the rate of revolution of the axle of the top about its mean
position, and is very easily observed.

* The instantaneous axis is not so easily observed. It revolves round the
invariable axis in the same time with the axis of , at a distance which is
very small in the case when , , , are nearly equal. From its rapid angular

motion in space, and its near coincidence with the invariable axis, there is no
advantage in studying its motion in the top.

* 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:

The form of the body of the instrument which I have found most suitable is
that of a bell (fig. 6). is a hollow cone of brass, is a heavy ring cast in

the same piece. Six screws, with heavy heads, , , , , , , work
horizontally in the ring, and three similar screws, , , , work vertically
through the ring at equal intervals. is the axle of the instrument, is a
brass screw working in the upper part of the cone , and capable of being
firmly clamped by means of the nut . is a cylindrical brass bob, which
may be screwed up or down the axis, and fixed in its place by the nut .

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
. 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 , , .

Suppose that the colour of the centre of motion, when farthest from the axle,
indicated that the axis of rotation passed through the sector , 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 in order to bring it nearer the axle . If it be the axis of least
moment we must lower the screw . 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 , , , 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 , , , if the top be spinning in the direction , , , 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 and farther from the axle
without altering the centre of gravity.

Let us suppose the bob screwed up so as to make the axle the axis of least
inertia. Then the mean axis is parallel to , and the greatest is at right
angles to 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 .

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 the least. The major axis of the ellipse described by
the invariable axis will now be perpendicular to , 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.

On the Earth’s Motion

We must remember that these motions of a body about its centre of gravity,
are not illustrations of the theory of the precession of the Equinoxes.
Precession can be illustrated by the apparatus, but we must arrange it so that
the force of gravity acts the part of the attraction of the sun and moon in
producing a force tending to alter the axis of rotation. This is easily done by
bringing the centre of gravity of the whole a little below the point on which it
spins. The theory of such motions is far more easily comprehended than that
which we have been investigating.

But the earth is a body whose principal axes are unequal, and from the
phenomena of precession we can determine the ratio of the polar and
equatorial axes of the “central ellipsoid;” and supposing the earth to have
been set in motion about any axis except the principal axis, or to have had its
original axis disturbed in any way, its subsequent motion would be that of the
top when the bob is a little below the critical position.

The axis of angular momentum would have an invariable position in space,
and would travel with respect to the earth round the axis of figure with a

velocity where is the sidereal angular velocity of the earth.
The apparent pole of the earth would travel (with respect to the earth) from

west to east round the true pole, completing its circuit in sidereal
days, which appears to be about 325.6 solar days.

The instantaneous axis would revolve about this axis in space in about a day,
and would always be in a plane with the true axis of the earth and the axis of
angular momentum. The effect of such a motion on the apparent position of a
star would be, that its zenith distance should be increased and diminished
during a period of 325.6 days. This alteration of zenith distance is the same

above and below the pole, so that the polar distance of the star is unaltered. In
fact the method of finding the pole of the heavens by observations of stars,
gives the pole of the invariable axis, which is altered only by external forces,
such as those of the sun and moon.

There is therefore no change in the apparent polar distance of stars due to this
cause. It is the latitude which varies. The magnitude of this variation cannot
be determined by theory. The periodic time of the variation may be found
approximately from the known dynamical properties of the earth. The epoch
of maximum latitude cannot be found except by observation, but it must be
later in proportion to the east longitude of the observatory.

In order to determine the existence of such a variation of latitude, I have
examined the observations of Polaris with the Greenwich Transit Circle in
the years 1851-2-3-4. The observations of the upper transit during each
month were collected, and the mean of each month found. The same was
done for the lower transits. The difference of zenith distance of upper and
lower transit is twice the polar distance of Polaris, and half the sum gives the
co-latitude of Greenwich.

In this way I found the apparent co-latitude of Greenwich for each month of
the four years specified.

There appeared a very slight indication of a maximum belonging to the set of
months,

March, 51. Feb. 52. Dec. 52. Nov. 53. Sept. 54.

This result, however, is to be regarded as very doubtful, as there did not
appear to be evidence for any variation exceeding half a second of space, and
more observations would be required to establish the existence of so small a
variation at all.

I therefore conclude that the earth has been for a long time revolving about an
axis very near to the axis of figure, if not coinciding with it. The cause of this
near coincidence is either the original softness of the earth, or the present
fluidity of its interior. The axes of the earth are so nearly equal, that a
considerable elevation of a tract of country might produce a deviation of the

principal axis within the limits of observation, and the only cause which
would restore the uniform motion, would be the action of a fluid which
would gradually diminish the oscillations of latitude. The permanence of
latitude essentially depends on the inequality of the earth’s axes, for if they
had been all equal, any alteration of the crust of the earth would have
produced new principal axes, and the axis of rotation would travel about
those axes, altering the latitudes of all places, and yet not in the least altering
the position of the axis of rotation among the stars.

Perhaps by a more extensive search and analysis of the observations of
different observatories, the nature of the periodic variation of latitude, if it
exist, may be determined. I am not aware of any calculations having been
made to prove its non-existence, although, on dynamical grounds, we have
every reason to look for some very small variation having the periodic time
of 325.6 days nearly, a period which is clearly distinguished from any other
astronomical cycle, and therefore easily recognised.

Note: Dimensions and Weights of the parts of the
Dynamical Top.

Part Weight
lb. oz.
I. Body of the top--
Mean diameter of ring, 4 inches.

Section of ring, inch square.
The conical portion rises from the upper and

inner edge of the ring, a height of inches
from the base.
The whole body of the top weighs 1 7
Each of the nine adjusting screws has its
screw 1 inch long, and the screw and head 9
together weigh 1 ounce. The whole weigh
II. Axle, &c.--

Length of axle 5 inches, of which inch at

the bottom is occupied by the steel point,
inches are brass with a good screw turned on
it, and the remaining inch is of steel, with a
sharp point at the top. The whole weighs
The bob has a diameter of 1.4 inches, and
a thickness of .4. It weighs
The nuts and , for clamping the bob and 1
the body of the top on the axle, each weigh

oz.
Weight of whole top
2

The best arrangement, for general observations, is to have the disc of card
divided into four quadrants, coloured with vermilion, chrome yellow, emerald
green, and ultramarine. These are bright colours, and, if the vermilion is
good, they combine into a grayish tint when the rotation is about the axle, and
burst into brilliant colours when the axis is disturbed. It is useful to have
some concentric circles, drawn with ink, over the colours, and about 12 radii
drawn in strong pencil lines. It is easy to distinguish the ink from the pencil
lines, as they cross the invariable axis, by their want of lustre. In this way, the
path of the invariable axis may be identified with great accuracy, and
compared with theory.

* 7th May 1857. The paragraphs marked thus have been rewritten since the
paper was read.

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