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 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 $A$, $B$, $C$ be the moments of inertia about the principal axes
through the centre of gravity, taken in order of magnitude, and let
$\omega_1$ $\omega_2$ $\omega_3$ be the angular velocities about them,
then the angular momenta will be $A\omega_1$, $B\omega_2$, and
$C\omega_3$.

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


The third figure shows the sections perpendicular to the axis of least
moment of inertia. From $e^2 = 110$ to $e^2 = 107$ the sections are
ellipses, $e^2 = 107$ gives two parallel straight lines, and beyond
these the curves are hyperbolas.


Figure: Figure 4


* 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: Figure 5


* 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 $l$, $m$, $n$, about
the axis of $x$ is


\begin{displaymath} \frac{\omega_1}{1 - l^2} - \frac{l}{1 -
l^2}(l\omega_1 + m\omega_2 + n\omega-3). \end{displaymath}


Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$, in terms
of $l$, $m$, $n$, and taking account of equation (3), this expression
becomes


\begin{displaymath} H\frac{(a^2 - e^2)}{1 - l^2}l. \end{displaymath}


Changing the sign and putting $\displaystyle l = \frac{\omega_1}{a^2H}$
we have the angular velocity of the invariable axis about that of $x$


\begin{displaymath} = \frac{\omega_1}{1 - l^2} \frac{e^2 - a^2}{a^2},
\end{displaymath}


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 $e^2$. 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 $yz$, are swept out at the
rate


\begin{displaymath} \omega_1 \frac{e^2 - a^2}{a^2}. \end{displaymath}


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


\begin{displaymath} \sqrt{\frac{e^2 - a^2}{b^2 - a^2}}
\hspace{1cm}\textrm{and}\hspace{1cm} \sqrt{\frac{e^2 - a^2}{c^2 -
a^2}}. \end{displaymath}


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--


\begin{displaymath} = \frac{a^2}{\sqrt{b^2 - a^2}\sqrt{c^2 - a^2}}.
\end{displaymath}


The projections of the spherical ellipses upon the plane of $yz$ 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
$x$, 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 $x$, $y$ and $z$ 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 $b = c$, then
the angular velocity $\omega_1$ 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


\begin{displaymath} \frac{a^2}{b^2 - a^2} \end{displaymath}


The motion is in the same direction as that of the rotation, or in the
opposite direction, according as the axis of $x$ 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 $x$ about the invariable axis in space is


\begin{displaymath} \omega_1\frac{e^2 - a^2l^2}{a^2(1 - l^2)},
\end{displaymath}


which is greater or less than $\omega_1$, as $e^2$ is greater or less
than $a^2$, and, when these quantities are nearly equal, is very nearly
the same as $\omega_1$ 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 $x$, at a
distance which is very small in the case when $a$, $b$, $c$, 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: 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.

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.

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 $\displaystyle = \omega\frac{C - A}{A}$ where
$\omega$ 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 $\displaystyle
\frac{A}{C - A}$ 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, $\frac{1}{3}$ inch square. The conical portion
rises from the upper and inner edge of the ring, a height of
$1\frac{1}{2}$ 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 together weigh 1 ounce. The whole
weigh   9 II. Axle, &c.-- Length of axle 5 inches, of which
$\frac{1}{2}$ inch at the bottom is occupied by the steel point,
$3\frac{1}{2}$ 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   $1\frac{1}{2}$ The bob $B$ has a diameter of 1.4 inches,
and a thickness of .4. It weighs   $2\frac{3}{4}$ The nuts $b$
and $c$, for clamping the bob and the body of the top on the axle, each
weigh $\frac{1}{2}$ oz.   1 Weight of whole
top 2 $5\frac{1}{4}$


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.