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+\documentclass[12pt]{article}
+\usepackage{graphicx}
+%\usepackage{amssymb}
+
+\textwidth = 6.5 in
+\textheight = 9 in
+\oddsidemargin = 0.0 in
+\evensidemargin = 0.0 in
+\topmargin = 0.0 in
+\headheight = 0.0 in
+\headsep = 0.0 in
+
+%\newtheorem{theorem}{Theorem}
+%\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{definition}{Definition}
+
+\renewcommand{\thefootnote}{\fnsymbol{footnote}}
+
+\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}
+\date{\small [From the \emph{Transactions of the Royal Society of Edinburgh},
+Vol. XXI. Part IV.] \\
+(Read 20th April, 1857.)}
+
+\begin{document}
+\maketitle
+
+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.~Poins\^ot 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.
+
+\section*{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
+co~ne 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. Poins\^ot 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 Poins\^ot 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.''
+(\emph{Trans. Cambridge Phil. Soc} Vol. x. Part I.)
+
+\footnote[1]{7th May 1857.  The paragraphs marked thus have been
+rewritten since the paper was read.}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 \emph{angular momentum} in
+direction and magnitude, and the permanence of the original \emph{vis
+viva}.
+
+\footnotemark[1]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.
+
+\begin{definition}
+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.
+\end{definition}
+
+\footnotemark[1]The angular momentum of any system about an axis is
+the algebraical sum of the angular momenta of its parts.
+
+As the \emph{rate of change} of the \emph{linear momentum} of a
+particle measures the \emph{moving force} which acts on it, so the
+\emph{rate of change} of \emph{angular momentum} measures the
+\emph{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.
+
+\footnotemark[1]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 \emph{angular velocity}
+and the \emph{moment of inertia} about that axis.
+
+\footnotemark[1]It is only in particular cases, however, that the
+\emph{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.
+
+\footnotemark[1]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{equation}
+\sqrt{A^2\omega_1^2 + B^2\omega_2^2 + C^2\omega_3^2} = H
+\end{equation}
+%
+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 \emph{invariable
+axis}.  It is perpendicular to what has been called the invariable
+plane.  Poins\^ot calls it the axis of the couple of impulsion.  The
+\emph{direction-cosines} of this axis in the body are,
+
+\begin{displaymath}
+\begin{array}{c c c}
+\displaystyle l = \frac{A\omega_1}{H}, &
+\displaystyle m = \frac{B\omega_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 \emph{vis viva} of a system of invariable form in
+which there is no friction.  The \emph{vis viva} of such a system must
+be constant.  We express this in the equation
+
+\begin{equation}
+A\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V
+\end{equation}
+
+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{equation}
+a^2l^2 + b^2m^2 + c^2n^2 = e^2
+\end{equation}
+%
+and the equation to the cone, described by the invariable axis within
+the body, is
+
+\begin{equation}
+(a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2 = 0 
+\end{equation}
+
+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.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{fig1.png}
+\caption{}
+\end{figure}
+
+\footnotemark[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.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{fig2.png}
+\caption{}
+\end{figure}
+
+\footnotemark[1]The second figure represents the sections made by a
+plane, perpendicular to the \emph{mean} axis.  They are all
+hyperbolas, except when $e^2= 107$, when the section is two
+intersecting straight lines.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{fig3.png}
+\caption{}
+\end{figure}
+
+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.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{fig4.png}
+\caption{}
+\end{figure}
+
+\footnotemark[1]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.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{fig5.png}
+\caption{}
+\end{figure}
+
+\footnotemark[1]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.
+
+\footnotemark[1]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.
+
+\footnotemark[1]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.
+
+\footnotemark[1]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.
+
+\footnotemark[1]The \emph{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.
+
+\footnotemark[1]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 \emph{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.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=0.8\textwidth]{fig6.png}
+\end{center}
+\caption{}
+\end{figure}
+
+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
+\emph{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 \emph{greatest} moment of inertia,
+we must \emph{raise} the screw $l$ in order to bring it nearer the
+axle $A$.  If it be the axis of least moment we must \emph{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 \emph{least} moment of inertia.
+
+The motion of the instantaneous axis will then be that of the point of
+contact of the stick with the \emph{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
+\emph{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.
+
+\section*{On the Earth's Motion}
+
+We must remember that these motions of a body about its centre of
+gravity, are \emph{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 \emph{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 \emph{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,
+
+\begin{center}
+\begin{tabular}{ccccc}
+March,	51.  & Feb.  52.  & Dec.  52.  & Nov.  53.  & Sept.  54.
+\end{tabular}
+\end{center}
+
+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.
+
+\section*{Note: Dimensions and Weights of the parts of the Dynamical Top.}
+
+\begin{tabular}[b]{|p{.73\textwidth}|r|r|}
+\hline
+Part & \multicolumn{2}{|c|}{Weight} \\ \cline{2 - 3}
+       & lb.  & oz.  \\ \hline\hline
+\multicolumn{3}{|l|}{\bf I.  Body of the top---} \\ \hline
+\hspace{3mm}Mean diameter of ring, 4 inches.  & & \\
+\hspace{3mm}Section of ring, $\frac{1}{3}$ inch square.  & & \\
+\hspace{3mm}The conical portion rises from the upper and 
+inner edge of the ring, a height of $1\frac{1}{2}$ inches from the base.  & & \\
+\hspace{3mm}The whole body of the top weighs &1&7 \\ \hline
+\hspace{3mm}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 \\ \hline
+\multicolumn{3}{|l|}{\bf II.  Axle, \&c.---} \\ \hline
+\hspace{3mm}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 & & $1\frac{1}{2}$ \\ \hline
+\hspace{3mm}The bob $B$ has a diameter of 1.4 inches, and a 
+thickness of .4.   It weighs && $2\frac{3}{4}$ \\ \hline
+\hspace{3mm}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 \\ \hline
+{\bf Weight of whole top} &2& $5\frac{1}{4}$ \\ \hline
+\end{tabular}
+
+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.
+
+\end{document}
+
+% End of Project Gutenberg etext of On a Dynamical Top...
\ No newline at end of file
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+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
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diff --git a/old/dytop10t.zip b/old/dytop10t.zip
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