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+The Project Gutenberg eBook of On a Dynamical Top, by James Clerk Maxwell
+
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online at
+www.gutenberg.org. If you are not located in the United States, you
+will have to check the laws of the country where you are located before
+using this eBook.
+
+Title: On a Dynamical Top
+
+Author: James Clerk Maxwell
+
+Release Date: June 1, 2002 [eBook #5192]
+[Most recently updated: January 21, 2021]
+
+Language: English
+
+Character set encoding: UTF-8
+
+Produced by: Gordon Keener
+
+*** START OF THE PROJECT GUTENBERG EBOOK ON A DYNAMICAL TOP ***
+
+
+
+
+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.
+
+
+
+
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+
+<div style='text-align:center; font-size:1.2em; font-weight:bold;'>The Project Gutenberg eBook of On a Dynamical Top, by James Clerk Maxwell</div>
+<div style='display:block;margin:1em 0'>
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online
+at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you
+are not located in the United States, you will have to check the laws of the
+country where you are located before using this eBook.
+</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: On a Dynamical Top</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: James Clerk Maxwell</div>
+<div style='display:block;margin:1em 0'>Release Date: June 1, 2002 [eBook #5192]<br>
+[Most recently updated: January 21, 2021]</div>
+<div style='display:block;margin:1em 0'>Language: English</div>
+<div style='display:block;margin:1em 0'>Character set encoding: UTF-8</div>
+<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Gordon Keener</div>
+<div style='margin-top:2em;margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK ON A DYNAMICAL TOP ***</div>
+
+<h1>On a Dynamical Top,</h1>
+
+<h3>
+for exhibiting the phenomena of the motion
+of a system of invariable form about a fixed point, with some
+suggestions as to the Earth&rsquo;s motion</h3>
+<h2>James Clerk Maxwell</h2>
+<P ALIGN="CENTER"><STRONG>[From the <i>Transactions of the Royal Society of Edinburgh</i>,
+Vol. XXI. Part IV.]
+<BR>(Read 20th April, 1857.)</STRONG></P>
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+Accordingly, we find Euler and D&rsquo;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.
+
+<P>
+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&rsquo;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&rsquo;s, though not borrowed from
+his.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+Lastly, I shall attempt to explain the nature of a possible variation
+in the earth&rsquo;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&rsquo;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.
+
+<H2>On the Theory of Rotation.</H2>
+
+<P>
+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&rsquo;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.
+
+<P>
+The method by which M. Poinsôt has rendered the theory more
+manageable, is by the liberal introduction of &ldquo;appropriate ideas,&rdquo;
+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.
+
+<P>
+An important contribution to our stock of appropriate ideas and
+methods has lately been made by Mr R. B. Hayward, in a paper, &ldquo;On a
+Direct Method of estimating Velocities, Accelerations, and all similar
+quantities, with respect to axes, moveable in any manner in Space.&rdquo;
+(<i>Trans. Cambridge Phil. Soc</i> Vol. x. Part I.)
+
+<P>
+<a href="#linknote-1" name="linknoteref-1" id="linknoteref-1"><big>*</big></a>
+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 <i>angular momentum</i> in
+direction and magnitude, and the permanence of the original <i>vis
+viva</i>.
+
+<P>
+<a href="#linknote-2" name="linknoteref-2" id="linknoteref-2"><big>*</big></a>
+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.
+
+<P>
+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.
+
+<P>
+Mr Hayward, in his paper already referred to, has made great use of
+the mechanical conception of Angular Momentum.
+
+<P>
+<P>
+<DIV><B>Definition 1</B>
+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.</DIV><P></P>
+
+<P>
+<a href="#linknote-3" name="linknoteref-3" id="linknoteref-3"><big>*</big></a>
+The angular momentum of any system about an axis is
+the algebraical sum of the angular momenta of its parts.
+
+<P>
+As the <i>rate of change</i> of the <i>linear momentum</i> of a
+particle measures the <i>moving force</i> which acts on it, so the
+<i>rate of change</i> of <i>angular momentum</i> measures the
+<i>moment</i> of that force about an axis.
+
+<P>
+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.
+
+<P>
+<a href="#linknote-4" name="linknoteref-4" id="linknoteref-4"><big>*</big></a>
+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 <i>angular velocity</i>
+and the <i>moment of inertia</i> about that axis.
+
+<P>
+<a href="#linknote-5" name="linknoteref-5" id="linknoteref-5"><big>*</big></a>
+It is only in particular cases, however, that the
+<i>whole</i> 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.
+
+<P>
+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.
+
+<P>
+<a href="#linknote-6" name="linknoteref-6" id="linknoteref-6"><big>*</big></a>
+Let <IMG
+ WIDTH="21" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img1.png"
+ ALT="$A$">, <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img2.png"
+ ALT="$B$">, <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img3.png"
+ ALT="$C$"> be the moments of inertia about the
+principal axes through the centre of gravity, taken in order of
+magnitude, and let <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$"> <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img11.png"
+ ALT="$\omega_2$"> <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img12.png"
+ ALT="$\omega_3$"> be the angular
+velocities about them, then the angular momenta will be <IMG
+ WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img13.png"
+ ALT="$A\omega_1$">,
+<IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img14.png"
+ ALT="$B\omega_2$">, and <IMG
+ WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img15.png"
+ ALT="$C\omega_3$">.
+
+<P>
+Angular momenta may be compounded like forces or velocities, by the
+law of the &ldquo;parallelogram,&rdquo; and since these three are at right angles
+to each other, their resultant is
+<BR>
+<DIV ALIGN="RIGHT">
+
+<!-- MATH
+ \begin{equation}
+\sqrt{A^2\omega_1^2 + B^2\omega_2^2 + C^2\omega_3^2} = H
+\end{equation}
+ -->
+<TABLE WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
+ WIDTH="397" HEIGHT="36" BORDER="0"
+ SRC="images/img16.png"
+ ALT="\begin{displaymath}
+\sqrt{A^2\omega_1^2 + B^2\omega_2^2 + C^2\omega_3^2} = H
+\end{displaymath}"></TD>
+<TD WIDTH=10 ALIGN="RIGHT">
+(1)</TD></TR>
+</TABLE>
+<BR CLEAR="ALL"></DIV><P></P>
+and this must be constant, both in magnitude and direction in space,
+since no external forces act on the body.
+
+<P>
+We shall call this axis of angular momentum the <i>invariable
+axis</i>. It is perpendicular to what has been called the invariable
+plane. Poinsôt calls it the axis of the couple of impulsion. The
+<i>direction-cosines</i> of this axis in the body are,
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \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}
+ -->
+
+<IMG
+ WIDTH="415" HEIGHT="46" BORDER="0"
+ SRC="images/img17.png"
+ ALT="\begin{displaymath}
+\begin{array}{c c c}
+\displaystyle l = \frac{A\omega_1}{H}, ...
+...ga_2}{H}, &amp;
+\displaystyle n = \frac{C\omega_3}{H}.
+\end{array}\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+Since <IMG
+ WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img18.png"
+ ALT="$I$">, <IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$"> and <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$"> vary during the motion, we need some additional
+condition to determine the relation between them. We find this in the
+property of the <i>vis viva</i> of a system of invariable form in
+which there is no friction. The <i>vis viva</i> of such a system must
+be constant. We express this in the equation
+
+<P>
+<BR>
+<DIV ALIGN="RIGHT">
+
+<!-- MATH
+ \begin{equation}
+A\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V
+\end{equation}
+ -->
+<TABLE WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
+ WIDTH="377" HEIGHT="33" BORDER="0"
+ SRC="images/img21.png"
+ ALT="\begin{displaymath}
+A\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V
+\end{displaymath}"></TD>
+<TD WIDTH=10 ALIGN="RIGHT">
+(2)</TD></TR>
+</TABLE>
+<BR CLEAR="ALL"></DIV><P></P>
+
+<P>
+Substituting the values of <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$">, <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img11.png"
+ ALT="$\omega_2$">, <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img12.png"
+ ALT="$\omega_3$"> in terms
+of <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">, <IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$">, <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$">,
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+\frac{l^2}{A} + \frac{m^2}{B} + \frac{n^2}{C} = \frac{V}{H^2}.
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="367" HEIGHT="48" BORDER="0"
+ SRC="images/img23.png"
+ ALT="\begin{displaymath}
+\frac{l^2}{A} + \frac{m^2}{B} + \frac{n^2}{C} = \frac{V}{H^2}.
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+Let <IMG
+ WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img24.png"
+ ALT="$1/A = a^2$">, <IMG
+ WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img25.png"
+ ALT="$1/B = b^2$">, <IMG
+ WIDTH="74" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img26.png"
+ ALT="$1/c = c^2$">, <IMG
+ WIDTH="95" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img27.png"
+ ALT="$V/H^2 = e^2$">, and this
+equation becomes
+<BR>
+<DIV ALIGN="RIGHT">
+
+<!-- MATH
+ \begin{equation}
+a^2l^2 + b^2m^2 + c^2n^2 = e^2
+\end{equation}
+ -->
+<TABLE WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
+ WIDTH="376" HEIGHT="30" BORDER="0"
+ SRC="images/img28.png"
+ ALT="\begin{displaymath}
+a^2l^2 + b^2m^2 + c^2n^2 = e^2
+\end{displaymath}"></TD>
+<TD WIDTH=10 ALIGN="RIGHT">
+(3)</TD></TR>
+</TABLE>
+<BR CLEAR="ALL"></DIV><P></P>
+and the equation to the cone, described by the invariable axis within
+the body, is
+
+<P>
+<BR>
+<DIV ALIGN="RIGHT">
+
+<!-- MATH
+ \begin{equation}
+(a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2 = 0
+\end{equation}
+ -->
+<TABLE WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
+ WIDTH="453" HEIGHT="33" BORDER="0"
+ SRC="images/img29.png"
+ ALT="\begin{displaymath}
+(a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2 = 0
+\end{displaymath}"></TD>
+<TD WIDTH=10 ALIGN="RIGHT">
+(4)</TD></TR>
+</TABLE>
+<BR CLEAR="ALL"></DIV><P></P>
+
+<P>
+The intersections of this cone with planes perpendicular to the
+principal axes are found by putting <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, <IMG
+ WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img31.png"
+ ALT="$y$">, or <IMG
+ WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img32.png"
+ ALT="$z$">, constant in this
+equation. By giving <IMG
+ WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img33.png"
+ ALT="$e$"> various values, all the different paths of
+the pole of the invariable axis, corresponding to different initial
+circumstances, may be traced.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="63"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD><!-- MATH
+ $\includegraphics[width=\textwidth]{fig1.png}$
+ --><img SRC="images/fig1.png" alt="Figure 1">
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<a href="#linknote-7" name="linknoteref-7" id="linknoteref-7"><big>*</big></a>
+In the figures, I have supposed <IMG
+ WIDTH="77" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img36.png"
+ ALT="$a^2 = 100$">, <IMG
+ WIDTH="75" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img37.png"
+ ALT="$b^2=
+107$">, and <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img38.png"
+ ALT="$c^2= 110$">. The first figure represents a section of the
+various cones by a plane perpendicular to the axis of <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, which is
+that of greatest moment of inertia. These sections are ellipses
+having their major axis parallel to the axis of <IMG
+ WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img39.png"
+ ALT="$b$">. The value of
+<IMG
+ WIDTH="23" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img40.png"
+ ALT="$e^2$"> 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 <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img41.png"
+ ALT="$e^2 = 107$"> would be two
+parallel straight lines (beyond the bounds of the figure), after which
+the sections would be hyperbolas.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="67"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD><!-- MATH
+ $\includegraphics[width=\textwidth]{fig2.png}$
+ --><img SRC="images/fig2.png" alt="Figure 2">
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<a href="#linknote-8" name="linknoteref-8" id="linknoteref-8"><big>*</big></a>
+The second figure represents the sections made by a
+plane, perpendicular to the <i>mean</i> axis. They are all
+hyperbolas, except when <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img41.png"
+ ALT="$e^2 = 107$">, when the section is two
+intersecting straight lines.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="72"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD><!-- MATH
+ $\includegraphics[width=\textwidth]{fig3.png}$
+ --><img SRC="images/fig3.png" alt="Figure 3">
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+The third figure shows the sections perpendicular to the axis of least
+moment of inertia. From <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img45.png"
+ ALT="$e^2 = 110$"> to <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img41.png"
+ ALT="$e^2 = 107$"> the sections are
+ellipses, <IMG
+ WIDTH="76" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img41.png"
+ ALT="$e^2 = 107$"> gives two parallel straight lines, and beyond
+these the curves are hyperbolas.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="76"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD><!-- MATH
+ $\includegraphics[width=\textwidth]{fig4.png}$
+ --><img SRC="images/fig4.png" alt="Figure 4">
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<a href="#linknote-10" name="linknoteref-10" id="linknoteref-10"><big>*</big></a>
+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.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="80"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD><!-- MATH
+ $\includegraphics[width=\textwidth]{fig5.png}$
+ --><img SRC="images/fig5.png" alt="Figure 5">
+</TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+<a href="#linknote-11" name="linknoteref-11" id="linknoteref-11"><big>*</big></a>
+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 <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">,
+<IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$">, <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$">, about the axis of <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$"> is
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+\frac{\omega_1}{1 - l^2} -
+           \frac{l}{1 - l^2}(l\omega_1 + m\omega_2 + n\omega-3).
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="434" HEIGHT="48" BORDER="0"
+ SRC="images/img50.png"
+ ALT="\begin{displaymath}
+\frac{\omega_1}{1 - l^2} -
+\frac{l}{1 - l^2}(l\omega_1 + m\omega_2 + n\omega-3).
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+Substituting the values of <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$">, <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img11.png"
+ ALT="$\omega_2$">, <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img12.png"
+ ALT="$\omega_3$">, in
+terms of <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">, <IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$">, <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$">, and taking account of equation (3), this
+expression becomes
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+H\frac{(a^2 - e^2)}{1 - l^2}l.
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="332" HEIGHT="50" BORDER="0"
+ SRC="images/img51.png"
+ ALT="\begin{displaymath}
+H\frac{(a^2 - e^2)}{1 - l^2}l.
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+Changing the sign and putting <!-- MATH
+ $\displaystyle l =
+\frac{\omega_1}{a^2H}$
+ -->
+<IMG
+ WIDTH="76" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img52.png"
+ ALT="$\displaystyle l =
+\frac{\omega_1}{a^2H}$"> we have the angular velocity of the invariable
+axis about that of <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+= \frac{\omega_1}{1 - l^2} \frac{e^2 - a^2}{a^2},
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="348" HEIGHT="50" BORDER="0"
+ SRC="images/img53.png"
+ ALT="\begin{displaymath}
+= \frac{\omega_1}{1 - l^2} \frac{e^2 - a^2}{a^2},
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+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 <IMG
+ WIDTH="23" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img40.png"
+ ALT="$e^2$">. 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.
+
+<P>
+<a href="#linknote-12" name="linknoteref-12" id="linknoteref-12"><big>*</big></a>
+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 <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img55.png"
+ ALT="$yz$">, are swept
+out at the rate
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+\omega_1 \frac{e^2 - a^2}{a^2}.
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="323" HEIGHT="48" BORDER="0"
+ SRC="images/img56.png"
+ ALT="\begin{displaymath}
+\omega_1 \frac{e^2 - a^2}{a^2}.
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+Now the semi-axes of the projection of the spherical ellipse described
+by the pole are
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \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}
+ -->
+
+<IMG
+ WIDTH="420" HEIGHT="55" BORDER="0"
+ SRC="images/img57.png"
+ ALT="\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}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+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--
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+= \frac{a^2}{\sqrt{b^2 - a^2}\sqrt{c^2 - a^2}}.
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="367" HEIGHT="52" BORDER="0"
+ SRC="images/img58.png"
+ ALT="\begin{displaymath}
+= \frac{a^2}{\sqrt{b^2 - a^2}\sqrt{c^2 - a^2}}.
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+The projections of the spherical ellipses upon the plane of <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img55.png"
+ ALT="$yz$"> 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 <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, 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 <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, <IMG
+ WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img31.png"
+ ALT="$y$"> and <IMG
+ WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img32.png"
+ ALT="$z$">
+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.
+
+<P>
+<a href="#linknote-13" name="linknoteref-13" id="linknoteref-13"><big>*</big></a>
+When two of the axes have equal moments of inertia, or
+<IMG
+ WIDTH="48" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img60.png"
+ ALT="$b = c$">, then the angular velocity <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$"> 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
+
+<P>
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+\frac{a^2}{b^2 - a^2}
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="311" HEIGHT="50" BORDER="0"
+ SRC="images/img61.png"
+ ALT="\begin{displaymath}
+\frac{a^2}{b^2 - a^2}
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+
+<P>
+The motion is in the same direction as that of the rotation, or in the
+opposite direction, according as the axis of <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$"> is that of greatest
+or of least moment of inertia.
+
+<P>
+<a href="#linknote-14" name="linknoteref-14" id="linknoteref-14"><big>*</big></a>
+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 <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$"> about the invariable
+axis in space is
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+\omega_1\frac{e^2 - a^2l^2}{a^2(1 - l^2)},
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="334" HEIGHT="53" BORDER="0"
+ SRC="images/img63.png"
+ ALT="\begin{displaymath}
+\omega_1\frac{e^2 - a^2l^2}{a^2(1 - l^2)},
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+which is greater or less than <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$">, as <IMG
+ WIDTH="23" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img40.png"
+ ALT="$e^2$"> is greater or less
+than <IMG
+ WIDTH="24" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img64.png"
+ ALT="$a^2$">, and, when these quantities are nearly equal, is very
+nearly the same as <IMG
+ WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img4.png"
+ ALT="$\omega_1$"> itself. This quantity indicates the
+rate of revolution of the axle of the top about its mean position, and
+is very easily observed.
+
+<P>
+<a href="#linknote-15" name="linknoteref-15" id="linknoteref-15"><big>*</big></a>
+The <i>instantaneous axis</i> is not so easily
+observed. It revolves round the invariable axis in the same time with
+the axis of <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, at a distance which is very small in the case when
+<IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img66.png"
+ ALT="$a$">, <IMG
+ WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img39.png"
+ ALT="$b$">, <IMG
+ WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img67.png"
+ ALT="$c$">, 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.
+
+<P>
+<a href="#linknote-16" name="linknoteref-16" id="linknoteref-16"><big>*</big></a>
+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.
+
+<P>
+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 <i>see</i> 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.
+
+<P>
+
+<P></P>
+<DIV ALIGN="CENTER"><A NAME="132"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
+</CAPTION>
+<TR><TD>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ $\includegraphics[width=0.8\textwidth]{fig6.png}$
+ --><img SRC="images/fig6.jpg" alt="Figure 6">
+</DIV></TD></TR>
+</TABLE>
+</DIV><P></P>
+
+<P>
+The form of the body of the instrument which I have found most
+suitable is that of a bell (fig. 6). <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img3.png"
+ ALT="$C$"> is a hollow cone of brass,
+<IMG
+ WIDTH="21" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img70.png"
+ ALT="$R$"> is a heavy ring cast in the same piece. Six screws, with heavy
+heads, <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$">, <IMG
+ WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img31.png"
+ ALT="$y$">, <IMG
+ WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img32.png"
+ ALT="$z$">, <IMG
+ WIDTH="22" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img71.png"
+ ALT="$x'$">, <IMG
+ WIDTH="21" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img72.png"
+ ALT="$y'$">, <IMG
+ WIDTH="21" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img73.png"
+ ALT="$z'$">, work horizontally in the ring,
+and three similar screws, <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">, <IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$">, <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$">, work vertically through the
+ring at equal intervals. <IMG
+ WIDTH="33" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img74.png"
+ ALT="$AS$"> is the axle of the instrument, <IMG
+ WIDTH="32" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img75.png"
+ ALT="$SS$"> is
+a brass screw working in the upper part of the cone <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img3.png"
+ ALT="$C$">, and capable
+of being firmly clamped by means of the nut <IMG
+ WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img67.png"
+ ALT="$c$">. <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img2.png"
+ ALT="$B$"> is a cylindrical
+brass bob, which may be screwed up or down the axis, and fixed in its
+place by the nut <IMG
+ WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img39.png"
+ ALT="$b$">.
+
+<P>
+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 <IMG
+ WIDTH="21" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img76.png"
+ ALT="$P$">. 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.
+
+<P>
+When the instrument is in use, a coloured disc is attached to the
+upper end of the axle.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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.
+
+<P>
+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
+<i>circumference</i>. 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 <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">, <IMG
+ WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img19.png"
+ ALT="$m$">, <IMG
+ WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img20.png"
+ ALT="$n$">.
+
+<P>
+Suppose that the colour of the centre of motion, when farthest from
+the axle, indicated that the axis of rotation passed through the
+sector <IMG
+ WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img77.png"
+ ALT="$L$">, then the principal axis must also lie in that sector at
+half the distance from the axle.
+
+<P>
+If this principal axis be that of <i>greatest</i> moment of inertia,
+we must <i>raise</i> the screw <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$"> in order to bring it nearer the
+axle <IMG
+ WIDTH="21" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img1.png"
+ ALT="$A$">. If it be the axis of least moment we must <i>lower</i> the
+screw <IMG
+ WIDTH="13" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img22.png"
+ ALT="$l$">. 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.
+
+<P>
+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 <IMG
+ WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img77.png"
+ ALT="$L$">, <IMG
+ WIDTH="27" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img78.png"
+ ALT="$M$">, <IMG
+ WIDTH="24" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img79.png"
+ ALT="$N$">, the colours will appear in
+the same order.
+
+<P>
+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.
+
+<P>
+By screwing the bob up farther, we may make the axle the principal
+axis of <i>least</i> moment of inertia.
+
+<P>
+The motion of the instantaneous axis will then be that of the point of
+contact of the stick with the <i>outside</i> of the hoop rolling on
+it. The order of colours will be <IMG
+ WIDTH="24" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img79.png"
+ ALT="$N$">, <IMG
+ WIDTH="27" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img78.png"
+ ALT="$M$">, <IMG
+ WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img77.png"
+ ALT="$L$">, if the top be
+spinning in the direction <IMG
+ WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img77.png"
+ ALT="$L$">, <IMG
+ WIDTH="27" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img78.png"
+ ALT="$M$">, <IMG
+ WIDTH="24" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img79.png"
+ ALT="$N$">, 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.
+
+<P>
+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.
+
+<P>
+Now let us make the axes of inertia in the plane of the ring unequal.
+We may do this by screwing the balance screws <IMG
+ WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img30.png"
+ ALT="$x$"> and <IMG
+ WIDTH="25" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img80.png"
+ ALT="$x^1$"> farther
+from the axle without altering the centre of gravity.
+
+<P>
+Let us suppose the bob <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img2.png"
+ ALT="$B$"> screwed up so as to make the axle the axis
+of least inertia. Then the mean axis is parallel to <IMG
+ WIDTH="36" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img81.png"
+ ALT="$xx^1$">, and the
+greatest is at right angles to <IMG
+ WIDTH="36" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img81.png"
+ ALT="$xx^1$"> 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 <IMG
+ WIDTH="36" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img81.png"
+ ALT="$xx^1$">.
+
+<P>
+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.
+
+<P>
+When the bob is screwed still farther down, the axle becomes the axis
+of greatest inertia, and <IMG
+ WIDTH="36" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img81.png"
+ ALT="$xx^1$"> the least. The major axis of the
+ellipse described by the invariable axis will now be perpendicular to
+<IMG
+ WIDTH="36" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img81.png"
+ ALT="$xx^1$">, 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.
+
+<P>
+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
+<i>position</i> 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.
+
+<P>
+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.
+
+<H2>On the Earth&rsquo;s Motion</H2>
+
+<P>
+We must remember that these motions of a body about its centre of
+gravity, are <i>not</i> 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.
+
+<P>
+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 &ldquo;central ellipsoid;&rdquo; 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.
+
+<P>
+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 <!-- MATH
+ $\displaystyle = \omega\frac{C - A}{A}$
+ -->
+<IMG
+ WIDTH="95" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img82.png"
+ ALT="$\displaystyle = \omega\frac{C - A}{A}$"> where
+<IMG
+ WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img83.png"
+ ALT="$\omega$"> 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 <!-- MATH
+ $\displaystyle
+\frac{A}{C - A}$
+ -->
+<IMG
+ WIDTH="63" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img84.png"
+ ALT="$\displaystyle
+\frac{A}{C - A}$"> sidereal days, which appears to be about 325.6 solar
+days.
+
+<P>
+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 <i>invariable axis</i>, which is altered only by external
+forces, such as those of the sun and moon.
+
+<P>
+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.
+
+<P>
+In order to determine the existence of such a variation of latitude, I
+have examined the observations of <i>Polaris</i> 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.
+
+<P>
+In this way I found the apparent co-latitude of Greenwich for each
+month of the four years specified.
+
+<P>
+There appeared a very slight indication of a maximum belonging to the
+set of months,
+
+<P>
+<DIV ALIGN="CENTER">
+<TABLE CELLPADDING=3>
+<TR><TD ALIGN="CENTER">March, 51.</TD>
+<TD ALIGN="CENTER">Feb. 52.</TD>
+<TD ALIGN="CENTER">Dec. 52.</TD>
+<TD ALIGN="CENTER">Nov. 53.</TD>
+<TD ALIGN="CENTER">Sept. 54.</TD>
+</TR>
+</TABLE>
+</DIV>
+
+<P>
+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.
+
+<P>
+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&rsquo;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.
+
+<P>
+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.
+
+<H2>Note: Dimensions and Weights of the parts of the Dynamical Top.</H2>
+
+<P>
+<TABLE CELLPADDING=3 BORDER="1">
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>Part</TD>
+<TD ALIGN="CENTER" COLSPAN=2>Weight</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>&nbsp;</TD>
+<TD ALIGN="RIGHT">lb.</TD>
+<TD ALIGN="RIGHT">oz.</TD>
+</TR>
+<TR><TD ALIGN="LEFT" COLSPAN=3><B>I. Body of the top--</B></TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>Mean diameter of ring, 4 inches.</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>Section of ring, <IMG
+ WIDTH="17" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img85.png"
+ ALT="$\frac{1}{3}$"> inch square.</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>The conical portion rises from the upper and
+inner edge of the ring, a height of <IMG
+ WIDTH="27" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img86.png"
+ ALT="$1\frac{1}{2}$"> inches from the base.</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>The whole body of the top weighs</TD>
+<TD ALIGN="RIGHT">1</TD>
+<TD ALIGN="RIGHT">7</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>Each of the nine adjusting screws has its screw 1 inch long,
+and the screw and head together weigh 1 ounce.
+ The whole weigh</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT">9</TD>
+</TR>
+<TR><TD ALIGN="LEFT" COLSPAN=3><B>II. Axle, &amp;c.--</B></TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>Length of axle 5 inches, of which <IMG
+ WIDTH="17" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img87.png"
+ ALT="$\frac{1}{2}$"> inch at the
+bottom is occupied by the steel point, <IMG
+ WIDTH="27" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img88.png"
+ ALT="$3\frac{1}{2}$"> 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</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="27" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img86.png"
+ ALT="$1\frac{1}{2}$"></TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>The bob <IMG
+ WIDTH="22" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img2.png"
+ ALT="$B$"> has a diameter of 1.4 inches, and a
+thickness of .4. It weighs</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="27" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img89.png"
+ ALT="$2\frac{3}{4}$"></TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365>The nuts <IMG
+ WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img39.png"
+ ALT="$b$"> and <IMG
+ WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ SRC="images/img67.png"
+ ALT="$c$">, for clamping the bob and the body
+of the top on the axle, each weigh <IMG
+ WIDTH="17" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img87.png"
+ ALT="$\frac{1}{2}$"> oz.</TD>
+<TD ALIGN="RIGHT">&nbsp;</TD>
+<TD ALIGN="RIGHT">1</TD>
+</TR>
+<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=365><B>Weight of whole top</B></TD>
+<TD ALIGN="RIGHT">2</TD>
+<TD ALIGN="RIGHT"><IMG
+ WIDTH="27" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
+ SRC="images/img90.png"
+ ALT="$5\frac{1}{4}$"></TD>
+</TR>
+</TABLE>
+
+<P>
+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.
+
+<HR>
+
+<P>
+<a name="linknote-1" id="linknote-1"></a> <a href="#linknoteref-1"><big>*</big></a>
+<a name="linknote-2" id="linknote-2"></a> <a href="#linknoteref-2"></a>
+<a name="linknote-3" id="linknote-3"></a> <a href="#linknoteref-3"></a>
+<a name="linknote-4" id="linknote-4"></a> <a href="#linknoteref-4"></a>
+<a name="linknote-5" id="linknote-5"></a> <a href="#linknoteref-5"></a>
+<a name="linknote-6" id="linknote-6"></a> <a href="#linknoteref-6"></a>
+<a name="linknote-7" id="linknote-7"></a> <a href="#linknoteref-7"></a>
+<a name="linknote-8" id="linknote-8"></a> <a href="#linknoteref-8"></a>
+<a name="linknote-10" id="linknote-10"></a> <a href="#linknoteref-10"></a>
+<a name="linknote-11" id="linknote-11"></a> <a href="#linknoteref-11"></a>
+<a name="linknote-12" id="linknote-12"></a> <a href="#linknoteref-12"></a>
+<a name="linknote-13" id="linknote-13"></a> <a href="#linknoteref-13"></a>
+<a name="linknote-14" id="linknote-14"></a> <a href="#linknoteref-14"></a>
+<a name="linknote-15" id="linknote-15"></a> <a href="#linknoteref-15"></a>
+<a name="linknote-16" id="linknote-16"></a> <a href="#linknoteref-16"></a>
+7th May 1857. The paragraphs marked thus have been
+rewritten since the paper was read.
+</P>
+
+<div style='display:block;margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK ON A DYNAMICAL TOP ***</div>
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+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
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@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #5192 (https://www.gutenberg.org/ebooks/5192)
diff --git a/old/5192-pdf.pdf b/old/5192-pdf.pdf
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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...
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+The Project Gutenberg EBook of On a Dynamical Top, by James Clerk Maxwell
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+Title: On a Dynamical Top
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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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