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diff --git a/5192-0.txt b/5192-0.txt new file mode 100644 index 0000000..4b29648 --- /dev/null +++ b/5192-0.txt @@ -0,0 +1,1113 @@ +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. 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