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diff --git a/5192-h/5192-h.htm b/5192-h/5192-h.htm new file mode 100644 index 0000000..0f3ce2e --- /dev/null +++ b/5192-h/5192-h.htm @@ -0,0 +1,2067 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> +<html> +<head> +<meta http-equiv="content-type" content="text/html; charset=utf-8"> +<meta http-equiv="Content-Style-Type" content="text/css"> +<title>The Project Gutenberg eBook of On a Dynamical Top, by James Clerk Maxwell</title> + +<style type="text/css"> + +body { margin-left: 20%; + margin-right: 20%; + text-align: justify; } + +h1, h2, h3, h4, h5 {text-align: center; font-style: normal; font-weight: +normal; line-height: 1.5; margin-top: .5em; margin-bottom: .5em;} + +h1 {font-size: 300%; + margin-top: 0.6em; + margin-bottom: 0.6em; + letter-spacing: 0.12em; + word-spacing: 0.2em; + text-indent: 0em;} +h2 {font-size: 150%; margin-top: 2em; margin-bottom: 1em;} +h3 {font-size: 150%;} +h4 {font-size: 120%;} +h5 {font-size: 110%;} + +/* mathematics styles */ +div.displaymath { } /* math displays */ + +.math { font-family: "Century Schoolbook", serif; } + +a:link {color:blue; text-decoration:none} +a:visited {color:blue; text-decoration:none} +a:hover {color:red} + +</style> + +</head> + +<body > + +<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’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’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’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. + +<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’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. + +<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’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 “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. + +<P> +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.” +(<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 “parallelogram,” 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}, & +\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’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 “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. + +<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’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> </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"> </TD> +<TD ALIGN="RIGHT"> </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"> </TD> +<TD ALIGN="RIGHT"> </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"> </TD> +<TD ALIGN="RIGHT"> </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"> </TD> +<TD ALIGN="RIGHT">9</TD> +</TR> +<TR><TD ALIGN="LEFT" COLSPAN=3><B>II. Axle, &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"> </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"> </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"> </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. 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