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are ellipses having their major axis parallel to the axis of h. The value of e*
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 6^=107 would be two parallel straight lines (beyond the bounds
of the figure), after which the sections would be hyperbolas.



ON A DYNAMICAL TOP. 253

*The second figure represents the sections made by a plane, perpendicular
to the mean axis. They are all hyperbolas, except when 6^=107, when the
section is two intersecting straight lines.

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

*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.

*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 I, m, n, about the axis of x is

Substituting the values of w^, w^, w,, in terms of I, m, n, and taking
account of equation (3), this expression becomes

Changing the sign and putting 1=^tt we have the angular velocity of
the invariable axis about that of x

_ o>, e' — a"

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*. 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



254 ON A DYNAMICAL TOP.

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

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






Dividing the area of this ellipse by the area described during one revo-
lution of the body, we find the number of revolutions of the body during
the description of the ellipse —



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 there-
fore 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 h — c, then
the angular velocity (o^ 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



ON A DYNAMICAL TOP. 255

The motion is in the same direction as that of 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
dimlulshing the difference of the moments of inertia. The angular velocity of
the axis of x about the invariable axis in space is

to.



a'(l-l')'

which is greater or less than Wj, as e* is greater or less than a\ and, when
these quantities are nearly equal, is very nearly the same as Wj 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, h, 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.



256 ON A DYNAMICAL TOP.

The form of the body of the instrument which I have found most suitable is
that of a bell (p. 262, fig. 6). (7 is a hollow cone of brass, i2 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, I, 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 (7, and capable of being
firmly clamped by means of the nut c. 5 is a cylindrical brass bob, which may
be screwed up or down the axis, and fixed in its place by the nut 7).

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
pillai' 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.



ON A DYNAMICAL TOP. 257

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 I, in, 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 I in order to bring it nearer the axle A. If it be the axis of least
moment we must lower the screw /. 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 wiU 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 difierence 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



258 ON A DYNAinCAL TOP.

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 repre-
sents 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 instantaneous axis travels round the hoop, is in this case the same as
that in which the hoop moves round the stick, so that if the top be spinning in
the direction i, 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 Z, 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^ 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 xt^, and the greatest is at right
angles to xdd^ 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 xo^.

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 alto-
gether from the neighbourhood of the axle. When the top is in this condition



ON A DYNAMICAL TOP. 259

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 a:x^ the least. The major axis of the ellipse described by
the invariable axis will now be perpendicular to ccx", 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
■vvith 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 prac-
tised, or the first attempt at spinning rapidly may end in the destruction of
the top, if not of 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 gra-
vity, 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



260 ON A DYNAMICAL TOP.

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 velo-

C—A

city = 0) -— : — where w 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

A

round the true pole, completing its circuit in jy — ^ 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 would 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 obser-
vations of stars, gives the pole of the invan-aUe 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.



ON A DYNAMICAL TOP. 261

Tliis result, liowever, 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 essen-
tially 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 prin-
cipal 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.



262 ON A DYNAMICAL TOP.

NOTK

Dimensions and Weights of the parts of the Dynamical Top.

I. Body of the top —

Mean diameter of ring, 4 inches.
Section of ring, | inch square.

The conical portion rises from the upper and inner edge of the ring, a
height of 1| inches from the base.

The whole body of the top weighs 1 lb. 7 oz.

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 | inch at the bottom is occupied by
the steel point, 3J 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^ „

The bob B has a diameter of 1'4< inches, and a thickness of •4. It weighs 2| „

The nuts b and c, for clamping the bob and the body of the top on the

axle, each weigh ^ oz. 1 „



Weight of whole top 2 lb. 5J oz.

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 revolution 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.



VOL. I . PLATE III.



riG 1



FIG. 2




PIG 4-



VOL. I . PLATE III




riG 6



[From the Philosophical Magazine, Vol. xiv.]

XVI. Account of Experiments on the Perception of Colour.

To the Editors of the Philosophical Magazine and Journal.

Gentlemen,

The experiments which I intend to describe were undertaken in order
to render more perfect the quantitative proof of the theory of three primary
colours. According to that theory, every sensation of colour in a perfect human
eye is distinguished by three, and only three, elementary qualities, so that in
mathematical language the quahty of a colour may be expressed as a function
of three independent variables. There is very little evidence at present for
deciding the precise tints of the true primaries. I have ascertained that a
certain red is the sensation wanting in colour-blind eyes, but the mathematical
theory relates to the number, not to the nature of the primaries. If, with Sir
David Brewster, we assume red, blue, and yellow to be the primary colours, this
amounts to saying that every conceivable tint may be produced by adding
together so much red, so much yellow, and so much blue. This is perhaps the
best method of forming a provisional notion of the theory. It is evident that if
any colour could be found which could not be accurately defined as so much of
each of the three primaries, the theory would fall to the ground. Besides this,
the truth of the theory requires that every mathematical consequence of assu m i n g
every colour to be the result of mixture of three primaries should also be true.

I have made experiments on upwards of 100 diiferent artificial colours, con-
sisting of the pigments used in the arts, and their mechanical mixtures. These
experiments were made primarily to trace the effects of mechanical mixture on
various coloured powders ; but they also afford evidence of the truth of the
theory, that all these various colours can be referred to three primaries. The



264 EXPEKIMENTS ON THE PERCEPTION OF COLOUR.



Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 23 of 50)