We must remember that these motions of a body about its centre of
gravity, are _not_ illustrations of the theory of the precession of the
Equinoxes. Precession can be illustrated by the apparatus, but we must
arrange it so that the force of gravity acts the part of the attraction
of the sun and moon in producing a force tending to alter the axis of
rotation. This is easily done by bringing the centre of gravity of the
whole a little below the point on which it spins. The theory of such
motions is far more easily comprehended than that which we have been
investigating.
But the earth is a body whose principal axes are unequal, and from the
phenomena of precession we can determine the ratio of the polar and
equatorial axes of the “central ellipsoid;” and supposing the earth to
have been set in motion about any axis except the principal axis, or to
have had its original axis disturbed in any way, its subsequent motion
would be that of the top when the bob is a little below the critical
position.
The axis of angular momentum would have an invariable position in
space, and would travel with respect to the earth round the axis of
figure with a velocity $\displaystyle = \omega\frac{C - A}{A}$ where
$\omega$ is the sidereal angular velocity of the earth. The apparent
pole of the earth would travel (with respect to the earth) from west to
east round the true pole, completing its circuit in $\displaystyle
\frac{A}{C - A}$ sidereal days, which appears to be about 325.6 solar
days.
The instantaneous axis would revolve about this axis in space in about
a day, and would always be in a plane with the true axis of the earth
and the axis of angular momentum. The effect of such a motion on the
apparent position of a star would be, that its zenith distance should
be increased and diminished during a period of 325.6 days. This
alteration of zenith distance is the same above and below the pole, so
that the polar distance of the star is unaltered. In fact the method of
finding the pole of the heavens by observations of stars, gives the
pole of the _invariable axis_, which is altered only by external
forces, such as those of the sun and moon.
There is therefore no change in the apparent polar distance of stars
due to this cause. It is the latitude which varies. The magnitude of
this variation cannot be determined by theory. The periodic time of the
variation may be found approximately from the known dynamical
properties of the earth. The epoch of maximum latitude cannot be found
except by observation, but it must be later in proportion to the east
longitude of the observatory.
In order to determine the existence of such a variation of latitude, I
have examined the observations of _Polaris_ with the Greenwich Transit
Circle in the years 1851-2-3-4. The observations of the upper transit
during each month were collected, and the mean of each month found. The
same was done for the lower transits. The difference of zenith distance
of upper and lower transit is twice the polar distance of Polaris, and
half the sum gives the co-latitude of Greenwich.
In this way I found the apparent co-latitude of Greenwich for each
month of the four years specified.
There appeared a very slight indication of a maximum belonging to the
set of months,
March, 51. Feb. 52. Dec. 52. Nov. 53. Sept. 54.
This result, however, is to be regarded as very doubtful, as there did
not appear to be evidence for any variation exceeding half a second of
space, and more observations would be required to establish the
existence of so small a variation at all.
I therefore conclude that the earth has been for a long time revolving
about an axis very near to the axis of figure, if not coinciding with
it. The cause of this near coincidence is either the original softness
of the earth, or the present fluidity of its interior. The axes of the
earth are so nearly equal, that a considerable elevation of a tract of
country might produce a deviation of the principal axis within the
limits of observation, and the only cause which would restore the
uniform motion, would be the action of a fluid which would gradually
diminish the oscillations of latitude. The permanence of latitude
essentially depends on the inequality of the earth’s axes, for if they
had been all equal, any alteration of the crust of the earth would have
produced new principal axes, and the axis of rotation would travel
about those axes, altering the latitudes of all places, and yet not in
the least altering the position of the axis of rotation among the
stars.
Perhaps by a more extensive search and analysis of the observations of
different observatories, the nature of the periodic variation of
latitude, if it exist, may be determined. I am not aware of any
calculations having been made to prove its non-existence, although, on
dynamical grounds, we have every reason to look for some very small
variation having the periodic time of 325.6 days nearly, a period which
is clearly distinguished from any other astronomical cycle, and
therefore easily recognised.
Note: Dimensions and Weights of the parts of the Dynamical Top.
Part Weight lb. oz. I. Body of the top-- Mean diameter of ring, 4
inches. Section of ring, $\frac{1}{3}$ inch square. The conical portion
rises from the upper and inner edge of the ring, a height of
$1\frac{1}{2}$ inches from the base. The whole body of the top
weighs 1 7 Each of the nine adjusting screws has its screw 1 inch
long, and the screw and head together weigh 1 ounce. The whole
weigh 9 II. Axle, &c.-- Length of axle 5 inches, of which
$\frac{1}{2}$ inch at the bottom is occupied by the steel point,
$3\frac{1}{2}$ inches are brass with a good screw turned on it, and the
remaining inch is of steel, with a sharp point at the top. The whole
weighs $1\frac{1}{2}$ The bob $B$ has a diameter of 1.4 inches,
and a thickness of .4. It weighs $2\frac{3}{4}$ The nuts $b$
and $c$, for clamping the bob and the body of the top on the axle, each
weigh $\frac{1}{2}$ oz. 1 Weight of whole
top 2 $5\frac{1}{4}$
The best arrangement, for general observations, is to have the disc of
card divided into four quadrants, coloured with vermilion, chrome
yellow, emerald green, and ultramarine. These are bright colours, and,
if the vermilion is good, they combine into a grayish tint when the
rotation is about the axle, and burst into brilliant colours when the
axis is disturbed. It is useful to have some concentric circles, drawn
with ink, over the colours, and about 12 radii drawn in strong pencil
lines. It is easy to distinguish the ink from the pencil lines, as they
cross the invariable axis, by their want of lustre. In this way, the
path of the invariable axis may be identified with great accuracy, and
compared with theory.
* 7th May 1857. The paragraphs marked thus have been rewritten since the
paper was read.