Perturbations, Periodic and Secular—Disturbing Action equivalent to
three Partial Forces—Tangential Force the cause of the Periodic
Inequalities in Longitude, and Secular Inequalities in the Form and
Position of the Orbit in its own Plane—Radial Force the cause of
Variations in the Planet’s Distance from the Sun—It combines with the
Tangential Force to produce the Secular Variations in the Form and
Position of the Orbit in its own Plane—Perpendicular Force the cause
of Periodic Perturbations in Latitude, and Secular Variations in the
Position of the Orbit with regard to the Plane of the Ecliptic—Mean
Motion and Major Axis Invariable—Stability of System—Effects of a
Resisting Medium—Invariable Plane of the Solar System and of the
Universe—Great Inequality of Jupiter and Saturn.
THE planets are subject to disturbances of two kinds, both resulting
from the constant operation of their reciprocal attraction: one kind,
depending upon their positions with regard to each other, begins from
zero, increases to a maximum, decreases, and becomes zero again, when
the planets return to the same relative positions. In consequence of
these, the disturbed planet is sometimes drawn away from the sun,
sometimes brought nearer to him: sometimes it is accelerated in its
motion, and sometimes retarded. At one time it is drawn above the plane
of its orbit, at another time below it, according to the position of the
disturbing body. All such changes, being accomplished in short periods,
some in a few months, others in years, or in hundreds of years, are
denominated periodic inequalities. The inequalities of the other kind,
though occasioned likewise by the disturbing energy of the planets, are
entirely independent of their relative positions. They depend upon the
relative positions of the orbits alone, whose forms and places in space
are altered by very minute quantities, in immense periods of time, and
are therefore called secular inequalities.
The periodical perturbations are compensated when the bodies return to
the same relative positions with regard to one another and to the sun:
the secular inequalities are compensated when the orbits return to the
same positions relatively to one another and to the plane of the
ecliptic.
Planetary motion, including both these kinds of disturbance, may be
represented by a body revolving in an ellipse, and making small and
transient deviations, now on one side of its path, and now on the other,
whilst the ellipse itself is slowly, but perpetually, changing both in
form and position.
The periodic inequalities are merely transient deviations of a planet
from its path, the most remarkable of which only lasts about 918 years;
but, in consequence of the secular disturbances, the apsides, or
extremities of the major axes of all the orbits, have a direct but
variable motion in space, excepting those of the orbit of Venus, which
are retrograde (N. 61), and the lines of the nodes move with a variable
velocity in a contrary direction. Besides these, the inclination and
excentricity of every orbit are in a state of perpetual but slow change.
These effects result from the disturbing action of all the planets on
each. But, as it is only necessary to estimate the disturbing influence
of one body at a time, what follows may convey some idea of the manner
in which one planet disturbs the elliptical motion of another.
Suppose two planets moving in ellipses round the sun; if one of them
attracted the other and the sun with equal intensity, and in parallel
directions (N. 62), it would have no effect in disturbing the elliptical
motion. The inequality of this attraction is the sole cause of
perturbation, and the difference between the disturbing planet’s action
on the sun and on the disturbed planet constitutes the disturbing force,
which consequently varies in intensity and direction with every change
in the relative positions of the three bodies. Although both the sun and
planet are under the influence of the disturbing force, the motion of
the disturbed planet is referred to the centre of the sun as a fixed
point, for convenience. The whole force (N. 63) which disturbs a planet
is equivalent to three partial forces. One of these acts on the
disturbed planet, in the direction of a tangent to its orbit, and is
called the tangential force: it occasions secular inequalities in the
form and position of the orbit in its own plane, and is the sole cause
of the periodical perturbations in the planet’s longitude. Another acts
upon the same body in the direction of its radius vector, that is, in
the line joining the centres of the sun and planet, and is called the
radial force: it produces periodical changes in the distance of the
planet from the sun, and affects the form and position of the orbit in
its own plane. The third, which may be called the perpendicular force,
acts at right angles to the plane of the orbit, occasions the periodic
inequalities in the planet’s latitude, and affects the position of the
orbit with regard to the plane of the ecliptic.
It has been observed, that the radius vector of a planet, moving in a
perfectly elliptical orbit, passes over equal spaces or areas in equal
times; a circumstance which is independent of the law of the force, and
would be the same whether it varied inversely as the square of the
distance, or not, provided only that it be directed to the centre of the
sun. Hence the tangential force, not being directed to the centre,
occasions an unequable description of areas, or, what is the same thing,
it disturbs the motion of the planet in longitude. The tangential force
sometimes accelerates the planet’s motion, sometimes retards it, and
occasionally has no effect at all. Were the orbits of both planets
circular, a complete compensation would take place at each revolution of
the two planets, because the arcs in which the accelerations and
retardations take place would be symmetrical on each side of the
disturbing force. For it is clear, that if the motion be accelerated
through a certain space, and then retarded through as much, the motion
at the end of the time will be the same as if no change had taken place.
But, as the orbits of the planets are ellipses, this symmetry does not
hold: for, as the planet moves unequably in its orbit, it is in some
positions more directly, and for a longer time, under the influence of
the disturbing force than in others. And, although multitudes of
variations do compensate each other in short periods, there are others,
depending on peculiar relations among the periodic times of the planets,
which do not compensate each other till after one, or even till after
many revolutions of both bodies. A periodical inequality of this kind in
the motions of Jupiter and Saturn has a period of no less than 918
years.
The radial force, or that part of the disturbing force which acts in the
direction of the line joining the centres of the sun and disturbed
planet, has no effect on the areas, but is the cause of periodical
changes of small extent in the distance of the planet from the sun. It
has already been shown, that the force producing perfectly elliptical
motion varies inversely as the square of the distance, and that a force
following any other law would cause the body to move in a curve of a
very different kind. Now, the radial disturbing force varies directly as
the distance; and, as it sometimes combines with and increases the
intensity of the sun’s attraction for the disturbed body, and at other
times opposes and consequently diminishes it, in both cases it causes
the sun’s attraction to deviate from the exact law of gravity, and the
whole action of this compound central force on the disturbed body is
either greater or less than what is requisite for perfectly elliptical
motion. When greater, the curvature of the disturbed planet’s path, on
leaving its perihelion (N. 64), or point nearest the sun, is greater
than it would be in the ellipse, which brings the planet to its aphelion
(N. 65), or point farthest from the sun, before it has passed through
180°, as it would do if undisturbed. So that in this case the apsides,
or extremities of the major axis, advance in space. When the central
force is less than the law of gravity requires, the curvature of the
planet’s path is less than the curvature of the ellipse. So that the
planet, on leaving its perihelion, would pass through more than 180°
before arriving at its aphelion, which causes the apsides to recede in
space (N. 66). Cases both of advance and recess occur during a
revolution of the two planets; but those in which the apsides advance
preponderate. This, however, is not the full amount of the motion of the
apsides; part arises also from the tangential force (N. 63), which
alternately accelerates and retards the velocity of the disturbed
planet. An increase in the planet’s tangential velocity diminishes the
curvature of its orbit, and is equivalent to a decrease of central
force. On the contrary, a decrease of the tangential velocity, which
increases the curvature of the orbit, is equivalent to an increase of
central force. These fluctuations, owing to the tangential force,
occasion an alternate recess and advance of the apsides, after the
manner already explained (N. 66). An uncompensated portion of the direct
motion, arising from this cause, conspires with that already impressed
by the radial force, and in some cases even nearly doubles the direct
motion of these points. The motion of the apsides may be represented by
supposing a planet to move in an ellipse, while the ellipse itself is
slowly revolving about the sun in the same plane (N. 67). This motion of
the major axis, which is direct in all the orbits except that of the
planet Venus, is irregular, and so slow that it requires more than
109,830 years for the major axis of the earth’s orbit to accomplish a
sidereal revolution (N. 68), that is, to return to the same stars; and
20,984 years to complete its tropical revolution (N. 69), or to return
to the same equinox. The difference between these two periods arises
from a retrograde motion in the equinoctial point, which meets the
advancing axis before it has completed its revolution with regard to the
stars. The major axis of Jupiter’s orbit requires no less than 200,610
years to perform its sidereal revolution, and 22,748 years to accomplish
its tropical revolution from the disturbing action of Saturn alone.
A variation in the excentricity of the disturbed planet’s orbit is an
immediate consequence of the deviation from elliptical curvature, caused
by the action of the disturbing force. When the path of the body, in
proceeding from its perihelion to its aphelion, is more curved than it
ought to be from the effect of the disturbing forces, it falls within
the elliptical orbit, the excentricity is diminished, and the orbit
becomes more nearly circular; when that curvature is less than it ought
to be, the path of the planet falls without its elliptical orbit
(N. 66), and the excentricity is increased; during these changes, the
length of the major axis is not altered, the orbit only bulges out, or
becomes more flat (N. 70). Thus the variation in the excentricity arises
from the same cause that occasions the motion of the apsides (N. 67).
There is an inseparable connection between these two elements: they vary
simultaneously, and have the same period; so that, whilst the major axis
revolves in an immense period of time, the excentricity increases and
decreases by very small quantities, and at length returns to its
original magnitude at each revolution of the apsides. The terrestrial
excentricity is decreasing at the rate of about 40 miles annually; and,
if it were to decrease equably, it would be 39,861 years before the
earth’s orbit became a circle. The mutual action of Jupiter and Saturn
occasions variations in the excentricity of both orbits, the greatest
excentricity of Jupiter’s orbit corresponding to the least of Saturn’s.
The period in which these vicissitudes are accomplished is 70,414 years,
estimating the action of these two planets alone; but, if the action of
all the planets were estimated, the cycle would extend to millions of
years.
That part of the disturbing force is now to be considered which acts
perpendicularly to the plane of the orbit, causing periodic
perturbations in latitude, secular variations in the inclination of the
orbit, and a retrograde motion to its nodes on the true plane of the
ecliptic (N. 71). This force tends to pull the disturbed body above, or
push (N. 72) it below, the plane of its orbit, according to the relative
positions of the two planets with regard to the sun, considered to be
fixed. By this action, it sometimes makes the plane of the orbit of the
disturbed body tend to coincide with the plane of the ecliptic, and
sometimes increases its inclination to that plane. In consequence of
which, its nodes alternately recede or advance on the ecliptic (N. 73).
When the disturbing planet is in the line of the disturbed planet’s
nodes (N. 74), it neither affects these points, the latitude, nor the
inclination, because both planets are then in the same plane. When it is
at right angles to the line of the nodes, and the orbit symmetrical on
each side of the disturbing force, the average motion of these points,
after a revolution of the disturbed body, is retrograde, and
comparatively rapid: but, when the disturbing planet is so situated that
the orbit of the disturbed planet is not symmetrical on each side of the
disturbing force, which is most frequently the case, every possible
variety of action takes place. Consequently, the nodes are perpetually
advancing or receding with unequal velocity; but, as a compensation is
not effected, their motion is, on the whole, retrograde.
With regard to the variations in the inclination, it is clear, that,
when the orbit is symmetrical on each side of the disturbing force, all
its variations are compensated after a revolution of the disturbed body,
and are merely periodical perturbations in the planet’s latitude; and no
secular change is induced in the inclination of the orbit. When, on the
contrary, that orbit is not symmetrical on each side of the disturbing
force, although many of the variations in latitude are transient or
periodical, still, after a complete revolution of the disturbed body, a
portion remains uncompensated, which forms a secular change in the
inclination of the orbit to the plane of the ecliptic. It is true, part
of this secular change in the inclination is compensated by the
revolution of the disturbing body, whose motion has not hitherto been
taken into the account, so that perturbation compensates perturbation;
but still a comparatively permanent change is effected in the
inclination, which is not compensated till the nodes have accomplished a
complete revolution.
The changes in the inclination are extremely minute (N. 75), compared
with the motion of the nodes, and there is the same kind of inseparable
connection between their secular changes that there is between the
variation of the excentricity and the motion of the major axis. The
nodes and inclinations vary simultaneously; their periods are the same,
and very great. The nodes of Jupiter’s orbit, from the action of Saturn
alone, require 36,261 years to accomplish even a tropical revolution. In
what precedes, the influence of only one disturbing body has been
considered; but, when the action and reaction of the whole system are
taken into account, every planet is acted upon, and does itself act, in
this manner, on all the others; and the joint effect keeps the
inclinations and excentricities in a state of perpetual variation. It
makes the major axes of all the orbits continually revolve, and causes,
on an average, a retrograde motion of the nodes of each orbit upon every
other. The ecliptic (N. 71) itself is in motion from the mutual action
of the earth and planets, so that the whole is a compound phenomenon of
great complexity, extending through unknown ages. At the present time
the inclinations of all the orbits are decreasing, but so slowly, that
the inclination of Jupiter’s orbit is only about six minutes less than
it was in the age of Ptolemy.
But, in the midst of all these vicissitudes, the length of the major
axes and the mean motions of the planets remain permanently independent
of secular changes. They are so connected by Kepler’s law, of the
squares of the periodic times being proportional to the cubes of the
mean distances of the planets from the sun, that one cannot vary without
affecting the other. And it is proved, that any variations which do take
place are transient, and depend only on the relative positions of the
bodies.
It is true that, according to theory, the radial disturbing force should
permanently alter the dimensions of all the orbits, and the periodic
times of all the planets, to a certain degree. For example, the masses
of all the planets revolving within the orbit of any one, such as Mars,
by adding to the interior mass, increase the attracting force of the
sun, which, therefore, must contract the dimensions of the orbit of that
planet, and diminish its periodic time; whilst the planets exterior to
Mars’s orbit must have the contrary effect. But the mass of the whole of
the planets and satellites taken together is so small, when compared
with that of the sun, that these effects are quite insensible, and could
only have been discovered by theory. And, as it is certain that the
length of the major axes and the mean motions are not permanently
changed by any other power whatever, it may be concluded that they are
invariable.
With the exception of these two elements, it appears that all the bodies
are in motion, and every orbit in a state of perpetual change. Minute as
these changes are, they might be supposed to accumulate in the course of
ages, sufficiently to derange the whole order of nature, to alter the
relative positions of the planets, to put an end to the vicissitudes of
the seasons, and to bring about collisions which would involve our whole
system, now so harmonious, in chaotic confusion. It is natural to
inquire, what proof exists that nature will be preserved from such a
catastrophe? Nothing can be known from observation, since the existence
of the human race has occupied comparatively but a point in duration,
while these vicissitudes embrace myriads of ages. The proof is simple
and conclusive. All the variations of the solar system, secular as well
as periodic, are expressed analytically by the sines and cosines of
circular arcs (N. 76), which increase with the time; and, as a sine or
cosine can never exceed the radius, but must oscillate between zero and
unity, however much the time may increase, it follows that when the
variations have accumulated to a maximum by slow changes, in however
long a time, they decrease, by the same slow degrees, till they arrive
at their smallest value, again to begin a new course; thus for ever
oscillating about a mean value. This circumstance, however, would be
insufficient, were it not for the small excentricities of the planetary
orbits, their minute inclinations to the plane of the ecliptic, and the
revolutions of all the bodies, as well planets as satellites, in the
same direction. These secure the perpetual stability of the solar system
(N. 77). However, at the time that the stability was proved by La Grange
and La Place, the telescopic planets between Mars and Jupiter had not
been discovered; but La Grange, having investigated the subject under a
very general point of view, showed that, if a planetary system be
composed of very unequal masses, the whole of the larger would maintain
an unalterable stability with regard to the form and position of their
orbits, while the orbits of the lesser might undergo unlimited changes.
M. Le Verrier has applied this to the solar system, and has found that
the orbits of all the larger planets will for ever maintain an
unalterable stability in form and position; for, though liable to
mutations of very long periods, they return again exactly to what they
originally were, oscillating between very narrow limits; but he found a
zone of instability between the orbit of Mars, and twice the mean
distance of the earth from the sun,[1] or between 1·5 and 2·00;
therefore the position and form of the orbits of such of the telescopic
planets as revolve within that zone will be subject to unlimited
variations. But the orbits of those more remote from the sun than Flora,
or beyond 2·20, will be stable, so that their excentricities and
inclinations must always have been, and will always remain, very great,
since they must have depended upon the primitive conditions that
prevailed when these planetary atoms were launched into space. The 51st
of these small bodies, which was discovered, and the elements of its
orbit determined, by M. Valz, at Nimes, has a mean distance of 1·88; so
it revolves within the zone of instability. It has a shorter periodic
time than any of those previously discovered, and a greater
excentricity, with the exception of Nysa. Its orbit cuts that of Mars,
and comes nearer to the earth than the orbits of either Mars or Venus, a
circumstance which would be favourable for correcting the parallax of
the sun, or confirming its accuracy. The telescopic planets, numerous as
they are, have no influence on the motions of the larger planets, for
Jupiter has a diameter of 90,734 miles, while that of Pallas, his
nearest neighbour, is only 97 miles, little more than the distance from
London to Bath. The diameter of Mars, on the other side of the small
planets, is 4546 miles, and that of the earth 7925-1/2 miles, so that
the telescopic group are too minute to disturb the others. M. Le Verrier
found another zone of instability between Venus and the sun, on the
border of which Mercury is revolving, the inclination of whose orbit to
the plane of the ecliptic is about 7°, which is more than that of any of
the large planets. Neptune’s orbit is, no doubt, as stable as that of
any other of the large planets, as the inclination is very small, but he
will have periodical variations of very long duration from the
reciprocal attraction between him and Uranus, one especially of an
enormous duration, similar to those of Jupiter and Saturn, and, like
them, depending on the time of his revolution round the sun, being
nearly twice as long as that of Saturn. Mr. Adams has computed that
Neptune produces a periodical perturbation in the motion of Uranus,
whose duration is about 6800 years.
The equilibrium of the system, however, would be deranged if the planets
moved in a resisting medium (N. 78) sufficiently dense to diminish their
tangential velocity, for then both the excentricities and the major axes
of the orbits would vary with the time, so that the stability of the
system would be ultimately destroyed. The existence of an ethereal
medium is now proved; and, although it is so extremely rare that
hitherto its effects on the motions of the planets have been altogether
insensible, there can be no doubt that, in the immensity of time, it
will modify the forms of the planetary orbits, and may at last even
cause the destruction of our system, which in itself contains no
principle of decay, unless a rotatory motion from west to east has been
given to this medium by the bodies of the solar system, which have all
been revolving about the sun in that direction for unknown ages. This
rotation, which seems to be highly probable, may even have been coeval
with its creation. Such a vortex would have no effect on bodies moving
with it, but it would influence the motions of those revolving in a
contrary direction. It is possible that the disturbances experienced by
comets, which have already revealed the existence of this medium, may
also, in time, disclose its rotatory motion.
The form and position of the planetary orbits, and the motion of the
bodies in the same direction, together with the periodicity of the terms
in which the inequalities are expressed, assure us that the variations
of the system are confined within very narrow limits, and that, although
we do not know the extent of the limits, nor the period of that grand
cycle which probably embraces millions of years, yet they never will
exceed what is requisite for the stability and harmony of the whole; for
the preservation of which every circumstance is so beautifully and
wonderfully adapted.
The plane of the ecliptic itself, though assumed to be fixed at a given
epoch for the convenience of astronomical computation, is subject to a
minute secular variation of 45ʺ·7, occasioned by the reciprocal action
of the planets. But, as this is also periodical, and cannot exceed 2°
42ʹ, the terrestrial equator, which is inclined to it at an angle[2] of
23° 27ʹ 28ʺ·29, will never coincide with the plane of the ecliptic: so
there never can be perpetual spring (N. 79). The rotation of the earth
is uniform; therefore day and night, summer and winter, will continue
their vicissitudes while the system endures, or is undisturbed by
foreign causes.
Yonder starry sphere
Of planets and of fix’d, in all her wheels,
Resembles nearest mazes intricate,
Eccentric, intervolved, yet regular,
Then most, when most irregular they seem.
The stability of our system was established by La Grange: “a discovery,”
says Professor Playfair, “that must render the name for ever memorable
in science, and revered by those who delight in the contemplation of
whatever is excellent and sublime.” After Newton’s discovery of the
mechanical laws of the elliptical orbits of the planets, that of their
periodical inequalities, by La Grange, is, without doubt, the noblest
truth in the mechanism of the heavens; and, in respect of the doctrine
of final causes, it may be regarded as the greatest of all.
Notwithstanding the permanency of our system, the secular variations in
the planetary orbits would have been extremely embarrassing to
astronomers when it became necessary to compare observations separated
by long periods. The difficulty was in part obviated, and the principle
for accomplishing it established, by La Place, and has since been
extended by M. Poinsot. It appears that there exists an invariable plane
(N. 80), passing through the centre of gravity of the system, about
which the whole oscillates within very narrow limits, and that this
plane will always remain parallel to itself, whatever changes time may
induce in the orbits of the planets, in the plane of the ecliptic, or
even in the law of gravitation; provided only that our system remains
unconnected with any other. The position of the plane is determined by
this property—that, if each particle in the system be multiplied by the
area described upon this plane in a given time, by the projection of its
radius vector about the common centre of gravity of the whole, the sum
of all these products will be a maximum (N. 81). La Place found that the
plane in question is inclined to the ecliptic at an angle of nearly 1°
34ʹ 15ʺ, and that, in passing through the sun, and about midway between
the orbits of Jupiter and Saturn, it may be regarded as the equator of
the solar system, dividing it into two parts, which balance one another
in all their motions. This plane of greatest inertia, by no means
peculiar to the solar system, but existing in every system of bodies
submitted to their mutual attractions only, always maintains a fixed
position, whence the oscillations of the system may be estimated through
unlimited time. Future astronomers will know, from its immutability or
variation, whether the sun and his attendants are connected or not with
the other systems of the universe. Should there be no link between them,
it may be inferred, from the rotation of the sun, that the centre of
gravity (N. 82) of the system situate within his mass describes a
straight line in this invariable plane or great equator of the solar
system, which, unaffected by the changes of time, will maintain its
stability through endless ages. But, if the fixed stars, comets, or any
unknown and unseen bodies, affect our sun and planets, the nodes of this
plane will slowly recede on the plane of that immense orbit which the
sun may describe about some most distant centre, in a period which it
transcends the power of man to determine. There is every reason to
believe that this is the case; for it is more than probable that, remote
as the fixed stars are, they in some degree influence our system, and
that even the invariability of this plane is relative, only appearing
fixed to creatures incapable of estimating its minute and slow changes
during the small extent of time and space granted to the human race.
“The development of such changes,” as M. Poinsot justly observes, “is
similar to an enormous curve, of which we see so small an arc that we
imagine it to be a straight line.” If we raise our views to the whole
extent of the universe, and consider the stars, together with the sun,
to be wandering bodies, revolving about the common centre of creation,
we may then recognise in the equatorial plane passing through the centre
of gravity of the universe the only instance of absolute and eternal
repose.
All the periodic and secular inequalities deduced from the law of
gravitation are so perfectly confirmed by observation, that analysis has
become one of the most certain means of discovering the planetary
irregularities, either when they are too small, or too long in their
periods, to be detected by other methods. Jupiter and Saturn, however,
exhibit inequalities which for a long time seemed discordant with that
law. All observations, from those of the Chinese and Arabs down to the
present day, prove that for ages the mean motions of Jupiter and Saturn
have been affected by a great inequality of a very long period, forming
an apparent anomaly in the theory of the planets. It was long known by
observation that five times the mean motion of Saturn is nearly equal to
twice that of Jupiter; a relation which the sagacity of La Place
perceived to be the cause of a periodic irregularity in the mean motion
of each of these planets, which completes its period in nearly 918
years, the one being retarded while the other is accelerated; but both
the magnitude and period of these quantities vary, in consequence of the
secular variations in the elements of the orbits. Suppose the two
planets to be on the same side of the sun, and all three in the same
straight line, they are then said to be in conjunction (N. 83). Now, if
they begin to move at the same time, one making exactly five revolutions
in its orbit while the other only accomplishes two, it is clear that
Saturn, the slow-moving body, will only have got through a part of its
orbit during the time that Jupiter has made one whole revolution and
part of another, before they be again in conjunction. It is found that
during this time their mutual action is such as to produce a great many
perturbations which compensate each other, but that there still remains
a portion outstanding, owing to the length of time during which the
forces act in the same manner; and, if the conjunction always happened
in the same point of the orbit, this uncompensated inequality in the
mean motion would go on increasing till the periodic times and forms of
the orbits were completely and permanently changed: a case that would
actually take place if Jupiter accomplished exactly five revolutions in
the time Saturn performed two. These revolutions are, however, not
exactly commensurable; the points in which the conjunctions take place
are in advance each time as much as 8°·37; so that the conjunctions do
not happen exactly in the same points of the orbits till after a period
of 850 years; and, in consequence of this small advance, the planets are
brought into such relative positions, that the inequality, which seemed
to threaten the stability of the system, is completely compensated, and
the bodies, having returned to the same relative positions with regard
to one another and the sun, begin a new course. The secular variations
in the elements of the orbit increase the period of the inequality to
918 years (N. 84). As any perturbation which affects the mean motion
affects also the major axis, the disturbing forces tend to diminish the
major axis of Jupiter’s orbit, and increase that of Saturn’s, during one
half of the period, and the contrary during the other half. This
inequality is strictly periodical, since it depends upon the
configuration (N. 85) of the two planets; and theory is confirmed by
observation, which shows that, in the course of twenty centuries,
Jupiter’s mean motion has been accelerated by about 3° 23ʹ, and Saturn’s
retarded by 5° 13ʹ. Several instances of perturbations of this kind
occur in the solar system. One, in the mean motions of the Earth and
Venus, only amounting to a few seconds, has been recently worked out
with immense labour by Professor Airy. It accomplishes its changes in
240 years, and arises from the circumstance of thirteen times the
periodic time of Venus being nearly equal to eight times that of the
Earth. Small as it is, it is sensible in the motions of the Earth.
It might be imagined that the reciprocal action of such planets as have
satellites would be different from the influence of those that have
none. But the distances of the satellites from their primaries are
incomparably less than the distances of the planets from the sun, and
from one another. So that the system of a planet and its satellites
moves nearly as if all these bodies were united in their common centre
of gravity. The action of the sun, however, in some degree disturbs the
motion of the satellites about their primary.