earth might be spheroidal before Newton or Huyghens turned their
attention to the subject. At a meeting of the Royal Society on the 28th
of February, 1678, a discussion arose respecting the figure of Mercury
which M. Gallet of Avignon had remarked to be oval on the occasion of
the planet's transit across the sun's disk on the 7th of November, 1677.
Hooke was inclined to suppose that the phenomenon was real, and that it
was due to the whirling of the planet on an axis "which made it somewhat
of the shape of a turnip, or of a solid made by an ellipsis turned round
upon its shorter diameter." At the meeting of the Society on the 7th of
March, the subject was again discussed. In reply to the objection
offered to his hypothesis on the ground of the planet being a solid
body, Hooke remarked that "although it might now be solid, yet that at
the beginning it might have been fluid enough to receive that shape; and
that although this supposition should not be granted, it would be
probable enough that it would really run into that shape and make the
same appearance; _and that it is not improbable but that the water here
upon the earth might do it in some measure by the influence of the
diurnal motion, which, compounded with that of the moon, he conceived to
be the cause of the Tides_." (Journal Book of the Royal Society, vol.
vi. p. 60.) Richer returned from Cayenne in the year 1674, but the
account of his observations with the pendulum during his residence
there, was not published until 1679, nor is there to be found any
allusion to them during the intermediate interval, either in the volumes
of the Academy of Sciences or any other publication. We have no means of
ascertaining how Newton was first induced to suppose that the figure of
the earth is spheroidal, but we know, upon his own authority, that as
early as the year 1667, or 1668, he was led to consider the effects of
the centrifugal force in diminishing the weight of bodies at the
equator. With respect to Huyghens, he appears to have formed a
conjecture respecting the spheroidal figure of the earth independently
of Newton; but his method for computing the ellipticity is founded upon
that given in the Principia.--_Translator_.
[29] Newton assumed that a homogeneous fluid mass of a spheroidal form
would be in equilibrium if it were endued with an adequate rotatory
motion and its constituent particles attracted each other in the inverse
proportion of the square of the distance. Maclaurin first demonstrated
the truth of this theorem by a rigorous application of the ancient
geometry.--_Translator_.
[30] The results of Clairaut's researches on the figure of the earth are
mainly embodied in a remarkable theorem discovered by that geometer, and
which may be enunciated thus:--_The sum of the fractions expressing the
ellipticity and the increase of gravity at the pole is equal to two and
a half times the fraction expressing the centrifugal force at the
equator, the unit of force being represented by the force of gravity at
the equator._ This theorem is independent of any hypothesis with respect
to the law of the densities of the successive strata of the earth. Now
the increase of gravity at the pole may be ascertained by means of
observations with the pendulum in different latitudes. Hence it is plain
that Clairaut's theorem furnishes a practical method for determining the
value of the earth's ellipticity.--_Translator_.
[31] The researches on the secular variations of the eccentricities and
inclinations of the planetary orbits depend upon the solution of an
algebraic equation equal in degree to the number of planets whose mutual
action is considered, and the coefficients of which involve the values
of the masses of those bodies. It may be shown that if the roots of this
equation be equal or imaginary, the corresponding element, whether the
eccentricity or the inclination, will increase indefinitely with the
time in the case of each planet; but that if the roots, on the other
hand, be real and unequal, the value of the element will oscillate in
every instance within fixed limits. Laplace proved by a general
analysis, that the roots of the equation are real and unequal, whence it
followed that neither the eccentricity nor the inclination will vary in
any case to an indefinite extent. But it still remained uncertain,
whether the limits of oscillation were not in any instance so far apart
that the variation of the element (whether the eccentricity or the
inclination) might lead to a complete destruction of the existing
physical condition of the planet. Laplace, indeed, attempted to prove,
by means of two well-known theorems relative to the eccentricities and
inclinations of the planetary orbits, that if those elements were once
small, they would always remain so, provided the planets all revolved
around the sun in one common direction and their masses were
inconsiderable. It is to these theorems that M. Arago manifestly alludes
in the text. Le Verrier and others have, however, remarked that they are
inadequate to assure the permanence of the existing physical condition
of several of the planets. In order to arrive at a definitive conclusion
on this subject, it is indispensable to have recourse to the actual
solution of the algebraic equation above referred to. This was the
course adopted by the illustrious Lagrange in his researches on the
secular variations of the planetary orbits. (_Mem. Acad. Berlin_,
1783-4.) Having investigated the values of the masses of the planets, he
then determined, by an approximate solution, the values of the several
roots of the algebraic equation upon which the variations of the
eccentricities and inclinations of the orbits depended. In this way, he
found the limiting values of the eccentricity and inclination for the
orbit of each of the principal planets of the system. The results
obtained by that great geometer have been mainly confirmed by the recent
researches of Le Verrier on the same subject. (_Connaissance des Temps_,
1843.)--_Translator_.
[32] Laplace was originally led to consider the subject of the
perturbations of the mean motions of the planets by his researches on
the theory of Jupiter and Saturn. Having computed the numerical value of
the secular inequality affecting the mean motion of each of those
planets, neglecting the terms of the fourth and higher orders relative
to the eccentricities and inclinations, he found it to be so small that
it might be regarded as totally insensible. Justly suspecting that this
circumstance was not attributable to the particular values of the
elements of Jupiter and Saturn, he investigated the expression for the
secular perturbation of the mean motion by a general analysis,
neglecting, as before, the fourth and higher powers of the
eccentricities and inclinations, and he found in this case, that the
terms which were retained in the investigation absolutely destroyed each
other, so that the expression was reduced to zero. In a memoir which he
communicated to the Berlin Academy of Sciences, in 1776, Lagrange first
showed that the mean distance (and consequently the mean motion) was not
affected by any secular inequalities, no matter what were the
eccentricities or inclinations of the disturbing and disturbed
planets.--_Translator_.
[33] Mr. Adams has recently detected a remarkable oversight committed by
Laplace and his successors in the analytical investigation of the
expression for this inequality. The effect of the rectification rendered
necessary by the researches of Mr. Adams will be to diminish by about
one sixth the coefficient of the principal term of the secular
inequality. This coefficient has for its multiplier the square of the
number of centuries which have elapsed from a given epoch; its value was
found by Laplace to be 10".18. Mr. Adams has ascertained that it must be
diminished by 1".66. This result has recently been verified by the
researches of M. Plana. Its effect will be to alter in some degree the
calculations of ancient eclipses. The Astronomer Royal has stated in his
last Annual Report, to the Board of Visitors of the Royal Observatory,
(June 7, 1856,) that steps have recently been taken at the Observatory,
for calculating the various circumstances of those phenomena, upon the
basis of the more correct data furnished by the researches of Mr.
Adams.--_Translator_.
[34] [Illustration]
The origin of this famous inequality may be best understood by reference
to the mode in which the disturbing forces operate. Let P Q R, P' Q' R'
represent the orbits of Jupiter and Saturn, and let us suppose, for the
sake of illustration, that they are both situate in the same plane. Let
the planets be in conjunction at P, P', and let them both be revolving
around the sun S, in the direction represented by the arrows. Assuming
that the mean motion of Jupiter is to that of Saturn exactly in the
proportion of five to two, it follows that when Jupiter has completed
one revolution, Saturn will have advanced through two fifths of a
revolution. Similarly, when Jupiter has completed a revolution and a
half, Saturn will have effected three fifths of a revolution. Hence when
Jupiter arrives at T, Saturn will be a little in advance of T'. Let us
suppose that the two planets come again into conjunction at Q, Q'. It is
plain that while Jupiter has completed one revolution, and, advanced
through the angle P S Q (measured in the direction of the arrow), Saturn
has simply described around S the angle P' S' Q'. Hence the _excess_ of
the angle described around S, by Jupiter, over the angle similarly
described by Saturn, will amount to one complete revolution, or, 360°.
But since the mean motions of the two planets are in the proportion of
five to two, the angles described by them around S in any given time
will be in the same proportion, and therefore the _excess_ of the angle
described by Jupiter over that described by Saturn will be to the angle
described by Saturn in the proportion of three to two. But we have just
found that the excess of these two angles in the present case amounts to
360°, and the angle described by Saturn is represented by P' S' Q';
consequently 360° is to the angle P' S' Q' in the proportion of three to
two, in other words P' S' Q' is equal to two thirds of the circumference
or 240°. In the same way it may be shown that the two planets will come
into conjunction again at R, when Saturn has described another arc of
240°. Finally, when Saturn has advanced through a third arc of 240°, the
two planets will come into conjunction at P, P', the points whence they
originally set out; and the two succeeding conjunctions will also
manifestly occur at Q, Q' and R, R'. Thus we see, that the conjunctions
will always occur in three given points of the orbit of each planet
situate at angular distances of 120° from each other. It is also
obvious, that during the interval which elapses between the occurrence
of two conjunctions in the same points of the orbits, and which includes
three synodic revolutions of the planets, Jupiter will have accomplished
five revolutions around the sun, and Saturn will have accomplished two
revolutions. Now if the orbits of both planets were perfectly circular,
the retarding and accelerating effects of the disturbing force of either
planet would neutralize each other in the course of a synodic
revolution, and therefore both planets would return to the same
condition at each successive conjunction. But in consequence of the
ellipticity of the orbits, the retarding effect of the disturbing force
is manifestly no longer exactly compensated by the accelerative effect,
and hence at the close of each synodic revolution, there remains a
minute outstanding alteration in the movement of each planet. A similar
effect will he produced at each of the three points of conjunction; and
as the perturbations which thus ensue do not generally compensate each
other, there will remain a minute outstanding perturbation as the result
of every three conjunctions. The effect produced being of the same kind
(whether tending to accelerate or retard the movement of the planet) for
every such triple conjunction, it is plain that the action of the
disturbing forces would ultimately lead to a serious derangement of the
movements of both planets. All this is founded on the supposition that
the mean motions of the two planets are to each other as two to five;
but in reality, this relation does not exactly hold. In fact while
Jupiter requires 21,663 days to accomplish five revolutions, Saturn
effects two revolutions in 21,518 days. Hence when Jupiter, after
completing his fifth revolution, arrives at P, Saturn will have advanced
a little beyond P', and the conjunction of the two planets will occur at
P, P' when they have both described around S an additional arc of about
8°. In the same way it may be shown that the two succeeding conjunctions
will take place at the points _q, q', r, r'_ respectively 8° in advance
of Q, Q', R, R'. Thus we see that the points of conjunction will travel
with extreme slowness in the same direction as that in which the planets
revolve. Now since the angular distance between P and R is 120°, and
since in a period of three synodic revolutions or 21,758 days, the line
of conjunction travels through an arc of 8°, it follows that in 892
years the conjunction of the two planets will have advanced from P, P'
to R, R'. In reality, the time of travelling from P, P' to R, R' is
somewhat longer from the indirect effects of planetary perturbation,
amounting to 920 years. In an equal period of time the conjunction of
the two planets will advance from Q, Q' to R, R' and from R, R' to P,
P'. During the half of this period the perturbative effect resulting
from every triple conjunction will lie constantly in one direction, and
during the other half it will lie in the contrary direction; that is to
say, during a period of 460 years the mean motion of the disturbed
planet will be continually accelerated, and, in like manner, during an
equal period it will be continually retarded. In the case of Jupiter
disturbed by Saturn, the inequality in longitude amounts at its maximum
to 21'; in the converse case of Saturn disturbed by Jupiter, the
inequality is more considerable in consequence of the greater mass of
the disturbing planet, amounting at its maximum to 49'. In accordance
with the mechanical principle of the equality of action and reaction, it
happens that while the mean motion of one planet is increasing, that of
the other is diminishing, and _vice versâ_. We have supposed that the
orbits of both planets are situate in the same plane. In reality,
however, they are inclined to each other, and this circumstance will
produce an effect exactly analogous to that depending on the
eccentricities of the orbits. It is plain that the more nearly the mean
motions of the two planets approach a relation of commensurability, the
smaller will be the displacement of every third conjunction, and
consequently the longer will be the duration, and the greater the
ultimate accumulation, of the inequality.--_Translator_.
[35] The utility of observations of the transits of the inferior planets
for determining the solar parallax, was first pointed out by James
Gregory (_Optica Promota_, 1663).--_Translator_.
[36] Mayer, from the principles of gravitation (_Theoria Lunæ_, 1767),
computed the value of the solar parallax to be 7".8. He remarked that
the error of this determination did not amount to one twentieth of the
whole, whence it followed that the true value of the parallax could not
exceed 8".2. Laplace, by an analogous process, determined the parallax
to be 8".45. Encke, by a profound discussion of the observations of the
transits of Venus in 1761 and 1769, found the value of the same element
to be 8".5776.--_Translator_.
[37] The theoretical researches of Laplace formed the basis of
Burckhardt's Lunar Tables, which are chiefly employed in computing the
places of the moon for the Nautical Almanac and other Ephemerides. These
tables were defaced by an empiric equation, suggested for the purpose of
representing an inequality of long period which seemed to affect the
mean longitude of the moon. No satisfactory explanation of the origin of
this inequality could be discovered by any geometer, although it formed
the subject of much toilsome investigation throughout the present
century, until at length M. Hansen found it to arise from a combination
of two inequalities due to the disturbing action of Venus. The period of
one of these inequalities is 273 years, and that of the other is 239
years. The maximum value of the former is 27".4, and that of the latter
is 23".2.--_Translator_.
[38] This law is necessarily included in the law already enunciated by
the author relative to the mean longitudes. The following is the most
usual mode of expressing these curious relations: 1st, the mean motion
of the first satellite, plus twice the mean motion of the third, minus
three times the mean motion of the second, is rigorously equal to zero;
2d, the mean longitude of the first satellite, plus twice the mean
longitude of the third, minus three times the mean longitude of the
second, is equal to 180°. It is plain that if we only consider the mean
longitude here to refer to a _given epoch_, the combination of the two
laws will assure the existence of an analogous relation between the mean
longitudes _for any instant of time whatever_, whether past or future.
Laplace has shown, as the author has stated in the text, that if these
relations had only been approximately true at the origin, the mutual
attraction of the three satellites would have ultimately rendered them
rigorously so; under such circumstances, the mean longitude of the first
satellite, plus twice the mean longitude of the third, minus three times
the mean longitude of the second, would continually oscillate about 180°
as a mean value. The three satellites would participate in this
libratory movement, the extent of oscillation depending in each case on
the mass of the satellite and its distance from the primary, but the
period of libration is the same for all the satellites, amounting to
2,270 days 18 hours, or rather more than six years. Observations of the
eclipses of the satellites have not afforded any indications of the
actual existence of such a libratory motion, so that the relations
between the mean motions and mean longitudes may be presumed to be
always rigorously true.--_Translator_.
[39] Laplace has explained this theory in his _Exposition du Système du
Monde_ (liv. iv. note vii.).--_Translator_.
APPENDIX.
(A.)
THE FOLLOWING IS A BRIEF NOTICE OF SOME OTHER INTERESTING RESULTS OF THE
RESEARCHES OF LAPLACE WHICH HAVE NOT BEEN MENTIONED IN THE TEXT.
_Method for determining the orbits of comets._--Since comets are
generally visible only during a few days or weeks at the utmost, the
determination of their orbits is attended with peculiar difficulties.
The method devised by Newton for effecting this object was in every
respect worthy of his genius. Its practical value was illustrated by the
brilliant researches of Halley on cometary orbits. It necessitated,
however, a long train of tedious calculations, and, in consequence, was
not much used, astronomers generally preferring to attain the same end
by a tentative process. In the year 1780, Laplace communicated to the
Academy of Sciences an analytical method for determining the elements of
a comet's orbit. This method has been extensively employed in France.
Indeed, previously to the appearance of Olber's method, about the close
of the last century, it furnished the easiest and most expeditious
process hitherto devised, for calculating the parabolic elements of a
comet's orbit.
_Invariable plane of the solar system._--In consequence of the mutual
perturbations of the different bodies of the planetary system, the
planes of the orbits in which they revolve are perpetually varying in
position. It becomes therefore desirable to ascertain some fixed plane
to which the movements of the planets in all ages may be referred, so
that the observations of one epoch might be rendered readily comparable
with those of another. This object was accomplished by Laplace, who
discovered that notwithstanding the perpetual fluctuations of the
planetary orbits, there exists a fixed plane, to which the positions of
the various bodies may at any instant be easily referred. This plane
passes through the centre of gravity of the solar system, and its
position is such, that if the movements of the planets be projected upon
it, and if the mass of each planet be multiplied by the area which it
describes in a given time, the sum of such products will be a maximum.
The position of the plane for the year 1750 has been calculated by
referring it to the ecliptic of that year. In this way it has been found
that the inclination of the plane is 1° 35' 31", and that the longitude
of the ascending node is 102° 57' 30". The position of the plane when
calculated for the year 1950, with respect to the ecliptic of 1750,
gives 1° 35' 31" for the inclination, and 102° 57' 15" for the longitude
of the ascending node. It will be seen that a very satisfactory
accordance exists between the elements of the position of the invariable
plane for the two epochs.
_Diminution of the obliquity of the ecliptic._--The astronomers of the
eighteenth century had found, by a comparison of ancient with modern
observations, that the obliquity of the ecliptic is slowly diminishing
from century to century. The researches of geometers on the theory of
gravitation had shown that an effect of this kind must be produced by
the disturbing action of the planets on the earth. Laplace determined
the secular displacement of the plane of the earth's orbit due to each
of the planets, and in this way ascertained the whole effect of
perturbation upon the obliquity of the ecliptic. A comparison which he
instituted between the results of his formula and an ancient observation
recorded in the Chinese Annals exhibited a most satisfactory accordance.
The observation in question indicated the obliquity of the ecliptic for
the year 1100 before the Christian era, to be 23° 54' 2".5. According to
the principles of the theory of gravitation, the obliquity for the same
epoch would be 23° 51' 30".
_Limits of the obliquity of the ecliptic modified by the action of the
sun and moon upon the terrestrial spheroid._--The ecliptic will not
continue indefinitely to approach the equator. After attaining a certain
limit it will then vary in the opposite direction, and the obliquity
will continually increase in like manner as it previously diminished.
Finally, the inclination of the equator and the ecliptic will attain a
certain maximum value, and then the obliquity will again diminish. Thus
the angle contained between the two planes will perpetually oscillate
within certain limits. The extent of variation is inconsiderable.
Laplace found that, in consequence of the spheroidal figure of the
earth, it is even less than it would otherwise have been. This will be
readily understood, when we state that the disturbing action of the sun
and moon upon the terrestrial spheroid produces an oscillation of the
earth's axis which occasions a periodic variation of the obliquity of
the ecliptic. Now, as the plane of the ecliptic approaches the equator,
the mean disturbing action of the sun and moon upon the redundant matter
accumulated around the latter will undergo a corresponding variation,
and hence will arise an inconceivably slow movement of the plane of the
equator, which will necessarily affect the obliquity of the ecliptic.
Laplace found that if it were not for this cause, the obliquity of the
ecliptic would oscillate to the extent of 4° 53' 33" on each side of a
mean value, but that when the movements of both planes are taken into
account, the extent of oscillation is reduced to 1° 33' 45".
_Variation of the length of the tropical year._--The disturbing action
of the sun and moon upon the terrestrial spheroid occasions a continual
_regression_ of the equinoctial points, and hence arises the distinction
between the sidereal and tropical year. The effect is modified in a
small degree by the variation of the plane of the ecliptic, which tends
to produce a _progression_ of the equinoxes. If the movement of the
equinoctial points arising from these combined causes was uniform, the
length of the tropical year would be manifestly invariable. Theory,
however, indicates that for ages past the rate of regression has been
slowly increasing, and, consequently, the length of the tropical year
has been gradually diminishing. The rate of diminution is exceedingly
small. Laplace found that it amounts to somewhat less than half a second
in a century. Consequently, the length of the tropical year is now about
ten seconds less than it was in the time of Hipparchus.