•i

SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE.

232

M E M O I E

SECULAR VARIATIONS OF THE ELEMENTS

DEBITS OF THE EIGHT PRINCIPAL PLANETS,

MERCURY, VENUS, THE EARTH, MARS, JUPITER, SATURN, URANUS, AND NEPTUNE;

WITH TABLES OP THE SAME;

TOGETHER WITH THB

OBLIQUITY OP THE ECLIPTIC, AND THE PRECESSION OP THE EQUINOXES
IN BOTH LONGITUDE AND RIGHT ASCENSION.

BY

JOHN N. STOCKWELL, M.A.

[ACCEPTED FOB PUBLICATION, DECEMBER, 1870.]

ADVERTISEMENT.

THIS memoir was presented to the Institution by Professor J. H. C. Coffin,
Superintendent of the American Nautical Almanac, and was afterwards submitted
to Prof. S. Newcomb of the Naval Observatory, both of whom recommended its
adoption as one of the Smithsonian Contributions to Knowledge. The appropria-
tion for printing at the time it was presented having been temporarily exhausted,
a friend of science, who does not allow his name to be mentioned, furnished the
means for its immediate publication.

JOSEPH HENRY,

Secretary.

SMITHSONIAN INSTITUTION,
May, 1872.

(iii)

PREFACE.

THE computations of the secular inequalities, the results of which are given in
the following Memoir, were commenced about ten years ago, and have been con-
tinued, during many interruptions, till the present time. In the spring of 1870
the calculations were so far advanced that the greater part of the Memoir was
put in form for the press ; but funds for printing it were not then available, and
the computations were suffered to languish till late in autumn, when provision was
made through the Smithsonian Institution for its publication. The work was then
resumed, completed, and forwarded to the printer without delay ; but an unex-
pected interval of leisure occurring during the process of publication, I availed
myself of the opportunity thus presented, and prepared an additional chapter con
taining tabular values of the elements of the planetary orbits, together with the
formulas necessary for their convenient application. It is believed that this addi-
tional chapter will contribute largely to the usefulness of the work, and be found
of great practical value in the researches of astronomers.

J. N. STOCKWELL.

CLEVELAND, May, 1872.

(v)

INTRODUCTION,

THE reciprocal gravitation of matter produces disturbances in the motions of the
heavenly bodies, causing them to deviate from the elliptic paths which they would
follow, if they were attracted only by the sun. The determination of the amount
by which the actual place of a planet deviates from its true elliptic place at any
time is called the problem of planetary perturbation. The analytical solution of
this problem has disclosed to mathematicians the fact that the inequalities in the
motions of the heavenly bodies are produced in two distinct ways. The first is a
direct disturbance in the elliptic motion of the body; and the second is produced
by reason of a variation of the elements of its elliptic motion. The elements of the
elliptic motion of a planet are six in number, viz: the mean motion of the planet
and its mean distance from the sun, the eccentricity and inclination of its orbit,
and the longitude of the node and perihelion. The first two are invariable • the
other four are subject to both periodic and secular variations.

The inequalities in the planetary motions which are produced by the direct
action of the planets on each other, and depend for their amount only on their
distances and mutual configurations, are called periodic inequalities, because they
pass through a complete cycle of values in a comparatively short period of time ;
while those depending on the variation of the elements of the elliptic motion are
produced with extreme slowness, require an immense number of ages for their
full development, and are called secular inequalities. The general theory of all the
planetary inequalities was completely developed by La Grange and La Place nearly
a century ago; and the particular theory of each planet for the periodic inequali-
ties was given by La Place in the Mecanique Celeste.

The determination of the periodic inequalities of the planets has hitherto
received more attention from astronomers than has been bestowed upon the secular
inequalities. This is owing in part to the immediate requirements of astronomy,
and also in part to the less intricate nature of the problem. It is true that an
approximate knowledge of the secular inequalities is necessary in the treatment of
the periodic inequalities ; but since the secular inequalities are produced with such
extreme slowness, most astronomers have been content with the supposition that
they are developed uniformly with the time. This supposition is sufficiently near
the truth to be admissible in most astronomical investigations during the compara-
tively short period of time over which astronomical observations or human history
extends ; but since the values of these variations are derived from the equations

(vii)

INTRODUCTION.

of the differential variations of the elements at a particular epoch, it follows that
they afford us no knowledge respecting the ultimate condition of the planetary
system, or even a near approximation to its actual condition at a time only com-
paratively remote from the epoch of the elements on which they are founded. But
aside from any considerations connected with the immediate needs of practical
astronomy, the study of the secular inequalities is one of the most interesting and
important departments of physical science, because their indefinite continuance in
the same direction would ultimately seriously affect the stability of the planetary
system. The demonstration that the secular inequalities of the planets are not
indefinitely progressive, but may be expressed analytically by a series of terms de-
pending on the sines and cosines of angles which increase uniformly with the time,
is due to La Grange and La Place. It therefore follows that the secular inequali-
ties are periodic, and differ from the ordinary periodic inequalities only in the
length of time required to complete the cycle of their values. The amount by
which the elements of any planet may ultimately deviate from their mean values
can only be determined by the simultaneous integration of the differential equa-
tions of these elements, which is equivalent to the summation of all the infinitesi-
mal variations arising from the disturbing forces of all the planets of the system
during the lapse of an infinite period of time.

The simultaneous integration of the equations which determine the instantaneous
variations of the elements of the orbits gives rise to a complete equation in which
the unknown quantity is raised to a power denoted by the number of planets, whose
mutual action is considered. La Grange first showed that if any of the roots of
this equation were equal or imaginary, the finite expressions for the values of the
elements would contain terms involving arcs of circles or exponential quantities,
without the functions of sine and cosine, and as these terms would increase inde-
finitely with the time, they would finally render the orbits so very eccentrical that
the stability of the planetary system would be destroyed. In order to determine
whether the roots of the equation were all real and unequal, he substituted the
approximate values of the elements and masses which were employed by astrono-
mers at that time in the algebraic equations, and then by determining the roots he
found them to be all real and unequal. It, therefore, followed, that for the parti-
cular values of the masses employed by La Grange, the equations which determine
the secular variations contain neither arcs of a circle nor exponential quantities,
without the signs of sine and cosine ; whence it follows that the elements of the
orbits will perpetually oscillate about their mean values This investigation was
valuable as a first attempt to fix the limits of the variations of the planetary
elements ; but, being based upon values of the masses which were, to a certain
extent, gratuitously assumed, it was desirable that the important truths which it
indicated should be established independently of any considerations of a hypothetic
character. This magnificent generalization was effected by La Place. He proved
that, whatever be the relative masses of the planets, the roots of the equations
which determine the periods of the secular inequalities will all be real and un-
equal, provided the bodies of the system are subjected to this one condition, that
they all revolve round the sun in the same direction. This condition being satisfied

INTRODUCTION. jx

by all the members of the solar system, it follows that the orbits of the planets
will never be very eccentrical or much inclined to each other by reason of their
mutual attraction. The important truths in relation to the forms and positions of
the planetary orbits are embodied in the two following theorems by the author of
the Mecanique Celeste: I. If the mass of each planet be multiplied by the product of
the square of the eccentricity and square root of the mean distance, the sum of all
these products will always retain the same magnitude. II. If the mass of each
planet be midtiplicd by the product of the square of the inclination of the orlit and
the square root of the mean distance, the sum of these products will always remain
invariable. Now, these quantities being computed for a given epoch, if their sum
is found to be small, it follows from the preceding theorems that they will always
remain so ; consequently the eccentricities and inclinations cannot increase inde-
finitely, but will always be confined within narrow limits.

In order to calculate the limits of the variations of the elements with precision,
it is necessary to know the correct values of the masses of all the planets. Unfor-
tunately, this knowledge has not yet been attained. The masses of several of the
planets are found to be considerably different from the values employed by La
Grange in his investigations. Besides, he only took into account the action of the
six principal planets which are within the orbit of Uranus. Consequently, his solu-
tion afforded only a first approximation to the limits of the secular variations of
the elements.

The person who next undertook the computation of the secular inequalities was
Pontecoulant, who, about the year 1834, published the third volume of his Theorie
Analytique du Systeme du Monde. In this work he has given the results of his
solution of this intricate problem. But the numerical values -of the constants
which he obtained are totally erroneous on account of his failure to employ a suf-
ficient number of decimals in his computation. Our knowledge of the secular
variations of the planetary orbits was, therefore, not increased by his researches.

In 1839 Le Verrier had completed his computation of the secular inequalities
of the seven principal planets. This mathematician has given a new and accurate
determination of the constants on which the amount of the secular inequalities
depends ; and has also given the coefficients for correcting the values of the con-
stants for differential variations of the masses of the different planets. This
investigation of Le Verrier's has been used as the groundwork of most of the sub-
sequent corrections of the planetary elements and masses, and deservedly holds
the first rank as authority concerning the secular variations of the planetary orbits.
But Le Verrier's researches were far from being exhaustive, and he failed to notice
some curious and interesting relations of a permanent character in the secular
variations of the orbits of Jupiter, Saturn, and Uranus. Besides, the planet Nep-
tune had not then been discovered ; and the action of this planet considerably
modifies the secular inequalities which would otherwise take place. We will now
briefly glance at the results of our own labors on the subject.

On the first examination of the works of those authors who had investigated
this problem, we perceived that the methods of reducing to numbers those
analytical integrals which determine the secular variations of the elements were

B June, 1872.

x INTRODUCTION.

far from possessing that elegance and symmetry of form which usually charac-
terize the formulas of astronomy. The first step, therefore, was to devise a
system of algebraic equations, by means of which we should be enabled to obtain
the values of the unknown quantities with the smallest amount of labor. It was
soon found to be impracticable to deduce algebraic formulas for the constants, by
the elimination of eight unknown quantities from as many linear symmetrical
equations, of sufficient simplicity to be used in the deduction of exact results.
It therefore became necessary to abandon the idea of a direct solution of the
equations, and to seek for the best approximate method of obtaining rigorous
values of the unknown quantities. This we have accomplished as completely as
could be desired, and by means of the formulas which we have obtained, it is now
possible to determine the secular variations of the planetary elements with less
labor, perhaps, than would be necessary for the accurate determination of a comet's
orbit. The methods and formulas are given in detail in the following Memoir.

After computing anew the numerical coefficients of the differential equations of
the elements, we have substituted them in these equations, and have obtained, by
means of successive approximations, the rigorous values of the constants corre-
sponding to the assumed masses and elements. The details of the computation
are given in the Memoir referred to, and it is unnecessary to speak of them here.
We shall, therefore, only briefly allude to some of the conclusions to which our
computations legitimately lead.

The object of our investigation has been the determination of the numerica.
values of the secular changes of the elements of the planetary orbits. These
elements are four in number, viz : the eccentricities and inclinations of the orbits,
and the longitudes of the nodes and perihelia. The questions that may legitimately
arise in regard to the eccentricities and inclinations relate chiefly to their magni-
tude at any time ; but we may also desire to know their rates of change at any
time, and the limits within which they will perpetually oscillate. In regard to the.
nodes and perihelia, it is sometimes necessary to know their relative positions
when referred to any plane and origin of coordinates ; and also their mean motions,
together with the amount by which their actual places can differ from their mean
places. With respect to the magnitudes and positions of the elements, together
with their rates of change, we may observe that our equations will give them for
any required epoch, by merely substituting in the formulas the interval of time
between the epoch required and that of the formulas, which is the beginning of
the year 1850. A tabulation of the various planetary elements, of sufficient extent
to supply the needs of the astronomer, is given at the close of the work. A
similar tabulation of the elements of the earth's orbit of sufficient extent to be
useful in geological investigations, does not come within the scope of our work ; and
we leave the computation of the elements for special epochs to those investigators
who may need them in their researches. We shall here give the limits between
which the eccentricities and inclinations will always oscillate, together with the
mean motions of the perihelia and nodes on the fixed ecliptic of 1850; and shall
also give the inclinations and nodes referred to the invariable plane of the plane-
tary system.

INTRODUCTION. xi

For the planet Mercury, we find that the eccentricity is always included within
the limits 0.1214943 and 0.2317185. The mean motion of its perihelion is
5". 463803 ; and it performs a complete revolution in the heavens in 237,197 years.
The maximum inclination of his orbit to the fixed ecliptic of 1850 is 10° 36' 20",
and its minimum inclination is 3° 47' 8" ; while with respect to the invariable plane
of the planetary system, the limits of the inclination are 9° 10' 41" and 4° 44' 27".
The mean motion of the node of Mercury's orbit on the ecliptic of 1850, and on
the invariable plane, is in both cases the same, and equal to 5".126172, making a
complete revolution in the interval of 252,823 years. The amount by which the
true place of the node can differ from its mean place on the ecliptic of 1850 is
equal to 30° 8', while on the invariable plane this limit is only 18° 31'.

For the planet Venus, we find that the eccentricity always oscillates between 0
and 0.0706329. Since the theoretical eccentricity of the orbit of Venus is a
vanishing element, it follows that the perihelion of her orbit can have no mean
motion, but may have any rate of motion, at different times, between nothing and
infinity, both direct and retrograde. The position of her perihelion cannot there-
fore be drtermined within given limits at any very remote epoch by the assumption
of any particular value for the mean motion, but it must be determined by direct
computation from the finite formulas. The maximum inclination of her orbit to
the ecliptic of 1850 is 4° 51', and to the invariable plane it is 3° 16'. 3; while the
mean motion of her node on both planes is indeterminate, because the inferior
limit of the inclination is in each case equal to nothing.

A knowledge of the elements of the earth's orbit is especially interesting and
important on account of the recent attempts to establish a connection between
geological phenomena and terrestrial temperatures, in so far as the latter is modified
by the variable eccentricity of her orbit. The amount of light and heat received
from the sun in the course of a year depends to an important extent on the eccen-
tricity of the earth's orbit; but the distribution of the same over the surface of
the earth depends on the relative position of the perihelion of the orbit with
respect to the equinoxes, and on the obliquity of the ecliptic to the equator. These
elements are subject to great and irregular variations ; but their laws can now be
determined with as much precision as the exigencies of science may require. We
will now more carefully examine these elements, and the cqnsequences to which
their variations give rise.

As we have already computed the eccentricity of the earth's orbit at intervals
of 10,000 years during a period of 2,000,000 years, by employing the constants
which correspond to the assumed mass of the earth increased by its twentieth part,
we here give the elements corresponding to this increased mass. We therefore
find that the eccentricity of the earth's orbit will always be included within the
limits of 0 and 0.0693888 ; and it consequently follows that the mean motion of
the perihelion is indeterminate, although the actual motion and position at any
time during a period of 2,000,000 years can be readily found by means of the
tabular value of that element. The eccentricity of the orbit at any time can also
be found by means of the same table.

The inclination of the apparent ecliptic to the fixed ecliptic of 1850 is always

xii INTRODUCTION.

less than 4° 41'; while its inclination to the invariable plane of the planetary
system always oscillates within the limits 0° 0' and 3° 6'. It is also evident that
the mean motion of the node of the apparent ecliptic on the fixed ecliptic of 1850,
and also on the invariable plane, is wholly indeterminate.

The mean value of the precession of the equinoxes on the fixed ecliptic, and
also on the apparent ecliptic, in a Julian year, is equal to 50".438239 ; whence it
follows, that the equinoxes perform a complete revolution in the heavens in the
average interval of 25,694.8 years ; but on account of the secular inequalities in
their motion, the time of revolution is not always the same, but may differ from
the mean time of revolution by 281.2 years. We also find that if the place of
the equinox be computed for any time whatever, by using the mean value of pre-
cession, its place when thus determined can never differ from its true place to a
greater extent than 3° 56' 26". The maximum and minimum values of precession
in a Julian year are 52".664080 and 48".212398, respectively, and since the length
of the tropical year depends on the annual precession, it follows that the maximum
variation of the tropical year is equal to the mean time required for the earth to
describe an arc which is equal to the maximum variation of precession. Now this
latter quantity being 4".451682, and the sidereal motion of the earth in a second
of time being 0".041067, it follows that the maximum variation of the tropical
year is equal to 108.40 seconds of time. In like manner, if we take the difference
between the present value of precession and the maximum and minimum values
of the same quantity, we shall find that the tropical year may be shorter than at
present by 59.13 seconds, and longer than at present by 49.27 seconds. We also
find that the tropical year is now shorter than in the time of Hipparchus, by
11.30 seconds.

The obliquity of the equator to the apparent ecliptic, and also to the fixed
ecliptic of 1850, has also been determined. The variations of this element follow
a law similar to that which governs the variation of precession, although the
maximum values of the inequalities are considerably smaller than those which
affect this latter quantity. The mean value of the obliquity of both the apparent
and fixed ecliptics to the equator is 23° IT 17". The limits of the obliquity of the
apparent ecliptic to the equator are 24° 35' 58" and 21° 58' 36"; whence it follows
that the greatest and least declinations of the sun at the solstices can never differ
from each other to any greater extent than 2° 37' 22". And here we may mention
a few, among the many happy, consequences which result from the spheroidal form
of the earth. Were the earth a perfect sphere there would be no precession or
change of obliquity arising from the attraction of the sun and moon ; the equinoc-
tial circle would form an invariable plane in the heavens, about which the solar
orbit would revolve with an inclination varying to the extent of twelve degrees,
and a motion equal to the planetary precession of the equinoctial points. The
sun, when at the solstices, would, at some periods of time, attain the declination
of 29° 17' for many thousands of years; and again, at other periods, only to 17°
1 7'. The seasons would be subject to vicissitudes depending on the distance of
the tropics from the equator, and the distribution of solar light and heat on the
surface of the earth would be so modified as essentially to change the character of

INTRODUCTION. xiii

its vegetation, and the distribution of its animal life. But the spheroidal form of the
earth so modifies the secular changes in the relative positions of the equator and eclip-
tic that the inequalities of precession and obliquity are reduced to less than one-quarter
part of what they would otherwise be. The periods of the secular changes, which,
in the case of a spherical earth, would require nearly two millions of years to pass
through a complete cycle of values, are now reduced to periods which vary between
26,000 and 53,000 years. The secular motions which would take place in the case
of a spherical earth are so modified by the actual condition of the terrestrial globe
that changes in the position of the equinox and equator are now produced in a few
centuries, which would otherwise require a period of many thousands of years.
This consideration is of much importance in the investigation of the reputed anti-
quity and chronology of those ancient nations which attained proficiency in the
science of astronomy, and the records of whose astronomical labors are the only
remaining monument of a highly intellectual people, of whose existence every other
trace has long since passed away. For it is evident that, if these changes were
much slower than they are, a much longer time would be required in order to pro-
duce changes of sufficient magnitude to be detected by observation, and we should
be unable to estimate the interval between the epochs of elements which differed
by only a few thousand years, since they would manifestly be so nearly identical
with our own that the value of legitimate conclusions would be greatly impaired
by the unavoidable errors of the observations on which they were based.

The duration of the different seasons is also greatly modified by the eccentricity
of the earth's orbit. At present the sun is north of the equator scarcely 1863 days,
and south of the same circle about 178f days; thus making a difference of 7f days
between the length of the summer and winter at present. But when the eccen-
tricity of the orbit is nearly at its maximum, and its transverse axis also passes
through the solstices, both of which conditions have, in past ages, been fulfilled,
the summer, in one hemisphere, will have a period of 198| days, and a winter of
only 166 1 days, while, in the other hemispheres, these conditions will be reversed;
the winter having a period of 198f days, and a summer of only 166^ days. The
variations of the sun's distance from the earth in the course of a year, at such
times, are also enormous, amounting to almost one-seventh part of its mean distance
— a quantity scarcely less than 13,000,000 of miles!

Passing now to the consideration of the elements of the planet Mars, we find that
the eccentricity of his orbit always oscillates within the limits 0.018475 and
0.139655; and the mean motion of his perihelion is 17".784456. The maximum
inclination of his orbit to the fixed ecliptic of 1850, and to the invariable plane of
the planetary system, is 7° 28' and 5° 56' respectively. The minimum inclination
to both planes being nothing, the mean motion of the node is indeterminate.

The secular variations of the orbits of Jupiter, Saturn, Uranus, and Neptune
present some curious and interesting relations. These four planets compose a sys-
tem by themselves, which is practically independent of the other planets of the
system.

The maximum and minimum limits of the eccentricity of the orbits of these four
planets are as follows:—

INTRODUCTION.

Maximum eccentricity. Minimum eccentricity.

Jupiter O.OGOS274 0.0254928

Saturn 0.0843289 0.0123719

Uranus 0.0779652 0.0117576

Neptune 0.0145066 0.0055729

The maximum and minimum inclinations of their orbits to the invariable plane
of the planetary system have the following values: —

Maximum inclination. Minimum inclination.

Jupiter 0° 28' 56" 0° 14' 23"

Saturn 1 0 39 0 47 16

Uranus 1 7 10 0 64 25

Neptune 0 47 21 0 33 43

The perihelia and nodes of their orbits have the following mean motions in a
Julian year of 365£ days: —

Mean motion of perihelion. Mean motion of node on the

invariable plane.

Jupiter + 3".716607 — 25".934567

Saturn +22 .460848 — 25 .934567

Uranns +3 .716607 — 2 .916082

Neptune +0 .616685 — 0 .661666

But the most curious relation developed by this investigation pertains to the
relative motions and positions of the perihelia and nodes of the three planets,
Jupiter, Saturn, and Uranus. These relations are expressed by the two following
theorems : —

I. The mean motion of Jupiter's perihelion is exactly equal to the mean motion of
tJte perihelion of Uranus, and the mean longitudes of these perihelia differ by exactly
180°. II. The mean motion of Jupiter's node on tlie invariable plane is exact!// equal
to that of Saturn, and Hie mean longitudes of these nodes differ by exactly 180°.

We also perceive that the mean motion of Saturn's perihelion is very nearly six
times that of Jupiter and Uranus, and this latter quantity is very nearly six times
that of Neptune; or, more exactly, 985 times the mean motion of Jupiter's peri-
helion are equal to 163 times that of Saturn, and 440 times the mean motion of
Neptune's perihelion are equal to 73 times that of Jupiter and Uranus. The peri-
helion of Saturn's orbit performs a complete revolution in the heavens in 57,700
years; the perihelia of Jupiter and Uranus in 348,700 years; while that of Nep-
tune requires no less than 2,101,560 years to complete the circuit of the heavens.
In like manner the nodes of Jupiter and Saturn, on the invariable plane, perform a
complete revolution in 49,972 years ; that of Uranus in 444,432 years ; while the
node of Neptune requires 1,958,692 years to traverse the circumference of the
heavens. The motions of the nodes are retrograde, and those of the perihelia are
direct ; thus conforming to the same law of variation as that which obtains in the
corresponding elements of the moon's motion.

We may here observe that the law which controls the motions and positions of
the perihelia of the orbits of Jupiter and Uranus is of the utmost importance in
relation to their mutual perturbations of Saturn's orbit. For, in the existing

INTRODUCTION. xv

arrangement, the orbit of Saturn is affected only by the difference of the perturba-
tions by Jupiter and Uranus ; whereas, if the mean places of the perihelia of these
two planets were the same, instead of differing by 180°, the orbit of Saturn would
be affected by the sum of their disturbing forces. But notwithstanding this favor-
ing condition, the elements of Saturn's orbit would be subject to very groat pertur-
bations from the superior action of Jupiter, were it not for the comparatively rapid
motion of its perihelion; its equilibrium being maintained by the very act of per-
turbation. Indeed, the stability of Saturn's orbit depends to a very great extent
upon the rapidly varying positions of its transverse axis. For, if the motions of
the perihelia of Jupiter and Saturn were very nearly the same, the action of Jupiter
on the eccentricity of Saturn's orbit would be at its maximum value during very
long periods of time, and thereby produce great and permanent changes in the
value of that element. But, in the existing conditions, the rapid motion of Saturn's
orbit prevents such an accumulation of perturbation, and any increase of eccen-
tricity is soon changed into a corresponding diminution. The same remark is also
applicable to the perturbations of the forms of the orbits of Jupiter and Uranus
by the disturbing action of Saturn; for the secular variations of Jupiter's orbit
depend almost entirely upon the influence of Saturn, because the planet Neptune
is too remote to produce much disturbance, and the mean disturbing influence of
Uranus on the eccentricity of Jupiter's orbit is identically equal to nothing, by
reason of the relation which always exists between the perihelia of their orbits.
We may here observe that the eccentricity of the orbit of Saturn always increases,
while that of Jupiter diminishes, and vice versd.

The consequences which result from the mutual relations which always exist
between the nodes of Jupiter and Saturn, on the invariable plane of the planetary
system, are no less interesting or remarkable with respect to the position of the
orbit of Uranus than those which result from the permanent relation between the
perihelia of Jupiter and Uranus are with respect to the form of the orbit of Saturn.
The mean disturbing force of Saturn on the inclination of the orbit of Uranus is
about four times that of Jupiter ; but as these two planets always act on the
inclinations in opposite directions, it follows that the joint action of the two
planets is equivalent to the action of a single planet at the distance of Saturn
and having about three-fourths of his mass ; so that the orbit of Uranus might
attfain a considerable inclination from the superior action of Saturn if allowed to
accumulate during the lapse of an unlimited time, at its maximum rate of variation
depending on the action of this planet. But such an accumulation of perturbation
is rendered forever impossible by reason of the comparatively rapid motion of the
nodes of Jupiter and Saturn, with respect to that of Uranus, on the invariable
plane. By reason of this rapid motion, the secular changes of the inclination of
the orbit of Uranus pass through a complete cycle of values in the period of
56,300 years. The corresponding cycle of perturbation in the eccentricity of
Saturn's orbit is 69,140 years. It is the rapid motion of the orbit with respect
to the forces in the one case, and the rapid motion of the forces with respect to
the orbit in the other, that gives permanence of form and position to the orbits of
Saturn and Uranus.

xvi INTRODUCTION.

The .mean angular distance between the perihelia of Jupiter and Uranus is
exactly 180°; but the conditions of the variations of these elements are sufficiently
clastic to allow of a considerable deviation on each side of their mean positions.
The perihelion of Jupiter may diifer from its mean place to the extent of 24° 10',
and that of Uranus to the extent of 47° 33'; and therefore the longitudes of the
perihelia of these two planets can differ from 180° to the extent of 71° 43'. The
nearest approach of the perihelia of these two planets is, therefore, 108° 17'.

In like manner the longitudes of the nodes of Jupiter and Saturn, on the invari-
able plane, can suffer considerable deviations from their mean positions. The
actual position of Jupiter's node may differ from its mean place to the extent of
19° 38'; while that of Saturn may deviate from its mean place to the extent of
7° 7'. It therefore follows that their longitudes on the invariable plane can differ
from 180° by only 26° 45'. Their nearest possible approach is 153° 15', while their
present distance apart is 166° 27'.

The inequalities in the eccentricity of Neptune's orbit are very small, and the
two principal ones have periods of 613,900 years, and 418,060 years respectively.
Strictly speaking, the periods of the secular inequalities of the eccentricities and
perihelia are the same for all the planets ; and the same remark is equally applicable
to the nodes and inclinations. But the principal inequalities of the several planetary
orbits arc different, unless they are connected by some permanent relation, similar
to that which exists between the perihelia of Jupiter and Uranus, or the nodes of
Jupiter and Saturn. Thus the principal inequalities affecting the inclination of
the orbits of Jupiter and Saturn have the same periods for each planet, and these
periods are, for the two principal inequalities, 51,280 years, and 56,303 years. In
like manner the principal inequalities in the eccentricities of Jupiter and Saturn
depend on their mutual attraction, and have a period of 69,141 years. The secular
inequalities of those orbits which have no vanishing elements are composed of
terms, of very different orders of magnitude ; and it frequently happens that two
or three of these terms are greater than the sum of all the remaining ones. In
such cases the variation of the corresponding element very approximately conforms
to a much simpler law, and the maxima and minima repeat themselves according
to definite and well-defined cycles. But with regard to the orbits of Venus, the
Earth, and Mars, the rigorous expressions of the eccentricities and inclinations
are composed of twenty-eight periodic terms, having coefficients of considerable
magnitude ; and this circumstance renders the law of their variations extremely
intricate.

The method we have adopted for finding the coefficients of the corrections of
the constants, depending on finite variations of the different planetary masses,
consists in supposing that each planetary mass receives in succession a finite incre-
ment, and then finding the values of all the constants corresponding to this
increased mass in the same manner as for the assumed masses. By this means we
have a set of values corresponding to the assumed masses, and another set corre-
sponding to a finite increment to each of the planetary masses. Then, taking the

INTRODUCTION.

xv

difference between the two sets of constants, and dividing by the increment, which
produced it, we get the coefficient of the variation of the constants for any other
finite increment of mass to the same planet ; but, on account of the importance of
the earth's mass, and the probable inaccuracy of its assumed value, we have pre-
pared separate solutions corresponding to the several increments of J^, ^, and ^V
of its assumed mass ; and a comparison of the values which the different solutions
give for the superior limit of the eccentricity of the earth's orbit has suggested
the inquiry whether there may not be some unknown physical relation between the
masses and mean distances of the different planets. The same peculiarity in the
elements of the orbit of Venus is also found to depend upon particular values of
the mass of that planet. But without entering into details in regard to the pecu-
liarity referred to, we here give the several values of the masses of these two
planets which we have employed in our computations, and the corresponding values
of the superior limit of the eccentricity of their orbits.

Mass mf
m'

For Venus.

Maximum e'
0.070G33

V) 0.074372

0.076075
0.075029
0.072098

For the Earth.

Mass m" Maximum e".

m"0 0.067135

0.069389
0.069649
0.068089

These numbers show that if the mass of Venus were to be increased, the supe-
rior limit of the eccentricity of her orbit would also increase until it had attained
a maximum value, after which a further increase of her mass would diminish that
limit; and the same remark is also applicable to the eccentricity of the earth's
orbit.

The above numbers indicate that the superior limit of the eccentricity of the

orbit of Venus is a maximum if the mass of that plant is equal to m'0 M -| — -);

» /^U /

or, if m' = -- of the sun's mass ; and the superior limit of the eccentricity of

3o449()

the earth's orbit is a maximum if the earth's mass is equal to m"0 M -| — -; \

A\J /

or, if m" =

tne sun's mass. But this value of the earth's mass corre-

sponds to a solar parallax of 8". 7 30, a value closely approximating to the recent
determinations of that element.

If, then, the mass of Venus is equal to -^-.-. — , and the mass of the earth is equal

Ot)^tTci/ \J

to — , it follows that the orbits of these two planets will ultimately become

more eccentric from the mutual attraction of the other planets, than they would for
any other values of their respective masses; and we may now inquire whether such
coincidence between the superior limits of the eccentricities and the masses of these
two planets has any physical significance, or is merely accidental.

C July, 1872.

xviii INTRODUCTION.

Since ' the mean motions and mean distances of the planets are invariable, and
independent of the eccentricities of the orbits, it would seem that there could be
no connection between these elements, by means of which the stability of the sys-
tem might be secured or impaired ; but a more careful examination shows that,
although the mean motions or times of revolution of the planets arc invariable,
their actual velocities, or the variation of their mean velocities, depends wholly on
the eccentricities ; and, were any of the planetary orbits to become extremely ellip-
tical, the velocity of the planet would be subject to great variations of velocity, —
moving with very great rapidity when in perihelion, and with extreme slowness
when in the neighborhood of its aphelion; and it is evident that when the planet
Avas in this latter position a small foreign force acting upon it might so change its
velocity as to completely change the elements of its orbit, by causing it to fall upon
the sun or fly off into remoter space. A system of bodies moving in very eccen-
trical orbits is, therefore, one of manifest instability; and if it can also be shown
that a system of bodies moving in circular orbits is one of unstable equilibrium, it
would seem that, between the two supposed conditions, a system might exist which
should possess a greater degree of stability than either. The idea is thus suggested
of the existence of a system of bodies in which the masses of the different bodies
are so adjusted to their mean distances as to insure to the system a greater degree
of permanence than would be possible by any other distribution of masses. The
mathematical expression of a criterion for such distribution of masses has not yet
been fully developed; and the preceding illustrations have been introduced here,
more for the purpose of calling the attention of mathematicians and astronomers
to this interesting problem than for any certain light we have yet been able to
obtain in regard to its solution.

MEMOIR

THE SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBITS OF THE

EIGHT PRINCIPAL PLANETS,

MERCURY VENUS THE EARTH MARS, JUPITER, SATURN, URANUS, AND NEPTUNE.

CHAPTER I.
ON THE SECULAR VARIATIONS OF THE ECCENTRICITIES AND PERIHELIA.

1. We shall assume as the basis of our computation the following differential
equations, which determine the instantaneous variations of the eccentricities and
places of the perihelia of the planetary orbits at any time. These equations are
demonstrated by LA PLACE, in Book II, Chapter VII, of the Mecanique Celeste;
and by PONTECOULANT, in Book II, Chapter VIII, of his Theorie Analytique du
Syst&me du Monde, and are as follows : —

ft = | (o,o+(o,8)+(o,8)+&c. } i -E]

* -

di~

dhf_ I
dt = \

§=-{(i,o)+<i,0+0,«)+&e.}A'-

&c.

If we denote by e, e1, e", &c., js, rf, -a", &c., the eccentricities and longitudes of

the perihelia of the orbits of Mercury, Venus, the Earth, &c., we shall have the

following equations for the determination of these quantities : —

h=e sin or, h'=el sin c/, K'=e" sin

I =e cos or, Z' =e' cos or*, Z" =e" cos GT",

Whence we deduce

— Ogn?" — [oTIJZ"' — &c.;

—rjl"—\l'"— &c.;

, (A)

r», &c., )
", &c. j

n

, &c.; tan »=y tan CT'=, &c. (2)

If h, Z, A', I', &c., are determined by the integrals of equations (A), for any time
whatever, and substituted in equations (2), we shall obtain the corresponding values
of e, e", &c., cr, cr7, &c.

May, 1871. \ ( 1 )

SECULAR VARIATIONS OF THE ELEMENTS OF

2. Now to find the integrals of equations (A), we shall suppose

7t=iV sin (flr«+/3), h'=N' sin (gt+Q) h"=N" sin («/<+/?), &c., ) g
I=N cos (0<-j-0)» ?=#' cos (gt+P) l"=N" cos (flrt-j-0), &c. ) ^
If these values be substituted in equations (A), they will become,

Ng ={(o,i)+(o,z)-j-(o,s)-f- &c.\ N— [oTHiV— [oTTj^V"— [oT^Y'"— &c.

»

/— 27^"'"— &c.

&c. JV>— To^—

&c.

If we suppose the number of planets whose mutual action is considered, to be i,
the number of these equations will be i; and by eliminating the constant quantities
N, N1, N"t &c., we shall obtain a final equation in g of the degree i.

3. The quantities (0,1) and (1,0), [0,1 1, [ I,Q| ; (0,2) and (2,0), |o,2|, 1 2 . o | ; (1,2) and
(2,1); |i,8| and |2,i|, &c., have some remarkable relations with each other, which
not only facilitate their computation, but render the equations resulting from the
elimination of N, N', N", &c., much shorter and more commodious. The general
expression for (0,1) is

In this equation n and a denote the mean motion and mean distance of Mercury,
m' denotes the mass of Venus and a' its mean distance from the sun. If we change
n, a into »', a', and m', a' into m, a, respectively, (0,1) will change it into (1,0), and
we shall have

Now since (a, a')'=(a', a)', equations (4) and (5) will give

m n'a'

we shall also have

&c.

na m'

na m

na m"

(6)

The same relations also hold with respect to the quantities [oTT], [To], [oTFI, [TTo],
&c., so that we shall also have

rri=|TT|~ n'a>

8 , 0 = I 0 , i

3,0 = 0,3

na m'

na m"

na m'"

&c.

(8)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

Therefore when we have computed the quantities (0,1), (°,2), (°,3), &c-> or the
coefficient for an interior planet, depending on the action of an exterior planet, we
shall obtain the corresponding coefficient for an exterior planet, depending on the
action of an interior planet, by means of equations (6), (7), and (8).

Equations (6) and (7) may be written as follows : —

, . in f , m
(1,0) = (0,1)

n'a' na

, m" , > TO

I 2 , 0) = (0,2)

' ri'a" ' na

&c.

We shall also similarly have

(\

TO" , ,

m' "

2, 1 1

' 1

n"o"

n'a'

M

TO'"

(1 ,3

TO'

n'"a'"

n'a'

(•\

TO'"

TO"

3,2)

^-— (2,3

/

ri"a'"

ri'a"

&c.
We shall therefore have

(3,0)(0,2)(2,3) = (0,3)(3,2)(2,0),

&c.
We shall also have the following products of four factors

&c
And of five factors we have

, O) = (l, 0)(0,4)(4,S)(S,

(9)

(10)

(H)

(12)

j. . (13)

(1,2)(2,4)(4,3)(3,5)(5,1) = (2,1)(1,a)(5,3)(3,4)(4,2),

&c. .

And in like manner we may form the products of six, or any number of factors.
Equations (11), (12), and (13) are very useful in reducing two terms to a single
one. We may in like manner form the following equations : —

(14)

SECULAR VARIATIONS OF THE ELEMENTS OF

• (15)

H] [Hil [HI] HH] — [HiD H3 [HE GUI] »

•ZEl Hill EH] E3 ~ E3 E3 EH] UH] >

0^1|l.8||«.8||g.4l[4.o| = |l.Q||o.4||4.S||3.a||a.l|,

0,1 1,2 a,* h,3 3,0 = 1,0 0,3 3,4 4,2 2,1 ,

1,2 2,4 4,3 3,5 3,1 = 2,1 1,6 5,3 3,4 4,2 I

4. The quantities (0,1), (0,2), (0,3), &c., (1,0), (1,2), (1,3), &c.; [uj], FT^l. 1^1, &c.,
[TTo], [T7T|, QTT], &c. ; depend on the masses and mean distances of the different
planets. The analytical expressions of (<M) and [ » , ' | are as follows: Mec. Gel.
[1076], [1082], Sowditch's Translation.

(18)

In these equations n denotes the mean motion of the disturbed planet ; m' the

mass of the disturbing planet ; a the ratio of the mean distances of the inner and

outer planets, b($ and b(^ depend entirely on a, and are given by equations [989]

Mec. C61. If we reduce the coefficients of the different powers of a to decimal

numbers we shall have the following equations to determine b($ and b(}k\

1 6i») = 1 + 0.25a2 + 0.015625a4 -f 0.003906249a6 + 0.001525878a8

-f- 0.0007476805a10 -f 0.0004205703au + 0.0002596378aw

+ 0.0001714015a16 + 0.0001 190288a18 4- O.OOOOSSggSSGa20

4- 0.00006414336a22 + 0.00004910978aM -f- 0.00003843058a26

-j- O.OOOOSOGSGGSa28 -j- 0.00002481567a80 4- &c.

ilV=— a + 0.125a3 + 0.015625a8 + 0. 0048828 12a7 + 0.002136230a9
4- 0.001 121521au + 0.0006608960a13 + 0.0004219114a15
4- 0.0002856691a" + 0.0002023490a19 + 0.0001485425a21
-j- 0.0001 122509a23 4- O.OOOOSGSSGSOa28 + 0.00006862602a27
+ 0.00005514591 a29 + 0.00004497838a81 + &c.
If we now multiply equation (19) by 1 -f a2, and equation (18) by a, and put

. | (l+a2)6[V+|a#.(J) | »=&(0), (20)

we shall have

6<°> = — 0.625a4 + 0.15625a6 + 0.024414061a8 + 0.008544920a10

+ 0.004005431a12 + 0.002202987a14 + 0.001 3424452aM
\ + 0.0008789820^ + 0.00()6070469a20 + 0.0004368899a23
4- 0.0003249368a24 + 0.00024824720,26 + 0.0001 939431a28
4- 0.0001 544085601"° -j- 0.0001 2493996as2 + &c.

(19)

(21)

(22)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 5

It will manifestly be unnecessary to compute b(V separately, since it is already
included in the value of &(0). Therefore taking logarithmic coefficients of equations
(19) and (21), we shall have the two following working formulae for the computation
of bm and b(V.
&<«>= — a4^ 0.625— [9.1938200]a2 — [8.3876400]a* — [7.9317080]a8

— [7.6026493]a8 — [7.34301 19]a10 — [7.1278964]a12

— [6.9439800]au — [6.7832222]a16 — [6.6403720]a18

- [6.51179S9]*20 — [6.3948844]a22 — [6.2876743]a24

- [6.1886714]a20 — [6.0967014] x28— &c. }

&(i>= _ a\ 1 — [9.0969100]a2 — [8.1938200]a4 — [7.6886700]a6

- [7.3296480]a8 — [7. 049 80 7 3] a10 — [6.8201332]a12
-[6.6252212]a14 — [6.4558633] x1G — [6.3061010]a18

- [6.1718509]a20 — [6.050 1898]a22 — [5.9389523]a24

- [5.8364888]a20 — [5.7415133]a28 — [5.6530038]a30— &c.

Then we shall have

3

c«.o= -

. (23)

5. To reduce the preceding formulae to numbers we shall assume the following
values of the invariable elements of the eight principal planets.

Invariable Elements of tlie Eight Principal Planets.

Masses.

Mean motions
in a Julian year.

Mean distances from
the sun.

Mercury ....

l+H
771 _ _ !__£ .

" 4865751

n = 538101 6".2

a =0.3870987

Venus ....

, — * + ^'

n' = 2106641 .438

a' =0.7233323

390000

The Earth . . .

1 + 1""

n" = 1295977 .440

a" = 1.0000000

368689

Mars

m •*• 1 /*'"

n'" = 689050 .9023

a'" = 1.5236878

= 2680637

Jupiter ....

IY

'' 1047.879

n'F= 109256.719

a'v= 5.202798

Saturn ....

v 1 + ^

nr == 43996 .127

av = 9.538852

3501.6

Uranus ....

m " ~T~ P

nw= 15424.5094

avl= 19.183581

24905

Neptune ....

"18780

nF"= 7873 .993

av"= 30.03386

6

SECULAR VARIATIONS OF T U E ELEMENTS OF

From these quantities we shall obtain

m log. 93.3128501 ;
m' log. 94.4089354;

log. 94.4333398 ;

log. 93.5717620;

log. 96.9796889 ;

log. 96.4557335 ;

log. 95.6037135;
mra log. 95.7263044;

m"

m'
mf
mr

the following logarithms

a log. 9.58782172;
log. 9.85933786;
log. 0.00000000 ;
log. 0.18289600;
log. 0.71623697;
log. 0.97949611;
log. 1.28292968;
log. 1.47761116;

a'

a"
a'"
a']
ar
ar

a

n

n'

n"

n"

n"

nr

nn

log. 6.7308643;
log. 6.3235906 ;
log. 6.1125974;
log. 5.8382513;
log. 5.0384481 ;
log. 4.6434145;
log. 4.1882114;
log. 3.8961950;

m —

- na

m' -

- n'a'

m" -

- n"a"

m'" -

- n'"a'"

m" -

- nrra'

mT '—

- nrar

mvi -

-nvla

m7"-

- nv"a

log. 86.9941641 ;
log. 88.2260070;
log. 88.3207424 ;
log. 87.5506147;
log. 91.2250039 ;
log. 90.8328229 ;
log. 90.1325724;
log. 90.3524982.

The values of a, a', a", &c., give the following values of a and 1 — a2.

For Mercury and Venus a=0.5351603 log. 9.72848386; 1

Earth a=0.3870987 log. 9.58782172; 1

Mars a=0.2540538 log. 9.40492572 ; 1

Jupiter a=0.07440202 log. 8.87158475; 1

Saturn a=0.04058127 log. 8.60832561 ; 1

Uranus a=0.02017864 log. 8.30489204; 1

«« Neptune a=Q.Q!2888U log. 8.11021056; 1

— a2 lot

-a- log. 9.8534569 ;
-a2 log. 9.9294980 ;
-a2 log. 9.9710237;
-a2 log. 9.9975892;
-a2 log. 9.9992842;
-a2 log. 9.9998231 ;
—a2 log. 9.9999279.

For Venus and Earth a=0.7233323 log. 9.85933786 ; 1— a2 log. 9.6783274;

Mars a=0.4747247 log. 9.67644186 ; 1— a2 log. 9.8890979 ;

J^er a=0. 1390276 log. 9.14310089 ; 1— a2 log. 9.9915235 ;

Saturn a=0.07583011 log. 8.87984175; 1— a2 log. 9.9974955 ;

Uranus a=(X03770580 log. 8.57640818; 1— a2 log. 9.9993822;

Neptune a=0.02408390 log. 8.38172670; 1— a2 log. 9.9997480.

For Earth and Mars a=0.6563025 log. 9.81710400; 1— a2 log. 9.7553161 ;

Jupiter a=0.1922043 log. 9.28376303 ; 1— a2 log. 9.9836522 ;

Saturn a=0.1048344 log. 9.02050389; 1— a2 log. 9.9952006 ;

Uran-as a=0.0521279 log. 8.71707032; 1— a2 log. 9.9988183 ;

Neptune a=0.03329575 log. 8.52238884; 1— a2 log. 9.9995183.

For Mars and Jupiter a=0.2928593 log. 9.46665903 ; 1— a2 log. 9.9610571 ;

Saturn a=0.1597349 log. 9.20339989 ; 1— a2 log. 9.9887751 ;

Uranus a=0.07942665 log. 8.89996632; 1— a2 log. 9.9972515;

Neptune a=0.05073232 log. 8.70528484 ; 1— a2 log. 9 9988808.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 7

For Jupiter and Saturn a=0.5454324 log. 9.73674086; 1— a2 log. 9.8466486;

Uranus a=0.2712110 log. 9.43330729; 1— a2 log. 9.9668195;

Neptune a=0.1732311 log. 9.23862581 ; I—a? log. 9.9867677.

For Saturn and Uranus <x=0.4972404 log. 9.69656643; 1— a2 log. 9.8766519;

Neptune a=0. 3 176033 log. 9.50188495; 1— a2 log. 9.9538216.

For Uranus and Neptune a=0. 6881 318 log. 9.80531852; 1— a2 log. 9.7723377.

6. If we now substitute these several values of a inequations (22) and (23),
we shall obtain the following values of the quantities 6(0) and by.

Mercury and Venus log. 6<°> = 98.6758747% ; log. 61V = 99.7120142% ;

Earth log. 6(0) = 98.1301668%;

" Mars log. 6<0> = 97.4084445% ;

" Jupiter log. 6(0) = 95.2816171%;

Saturn log. 6(0) = 94.2290035% ;
Uranus log. 6<°> = 93.0154041%;

log. 61V = 99.5794468%;
log. 61!' = 99.40 13786%;
log. 61V = 98.8712839%;
log. by = 98.6082362% ;
log. by = 98.3048699% ;

Neptune log. 6(0) = 92.2367042% ; log. by = 98.1102016%.

Venus and

= 99.8275394»;

= 99.6636495^;

Earth log. 6<°> = 99. 1656339%; log. 61

Mars log. 6<°> = 98.4754676% ; log. 6^

Jupiter log. 6<°> = 96.3661735% ; log. 61!

Saturn log. 6<°> = 65.3146217% ; log. 61't> = 98.8795292% ;

Uranus log. 6<°> = 94.1013583% ; log. by = 98.5763310% ;

Neptune log. 6(0> = 93.3227239% ; log. by = 98.3816952%.

TheEarth and Mars log. 6<°> = 99.0105934% ; log. by = 99.7915113% ;
Jupiter log 6(0) = 96.9268788% ; log. 61V = 99.2817435% ;
« Saturn log. 6(0) = 95.8766987%;
Uranus log. 6(0) = 94.6638660% ;

log. 61V = 99.0199061% ;
log. 6lJ> = 98.7169227%;

Neptune log. 6(0) = 93.8853150% ; log. 61V = 98.5223286%.

(C

u

97.6529713% ; log. 61V = 99.4619255% ;

96.6066892% ; log 61j> = 99.2020080% ;

95.3950591% ; log. 61V = 98.8996234% r

Neptune log. 6(0) = 94.6167398% ; log. 6<V = 98.7051450%.

Mars and Jupiter log. 6(0>
Saturn log. 6(0)

Uranus log.

Jupiter and Saturn log. 6(0) :
Uranus log. 6(0) :
Neptune log. 6(0) ;

a
»

: 98.7074558%;

; 97.5209527%; log.

: 96.7470972%; log.

' = 99.7195915%;
= 99.4292578%;
1 = 99.2369875%.

Saturn and Uranus log.
" Neptune log.

= 98.5532200% ; log. 61V = 99.6824668% ;
= 97.7921436% ; log. 61V = 99.4963019%.

Uranus and Neptune log. 6(0) = 98.9667121% ; log. by = 99.7812070%.

8

SECULAR VARIATIONS OF THE ELEMENTS OP

7. We must now substitute the values of #,', and the corresponding values of a,
in equation (24) ; and change m', successively into m", m'", m'r, &c., and we shall
obtain the values of (o,i)> (°.2), (°'3)> &c-» or the coefficient of the action of each
of the planets on Mercury. The characters 0, 1, 2, 3, 4, 5, 6, 7, refer respectively
to Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. Then
changing n into ri, and m', into m", m'", &c., we shall obtain the values of (1,2),
(1,3), &c., or the action of each outer planet on Venus. And in like manner we
shall obtain (2,3), (2,4), (2,5), &c. ; (3,4), (3,5), &c., or the action of each outer planet
on each inner one. We shall then obtain (1,0), (2,0), (2,1), &c., or the action of an
inner planet on an outer one by means of the equations (6), (7), (10), &c.

In this manner we have obtained the following results : —

(0,1) =

(0,7) =

(2,0) =
(2.1)=

(•'«)=
(2,6)=

(3,0) =

). 2".9986729

). 0.8617070

). 0.0279815

'). 1.6028375

). 0.0772642

r). 0.0013324

rO. 0.0004603

). 0". 1758273

). 6.6305873

). 0.1020355

r). 4.2028443

). 0.1988873

r). 0.0034100

r/). 0.0011765

). 0".0406239

). 5.3310972

). 0.2982001

'). 7.0682646

). 0.3265163

'). 0.0055565

rO. 0.0019144

). (T.0077700
). 0.4832186
). 1 .7564488
(,,4) = (1-fy ).14 .6598964
(i,. ) = (!+!«'' ). 0.6313987
(i, e) = (!-{-/'). 0.0105215
(3, 7) = (1-|-/"). 0.0036105

log. 0.4769291;
log. 9.9353596 ;
log. 8.4468702;
log. 0.2048895;
log. 8.8879781 ;
log. 7.1246469;
log. 6.6629969.

log. 9
log. 0
log. 9
log. 0
log. 9
log. 7
log. 7.

log. 8.
log. 0.
log. 9.
log. 0.
log. 9.
log. 7.
log. 7.

2450862;
8215520;
0087513;
6235433;
2986071;
5327484;
0706089.

6087813;
7268166;
4745078 ;
8493128;
5139049;
7447989 ;
2820328.

log. 7.8904196 ;
log. 9.6841436;
log. 0.2446355 ;
log. 1.1661309;
log. 9.8003037;
log. 8.0220791 ;
log. 7.5575701.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

). 0".0000942.0
). 0.0042125.6
(4,j)=(l-(-/ )• 0.0088115.3

(4,s)=(l-|y
(4,5)=(l-j-/

log. 5.9740497 ;

log. 7.6245464;

log. 7.9450513;

log. 7.4917417;

log. 0.8690189 ;

log. 8.8794563 ;

log. 8.3805175.

log. 5.0493193 ;

log. 6.6917912;

log. 7.0018244;

log. 6.5180955;

log. 1.2611999;

log. 9.4444852 ;

log. 8.8372088.

log. 93.9862386 ;

log. 95.6261830;

log. 95.9329689 ;

log. 95.4401214;

log. 99.9718878;

log. 0.1447357;

log. 99.6367457.

). 0".0000002.0168 log. 93.3046628 ;
). 0.0000087.926 log. 94.9441177;
). 0.0000177.94 log. 95.2502770 ;
(7,3)=(14-^"' ). o .0000056.975 log. 94.7556866;

log. 99.2530232;

log. 99.3175335;

log. 99.4168199.

In like manner formulas (25) and (8) will give the following values of

0 .0031027.1
7 .3963746.3
0 .0757628.5
0 .0240169.3
0".0000112.0
0 .0004918.0
0 .0010042.1
(5,3)=(l_)-ft'" ). 0.0003296.8
(5,4)=:(l-|-^jr).18 .2473541
(«,<>)=(l-h*TJ)- 0.2782820.5
"). 0.0687398.9

). 0".0000009:688

). 0.0000422.85

). 0.0000856.98

). 0.0000275.50

'). 0.9373198.2

). 1.3955188.2

'). 0.4332570.2

). 0.1790701.5
(7,«)=(14.#t'r ). 0 .2077464.0
). 0.2611078.3

7i]=(l+^ ).1".926868
Ts]=(l-f/i' ).0. 4087579
78l=(l-j-p* ).0. 008812816
I]=(l-)_/r).0. 1489646
).0. 00391854
).0. 0000336068
-0- 00000741495

).0."1129820
)-5. 520785
).0. 0587105

[Hl]=(l+iaiy ).0. 7286137
QT3=(l+iit1' ).(). 0188385
QT3=(l-j-ftw).0. 00016069

.0. 0000354172

log. 0.2848519;
log. 99.6114662;
log. 97.9451147;
log. 99.1730832;
log. 97.5931242;
log. 95.5264270;
log. 94.8701084.

log. 99.0530090 ;
log. 0.7420008;
log. 98.7687157;
log. 99.8624973 ;
log. 98.2750461 ;
log. 96.2059893;
log. 95.5492142.

July, 1871.

10

SECULAR VARIATIONS OF THE ELEMENTS OF

]=(1+^

!=(!+/

. 0".0192703

log. 98.2848879 ;

. 4.438798

log. 0.6472654;

. 0.2293041

log. 99.3604119;

. 1 .690254

log. 0.2279520;

. 0.0427287

log. 98.6307197;

. 0.000361930 -

log. 96.5586316;

r 0 .0000796657

log. 95.9012715.

. 0".00244717

log. 97.3886641 ;

. 0.2780404

log. 99.4441080 ;

. 1 .350640

log. 0.1305396;

, 5.307483

log. 0.7248886;

. 0.1256652

log. 99.0992151 ;

. 0.001043788

log. 97.0186122;

. 0.000228889

log. 96.3596252.

. 0".0000087547

log. 94.9422434 ;

. 0.00073030

log. 96.8635004 ;

, 0.00210713

log. 97.3236905 ;

. 0.00112331

log. 97.0504994 ;

, 4.835390

log. 0.6844315;

. 0.0254429

log. 98.4055666 ;

. 0.00518090

log. 97-.7144056.

. 0".00000056815

log. 93.7544654;

. 0.0000465833

log. 95.6682302;

. 0.000131413

log. 96.1186392;

. 0.0000656156

log. 95.8170069;

.11 .92923

log. 1.0766125;

. 0.1671612

log. 99.2231355;

. 0.0269346

log. 98.4303106.

. 0".000000024435

log. 92.3880187;

. 0.0000019926

log. 94.2994239 ;

. 0.00000558215

log. 94.7468016;

. 0.0000027331

log. 94.4366545 ;

. 0.3147736

log. 99.4979981 ;

. 0.8382741

log. 99.9233860 ;

. 0.3255696

log. 99.5126438.

. 0".00000()003249

log. 91.5117743;

. 0.00000026468

log. 93.4227230;

. 0.000000740484

log. 93.8695157;

. 0.000000361195

log. 93.5577417;

. 0.0386288

log. 98.5869113;

. 0.0814020

log. 98.9106353;

. 0.1962086

log. 99.2927180.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 11

8. The values of (0,1), (0,2), &c., [oTT], [oTa], &c., (1,0), (1,2), &c., QTo], [T^\, &c.,
being substituted in equations (B), supposing ^, ^', ^", &c., to be equal to nothing,
we shall have a series of numerical equations which are perfectly symmetrical in
form, between g and the unknown quantities N, N', N", &c. If we then eliminate
N, N', N" from these equations, we shall obtain a final equation in g, of a degree
equal to the number of original equations. The construction of this final equation
in g is the most delicate and intricate problem connected with the actual determi-
nation of the secular inequalities. Theoretically speaking, this equation may be
formed by eliminating the quantities N, N', N", &c., by the ordinary methods of
elimination in algebra. But this method, though direct and simple in theory, leads
to impracticable operations when we attempt to apply it to the formation of the
equation of the eighth degree which is necessary in the simultaneous determination
of the secular inequalities of the eight principal planets. The only actual merit
of this method seems to be that of leaving the algebraic values of N',N", N'", &c.,
in the successive eliminations, in very good form for computation, when the value
of g has been determined ; while its defects are twofold, as follows : First, it intro-
duces foreign facts depending on g, which raise the final equation in g, to a very
high degree ; and secondly, it necessitates the employment of a very great number
of decimals in the successive eliminations, in order to obtain a near approximation
to the correct value of the final equation. The method of determinants enables us
to form the final equation in g without actually performing the eliminations of the
unknown quantities N, N', N", &c. It also enables us to estimate, in advance, the
exact amount of labor necessary for forming the final equation arising from any
number of linear symmetrical equations. In the year 1860, we published a short
paper on this interesting branch of analysis in GOULD'S Astronomical Journal, Vol.
VI. From the explanation and formulae there given, it follows that each of the
given equations contains one binomial term of the form g — a, and that each term
of the final equation contains one factor from each of the given equations ; and
also that the whole number of terms in the final equation is equal to the continued
product of the numbers 1, 2, 3, 4, &c., to n inclusive; n denoting the number of
given equations. In the present case n is equal to eight ; there being one equation
corresponding to each of the eight principal planets. The whole number of terms
in the equation of the eighth degree is therefore equal to 1.2.3.4.5.6.7.8 =
40320. There are therefore 40320 distinct terms in the equation of the eighth
degree, each of which contains eight factors which are either monomial or binomial.
They are distributed in the following order : —

1 term having 8 binomial factors producing 9 monomial terms.
28 terms " 6 " " " 196

112 " " 5 " " " 672 " "

630 " " 4 ". " " 3150 " "

2464 " " 3 " " " 9856 " "

7420 " " 2 " " " 22260 "

14832 " 1 " " " 29664

14833 " without binomial factors " 14833

Total 40320 80640

SECULAR VARIATIONS OF THE ELEMENTS OF

The equation of the eighth degree when completely developed is therefore com-
posed of 80640 distinct monomial terms, each of which contains eight factors. The
actual formation of this equation could therefore with difficulty be brought within
the compass of an ordinary lifetime ; and we must, therefore, seek for a shorter and
more expeditious method of attaining results which seem to necessarily involve such
an immense expenditure of labor.

9. For this purpose we shall resume equations (B) of § 2, and shall suppose

, 3)+ &c. £ ;

&c- !

&c.

By this means equations (B) will be reduced to the following : —

Now since the coefficients of Nir, N\ NVI, and NT" of the first four of the
equations (B') are independent of y, and also the coefficients of N, N\ N" and Nm,
in the last four of equations (B'), we may divide them into two distinct groups, and
determine the values of g, N\ N", N'", &c., by successive approximations. We shall
therefore suppose

(27)

(28)

Substituting these quantities in equations (B'), they will become

(B")

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

13

417jA-'-|-61=0,

3^w-fia=o,

If we now suppose b, b, b", &c. to be equal to nothing, and eliminate IV, N", and
N" from equations (B"), and N\ N", and JVr// from equations (B'"), the resulting
equations will be divisible by N and N'v respectively ; and we shall have

+[iTT][TT7][T]nii^+ir^r^r^

, -2 2 , ii o , 3 —

3,2 2,0 0,1 1,3 — 3,0 0,1 1,2 2,3 + 3, 2 2,1 1,0 0,3

4,4 .I,* 8,6 7,7 — 7,6 6,7 4,4 5,5 — 7,8 5,7 4,4 8,6 — 7,4 4,7 a, 5 6,

6,05,64,47,7 6,44,65,57,7 .',,44,56,67,7

7,b6,44,S-,,7 7,4 4.5 -,,66,7 -4-7,6 6,5 5,4 4,7

Each of these equations is evidently of the fourth degree in g, and consequently
has four roots, which may be found by any of the ordinary methods of finding
the roots of numerical equations. These roots will be only approximate, because
we have neglected b, b\ b", &c., in the determination of equations (29) and (30).
If we substitute the approximate roots derived from equation (29) in any three of
equations (5"), we can find by elimination the values of N\ N", and N'" in terms of
N, which remains indeterminate. When N', N", and N'" have been thus deter-
mined we must substitute their values in equations (28), and we shall obtain the
values J1? 525 ^3, and b^ in terms of N; and these quantities are then to be substituted
in equations (-B'"), together with the corresponding value of g; and we shall then
obtain the values of N'\ Nr, NVI, and NT", in terms N. But instead of performing
this operation separately for each of the roots, in the manner described above, it is
better to deduce a system of algebraic equations, not only for the purpose of facili-
tating the numerical calculations, but also for the purpose of devising checks to
the accuracy of the different parts of the computations.

10. If we now assume the following quantities

(32)

14

SECULAR VARIATIONS OF THE ELEMENTS OF

TO] I 0 , 1 I 1 . 3 | I 2. ill 1 ,0|[0.3| |

j

o7l|[l,2|riT3| Io,2|[-J.l||l.3| I

L—J I0-3)

: )

ED f(;

[3, 2j [ 2.1 ] ^0,2|[3,3][3.l| I

i~ . r° • * i

J

5,4|j4,8|[6,:| [i, 6 [[6, 4][ 4. 7 [ 1

Frn~~ ~ E3 r(;

1

(45)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

15

,(46)

B

„ J I 4,lil| 5, fl | . ) , „ f | 4 , 5 1 1 li , f] | . ) ,

1 (50)

,(49)

=B +C +E ;

?';1
fr=B"-\-C'"+F'"; j

iTJ:; I (52)

/.==j

we shall have the following system of equations for the rigorous determination of
g, N', N", JV'", &c.— observing that the terms of equations (29) and (30), which are
inclosed by braces, are reduced to single terms by means of equations (14) and (15).

6,4)[4,6|[s,a]| 7,7|

=(£4, J&, ^6,^7); (54)

16

SECULAR VARIATIONS OF THE ELEMENTS OF

/ D3 /) /•

I ^ | ^_

f [HO

I HTTl

I EH]/) / Ci3n /• I . J EZ]

= s |^=-^3/3 — p^M.A r ~=~ 1 jYYi —

J EZ

- rHE

1

f -*•

/En]

1 pj— n

moi M-

T71 I
Z11

m I

\r'

A'N+f

- (57)

'

" ±±Lly ; (58)

"

(55)

(56)

^RTU"Z>"— |2,i||o,:,M'

; (60)

-

; (61)

Nr"=,

'1/4— rr»ipr?i

3Z?3— |4,V||0,S|

. (64)

11. Equations (53) to (64) are entirely rigorous. They are, moreover, under a
very simple and convenient form for the computation of JV, N", N'", &c. • and, as
there are duplicate and independent formulae for all these quantities, any error that
may accidentally creep into the computation of one formula is at once detected by
computation of the other. They have also this additional advantage : sometimes
one of the formula? for N, N", N", &c., gives value for these quantities of the form

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 17

N=--' „ --> in which a is very nearly equal to a', and the computation of these

(t

quantities cannot be readily effected by logarithms with sufficient precision to give
their difference correct to more than three or four significant figures; and, in all
these cases, the other formula for the same quantity gives a value which is free
from this source of error.

The computation of the successive approximations to the values of the required
quantities is then arranged as follows: —

We first find the roots of equation (53), on the supposition that £ is equal to
nothing. We shall designate these roots by </, glt <jr2, and ga, and the correspond-
ing values of the second member by £, £15 £2, and £3. The roots of equation (54)
are also designated by g±, g6, g$, and g7, and the corresponding values of the second
member by £4, £6, £„, and £7. When g, g^ gz, &c. have been determined, we must
transform equation (53) into others whose roots shall be smaller by the values
<]•> ffii <Jzi an(l ffz- Then if we denote the corrections to be applied to </, g±, &c., in
order to obtain their correct values by 8g, %gu &g2, and ^3, we shall have a system
of equations for the determination of &g, <5<71? &/2, and ^3, of the following form: —

2+c 3# =% ;

' • (65)

0. (66) [Equation of condition.]

The equations for the determination of £</4, ^6, ^G, and <?<77 are entirely similar.
Then, having determined the approximate values of g, g^ g^ &c., we must substi-
tute them in succession in equations (31 — 42) inclusive, and we shall obtain the
values of A, A', A", &c., D, D, D'\ &c., which are to be substituted in equations
(57 — 59), and we shall obtain the approximate values of N', N", and N". These
quantities are then to be substituted in equations (28), and we shall get the values
of &j, 625 ^31 and bi; which quantities, together with the value of g, are to be sub-
stituted in equations (47 — 50) inclusive, and we obtain -B1? J52, S5, C^ C2, &c., &c.
Then equation (52) will give /15 /2, &c., which are to be substituted in equations
(61—64), and we shall obtain N'\ N\ Nrl, and Nv". Equations (27) will then
give ft, b', b", and &'", which are to be substituted along with g in equations (43 — 46).
Then equations (51) will give/, /',/", &c., which being substituted in equations
(54) will give the value of £, on which the value of $g depends in the first of
equations (65). When $g has been determined we must add it to the approximate
value of f/, and repeat the whole computation with the corrected value of g, and by
this means we shall obtain the correct values of g, N', N", N'", &c. In like manner
we shall obtain glt NJ, Nf, Nf, &c. ; gra, N2, N,H, &c.

12. We shall now reduce the preceding formulse to numbers, and illustrate by a
numerical example the extreme simplicity of the formula; (which appear so unwieldy
in their algebraic form), and the comparative facility with which the required
quantities can be obtained.

3 August, 1371.

18

SECULAR VARIATIONS OF THE ELEMENTS OF

Substituting the values of (0,1), (0,2), (0,3), &c., given in §7, in equations (26),
we shall get the following values of |Q.Q|, [_'.i|» [~.'-|, &c. : —

[o7o]=0— 5".5702558;
[7771=0— 11 .3147682 ; ;
rr«1=g— 13.0721730;
HTTI =0—17.5528645;

[I7T]=0— 7".5123754;
[»7a1=<y— 18.5962129;
[77jf]=0— 2.7662522;
[777] =0— o .6479569.

These give

(67)

[oTo]QTr]=02— 16.8850240.0+ 63.02615319170556 ;
[oTojjTTJ =02— 18.6424288.0+ 72.81534747185340 ;
[073 [373] =02— 23.1231203.0+ 97.77394528773910;
[777] pm =02— 24.386941 2.0+ 1 47.90860736529860 ;
[77JJ[37j]=:02— 28.8676327.0+198.60659306350890;
[I7?1 [77^=0*— 30.6250375.0+229.45408138955850;

]=02— 26.1085883.0+139.70173232312266;

]=<7'— 10.2786276.0+ 20.78112497747588 ;
[J77]=02— 8.1603323.0+ 4.86769547582026;
[T7J]=03— 21.3624651.0+ 51.44181484629338 ;
[7JJ=02— 19.2441698.0+ 12.04954446242401 ;

]=03— 3.4142091.0+ 1.79241220013018;

]=0*— 47.5100615.0S+809.584727769664.02 ) .
-5804.515976137376.0+14461.60808412039 j ;

]=0«— 29.5227974.03+230.634324285266.02)
—523.768277980466.0+250.403089395286 j '

(68)
(69)
(70)
(71)
(72)
(73)

(74)
(75)
(76)
(77)
(78)
(79)

13. We shall now give the computation of equations (31 — 50), and also of (53)
and (54) in full, because we can then readily correct them for any assumed changes
in the adopted masses; and observe at the same time that the coefficients in
equations (55 — 64) are independent of the masses of the planets, and consequently
are not affected by their supposed variation.

Sum of terms

Computation of A.

=ga-— 18.642428800.0+ 72.815347472
= —21.562395123.0+120.10805649
[:];]= — 0.016959303.0+ 0.22169494
= + 0.1803729465

!!B' 4- o.oi596936

— 0.00787688
= <f— 40.22178322.0+193.3335643

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS

19

Computation of A.

l=0'— 23.123120300..y+97.77394529

— HHlCnilEI]— [H3= - o.oi4363307.#+ O.OSOOOTS

_[sTg[T7o]|]>;T|— [772]=: — 0.008365 164.0+ 0.1468326

+ ED DUE [EH— E3= 4- 0.0002436

4-[T7s][37olEE— [TT3= + 0.0000106

_[sTo]^7?] - 0.0000216

Sum of terms A'= g~— 23.14584877.0+98.0010178

Computation of A'.

[oTo] Q7T]=02— 16.885024000.0+63.02615319
_ [oTojjTTjQT]?] — [|T3]= . 1.136499372.^+ 6.33059222
— [ZZIUZKIZ]— HH]= — 0.0.00740612.^+ 0.00837985

+HUE3EH]— Hil]= + 0.00950700

+[]TT] [173 [»7T] _[?:¥]= -}- 0.01927424

_[TTo][77T] — 0.21770124

Sum of terms 4"=0a— 1 8.022263984.^+69. 1 7620526

Computation of D.

24.386941200.^+147.908607365
= — 10.635634399#.+120.339738
= — 12.836685468.(/+167.803373

+276.789684
+ 12.087392
' 24.5056485
D=f— 47.859261067.^+700.423146

Computation of V.
=<72— 28.867632700.^+198.606593
:= - 0.0291 19780.(/+ 0.329484
= -26.024746029.^+456.808841
=: + 0.373801

= + 0.033095

0.016324
^=<72— 54.92149851.^+656.135490

Computation of 17.
=ga— 30.625037500.^+229.4540814
^ — 0.001271660.J/+ 0.0166233
j^ _ 0.941628729.^+ 16.5282815
= + 0.0274200

+ 0.0135249
0.3097073
= g1— 31.56793789.0+245.7302238

Sum of terms

Sum of terms

—
Sum of terms

20

SECULAR VARIATIONS OF THE ELEMENTS OF

Computation of A^
(T7T][77T|=(y2— 10.278627600.0+20.78112498

— E3E3E3— ELN • 2.020545804.04-15.17909859

— [E3E3E3— EjQ= - 2.294602698.0-j- 6.34744976

+ |s,4|[4,«'][o7r] _ [777]=: + 3.66870084

-f E3E3EZ]— E3= + 0.01012112

_[7T4][4T6j = — 0.00800875

Sum of terms ,41=02-14.593776102.0+45.97848654

Computation of A2.
=02— 8.160332300.0+4.86769548

I]= — 0.031614994.^-|-0.23750370
=: -1.815697936.^-j-1.17649401
= -(-0.07254385

= +0.00015836

-0.00020013

At=gt— 10.00764523.^+6.35419527

— EZHI
Sum of terms

Computation of A3.

T^|=<72— 26.108588300.<7+139.70173232
Z]— [13= — 0.069351012.^+ 0.52099084
l— [13= — 0.005009102.^+ 0.09315032
HKH3HII]— [H]= + 0.12592049

+|7:7][5TT][477]_ [777] = + 0.15913300

—(TTTJITrs] — 57.68249007

Sum of terms A^g1— 26.182948414.0+ 82.91843690

Computation of D^

(7T]][67I]==,7!!_21.362465100.<7+ 51.44181485
7]-!-[477j== — 1.598839244.^-f- 29.732355

— [T7T]= —25.138334450.0+ 69.538973

— [77T|= + 50.793156
_ |77fJ= 4- 0.110881

0.140127
Sum of terms Dl=g2— 48.09963879..(/ +201.477053

Computation of Z>2.

77r\=f— 19.244169800..<7+12.04954446
—[TTT] [777] [776] -^[176]= _ 0.039953699.^4- 0.7429875
]-:-[473= —31.768769968.^4-20.5847937

+ 1.0043694
+ 0.0027708

— [77J](TrfJ — 0.0021925

Sum of terms A=92— 51.052893467.//+34.3822734

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

21

Sum of terms

Computation of Dz.

77]=/— 3.4142091000.^+1.7924122001
— [Ts]= -0.0000872187..7+0.0002412688
— [HI = —0.00441 08379.^+0.0028580329

13= +0.0001394486

—0.0638795429
Da=<f— 3.4187071566.^+1.7319476368

Computation of B, B', and B'.

r?7*1=/7— 13".072173Q
— [rTJO[JJ][]^[i7jJ = — 21 -5623951
Sum =B^b=ff—34: .6345681 ;

prm=/7— 17".55286450Q
— |77jg[Trj]-^[T7J] = — 0.014363307
Sum =B-^b=g— 17.567227807;

[T7T|=g— 11".31476820

— [H3[T7jr]-^[T:?]= — 1 .13649937

Sum =B"^b=g—12 .45126757.

Computation of C, C', C", and C'".

— [7751=13".0721730— . 0
+[oTJ][27s]-i-[oTJ]=10 .6356344

Sum =23 .7078074—^

log. j [oTsJ-i-ETs] \ = [9 .176390].
Therefore C'= \ 23". 7078074— g\ x
[9 .1763990]6'.

—[373]= -17 .55286450—?
+[o73][372]-:-[oT2]= 0.02911978

Sum = 17 .58198428— (/

log. S[«Z]-^[III]! = [ 8.8694654]
Therefore C'= \ 17".58198428— g \ X
[8 .8694654]6'.

+^Tj[27r|H-[I73]=+ 0.170596
— HZO =— 1.926868

Sum =C"^-V=— 1.756272
Therefore C"=— [0.24459 17]6'.

+[o7J][37T]^-[37g=+ 0.084146
— Pvr| =_ 1.926868

Sum =Cm-±-V=— 1.842722
Therefore C""=— [0.2654598]?/.

Computation of J5lf J52, and Bz.

E3=(7— 2".7662522
-Q7J][«7J]-T-[a7J]= — 2.0205458
Sum =B^-b1=g— 4.7867980;

[777]=^— 0".64795690
— EZHHI]-^EII]= — 0.0316149937
Sum =B2+-bi=g— 0.6795718937;

H7J]=*7— 18 .5962129
— [«7»] [777] -=-[777]= — 0.069351012
Sum =Bt-t-b1=ff—l8 .665563912.

Computation of C±, Cy, C3, and C^

— ^7J= 2". 7662522 — g
+ E3EZI^-EZI= 1 .598839244

Sum = 4.365091444—^

log. \ [i7J]-T-[77jJ ( = [9 .2840950]

Therefore <71= \ 4".36509 1444— -g \ x
[9. 2840950] 62. "

—[777]= 0.6479569 — g
+ [47JJ[T7J]H-|T3= 0.0399536996

Sum ^= 0 .6879105996— y

log. iES^-EIJNC9 -1824311]
Therefore <72= \ 0".6879 105996— •g \ X
[9 .1824311]J2.

(T77J[T7i]-^[]T7]=+ 0.01333975
— \TJ\ =— 4.835390

Sum =Cs-^b2=— 4.822050
Therefore C3=— [0.683231 7]52.

UJ3CIZ]— QZ1=+ 0.01055562
— pr^l =— 4.835390

Sum =C^bf=— 4.8248344
Therefore (74=— [0.6834824]?;,.

SECULAR VARIATIONS OF THE ELEMENTS OF

Computation of E, E', E", and E'".

+GJZ][I!3-HI1I]=+ 0.8287048
_[o75] =— 0.4087579

Sum =E-+V'=+ 0.4199469
Therefore E=-{- [9.6231944]Z>".

— HZ1=+ H".3147682— g
4-EZ)IIZI^-C2Zl:=+ 12.8366855

Sum =+ 24.1514537—.?

log. { [oTJ] -=-[T]£]|= [ 8.5847028]
Therefore E'= J24".1514537— g\ X
[ 8 .5847028J6".

— [sTs]=+ 17".5528645— g

+E3EIKEII]=:+ 0.00127166

Sum =+ 17 .55413616— ^

log.|[oTT]-T-[TTl|= [9.6375865]
Therefore E"= |17".55413616— g\ X
[9 .6375865]6".

EIDlII!]-5-|IZl=+ 9-360164
— Q>TTJ=— 0.408758
Sum =Em-±-V'=+ 8.951406
Therefore ^"'=+[0.9518913]J".

Computation of F, F', F", and F'".
]=+ 0.004346915
- 0.008812816

Sum =F+V"—— 0.004465901
Therefore F=— [7.6499091]6'".

EII][!:I]H-IIID=+ 0.09954018
-[Ell - 0.00881282

Sum =.P-f-Zr=-f- 0.09072736
Therefore F'=+ [8.957738,3]6'".

-|im=+13".072173QO— g
]^-En]=-i- 0.94162873
Sum =+14.01380173— g

log. \ (oTII-5-ED \ = [° -8407439]
Therefore F"= |14".01380173— g\ X
[0 .8407439]6".

— rr7]=+ir.3I47682— a

.0247460

Sum =+37 .3395142—^

(13 -^-[E3 1 = [9.4809266]'
Therefore F"= |37".3395142— g\ x
[9 .4809266]^"'.

Sum
Therefore

Sum

Computation of E^ Es, Es, and E^

0.03215367
0.02544-290
0.00671077
El— ['i.b'M I723]63.

=+18'.5962129— g
N +25 .1383344
=+43 .7345473—.?

log- IEI]-5-EIIl!= [8.2017618]'
Therefore E2= \ 43 .7345473— </\ X
[8 .'2017618]&a.

— QTI]=+0 .6479569—^
+[47Tj|77T]-:-[4]T]— -)-() .000047219
Sum =+0 .648004119— <j

log. {[T6|-!-[8T»]}= [0.7610455]
Therefore E3= J0".648004119— g\ X
[0 .7610455]53.

—

Sum =Fl-
Therefore

=— 0.025443
Sum =E4-f-63=:+11.629608
Therefore Et= [1.0655652]6S.

Computation of Flt F2, F3, and Ft.

0.004099602
=_ O.OQ5180905
-+-bt=— 0.001081303
F:=— [7.0339474]&4.

1-8779726
=— 0.0051809
-64=+ 1.8727917
JP2=+[0.2724895]64.

. 7662522— ^
-004410838
=-|-2 .770663038— g
= [1 .7737902]
= { 2¥.77Q663038— g\ X
.7737962]64.

:g=+18 .5962129— g

.76876997

=+50 .364982867— 5f
! = [ 9.11284867]
4= |50*.3649829— ^} X
[ 9 .1128486]&4.

—

Sum =j
Therefore

Sum

log. | [73 j-QT

Therefore Fa

Sum

log. {EH -HI
Themforc

THE ORBITS OP THE E I G U T P II I N C I P A L PLANETS.

23

Computation of the Equations of the Fourth Degree.
[T^Qg]^— 24.50564851./+ 566.64705848.,/— 2396.01393568

0.30970734.</2+
0.21 7701 24./+
0.01632389.</2+

0.00787688.^+

- [^[rng[rT|[][71]=- 0.00002157./+

— 2 (irri nrsi r^i r^n =

5.22941589.^—
6.66710860.^-
0.30431699.^-
0.22738681. g-
0.00052594..;;—
0.70396439.^-
0.40998748.^—
0.00055368.r/—
0.00045875.</—

19.51966234
49.95243773
1.18862983
1.56439984
0.00318986
3.92126177
7.19645470
0.00723785
0.00519060
0.06742367
0.00052850
0.00012858
0,01193875
0.'00588878
0.00052136

Sum of preceding terms= - 25.05727943./+ 580.19077701.^+ 2479.32266834

/— 47.5100615./+809.58472777./— 5804.51597614.^— 14461.60808412
Sum is the value of equation (53).

Whence we get <f— 47.5100615./ + 784.52744S34.// ) , 82,

-5224.32519913.^ +H982.28541576' } =

In like manner,

=— 57.682490()72.92+196.940082515.<!7— 103.390798939
0.140126895./+ 1.143482030.^- 0.682095055
1.667804687.^— 0.924082807
0.154121732.^—

0.063879543./+
0.008008749./+

0.0021 92S32./+ 0.022536218.^—
0.000200 132./+ 0.0()4275316.<7—
0.508856230.^7—
0.010061979.y—

0.096501781
0.045563277
0.010295162
0.329716905
0.027833971

0.008860222.^— 0.066561313
0.000639958.^- 0.011900801
+ 3.684731101
+ 0.000028044
+ 0.000017559

— 0.020330689
0.016087486

— 0.000044382

Sum of preceding terms=— 57.896897923.^+200.460720887.^— 109.937035864

/— 29.5227974.<78+230.634324285./— 523.768277980.^+250.403089395
Sum is value of equation (54).

Whence we get / — 29.S227974./ +172.7374263G2./ ) ,

—323.307557093.^+140.466053531 [ "

24

SECULAR VARIATIONS OF THE ELEMENTS OF

Equations (55-64) reduced to numbers are as follows, in which the numbers
inclosed in brackets are loarithms.

N' =

#*=[!. 4152972]

=[1.1305346]

=[0.36241 35] -"

(84)

(85)
(86)

~.

[9.1 763990]Z>/ ^[8.8694654] £/_
~[8M94654]Air — [9.1763990]^'^" ;

[8.5847028]Z>/" — [9.6375865]Z7/^
~[9.6375865]1''Z>"— [8.584702^' A ;

[0.8407439]:^r— [9^4809266]A/vr
~[9.4809266]^Zy —[0.8407439] A" D" '

; (90)

Nri =

Da

; (91)
; (92)

/r_[9.2840950]A/2 -[9.1824311] A/i .

" '

/r_[8.2017618]A/4 -[0.7610455]7).,/3 .
[0.7610455]13JD3— [8.2017618] AA'
,r_[1.7737962]A/« -[9.1 128486] A/6 . ,«*•.
- '

[9.9882719]£/ -[0.2952055]Z>/ -L

{ [8.9873765]Z>/" — [0.0402602W/" 1 l

J N

[8.6595749] A/"'"— [0.0193922]A/r } —

[0.5787943] A/, —[0.6804582] A A } A

| [7.4419161] A/* -[0.001 1998] A/3 } ^

[7.3400015]A/6 — [0.0009491]A/o

=(& &>Z»Z*)'' (96>

We shall here repeat and number the equations which we have computed, for
convenience of future reference. By this means we shall obtain the following

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

25

Fundamental Equations for flic
A =f— 40.22178322 .g+ 193.
A'=y*— 23.14584877 .0-j- 98.
A'=<f— 18.02226398 .g+ 69.
Av=(f— 14.593776102 .g\- 45.
-4,=^— 10.00764523 .g+ 6.
A3=f— 26.182948414 . 82.

D =£2— 47.859261067 .^+700.
D=(f— 54.92149851 #-{-656
Z>"=<?2— 31.56793789 .<7+245
A=0*— 48.09963879 .</+201
A=<72— 51.052893467 #-j- 34
D3=(f— 3.4l87071566.r/-|- 1

Adopted Masses.

3335643;

0010178;

17620526;

97848654;

35419527;

91843690;

423146;

135490;

7302238;

477053;

3822734;

,7319476368;

B=\g— 34.6345681 \l- # = {0—17.567227807 \b; \

; J

C=
C' =
C"=—

C'"=—

E'=

{23.
{ 17
\_ 0
[ 0.

9.

^24.
\ 17.

0.

£"=,7—12.451267577216

7078074 — # |[9.1763990]i' ;
58198428— ^[8.8694654]Z/;
2445917]6';
2654598J&';

( 98)

( 99)
(100)
(101)
(102)
(103)

(104)
(105)
(106)
(107)
(108)
(109)

(110)

F =—[ 7

F"=

F"=

\ 14
37

.6231944]&»

.1514537 — </}[8.5847028]6";

.55413616— #j[9.6375865]/f/;

.9518913]&";

.6499091]6"';
.9577383]^'";

.01380173— g \ [0.8407439]i'";
3395142 — # j[9.4809266]Z/";

Bt=|j--4i7867980}6l; B,= \g— 0.6795718937;

B3=\g— 18.665563912 j

G!= I 4.365091444 — <7J[9.2840950]62;

C2= { 0.6879105996— <7J[9.1824311]52;

Cfa=— [ 0.68823 17] J2, •

(7*=— [ 0.6834824]62;

7.8267723]&3;

{43.7345473 -0} [8.201 7618]*,;
j 0.648004119— r/ j [0.761 0455]&3;

1.0655652]6S;

FI=—[ 7.0339474] J4;

J'a=+[ 0.2724895]64;

^3= { 2.770663038— g \ [1.7737962]64;

Ft= {50.3649829 -^/j[9.1128486]i4;

August, 1871.

E2=
E3=

(Ill)

(112)
(113)

£!«"*

(115)
(116)
(117)

26 SECULAR VARIATIONS OF THE ELEMENTS OP

b = 1 0.1489647 . . . [9.1730832] ^"+[7.5931242].Vr+[95.5264270].Vr/

+[94.8701084].V"'

I' =^0.7286137 . . . [9.8624973]^+[8.2750461].V+[96.2059893].Vr'

+[95.5492142],V" , n ,

b" = \ 1.690254 . . . [0.2279520] | ArJT+[8.6307197].r+[96.5586316]iVr'

+[95.9012715]^™

i"'={5.307482 . . . [0.7248886]| jy/r+[9.0992151]ATr+[97.0186122]iV"

(119)

b^= \ 0.000008754742 . . . [94.9422434] } jV+[96.8635004]^V'

+[97.3236905] AT"+ [97.0504994]^'";

bz= \ 0.0000005681531 . . . [93.7544654] \ ^+[95.6682302]^'

+[96.1186392] ^"+[95.81 70069]iV";

b3= \ 0.00000002443536 . . . [92.38801 87] ^+[94.2994239]^'

+[94.7468016] AT"+[94.4366545]JV'";

^={0.000000003249184 . . . [91.51 17743] j JV+[93.4227230]^V'

+[93.8695157] ^"+[93.557741 7] AT/";
We have given the natural and logarithmic coefficients of N and Nir, in the
values of 6, b\ b", &c., because the values of the other quantities arc determined in
functions of these, and they will therefore be wanted.

14. If we now suppose the second members of equations (82) and (83) to be
equal to nothing, we shall obtain the following values of g, g^ g2, &c. : —
0 = 5".46370645 ; gt= 0",61668516;

ffl= 7.24769852; g,= 2.72772365;

gt= 17 .01424590; g6= 3.71780374;

<73=17 .78441063 ; J77=22 .46058485.

We must now transform equation (82) in four others whose roots shall be re-
spectively less by g, <jlt g2, and g3. Putting $g for the root of the first transformed
equation, we shall have the following equation to determine $g.

W— 25.6552V+184.8969V— 253.88 125y+z=0.

But since &/, &7j, &c. are very small quantities, we may neglect &g3, fy/4, in these
transformed equations, and we shall then get by dividing by the coefficients of Sg

to— - 4-

g~ 253.8812^ 253.8812 '

We may first neglect the- last term of this equation, and we shall obtain a first
approximation to the value of $g, with which we can compute the last term of the
equation. If we perform the same process with the other roots, we shall obtain the
following equations for determining $#, $glt fy/<2, &c.

tg =+[7.59537 ]x +[9-8623 ^ . (120)

«^i= -[7.73616 ]^-[9.5602 ]^2; (121)

^2=+[8.061074]^+[0.04511]^2a; (122)

%3=— [8.00000 -]%a— [G.16856^32; (123)

^=+[7.84466 ]^4+[9.92529%42 ; (124)

$74=— [8.38464 ]^6+[9.76863%62; (125)

fye=+[8.23997 ]^-[0.10692]^0a; (126)

^7=— [6.09265 ]^7— [9.17554]^72. (127)

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

27

15. We shall now give the computation of N1, N", &c., for the root g. Using
#=5".46370645, we get #2=29.8520882. Substituting these in equations (98-109),
we get the following values of A, A', &c., D, D1, &c.

A=+ 3.425636;
A'=+ 1.390983
A"=+ 0.5599336
A!=— 3.9055338
J2=— 18.472552
J3=— 30.285419

log. 0.5347412;
" 0.1433218;
" 9.7481365;
" 0.5916804n;
" 1.2665269n;
" 1.4812336«;

D =+468.78628
#=+385.91263
Z>"=+103.10436
D1=~ 31.473166
A=— 214.70366
Z>3=+ 12.899760

log. 2.6709749 ;
" 2.5864890 ;
" 2.0132764;
" 1.4979404n;
" 2.3318394n ;
" 1.1105816.

Substituting A, A', A", D, D, B', in equations (84-86), and neglecting/,/', &c.,
we shall get the following computation of N',N", and N'".

Computation of N' & N". Computation of N' & N'". Computation of N" & N'".

constant log. 0.3624135

constant log. 0.8236010

constant log. 1.1305346

A

!-=-£>

A"
constant

N'
N"

tt
u
1C

u

u

it

0.
7.
9.
1.

5347412
3290251
7481365
4152972

A
l-^D

A'
constant

N'
N'"

u

u
M
M

u
M

0.1433218
7.4135110
9.7481365
0.5190734

A

1-KD"
A

constant

N"
N'"

n

M

K
it

1C
1C

0.1433218
7.9867236
0.5347412
9.1592561

8.

8.

6873673
4924588

8.6873674
7.6807209

8.4924589
7.6807209

The computation of these quantities is thus seen to be correct, since independent
formula give the same values. We may now use these values of N\ N", and Nm,
in equations (119), and we shall obtain the following values of J,, &2> b& an(l ^4-

^=+0.0001151786^
^=+0.000007234609^

J3=+0.0000003080273^
64=+0.00000004087918JV

Equations (114-117) will now give,

^=+0.00007796538^
_B2=+0.0005510300JV
£3=— 0.001520571^V

(71=— 0.00000152882^"
<72=— 0.00000525886^
^=—0.0000348856^
Ct=— 0.0000349058^

^=+0.000000002067JV
^=+0.000000187594^
E3=— 0.00000855646^
^=+0.00000358228JV
^=—0.0000000000442^
^=+0.000000076558^
^3=— 0.00000653946^
^4=+0.000000238018^V

log. 96.0613718;
" 94.8594150;
" 93.4885892;
" 92.6115022;

95.8919018;

96.7411752;

97.1820068»;

94.1843555w;

94.7208918n;

95.5426467«;

95.5428974n;

91.3153615;

93.2732190 ;

94.9322943»;

94.5541593;

89.6454496n ;

92.8839917;

94.8155417n;

93.3766095.

28 SECULAR VARIATIONS OF THE ELEMENTS OF

Substituting these quantities in equations (52) we get,

^=-(-0.00007643863^ log. 95.8833129;

/2=4-0.0005457711JV " 96.7370106;

/,=— 0.001555269^" " 97.1918055n;

/4=+0.0005425501# " 96.7344399 ;

^=4-0.00007500820^ " 95.8751088;

/6=— 0.001555239^" " 97.1917972?*.

With these values of /„ /2, &c., and A» A2, As, Du Z>2, and D3, either of the
equations (93-95) will give

JV"=_0.00005102365JV log. 95.7077716«.

Therefore we shall easily find
^#""=4-0.0001992746^, J2tf'"=4-0.0009425372iV, J3Ar/r

Then

AzN'r+fz=— 0.000009996^";

A3N'r+f6=— 0.000009966M

Equations (90-92) will now give the following : —

constant log.

A^ir+/. "

1-s-A "

A*N"+fz »

constant "

Nr "

Nn "

Nr and Nri.

Computation of Nr and Nr".

0.7159050

constant

log. 0.8175689

96.4404576

A2Nir-\-f

; " 97.1726929

98.5020596n

1-4- A

" 97.6681606»

94.9998262n

J^y'-j-/

i " 94.9985209n

1.7982382

constant

" 0.8871514

95.6584222»

Nr

" 95.6584224»

95.3001240

Nr"

" 93.5538329

Computation of N" and Nv
constant log. 99.2389545
" 97.1717519
" 98.8894184
" 96.4381985
" 98.2262038

constant

N"
Nr"

" 95.3001248
" 93.5538207

The small differences in the different values of N TI and Nv", in this example,
are owing to the circumstance that A3N'V is nearly equal to /3 and /6, and has a
contrary sign, which renders AzN'r-\-fs and J3^V/r-f-/6 very small quantities. We
must therefore reject these values and use those depending on /4 and /s.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 29

Having found the values of Nir, Nr, Nrt, and N"1, we must now substitute them
in equations (118), and we shall get the following values of b, V, I", and V".
b =—0.000007745108^ log. 94.8890274»;
Z/ =—0.00003787069^ " 95.5783033» ;
If — _ 0.00008781746 JV " 95.9435809%;
b'"=— 0.0002754383JV " 96.4400243«.

Substituting these quantities, together with #, in equations (110-113), we get

B =+0.0002259314^ log. 96.3539766 ;

B' =+0.00009374307JV " 95.9719391 ;

B" =+0.00005411940JV " 95.7333530;

C =— 0.00010371 10JV " 96.0158248%;

C' =— 0.00003397893JV " 95.5312097%;

C" =+0.00006651123JV " 95.8228950;

e""=+0.00006978516JV " 95.8437631 ;

E —— 0.0000368787^ " 95.5667753%;

.#' =_ 0.00006307260JV " 95.7998407%;

E'= —0.0004609025 N. " 96.6636091«;

E"=— 0.0007860900^" " 96.8954722?i;

F =+0.00000123008^" " 94.0899334;

^'^—0.00002498980^ " 95.3977626«;

F"=— 0.01632072iV " 98.212739471 ;

^'"=—0.002657125^ " 97.42441 19n.

These quantities being substituted in equations (51) we shall obtain

/ =+0.000085341 IN log. 95.9311615;

/' =+0.00006099422^ " 95.7852886;

/" =+0.00005755803^V " 95.7601059 ;

/'"=— 0.0003921 492iV " 96.5934513n;

//r=— 0.01688088^" " 98.2273951«;

/^==_0.002533220Ar " 97.4036729».

Now substituting /, /', &c., D, D, D\ in equations (96), each one of them will
give x =+0.0243677.

This value of £ is to be substituted in equation (120), and we shall find

and consequently #= 5".4638024.

We have thus obtained ar first approximation to the values of the required
quantities. In order to get a closer approximation we must repeat the whole
computation by using the corrected value of </, and the values of /, /', /", &c.,
already found. But instead of computing new values of A, A\ A", &c., D, D, D", &c.,
by the formulae (98-109), it is much more convenient, as well as less laborious, to
compute corrections to these quantities depending on tig by formulae similar to

the following: —

U =2^+^-^40.22178 ....;)
— ^23.14584 ....;]

30 SECULAR VARIATIONS OF THE ELEMENTS OF

Then we shall have

Corrected A =A -\-&A ; ~|

A'=A'+&A; V (129)
&c. J

In this manner we shall obtain for the root </,

<j= 5".4638027 ;
N' =+0 .04864325JV
N" =+0 .03104381JV
Nm =+0 .004766696iV
N'T=— 0 .00005096246 JV
, NT =—0 .00004548871JV
N"=+0 .00001992532JV
JVF"=-|-0 .000000357473^

log. 98.6870226 ;

" 98.4919751 ;

" 97.6782174;

" 95.7072504»;

" 95.6579036?i ;

" 95.2994053 ;

" 93.5532432.

By performing similar calculations with reference to g^ #2, gz, &c., we shall obtain
the following values : —

gl= 7".2484269 ;
N,' =—0.7493120^
Ai" =—0.5714185^
N™ =—0 .0946700JV,
^"=+0 .0003959309 JVi
NS =+0 .0004045142JV;
^ir/=— 0 .0001020591^

log. 9.8746627n;
" 9.7569543n;
" 8.9762124?i ;
" 96.5976194;
" 96.6069338 ;
" 96.0088516w;

N1T"=— 0 .000004173243JVi
02= 17".0143734 ;

Njt' =+ 7.708429JV2
W =4-15 .383402v;
Natr=— 0.0007220048^2
N,r =— 0 .004362647^2
JV//=+ 0.000267212^
N™=+ 0 .00001963376^

g3= 17".7844562;
NJ =— 8.277302JV3
JV3" =+10 .207100JV3
N3m =—49 .61991JV3
JV3/r=+ 0.0006685073^
Nar =4- 0 .006909513^3
N3T'=— 0 .00039271 15NS
N3r"=— 0 .00002909962^,

94.6204737».

log. 0.8833640n;

" 0.8869658 ;

" 1.1870524;

" 96.8585401»;

" 97.6397511n;

" 96.4269521 ;

" 95.2930035.

log. 0.9178888«;

1.0089024;
" 1.6956560»;
" 96.8251062;
" 97.8394474;
« 96.5940804n ;
" 95.4638874«.

TUB ORBITS OF THE EIGHT PRINCIPAL PLANETS.

31

r4= 0".6166849;

=+ 0.1213144^'

=4- 0 .1843578^

=4- 0 .2135417^7"

=-f 0 .3454671AVr

=4- 1 .127803^4/F

.8892$,

<76= 2". 7276592;
N& =+ 0.2925180^
N; =4- 0 .2866292#6/r
JV5" =+ 0 .3000735 Ar/F
Nf =4- ° .3995364^/F

^5r =+ C
^5"=+15.29769iV5J

^sr//=- 1

,= 3".7166075;
=+0.5675117^

=4-° .3847378^6/^
=+0 .

JV6r =-0 -
Nri=— 1 .0394174

g,= 22".4608479;
• __ 0.006236689^"
•' =+ 0.02030250^

=— O.J

- 3.091803^/F
'=+ 0.1154716^/r
'=4- 0.00872852^

log. 9.0839135;

" 9.2656615;

" 9.3294826;

" 9.538406T;

" 0.0522322;

" 1.389161T;

" 2.1983503.

log. 9.4661526;

" 9.4573205 ;

" 9.4772276;

" 9.6015563;

" 9.9592151 ;

" 1.1846258;

" 0.1753554n.

log. 9.7539748;

" 9.5851649 ;

" 9.5782040 ;

" 9.6389698;.

" 9.8976854;

" 0.0167908«;

" 8.5173628.

log. 97.7949541n;

" 98.3075494;

" 99.1820317n;

" 99.9829761w;
" 0.49021 18/i ;

" 99.0624752;

" 97.9409405.

16. We have thus determined all the roots appertaining to the equation of the
eighth degree, together with the ratios of the constant quantities N', N", &c., to N;
NI, JVy, &c., to Ni ; and the ratios of the similar quantities relative to each of the
other roots. The complete integrals of the equations (A) will therefore be the
sums of all the corresponding terms depending on g, g^ g2, &c., and we shall there-
fore have

32

SECULAR VARIATIONS OF THE ELEMENTS OF

h —N sin (qt-\-^-\-Nt sin (<7i<+A)+^2 sin
&c.;

(CD

I =N COS (</<+/2)+^l COS (f/ii!+/3i)+Ar2 COS '

?=W'cos

r=i

&c.

|3, /?u /?2, &c., being arbitrary constant quantities. These equations contain twice
as many constant quantities as there are roots g, g^ g2, &c., of the differential
equations (A). These constant quantities are indeterminate by analysis ; but they
may be deduced from the values of 7t, h', 7t", &c., I, l\ I", &c., at any given epoch.
If we suppose t=0, in equations (C), and also suppose the first members to be
known quantities, the preceding equations will give the values of these constant
quantities by direct elimination. But in order to determine them more conveniently,
we shall resume the differential equations (A), and we shall multiply the first, third,
fifth, &c., by Nm-r-na^ N'm'-z-n'a', N"m"-z-n"a", &c., and they will become

771

771

m ( _

net I

-*w{E?

„ m" d_7^_ vrf_ f
7i ci dt n '&'' I

*«" r

—2r— I r^

n'a" 1 L

,}

, }r

,}

(130)

If we add these equations, and in their sum substitute the values of JV|(o,i
(o,»)-(-(o,3)-f-&c.|, JVj(i,0)-f-(i,2)-f (i,3)-f-&c.|, &c., deduced from equations (B),
we shall obtain

na dt n'a' dt ' n"a" dt

-f &c.

".

no1

OT' .\

"" w'a' "^ /

(131)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

33

But each of the factors of Z, I', I', &c., in the terms of the second member of this
equation, which are independent of y, becomes nothing by means of equations (9) ;
and we shall therefore have

m all

na dt

,m' dh^
n'a' dt

m

ri'a" dt

na

n'a

n a

n a

(132)

If we substitute in this equation the preceding values of h, A', 7t", &c., Z, T, I", &c. ,
we shall have, by comparing the coefficients of the same cosines,

HIT.

na
m
na

na

n a

f+ &o.=0 ;

l+ &c.=o ;

&c.

If we now multiply the preceding values of h, 7t', 7t", &c., respectively

' — — ,, N" „ „, we shall have by means of equations (133),
n'a' n a

h + N' "

(133)

N-"1,

na

y (134)

In like manner the preceding values of Z, Z', Z", &c., will give,

> (135)

If the origin of the time t, be fixed at the epoch for which the values of h, I, 7t', ?,
&c., are supposed to be known; the two preceding equations will give,

m ,

n

N

tan 3 =-

h + N' 7,'
na na

N'mrr
na

,
na

n a

n a

Z'"+&c.

(136)

Since N', N", N'", &c. are given in terms of N, this indeterminate quantity will
disappear from the expression of tan /3. Having found /?, we may substitute it in
either of equations (134) or (135), and we shall obtain the value of N; and conse-
quently N\ N", N'", &c. will be known. We shall thus have the system of con-
stant quantities corresponding to the root g. Changing in the preceding equations
the root g successively into g^ g.,, g3, &c., we shall obtain the values of the constant
quantities, corresponding to each of these roots.

5 September, 1871.

34

SECULAR VARIATIONS OF THE ELEMENTS OF

If we now put the first members of equations (134) and (135) equal to x and y
respectively and the coefficients of sin or cos (gt+P) in the same equations equal
to z, we shall have,

z=z sin (0H-/S) ; y=z cos (gi+fl. (137)

Whence tan (gt+^=x-r-y. (138)

17. In order to find the values of a; and y, it is necessary to know the values of
It and I. We shall therefore suppose that at the epoch of 1850, the eccentricities
and places of the perihelia of the eight principal planets have the following values,

Mercury, e
Venus, e'
The Earth, e'

=0.2056179

=0.00684184

=0.01677120

Mars,

Jupiter,

Saturn,

Uranus,

Neptune,

e'" =0.0931324
e'r =0.0482388
ev =0.0559956
e" =0.0462149
er"=0.00917396

log. e =9.3130610;
log. e' =7.8351730;
log. e" =8.2245642;
log. e'" =8.9691008;
log. e" =8.6833965;
log. er =8.7481539;
log. eri =8.6647821;
log. er"=7.9625568 ;

cj = 75° 7' 0".0;
d =129 28 51.7;
vf =100 21 41.0;
CT'"=333 17 47.8;
vs'r= 11 54 53.1;
crr = 90 6 12.0;
*jr'=170 34 17.6;
tffr"= 50 16 39 .1.

Now since h=e sin or, l=e cos or, 7t'=^ sin rs\ l=e' cos cy', &c., we shall obtain
the following values

h =+0.198720, log. K =

li =+°-00528°8, log. h' =
h" =+0.0164977,
h" =—0.0418510,
h'r =+0.0099592,
hr =+0.0559955,
h" =+0.0075707,
7ir"=+0.0070561,
I

=+0.052813,
7' =—0.0043502,
f =—0.0030164,
r =+0.083199,
I'7 =+0.0471996,
lr =—0.00010988,
r'=— 0.0455906,
Zr"=+0.0058628,

log. h"
log. h'"
log. h"
log. hv
log. hn
log. 7tr"
log. Z
log. Z'
log. T
log. T
log. r
log. 1T
log. Z"
log. Zr//

9.2989408 ;

7.722.6976;

8.2174237;

8.6217066rc;

7.9982241 ;

8.7481532;

7.8791377;

7.8485672;

8.7227434;

7.6385092ra;

7.4794898ra;

8.9201199;

8.6739377;

6.0042715n;

8.6588752»;

7.7681049.

We have already given the logarithms of m-~-na, m'-t-ria', m"-r-n"a", &c., in § 5 ;
and if we add them successively to log. h and log. I, log. 7t' and log. Z", &c., we
shall obtain the following values of the logarithms of the constants, for the given
epoch, which enter into the values of x and y.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 35

log. hm =86.2924049 ; log. I— =85.7169075 ;

na na

n-rt' fn-^

log. h'-- =85.9487046 ; log. 7— =85.8645162n;

7i ft ftd

m" m"

log. 7i"-,-,,=86.5381661 ; log. ?SrT,=85.8002322«;

it Ct> /v CL

log. 7i'"-^,=86. 17232137*; log. r^,=86. 4707346;

Y& to 7tf Ct

IV IV

log. 7ijr^-jr=89.2232280; log. ^-^=89.8989416;

IV \A> 79 W

V V

1TI 1YI

log. 7ir-r-F =89.5809761 ; log. r-^-r=86.8370944w;

7i CL ¥1 Ct

log. 7i"4^=88.01 17101; log. r'4^=88.7914476ra;

fL (Z ?i (Z

mv" mv"

log. 7^-^-^^88.2010654; log. Z"S«- ^=88.1206031.

fit \JU IV Ct

18. These quantities are now to be substituted in equations (136) and (137), in

connection with the values of N\ N", N'", &c., corresponding to the different roots.
For the root g=5". 4638027, we find

,1847539,. ,64178.9.. . 10467760 „

-r-w 2/==^

Whence ^=88° 0' 37".8, and log. ^V=9.2470063.

Therefore for root #, we have the following values,

jV =-|^0.1766064, NIV=— 0.0000090002,

N' ^-(-0.0085906, Nr =—0.0000080335,

N" =4-0.0054825, Nri =+0.0000035 168,

Nm =+0.00084182, ^m=-(-0.0000000631-31.

In like manner we shall find for the root #1=7".2484269,

1654860.4.. 4347600.6 173037154 .

Whence ft=20° 50' 19".4, and log. ^,=8.4294913.

Therefore for the root gl we have

N, =+0.02688384, N^ =+0.000010644,

NJ =—0.02014438, N^ =+0.000010875,

Nf =—0.01536192, N," = — 0.000002744,

Nf =—0.00254509, ^"=—0.0000001122.

For the root ^2=17".0143734, we find,

18893046,, , 40873423 .- . 3.068840 ._.

- --- -

/32=335° 11' 31".4, log. ^2=7.1665150.

N2 =+0.001467287, N2IV =—0.000001059,

^2'=— 0.01121706, W =—0.000006401,

Nf =+0.01 131047, Ntn =+0.000000392,

^'"=+0.02257186, N^"= +0.0000000288.

36

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root #,=17".7844562, we find,

, 0.01310358 ... 0.014106133 12.083015

a*=H ---- 10io -&**?= "To>»~ 10r°

&=137° 6' 36".5, log. ^3=7.2023283.
Na =+0.001593413, N," =+0.0000010652,

Ns' =-0.01318916, Nsr =+0.0000110097,

^"=+0.01626412, N3ri =-0.0000006258,

^"=-0.0790650, ^''"=-0.00000004636.

For the root 04=0".6166849, we find,

. 3.3572052 ,r 1.360283 56970.44

** ' «*- --

ft=67° 56' 34".9, log. JV/r=95. 8033371.

JV4 =+0.000007714, Ntir =+0.000063582,

^'=+0.000011722, NS =+0.000071708,

N{ =+0.000013577, NJ1 =+0.00155775,

^=+0.000021946, ^/"=+0.01003893.

For the root ^5=2".7276592, we find,

, 0.6475797 ..„ 0.1743025 345.0396

«3=H --- -.O -- ^6 , 2/5= - ----- ^ 2= -•"• •

/?6=105° 3 52".9, log. ^57r=9

Ns =+0.000568545, N6'r =+0.001943624,

Nb' =+0.000557100, N6r =+0.00176940,

N: =+0.000583230, ^6" =+0.02973295,

^6"'=+0.000776548, N™= -0.0029 10500.

•

For the root #6=3".7166075, we find,

, 0.4583186 v . 0.8566965 tr . 22.5112

x«=-\ --- W -- W« » 2/6= IO -- ^6 . ZG— ~*~~

/88=28° 8' 45".8, log. N,ir =98.6350825.

JV6 =+0.02449386, JV8/r =+0.04316011,

JV6' =+0.01660532, N6r =+0.03410106,

N," =+0.01634130, JVaF/ =—0.04486144,

^=+0.01879544, N™= +0.00 14205 13.

For the root 07=22".4608479, we find,

1.0095026 v/r , 0.7872144 AT/r 81.86052

1010 — ' ^T — ' ~ 10'" f ' 7~ 1()10

^=307° 56' 50".l, log. ^/r=98.1941887.

If, =—0.00009753, JV/r =+0.01563827,

Nj' =+0.00031 750, N,r =—0.04835044,

JVT" =—0.00237805, N7" =+0.001805776,

N7"=— 0.01503712, JV7v//=+O.Oj00136499.

T1IE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 37

If these values be substituted in equations (C), we shall have the complete values
of h, h', h", &c., /, l\ I", &c., from which we can obtain the eccentricity and place
of the perihelia by means of the formulas,

tan or =7t-f-Z ; e=-j/7t2+Zz=7j cosec c*=Z sec or. (139)

19. If, in equations (C), we put, instead of Ji and Z, their values, e sin «j, and
e cos or, we shall get

e sin a=N sin (</<+/3)+ JV; sin gJ+pJ+N, sin (#,*+&)+ &c. (140)
e cos s= JV cos (<7<+/3)+AT' oos fcH-jffi)-j--ft| cos (Jtf+ftH- &c- (141)

Multiplying equation (140) by cos (gtf+/3)> and (141) by sin (</<+/3), the differ-
ence of their products will give
e sin (s-j7«-/8)=^i sin | (fc-^+ft-/? } +iV2 sin { (gt-g)t+pa-p \ +&c. (142)

If we multiply equation (140) by sin (gt-\-(3), and (141) by cos (#<+/3) the sum
of their products will give

e cos (a— gt— /3)= 1 M43}

JV+JV-1cosl(i7l-^+/31-/31+^cos|(flr2-^+l82-/3|+&c. j k

Dividing equation (142) by (143), we shall eliminate e, and get,

^ sin | (ft-^+ft-ff | + JV2 sin 1 (ga-g)<+ft-ff | +&c

a(a-flrt-^-JV+ JVTCOS j fa-gy+pt-p | + Aicos 1 (gt-gyt+fr-p \ +»* **'

When the sum JVr1-(-^r2-}--^r3+&c., of the coefficients of the cosines of the denomi-
nator, all taken positively, is less than N, tan (or — j7< — /3) cannot become iniinite ;
the angle is — gt — (3 cannot therefore become equal to a right angle; consequently,
the mean motion of the perihelion will be in that case equal to gt. It is also easy
to see that when all the cosines of the denominator of tan (EJ — gt — /3) become equal
to ±1, Nlt N2, Ns, »fec., being supposed positive, the denominator will be either a
maximum or a minimum, and the numerator will be equal to nothing ; tan (ts—gt
— /3) will then be equal to nothing; or— gt — /? will therefore become equal to
nothing, and equation (143) will become,

e=N± I #i+JV2+ JV3+&c. I ; (145)

Consequently, the maximum and minimum values of e will be

maximum e=N-{- Nl-\-N2-\-N3-\-&c.; ) fid.R-\

and minimum 6=N—\Nl-\-N2-\-N3-\-8cc.\ J

We shall now substitute the numbers which we have already computed, in these
equations, for the purpose of determining the maximum and minimum values of
the eccentricities, and the mean motions of the perihelia.

20. For the planet Mercury, we have,

Maximum 6=^+^+^4-^3+^+^+^+^=0,2317185. One-half of this
is 0.1158593, which being less than N, it follows that N exceeds the sum of all the
remaining terms ; consequently, the mean annual motion of Mercury's perihelion
is equal to g or 5".4638027, and performs a complete revolution in 237197 years;
and the minimum value of the eccentricity is 0.1214943.

38 SECULAR VARIATIONS OF THE ELEMENTS OF

For the planet Venus we have,

Maximum e'=JV+JV1'+Ar2'+&c.=0.0706329. One-half of this is 0.0353165.
As this number exceeds any one of the coefficients, N', NJ, NJ, &c., it follows that
the perihelion of the orbit of Venus has no mean motion, and that the minimum
value of its eccentricity is zero.

For the Earth we have,

Maximum e"=JV"+Jtfi"+Ay'+&c. =0.0677352. One-half of this is 0.0338676.
As this number exceeds any one of the coefficients N", N^', N2", &c., it follows that
the perihelion of the Earth's orbit has no mean motion, and that the minimum value
of the eccentricity is zero.

For the planet Mars we have,

Maximum tr=2r+Nl"+W+&c. =0.1 396547. One-half of this is 0.0698274.
As this number is less than NJ", it follows that the perihelion of the orbit of Mars
has a mean annual motion equal to g3 or 17".7844562; and that the minimum
eccentricity of his orbit is equal to 0.0184753. We shall here observe that a small
variation in the assumed mass of the Earth would produce a considerable variation
in the limits of eccentricity and mean motion of the perihelion.

For the planet Jupiter we have,

Maximum e'r=JV/r+Ar1/r+JV/r+&c.=0.0608274. One-half of this is 0.0304137.
As this number is less than N6'T, it follows that the perihelion of the orbit of Jupi-
ter has a mean annual motion equal to g6 or 3".7166075; and that the minimum
value of the eccentricity is equal to 0.0254928.

For the planet Saturn we have,

Maximum er=Wr+Jv7-f-JV2''-f &c,=0.0843289. One-half of this is equal to
0.0421644. As this number is less than N7r, it follows that the perihelion of Saturn's
orbit has a mean annual motion equal to #, or 22".4608479 ; and that the minimum
value of the eccentricity is equal to 0.0123719.

For the planet Uranus we have,

Maximum eri=Nv/-\-N1"-\-N2r'-\-&c. =0.0779652. One-half of this is 0.0389826.
As this number is less than N6TI, it follows that the perihelion of the orbit of Uranus
has a mean annual motion equal to <?„ or 3".7166075 ; and that the minimum value
of the eccentricity is equal to 0.0117576.

For the planet Neptune we have,

Maximum er"=Nr"-\-Nlr"-}- N2v"-\-&c. =0.0145066. One-half of this is
0.0072533. As this number is less than JV/", it follows that the perihelion of
Neptune's orbit has a mean annual motion equal to g4 or 0".6166849 ; and that the
minimum value of the eccentricity is equal to 0.0055712.

21. -We see, by the preceding article, that the mean motions of the perihelia of
Jupiter and Uranus are exactly equal. It follows from this circumstance that the

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 39

mean relative positions of their perihelia will always be the same ; and we shall
now inquire what their mean relative positions are. For this purpose we shall
resume equation (144), and substitute in it the values corresponding to these two
planets. By this means we shall get the following equations,

tan(sr/r — g6t — /36) =

-frl +&C.'

tan(orrj-<70<— ,36)=

Now, since the mean values of the numerators of these equations are each equal
to nothing, and the signs of the denominators depend wholly on N6'r and N6"; it
follows that cr/r will always be equal to cr", if JV6/r has the same sign as N6"; and
CTJF will always differ from o" by two right angles if Nair and ^V6" have different
signs. According to the numbers which we have calculated, N6IV and N6yi have dif-
ferent signs ; consequently, the mean longitudes of the perihelia of the orbits of
Jupiter and Uranus differ by a semicircumference.

For the purpose of determining the maximum values of tan (rar/r — g^t — /36) and
tan (csvl—g6t — /3e), we may suppose them to be of the following form,

sin a.

Bm a.
1 cos a' (l<

the coefficients of cos a being supposed positive. These equations evidently attain
their maximum values when cos a is equal to the quotient of its coefficient divided
by the constant term of the denominator, taken negatively. If we reduce them to
numbers, they will become

_ 0.0176673 sing

-&<-&)= -o6431601+6:oi76673 cos a
F/ _ _ 0.0331038^sina _

—9tf— Pe)~ _o.04486 f4^07)33T038 cos a' ^ '

The first of these evidently attains a maximum value when a=±114° 10'; and
the second one when a=±42° 27'. Consequently, we shall find
Maximum value of (a'"— g6t— /36)=±24° 10';
and " " " (crr/— gj— (3 )=180°+47° 33'.

The nearest approach of the perihelia of these two planets wjll therefore be
108° IT.

40 SECULAR VARIATIONS OF THE ELEMENTS OF

The mean annual motions of the perihelia of the four planets Jupiter, Saturn,
Uranus, and Neptune being

Jupiter and Uranus 3". 7 166075;

Satvrn 22 .4608479 ;

Neptune 0.6166849;

it follows that the mean motion of Saturn's perihelion is very nearly six times that
of Jupiter and Uranus ; and this latter quantity is very nearly six times that of
Neptune; or, more exactly, 985 times the mean motion of Jupiter 's perihelion is
equal to 163 times that of Saturn; and 440 times the mean motion o.f Neptune's
perihelion is equal to 73 times that of Jupiter and Uranus. It also follows that
the perihelion of Saturn performs a complete revolution in the heavens i'n 57700
years ; that of Jupiter and Uranus in 348700 years ; while that of Neptune requires
no less than 2101560 years to perform the circuit of the heavens.

22. Having determined the numerical values of all the constants which enter into
the integrals of the differential equations (A), corresponding to the assumed masses,
it now remains to determine the changes which would be introduced into these con-
stants, on the supposition that the masses were changed from m to m(l--\-u), in' to
m'( 1+/"')) &c-> °r> on the supposition that the masses of all the planets were multi-
plied by a factor (l-|-a), a being any supposed correction of the mass of the planet,
and greater than — 1. If we suppose a, or /t, u', u", &c. to be equal to — 1, it is
evident the whole system of differential equations would vanish. The effect of
changing all the masses, in the ratio of 1 to 1-f-a, on the equation of the eighth
degree, would be equivalent to multiplying the coefficients of the different powers
of g by (l-|-a)8~n, n denoting the exponent of g in the given term. Consequently,
the coefficient of <f would remain unaltered ; that of (f would be multiplied by
1-j-a; that of g6 would be multiplied by (l-f-«)2> &c. ; while the term of the
equation which is independent of g would be multiplied by (1-j-a)8. It is evident
that these changes are such as would be produced by multiplying each of the roots
of the equation by 1-f-a ; consequently, we shall have the following theorem : —

If (lie masses of all tlie planets be simultaneously increased in the ratio of i W\ -|-a,
all the roots of the equation in g will be increased in the same ratio.

It is also evident that, if the masses of all the planets be multiplied by l-|-a, the
values of A, A', A", &c., D, D, Lf', &c. will all be multiplied by (1-f a)2; and, as
they are all multiplied by the same quantity, it is manifest that the ratios of the
quantities N, N, N", &c., will remain unaltered. And since the ratios of N, N', N",
&c., remain unaltered, it is evident that tan (3 will be unchanged, and consequently
the values of N, N\ N", &c. will not be changed. It therefore follows that, if the
masses of all tfa planets be simultaneously increased in any given ratio, tlie magni-
tudes of the secular inequalities will remain unchanged.

To illustrate,, we shall observe that if the masses of all the planets be supposed
to be doubled, the intensity of the disturbing forces would be doubled ; but, accord-
ing to the preceding theorem, the roots g, #„ g,, &c. would be doubled ; and conse-
quently the disturbing forces would operate in the same direction during only one-

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 41

half of the time ; and since a double force acting during one-half the time produces
the same effect as a single force acting during a double interval, it follows that the
magnitude of the resulting inequalities will remain unchanged.

23. The rigorous determination of the separate effects of the corrections of the
masses fi, fi, ft", &c. on the values of the constants which determine the secular
variations of the elements, when the masses simultaneously vary, is a much more
difficult and intricate problem than that of the determination of the secular inequali-
ties themselves. For, if we employ the masses in indeterminate forms, m(l-\-ft),
m'(l-f ft), m"(l-\-ft"), &c., instead of m, m', m", &c., it is evident that the solutions
of the differential equations would contain terms depending on ft, ft', ft", &c., and
on all the powers and products of these quantities up to [S, ft1, ft"7, inclusive, in
addition to the terms already calculated. And if we neglect the powers and pro-
ducts of ft, ft, ft", &c., above the first, it is evident that our solution would be very
imperfect, unless fi, ft', fi", &c. were very small quantities. Unfortunately, this is
not the^case, for the masses of some of the planets are still very imperfectly known;
and consequently the terms depending on the powers and products of ft, ft, fi", &c.
ought not to be neglected. There seems to be only one practicable method of
determining the effects of the corrections of the masses on the values of the con-
stants which we have already determined. And this method consists in supposing
the mass of each planet, in succession, to be increased by a finite quantity, //„, and
then determine anew all the constants in the same manner as with the assumed
masses. If we then subtract the values of the constants which depend on the
assumed masses, from the values of the constants which result from the corrected
mass of the planet, we shall obtain the coefficient of the correction depending on
ft, by dividing the difference of the constants by fi0, or the finite variation of the
mass of the planet. In this way we get the whole variation resulting from the
assumed variation of mass, or, in other words, we retain the terms depending on
all the powers ft0, fi02, jt/03, &c., and neglect only the terms depending on the products
ft, ft, ft", &c., when they simultaneously vary. As this is the method which we
have adopted, we shall here give the resulting fundamental equations, together with
the values of the constants determined by their solution.

24. We shall now suppose the mass of Mercury to be increased to two and one-
half times its assumed value. In this case fia will be equal to 1|, and OT—

4H6575l~ 1946300 4' value of the mass of Mercury is very nearty tne

same as that employed by astronomers, during the early part of the present century,
and is doubtless considerably larger than the actual value. But as the perturbations
produced by this planet are very small, a considerable variation of its mass will
produce only a small variation in the values of the fundamental equations. We
shall now compute the effect which this change in the mass of Mercury produces
in the fundamental equations.

6 October, 1871.

42

SECULAR VARIATIONS OF THE ELEMENTS OF

If we now put 1+^=2.5 in the values of (i ,o), (2,0), (3,0), &c., Q7_o], [To], QjTo], &c.,
and substitute the resulting numbers in equations (26), we shall get the following

values of ["7^1, |TT1, [J3> &c-

[TT4]==0— 7".5125167;

[T7T|=0— 18.5962297;

«7s]=0— 2.7662536;
777] =<7— 0.6479572.

[oTo]=0— 5".5702558;
(TT71=0— 11.5785090;
[Tg] =0—13.1331088;
pTT|=0— 17.5645194;

These values give

^lT7T\=g2— 17.1487648.0+ 64.4952569126 ;
EI!][I3=^— 18.7033646.04- 73.1547754652 ;
[oTo][3:i]=02— 23.1347752.04- 97.83886606206 ;
[TTTI [17^1=0'— 24.7116178.04-152.06181843878;
|TTT|=r/2— 29.1430284.0+203.37094595357;
]=/— 30.6976282.04-230.67674429991;

[T^/— 26.1087464.0+139.70448617829;
]=02— 10.2787703.04- 20.78152636644 ;
]=02— 8.1604739.04- 4.86778928589;
]=0a— 21.3624833.,v-(- 51.44188735405 ;
]=02— 19.2441869.04- 12.04956092697 ;
1=4"— 3.4142108.04- 1.792413937146.

]=0*— 47.8463930.0H- 821.59840712
-5935.67265019.04-14877.555887385

]=0*— 29.5229572.03+230.63766404.0
—523.77824644.0 4-250.40826811

(153)

(154)
(155)
(156)
(157)
(158)
(159)

(160)

(161)'

(162)

(163)

(164)

(165)

f } (166)
} (167)

The difference between these values and the similar quantities depending on the
assumed masses being denoted by A, we shall find the following values,
A[o7o]= 0. AE3=— 0.0001413, "

A[T7|=:— 0.2637408, A FT1=— 0.0000168,

A[J3=— 0.0609358, A [^1=— 0.0000014,

A[^\=— 0.0116549, A [Tm=— 0.0000003.

(168)

-0.2637408.^+1,
-0.0116549.^4-0
-0.2753957.^+4
-0.0609358.^4-0
-0.3246766.^4-4
-0.0725907.^+1

-0.0001581.^+0
-0.0001416.^4-0
-0.0000171.<7+0
-0.0001427.*9+0
-0.0000182.^+0
-0.000001 7..J7+0.

4691037;
06492077 ;
7643529 ;
33942800 ;
1532110 ;
2226629 ;
00275386 ;
00009381 ;
.00001647;
.00040139;
00007250 ;
0000017370.

(169)

THE ORBITS OF THE EIGHT PRINCIPAL PLANET 8.

43

The following calculation of AA must serve as an example for the computation
of all similar quantities.

— 0.06093580.0+ 0.33942800
0.02543896.</+ 0.33254241
+ 0.00258360
0.28269800
0.95725201

___

Sum of last three terms of A X [*= +

Sum of terms = &A= - 0.08637476.^+

Computed A=tf— 40.22178322.^+193.3335643

Corrected A=gz— 40.30815798.^+194.2908163.

If we proceed in like manner with A', A", &c., we shall obtain the following

Q

system of fundamental equations for ^=+^.

<Q

3 1

Fundamental Equations for fi=^ ; or for tJie case where wi=—

1946300.4
(170)
(171)
(172)
(173)
(174)
(175)

(176)
(177)
(178)
(179)
(180)
(181)

(182)

(183)

(184)

F =— [7.6499091]&'";

*"=+[8.9577383]&'"; (185)

F" = \ 14.07473753— # } [0.8407439]&'" ;

Fa= J37.6032551 — /j[9.4809266]ft'".

t = 1^—4.7867994 1 6,; Bf=\g— 0.6795721937 16, ;) (lg6)

£3=^—18.665580712 (Bj; j

A =/— 40.30815798.0 +194.2908163 ;
A'=g*— 23. 17005 142.«/ + 98.2867801;
A"=g*— 18.28711570.^ + 70.37498708;
Ai=f— 14.59391880.<7 + 45.9791766;
A*=f— 10.00778683.-7 + 6.35429409 ;
A3=f— 26.183106514.(/ + 82.92120064;

D =/_48 .18393767 .g +708.16361;
D—f— 55.19689421.0 +661-210840;
D"=f— 31.64052859.0 +246.9639388 ;
A=0"— 48.09965699.0 +201.477187;
D2=f— 51.05291057.0 + 34.3823000;
D3=y2— 3.4187088566.0+ 1.7319493752.

* = j^_34.6955039 } 6 ; B = {^-17".578882777 } b :

S>= j</— 12 ..71500846 \b :

C = ^23.7687432 — ^^[9.1763990]&';
C' = { 17.59363918— #j[8.8694654]6';
C'=— [0. 24459 17]fe';
C'"=— [0.2654598J6'.

E =+[9.6231944]6";

E'= J24.4151946 — ^|[8.5847028]6";

E"= \ 17.56579106— /j[9.6375865]6";

SECULAR VARIATIONS OF THE ELEMENTS OP

d= J4.365092844 — ^}[9.2840950]ft2;
C,= 10.6879108996— ^|[9.1824311]62;
Ct=— [0.68323 17]Z>2;
(74=— [0.6834824]i2;

^=+[7.8267723]^;

E2= |43.7345641 -^|[8.2017618]&3;

E3= 1 0.648004419— #j[0.7610455]&3;

j&4=+[1.0655652]63;

^1=_[7.0339474]&4;

^3= \ 2.770664438— g \ [1.7737962]64;
<= j 50.3649997 — ^}[9.1128486]64;

(186)

(187)

(188)

(191)

0«_47.8463930./+796.2()272817.<f-5344.10960488.<7 \ , ,.

+ 12307.3911771 j "
^-29.5229572./+172.74076612./- 323.31739720.flr) ' ,

+140.47094066 j "

And lastly, the values of bL, b2, b3, and &4 become,

bl=\ 0.00002188686 [95.3401834] ^+[96.8635004]Ar'

+[97.3236905] ^"+[97.0504994]^'" ;

b,= \ 0.00000 1420383 .... [94.1 524054] |# +[95.6682302]^'

+[96.1186392] ^"+[95.8170069]^'";

68= j 0.0000000610884 . . . [92.7859587] \N +[94.2994239]^'

+[94.7468016] ^"+[94.4366545]^'" ;

14=\ 0.0000000081 2296 . . [91.9097143] {# +[93.4227230]^'

+[93.8695157] ^"+[93.5577417]^'".

The values of 6, V, J", and V remain the same as in equations (118).
Equations (189) and (190) give, when the second members are put^equal to 0,

g= 5".34449540; gt= 0".616685510; %

ffi= 7.53805074; gb= 2.72773208;

^=17.12785195; g,= 3.71791243;

#,=17 .83599491 ; #=22 .46062783.

The solutions of equations (170-191) will now give the following values, re-
membering that the coefficients of equations (84-97) remain unchanged. For the
root g, we get,

(192)

^=5".3446763 ;
N1 =+0.10318901^
N" =+0.0652640tf
N" =+0.01001010^
N" =—0.0001 170589 N
Nr =— 0.0001036091^
tf r/ =+0.0000476249JV
JVr"=+0.000000750320#

log. 99.0136340;

" 98.8146736;

" 98.0004383 ;

" 96.0684043«;

" 96.01 53978n;

" 95.6778341 ;

" 93.8752464,

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.
For the root </ls we get the following values,

log. 99.9418576«;

" 99.8388736w;

" 99.0658468w;

" 96.6369257;

" 96.6566098;

" 96.0281973w;

" 94.6661815w.

45

NJ =— 0.8746970JVi
Nf =—0.6900382^
W =—0.1163716^
jV/r =+0.0004334367^
NS =+0.0004535340JVi
NJZ =—0.0001067081 Ai
^"'=—0.000004636406^

For the root </2, we get the following values,

#2= 17". 1279859;
Nj =— 7.663670.^
NJ =+ 7.458903JV2

IV=— 0.0007514405^2

r =— 0.00483281 8JV2

"=-}- 0.0002927626^2

r//= 0.00002153572JV^

For the root #,, Ave get the following values,

^3=17".8360300;
^a' =— 8.244660JV

iV3" =+
JV3'" =— 39.935747V3
N3ir=+ 0.0004835988^3
N,r =+ 0.005252281^,
JV3"=— 0.0002970608^3
N™=— 0.00002202336^3

For the root gt, we get the following values,
#4=0".6166852 ;

p; =+ 0. 18071 UN,1
V =+ 0.21 14780 Ntl
V™ =+ 0.3450307W;'

'=+ 24.500742V;7

log. 0.8844368/i;
" 0.8726750;
1.2600767;
" 96.8758946M;
" 97.6842004w;
" 96.4665156;
" 95.3331593.

log. 0.9161728w;

" 0.9875922;

" 1.6013617w;

" 96.6844852;

" 97.7203480;

" 96.4728453«;

" 95.3428835w.

log. 9.0781956;

" 9.2570000 ;

" 9.3252650;

" 9.5378578;

" 0.0522403 ;

" 1.3891794;

" 2.1983687.

46

SECULAR VARIATIONS OF THE ELEMENTS OF

6/r
"

For the root gb, we get the following values,

#5=2". 7276680;
Nt =+ 0.2886746#6/
N; =+ 0.2816212^
#6" =+ 0.296961 lN6ir
#/' =+ 0.3989168#6/r
N6r =+ 0.9103832#6'F
#6"=+15.29957#/r
#6F"=— 1.497633#5/r

For the root gw we get the following values,

£6=3".7167141 ;
N6 =+0.5652238#/r
NJ =+0.3828501#6'r
JV6" =+0.3770050#a'r

=+0.7901118

W;F//=± +0.03290417JV/r

For the root g7, we get the following values,

^7=22".4608918;
N, =— 0.006358233JV/
N,' =+0.02171584^'
^7" =— 0.1536641 N7ir
Nj" =— 0.9634742#7/F
JV/ =— 3.091784JV/r

r/r
r/r

JV7r'/=+0.00872848JV/

log. 9.4604086 ;

" 9.4496654 ;

" 9.4726996;

" 9.6008823 ;

" 9.9592242;

" 1.1846793;

" 0.1754053M.

log. 9.7522204;

" 9.5830288;

" 9.5763472;

" 9.6385614;

« 9.8976886;

" 0.0167438w;

« 8.5172509.

log. 7.8033365n;
« 8.3367766;
« 9.1865723w;

9.9838401n;

0.4902090» ;

9.0624732;

7.9409386.

25. We must now substitute the numbers we have computed, in the last article,

in equations (134) and (135) ; making use of the logarithms of 7/*-/t , —J, — , &c.,

7fr vv /» \A/ it/ U

which were used for the adopted masses, but using the following values of log.

m
na

m-

1 7« I I '" 7

log. _ft, log. --z.

na

m

na

log. —=87.3921041; log. ^^=86.6903449; log. ^=86.1148475.
1 na na na

For the root g=5". 344676, we get,

x=-

4618325

„

N; y=

259345

27356380

-JQM L' » it ~1 \()20 JO20

Whence /3=86° 47" 9" 2; and log. #=9.2281141.

# =+0.1690885; #'r= -0.0000198;

#-=+0.0174481; #"=—0.0000175;

#" =+0.0110355; #"=+0.00000805;

#" =+0.0016926; #"=+0.000000127.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 47

For the root gl—T.538699, we get,

4456102^. 5531411 253588600

xi— +~ 10-" *' Jl~ ' UP" "i --- 10* -- • * J

Whence £=38° 13' 16".2; log. JVi=8.4434891.

JV, =+0.0277645; Nf =+0.00001203;

Nl'=— 0.0242855; . #/ =+0.00001259;

#,"=— 0.0191586 ; Ai" =—0.000002963 ;

.AY"=— 0.0032310 ; #/"=— 0.0000001287.

For the root g2= 17". 127986, we get,

22856900 ,r , 49895050 Ar , 3.332267 ,

*2= ~~ *' y2==+' 2° 2; **=+- io N*>

Whence ^2=335° 23' 35". 1 ; log. JV2=7.2167768.

^ =+0.0016473; ^"=—0.00000124;

NJ =—0.0126245 ; Nar =—0.00000796 ;

N," =+0.0122872; Nz" =+0.000000482;

^"'^+0.0299815 ; JV2m=+0.0000000355.

For the root gz= 17". 836 030, we get,

, 111304510 A7 112878600,r . 8.789905

^3=-H --- 1(^5 -- f»i ^3= i(p ---- N*> %=-\ --- JQ10

Whence /33=135° 24' 8".2 ; log. V,=7.2561145.

N3 =+0.0018035 ; N3tr =+0.000000872 ;

NJ =—0.0148692 ; N3r =+0.00000947 ;

JV3" =+0.01 75270; #,"=—0.000000536;

#3'"=— 0.0720238 ; N3r"=— 0.0000000397.

For the root #4=0".6166852, we get,

. 3.358365 „„ , , 1.360318 V/F. 56975.27

SB*=H --- IO - ^* ' 2^*= ~ - ^ ' 2^=

/34=67° 56' 57".9 ; log. AT4/r=95.8034310.

N, =+0.00000761 ; N" =+0.00006359 ;

#; =+0.00001098 ; #4F =+0.00007172 ;

JV4''=+0.00001345; NJ1 =+0.00155816;

^'"=+0.00002194 ; JV4r;/=+0.0100415.

For the root #5= 2". 727688, we get,

. 0.6475863 v/r 0.1743984 v/r. 345.0960 ^

X;>~ "ioin — ' ^6= "lo70"" 6"~ "lo10 6 '

Whence &=105° 4' 20".9 ; log. AVr=7.2885615.

N, =+0.00056101 ; NbIV =+0.0019434 ;

#; =+0.00054730 ; N* =+0.0017692 ;

JV6" =+0.00057711 ; N™ =+0.0297331 ;

^6"'=+0.00077525 ; Ntm=- 0.0029105.

48 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root gr6=3".716714, we get,

0.8567341 „„. 22

io

0.4584871

„. , ,0.8567341

• ; 9*=+- - - -

22.51143_

"".to "t •

Whence #,=28° 9' 13".5 ; log. #6'r=8. 635 1298.

#6 =+0.0243977 ; #fl'r =+0.0431648 ;

#8 =+0.0165257; #/ =+0.0341050;

#; =+0.0162732; #u" = -0.0448615;

#8"=+0.0187798 ; #„" '=+0.0014203.

For the root ^7=22".460892, we get,

1.009497 .,„ , 0.7872134 „„. 81.35929 r,

r— ^7 ;yr= -,-TVO ^ ' '— ~r" 10'° 7 '

10"

10"

Whence ft=307° 56' 50".6 ; log. #/r=8.l941936.

#T =—0.0000994; #T/r =+0.0156384;

#7 =+0.0003396; #/ =—0.0483507 ;

#7 =—0.0024030 ; #/' =+0.0018058 ;

#/=-0.0150672 ; #7m=+0.00013650.

The difference between the values here given, and the values depending on the
adopted masses, manifestly measures the increment arising from the supposition
that fi=| ; and two-thirds of this difference is the coefficient of //, in the expression
for the values of the constants corresponding to any other value of //.

26. We shall now suppose that n'
=l -4-371428.6.

Ap^n=— 0'. 1499336;

Arrq=— 0.2665548 ;
A[I^1=— 0.0241609;

Whence,

[oTo]=i7— 5".7201894;
(TT)=0— 11 .3147682;
[Tnn=<7— 13.3387278;
|1TT1 =(7—17.5770254;

These quantities give,

I— 1*— 17.0349576.
1 =</'— 19.0589172.
[sTTj =g>— 23. 29721 48,
l—y2— 24.6534960
=p»— 28.8917986
l=<y»— 30.9157532

+^, and the mass of Venus will become
Using this mass, AVC shall find,

A[TT]=— 0".0002106; 1

A^Tj] =— 0 .-0000246 ; I , 9.}.

A|7TT|=— 0 .0000021 ; j

AH7T|=— 0 .0000004. J

j7— 7".5125860;
«/— 18.5962375;
g— 2.7662543;
<7— 0.6479573.

64.7226171211; (195)

76.3000493710; (196)

(7+100.5439143766; (197)

i/+150.9246131399; (198)

•7+198.8799680465; (199)

,<7+234.4551573443; (200)

(194)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

49

[I3[II!]=<72— 26.1088235.0+139.7058334952; (201)

[HH{HIl=f— 10.2788403.0+ 20.7817233266; (202)

jTTTlfTTT]^-- 8.1605433.0+ 4.8678349406; (203)

[EI]IM!=rf— 21.3624918.0+ 51.4419219482; (204)

ESEIINj/2— 19.2441948.0+ 12.0495678407; (205)

[67«][777]=:0»_ 3.4142116.0+ 1.79241466734. (206)

47.9507108.03+ 825.82631940.02 )

5994.8821218.0+15174.5513808 ' J ; (

- 29.5230351.03 +230.63929622.02
-523.78311550.0+250.41078507

We shall therefore obtain the following

Fundamental Equations for ju'=+— -; or for m'=

1

j '

A =f— 40.63827162.<7
A=gz— 23.31994327.0
A"=f— 18.22902235.0
J,=<72— 14.5939888.0
Js=flr8— 10.00785623.0

— -
20

+200.0557143 ;
+100.7733425;
+ 71.3586722;
+ 45.9795152;
+ 6.35434212;

J3=02— 26.183183614.0 + 82.92255281;

D =#2— 48.76765014.0 +728.640657;
D'=02— 56.24689671.0 +679.929057;
Z>"=02— 31.8586536.0 +250.7543893;
X>1=02— 48.09966549.0 +201-477252;
A=^2— 51.052918467.0 + 34.3823105;
Ds=g2— 3.4187096566.0+ 1.731950106.

— - - .
371428.6

(209)
(210)
(211)
(212)
(213)
(214)

(215)
(216)
(217)
(218)
(219)
(220)

5=^0—34

.9011230(6; B = \g— 17.591388707J6; ) „„..
B"=\g— 12.508092541 \b; J (

C = 1 23.9743622 — 0|[9.1763990]&'; 1
C"= S 17.60614518— 0| [8.8694654]&;
C*=— [0.265 7810]6';
(7'"=— [0.2866491]&;;

E =+[9.6231944]J"; 1

^'= ^24.7932880 — 0|[8.5847028]6"; |
^"= S 17.57829706— 0j [9.6375865J&";

.»

F =— [7.6499091]i"f;

JF'=+[8.9577383]6"';

.F" = \ 14.28035653—0 j [0.8407439]i'

F'"= |38.6407515 — 0j[9.4809266]//

October, 187L

50 SECULAR VARIATIONS OF THE ELEMENTS OF

R = {7— 4.786800116,; £,= ^—0.6795722937(6,;)

B3= ^-18.665588512*6,; J

C,= J4.365093544 -<7J[9.2840950]6,;
C2= {0.68791009996— <7J[9.1824311]&S;
C3=-[0.6832317]62;
C4=— [0.6834824]62;

.#1=+[7.8267723]63;

Ef= 543.7345719 — #j[8.2017618]63;

E3= | 0.648004519— j7J[0.7610455]63;

^=-f[1.0655652]63;

Fl=— [7.0339474]64;

.F2=+[0.2724895]&4; (228)

F3= \ 2.770665138— flrj[1.7737962]54;

Ft= \ 50.36500746 — «7}[9.1128486]64;

o4-47.9507108y+799.53205638.(/2— 5381.3548732.j7 ) , v

+12499.1914947 J "

^-29.5230351.^+172.74239830.^— 323.32220515.^ ) _,

+140.47332174 j

The values of b, J', 6", and lm are given by equations (118); and the values of
6,, 62, 68, and 64 are given by equations (119), by simply adding log.
[0.0211893] to the coefficients of N'.

Putting equations (229) and (230), equal to nothing, they will give,
g = 5".59773937 ; g,= 0".61668564;

gl= 7.25215980; gt= 2.72773622;

j72=17. 20072233; g.= 3.71796565;

&=17 .90008930 ; </7=22 .46064759.

The equations just computed will give the following values : —
For the root g, we get,

0=5".5978504;

JV =+0.05341742^" log. 98.7276830 ;

JV" =+0.03480002^ " 98.5415794;

N" =+0.00540574^ " 97.7328551 ;

Nir =— 0.00005306593JV " 95.7248158»

Nr =— 0.00004782067JV " 95.6796157«

N" =+0.00001988778JV " 95.2985863;

tfr"=+0.0000004029593W " 93.6052612.

(231)

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

51

For the root r/l5 we get,

^=7".2528745 ;
W =— 0.6557708JV,
Nf =— 0.-5021007JV;
W =— 0.08380783JV;
N^ =+0.0003520860^
NS =+0.0003598465^
W =—0.00009068952^
JV;r/'=— 0.0000037120502V;

For the root g2, we get,

#F=17".2008675;
NJ —— 7.2870812^2
N," =+ 7.52192987V2
TV/' =+21.345970^2
N2'r=— 0.0008132064^2
N,r =— 0.0054553222^2
^2"=+ 0.0003281043W2
^2"'=+ 0.00002415603^2

For the root g3, we get,

N3' =— 7.829047V3

7V3" =+ 9.673137V3
TV/' =— 33.378157V3

7V/r=+ 0.0003476427^3

N3T =+ 0.004032944^,

N3"=— 0.0002267157V3

NS"=— 0.00001681933JV3

For the root g4, we get,

^4=0".61668534;
N4 =+ 0.1207828^""
TV,' =+ 0.18354447V/r
N4" =+ 0.21197707V4/r
7V4'" =+ 0.34498807V4/r

For the root gb, we get,

#6=27.7276716;
7V6 =+ 0.28574457V/

=+

0.2850941JV/

JV6" =+ 0.2978305JV5/r
y," =+ 0.3989201JV/"
JV6r =+ 0.9103906^"
^"=+15.80032^;"

log. 99.8167520n;
" 99.7007908n;

" 96.5466488 ;

" 96.5561173;

" 95.9575571«;

" 94.5696138/1.

log. 0.8625536M ;

" 0.8763293;

" 1.3293159;

" 96.9102008n;

" 97.7368204/z ;

" 96.5160119;

" 95.3830256.

log. 0.8937084ra;

" 0.9855672;

" 1.5234623n ;

« 96.5411317;

" 97.6056222;

" 96.3554809«;

" 95.2258087«.

log. 9.0820049 ;

" 9.2637411;

" 9.3262888;

" 9.5378038;

" 0.0522431 ;

« 1.3891867;

" 2.1983761.

log. 9.4559779 ;

« 9.4549882 ;

" 9.4739691 ;

» 9.6008859 ;

" 9.9592278;

•« 1.1847004;

" 0.1754251n.

52 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root #„, we get,

fle=3*.7167656 ;

Nt =+0.54047 W log. 9.7327733;

JV6' =+0.3822413tf6'v " 9.5823376;

N; =+0.3758324^" " 9.5748787;

N6" =+0.4347748tf6'r " 9.6382644 ;

Ntr =+0.7901 155N6IV " 9.8976906;

#8"=— 1.039253^" " 0.0167214n;

JV0"=+0.03290012JV6"' " 8.5171974.

For the root «7,, we get,

<77=22".4609133;

N, =— 0.006641048JV/r log. 7.8222366rc ;

JW =+0.02354937^"" " 8.3719794;

N," =— 0.1 585801 NT " 9.2002488n;

jff =— 0.96475442V/'' " 9.9844168?i;

^7" =— 3.091771JV/r " 0.4902072ft;

JVr7"=+0.1154698JV/r " 9.0624686;

JVJr"=+0.008728388^7/r " 7.9409340.

27. These numbers are now to be substituted in equations (134) and (135). We

t t t

must also add the logarithm of (1+^') to the logarithms of -^ --K, and --/,

7Z tt 72 Ct lid

which were used for the adopted masses ; and by this means they will become,
log -^=88.2471963; log. -^h' =85.9698939; log.-^ T=86.8867Q56».

We shall therefore get for the root gr=5".5978504;

1853795.. , 41624.4 v 10625807

K y=-\ — M-^' 2=

Whence ^=88° 421 49".4 ; log. JV=9.2418092.

N =+0.1745056; ^"=—0.0000092603;

N' =+0.0093216; Nr =—0.000008345;

N" =+0.0060728; Nri =+0.000003470;

N" =+0.00094333 ; JVr//=+0.0000000703.

For the root flr!=7".252874, we get,

1689761 3937391, 138888320..

" 2l= ~~» — l

Whence ^,=23° 23' 37".0 ; log. ^=98.4892507.

AJ =+0.0308497; ^,^=+0.000010862;

JV,' =—0.0202303; JV/ =+0.000011101 ;

N' =—0.0154897; ^"=—0.0000027974;

Ni" =—0.00258544; JV/"=— 0.000000 11452.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 53

For the root #,= 17". 20086 7, we get,

32718590 A7 , 57870410 3.742550

2= " - 2' =

Whence &=330° 31' 1".9 ; log. JV2=97.2495186.

N, =+0.00177621; ^/r=— 0.0000014445 ;

Nj =—0.0129441 ; Nzr =—0.0000096903 ;

#,"=+0.0133612; Nzrt =+0.00000058281 ;

#2'"=+0.0379170 ; N™= +0.00000004291.

For the root </3=17".900114, we get,

93612050 95375820 _ 7.000929

2/3— 20 •"« Z3 - " 3 '

- JQ20 •"» 3— JQ20

Whence /?3=135° 32' 5".0 ; log. ^,=97.2807821.

N3 =+0.0019089 ; N3'r =+0.00000066361 ;

N3' =—0.0149448 ; N3r =+0.00000769847 ;

JV3" =+0.0184650; ^s^/=— 0.00000043278;

^3'"=— 0.0637154 ; N3"'=— 0.00000003210.

For the root #4=0".6166853, we get,

, 3.357379 v 1.360319 ,. 56977.20

*- -1^)10— ' *- -ioio— -. * 10io

Whence /?4=67° 56' 36".7 ; log. ^4/r=95.8033066.

JV4 =+0.000007679; JV4/r =+0.000063578;

JV4' =+0.00001 1669; JV4r =+0.000071705;

N<!' =+0.000013477; NJ1 =+0.00155773;

JV4"'=+0.000021934; ^

v

^ •

For the root #6=2".727672, we get,

, 0.6476120 v 0.1744696 „„ 345.1504

r

Whence ^5=1050 4' 40".0 ; log. ^^=97.2885211.

Nb =+0.00055526; Nb1T =+0.00194322;

Ns' =+0.00055400 ; N&r ^+0.00176909 ;

Nf =+0.00057875; NSVI =+0.0297318;

7V6'"=+0.00077519 ; #5r"=— 0.00291036.

For the root #6=3".716766, we get,

0.4683190 0.8566834 _ 22.51091

xe— 1" -- ' — ~"10 '

Whence ,86=280 8' 47". 2 ; log. N6'r =98.6350843.

NR =+0.0233269 ; Neir =+0.0431603 ;

Nt' =+0.0164976 ; N6V =+0.0341016 ;

JV6" =,+0.0162167; JVa" =—0.0448545 ;

^'"=+0.0187650 ; JV6r//=:+0.0014200.

54

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root #T=22".460913, we get,

1.009492

,TIT

•"! » 2/7 -

0.7872137

~ 1Q10

81.858790

iV7 •

Whence /?7=307° 56' 51".l ; log. 7V/r=98.1941950.

N, =—0.0001039; N,IY =+0.0156385;

Nj' =+0.00036828 ; N,T =—0.0483506 ;

N,"=— 0.00247996 ; N,n =+0.0018058 ;

^"=—0.0150873 ; JV7r"=+0.0001365.

28. We shall now suppose n"=-{-s\ ; or, the mass of Earth to be
-=-368689=l-j-351132.4. Using this mass, we shall find,

A[oTo]=— 0".0430854;
A|T7T1=— 0.3315294;

A[sT3]=— 0.0878224;

Whence we get,

\^\=g— 5".6133412;
|j7r]=£_ll .6462976;
[T^]=ff— 13.0721730;
f!TT| =g— 17.6406869;

[474]=— 0".0004406 ;
FT^1=— 0.0000502;
f^1=— 0 .0000043 ;
|77T1=— 0.0000009.

]=£_ 7".5128160;
]=^_18 .5962631 ;

—g— 2.7662565;

=_ 0.6479578.

These quantities give the following equations,

]=/— 17.2596388.^+ 65.3746421455 ;
=0"— 18.6855142.^+ 73.3785672744;
]=/— 23.2540281.^+ 99.0231945721 ;
p7n|Tri1=(72— 24.7184706.<y+152.2424170367;
[777] |T^l=(y2—29.2869845.|(/-i- 205.4486895058;
[iniFTTI— <7a— 30.7128599.(y+230.6021 109956;

QZIEd]=/— 26.1090791.^+139.7103029579;
Q7I][«T8]==^2— 10.2790725.^r+ 20.7823760933 ;

]=/— 8.1607738.^+ 4.8679877272;

]=/— 21.3625196.^+ 51.4420336761 ;
y_19.2442209.#+ 12.0495937275 ;

]=/— 3.4142143.?+ 1.79241747598.

]=/- 47.9724987.^ + 826.06962154./
-5987.951367523.<7 +15075.53048434*

]=94— 29.5232934,<73 +230.64471 166.</2
-523.79928387.^ +250.41918860

We shall therefore obtain the following

(232)

(233)

(234)

(235)

(236)
(237)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

55

Fundamental Equations for /t" =-{-o ; or, for m"=l-:-351132.4.

A =(f— 41.34298838.0 +200.8870856;. ]
A=<f— 23.27675657.0 + 99.2516206;
A"=g*— 18.39687878.0 + 71.57390627;
A=02— 14.594221002.0 + 45.9806378;
-42=02— 10.00808673.0 + 6.35450308 ;
A3=(f— 26.183439214.0 + 82.92703835;
D =03— 48.72257219.0 +727.694841 ;
D'=02— 55.34085031.0 +665.272796;
Z>"=02— 31.70284173.0 +247.7780603 ;
A=<72— 48.09969329.0 +201.477460 ;
A=<f— 51.052944567.0 + 34.3823533;
i>3=02— 3.4187123566.0+ 1.7319529170.

B= {#—35.7126879 } b ; B = \g— 17.655050207 1 b ;

B"=\q— 12.782796972 6

I*/

C = {24.2395891 — ^[9.1763990]&';
C" = {17.66980668— g \ [8.8694654]6';
C"=— [0.2445917]&';
C'"=— [0.2654598]6'.

E =+[9.6443837]6";

E'= {24.4829831 — ^|[8.5847028]5";

E"= {17.64195856— g \ [9.6375865J6";

(238)

; 1

> I

F =— [7.649909 1]5'";

F' =+[8.9577383]6"';

F'= {14.06088317— yj[0.8407439]6'";

F"= {37.6710436 — ^|[9.4809266]i'".

: {^—4.7868023 }6,; B^=\g— 0.6795727937^6
^={^—18.665614112 \b

Ci= {4.365095744 — ^[9.2840950]62;
(72= {0.6879114996— #j[9.1824311]&2;
(73=— [0.68323 17]62;
(74=— [0.6834824]&a;

^=+[7.8267723]^;
Et= {43.7345975 -^|[8.2017618]53;
E3= \ 0.648005019— «7j[0.7610455]&3;
^4=-j-[1.0655652]J8 ; *

Fl=— [7.0339474]&4 ;

F3= \ 2.770667338— g \ [1.7737962]64;
Ft= {50.3650331 —

(239)

(240)
(241)

(242)

(243)
(244)
(245)

(246)
(247)

56 SECULAR VARIATIONS OF THE ELEMENTS

</-47 9724987.o3+799.77118048./-5375.58619284.5r 1 } 24g)

+ 12441.50566776 j
<,«-29 5232934./+172.74781374.,2- 323.33816841., ) = , 24g)

+ 140.48125235 j

The values of J, A', 6*, and A'" arc given by equations (118); and the values of
A,, A2, A,, and 64 are given by equations (119), by simply adding log. (1+^)=
[0.0211893], to the coefficients of N".

Putting equations (248) and (249), equal to nothing, they will give,
g= 5".5094502; 9*= 0".61668624; "

L= 7 3146241; fr= 2.7277498;

fl,=17. 2173969; ^8= 3.7181422;

^3=17 .9310275 ; fr=22 .4607151.

The equations just computed, will now give the following values:
For the root g, we get,

£=5".5095453 ;

(250)

=+0.04709515JV
N» =+0.03032428JV

N"' =— 0.0000499083JV
NT =— 0.00004469 284 JV
Nrt =+0.00001922807JV

For the root </„ we get,

^=7.3153801 ;
JV/ =—0.7533911^
Nf =—0.5814061^
N™ =—0.0996916^
Nt" =+0.0004072400^Vi

N" =— 0.0001038256^
^""=—0.000004308711^

For the root ,2, we get,

^2=17".2175318;
NJ =— 7.71 3321 ^2
NJ =+ 7.200735^

N3'r=^— 0.0007473015J\T2
JV2F =- - 0.005061101^
JV2r'=+ 0.0003038889^2
^"=+ 0.00002237723^2

log. 98.6729762 ;

" 98.4817906;

" 97.6815864;

«* 95.6981728re;

« 95.6502379?i;

" 95.2839357 ;

" 93.5568171.

log. 9.8770206»i;

" 9.7644796?i;

" 8.9986586»;

" 6.6098504;

" 6.6215340;

" 6.0163044«;

" 4.6343474».

log. 0.8872414n ;

0.8573768;
" 1.2801108;
" 96.8734959ri ;
" 97.7042450ra;
» 96.4827148;
" 95.3498063.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

For the root </3, we get,

<73=17".9310572;
JVj' = — 8.306614/V3
JV," =-|- 9.36934^
Nam =—37.82400^3
Na"—+ 0.000407767JV,
N3r =4- 0.00487549^3
N3VI=— 0.0002732457^3
N3ra=— 0.00002027655JV3

For the root g^ we get,

(71=0".6166859;
Nj, =+ 0.1210308JV/r
JV4' =+ 0.1 8409 70^4/r

/ =+ 1.127857JV4/
J'=+ 24.50277^4/r

For the root #6, we get,

(75=2".7276841 ;
N, =+ 0.2888086JV5/r
JV5' =-j- 0.284631 1JV5/F
iV5" =4- 0.2991 853 JVB/r
Ns" =-{- 0.3984217^V5/r

A;W=— 1.497941iV5j
For the root <76, we get,

A;' =+0.3799388^"
^6" =-j-0.3762970JVa"
^6'" =-i-0.4342462iV6"
Ner =+0.790124^/r
N™=— 1.0390902V."

For the root gr7, we get,

#7=22".4609846 ;
N7 =—0.006671589^
N7' =+0.02618465JV/r

Njr =— 3.091724JV/"
JV/'=-f0.1154676A7r

log. 0.9194240n;
" 0.9717089;
" 1.5777676?i ;
" 96.6104120;
" 97.6880184;
" 96.4365534n;

log. 9.0828960 ;

" 9.2650466 ;

" 9.3292589 ;

" 9.5372136;

" 0.0522542 ;

" 1.3892152;

" 2.1984055.

log. 9.4606101;

" 9.4542824 ;

" 9.4759404;

" 9.6003430 ;

" 9.9592404 ;

" 1.1847750;

« 0.1754948n.

log. 9.7430272;

" 9.5797136;

" 9.5755308 ;

" 9.6377360;

" 9.8976952;

« 0.0166530^;

« 8.5170337.

log. 7.8242293w;

" 8.4180469 ;

" 9.1888659n;

« 9.9896951n;

" 0.4902020ra ;

" 9.0624601 ;

" 7.9409255.

8 November, 1871.

58 SECULAR VARIATIONS OF THE ELEMENTS OF

29. These numbers are now to be substituted in equations (134) and (135). We

ffn tJt ft I

must also add the logarithm of (1+/O to those of -„-„, — li", and _ /, in order

rfL Ct 7i ft ?Z> Cb

to obtain the numbers which are to be used in this computation.

For the root 0=5".5095453, we get,

1853601 „ J3698.3,, _1Q443191

X— 1Q20 "' y 1 1Q20 Z~

Whence 0=87° 43' 23".3 ; log. ^=9.2495260.

N =+0.1776340 ; N'r =—0.000008865 ;

N" =+0.0083657 ; NT =—0.000007939 ;

N" =4-0.0053866; NTI =+0.000003416 ;

^'"=-(-0.00085332 ; ^"'=+0.0000000640.

For the root ^=7". 3 15380, we get,

1595251 4450991 180052670

&\ - H --- TQM •"!» 2^1 - I JQiM •"!» Zl - I 1Q20 **

Whence fr=190 43' 4".3 ; log. ^=98.4192990.

JVt =+0.0262603; N^ =+0.000010694;

^'=—0.0197843; NJ =+0.000010986;

^"=—0.0152679 ; Nj" =—0.0000027265 ;

JVi"'=— 0.00261793 ; ^^"=—0.00000011315.

For the root <72=17".217532, we get,

_ 27629730 51666060 __a432446

•"•• 2/2 - + 20 "t» 22— iO •

1— JQ20

Whence /?2=331° 51' 47".6 ; log. ^2=97.2322182.

Nt =+0.00170694; ^"=—0.0000012756;

JV2' =—0.01 31662 ; N2r =—0.000008639 ;

Ai" =+0.0122912; N," =+0.0000005187;

^'"=+0.0325334 ; ^2m=+0.0000000382.

For the root j73=17".931057, we get,

__ ^104020250 108060520 _ 8.174639

X*~ » 3; y*~ 20 3* z*~ ~~ s<

Whence &=136° 5' 29".0 ; log. JV3=97.2635963.

N9 =+0.0018348; N3rr =+0.0000007482;

JVS' =—0.0152412 ; N3r =+0.000008946 ;

2V3" =+0.0171912; N3VI =— 0.0000005014;

AV=— 0.0694007 Nsr"=— 0.0000000372.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. £>9

For the root </4=0".6 166859, we get,

3.357579 1.360360 /r 58981.24

Whence /34=67° 56' 38".9 ; log. ^/r=95.8033000.

N4 =+0.000007695; JV/r =+0.000063577 ;

A7 =+0.00001 170; NJ =+0.000071706;

^"=+0.000013569 ; Ntn =+0.00155781 ;

JV»'"=+0.000021904 ; JV4 "'=+0.01003935.

For the root <76=2". 727684, we get,

0.6476505 r 0.1746351 345.2619

_ . r . _

X&~ 10'° ' ^5~~ 10 ° ' 2fi —

Whence &=105° 5' 26".0 ; log. JV5/r==97.2884328.

N, =+0.00056110; N617 =+0.00194282;

#,' =-{-0.00055299; N* =+0.00176878;

JV6" =+0.00058126 ; N&rl =+0.02973089 ;

JV5"'=+0.00077406 ; JV6F//=— 0.00291023.

For the root #6=3".7 16923, we get,

_0.4583310 A7/r , 0.8566739 AT/r 22.51056 F2

— — "~~ -"e » 2e — ~~~ I -" •

Whence /26=28° 8' 50".4 ; log. NJ "=98.6350898.

N6 =+0.0238845; Neir =+0.04316083 ;

N,' =+0.0163985 ; N6V =+0.03410240 ;

JV6" =+0.0162413 ; NeVI =— 0.04484797 ;

iV6'"=+0.0187424 ; ^6"'=+0.00141946.

For the root #7,=22".460985, we get,

L009476 v f , 0.7872105 ^r,Y , 81.85731 Affr2

^ — TTMO •"» ' y^ — + inio -"T > zi — + imo •"! •

Whence ^r=307° 56' 52".2; log. JV/r=98.1941978.

Nr =—0.0001043 ; N,IV =+0.01563860;

Nj' =+0.0004095 ; N,r =—0.04835037 ;

N,"=— 0.0024158 ; N," =+0.00180575 ;

JV/"=— 0.0152719 ; N7T"= +0.000136497

30. The assumed mass of the earth was obtained on the hypothesis that the
mean equatorial parallax of the sun is only 8".50. Recent investigations indicate
that the sun's parallax is greater by at least one- thirtieth part. Consequently the
assumed mass of the earth is too small by its one-ninth or one-tenth part; and on
account of the importance of this element in most astronomical theories, we shall
here give another determination of the constants corresponding to an increment of
V to the earth's mass. This new determination will be useful as an indication to

60

SECULAR VARIATIONS OF THE ELEMENTS OF

what extent the variation of the constants is proportional to the variation of the
masses ; and will also enable us to interpolate the constants for intermediate values
of the earth's mass.

And by using

If we now suppose ^"=+™ we shall find m"—i^2:-^^=f

this value of the mass we shall find,

A[o7o]=— 0".0861707,
A[T7J]=— 0 .6630587,

Af87J]=— 0 .1756449,

1
335172'

T7T1==_0".0008812,
77]:=— 0 .0001004,
Tcj^—O .0000086,

777]=— 0 .0000018,

Whence we get,

[J7o]=ff— 5".6564265;
JT7n=«7— 11.9778269;
[T7I1=ff— 13.0721730;
[T7?1 =ff— IT .7285094;

=g— 7".5 132566;
]=flr— 18.5963133
=ff— 2.7662608
=ff— 0.6479587.

These quantities will give the following equations,

[o7JJ[77T]=ff2— 17.6342534.ff+ 67.75169748957285 ;
[oTJJUTJ^ff2— 18.7285995.ff+ 73.94178576978450 ;
|77^|lT71=ff2— 23.3849359.ff+100.28001037565910;
[77iirm=.g>— 25.0499999,9+156.57622540085370;
[177117771 =ga— 29.7063363.ff+212.34901678822286;
fam|T73l=,ji'— 30.8006824,^+231.75014190892620.

}=<f— 26.1095699.ff+139.71887363689278;
}=/— 10.2795174.«7+ 20.78362721292128 ;
]=ff2— 8.1612153.ff+ 4.86827997930242;
]=ffa— 21.3625741.ff+ 51.44225250630864 ;
}=g>— 19.2442720^+ 12.04964299066071 ;
]=g*— 3.4142195.ff+ 1.79242275182896.

|773]=ff*— 48.4349358.ff«+842.64887773.ff2 1
—6173.539244345.^ +15701.465507779 J

r5.a||6.6|r777|=ff4— 29.5237894.(73+230.655099078.ff8)
—523.830290018.^+250.4352879666 J

(251)

(252)

(253)

(254)

(255)

(256)

We shall therefore obtain the following

Fundamental Equations for (i"=-\-^; or for 7n'=l-f-335172.
A =/— 42.46419343.*? +208.5335061 ;
A'=tf— 23.40766437.ff +100.5097899 ;
A"=g*— 18.77149338.ff + 74.00017359;
A\=f— 14.594665902.ff+ 45.9827890;
A,=/— 10.00852823.ff + 6.35481085;
-43=/— 26.183930014.ff+ 82.93563982 ;

(257)

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

61

D =</2— 49.585883207.0 +755.319136 ;
D'=f— 55.7602021 1.0 +674.468336 ;
Z>"=/— 31.83774566.0 +249.8341677 ;
Z^^— 48.09974779.0 +201.477866 ;
A=/— 51.052995667.0 + 34.3824324;
Z>3=y— 3.4187175566.0+ 1.7319581971.

B=\g— 36". 7908076^; # = ^0—17" 742872707 \b; \

J3"= 1 0—13 .114326272 |&;J

C = J24.7713708— g \ [9.1 763990]6';
C' = \ 17.75762918— # j [8.8694654J&' ;
€"=.— [0.2445917]6';
C""=— £0.2654598]6' :

E =+[96645871]Z>";
E' => |24.8145124— #J[8.5847028]6";
E" = \ 17.72978106—0 } [9.6375865]6" ;
E"= +[0.9932840]6";

F =— [7.649909 !]&'";

F"= {14.10796460— 0j[0.8407439]Z>'";
F»= 1 38.0025729—0 1 [9 4809266]6'".

5,= {0—4.78680657^; 52={0— 0.6795736737 J&!

53=J0— 18.665664332 1 &!

Ci= 14.365100014— 0j[9.2840950]52;
C,= J0.6879123796— 0|[9.1824311]52;
(7S=— [0.6832317]52;
(74=— [0.6834824]62;

^=+[7.8267723363;

Et= |43.7346477— 0| [8.2017618]53;

E3= \ 0.648005897— 0}[07610455]&3;

Fl=— [7,0339474]&4 ;

F3= \ 2.770671608— 0| [1.7737962]64;
F,= |50.3650833— 0^[

(258)

(259)

(260)

(261)

(262)

(263)

(264)

(265)

(266)

0<-48.4349358./+815.10927504./ 1
—5528.66691090.^+12909.37803021 | k

}
Z *• Z

SECULAR VARIATIONS OF THE ELEMENTS OP

,,<-29.5237894./+172.758201154./ 1 }

-323.368779974..</ +140.4964513342 J l

62

The values of J, &', b", and Z/" are given by equations (118); and the values of
&i, ftfc, fes, and &4 are given by equations (119), by simply adding log.
0.0413927 to the coefficient of N".

Putting equations (267) and (268) equal to nothing, they will give,

g= 5".554906079, g,= 0".616687321, '

gf= 7.378294649, gb= 2.727775991,

#=17.403271601, #,== 3.718480731,

^8=18 .098463471, <?7=22 .460845357.

(2681)

Equations (257-268), in connection with equations (84-97), will now give the
following values : —
For the root g, we get,

#=5".5550002,

N1 =+0.04569346^"
N* =+0.02968146^
N" =+0.00482527^
JV/r=— 0.00004897096JV
NT =—0.00004399572^
jVr/=+0.00001859827JV

For the root glt we get,

^1=7".3790776,

JV," =—0.590047^
Nf =— 0.1045547JV;
N^ =+0.00041 79936 JV1
Air =+0.0004316384^
Ntn =—0.0001054817^
Nlm=— 0.000004437763JV1

For the root gt, we get,

<;2=17".4034121,
Ai' =— 7.766638^
Nl =+ 6.68909()^2
N," =+23.91929JV2
N," =— 0.000785956A",
Ntr =— 0.005986225^
Ntri =+ 0.0003529385A;
#/"=+ 0.00002604318^

log. 98.6598541 ;

" 98.4724852 ;

" 97.6853181 ;

" 95.6899385n ;

" 95.6434103re;

" 95.2694725 ;

" 93.5602333.

log. 99.8784552/1 ;

" 99.7708867w;

" 99.0193438w;

" 96.6211696;

« 96.6351200;

" 96.0231771«;

" 94.6471640w.

log. 0.8902330»;
" 0.8253671 ;
1.3787482;
» 96.8953983w;
" 97.7771530n;
" 96.5476990 ;
" 95.4156940.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

63

For the root gz, we get,

#,=18".0984790,
Na' =— 8.350533^3
N," =+ 8.67378(W3
Nam =—28.56700^3
JV3/F=+ 0.0002228091.M,
N3r =4- 0.003251222^
^3"=— 0.000179346^3
NSVI'=— 0.00001332988^;

For the root g^ we get,

#4=0".6 166870,
Nt =+ 0.1 207555 jtf/r

ft' =+
Nt" =+ 0.3435830^'"
JVi" =+ 1.127906W/r

JV4r/=+ 24.50543JV4/r

For the root g&, we get,

gb=2". 7277089 ;
N5 =+ 0.285290iV/r
JVi' =-f 0.2827626^'"

^6" =+
JV5'" =+
N,r =+ 0.91047W

For the root.#6, we get,

6" =-(-0.3741537^6/F
6'" =+0.4330305^^
/ =+0.7901418JV6/F

For the root g^ we get,

N7 =— 0.007172393JV/r
N7' =+0.0327495W/F
N," =— 0.1572018^/v
Njm ==— 0.9919734#7F

log. 0.9217142w;

" 0.9382084;

" 1.4558646*1 ;

" 96.3479330;

" 97.5120466;

" 96.2536923w;

" 95.1248263w.

log. 9.0819069 ;
" 9.2644574;

9.3290446 ;

9.5360316;

0.0522730;

1.3892623;

2.1984543.

IV

ir

log. 9.4552860 ;

" 9.4514220;

" 9.4747313;

" 9.5991402;

" 9.9592660;

" 1.1849243;

" 0.1756342».

log. 9.7325934;

" 9.5746276;

" 9.5730500;

" 9.6365185;

" 9.8977050 ;

" 0.0165127n;

" 8.5166994.

log. 97.8556641«;

" 98.5152046;

" 99.1964576w;

" 99.9965000w;
« 0.4901920w ;

" 99.0624468 ;

" 97.9409123.

64 SECULAR VARIATIONS OF THE ELEMENTS OF

31. These numbers are now to be substituted in equations (134) and (135). We
must also add the logarithm of (1+/O or 0.0413927 to those of ™,,, ~Ji"t and

?Z (I ?i Ct
tt

~ii—al", in order to obtain the numbers to be used in this computation.

For the root #=5". 5550002, we get,

1859252 982149.1 _10422045

1020 ' y~ "TO20 ' 2 1()20 '

Whence /3=87° 28" 12".8, log. #=9.2518088.

# =+0.1785701; N" =-0.000008745 ;

#'=+0.0081595; #r =—0.000007856;

#" =+0.0053002; #"=+0.000003321;

Nm =+0.00086522; #"'=+0-0000000649.

For the root ^=7".3790776, we get,

. 1535521 , 4548921 186590940

, _

> 2/1— ~~ 20 '• Zi~

"1 I 1020 > yl~ 1020

Whence /3,=18° 39' 8".9 ; log. #!=98.4104497.

#, =+0.0257306 ; #1'r =+0.00001075$ ;

#/ =—0.0194493 ; #jr =+0.000011106;

#,"=— 0.0151823; #"=— 0.000002714;

#^=—0.00269026 ; #/"=— 0.0000001142.

For the root gr2=17".4034121, we get,

__?9_188_05J) .65870040 4.079210

10io

Whence ^2=329° 15' 1".2; log. #2=97.2739118.

#2 =+0.0018789 ; #2/r =—0.000001477 ;

#2' =—0.0145930 ; #/ =—0.000011248 ;

#2" =+0.0125684; #2" =+0.0000006631 ;

#2'"=+0.0449428 ; #2"'= +0.00000004893.

For the root <73=18".09 84790, we get,

__j_82704460 _8JL987110 _5.806087

2/3 into 5 23— fmo •

Whence /3S=134° 45' 1".6 ; log. JV8=97.3022770.

N3 =+0.0020058 ; N^ =+0.0000004469 ;

JV3' =—0.0167491 ; Nsr =+0.0000065211 ;

^"=+0.0173974; ^"=—0.0000003597;

W=— 0.0572983 ; #8r"=— 0.00000002674.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 65

For the root #4=0".6 166870, we get,

3.357912_ 1.360959 56997.71

1010 1010

Whence &=67° 56' 42".3 ; log. #4/r=95.8032384.

JV4 =+0.000007676 ; JV4/F =+0.000063565 ;

JV4' =+0.000011687 ; NJ =+0.000071695 ;

Nt' =+0.000013561 ; ^"=+0.00155764;

^'"=+0.000021841 ; JV4" '=+0.01003850.

For the root <75=2".7277089, we get,

0.6477216 0.1749684 _345.4842

x* — I TTiTo > 2/5 five ' 25 inio

10
Whence &=105° 6' 59".3 ; log. ^=97.2882538.

Nb =+0.00055404; N&'r =+0.0019420;

Nj =+0.00054913 ; Ntr =+0.0017682;

JV6" =+0.00057941 ; ^"=+0.0297289;

JV5'"=+0.00077160 ; N6™'=— 0.00290997.

For the root r7,j=3".7172386, we get,

0.4583439 0.8566511 _22.50983

x*~ "UF ' 2/6~ "TO1" ' Z°~ ~1015 '
Whence /S6=28° 8' 55". 1 ; log. ^8"=98.6350978.

N6 =+0.0233180; N6ir =+0.0431616 ;

NJ =+0.0162078 ; Ntr =+0.0341038 ;

N,' =+0.0161491 ; JV6r/ =— 0.0448343 ;

JVa'"=+0.()186903 ; JVar/J=+0.00141840.

For the root <77=22".46H216, we get,

1.009450 , 0.7872062 81.85450

1010 ; Vn~ 1010 ; 2''~ xv

Whence /37=307° 56' 54".3 ; log. ^/1=98.1942049.

N, =—0.00011217; N,ir =+0.0156388;

AY =+0.0005 1216; ^TTF =—0.0483500;

JV7"=— 0.0024585; N™ =+0.0018057;

N™=— 0.0155133 ; .#7"=+0.00013650.'

32. We shall now suppose ^"'=+1 ; and the mass of Mars will become,

m,= i+i_ j_

"2680637~ 1340318.5'
Using this mass, we shall find,

AFT]D=— 0".0279815 ; AR3=—0". 0031027 ; ]

. !

AQTTj^—0. 1020355; A^^— ° -0003297 ; , (26g.

AH7g=— 0 .2982001; A[«T«]=—0 .0000275; |

A[sTs]= 0. ; AQT3=— 0 .0000057. J

9 November, 1871.

(Jfi

SECULAR VARIATIONS OF THE ELEMENTS OF

Whence we get,

[o7o]=^— 5".5982373;
[77T]=0--11 .4168037;
[272]=0— 13 .3703731 ;

ri~r]=(y— 17.5528645;

[TT]=0- 7".5154781;

[«76]=(jr— 18 .5965426 ;

[67e]=0— 2.7662797;

=— 0.6479626.

(270)

From these quantities we get the following equations,

]=02— 17,0150410.04- 63.9139763201 ;
0»— 18.9686104.^4- 74.8505214033 ;
^— 28.1511018.04- 98.2651007657 ;
02— 24.7871768.0+152.6469250785;
g*— 28.9696682.0+200.3976083692;
0*— 30.9232376.J/4- 234.6883473387;

l=fl«— 26.1120207.0+139.7619086460;
|=0s— 10.2817578.04- 20.7899145038 ;
]=^— 8.1634407.0+ 4.8697487299;
] =0»— 21.3628223.0+ 51.4432382846 ;
]=02— 19.2445052.04- 12.0498640941 ;
]^— 3.4142423.04- 1.79244578674.

(271)

(272)

=04— 47.9382786.03+824.7624793..72 ) ,,,-gv
-5969.6589279.^+14999.865474416 ; j

•f\

99. j

—523.98540191 .^+250.515644299
We shall therefore obtain the following

Fundamental Equations for ^'"=+1; or, for mw=l-:-1340318.5.

A =y2— 40.54796482.0 +195.9771436 ;
A'=rf— 23.18819358.0 + 98.5732170;
A"=<f— 18.15228098.7 -\- 70.09590592;
Af=f— 14.596906302.0 + 45.9936083;
Af=<f— 10.01075363.0 -j- 6.35635695 j
A3='f— 26.186380814.0 \- 82.97883004;

(275)

D =0Z— 48.259496667.0 +710.074578 ;
D =tf— 55.05265379.0 4-658.652503 ;
rr=g*— 31.86740965.7 4-250-7131091 '
Z>1=02_ 48.09999599.7 4-201-479694 5
Z)a=0*_5 1.053228867,7 4- 34.3827873;
A=«f— 3.4187403566.04- 1.7319812509.

(276)

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

G7

5=^—34.9327682^; B = \g— 17.581591114J6'; 1 „„.

Jf= \ff— 12.553303072 J6; J

C = \ 24.0060075 — ^[9.1763990]&';
C" = \ 17.61110406— ^[8.8694654]6';
C*=— [0. 24459 17J6';
C""=— [0.2654598]6';

E =+[9.6231944]i" ;

E'= {24.2634892 — <7j[8.5847028]Z>";

E" = \ 1 7.55540782— g \ [9.6375865]6" ;

F =— [7.9509391]//";

l<"=+[9.2587683]6"';

F"= 114.31200183— ^|[0.8407439]i"';

F'"= 537.4415497 — (/([9.4809266]//";

C,= ^4.365118944 — ^?[9.2840950]&2;
C2= 1 0.6879162996— g\ [9. 182431 l]i, ;
(73=— [0.683231 7]S2;
C<=— [0.6834824]62;

.#1=+[7.8267723]63 ; 1

E2= |43.7348770 — g\ [8.201 761 8]i3; '
Ea= \ 0.648009819— ^i[0.7610455]63; j

J

^=—[7.0339474]^;

JFi=+[0.2724896]64;

F3= \ 2.770690538— ^}[1.7737962]54;

F,= J50.365313 — flr}[9.1128486y>4;

(278)

(279)

(280)

*,={</— 4.7868255 }5i; ^=^—0.679577593716,;) 2fin

^={^—18.665893612^; )

(282)

(284)

^-47.9382786.^ +799.3791471.^ ) r- rO' (286)

-5382.3863166.J7+12482.1014329 j "

(f— 29.5262630../8 +172.810222X2.^ ) , ,28fix

-323.52210151.^+140.57248634 j "

The values of 6, Z/, b", and V are given by equations (118); and the values of
i,, fy>, f>3, an(J ^4 are given by equations (119), by simply adding [0.3010300] to
the coefficients of N'".

68

SECULAR VARIATIONS OF THE ELEMENTS OF

Putting .equations (285) and (286), equal to

0 = 5".4982776; gt=

01= 7.4032880; gb--

02=17.0209128; g<-

03=18.0158003; g,--

The equations just computed will now give

For the root </, we get,

0=5".4983682;
A' =+0.04574900A
A" =+0.02849177A
A* =+0.004428614A
A/r =—0.00004876604^
A" =— 0.0000436362-7A
A" =+0.00001885617A

nothing, they will give.

0".616692122;

2.7279047;
= 3.7201830;
=22 .4614832.

(287)

For the root g» we get,

^=7". 4040670 ;
Ai' =— 0.807831 7 A;
N* =— 0.6055137JV,
Ar =— 0.10224573 Ai
Nf =±+0.0004271000^,
=-i-0.0004419796JV,
=— 0.0001073512JV,
^,m=— 0.000004540672V,

For the root >j2, we get,

=— 7.637284JV2
=+

Nt'v=— 0.0011 16683#a
Nt* =— 0.006781928JV2
2V,W=+ 0.0004152514#2
N2r"=+ 0.00003050897^,

For the root 173, we get,

£8=18".0158631 ;
JV3' =— 8.466438JV;
Na" =+10.65257^3
Nam =— 26.00480 Na
N3'r =+ 0.0005716405^s
NS =+ 0.007478025^8
Na"=— 0.00041568153T,
^r"=— 0,00003086801^,

the following values :—

log. 98.6603816;

" 98.4547195 ;

" 97.6462679 ;

" 95.6881175n;

" 95.6398476n ;

" 95.2754534;

" 93.5437656.

log. 9.9073209n;

" 9.7821240n;

" 9.0096452n ;

" 6.6305296 ;

" 6.6454022 ;

" 6.0308069n ;

« 4.6571202».

log. 0.8829390N ;

« 0.8702282 ;
" 1.1714047;

" 97.0479297/1 ;

" 97.8313532»;

" 96.6183110;

" 95.4844276.

log. 0.9277008»;

« 1.0274544;

" 1.4150536n;

" 6.7571230;

« 7.8737870 ;

" 6.6187606n ;

" 5.4895086rt,

THE ORBITS OF THE EIGUT PRINCIPAL PLANETS.

69

For the root <;4, we get,

r/4=0".6 1669 170;
JV4 =+ 0.1217372^'"

N>" =+ 0.2150961^V4/F
N" =+ 0.345612iV/F
NJ =+ 1.128136iV47
^4"=+ 24.51803A7

nr

'IV

For the root #5, we get,

#,=2".7278184 ;
JV5 =+ 0.29()3833^"6/F
JV5' =+ 0.2
Nb" =+ 0.2

5" =+ 0.

'=— 1.500545^"

For the root yti, we get,

^i=3''.7186222;

; =--0.3754068JV6/r
f =-j-0.4351473^6/ir
S =-]-0.7902211JV6JF
6"=— 1.0372986We/F

For the root </7, we get,

N, =— 0.00561240^/F
Nj' =+0.01444352^
N7" =— 0.1296412^/F
N,'" =— 0.967188A7/F
N7T =— 3.091 169iV/F

log. 9.0854233 ;

" 9.2678704;

" 9.3326326 ;

" 9.5385887;

" 0.0523616;

" 1.3894856;

" 2.1986843.

log. 9.4629717;

" 9.4561671 ;n

" 9.4770622;

" 9.6015396;

" 9.9593780;

" 1.1855820;]

" 0.1762490?».

log. 9.7456734;

" 9.5804642;

" 9.5745022;

" 9.6386363 ;

" 9.8977486;

" 0.0159038«

" 8.5152447.

log. 7.7491486/1;

" 8.1596730;

" 9.1127430»;

" 9.9855109n ;

" 0.4901228n;

" 9.0623510;

" 7.9408170.

33. These numbers are now to be substituted in equations (134) and (135). We

must also add the logarithm of (2=1+^") to those of _"1"" 7"' V, and — T, in

n'"(t'" n" ri" nma'"

order to obtain the numbers which are to be used in this computation.

70 SECULAR VARIATIONS OF THE 'ELEMENTS OF

For the root 0=5".4983682, we get,

1840691 Ar , 98053.1 10390430

^; y=+— -»— ^ z= -

Whence /3=86° 57' 2". 7 ; log. JV=9.2489632.

tf =+0.1774036; #"=—0.000008651 ;

N' =+0.00811605; Nr =—0.000007741 ;

N" =+0.0050546 ; N" =+0.0000033452 ;

N'n =+0.00078565 ; #""'=+0.00000006205.

For the root #1=7".404067, we get,

, 1842718 „ . 4337257 _r 197199700 AT2

~— -

Whence £,=23° 1' 6".9 ; log. #=8.3783427.

# =+0.0238970; # 7" =+0.000010206 ;

^' =—0.0193047; #"=+0.000010562;

^" =-0.0144699; #"=—0.000002565;

Nf =—0.0024433 ; ' # T"=— 0.0000001085.

For the root ^2=17".021165, we get,

_ 51010670 _

X* - ---- VM - -"2» 2/2 - I — 20 Z2~ ^'

51010670 80108830 3.698921 ,

Whence /?2=327° 30' 44".5 ; log. #=97.4095166.

# =+0.00256754 ; #'" =—0.0000028671 ;
#' =—0.0196090 ; #" =—0.000017413 ;
#" =+0.0190434 ; #"' =+0.0000010662 ;
#'"=+0.0380997 ; #m= +0.00000007833.

For the root #,=18".015863, we get,

137961060 149026100 8.388122

X3 +' ""1020 ""a! 2/3— JQ20 -"3* Z3— JQ10

Whence /?3=137° 12" 28".8; log. #=97.3840054.

# =+0.00242106; #"=+0.0000013840;
#' =—0.0204978 ; #r =+0.000018105 ;
#"=+0.0257905 ; #"' =—0.0000010064 ;
#'"=—0.0629592 ; #""=—0.00000007473.

For the root 04=0".6166917, we get,

__j_ 3.359379 ^.360856 r ^57058.06 /F2

A* ; 2/4 + JQ1U •"* » 2*~ 1Q10 4

Whence /34=67° 56' 51".2 ; log. #'"=95.8029370.

# =+0.000007733 ; #" =+0.000063524 ;

#' =+0.000011770 ; #r =+0.000071663 ;

#" =+0.00001 3664 ; #"' =+0.00155748 ;

#"=+0.000021955 ; #""=+0.01003740.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.
For the root </5=2".727818, we get,

.1763111

346.4665

- j_ 0-6-479474 /r

-10.0— ^5 ; 2/5--=

Whence 04=105° 13 19".4; log. #,"=97.2873891.

JV5 =+0.000562809; N6ir =+0.00193816;

JV5' =+0.000554059 ; N6r =+0.00176509 ;

NS" =+0.000581368; ^"=+0.0297147;

JV5'"=+0.000774334 ; #5m=— 0.00290829.

rz

6 *

For the root <76=3".718622, we get,

, 0.4583136 ,. , 0.8566921

ZA=-

22.50708
To7*

1Q10 > V6- JQ10

Whence /?6=28° 8' 45".3 ; log. N6'v =98.6351607.

^=+0.0240344; N6tr =+0.0431679;

Ne' =+0.0164295 ; N6r =+0.0341122;

JV6" =+0.0162055 ; N6rr=— 0.0447780 ;

JV6'"=+0.0187844 ; JV6r//=+0.0014139.

For the root g7=22". 461946, we get,

1.009108

r

; 2/7=

0.7869276

81.83667

1010 1010

AVhence /?7=307° 56' 52".8 ; log. N7ir=9S. 1941499.

N7 =—0.0000878; N7IV =+0.0156369;

Nj' =+0.0002258 ;
JV7"=— 0.0020272;
^7"'=— 0.0151238;

NS =—0.0483362 ;
N^ =+0.0018051;
JV7m=+0.0001364.

71

34. We shall now suppose ^/r=+T

; and the mass of Jupiter will become,

Using this value of Jupiter's mass' we

shall find,

A[oTo]=—0".01 60284;
A[TT]=— Q .0420284 ;
Af^1=— 0 .0706826 ;
A[s73]=— 0.1465990;

Whence we get,

[o7o]=<7— 5".5862842;
Q7[l=#— 11 .3567966;
\j7j\ =ff— 13.1428556;
[3TT]=^— 17.6994635;

E3= 0". ;
[1711=— 0.1824735;
[?7J]=— 0.0093732;
r?771=— 0 .0017907.

GT4]=#— 7".5123754;
[ITI1=/7— 18.7786864;
[67J]=^— 2.7756254;
[7^]=a— 0.6497476.

(288)

(289)

72

SECULAR VARIATIONS OF THE ELEMENTS OF

From these quantities we get the following equations,

]^— 16.9430808.^+ 63.4422934092 ;
^8— 18.7291398va+ 73.4197265812 ;
|=^— 23.2857477.J4- 98.8742332985 ;
[272]=/— 24.4996522.^-j- 149.2607377924 ;
3=^— 29.0562601.^-j- 201.0092068986;
==.92— 30.8423191.^+232.6214929780;

(290)

]=02— 26.2910618.^+141.0725417557;
]=02— 10.2880008.0+ 20.8515399746 ;
[4TT][77f]=!72— 8.1621230.0+ 4.8811478864;
]=0'— •21.5543118.0+ 52.1225989505 ;
-f-r 19.4284340.0+ 12.2014064196;
]=02— 3.4253730.0+ 1.80345594215.

]=04— 47.7853999.03+ 818.62769096.02 )
—5898.0322091.0 +14758.0410108 ; j

]=04— 29.7164348.0s +232.93269093.02 )
—530.6408472.0 +254.418113702. j

We shall therefore obtain the following

Fundamental Equations for [iir=-\-— — ; or for mrr=

• (291)

100

^ =/_40.3084942.<7 +194.2847527 ;
^'=i72— 23.30847617.<7 + 99.1027623;
A"=./2— 18.08032078..; + 69.61059281;
Al=f— 14.626095329.<7 + 46.1708070;
J2=j72— 10.02759291.^ + 6.38342153;
A3=g2— 26.365472005.^ + 83.71712664;

D =/— 47.971972067.<7 +703.129607 ;
D=fft— 55.11012591.^ +662-354-530 ;
IT=(f— 31.78521949..7 +249.0357671 ;
Z)i=i72_48.29148549..7 +202.685210 ;
A=i/2— 51.237157667.«7 + 34.5983143;
D3='f— 3.4298710566.;/+ 1.7430000948.

(294)

(295)

B= }j/— 34.7052507 |ft;

# = ^—17.713826807(6; )
B"= \ (/—12.49329597 } 5; j

C = 1 23.7784900 — ^|[9.1763990]fe';
C' = \ 17.72858328— ^|[8.8694654]6'f
0'=— [0.24459 17]6';
C"=— [0.2654598]6'.

(296)

(297)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 73

E =+[9.6231944]6";
E'= J24.1934821 — #j[8.584702S]6";
K'= \ 17.70073515— #j[9.6375865]6";
,#'"= -(-[0.951 89 13]Z>".

F =_ [7.649909 !]&'";

JF'=+[8.9577383]i'";

F" = \ 14.08448433— # } [0.8407439]6'" ;

1™= )37.3815426 — #j[9.4809266]Z>'".

(298)

(299)

^7-4.7961712^; £,= )</- 0.6813625937 |&t; 1 g()())

}i > '

Cf=

3= {0—18.848037412

{4.374464644 — <^[9.2840950]&2;

{0.6897012996— #j [9. 182431 1]Z>2;
-[0.6832317]62;
— [0.6834824]62 ;

(301)

j 43.9170208 -^[8.2017618]&3;
\ 0.649794819— <7J[0.7610455]63;

(302)

(303)

^=—[7.0339474]^;

jF2=+[0.2724895]64;

F,= { 2.780036238— 0} [1.7737962]64;

Ft= {50.5474564 -0j[9.1128486]64; j

0*— 47.7853999./+793.57041154.02 1 v

-5313.7879554.*7-f 12250.8363744 J =

tf— 29.7164348.0»+174.4588860.^' ) _, x

-327.5400877.^ +142.7433842 j =

The values of 6,, Z>2, &3, and 64 are given by equations (119); and the values of
6, U, b", and 11" are given by equations (118), by simply adding log. (l+//r)=
[0.004321,4], to the coefficients of N".

Putting equations (286) and (287), equal to nothing, they will give,

g= 5".48175018;
9l= 7.29977391;
^2=17 .09470611 ;
#3=17.90916970;

10 November, 1871.

gt= 0".61 8590378;
#6= 2.73702595;
#6= 3.72450143;
#,=22.63631704.

(306)

74 SECULAR VARIATIONS OF THE ELEMENTS OF

The equations just computed will now give the following values:
For the root g, we get,

^=5".481 84344;

N' =+0.04775507JV log. 98.6790194 ;

N" =+0.03031 1511V

N'" =+0.0046 156637V

N'T =— 0.00004974188JV

Nr =— 0.000044294982V

N" =+0.00001941068iV

JVm=+0.000000351368JV

98.4816076;
" 97.6642342;
" 95.6967222/»;
" 95.6463545» ;
" 95.2880406 ;
" 93.5457621.

For the root #,, we get,

</1=7".3005026 ;
Ai' =— 0.76574607V,
JV," =— 0.5821200JV,
Nf =— 0.09563476JV,
Ai" =+0.00039969327V,
JV,r =+0.0004077858JV,
. ^"=—0.00010221725^,
Nlm=— 0.000004220228JV;

For the root <jr2, we get,

Ni =— 7.6978803T,

N,' =+ 7.832065^a
N2m =+13.95241 ^2

NJr^— 0.0007072570JVj

N,r =— 0.004093204^

JVra"=+ 0.0002502615JV2

JV2r//=+ 0.00001841771JV;

For the root gM we get,

N3 =—

N3" =+10.511163^
N3m =— 56.628832V3
J\ri/r=+ 0.00081 58954JV3
N3r =+ 0.008084869^3
N3rt=— 0.0004567941 N3
-#3"'=— 0.00003392040JV3

log. 99.8840847ft;

" 99.765()126?t;

" 98.9806 158»i;

" 96.6017268;

" 96.6104318;

" 96.0095242n ;

" 94.6253359«.

log. 0.8863712ft;

" 0.8938764 ;

" 1.1446493;

" 6.8495773?i;

" 7.6120634ft;

" 6.3983940;

" 5.2652356.

log. 0.9225281ft ,

" 1.0216508;

" 1.7530376ft;

" 6.9116345;

" 7.9076730;

" 6.6597205w ;

" 5.5304609?i.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

75

For the root g±, we get,

#4=0". 6 1859008;
jV4 =-\- (
N; =+ 0.1844663A7'
N! =+ 0.2137640iV4"
N" =+ 0.3456726JV4"
NJ =+ 1.1 25699 NI'Y

ir

V

For the root </5, we get,

<75=2".7369612;
N, =+ 0.2919622JV5/r

JV5' =+ 0.286
JV5" =+ 0.299
JV5'" =+ 0.

N,r =-}- 0.9082830iV5/
^"==+15.32706^,"

Ai"^— 1.494938^

For the root ga, we get,

8r =+0.7887484.yar

For the root </7, we get,

^7=22".6365781 ;
N7 =—0.006141989^7'.

N™ =

— 0.1513593JV/

— 0.9658338JV/
3.128146JV7/F

log. 9.0839733 ;

" 9.2659172;

" 9.3299345;

" 9.5386650;

" 0.0514222;

" 1.3907546;

" 2.2015132.

log. 9.4653267 ;

" 9.4565760;

" 9.4766953;

" 9.6013642;

" 9.9582212;

" 1.1854589;

" 0.1746230».

log. 9.7502802;

« 9.5825976;

" 9.5760403;

" 9.6381695 ;

" -9.8969386;

" 0.0189238ra;

" 8.5197888.

log. 7.7883092M;

" 8.2822223 ;

« 9.1 800092/z ;

" 9.9849024»;

« 0.4952871«;

" 9.0640050 ;

" 7.9431488.

35. These numbers are now to be substituted in equations (134) and (135). We

must also add the logarithm of (l-\-u'T) to those of ,„, —.-v—Ji'7, and Zfr, in

n a n a n a

order to obtain the numbers which are to be used in this computation.

76 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root <7=5".481S43, we get,

,1850169 70884.5 10443875

*=+ " 1020 ; J" 1020 1030
Whence (3=8T 48' 21".4; log. A=9.2486682.

N =+0 1772834; N'r =-0.000008818;

A- =+0.0084662; Nr -. -0.000007853;

N" =+0.0053737 ; NT1 =+0.000003441 ;

^"=+0.00081828 ; #"'= +0.00000006229.

For the root (/i=7".3()0503, we get,

,1630157Ar ,4425088V 179817860

*i=+ » --wi; 2/1= ~i0B— l5 1= io20

Whence ^=20° 13' 23".8 ; log. ^=8.4187229.

JVi =+0.0262254; N,ir =+0.000010482;

JV/ =-0.0200820 ; N,T =+0.000010694 ;

Ai" =-0.0152664 ; W =-0.000002681 ;

^,"=-0.00250806 ; JV/"=-0.0000001 1068.

For the root #2=17".094828, we get,

15347790 .- .36663280 2.973721

*2=- J-Q-O - ^fJ 2/2= 1(p ---Nt; z*= 10io «••

Whence /32=337° 17' 6".3; log. A2=7.1259941.

Nt =+0.0013366 ; Nz'v =-0.0000009453 ;

Ni =-0.01 02888 ; #/ =-0.0000054709 ;

Nf =+0.0104682 ; W =+0.00000033449 ;

J\r;'=+0.0186485 ; JV/"=+0.00000002462.

For the root j73=17".909224, we get,

, 147160530,. 160644600 14.88588

«3= 20 A3! 1/3— » 1()10

Whence &=137° 30' 30".l ; log. #3=7.1654027.

N3 =+0.0014635 ; N3'r =+0.0000011941 ;

N,=— 0.0122442; N,r =+0.000011832;

As" =+0.0153834; N3r' = -0.00000066853 ;

N3m=— 0.0828782 ; N3r"=— 0.00000004964.

For the root #4=0".6 185901, we get,

_ 3.376737^,, l,.377873^,r 2^5^2.14^^

Whence ^4=67° 48' 7".8; log. ^"=95.7999963.

JV4 =+0.000007655 ; ^" = ,+0.0000630952 ;

AT; =,+0.000011639 ; Ntr -- ,+0.000071026 ;

Nt'=-. +0.0000 134875; JV/'=,+0.00155150;

^•= +0.000021 810 ; # "'=+0.01003483,

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

For the root </6=2". 736961, we get,

0.6488003_ . 0.1681608 _346.3853_

io10 io10 6 ' Zf>~ "To15 6 •

Whence &=1()4° 31 50" 1 ; log. ^"=97.2866700.

Nb =+0.00056493; N" =+0.00193495 ;

AV =+0.00055366 ; - Ntr =+0.00175748;

Ay =+0.00057992 ; Ns" =+0.0296571 ;

Ay"= +0.00077274 : A7/'^— 0.00289263.

For the root r/6=3".723307, we get,

_0.4594222 , 0.8649404 r i ??167899_ AT^Z

^o — jmu •"« > y^ — "I JQIO •**« ' Z6 — ~T JQIO -*^« •

Whence &=27° 58' 31".7 ; log. AVr=8.6353288.

A; =+0.0243002 ; Ne'r =+0.0431846 ;

Na' =+0.0165168 ; N6r =+0.0340618 ;

^"=+0.0162693; N6ri =— 0.0451079;

Ar6'"=+0.0187715 ; Ar6F"=+0.0014293.

For the root r/7=22".636578, we get,

1.021673 ,. 0.7951379 , 83.56643 Ar/,.2

^7 — T?no •"' ' 2/7 — T f?uo "T ' Z7 — H T7\io *M •

Whence /37=307° 53' 32".2; log. JV/ r=8J901156.

N7 =—0.0000952; N7!T =+0.0154923;

N7' =+0.0002967 ; N/ =—0.0484622 ;

iV7"=— 0.0023449 ; N,VI =+0.0017952 ;

N7'"=— 0.0149630 ; A7"=+0.0001359.

36. We shall now suppose that /ur=-| — ; and the mass of Saturn will become,

14- L 1

v=:0-n ^-=-oTfcTKc"» Using this value of Saturn's mass, we shall find,
oOUl.b

AU7o]=— 0".0019316,
^QZ1=— ° -0049722,
A[|72] = — 0 .0081629,
A[»Ts]=— 0 .0157850,

Whence we get,

I^o]=j7— 5".5721874;
03=^—11 .3197404;
Q3=flr— 13.0803359;
pTTI =/7—17.5686495;

rTl = —0". 1849094,

[erg =—0.0348880,
|T771=— 0.0051937.

[TT4]=«7- 7".6972848;
[771]=, ff— 18.5962129;
[°^]=g— 2.8011402;
[777]=^— 0.6531506.

(307)

(308)

78

SECULAR VARIATIONS OP THE ELEMENTS OF

From these quantities we get the following equations,

[o7_o][r;T|=02— 16.8919278.0+ 63.0757148282 ;
[o7o][27J]:=02— 18.6525233.0+ 72.8860828897 ;
[o7o][17T]=02— 23.1408369.0+ 97.8958073789 ;
[I7jf]=02_24.4000763.0+148.0660()67328;
r7J]=02— 28.8883899.0+198.8725515186;
[71]=^— 30.6489854.0+229.8038367694.

]=#*— 26.2934977.?+143.140;3468927;
3=^— 10.4984250.^+ 21.5611738841 ;

]=/— 8.3504354.^+ 5.0274861855;

]=g*— 21.3973531.^+ 52.0905995219 ;

\=f— 19.2493635.^+ 12.1461276134 ;
3=^— 3.4542908.^7+ 1.8295664023.

-47.5409132.03+810.60000012.0-
-5815.0364817.<7+14495.04127448

74— 29.7477885..73+235.7953005.r/
—542.55408336.0+261.88476948

We shall therefore obtain the following

(309)

(310)

(311)
(312)

Fundamental Equations for |Ur=+j^; or, for wr=-~- -— .

A =03— 40.2318777.0 +193.4460881 ;
A =02— 23.16356537.0 + 98.1230397;
A"=g2— 18.02916778.0 + 69.22796584;
,li=^_U.813573502.0+ 47.2122074-;
A3=g*— 10.19774833.0 + 6.52926208;
A3=g*— 26.369591592.0+ 84.94828460.

j) =02—47.872396167.0 +700.738213 ;
Z)'=02— 54.94225571.0 +656.812394;
I7=g*— 31.59188579.0 +246.0948532;
Z>1=02— 48.76298515.0 +206.032362 ;
Dt=g"-— 51.852306416.0 + 35.1877225;
D3=<f— 3.4587888566.0+ 1.7691277905.

B= ^-34.6427310 \b; ^ = {^-17.58301281 } 5; )

B"=\g— 12.4562398|&; )

C = J23.7159703— g\ [9.1763990]6 ;
C' = \ 17.5977693— g\ [8.8694654]&';
C"=— [0. 24459 17]6';
Cm=— [0.2654598]i';

(313)

(314)

(315)

(316)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

79

E =+[9.6231944]Z>";

E' = \ 24. 1564259— g \ [8.5847028]Z>" ;

E"= j 17.569921 16— g \ [9.6375865]6";

F =—[7.649909 !]&'";

JF'=+[8.9577383]V;

F"= \ 14.02196463— g\ [0.8407439]&'";

F'"= j 37.3444864— gr | [9.4809266]6'".

4.8216860 }6i; Bt=\g— 0.6847655937 }6i; )
Ba={y— 18.667297687}^; j

Q= { 4.399979444— g j [9.2840950]62;
<72= 1 0.6931042996— </J[9.1824311]Z>2;
Q=— [0.6939556]62;
C±=— [0.6942063]&2;

^=+[7.8267723]^;
E^= |44.3630057— g \ [8.2017618]63;
E3= \ 0.653197819— <7J[0.7610455]Z>3;
^=4-1:1. 0655652]63;

(317)

(318)

(320)

(321)

^=—[7.0339474]^;

(322)
F3= .\ 2.805551038— gr j[1.7737962]64;

F±= J51. 159202116— #j[9.1128486]Z>4;

j/4— 47.5409132./+785.54272070./ )
-5234.4038677.^+12012.62971359 j (

/— 29.7477885./+176.4527823.</2 1 , >

-334.71804394.^ +146.97863935 j ^~~ *4' ^6' ^ Xl)'

The values of fta 52, 63, and 54 are given by equations (119); and the values of
6, V, I", and I'" are given by equations (118), by simply adding log.
[0.0107239], to the coefficients of Nr.

Putting equations (323) and (324) equal to nothing, they will give,

g= 5".46587505; g,= 0".622281636;

'g,= 7.25377728; gb= 2.76237256;

gz= 17 .02345660; g,= 3 .7S742019;

^=17 .79780427; ^7=22 .57571411.

(325)

80

SECULAR VARIATIONS OF THE ELEMENTS OF

The equations just computed will now give the following values : —
For the root g, we get,

log. 8.6860778 ;

" 8.4907565 ;

" 7.6765512;

" 5.7205493n;

" 5.6713710?*;

" 5.3258490 ;

" 3.5604147.

log. 9.8757588/1 ;

" 9.7578974w ;

»« 8.9767704n ;

" 6.6034387 ;

" 6.6130658;

" 6.0257136w;

" 4.6315468/1.

log. 0.8837032«;

" 0.8877358 ;

" 1.1826643;

" 96.849441 In;

« 97.6314346/t;

" 96.4295512;

" 95.2944192.

log. 0.9183798n;

" 1.0102164;

" 1.70l6403n;

" 7.8226985;

" 7.8398536 ;

" 6.60541 10»;

" 5.4741710«.

N' =+0.04853754^
N" =+0.03095683^
Nm =+0.004748442JV
N" = — 0.00005254717JV
Nr =— 0.00004692140JV
tf" =+0.000021 17625 JV
#"'=+0.000000363425^

For the root «/„ we get,

«7,=7".2545148,
JV; =— 0.7512055W,
Nf =— 0.5726608JV,
NI" =— 0.09479172JV,
N,'r =+0.0004012718^,
W =+0.0004102662^,
Nirt =—0.0001060996^,
N™=— 0.000004281016^,

For the root

N,

we get,

^2=17".0235807,
' =— 7.650735JVj
" =+ 7.722107iV2
=+15.22875^2
=— 0.0007070363^a
^ =— 0.004279910JSr2
TI =+ 0.0002688755^V2
"= 0.00001969787^V

For the root

, we get,
^8=17".7978505,

JV3' =— 8.286660^8
Nt* =+ 10.238030 JV3
N3m =— 60.30837iv;
^3/r=+ 0.0066481 ISJVg
N3r =+ 0.006915978^,
N3r'= - 0.0004030984JV.
#,"'=— 0.00002979690^3

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

81

For the root g±, we get,

#4=0".62228135;
jV4 =4_ o. 1215129 #4'r
N{ =4- 0.1845219^'"
JV4" =4- 0.2136944JV/r
Nt" =+ 0.3455668^
NJ =4- 1.133130JV/"
Nt"=+ 24.593006^4/^
JV4r//=4-160.63011JV4/r

For the root f/6, we get,

N, =+ ().2979185JV/r
^V5' =4- 0.2890480^"
^V5" =4- 0.3020508JV6;r
N,'" =-)- 0.4005457JV6'r
Ntr =4- 0.9164754^5'r
JZV6"=4-15.69060W5/r
N™=— 1.514217^6/r

For the root gw we get,

N6 =

=+().3937991^V6J
=+0.3856959^
=4-0.4382820^
=4-0.794009JV67F

For the root <JTT, we get,

^7=22".5759799 ;
N7 =—0.0061 16321 JV/
N,' =-f0.01912057JV/r
N7" =— 0.1503075JV/F
Nj" =— 0.9432520JV/'

^"=+0.1143927^

^"'=-1-0.00864081^

log. 9.0846223 ;

" 9.2660478;

" 9.3297932;

" 9.5385321 ;

" 0.0542796 ;

" 1.3908116;

" 2.2058270.

log. 9.4740975 ;

" 9.4609699 ;

" 9.4800800 ;

" 9.6026522;

" 9.9621209;

" 1.1956394;

" 0.1801882n.

log. 9.7784954;

" 9.5952747 ;

" 9.5862450;

" 9.6417536;

" 9.8998255;

" 0.0095448n

" 8.4848952.

log. 7.7864903w ;

" 8.2815008;

" 9.1769806«;

»' 9.9746277w ;

" 0.4774559n ;

" 9.0583982 ;

" 7.9365546.

37. These numbers are now to be substituted in equations (134) and (135). We

r, in order

must also add the logarithm of l4-u\ to those of- ~r , - -_ hv, and -^-

n a na n a

to obtain the numbers to be used in this computation.

11 December. 1871.

82

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root 0=5".465974, we get,

1834742,., .50945.0 10464937

x=+— -J5- #; V= -j^r- -10s r"

Whence /3=88° 24" 34".2, log. ^=9.2440059.

N =+0.1753904; #'"=-0.000009216;

N1 =+0.0085130; JT =-0.000008230;

N" =+0.0054295; #"=+0.000003714;

N"' =+0.00083283; #"'=+0.00000006374.

For the root 01=7".254515, we get,

1718569 4394108 173815680

*i=+ io-° ~ l! Jl~ 10-° 1020

Whence ^=21° 21' 39".0 ; log. ^=8.4336898.

Nt =+0.0271450; #/'' =+0.000010892;

N, =-0.0203915; N,r =+0.000011137;

#/' =-0.0155449 ; #," = -0.0000028801 ;

#!'"=— 0.00257312; #T" -0.00000011621.

For the root Sr2=17".023581, we get,

18688360,. .40529780 _3.058100 v,

«*= 1020 - ^5 2/2= "^Q20 - ^25 Z2- — JQIO

Whence &=335° 14' 43".7 ; log. ^2=7.1641841.

N, =+0.0014594; N,ir =-0.000001032;

Ai' =-0.01 1 1657 ; ^/- = -0.000006246 ;

N,' =+0.01 12699 ; N2V1 =+0.0000003924 ;

JV.,"'=+0.0222253 ; JV2"'=+0.00000002875.

For the root &=17". 797850, we get

.. .

B; z*= 15"

132834470 .- 143133400.. 12.343316

Whence /?3=137° 8' 14".l ; log. ^=7.1992139.

N3 =+0.00158203; N," =+0.0000010517;

JV3' =—0.0131097 ; JV3r =+0.000010491 ;

JV/ =+0.0161968; Ntn =-0.0000006377;

JVr/"=-0.0795892 ; JV3"'=-0.00000()04714.

For the root j/4=0".6222813, we get,

. 3.414548 __,„ . 1.395571 „ 58943.27

*4-+ i(ya—K'>Vt—+--joSS—Nt 'z^- io'»"^4 •

Whence £4=67° 46' 10".2 ; log. ^"=95.7964432.

NI =+0.000007604 ; Nfr =+0.000062581 ;

JV4' =+0.000011548 ; #/ =+0.000070912;

JV4" =+0.000013373 ; NJ' ==+0.00153906 ;

JV/'= +0.00002 1626 ; JV4"/=+0.0100524.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

83

For the root #=2". 762311, we get,

0.6624109
1010

' 2/6=-

0.1988506
10'°

_36J..8939 r2
Z6— -^QIO *X6 "•

Whence &=106° 421 33".6 ; log. J\Vr=7.2812824.

Ns =+0.00056935; N,'r =+0.00191110;

Ns' =+0.00055240 ; N&r =+0.00175147 ;

JV5" =+0.00057725; Nbri =+0.0299862;

^'"=+0.00076548;
For the root ^=3". 786 209, we get,

JVr.ir"=— 0.00289381.

X6 — +

0.4675181

r

^'s ; z/o —

0.8555864

22.61242

,

\'

•"

T_,

' * «

Whence &=28° 39' 12".7 ; log. ^7 r=8.6346522.

N6 =+0.0258909 ; #6/r =+0.0431174;

JV8' =+0.0169796 ; N6r =-\-Q.03±2356 ;

Ne" =+0.0166303; JV6" =— 0.0440755 ;

JV6'"=+0.0188976 ; 2V6m=+0.0013169.

For the root gr7=22".575980, we get,

1.004018 , , 0.7872757

JQ1

JQ1

r _

•"I > Z7 —

79.68195

10

io

Whence /37=308° 6' 3".2 ; log. ^/r= 8. 2044480.

N7 =—0.0000979;
Nj' =+0.0003062 ;
JV7"=— 0.00240674;
N,'"=— 0.0151034 ;

38. We shall now suppose that ^"=+^5 and the mass of Uranus will become

N,'r =+0.0160121 ;
N,r =—0.0480733 ;
N7rr =+0.0018317;
tfm=:- 0.0001384

1 i TI 1
ir/=^j~^=HoYTq- Using this value of the mass of Uranus, we shall find,

A[o7o]=— 0".0000666;
A[TTT1=— 0.0001705;
A[UJ=— 0.0002778 ;
AQ73]=— 0 .0005261 ;

Whence we get,

{^]=g— 5".5703224;
OIO=«7— 11 .3149387;
[^}=(j—l3. 0724508;
[373=^—17.5533906;

[47*1 =— Off.OQ37881;
1775]=— 0.0139141;

=_0 .0130554.

71]=^- 7".5161635;
^\=g—18. 6101270;
7e]=i7— 2.7662522;
T=^— 0.6610123.

(326)

(327)

84 SECULAR VARIATIONS OF THE ELEMENTS OF

From these quantities we get the following equations :—

J=03— 16.8852611.0+ 63.0278564952 ;
[£3=^— 18.6427732.0+ 73.8177655141 ;
J75]=02— 23.1237130.0+ 97.7780448551 ;
r^n^ff2— 24.3873895.0+147.9139794608;
]=02— 28.8683793.0+198.6155386162;

(328)

]=02— 30.6258414.0+229.4658349917;

|4,4||5_,i|=0s— 26.1262905.0+139.8767572878;
g]=02— 10.2824157.0+ 20.7916038174 ;
]=09— 8.1771758.0+ 4.9682765223;
]=/— 21. 3763792.0+ 51.4803047560;
F77]=^_19.2711393.0+ 12.3015228516;
ET3Q7H0*— 3.4272645.0+ 1.82852672910.

04— 47.5111025.03+809.61901994.0a
-5804.87167416.0 +14462.73971842 ;

=y*— 29.5535550.03+231.24699197./ |
—527.16726514.0+255,76838948. J

(329)

(330)

We sliall therefore obtain the following

Fundamental Equations for //'"'=+—; or, for w "=237 19*

A =g*— 40.22212762.<7 +193.3374230 ;
A=tf— 23.14644147.J/ + 98.0051228;
A'=f— 18.02250108..7 + 69.17798438;
^,=^—14.698591492.^ + 46.9394976;
Ai=ff— 10.02448873.<7 + 6.47860075 ;
^3=/—26.200650614.^ + 83.09379428;

D =g*— 47.859709367.j7 +700.433897 ;
Lf=ff— 54.92219511.^ +656.158133;

Lr='f— 31.56874179.^ +245-7424731 -
Z>,=/72— 48.19349485.j7 +205.563726 ;
D^y*— 51.079862967./7 + 35.0495614;
D3=g*— 3.4319830985.J7+ 1.7650873387.

7_ 34.6348459 16 ; ^=10—17.56775390716; )
S"=\g— 12.45143805 \b- 1

C = |23.7080852 — 0|[9.1763990]i'; 1
f" = {17.58251038— 0|[8.8694654]ft';
f"=— [0.24459 17]6';
C"*==— [0.2654598JV;

(332)

(333)

(335)

THE 0KB ITS OF THE EIGHT PRINCIPAL PLANETS. 85

E =+[9.6231944]Z>";

E '= J24.1516242 — #j[8.5847028]Z>";

E"= j 17.55466226— 0j[9.6375865]i"; [

JEW=+[0.9518913]6";

F =— [7.649909 1]6";

F" == 1 14.01407953— g \ [0. 840743 9]ZT ;
F'"= {37.3396847 — g j [9.4809266]//" ;

J3,={0— 4.88782529 }&,; B,=\g— 0.692627293 \b,;

B3=\g— 18.679478012 j 6, ;

C,= J 4.445033406 — 0j[9.2840950]Z>2;
C2= 1 0. 7009659996—# j [9. 182431 1J6, ;
Cs=— [0.68323 17]i2;
C4=— [0.6834824]Z>2;

.1

E2= S43.7484614 — r7J[8.2017618]63; .

Ey= \ 0.661059519— #j[0.7610455]63; .'

£4=4-[1-0867545A; " J

F,=— [7.0339474]64; 1

(336)

(337)

(338)

(339)

(340)

\0^ i /341\

F3= \ 2.77088358— 0j[1.7737962]&4;
F,= j 50.3788970 — ^|[9.1128486]64;

r/4— 47.51 11025.r/ + 784. 561 74052.J/1 ) _, y

-5224.6661 1233.#-|-- H983.3131284 j =

^_29.5535550.?s +173.33949329.<72 ) _(

-325.77538456.^+143.38934238 j ~^4' ^6' ZM ZT)'

The values of &j, 62, 53, and 54 are given by equations (119); and the values of
fe, 6', i", and b'" are given by equations (118), by simply adding log.
[0.0211893], to the coefficients of Nri.

Putting equations (342) and (343), equal to nothing, they will give,

g= 5".46378145;
9l= 7 .24790553 ;
#2=17.01456031;
^3=17 .78485521 ;

g4— 0".628004507 ;
&= 2.72658456;
ge= 3.72632806;
o7=22 .47263787.

(344)

S6

SECULAR VARIATIONS OP THE ELEMENTS OF

The equations just computed will now give the following values :— -
For the root </, we get,

<7=5".4638782;

A" =+0.04863950^V

N' =+0.03104071JV

JV" =+0.004765953.V

N'r =—0.0000511782^

Nr =— 0.0000456230JV

Nrl =+0.00001999627JV

JVra=+0.000000318386iV

For the root glt we get,

-<7l=7".2486360;
NI =— 0.749378JVi
JW =— 0.5714613JV,
Nf =—0.0946740^
W =+0.0003968524^
NS =+0.0004049153JV;
Ai" =— 0.0001021 7875 JV,
ft r"=— 0.000004032583^,

For the root //2, we get,

^2=17".0146874;
Nj =— 7.644965 N,
NS =+ 7.708896^,
A7 =+15.37836JV2
JV2/r=— 0.0007260315A;
Ntr =— 0.004354339^2
Ntn=+ 0.0002668697JV2
JV2r//=+ 0.00001945905W,

For the root </„ we get,

£3=17".7849019 ;
JV8' =— 8.277612JV3
N3" =+10.208104^
W =— 49.64280JVS
N3'r=+ 0.0006759756JV3
N3T =+ 0.006899148JV3
N3"=— 0.0003922849A;
NST"=— 0.00002886839JV,

log. 8.6869890 ;

" 8.4919316;

" 7.6781498;

" 5.7090852»»;

" 5.6591837?*;

" 5.3009490;

« 3.5029538.

log. 9.8747004n ;

" 9.7569868«;

" 8.9762302/? ;

" 6.5986290 ;

" 6.6073642;

" 6.0093606n ;

« 4.6055833/4.

log. 0.8833755»i ;

" 0.8869922 ;

" 1.1869102;

" 6.8609555n ;

« 7.6389222» ;

" 6.4262993;

" 5.2891217.

log. 0.9179050n;

" 1.0089451 ;

" 1.6958563n;

" 6.8299311 ;

« 7.8387954;

« 6.5936016n ;

" 5.4604225rt.

THE ORBITS OF THE EIGHT JPUINCIPAL PLANETS.

87

For the root

^ we get,
c/4=0".628()0419 ;

Nt =+
iV4' =4-
Nt" =-{-
Nt" =+
tf4r =+

0. 121 7541 ^i
0.1847141jV/
0.2138538.V/
0.3456976.V,

1. 127368 NI"

=+ 23.96983.V/

For the root gb, we get,

//5=2". 7265 1823 ;
TV; =+ 0.2923210^5/r
JV7 =+ 0.286536^6/r
JV6" =+ 0. 299996 W
Nf =+ 0.3994943^'"
7V8F =+ 0.91 07594 N,,'r
^"=4-14.68646^"
Nar"=— 1.519462JV5'r

For the root r/6, we get,

/r

=+0.5714777#6

JVa" =

'" =+().4358324JV;/r
" =+0.7893470^a/|r
" =— 1.030466 N6'r

For the root fa, we get,

#7=22".4729002;
N, =— 0.0()622292JV/r
JV7' =+0.02016987JV/r
JV7" =— 0.1518602JV7/r
A7'" =— 0.9593122AV
JV/ =— 3.093542iV/r

log. 9.0854837 ;

" 9.2665000;

" 9.3301171;

" 9.5386963 ;

" 0.0520657 ;

" 1.3796650;

" 2.1862708.

log. 9.4658600 ;

" 9.4571790;

" 9.4771160;

" 9.6015106;

" 9.9594036 ;

" 1.1669170;

" 0.1816899».

log. 9.7569992;

" 9.5864288;

« 9.5792102;

" 9.6393195;

" 9.8972680;

" 0.0130335n;

" 8.5527582.

log. 7.7939944rt;

" 8.3047032 ;

" 9.1814440n;

" 9.9819600n;

" 0.4904560n ;

" 9.0624906 ;

" 7.9386915.

39. These numbers are now to be substituted in equations (134) and (135). We

VI VI VI

must also add the logarithm of 1+u", to those of , 7i", and - '—f.lrr, in

n a n a n a

order to obtain the numbers which are to be used in this computation.

88 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root 0=5".46387S, we get,

, 1846778 A7 .61805.2,, 10467663...

*=+ 10»-^ ?=+— jo» -~N> Z= ' KP ^*

Whence (3=SS° 4' 59".6 ; log. #=9.2468080.

N =+0.1765257; #"=—0.0000090343;

N> =+°-0085861 5 ^ =—0.0000080536;

#" =-(-0.0054795 ; NVI =+0.00000:35299 ;

N" =+0.00084131 ; #"'=+0.0000000562.

For the root ^=7".248636, we get,

1657215 _4358217 _173063980

Xl - H -- /Sf—-«lJ 111— 20 •"!? Zl— ~"20" ~~-"l •

Whence /3,=20° 49' 9".6 ; log. #,=8.4304273.

JVi =+0.0269418; #/r= +0.000010692;

JVY =—0.0201896; #,r =+0.000010909 ;

Nf =—0.0153962; ^i"=— 0.000002753 ;

JV,™ =—0.00255069; Nlr"=— 0.0000001086.

For the root sr2=17".014687, we get,

18857840 40818170 _ 3.068621 xr2

^2~ 20 Z2~ 10 '

1020 2~ 101

Whence /32=335° 121 11".5 ; log. #2=7.1659197.

#2 =+0.0014652; #/" ==— 0.0000010637;

#2' =—0.0112020 ; N,r =—0.0000063803 ;

N2" =+0.01 12957 ; NtT' =+0.00000039104 ;

#a'"=+0.0225336 ; #2F7/=+0.0()000002851.

For the root 03=17".784902, we get,

131044100 141058200 _ 12.091655

'"•> 2/3— I'JO **•» 23— 1"

Whence /33=137° 6' 27".7 ; log. #3=7.2020253.

#3 =+0.0015923; #3'r =+0.000001 0764;.

#3' =—0.0131805 ; #3r =+0.0000109855 ;

#3" =+0.0162544 ; #3r/ =—0.0000006246 ;

#T=— 0.0790463 ; #/"=— 0.0000000460.

For the 'root 0r4=0".6280042, we get,

3.295099 r 1.261747 53937.18

z«=-r inio ^4 ; 2/4= ,mo ^4 ; «.=- imo ^4 •

Whence /34=69° 2' 50".4 ; log. #/r=95.8156910.

#4 =+0.000007965 ; #4'r =+0.000065417 ;

#4' =+0.000011808 ; #/ =+0.00007374!) ;

#4" =+0.000013989; #4" =+0.00156804;

#,"=+0.000022614; #4r'7=+0.01004537.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.
For the root ^6= 2". 7 265 18, we get,

89

0.6486453
~10'«

0.1822083
-10w

337.9588

Whence £.=105° 41' 25".3 ; log. #"=7.2996358.

N& =+0.00058277; #'"=+0.00199359;

jV5' =+0.00057135 ; #r =+0.00181568 ;

#" =+0.00059807 ; #rf =+0.0292788 ;

#'"=+0.00079643 ; #"'=— 0.0030292.

For the root j70=3".725131, we get,

. 0.4579378 „„ . 0.8593662 r 22.55046 n

*e=H — YO>« — ° ' y«=~l io10 " 8 ' 2<i= "lo10 — 6 '

Whence ^,=28° 3' 7".9 ; log. ^c/r =8.6352986.

JVa =+0.0246773; N,'r =+0.0431816 ;

NJ =+0.0166620 ; N,r =+0.0340853 ;

JVe" =+0.0163874; JV6"=— 0.0444971 ;

^"'=+0.0188199 ; JV6m=+0.00154190.

For the root ^7=22".4729002, we get,

__1.010_089 /F 0.7868583

- y, ; 2/7—+- iow

81.93419

/FJ

'

1Q10 *-'7 ' ill I .1010 1Q10

Whence &=307° 55' 6".8 ; log. #'"=8.1938807.

# =—0.0000972; #/F =+0.0156272;

#' =+0.0003152 ; #" =—0.0483433 ;

#"=—0.0023732; #"=+0.0018046 ;

#'"=—0.0149913 ; #""=+0.0001357.

40. We shall now suppose ^r":=-\-^5 ; and the mass of Neptune will become.

'"— ii" — _ Using this value of Neptutie's mass, we shall find,

~18780~ 17072.73

A[oTo]=— 0".0000460;
A|T7n=— 0.0001177;
A^3=— 0.0001914;
A{TT3]=— 0.0003611;

Whence we get,

[oTo]=?— 5".5703018;
[TTTj=<?— 11 .3148859;
[273 =^—13.0723644;
[775] =(7— 17.5532256;

pTTT|=— 0'.0024017;
[IT?] =:_(). 0068740;
jTTJ] = —0.04333257;

rrri= o.

T4]=4— 7".5147771 ;

*7J]=^— 18.6030869;

3=^~ 2.8095779;

77=— 0.6479569.

(345)

12 December, 1871.

1)0

SECULAR VARIATIONS OF THE ELEMENTS OF

From these quantities we get the following equations,
jT7o][TT3=03— 16.8851877.0+ 63.027
[T^][7:7|=02— 18.6*426662.0+ 72.8170149476 ;
:02— 23.1235274.0+ 97.7767641555 ;
[]T]]=02— 24.3872503.0+147.9123116292;
QT]]=/— 28.8681115.0+198.6127448410;
\h^\EI\=^— 30.6255900.0+229.4621614386;

14.4115^=^—26.1178640.0+139.7980514254;
iTT]=^—10.3243550.0+ 21.1133516636 ;
(77J]=^_ 8.1627340.0+ 4.8692516739;
[6TT][676]=0?— 21.4126648.0+ 52.2668218260 ;
II3IIZM</2— 19.2510438.0+ 12.0539985182;
E3(IZ]=^— 3.4575348.0+ 1.82048538639.

r7s]=£4— 47.51 07777.03+ 809.60832632.0- 1
—5804.7608117.0 +14462.38720986 ; j

(347)

(348)

[77]=^— 29.S753988.03 +231.92196050.03
—530.90381750.0+254.50030966.

} (350)

We shall therefore obtain the following

Fundamental Equations for [ir"=-\--~; or for mr"=T^ .

10 1 iU ( 4. to

A =^2— 40.22202062.0 +193.3362269 ;
A=gl— 23.14625587.0 + 98.0038404;
A'=<f— 18.02242768..'/ + 69.11743373;
41==^_14.639503502.c/ + 46.4149812;
^2=^_10.()1320843.0 + 6.38683557;
As=g2— 26.192224114.0 + 83.01495699;

D =02— 47.859570167.0 +700.430559 ;
I/=(/2— 54.9219773.0 +656-1510425
ZX'=02— 31.56849039.0 +245 7386441 ;
A=02— 48.14983849..7 +203-40'2186 '>
A=02— 51.063762837.0 + 34.5618232;
Dz=<f— 3.4620415785.0+ 1.7536927202.

£=1^7—34.6347594^; B = \g— 17".567588907|6; )

B"=\g— 12 .451385272 \b; j

(351)

(352)

C = |23.7079988 —

C' = [17.58234538—. g j[8.8694654]A';

C"=— [0.24459 17]V;

Cm=— [0.2654598]//;

(353)

(354)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS 91

E =+[9.6231944]6";

E'= $24.1515714 —^[8.5847028]//;

E"= 1 17.55449726— #j[9.6375865]6";

F =— [7.6499091]6'";

F' =+[8.9577383]6'";

F" = \ 14.01399313— ^[0.8407439]&"';

F*= j 37.3396319 — #j[9.4809266]//";

= |£— 4.8301237 } Z>, ;

f=\g— 0.682733393(6,
3=\g— 18.672437912 jfti;

\ 4.408417144 — ^}[

\ 0.6919059696-r- g\ [9.182431 !]&,;
— [0.683231 7]&2;
—[0.6834824]^ ;

Ci=
C2=
C3=
^=

Ez= \ 43.7414213 -^[

E3= \ 0.648008841— g\ [0.7610455]&3;

"

^=—[7.0753401]^;
JP2=-|-[0.3138822]&4;
^3= { 2.813958738—
F<= 150.37185686 —

(355)

(356)

(358)

(359)

(360)

/-47.5107777./+784.55104690y ) 361

-5224.5598798.0+11982.99308371 j

#4— 29.575398S./+ 174.01 843537.0* ) _ . (^^

— 327.77137981.0 +142.39163768 j ~

The values of bt, &2, 53, and 64 are given by equations (119); and the values of
fc, 6', I", and Z/" are given by equations (118), by simply adding log.
[0.0413927], to the coefficients of Nv".

Putting equations (361) and (362), equal to nothing, they will give,

• (363)

g= 5". 463758326 ;
gl= 7 .247841113;
gr*= 17 .014463042;
#,=17,784715218;

fft= 0".6142757037 ;
lgb= 2.770720524;
gs= 3.723848045;
^7=22 .466554527.

SECULAR VARIATIONS OF THE ELEMENTS OP

The equations just computed will now give the following values:
For the root _</, we get,

log. 98.6869996 ;

" 98.4919453;

" 97.6781717;

" 95.7081905n;

" 95.6585913?*;

" 95.3071375;

" 93.5376972.

N' =+0.04864067JV
-N" =+0.03104168JV
N'" =+0.004766193^
N'r =—0.00005107290^
y =—0.0000455608^
Nri =+0.000()2028325JV
jVm=+0.0000003449032^V

•

For the root g}t we get,

^=7".2485708 ;
M' =—0.7476334^,
Nf =— 0.5714480^
Nf =— 0.09467274JV,
JVj7" =+0.0003964014^;
NS =+0.000404752^;
Nj" =—0.0001031014^
^""=—0,000004 1 4784W;

For the root //2, we get,

W =— 7.644900^a
N; =+ 7. 708750 Na
N2m =+15.37999JVa
&,"=— 0.0007239023^2
Ntr =— 0.004358588^
JV2"=_(_ 0.0002678406JV2
JV2r//=+ 0.0000195659JVa

For the root </;, we get,

9r.5=17''.7847617;

A73" =+10.20779JV8
W =— 49.63566JV3
N3'r=+ 0.0006721 358 N3
^3r = z+ 0.006905492JV;
#3"=-- 0.0003936316 JV;
Nar"=— 0.00002907741^3

log. 9.8746886«;

" 9.7569768?i;

" 8.9762250/1 ;

" 6.5981351 ;

" 6.6071885;

" 6.0132647«;

" 4.6178221«.

log. 0.8833718ra ;

" 0.8869839;

" 1.1869559;.

" 6.8596800n ;

" 7.6393458n;

" 6.4278763 ;

" 5.2924994.

log. 0.9179000n;
" 1.0089319;
" 1.6957938«;
" 96.8274570;
" 97.8391946;
" 96.5950900»i ;
« 95.4635558w.

THE ORBITS OP THE EIGHT PRINCIPAL PLAJMETS.

For the root g^ we get,

<74=0".6142754;
N, =+ 0.1212222^'
Nl =+ 0.1842843#4'
N; =+ 0.2134164iV4/
iV/' =+ 0.3454 170iV4'
tf/ =+ 1. 127956 Nt"

Arr"= -4-145.26667

For the root </5, we get,

<76=2".7706502;

N, =-f

«-f 0.2898491JV5/r

JV/' =- 0.4009023^6/
Nbr =+ 0.9054583^57
lf,w =4-14.64440^,"

JV5r//=- 1.406558^

For the root ^6, we get,

^6=3".7226566 ;
Ne =+0.5703212^"
JV6' =+0. 3855324^"
JV6" =-(-0.3792427JV6/r
N,f" =+0.4357298^"

^V8"=— 1.083630^
JV6m=+0.0356831JV6

"

/r

For the root <77, we get,

^7=22".4668172;
N7 =^—0.006229914^
N7' =+0.02023760^/r
NI" =— 0.1519652JV/r
Nf =— 0.9604665^V7/F

log. 9.0835822;

" 9.2654883 ;

" 9.3293498 ;•

" 9.5383437 ;

" 0.0522920;

" 1.3851038;

" 2.1621660.

log. 9.4765344;

" 9.4621720;

" 9.4810226;

" 9.6030386 ;

" 9.9568685;

" 1.1656714;

" 0.1481577».

log. 9.7561195;
" 9.5860609 ;
" 9.5789172;
" 9.6392173;
" 9.8973944 ;

" 8.5524627.

log. 7.7944821M ;

" 8.3061590;

" 9.1817441n;

" 9.9824823n ;

" 0.4903156n;

" 9.0633640 ;

" 7.9408532.

41. These numbers are now to be substituted in equations (134) and (135). We

r// mrfl mv"

must also add the logarithm of l-f-ft"7, to those of VII VIf V,,-VIJ^"-, and ~rif VIf"->

it/ CL it/ (t iv "

in order to obtain the numbers which are to be used in this computation.

94 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root g=S'. 463855, we get,

,1847131,. ,63087.5 10467693

~

Whence /3=88° y 37".9 ; log. #=9.2468998.

N =+0.1765630 ; N'r =-0.00000901 76 ;

#=+0.0085881; Nv =—0.0000080444;

#"=+0.0054808; NTf =+0.0000035813 ;

#'"=+0.00084153 ; #"'=+0.0000000609.

For the root gr1=7".248571, we get,

, 1656245 ,7 . 4351968 173055600

aH=+ 10,0 NI> fc=+— r- *J 2'=

Whence ft=20° 50' 7".8 ; log. #=8.4298727.

NI =+0.0269075 ; N," =+0.000010666 ;

N> =_().0201633 ; N,T =+0.000010891 ;

#/' =-0.0153761 ; #"=-0.00002774;

^'"=-0.0025474 ; N™= -0.0000001116.

For the root 02=17".014590, we get,

18874270 A7 .40848060 3.068734

*»=- i(^r— -^«! ^=H --- KP — ^; z^- IQW

Whence &=335° 12' 0".8' ; log. #2=7.1662319.

#2 =+0.0014663 ; Wv =-0.0000010862 ;

#2' =-0.01 12099 ; Ntr =-0.000006540 ;

#2" =+0.01 13035 ; NS' =+0.0000004019 ;

#2'"=+0.0225521 ; #2"'=+0.0000000294.

For the root &= 17". 7 84 76 2, we get,

, 131051610.. 141079200 12.088926

- — ^i; 2/3= 10* iow

Whence /33=137° 6' 45".3; log. #3=7.2021538.

#3 =+0.0015928 ; Wr =+0.0000010705 ;

#3' =-0.0131842 ; NJ =+0.000010999 ;

#3" =+0.0162587 ; ^P/ =-0.000000627 ;

#3^=-0.0000000463.

For the root ^4=0".6142754, we get,

, 3.385189 ,_,r ,1.399525 53091.32

*4=+- 1010 W;y*=+- ^ - z*- '»

Whence /34=67° 32" 18".7; log. #/r=95.8388230.

#4 =+0.000008364; N4'v = -- +0.000068996 ;

#4' =+0.000012715 ; #4r = =+0.000077824 ;

Nt' = +0.0000 14729 ; #"=,+0.00167466 ;

#"=-|-0.000023832 ; #4 -=+0,01002282.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 95

For the root r/5=2".7706502, we get,

0.15:382110 0.1345391 _318.2862

10 ° ' io10 ' "~io" — '

Whence p6— 101° 54' 14".5 ; log. #5/r=7.3115881.

N, =+0.00061393; Nb'r =+0.00204922 ;

JV5' =+0.00059396 ;. N,r =+0.00185548 ;

#5" =+0.00062031 ; N," = +0.0300096 ;

N™= +0.000821 54 ; #5r"=— 0.00288234.

For the root #0=3". 722656, we get,

_(U577640^ 0.8595159 r 2J2.63367 r2

IO10 ~1(P~~ 10'°
Whence /36=28° 2' 20".5 ; log. #6'r=8.6337215.

#8 = =+0.0245381 ; N6'r =+0.0430251 ;

Na' =+0.0165876 ; N6r =+0.0339716 ;

JV6" =+0.0163169; JV6"=— 0.0466232;

#8'"=+0.0187473 ; #6r/'=— 0.0015353.

For the root #7=22".466817, we get,

1.009767 0.7872124 r 81.89131 /r2

xi — InTo •"! > y^ — I iruo "•"'l ' ^ — H i Vuo ^7 •

Whence /?7=307° 56' 23".7 ; log. #/r=8. 1940957.

N7 =—0.0000974; JV/F =+0.0156349;

Nj' =+0.0003164 ; N,r =—0.0483516 ;

Nj'= -0.0023759 ; N7ri = +0.00 18091 ;

#,'"=—0.0150168 ; #7r//=+0.0001364.

42. We have thus obtained the values of all the constants, corresponding to the
separate variations of the planetary masses. If we now subtract the values of the
constants which correspond to the assumed masses from the values which result
from the supposition that each planetary mass receives in succession a finite incre-
ment, and divide the difference of the constants by the supposed increment of
mass, and connect together the different results, we shall have the following system
of equations for the determination of the constants which correspond to any other
assumed finite variation of the masses. The unit of the coefficients of ^u, [i, fi", &c.,
in the values of N, N', N", &c., Nlt #/, #/', &c., are the seventh decimal place of
these coefficients.

96 SECULAR VARIATIONS OF THE ELEMENTS OF

«=5".463803-0".079418«+2".68094iu'+0".91484«"+0".034565/.'"+l".8040;u'r

+0".08684u'+0".00150//'+0".00052um;

£=88° 0' 38"-2939^+50632V-20688y-3815y-73640yr+57456y

+5236y/+1202y";

J^=+0.1766064-50121iu-420200la'+205480i/+ 7970 '"+676800^

-s-486480 / — 1 6 1 8()«' ''— 4360^r" ;

^'=+00085906+59050iu+146200«'-44980X'-4746u'"-124400u"'-31040iWr

— 900;."— 250/.r//;

=+0 .0008418+5672^+20302 :'— 2300-/— oGl.T.a"— 23540u/r—

— 10

— 72^— 52/i'+27jtt"+3.5/'+18V— 86.4^"— 6.8^"— 1.

^"=+0.0000035+30^— 9^'— 20/— 1.7(u'"

^r//= +0.0000000.6+0.4^+1.4^+0.2^— 0.0^'"— 0.8u/F— 0.2lUF— 1.4^"— 0.2^".

^1=7".248427+0".193515/u+0".08894u'+l"33906/+0".155640i(/"+5".2076i«/r

+0". 2429^ r+0".0041 8:< "+0".001 44 u T" ;

/31=200 50' 19"+41718>+171946V— 86900>"+7848V— 221553'Xr+'''5188>r

-1396>p/— 115>F//;

^=+0.0268838+5871^+793180^'— 124700^*— 29868^'"— 658400//r

+104480^ r+11600^F/+2370m;

JV/=— 0.0201444— 27607^— 17180^'+ 72020/+839 V+62400/r— 98840/z"

-9120/1"— 1890 r";

^"=—0.0153619— 25311^— 25560^'+18800/+8920;u"'+95500/r— 73200^"

-6860/z"'— 1420r";

^'"=—0.0025451— 4573/u—8070u'—14568/+10181u'"+3730/r—11212ar

-1120//"— 23V";

JV1/r=+0.0000106+9.3/u+43.6^'+10//"— 4.4u"'— 162iu/r+99.4lur+9.61u"
^"=+0.0000109+11.4^+45.2^+22. V—3.1/r—181/p+104.8^r+6

Nlrr=— 0.0000027— 14.6.7— 10.6u'+3.6lu*+1.8-/'+63u/r— 54.4«r— 1.8/tw

-3.0,/"

JVi"^— 0.0000001— 0.1^—0.5iu'—0.2/+0.0//"'+1.5/p—1.6//+0.7/u"+0.1lury/.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 97

5r2=17".014373+Ow.075742iu+3".72988/t'+4''.06318M''4-0".0067921u"'+8".0455/t/F

+0".36832«F-fO".00628,uF'+0".0021V";

&=335° ll'3r+483^— 336590y— 239676y— 27647y"+753490y/F+7692y

+802y'+294'y";
^=+0.0014673+1200ia+61784iu'+47930/[i"+11002la"'— 130700^F— 3160/tF

_420^"— 90^";
N2'= -0.0112171— 9383^—345400^'— 339820lu"—83919,/"-f928300^F+20560;uF

-Ay'=+0.0113105+6512^+410140/u'+196140/+77329;u'"— 842300^^-

l!jm— 3928400/
-138640lMF— 7660^"+1980^r";

NJV=— 0.0000011— 1.2w— 77^'— 43.4uff— 18.1^'"-f-114"'+10.8;u "—!%"— 2.7^m;
Nar= -0.0000064— 10.4a—658/—447.6/—110.1la'"4-930/F+621a "+4.2^"

-13:9^"J;

^2F7=+0.0000004— 0.6i«+38;/+25.4't"+€.7^"f— 58,u
JVr2"'=:+0.0000000.3+0.0/u+2.8iu'+1.9iu''+0.5lu"'— 4.2/i

£73=17".784456+0".0343831u+2''.31316i«'+2".93202//'+0.2314071u"'+12".4768/t/|r

-fO".53576/iF+0".00892^r/+0".00306ur";
&=137° 6' 36".5— 4099>— 1 13430y— 73350>"+352'>'"+143360y F+3904>F

-ney^+ssy";

JVr3=-)-0.0015934-)-1401^+63100/<'-|-482801u"+8277^"'— 129900,u/F— 4560ittF

_220/'— 60^™;
^3'=— 0.0131892— 11200u—351120^—410400ia"—73086//'+945000/F

t«'"— 880700/r
-26920^ F— 1940 F/—540,«r//;
3"=— 0.0790650+46942y+3069920^'+1932860/+161058i«"T— 3813200/F

-209680«F+3740i;<F/+6701«F";
3'F=+0.0000011— 1.3w— 80.4-7— 63.47-

— 10.3«— 662.2X— 4131u"+71|[i"'+8237F— 207^F— 4.

N3r"=— 0.0000000.5+0.0/i+2.8/t'+1.8/— 0.3/'— 3.3^/F— O.

13 January, 1872.

98 SECULAR VARIATIONS OF THE ELEMENTS OF

04=0\6166849+Olu+0//+0^00002/+0".000007u'''+0''.1905/r+0''.22384:tF

+0".22638l«"— 0".()24V';
ft=67° 56' 35"+16>+34y+78y+16y-50700yr+25000y+79500y'

— 14560>F";

Ai =+0.0000077- l(i— 6fi'- 4/+ <y— 58/F- 44<u F+
AV =+0.0000117- 5fi—llfi'— V+ O/'-- 83^r- 70^r+
N: =+0.0000136- 1^— 20/*'— 2,a"+ 1^'"— 90/r- 83^ r+ 82^ "+11 5^ v";
NC =+0.0000219— 0^— V—
N4'T= +0.0000636+
JV;r =+0.0000717+
JV/' =+0.0015578+ 3/1— V+12/— 3/'— 6250^— 7476^r+2058^

jV4r/=+0.0100389+17iW— 22u'+84t"— 15/'—

^=2".727659+0".000019lu+0".00026ia'+0".00050l«"+0".00016/(i"'+0".9302lu/r

+l".38608i(tF— 0".02282^"+0".42991;um;
ft= 1 05° 3' 53"+ 19>+940y + 1 860>" +566 >'"— 192300 >' r+236840>r

+45040>F/— 1 13780V" ;
JV6 =+0.0005685— 50/1— 2656X—1488/— 57^'"— 3612iu/r+322iuF+2844iU"

=+0.0005571— 65^— 620^— 822^"— 30^'"— 3436a/F—1881;tr+ 2850^
6" =+0.0005832— 41iu—896X—394fi"—19/'—3314;/F—

+3708/tr//;
™ =+0.0007765— 9/£— 272^—498^"— 22^'"— 3807^F—4428(aF+3976^F/

^'"=+0.0019436—1^—80^'— 160^"— 55/'—8674/F—13008i« "+9994^"

+ 10560/";
JV/ =+0.0017694— l^—62u'—124/— 43/'—11920/F— 7172^ F+9256/tF/

+8608/ir//;
^6r/ =+0.0297330+1^— 240,./— 420^"— 182"'— 75800,u'r+101280^F— 90840,uF/

+27660/";
JV6F"=+0.0029105+0/«+281u'+54a"+22;/'+17870ia'p+6676/uF— 23736^"

+2816,um.

flf8=3".716607+0".0()0071/i+0".00318iu'+0".00632iu"+0".002015/'+0".6700/F

2".78408juF+0".17048/r/+0".06049i(iF//;
,38=280 8' 46"+19>+24>'+90y— 0".5^'"— 61400yF+73080VF— 6760>"

-385()y";

N9 =+0.0244939— 642«— 233400^'— 121880/— 4595^'"— 193700/F+558800^r

36680^ "+4420^"';
Nj =+0.0166053— 531(U— 21540u'— 41360/— 1758//'"— 88500/F+149720ur

+ 11340u"+1770ar";

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 99

6" =+0.0163413— 454u— 24920^'— 20000/— 1358^"— 72000/w/r+115600/ur

+9220p"— 2440^"';
a'" =+0.0187954— 104^— 6080^'— 106(%"— 110^'"— 23900lu/r+40840/zr

V/F =+0.0431601 +31^+40^'+ 140^"+78/'+24500^/r— 17080^^4300^

-1350
J =+0.0341011+26^4-100^'+260/+llV"— 39300/F+53800^r— 3160,u7

V6r/=— 0.0448614— 01«+1380la'+2680/+834iu'"—

+72860^"— 176180^r/J;
JV6F//=+0.0014205— 1^—100^—200^"— 66^'"+8800(U/F—41440^r+24280i«r/

+11480,ur".

/r

^7=22".460848+0".000029«+0".00130iu'+0".00274/+0".001098u'"+17".5730ju/

4".60528/ur+0".24104;tr/-fO".05969(tim;

/?7— 307° 56' 50"+OV+20y+42"/tt"+2".7^"— 19800y+22120>r— 2060>r/ ,

-265V"

, =—0.0000975— 13^— 1280^'— 1360lu"+97X"+2300/r—160^r+60//F/

7' =+0.0003175+147/i+10160^'+18400/— 911/'— 20800/"— 4520^ p

— 460,a"— HO/";
," : -0.0023780— 167^— 20400^'— 7560/[i"+3508/z'"+33100ia/r—11480^r

VS =—0.0150371— 201^— 10040^'— 46960^"— 867/t/'+74100f/F— 26520ur

+9160^"+2030^F//;
^/F =+0.0156383+l^+40^'+60//"— 14/'— 146000^/r+149520^F— 2220/'

-340^r//;

V7F =—0.0483504—2^-

r+ 10360/1 F— 240,u "+330^ F//;

100 SECULAR VARIATIONS OF THE ELEMENTS OF

CHAPTER II.

ON THE SECULAR VARIATIONS OF THE NODES AND INCLINATIONS

OF THE ORBITS.

1. THE secular variations of the nodes, and the inclinations of the orbits, are
determined by the integration of a system of differential equations which are
entirely similar in form to those from which the eccentricities and perihelia were
obtained.

If we denote by <J>, <£>, $", &c., the inclinations, and by 0, 0', 6", &c., the longi-
tudes of the nodes of the planets, Mercury, Venus, the Earth, #c., and put
tan $ sin 6=p, tan $>' sin 6'=p' tan <£" sin 6"=p" &c., ) /%Q±\
tan <p cos 6=q, tan <£>' cos 6'=qf tan <£>" cos 6"=q" &c. ; j

we shall have the following system of differential equations for the determination
of p, p',p", &c., q, q1, q", &c.

&C

. J

&C.

To integrate these equations, we shall suppose

, P'=N' sin (gt+p), /=#" sin (gt+(3), &c., ) ,g65

, &c. j v

q=Ncos (gt-\-p), </= W cos (</<+/?), q"=N" cos i

If we substitute these values of p, q, p', q", &c., in equations (E), they will become

"— &c.; "
-&c.; I (E;^

= K3'V~M.!i'V-h2's-focc-l-iV — ^<°)^ — (*,*)& '— ^.»;-iv — &c.;
&c.

These equations are similar in form to equations (B), except that g is negative.
They will produce, by eliminating N', N", &c., an equation in g, of the eighth degree.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 101

One of the roots of this equation will evidently be equal to nothing, since equations
(E') will be satisfied by supposing #=0, and N=N'=N", &c.

'2. We shall now suppose,

(366)

(2,2)=,/+(2,o)-H2,1)-j-(2,3).f&C.,

&C.J

— b =o,

-f(S , 0

I, =(4 , 0) JV -f (4 , !)#' +(4 , 2)#" -|-(4 ,
^=(»,0)# -f(5,l)^'+(5,2)JVr" + (3,3)
63 =(6 , 0) JV +(6 , l) #' -j_(6 , 2)^" _j_(. , 3)

(367)

(368)

If we now substitute these quantities in equations (E'), they will become

=0,

(E")

(3,0)JV (3,1)^' (3,5

(5 , ,)2V' ._ (8 , 4) Nir—(* , *)N "— (• , 7)
(e,6)JV'w^(^4)^-^^9,«)jf^^(,,r)

(7 , 7)iVm— (7 , 4) JV/r— (7 , 5) JV r —(7 , G)

2 =0,
j=0,

4 =0.

(E'")

These equations are similar to equations (B") and (B'") of § 9; and we may make
use of equations (31-64) for their solution, if we suppose [Q,O]=(O,Q), [i,i|=(i.i),

&c.; |»,ij^^ — (".O) |°.2I= — (°.2), &c.; [TTol^ — (1'°)» i1-2]^ — 0'2)» ^c-» in these
equations. We have given the values of (o,i)» (°'2)> ^CM O-'Oi C1*2)? ^c-» ^n § ^- '^ne
values of (0,0), (0,1), (2,2), &c., are given by means of the corresponding values |Q.Q|,
|i.i|, [2,2), &c., in equations (67), by simply changing the sign of the numerical
terms of the second member.

3. We shall now reduce equations (31-64) to numbers. The values of the pro-
ducts (o,o)(i,i)i (°.°)(2<2), &c. are given by means of equations (68-79) by simply
changing the sign of the coefficients of y.

102

SECULAR VARIATIONS OF THE ELEMENTS OF

Computation of A.

(o o)(a,8)=^+18.6424288.»7+ 72.81534747
_(o 0)0,2)(2,3M-0,3)= +19.3779798.J7+107.9403044
422 ,,0(0,3)^0,3)= + 0.0482175.0+ 0.6303080

+ (.;. ... X'.0-K'.0= + °-4437946

+0,. 2,0)0,3)^0,3)= + 0.0738673

_/. Ovo ,) = 0.0350051)

Sum of terras ^=^+88.0686261.^+181.^67616

Computation of A.

(0,o)(3,3)=02+23.1231203.</+97.7739453
-fo o)0,3)(3,2)--(i,2)= + 0.0270293../+ 0.1505601
1 ,VI,.V«M)-HI,»)= + 0.0228504.J/+ 0.4010900

',o 0,3) (.,.)-KI.O= +

l^'lvo'?/0'2 l'2 = - 0.0002174

Sum of terms ^'=02+23.1730000.0+98.3267843

Computation of A.

(o,o)0,0=<72+16-8850240-.H-63.0261532

_(0 o)(2,0(i,3)-0,3)= + 1.8241484.0+10.1609735

0 0(2 o)(o s)--(2 3)= + 0.0038119.0+ 0.0431310

+(,;.)(.',,xi>,o-k«,«)= + 0.0416825

H« 00,0 0,3-4,3)= + 0.0879559-

_(i'o)(o 0 = -0.5272484

Sum of terms ^"=02+18.7129843.0+72.8326477

Computation, of D.

(i,0(2,2)=02+24.3869412.0+147.9086074
-0 0(o,*)(*,3)-Ko,3)= + 9.1832668.0+103.9065348

(2 2)(« 00 »)-(o,»)= +10.9347838.0+142.941385

+(»'o0.2X2-3)-^(0-3)= +211.894020

Ho,2)(2,00.3)-K°'3)= + 16.751639

3)^,0,^) = - 35.348315

Sum of -terms D=02+44.50499 18.0+588.053871

Computation of D.

(l,0(3,3)=02+28.8676327.0+198.606593
_(,,0(o,3)(s,2)-4-(o,2)= + 0.0570369.0+ 0.645360
_X 3)(o i)(i 2)^(0,2)= +23.0739263.0+405.013500
00 3)(3 2)— (0,2)= + 0.623672

3)(3 00 2)— («'.«)= + °-104041

(3^)0,3) = — 0.049305

Sum of term8 77=02+51. 9985959.0+604.943861

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 103

Computation of _D".

(2,2)(3,3)=0a+30.6250375.flr+229.4540814
_(2,2)(o,3)(s,i)-:-(o,i)= + 0.0045090.^+ 0.0589430
_(3,s)(o,s)(2,i)-:-(o,i)= 4 1.5319589.0+ 26.8902683
+(o,s)(s,s)(2, !)-=-(<», i)= + 0.0873762

-J-(0,2)(2,3)(3, 1)-1-(0,1)= + 0.0414048

_(3,2)(2,3) — 0.5237732

Sum of terms JD"=02+32.1615054.0+256.0083004

Computation of A}.

(4,4)(G,o)=02+10.2786276.0+20.7811250
— (4,4)(o,6)(e,-)-^,7)= + 1.7539697.0+13.1764792
_(6,e)(S,4)(4,7)^-(»,7)= 4 6.3754170.0+17.6360114

+ (S,4)(4,6)(6,7)-:-(5,7) = + 8.7135093

-|_(«,8)(«,4X4.0-K»,0 = + 0.0911343

_(8,4)(4,8) - 0.0710140

Sum of terms J1=02+18.4080144.0+60.3272452

Computation of A.2.

(4 , 4)(7 , 7)^03+8. 16033230.0+4.8676955
—(4 , 4)(i , -)(? , 6)^(5 , 6)= +0.06449760.0+0.4845300
_(7,7)(4,4)(4,8)-=-(s,8)= +4-96787889.0+3.218971

+ (5,4)(4,7)(7,8)-:-(s,6)= +0.411199

+ (5 , 7)(7 , 4)(4 , 6)-l-(3 , 6)= +0.003351

_(7,4)(4,7) —0.004301

Sum of terms ^12=02+13.1927088.0+8.981445

Computation of Az.

(4, 4)(^,.0=02+26.1085883.0+139. 7017323
— (4, 4)(8,a)(a, ?)-=-(«, 7)= + 0.2214108.0+ 1.663321
_(.,,5)(6,4)(4,7)-^(6,7)=: + 0.0519589.0+ 0.966238

+ (6,4)(4,5)(5,7)-!-(6,7)= + 1.099942

+(e,*)(a,4)(4,7)-:-(e,7)= 4 1-4H586

_(S,4)(4,S) -134.964267

Sum of terms J3=02+26.381 9580.0+ 9.878552

Computation of D,.

(a , s)(e , 6)=</2+21.3624651.(7+ 51.441815
— (s,s)(4,8)(8,7)-:-(4,7)= + 1.3667356.0+ 25.416106
— (8,8)(4,«)(«,0— 0-0= +21.1694805.0+ 58.560122

+ (4,*)(S,8)(8,7)_ (4,7)= 4 37.130625

+ (4,8)(8,S)(5,7)— (4,7)= + 0.302610

-(8,0(«,«) 0.388348

Sum of terms JD.=/+43.8986812.0+172.462930

104

SECULAR VARIATIONS OF THE ELEMENTS OF

Computation of D.2.

(5,5)(7,7)=£s+19.2441698.#+12.0495445
_(i)5)(4,7)(7,6)-:-(4,6)= -J- 0.0827715.17+ 1.539238
— (',')(4'0(S'0-K4'0= +27. 1673827.^+ 17.603294
0(«.0('.0-K*'0- + 1-752231

7)(7,5)(5,6)-i-(4>6) = + 0.018327

—(7,6)0-0 - 0.014281

Sum of terms A^+46.4943240.^+32.948353

Computation of D3.

(6, 6)(7,7)=/+3.4142091.0+l. 79241220
—(6 , 6)(4 , T)(T , 5)-=-0 , s)= +0.0006746.0+0.00186605
_(7,7)(4,6)(6,s)-h(4,5)= +0.0142946.0+0.00926231
+(4-0(7-0(6-0-K4'0= +0.00118319

-j-(4,e)(e,7)(7,6)-:-(4,.)= +0-00092197

— ('.«)(«, 0 —0.11312682

Sum of terms Z>3=02+ 3.4291782.0+1.69251890

Computation of B, B, and B1.

(2,2)=0+13.072173

— (i,i)(«,s)-t-(i,s)= +19.37798

Sum =JB-i-&=0+32.45015

(3, 3)=0+l 7.5528645

— (.,.i)(3,2)-^-(i,2)= + 0.0270293

Sum =B+b=g+ 17.5798938

(i, 0=^+11.314768

_(1>.)(i,3)^-(2,3)= + 1.824148

Sum =5*^-6=^+13.138916.

Computation of C, C", C", and C'".

— (2,2)=— g— 13.0721730
(o,i)(i.i)-s-(o,j)= - - 9.183266
Sum =—|^+22.255439 1
log. \ (0,3) -±-(1,3) | = [9.4381 189]
'Therefore C'=— |gr+22.255439| x

[9.

-(>.») =-^—17.5528645
(o,3) (3, 2) -5- («.«)= - - 0.0570356
Sum =—^+17.60990011
log. j(o,«)-s-(«,i)|=[9.113807P]
Therefore C'=— \g-\-\ 7.6099001 } x

[9.1138076]//.

Computation of J515 B2, and B3.

(6, e) =<7+2.7662522
-(••0 («, »)-=-(», T)= +1.7539698
Sum =B,H-&1=«/+4.5202220;

(7, 7) =#+0.6479569
-_(«,T)(T,t)-t-(«,«j +0.0644976
Sum =52-=-Z>1=(7-)-0.7124545;

(S)5)=5r+18.5962120

_(«,,)(,,T)_!-(«,T)= + 0.2214108

Sum =53^-61=^+18.81 76237.

Computation of C^ Ct, C3, and Ck.

— (6,6)=— g— 2.7662522
(4,e) (0,7) -^(4 ,7)= - - 1.3667356

Sum =— \g+ 4.1329878|
log. {(«, 7)^-(«, 7) (=[9.5433087]
Therefore Ci=— \g+ 4.1 329878 jx

[9.5433087]52.

— (7,7)=— g— 0.6479569
(4,T)(7,6)-i-(4,.)= -- 0.0827715

Sum =— j«/+ 0.7307284 }
log. {(4,«)-5-(».e)|=s[9.484971l]
Therefore (7a=— \g-\- 0.7307284 } x

[9.434971 1]/>Z.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 1Q5

(o,»)(i,i)-f.(j,»)=— 0.5002407

— (0,1) =+ 2.9986729
Sum =C"-t-b'=4- 2.4984322
Therefore C"'=+[0.3976676]&'.

(o,j)(s1i)-j-(s,i)=— 0.2370651
-(•,i)=+ 2.9988729
Sum =G'"^-V=-\- 2.7616078
Therefore <7"=+[0.4411620]Z>'.

Computation of E, E', E", and E'".

(o,s)(i,2)-i-(i,s)=_ 1.818323
-(0,2)=+ 0.861707
Sum =E-^b"=— 0.956616
Therefore E=— [9.9807377J5".

-(1,1)=— 0—11.3147682
+(o,i)(i,s)^-(o,s)= —10.9347838
Sum =— {0+22.2495520 j

log. {(o.3)-i-(2,3)j = [8.9723624]
Therefore E'=— j 0+22. 249552 1 X

[8.9723624J&".

-(3,3)=— 0—17.5528645
+(o,s)(s,i)-i-(o1i)= - - 0.0045090
Sum = — |0+17.5573735 1

log. {(«,i)-s-(s.i)} = [9.7501125]

Therefore E"=— \ 0+17.5573735 1 x
[9.7501 125J6".

(o,i)(s,s)^-(31i)=_10.89986
-(o,s)=+ 0.86171
Sum —E'"^b"=— 10.03815
Therefore E'"=— [1.0016537J&".

Computation of F, F\' F", and F'".

(o.s)(i,s)-i-(i,j)=_ 0.01326047
-(o)3)=+ 0.02798148
Sum =F-+V"=-\- 0.01472101
Therefore JP=+[8-1679376]6'".

(o,1)(2,s)j-(2,i)=_ 0.1677337
-(0,3)=+ 0.0279815
Sum =F-±-V=— 0.1397522
Therefore F'=— [9.1453586J6'".

14 February, 1872.

(«,7X«.«)-K8-7)=— 0.0773584
-(4,5)=+ 7.3963746
Sum =C»-i-6,=+ 7.3190162
Therefore <73=+[0.8644527]&2.

(4,6)(7,6)-5-(7,6) = -. 0.0602795

-(4,8)=+ 7.3963746
Sum =(74-7-62=:+ 7.3360951
Therefore (74=+[0.8^54649]62.

Computation of Elt E2, E3, and E4.

(4, 7)(j, •)-!_(«, 7)=— 0.09722854
-(4,6)=+ 0.07576285
Sum El-^-b3=— 0.02146569
Therefore El=— [8.3317448]63.

—(5,5)— —0—18.5962129
(«.OC<'.7)^-(4'7)= —21.1694805
Sum =— J0+39.7656934|
log. {(4, 7)^(8, ?) I = [8. 74377 18]
Therefore^,=— ^0+39.7656934| x

[8.7437718]63.

— (7,7)=— 0— 0.6479569
(«,0(7.«)-5-(*'a)= -- 0.00067458
Sum =— 10+ 0.64863148|
log. I (4, s)-K«,«)| =[0.7242832]
Therefore^,=— ^0+0.6486315 } x
[0.7242832]63.

6)=— 9.296196
-(4,6)— +0.075763
Sum =Et-r-bf=— 9.220433
Therefore E<=— [0.96475 14]5S.

Computation of F^ F2, Fs, and F+

(4,eX«,»)-5-(«,«)=— 0.01871457
-(4, 7)=+ 0.02401693
Sum =Fl-^bt=4- 0.00530236
Therefore JF1=+[7.7244692]64.

(4,6X«.')-5-(6-0=— 2.296302
-(4,7)=+ 0.024017
Sum =F2+bt=— 2.272285
Therefore*^— [0.3564628]i4.

106

SECULAR VARIATIONS OF THE ELEMENTS OF

—(2,2)=:— g— 13.072173
(o, :,)(„, ,)_!_(„, i)=_. 1.531958
Sum =—{0+14.604131!
log. \ (o.O-K3- i)} = [0.7927855]
Therefore^*=- 1«7+14.604131! X

[0.7927855]6m.

_(i,0=— 0—11.3147682
(o,i)(i,«)-t-(o,i)= —23.0739263
Sum =-{0+34.3886945|
log. { (o.O-M3- 2) | = [9.6907241]
Therefore Fm=—\ 0+34.3886945 } x

[9.6907241JS'".

— (6,6)=— 0+ 2.7662522

(4,6X«,0-K*.0= + 0.0142946
Sum =—{0+2.7805468!

log. { (4, 0-K7. Oj =[1.5514854]
Therefore F3=— {0+2.7805468! X
[1.5514854]64.

-(5,5)= —0—18.5962129

(4.,0(a,0-H>,«)= —27.1673827
Sum =—{0+45.7635956!
log. {(4,6)^(7, 6)| [9.4626364]

Therefore JP4=— {45.7635956+0! X
[9.4626364]Z>4.

Computation of the Equations of the fourth dejree.

— (3,i)(i,2)(o,o)(3,3)=— 35.3483155.04— 817.363351.0— 3456

- (3,2)(2,s)(o,o)(i,i)=— 0.5237732.02- 8.843924.0- 33

- (i,o)(o,i)(2,2)(3,s)=— 0.5272484.02- 16.147000.0- 120,

- (s.i)(i,3)(o,o)(2,2)=— 0.0493054.02- 0.919173.0- 3,

— (2,0)(0,2)(1,1)(3,3) = — 0.0350059.02-

- (3,o)(o,3)(i,i)(2,2)=— 0.0002174.02-

+2(3, 2)(2,l)(l, 3)(0, 0) =
+2(2,0)(0,1)(1,2)(3,3)= -.

+2(3, OO.oXo, 3X2,2)= -

+ 2(3,0)(0,2)(2,3)(1,1)= —

+ (,,0X0,0(3,0(2,3) =

1.010537.0—
0.005302.0-
1.910880.0-
1.615446.0-
0.004755.0-
0.003993.0-

(2,0)(0,2)(3,00,0 =

(i

0

10
28
0

0
0
0

0

— 2(3,0)(0,2)(2,0(1,0= — 0

Sum of preceding terms =— 36.483865S.02— 847.824361.0— 3659
Add (o,o)(i, 0(2, 0(3-0=

04+47.5100615.03+809.5847278.02+5804.515976.0+14461,
Sum is the value of equation (53).
Whence we get

0<+47.5100615.03+773.1008620.02) , (

+4956.691615.0 +10801.93370 J "
In like manner \ye get

- (», 0(4, 0(«, 0('. 0=—1 34.9642669.0'— 460.796228.^— 241.911599

- (•.0(i,0(4,4)(T,7)=_ 0.3883479.02- 3.169047.^- 1.890359

— (7, 0(6, 7)(4,4)(«,»)=- 0.1131268.02- 2.953581.^— 15.804014

— («,0(<-")(s,«)(7'7)=- 0.0710140.ry2- 1.366608,9- 0.855687

- ('.0(6-7)(«'4X6'6)=— 0.0142805..72- 0.146774.'^- 0.296764

— ('.00.0(s'8)(".«)=— 0.0043007./— 0.091874.'^— 0.2212^7

.144265

.01142

,97930

,59019

,95240

.03216

,64409

35570

,06216

.04518

,27616

00768

00173

,09214

04366

00728

67438

60808

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 107

4)(4,*)(»,«)(7,»— — 3.858530.0— 2.500161

+2(7,5)(S,4)(4,7)(6,6)= 0.182088.0- 0.503701

6)(e,;)(5,7)(4,4)= — 0.050095.0— 0.376332

4)(4,6)(6,7)(a,a)= — 0.011756.0- 0.218615

-f- (S,4)(4,.)(7,o)(e,7)= + 15.268077

_)- (6,6)(S,6)(7,4)(4,7) = -f 0.001670

4- (o,4)(4,e)(7,8)(e,7)= -f 0.001014

— 2(7,4)(4,.)(S,8)(6,7)= 0.319377

— 2(7,s)(5'4)(4'8)(6-7)= — 0.248866

— 2(7, 4)(4 , •)(« , »)(a , 7)= 0.002603

Sum of preceding terms =— 135.5553368.02— 472.626581.0— 249.878554

04+29.5227974.03+230.6343243.02+523.7682780.0+250.403089
Sum is value of equation (54).
Whence we get

(f— 29.5227974.03+95.0789875.22 \ , (3~Q.

+51.141697.0+0.524535 j =

Equations (55-64) reduced to numbers are as follows, in which the numbers
inclosed in brackets are logarithms.

JT =[0.5618811]— lDL=rp.8861924]^±lJi-; (371)

D Jj

^"=[1.0276376] ^!^b^'=[0.2498875] —^f^- ; (372)

^'"=[9.2072145] - .y =[0.3092759] - ZgL. • (373)

[9.4381189] Df — [9.1138076] Df m m±.

~[9.1 138076] AD — [9.4381 189] AD '

[8.9723624]J>/W — [9.7501125]jy . ,^^

~[9.7501125]^"Z)"— [8.9723624]A'Z> '

[0.7927855]Z>'/r— [9.6907241]Z>//r. (3~6)

~ [9.6907241] AD —[0.7927855]^"!)"'

Nr =[0.4566913] -llY tv1— [0.5650289] -^ ^2; (377)

^"=[1.2562282] ^^!±^=[9.2757168] A*N ~™; (378)

JVr"=[8.4485146] ^ilv tV6^[0.5373636] f M '~™; (379)
Ar,r_[9.5433087]A/2-[9.4349711]A/1. (38Q.

•^ ~~m AiUf\nii-\ A T\ pn a Af>r>t\OT\ A n ' V '

A7/r_[8.7437718]D,/4-[0.7242832]A/3. /381)

[0.7242832] ASD3— [8.7437718]^^'

108

SECULAR VARIATIONS OP THE ELEMENTS OP

_J
"

r[9~.46~26364]AA— [1-5514854]^3A'
[9.9546226]ZX/ —[0.2789339]0/ ~
= I [9.3015115]Z>/"-[0.0792616]/)'/" 1 JL
= | [8.9337058] £/'r— [0.0357672]D'/'r 1 JL

[0.5477106] A/i -[0.6560482]A/2 } ~n

(382)

; (383)

= [8.0240547] A/4 - [0.004566 1]£>3/3
= [7.9147050]Z)2/5 -[0.0035540]^^

). (384)

If we repeat and number the formulae which we have computed, we shall have
the following

Fundamental Equations for the adopted masses.

A =0»+38.0686261.0r+181.867616 ;
A=g*+ 23.1730000.^+ 98.3267843;

72.8326477 ;

3.4080144.0+ 60.3272452;
8.981445 ;
9.878552;

D =^2+44.5049918.^+588.053871 ;
.9985959 .(7+604 .943861 ;

32.948353;
g=^+ 3.4291782.^+ 1.69251890;

6; )

; j

C =—^+22.255439 1 [9.4381189J&';
C"=— j^+17.6099001|[9.1138076]i';
Crff=+[0.3976676]&';
C""=-i-[0.4411620]&';

E =— [9.98()7377]iff;

E' =—^+22.2495521 [8.9723624]iw;

E" -\\ 7+17.5573735 \ [9.

E'"=— [1.0016537]&";

(385)
(386)
(387)
(388)
(389)
(390)

(391)
(392)
(393)
(394)
(395)
(396)

(397)

(398)

(399)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

(400)

109

F =+[8.1679376]//';

F'=— [9.1453586]//";

F"=— {0+14.604131 J [0.7927855]ZT;

F'"=— 10+34.3886945 } [9.6907241]^'";

A= {0+4.5202220^,; B3=\g+ 0.7124545 1 6,;

Ci=— {0+4.1329878 } [9.5433087]S.2 ; ]

c*=~ {0+0.7307284 j[9.4349711]Z>2; I
C3=+[0.8644527]62;

C4=+[0.8654649]ftj;

-E1=-[8.3317448]/;3; 1

Ef=— \ 0+39.7656934 } [8. 7437718]&3 ;
Ea=—\g+ 0.6486315 j[0.7242832]63;
EI=— [0.964 7514]63;

(401)
(402)

(403)

(404)

—V=\ 4.2028443

-V = \

-5'"= { 14.659896 .

-^ = {0.000094200
-b.2 = 1 0.0000 11 2026
-13 = 0.00000096881

(405)

F2=— [0.3564628]64;
•^3= -{0+ 2.7805468 }[1.5514854]64;
f\=— {0+45.7635956 } [9.4626364]/>4.
We shall also have

-b =\ 1.6028375 .... [0.2048895] ^V/r+[8.8879781]^r
+[7.1246469] ^F/+[6.6629969]^r";
. . [0.6235433] J^V/r+[9.2986071]^r
+[7.5327484] ^"+[7.0706089]JVr//;
7.0682646 .... [0.8493128] j^'r+[9.5139049]^F
+[7.7447989] ^r/+[7.2820328]^r";
. . [1.1661309] | ^/r+[9.8003037]^F
+[8.0220791] ^r/+[7.5575701]^r//.

.... [95.9740497] \N +[7.6245464]^'
+[ 7.9450513] ^"+[7.4917417]^'";
, . . [95.0493193]}^ +[6.6917912]^'
+[ 7.0018244] ^"+[6.5180955]^'";
, . . [93.9862386] }JV +[5.6261830]^
+[ 5.9329689] ^"+[5.4401214]^'";

-64 = { 0.00000020167.. . . [93.3046628] \N +[4.9441177]^'

+[ 5.2502770] ^"+[4.7556866]^'".

4. If we now suppose the second members of equations (369) and (370) to be
equal to nothing, we shall obtain the following values of g, g^ g2, &c.

0 =— 5".1223003, 04=— 0".01045922, '

gl=— 6 .5863879, 0B=— 0 .66298299,

g2= -17.3924594, g,=— 2.91695653,

ffa=— 18 .4089138, 07=— 25 .93239866.

(406)

(407)

110

SECULAR VARIATIONS OF THE ELEMENTS OF

By means of equations (385-407) and (371-384), we now obtain the Mowing

values.

For the root g, we get,

g=—5". 126 11 2,
N' = +0.1 2279 19 N
N" =4-0.08795864^
N" = 4-0.01 757422W
.N'r =— 0.0002076885JV
N "=— 0.0002642292V
N™ =+0.0002315532V
N m=4-0.000006403252V

For the root g» we get,

9l=— 6".592128,
W =— 0.27646362V;
M" =— 0.22293927V;
Nf =— 0.04673056 ^
Nt'r =+0.0003357396^
NJ =4-0.0004742548^
JVi" =—0.0002454315^
Nlr"=— 0.00001482107^,

For the root gz, we get,

g-=— 17'.393390,
Ni =— 5.563101^2
JV2" =+ 4.563350^2
Nf =4-33.24578^
Nt'r=— 0.001649287^2
NS =— 0.01403826^2
#/'=+ 0.001367028JV3
JV/"=+ 0.000157230^2

For the root #,, we get,

#,=— 18'.408914,
NJ =— 6.098710^
.JV3" =+ 6.6558882V3
Nf =—10.22309^,
N3'r=— 0.0001958278^,
N3r =— 0.0001514003^3
N,rt=+ 0.000012606272V3
N5™=-{- 0.00000140198^

log. 9.0891696,

" 8.9442784,

« 8.2448760,

« 6.3174124«,

» 6.421980n,

« 6.364650,

« 4.806400.

log. 9.4416380n,

" 9.3481864n,

" 8.6696010«,

« 6.5260026,

" 6.6760116,

« 6.3899302n,

" 5.1708794n.

log. 0.74531 70«,

" 0.6592838,

» 1.5217364,

" 7.2172964n,

" 8.1473134n,

» 7.1357772,

" 6.1965360.

log. 0.7852380n,

" 0.8232060,

« 1.0095825n,

" 5.2918731n,

" 6.1801268,

« 5.1005864,

« 4.1467414.

For the root g» we get,
^=0, and

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. HI

^ we get,
g,=— 0".661G66,
5 =+1.232212JV/

For the root

JV5" =+1.108223^."
N™ =+1.049477 A;"
N5r =+0.9653287 A/'
Nt"= — 0.9379252 N&'r
Ntru=— 9.829270^'"

For the root g6, we get,

<76=— 2".9 16082,
JV6 =+ 3.557327A6'r
AV =+ 2.059 163 A~0jr
A7 =+ 1.845317AVr
JV6'" =+ 1.314187JV6'r
Nar =+ 0.8164588W8/r
JVr/=— 20.11300W/r

log. 0.0906854,
0.0535832,
0.0446270,
0.0209706,
9.9846752,
9.9721682«,
0.9925212«.

log. 0.5511238,

" 0.3136906,

" 0.2660709,

" 0.1186570,

" 9.9119342,

" 1.3034768n,

" 0.3347880.

log. 8.6240604»,

" 8.6677625n,'

•" 9.6363826?i,

" 0.1667598n,

" 0.3963230n,

" 9.0387886,

" 8.0882343.

5. Having thus determined all the roots of the equation of the eighth degree,
together with the ratios of the constant quantities N', N'\ N"\ &c., corresponding
to each root, the complete integrals of equations (E) will be

For the root </7, we get,

fr=— 25".934567,
N7 =— 0.04207851aV
JV,' =— 0.04653315JV
N7" =— 0.4328950JV
N7" =—1.468114
Njr =— 2.490709JV

r
"

'
"

=N cos

cos

cos

&c.;

p =N sin

&c.

sin

sn

The analysis of § 16 will conduct us to the following equations for the deter-
mination of the arbitrary constants corresponding to each root.

112

SECULAR VARIATIONS OF THE ELEMENTS OF

m
na

na

na

n a

n'a"

&c. sin

(408)

(409)

And since, for the root #4, we have Nt=N^=:Nt"=Ntm, &c., the system of equations
similar to equations (133) will be divisible by JV4 NJ &c., we shall have the follow-
ing equations of condition : —

na

a

n a

&c.=o

&c-=° ;

'na

na'

n' a "

&c.

(410)

Now putting the first members of equations (408) and (409), equal to x and y
respectively, and the coefficient of sin or cos of (gt-\-(3), in the same equations
equal to 2, we shall have

x=z sin (gt+(3) ; y= cos (gt-\-@) ; 1
whence tan (gt-\-@)=x-i-y. j

(411)

6. In order to find the values of x and y, which are to be used in equations (411),
we shall assume the following values of <£>, $', $", &c., 6, 6', 6", &c., corresponding
to the beginning of the year 1850, at which epoch t—0.

Mercury,

<p =7°

O7

8".2,

Venus,

, r O
m r~~o

23

34.4,

The Earth,

<?>" =0

0

0.0,

Mars,

«r=i

51

2.3,

Jupiter,

^'=1

18

40.3,

Saturn,

^ =2

29

22.4,

Uranus,

$r/=0

46

29.9,

Neptune,

<?>m=l

47

0.9,

6 = 46°
& = 75
V = 0
0" =^ 48
0/r = 98
0" =112
0r/ = 73
0r"=130

33'
20
0
23
54
19
14
7

3".2;
42.9;
0 ;
36.8;
20.5;
20.6;
14.4;
45.3;

Now, since ^=tan <£> cos 0, and p=:tan $ sin 0, we find,

p =+0.0891690, log.jp =8.9502138;

p' =4-0.0573577,

p" = 0.

pm =+0.0241597,

log. p' =8.7585915;

log. p" = — QO

log. pm =8.3830911;

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 113

p'T =+0.0226127, log. p" =8.3543534 ;

pr =+0.0402201, log.pr =8.6044432;

pvi =+0.0129519, log. p" =8. 1123323;

^"'=+0.0238090, log. ^""=8.3767411 ;

q =+0.0844678, log. q =8.9266911;

q1 =+0.0149991, log. q' =8.1760651;

q" = 0. log./ =-oo ;

q" =+0.0214548, log. q'" =8.3315249;

q'r =—0.0035434, log. q'T =7.5494159n;

qv =—0.0165138, log. qv =8.2178478«;

qvi =+0.0039012, log. qri =7.591 1972;

qr"— —0.0200698, log. ?r"=8.3025434rc.

Now, adding the logarithms of m-^-na, m'-^-n'a', &c., which are given in § 5,
we shall obtain the following values of the logarithms of the constants for the given
epoch, which enter into the values of x and y.

log. »_^_ =85.9443779 ; log. q— =85.9208552 ;

na na

log.p'^L =86.9845985; log. q'^ =86.4020721;

n a na

P"^, —oo log.^ =-00

m'" m'"

p'"^- =85.9337058; log. q" ™- m =85.8821396;

p'T-^jr =89.5793573 ; log. q"^~v =88.774419871 ;

pr-^ =89.4372661; log. qr-^-f =89.0506707w;

log. pr! -"L_=88.2449047 ; log. qvi- —=87.7237696 ;

n a'
p™-^-88. 7292393 ; log. 9"'-^,=88.6550416n.

7. These quantities are now to be substituted in equations (408) and (409), in
connection with the values of N', N", N"1, &c., corresponding to the different roots.

For the root gr=— 5M26112, we find,

_ 0.612517 1.5865662 _ 14.046564

TO" ' y~ TOlT^ ~T6""~

Whence (3=211° 6' 26".8; and log. #=9.0830567.
Therefore, for the root g, we have the following values,

JV =+0.121076, N'r=— 0.00002517,

N' =+0.0148671, Nr =—0.00003200,

N" =+0.0106496, Nrr =+0.00002804,

N'" =+0.0021278, JVr//=+0.000000775.

15 February, 1872.

1H SECULAR VARIATIONS OF THE ELEMENTS OF

In like manner we shall find for the root gr1=6".592128,

, 0.692850 ._ 0.6389472 33.24230

a^+'-jpr— #,; y,= -JQH *iJ'%= -10u ^i-

Whence ^=132° 4tf 57".8; and log. #,=8.4525867.

Therefore, for the root #„ we have,

Ai =+0.028352, #/r =+0.00000952,

N{ =—0.007838, #/ =+0.00001345,

#/ =—0.006321, #/'=— 0.00000696,

#,* =—0.001325, #/"=— 0.00000042.

For the root <72= — 17".393390, we find,

6.863228 v . 2.889470 ... 488.6204 .

**=- -ir— N» v*=-\ — 13 — 2' Za= ~" — 2

Whence /?2=292° 49' 53".2; and log. N2=l. 1829906.

^=+0.001524, N2'r=— 0.00000251,

#; =—0.008478, NS =—0.00002140 ,

JV2" =+0.0069546, #/' =+0.00000208,

JV2"=+0.0506672, A2r//=+0.00000024.

For the root ff3=— 1 a". 4089 14, we get,

6.724398 .. 2.217061 , 1925.3617 ..

Xa= "~"^»; y*= 10" *; z*= "To15 — 3

Whence £,=251° 45' 8".6; and log. #,=7.5655490.

#3 =+0.003677, #,^=—0.000000072,

#,' =—0.022428, #3r =—0.000000557,

#3"= +0.024477, N3ri =+0.0000000463,

#,"=—0.0375951, #3F//=+0.000000005.

For the root ^=0", we get,

_ 0.7256453 „ 0.2113461 _ 27.24403 ,

** - +~ ~~f0IO "*• &*— JQ10 "tt Z*— IQW '™f

Whence /?4=106° 14' 18".0; log. JV4=8.4431335.

And #4=#;=#/=#;''=#/r=#4r=#/ '=#/"= +0.027741 73.

For the root g6=— 0".661666, we get,

. 0.1016992 , 0.2717211 ,77r 24.19165 ..„

*»=H --- IQIO -- ^;yb= ~w, -- #,^5^= w ^.

Whence /?6=20° 31' 24".6 ; and log. #6/r=7.0789383.

#,=+0.001478, N^ =+0.001 1994,

#; =+0.001 357, #6r =+0.0011577,

#,"=+0.001329, #6r/=— 0.00112485,

#."=+0.001259, #,"'=-0.0117882.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 115

For the root g,=— 2".916082, we get,

,0.3678921 /r. _°^54480lv-. _580.9484 Ar/r2

•Ce 1 nlO 6 ' "fl -I mo •''6 > Z0 •• "'" •"« •

JQ1

1011

Whence /?6=133° 56' 10".8; and log. ^'"=6. 9441 833.

Ne =+0.003128, N6ir =+0.0008794,

#„' =+0.001811, Ntr =+0.0007180,
JV6" =+0.001623,
JV6'"=+0.001156,

=— 0.0176872,

2V0m=+0.0019010.

"or the root #7= — 25".934567, we get,

0.2996623 __.„. , 0.2203054 AT,r 59.03157

w* . /V • i/ — 1 Ar • 5* —

"~JQ10 ^'7 > 111 H 1Q10 -tv? ' ^— 1Q1

„.

Xfl

Whence ^=306° 19' 21".2; log. ^/r=7.7993771.

N7 =—0.0002652, JV7/F =+0.00630053,

JV7' =—0.0002932, N,r =—0.0156928,

JV7" =—0.0027275, N7TI =+0.0006890,

JV7'"=— 0.0092499, JV7r/'=+0.00007720.

If these values be substituted in equations (F), we shall have the complete values
of q, q', q", &c., p, p', p", &c., from which we can obtain the inclination of the orbits
of all the planets to the fixed ecliptic of 1850, and the longitudes of the nodes, on
the same plane and referred to the equinox of 1850, by the formulae

tan <2>=n

tan 6=-^.

(412)

8. If we how substitute in equations (F), the values of q and p, we shall get
9=tan $ cos 0=^cos (^+/3)+^1 cos fat+PJ+N, cos (g^+&)+ &c.; (413)
p=tan <p sin e=Nsw (gt+^+Ni sin (gj+fid+Nt sin (^+ft)+ &c- (414)

Multiplying equations (413) by sin (#<+/3), and (414) by — cos (#<+/3), we shall
get, by adding their products, and reducing

tan (?) sin (0—gt—(3)=Nl sin { (gl—g)t-{-(3l—(3 \ +N2 sin {(gz—g)t ) ...

If we multiply (413) by cos (gt-\-(3), and (414) by sin
adding their products, and reducing
tan q> cos (6 — gt — /3)=

, we shall get, by

cos —

cos z—

Dividing equation (415) by (416) we eliminate tan$, and find,
tan(0— gt— /3)=

sn -

sn t-

- &c.

cos r1_

cos z-

-3 &c

:} <417)

When the sum ^+^+^3+ &c. of the coefficients of the cosines of the deno-
minator, taken positively, is less than N, tan (6 — gt — /3) cannot become infinite ;
the angle (0—gt—(3) cannot become a right angle: consequently, the mean motion

116 SECULAR VARIATIONS OF THE ELEMENTS OF

of the node will, in this case, be equal to (jt. The analysis of § 19 being applied
to equation (416), will show that

maximum tan q>—N-\- -ZV1-j-JYjj-|-.ZV3+ &c. ; )
and minimum tan<^=^V — j-ZV^-j--^-)-.^--}- &c. )

We shall now substitute the numbers which we have already computed, in these
equations, for the purpose of determining the maximum and minimum values of
the inclinations of the different orbits to the fixed ecliptic of 1850, and the mean
motions of the nodes of the different planets on that plane.

9. For the planet Mercury, we have,

Maximum tan <£ = N + JVi + -ZV2 + N3 + &c. =0. 1 87242. One-half of this is
0.093621, which being less than N, it follows that A7" exceeds the sum of all the
remaining terms ; consequently, the mean motion of Mercury's node is equal to g,
or — 5*. 126 11 2. The maximum inclination of his orbit to the ecliptic of 1850 is
10° 36' 20"; and the minimum inclination is 3° 47 8".

The substitution of the numbers for the other planets shows that the minimum
inclinations of all the other planetary orbits to the ecliptic of 1850 are equal to
nothing; consequently, the mean motions of the nodes on that plane are indeter-
minate. The maximum inclinations of the different orbits are as follows: —

Max. inclination. Max. inclination.

Venus, 4° 51' Jupiter, 2° 4'

Earth, 4 41 Saturn, 2 36

Mars, 7 28 Uranus, 2 42.5

Neptune, 2 22.7

Having thus given the solution of the fundamental equations for the assumed
masses, it now remains to determine the coefficients depending on the variation of
the masses. This we shall do by using the same finite variations of the masses as
were employed in finding the similar coefficients of the variations of the constants
on which the eccentricities and perihelia depend.

10. If we now suppose that ^=+1.5, we shall obtain the values of the funda-
mental quantities which are to be used in the computation by simply making all
the terms of equations (153-169) positive. We shall then obtain the following

O 1

Fundamental Equation/I for «=-(-_: or for m= — _

T2 1946300.4 '

A =£2+38.201888.<7 +183.882325;

A' =0«-|- 23.2189305.04- 98.9957893
A"=<7S+18.9824430.<7+ 73.7725468
^1=/-j-18.4081571.«7-(- 60.3279033
42=<72-fl3.1928504./7+ 8.981549;
A— 0M-26.3821 161. g+ 9.881338;

(419)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 117

D =<72+44.8296684.0+595.295403;
Z>'=02+52.2739916.<7+609.992181 ;
JD"=:02+32.2340961.<7+257.2490518;
A=<72-|-43.8986994.,7+172.463056 ;
D2=<72+46.4943411.0+ 32.948378;
A=i/2+ 3.4291799.0+ 1.69252064.

£=10+32.51109 j&; JB'=)0+17.5915487J6; \

B"= J0+13.402657 \l- \

C =—^+22.316375 j [9.4381 189]6'; 1
C" =— |<H-17.6215550j[9.1138076]&'; [
<7"=+[0.3976676]6';
C""=+[0.4411620]6';

E =— [9.9807377]6";
E'=— 10+22.5132928 j [S.9723624]//;
E" = — J0+17.5690284 } [9.7501125J6";
E"=— [1.0016537J6";

F =+[8.1679376]6'";

!"=— [9.1453586]6'";

F"=— 10+14.665067 |[0.7927855]6'";

Jp""=_^+34.6524353|[9.6907241]6'";

(420)

(421)
(422)

(423)

!,= |0+4.5202234 J&i; B2=\g+ 0.7124548^; 1

B3= ^0+18.8176405 j^; J

Ci=— 10+4. 1329892 } [9.5433087]^;
(72=_^+0.73072874J[9.4349711]&2;
(73=+[0.8644527]62;
6'4=+[0.8654649]&2 ;

E!=— [8.331 7448]63;
E,=- 10+39.7657102 |[8.7437718]63;
Et=—\g+ 0.64863178 } [0.7242832]Z>3;
E,=— [0.9647509]&3;

JF1=+[7.7244692]54;

Ft=— [0.3564628]64;

F3=— ^0+27.805482 } [1.5514854]64;

Ft=— 10+45.7636124 1 [9.4626364]&4.

^+47.8463980^+784.270833 V I =(r, - «„ y3) ;

+5058.994655.0+10978.29258 j

/+29.5229572.03+95.0823272.02 1 =( }<

+51.151359.^+0.529116 |

(424)

(425)

(426)

(427)

(428)

(429)

118

SECULAR VARIATIONS OF THE ELEMENTS OF

The values of 6, &', &", and W are given by equations (405), and the values of
4,, 42, &,, and 14 are given by equations (406), by merely multiplying the coefficients
of .Vby 1+^=2.5.

If we put equations (429) and (430) equal to nothing, they will give

g,=— 4".8105312, 04=— 0".0105499,

g,=— 7.0595858, gb=— 0.6629939,

&=— 17 .4356542, 96=— 2.9169622,

gt=— 18 .5406218, fr=— 25 .9324509.

The solutions of equations (419-430) will now give the following values — remem-
bering that the coefficients of equations (371-384) remain unchanged.

For the root g, we get,

•j =— 4".815328,
N' =+0.2099057#
N" =+0.1471310.ZV
N'" =+0.0292551^
N'r =— 0.000390605AT
2?' =—0.000486221^
N" =+0.000495923 /V
N r//=-|-0.000008719136^

log. 9.3210242,

" 9.1677040,

" 8.4662020,

" 6.5917376»,

" 6.6868340/?,

" 6.6954144,

" 4.9404735.

For the root g^ we get,

ffl=— 7".064535,
Ni =— 0.400690 N!
N," =—0.3382646^
Ai" =—0.0725079^,
^"=+0.000451081^
NS =+0.000660230JVi
NS1 =—0.000300091^
Nlr"=— 0.0000202801^

For the root _</2, we get,

gt=— 17".436558,

N, =— 5.522000JV2
NS =+ 4.240878iV;
-Iff =+37.23564^V2
N1'T=— 0.001749727JVS
NtT =— 0.0152271 lNt
Nt"=+ 0.001476315iV2
N,r"=+ 0.0001698568^,

log. 9.6028084w,

" 9.5292564w,

" 8.8603854n,

" 6.6542543,

" 6.8196951,

" 6.4772529«,

" 5.3070710».

log. 0.7420964n,

" 0.6274557,

" 1.5709589,

" 7.2429702w,

" 8.1826175n,

" 7.1691790,

«' 6.2300830.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 119

For the root </3, we get,

08=— 18". 540625,
N3' =— 6.10629JVa
N," =+6.47848iV3
Na" =— 8.653764JV,
N3ir =—0.0000396789^3
NS =— 0.000582834 /V3
N3n =+0.0000498987iV,
JV3r"=+0.00000572183JVs

For the root #t=0", we get,

log. 0.7857773ft,

" 0.8114730,

" 0.9372052«,

" 5.5985600w,

" 6.7655450«,

" 5.6980890,

" 4.7575350.

For the root g.a, we get,

g&=— 0".661636,

=+1. 049956 JV6/

^— 9.829980JV/

For the root g§, we get,

<76=-2".916041,
, =+ 3.695208iV6/r
6' =4- 2.162242^8^
6" =+ 1.91106JV6/F
J" =+ 1.327425^6IV
6r =+ 0.816341JV6r/

=+ 2.162624JV/r

For the root fa, we get,

g,=— 25".934626,
N7 =—0.0421 230^
N,' =—0.0455 785 A'/r
^•T" =—0.4351 129 N7ir
NJ" =— 1.469776JV/"
>/ =— 2.490698JV7/r

log. 0.0916830,

" 0.0551040,

" 0.0455402,

" 0.0211710,

" 9.9846725,

" 9.9722356/j,

" 0.9925526/z.

log. 0.5676388,

" 0.3349042,

" 0.2812740,

" 0.1230098,

" 9.9118716,

" 1.3036368w,

" 0.3349790.

log. 8.6245191 w,

" 8.6587603»,

« 9.6386020«,

" 0.1672216»,

" 0.3963210H,

" 9.0337842,

« 8.0882305.

1'20 SECULAR VARIATIONS OF THE ELEMENTS OF

11. Substituting these values in equations (408) and (409), we shall get the
following values: —

For the root g=— 4".815328, we get,

, 1.524075 .. . 3.435486 Ar 36.65228 .72

T" N; z=

10" -

Whence /?=23° 55' 15".6; and log. #=9.2108901.

#=+0.162514, #'F=— 0.0000635,

# =+0.034037, NT =—0.0000790,

#"=+0.023913, #r/ =+0.0000806,

#"=+0.004754, #F/'=+0.0000014.

For the root g,=— 7".064535, we get,

1/725880 46 To.88155

&1 — H --- JQM **!• 2/1 — ~~T(P* ' 2' — ' ""10"" ''

Whence £,=90° 0' T.I ; and log. #,=8.3568764.

#i =+0.022744, #/r =+0.0000103,

N1'=— 0.009114, Air =+0.0000150,

#t" =—0.007694, Air/ =—0.0000068,

N1"=— 0.001649, N1r"=— 0.000000046.

For the root gz=— 17".043656, we get,

6.709998 3.568591 _ 5820.032

-"a > Z2— " ~ 2 '

Whence ^=298° 0' 19". 7 ; and log. JV2=7. 1158842.

Nt =+0.001306, #2'r=— 0.00000228,

#; =—0.007211, NS =—0.00001988,

Nf =+0.005538, N2" =+0.00000193,

#2"=+0.048623, Ni"= +0.00000022.

For the root </3=— 18".540625, we get,

_ 6.589852 Ar 1.924650 X7 _1774.3737

X3 — JQ13 "81 2/3— JQ13 *»•? 23— JQ13 •«••

Whence /?8=253° 43' 8".0 ; and log. #,=7.5876056.

#3 =+0.003869, #3"=— 0.000000154,

#3' =—0.023626, #3r =—0.000002255,

#3" =+0.025066, #/' = +0.0000001931,

#3W=— 0.033482, #s" =+0.0000002213.

For the root <74=0, we get,
. 0.725777

4= To15

Whence ^==106° 13' 35".2; W"4=+0.0277436, log. #4=8.4431625.

. 0.7257772 ._. 0.2112211. 27.24551,

4= 15 — ^4; ?/4== ---- io -^; Zi=~~^~ 4*

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 121
For the root #6=— 0".661636, we get,

;

10'° HP

Whence /35=20° 32' 21".0 ; and log. JV6'r=7.0792052.

N, =+0.001482, Nb'r =+0.00120007,

NJ =+0.001362, JV7 =+0.00115846,

N," =+0.001333, N,TI=— 0.00112554,

JV6'"=+0.001260, N™=— 0.0117964.

For the root #6=— 2".916041, we get,

0.3540483 _ 581.3949

-^*6 » ^6 •• "in •"«

_ 0.3683810

To

IO10 IO10

Whence /?6=133° 51' 48".4; and log. N^ =6.9438946.

N, =+0.003247, Nt" =+0.0008788,

Nt' =+0.001900, Ntr =+0.0007174,

JV6"=+0.001679, &«"=— 0.0176820,

^6F//=+0.0019005.

59.03124

For the root g7=— 25".934657, we get,

0.2996641 , , 0.2202989

10

,10

Whence /37=306° 19' 17".6; and log. ^V/r=7.7993764.

N7 =—0.000265, N,ir =+0.00630052,

Nj' =—0.000287, N7r =—0.0156923,

JVT"=— 0.002741, N," =+0.0006891,

^'"=—0.007355, J/V7m=+0.00007720.

12. For an increment of ^, in the mass of Venm, we have the preliminary com-
putations by merely making all the coefficients positive in equations (193-208).
We shall then obtain the following

Fundamental Equations for ^'=+-_; or for m'= ^.

20 o714^8.o

A =^+38.4851145.^+188.270582;
^'=^+23.3470945.^+101.1013581 ;
^1"=/+18.9541253.#+ 75.3044562;

60.3282264;

8.981600;
^3=/+26.3821932.^+ 9.882701.

D =^^45.3182858.^+610.942268 ;
J>'=/+53.1764531.#+626.087193 ;
D"=/+32. 4522201.^+261. 0475918 ;
^=^+43.8987079.^+1 72. 4631 15 ;

Z>2=^2+46.4943490.^+ 32.948389 ;
Z>3=^+ 3.4291807.^+ 1.69252138.

(431)

(432)

16 February, 1872.

122 SECULAR VARIATIONS OF THE ELEMENTS OF

r> c i nn 11 ctn.) T. . TV t „ I 1 1 K(\ iD^ATI A .1

(433)

= {0+32.71670 j&; # = 10+17.6040547 J6; )
B"=\g-\- 13. 230 123 \b- j

C =— J0+22.521994 1 [9.4381 189]^';

c'=— j0+i7.63406io£[9.ii38076]&';

(7"=+[0.4188569]6';
<7"=+[0.4623513]&';

E =— [9.9807377J&";
E'=— { 0+22.7962912 j [8.9723624]i";
^"=—10+17.58153441 [9.7501125]6";
E'"=— [1.0016537J6";

F =+[8.1679376]6'";
F' =— [9.1453586]6'";

F'"=— I 0+35.542391 } [9.6907241JZT;

!; Bt=\g+ 0.7124549 \\; \
Bz= 1^+18.8176481 16,; j

Ci=— 1^+4. 1329899 } [9.5433087]62;
O2=— \g-\-0. 7307288 } [9.43497 ll]ia;
(78=+[0.8644527]J2;
(74=+[0.8654649]&2;

E,=— [8.331 7448]&3;

(434)

(435)

(436)

(437)

(438)

(439)

(440)

E4=— [0.9647509]&3;

Jl=+[7.7244692]64;
^=—[0.3564628]^;
J?;=—[0+2.7805489j[1.5514854]&4;
FI=— [0+45.763620 \ [9.4626364]64;

^+47.9507108.08+787.5462101.02 )
+5098.436146.0 +11226.89493 j "

94+29.5230351.08+95.0839594.r/2
+51.156083.0 + 0.531357

The values of J, &', i", and b'" are given by equations (405) ; and the values of
^> &2» ba, and 54 are given by equations (406), by merely multiplying the coefficients
oftf'by 1+^=1.05.

If we put equations (441) and (442) equal to nothing, they will give
g=— 5".1965445, g<=— 0".0105949,

ff,=— 6 .6295555, g,=— 0 .6629993,

^=—17.4583971, ^6=— 2 .9169649,

9t=— 18 .6662138, g->=— 25 .9324761.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

now give the following values:

123

The solutions of equations (431-442) will
For the root </, we get,

</=— 5".201065,

N' =+0. 13743982V
2V" =+0.10005562V
IV"' - -+0.02016892V
Nir =— 0.0002330312V
Nr =— 0.0002980262V
2V"=+0.0002525282V
2V"'=+0.000007542082V

For the root git we get,

</!=— 6".6347695,
2V/ =— 0.23729002V!
2V," =—0.19356762^
2V/" =—0.04095122^
2V/r =+0.0002920342V,
NS =+0.0004138182^
2Vxr/=— 0.0002115382V,
N1V"=— 0.00001292222^

For the root g^ we get,

&=— 17".459300,
2V2' =— 5.2349372V2

2V2"' =+40.010222V2
N2lr=— 0.001842382V2
2V2r ==— 0.016221342V2
2V2"=+ 0.0015690542V2
2V2"r=+ 0.0001805562V2

For the root gs, we get,

=—5.839411^3
=+6.569047JV3
=—7.879027^3
=—0.0000520881^3

=— 0.0009631222V;
=+0.0000818396^

N3r
N3TI
iV3rj/=+0.000009421622v"3

log. 9.1381124,
" 9.0002416,
" 8.3046820,

" 6.4742546»,
" 6.4023092,
" 4.8774912.

log. 9.3752793«,

" 9.2868326n,

" 8.6122666n,

" 6.4654333,

" 6.6168096,

" 6.3253889»,

« 5.1113360w.

log. 0.7189114n,

" 0.6243470,

" 1.6021710,

" 7.2653793n,

" 8.2100866«,

" 7.1956378,

« 6.2566120.

log. 0.7663690«,

" 0.8175024,

" 0.8964726«,

" 5.7167383»,

" 6.9836813«,

« 5.9129634,

" 4.9741254.

For the root g^ we get,

04=0", and Nt= JV4'=iV4"=, &c.

124 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root <76, we get,

ft=— 0".661664,

N6 =+1.229553JV5"r log. 0.0897471,

NJ =+1.131670^'" " 0.0537194,

N6" =+1.1 08860^" " 0.0448769,

Nf =+1.049663^ " 0.0210502,

jV6r =+0.9653253^6/ir " 9.9846737,

NJ'=— 0.9380050^" " 9.9722051»,

^6r//=— 9.829623^6/|r " 0.9925368».

For the root gw we get,

0e=— 2".9160753,
N6 =+ 3.484775JV/r
JV6' =+ 2.0637627V6/r
N; =+ 1.853037^r

6T =+ 0.

log. 0.5421747,

" 0.3146597,

" 0.2678842,

" 0.1194064,

" 9.9119207,

" 1.3035125?*,

" 0.3342872.

For the root «77, we get,

^7=— 25".934665,
#7 =— 0.0423304^•/^
^TT' =— 0.0418973^/r
Nf =— 0.4430757JV/r
W;'" =— 1.470350JV/r
Njr =— 2.490680 N,"

N, w=+0.

log. 8.6266524n,

" 8.6221861»,

" 9.6464778n,

" 0.1674208»,

" 0.3963178^,

" 9.0387805,

" 8.0882264.

13. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities: —

For the root g=— 5".201065, we get,

. 0.638032 v 1.6964973 ._ 15.329701 ._

X — —I- AT- 11 — - N- 7 — . AT*

JQ14 iT> y— ]QU JQ14

Whence /3=20° 36' 29".8 ; and log. ^=9.0727385.

N =+0.118233, #'"=— 0.00002755,

N' =+0.016251, #r =—0.00003524,

^"=_)-0.011804, Nrr =+0.00002986,

Nm =+0.002385, Nr"= +0.00000089 18.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS 125

For the root <h= — 6".634795, we get,

0.637032 0.4606865

27.744169

Whence ^=125° 52' 26".4; and log. iV;=8.4523391.

N,. =+0.028336, N,ir =+0.000008275,

N,' =—0.006724, JV/ =+0.000011726,

Nf =-0.005485, 2V/' =-0.000005994,

2VV" =—0.001160, N1va=— 0.0000003662.

For the root #2=— 17".459300, we get,

6.884438

.

2= -\

3.578728

%=

6546.046

Whence /?2=297° 28' 0".l ; and log. iV2=7.0738291.

jV2 =+0.001185,
J\T2' =—0.006205,
JV2" =+0.004991,
^'"=+0.047424,

N3ir=— 0.000002184,
N,r =—0.000019227,
NZYI =+0.000001 860,
^2m=+0.000000214.

For the root g3=— 18".666222, we get,

6.787567 m. 1.953449

1727.1619

Whence /33=253° 56' 39".4; and log. 2^=7.6116607.

2V3 =+0.004089,
2V3' =—0.023880,
2V," =+0.026863,
#,"=—0.032221,

For the root <74=0", we get,
. 0.7256935 ,.

N^"—— 0.000000213,
N3r =—0.000003939,
N3TI =+0.000000335,
N3T"= +0.0000000385.

0.2113335 Af
2/4 — Tmo •"•»

10io y*— i0io

Whence /?4=106° 14' ll".l ; and log. 2V4=8.4431447.

27.24487
^Oio •

For the root &=— 0".661664, we get,

0.1017327

0.2717512

241.9334

10

,10

Whence &=20° 31' 39".5; and log. 2V/r= 7.0789678.

JV5 =+0.001475, Nb'v =+0.001 19941,

2V6' =+0.001 357, 2V6r =+0.00115783,

2V5" =+0.001330, 2V5F/ =— 0.00112485,

2V5'"=+0.001259, 2V6r//=— 0.0117895.

126

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root #„=— 2".916075, we get,

_ 0.36782CM

X6 H 1|UO '

; 2/0=

0.3543521

581.0177

uo

10io 1010

Whence #,=133° 55' 53".6 ; and log. #0'r=6.9440119.

#6 =+0.003063, #/' =+0.000879047,

#.' =+0.001814, #6r =+0.000717683,

#.' =+0.001629, #6r/ = -0.0176817,

#6'"=+0.001157, #."'=+0.00189801.

For the root 07=— 25".934665, we get,

0.2996518 . . 0.2202706

•N"; "~

59.03076

Whence /?7=306° 19' 9".l ; and log. #/r=7.7993490.

#T =—0.0002667, #T" =+0.00630012,

#/ =—0.0002640, NS =-0.0156916,

jV;' =,—0.0027914, #7" =+0.00068886,

#7'"=-0.0092634, #/"=+0.000077193

14. For an increment of -fa, to the mass of the earth, we have the preliminary
computations by merely making all the coefficients positive in equations (232-237).
We shall then obtain the following

Fundamental Equations for ^"=+— ; or for m"=l-r-351 132.4.

A =0«+39.080610.0 +188.728586 ;
^'=02+23.3039078.<7+ 99.5792048;
4"==^+19.0875991.<7+ 75.2609947:
^1=^2+18.4084593.(7+ 60.3292965 ;
J2=*72+13.1931503.(7+ 8.981769;
J3=<72+26.3824488.(7+ 9.887223 ;

D =^+45.2956845.0+610.444623;
JJ =/+52.4179477.<7+613.831274 ;

(443)

i>1=^+43.8987357.0+172.463309 ;
#,=^+46.4943751.0+ 32.948430;
Z)3=02+ 3.4291834.0+ 1.69252419.

# = ^+17.667716216;)
S"= j 0+13.490445^6; j

C =— 10+22.7146021 [9.4381189]6';
C'=— J0+17.6977225 1[9.1138076]6';
C"'=+[0.3976676]6';
C""= +[0.441 1620]6';

(444)

(445)

(446)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

127

E =— [0.0019270]6";

JE'=— {0+22.6010814}[8.9723624]&";

E'"=— [1.0228430J&";
F =+[8.1679376]6'";

F"=— {,7+14.680729^ [0.7927855]Z/";
J""=— !$r+34. 7402239 } [9.6907241>'";

: {,7+4.5202263 } &,; B.>=\g+ 0.71245534^,; )
^=1^+18.8176739 }&,; J

C,=— {0+4. 13299,21 1 [9.5433087]62; ]
C2=— j«7+0. 7307293 j [9.434971 1]62;
C3=+[0.8644527]52;

1

S ^+39.7657436 } [8.7437718]&3 ;
\g-\- 0.6486324 } [0.7242832]63;
[0.9647509]&3;

j/;=

— [0.3564628]54;

— \g-\- 2.7805511 }[1.5514854]64;
_j^_)_45.7636454j[9.4626364]64.

(447)

(448)

(449)
(450)

(451)

(452)

^+47.9724987.<f+787.7904009./ ) , 45g

+5093.460923^+11190.32489 j"

^+29.5232934.^+95.0893749.^ ) ,

+51.171764.^ + 0.538790 j "

The values of 6, V, b", and V" are given by equations (405) ; and the values of
Jt, 62, &3, and &4 are given by equations (406), by merely multiplying the coefficients
of JVby l+j«"=1.05.

If we put the equations (453) and (454) equal to nothing, they will give

g =— 5". 1662610, g,=— 0".0107428,

9l=— 6 .6253960, g,=— 0 .6630168,

02=— 17 .5129033, 9*=— 2.9169739,

g3=— 18 .6679386, flf7=— 25 .9325598.

128 SECULAR VARIATIONS OF THE ELEMENTS OF

The solutions of equations (443-454) will now give the following values:
For the root g, we get,

<7=— 5". 170230,

N' = +0.1 21 3375 N
N" =+0.0874828^
N" =+0.01800280^
Ntr =— 0.0002097093JV
N7 =— 0.0002676246JV
Nri =+0.0002299092^
N "'=+0.00000665906^

For the root g^ we get,

#!=— 6".631323,
Ai' =— 0.2725185A;
Ai* =— 0.2210735JV,
Ai" =—0.04781081^
JV/r =+0.000340372Ai
Air =+0.000482200Ai
N,ri =—0.0002467402^
A1r/7=— 0.0000150588A;

For the root g^ we get,

g.2=— 17".5137898,

NJ =— 5.562224^
N2" =+ 4. 103496 N2
N? =+38.24098JV2
Ns'r=— 0.001733292^2
W =— 0.01569197^
Nt"=+ 0.001509355JV2
Ntr"=+ 0.0001737467^2

For the root g^ we get,

ga=— 18".667944,
2Vg' =—6.186960^3
N3" =+6.338887Aa
N3m =— 8.476533A3
N^ =—0.00004491846^
N3T =— 0.0007949226JV3
N3" =+0.00006740752JV3
JVr3"/=+0.00000774975A8

For the root #4, we get,
^=0", and

log. 9.0839952,

" 8.9419226,

" 8.2553402,

" 6.3216176»i,

" 6.4275260»z,

" 6.3615564,

" 4.8234126.

log. 9.4353960n,

" 9.3445366n,

" 8.6795260?i,

" 6.5319537,

" 6.6832273,

" 6.3922400n,

" 5.1777901w.

log. 0.7452485n,

» 0.6131540,

" 1.5825290,

" 7.2388717n,

" 8.1956776n,

« 7.1787913,

" 6.2399165.

log. 0.7914773n,

" 0.8020130,

" 0.9282183w,

" 5.6524248n,

" 6.9003248n,

« 5.8287083,

" 4.8892878.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

129

For the root g^ we get,

gt=— 0".G61665,

Nt' =+1.130316JV5"p
N," =+1.107793iVr,/F
Ns" =+1.049700JV5/r
Nar =4-0.96532472V5/r
#5"=— 0.9380148iV5/r
Ntra=— 9.829793i\r6

5
'r

For the root g^ we get,

</6= — 2".916080,
2V6 =+ 3.5096032Va"
2V6' =4- 2.0433762V6'r
2V6" =+ 1.8368842V."
2V6'" =4- 1.3157522V/r
2Var =-j- 0.81644042Va'r

2V8F"=-|- 2.161811^"

For the root g7, we get,

07=— 25".934782,
N, =— 0.04268995iV/F
2V/ =— 0.035352232V/r
2Vr" =— 0.43727582V/r
2V/" =— 1.4787182VT"
2V/ =— 2.49()6482V/r

log. 0.0900640,

" 0.0531999,

" 0.0444583,

" 0.0210647,

" 9.9846734,

" 9.9722097n,

" 0.9925444».

log. 0.5452580,

" 0.3103480,

" 0.2640819,

" 0.1191740,

" 9.9119245,

" 1.3035012«,

" 0.3348177.

log. 8.6303257n,
" 8.5484168n,
" 9.6407554n,

" 0.3963122^,
" 9.0387706,
« 8.0882161.

15. If we ndw substitute these values in equations (408) and (409), we shall
obtain the following quantities : —

For the root g=— 5". 1701 89, we get,

, 0.581731 . , 1.5880256 ,. 14.050229 ,„

*=H — JQU — N< y=-\ — 10^ — ; z= — I7m^

Whence /3=20° 7' 8".3, and log. 2V=9.0805176.

N =-(-0.120370, 2V/r=— 0.00002524,

2V =4-0.014605, Nv =—0.00003221,

N" =+0.010530, NVI =+0.00002767,

2V" =4-0.002167, 2Vm=+0.00000080l5.

17 February, 1872.

130 SECULAR VARIATIONS OF TIIE ELEMENTS OF

For the root <h=— 6".631323, we get,

, 0.768973 ... 0.6414600 Ar 33.22066 A72

A/ * ni /V • y /V &

L4 •**!> i'l — in11 l9 : — inu IB

Whence ^=129° 50' 4".4; and log. ^=8.4791971.

M =+0.030144, ^=+0.00001026,

W =—0.008215, NS =+0.00001454,

Nl"=— 0.006664, N™ =—0.000007438,

^'"=—0.001441, N1T"=— 0.0000004539.

For the root g2=— 17".5137898, we get,

6.914692 , 3.461327 AT _6089.390

xn— ~i3 2' y* — ' 2' Zz — *5

Whence /^2=2960 35' 29".2; and log. ^=7.1037540.

N2 =+0.001270, N2'r=— 0.000002201,

JVj' =—0.007063, NZT =—0.000019926,

Nf =+0.005211, Ai" =+0.0000019167,

A2'"=+0.048560, N.2 "'=+0.0000002206.

For the root ga=— 18".667944, we get,

2.032363 1783.3974

A7 _ .

«— 1013 3~ jo13 3ii~ i013 •"•••

Whence /38=253° 27' 40". 1 ; and log. JV3=7.6024178.

N8 =+0.004003, ^^=—0.00000018,

JV3' =—0.024768, N3r =—0.00000318,

JV3" =+0.025376, N3ri =+0.00000027,

^'"=—0.033934, A"3r//=+0.000000031.

For the root #4=0", we get,

0.7256453 0.2113461 _ 27.24508

a4 - +- ~~JO~10 •"*» 2/4— ~W* *' 2*~ 1016

Whence /?4=106° 14' 18".0; and Nt=Nj=Nt"=&c.,
—+0.02774066, log. 8.4431168.

For the root g&=— 0".6616647, we get,

0.1016673 . _ 0.2717444 _24JL19412

** - T~ ~W 1V6 ' 2/5 - ~T 10 iV6 > Z6— 10 * '

Whence /38=20° 30- 57".6 ; and log. JV8/r=7.0789098.

N6 =+0.001476, N,'T =+0.001 19925,

Nt' =+0.001356, N* =+0.00115768,

^"=+0.001329, N6rf = — 0.00112471,

JV6"'=+0.001259, ^""=—0.01178812

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 131

For the root 06=— 2".916080, we get,
, 0.3678561 ,

0.3544989

tftr.

-i'R , Z,;

581.0130

Whence /36=133° 56' 26".3; and log. #,"=6.9441240.

Ne -=+0.003086, Na" =+0.000879273",

JV6' =+0.001797, N6V =+0.000717874,

#6" =+0.001615, N6ri=— 0.0176858,

JV6'"=+0.001157, #0r"=+0.00190082.

For the root 07=— 25".934782, we get,

0.2996357

.

N"; 2/7=

0.2203002

59.02984

Whence /3T=306° 19' 27".6 ; and log. JV/r=7.7993611.

JV7 =—0.0002690, N,ir =+0.00630030,

NJ =—0.0002227, N7T =—0.01569181,

^"=—0.0027550, N,™ =+0.000688861,

JV/"=— 0.0093164, JV/"= +0.0000771929.

16. For an increment of ^ to the mass of the earth, we have the preliminary
computations by merely making all the coefficients positive in equations (251-256).
We shall then obtain the following

Fundamental Equations for ^"=-1 ; or /orm"=l-r-335172.

10'

4=^+40.092595.0 +195.673045;
A =#2+23.4348156.#+100.8391921 ;

4"=02+19.4622137.(7+ 77.7179078;
4!=^+ 18.4089042.0+ 60.3313475;
4j=02+13-1935918-#+ 8.982096;
43=02+26.3829396.0+ 9.895893.

D =02+46.0863772.#+633.1 39825;
Z7=02+52.837 2995.^+622.776921 ;
j9"==/+32.4903462.^+261.2498776 ;
A^+43.8987902.^+1 72.463686 ;
D2=02+46.4944263^+ 32.948508 ;
Z>3=02+ 3.4291886.^+ 1.69252948.

B= |0+34.38795 16; B' = ^+17.7555387} b )

B"= ^+13.801975 1 b j

C =—^+27.173766| [9.4381 149]5';
C"=— J0+17.7855450 } [9.1138076]6';
6"'=+[0.3976676]6';
C""=+[0.4411620]6'.

(455)

(456)

(457)

(458)

132

SECULAR VARIATIONS OF THE ELEMENTS OF

E =— [0.0221304]6";
E'=— {£+22.6010814 } [8.9723624J6*;
E"=— j^+17.7330184 j [9.7501 125]6";
E"=— [1.0430464J&".

F =+[8.1679376]6'";
F'=— [9.1453586J6'";
^"=—^+14.7573271 [0.7927855]6";; j
F"=— {^+35.0517532 \ [9.6907241J6". j

: {0+4.5202306 } 6,; Bt=\g+ 0.7 1 24563 } bt; \

uiv,/

<71=_ |^+4. 1329964 1 [9.5433087]&2 ;
(7,= — {^+0.7307302} [9.434971 1]JS.

<74=--[0.8654649]&.2.

E,=— 1^+39.7657938 } [8.7437718& ;
E3=— \g-\- 0.6486333 1[0.7242832]&3;

(459)

(460)

(461)

(462)

(463)

(464)

J\=-f[7.7244692]&4 ;

F2=— [0.3564628]64;

JT3=_|^+ 2.7805554 j[1.5514854]54;

F4=— |V-|-45.7636960|[9.4626364]64.

^ +48.4349358*- +802.5743024.^ ) =

+5231.898890.^+11585.83042 j

^+29.5237894^ +95.0997623.^ 1 =( }. (466)

+51.201831.^ + 0.553045 j

The values of &, ?/, &", and &"' are given by equations (405); and the values of
4,, ft,, ia, and Z>6 are given by equations (406), by merely multiplying the coefficients
of N" by 1+^=

If we put equations (465) and (466) equal to nothing, they will give,

g=— 5".2095599; g,=- 0".0110263;

g,=— 6 .6631448 ; gt=— 0 .6630507 ;

£2=— 17. 6257463; ge=— 2.9169913;

g3=— 18 .9364848 ; 9i=—^ .9327210,

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 133

Equations (455-466) will now give the following values: —
For the root g, we get,

g=— 5".2136546 ;

2V' =+0.12006842V log. 9.0794288;

2V" =+0.08711802V " 8.9401080;

2V'" =+0.018442172V " 8.2658120;

2V'"=— 0.00021192252V « 96.3261770rc;

2V r =— 0.0002712742V " 96.4334082n;

2V"=+0.0002285892V " 96.3590546;

2V "'=+0.0000069 104722V " 94.8395077.

For the root glt we get,

9l=— 6".6692717;
2VJ =— 0.26830562V,
2V/' =— 0.21891682V1
2VT =—0.04878432^
2V,'r =+0.00034441462^
2V/ =4-0.0004893l02T,
N," =— 0.000247679 2Vj
#,"'=— 0.00001526832^

For the root g^ we get,

^2= — 17".6265859;
Nj =:— 5.55453^2

NJ =+

N2ir=— 0.00179671^,
N,T =— 0.0172969^2
N2"=+ 0.00164470JV2
Nzv"=+ 0.000189469JV2

For the root gs, we get,

«/3=— 18".9364959;
N3' =— 6.27941 JV3
N3" =+6.067662^
2V3'" =—7.198122^
2V3jr=— 0.00004612212^3
N3r =—0.00129815^
2V^3r/ =+0.0001076252Vr3
2V7'=:+0.0()001241762V3

log.

9.4286296n;

9.3402791n;

8.6882800?*;
96.5370815;
96.6895841;
96.3938898n;
95.1837912n.

log.

log.

0.7446477n;

0.5654561;

1.6342060;
97.2544790»;
98.2379690n;
97.2160878;
96.2775388.

0.7979189n;

0.7830214;

0.8572192«;
95.6639090n;
97.1133248?i;
96.0319138;
95.0940376.

For the root g^ we get,

1:34

SECULAR VARIATIONS OP THE ELEMENTS OF

For the root g&, we get,

#>=— 0".66 16623;

Nt =+1.228747#6/r

r

log. 0.0894623;

" 0.0528360;

" 0.0442980;

" 0.0211576;

" 9.9846716;

" 9.9722493n;

" 0.9925608».

log. 0.5396342;

" 0.3072002;

" 0.2622141;

" 0.1196872;

" 9.9119142;

" 1.303530071;

" 0.3348464.

log. §.6375619n;

" 8.3659300»;

" 9,6454670n;

" 0.1730168«;

« 0.3962995?i;

" 9.038753Q;

" 8.0881970.

17. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities :~

For the root g=— 5".2136546, we get,

N; =+1.12937JV
#6" =+1.107383#8/r
N^ =+1.049923Ar5'r
. Ntr =+0.965320Ar6*r
N&"=— 0.938100^
N6T"=— 9.8301 !Nt'r

For the root gM we get,

g6=— 2".9160771;
N6 =+ 3.46445JZV6'F
Nj =+ 2.02862^6/r
N9' =+ 1.82900JV/"
JV6'" =-j- 1.317308^
N,r =+ 0.816421^
Ntn=— 20.11546^

= 2.161956iV6/

For the root

we get,
^7=— 25".9350099;

N, =— 0.0434072^V/F
^' =—0.0232236^"

N," =— 0.442046 JV/r
JV7W =— 1.489419JV7Jir
JV/ =— 2.490578^"
JV/'=-|-0.1093334JV7/ir

*=+

5.51372

15.90391^

140.64952
1016 '

10" 1016

Whence /3=19° 7' 7".0 ; log. #=9.0780045.

#=+0.1196753; #'r=— 0.000025364;

IT =+0.0143703 ; Nr =—0.000032468 ;

#"=+0.0104268 ; #"=+0.000027359 ;

#"=+0.00220757 ; #r"=+0.00000082705 .

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 135

For the root <7i=— 6".6692717, we get,

8.43328 6.419195 _ 331.3455

10i5 ; 2/i- 10i» J «i- -iQir— •

Whence ^=127° 16' 40". 5 ; log. ^=98.5049615.

JVi =+0.0319861; ^i/r =+0.000011017;

N,' =—0.0085822 ; NS =+0.000015651 ;

Nf =—0.0070024 ; N^ =—0.000007922 ;

Nf=— 0.0015604 ; Nlr"=— 0.0000004884.

For the root fjz—— 17".626586, we get,

6.947485 4.015769 _ 7425.592

X~2— '13 » 2/2 + l3 5 Z2— U •

Whence /?2=300° 0' 37".6; log. #2=97.0339306.

N2 =+0.0010813; NJr=— 0.000001943;

#,' =—0.0060059 ; NZT =—0.000018702 ;

JV2" =+0.00397545 ; #2" =+0.000001 778 ;

JV2" '=+0.0465733 ; #,""= +0.0000002049.

For the root ga= — 18".9364959, we get,

6.960810 1.901617 _1696.1782

**- To5 ; yz~ "To13 ; Zs~ "To13 '

Whence /?3=254° 43' 12".9 ; log. #,=97.6288181.

N3 =+0.0042542 ; N3ir =—0.0000001962 ;

#,' =—0.0267139 ; N3r =—0.0000055201 ;

#3" =+0.0258130 ; NSTI =+0.0000004576 ;

#,"=— 0.0306223 ; #3m=+0.0000000528.

For the root <74=0", we get,
, 0.7256453

X* - H JQ10

Whence ^4=106° 14' 18".0; log. JV4=98.443 1002.

, 0.7256453 021134(51, _ 27.24612

- H JQ10 ' 2/4— IO10 ' Z*~ 10l°

For the root g^=— 0".6616623, we get,

_ 0.1016437 0.2717608 _ 241.9590

~ ~ ' 2/6 — T 10 ' Zf>~ IO10

Whence /35=20° 30' 37".9 ; log. JV/P=97.0788886.

N5 =+0.0014735 ; Nbir =+0.001 1992 ;

Nj =+0.0013543 ; ^ =+0.0011576 ;

^"=+0.0013280; JV6r/=— 0.0011248;

JV6'"=+0.0012591 ; Nbv"=— 0.0117880.

136

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root 06=— 2".9160771, we get,

, 0.3678168 0.3545173
x'= + - Tmo . 2/6=

581.0884

Whence #,=133° 56' 42".7 ; log. N6ir =96.9440543.

#.=+0.0030457; N6'r =+0.0008812 ;

JN,' =+0.0017834 ; N0r =+0.00071774 ;

JV8"=+0.0016079; JV6r/=— 0.0176842;

^'"=+0.0011581;
For the root 07=— 25".9350099, we get,

85,=

0.2996051

7=+

0.2202944

NJ"=— 0.0018963.

59.02778

1Q10 y I 1010 JQ10

Whence /3T=306° 19' 35".l ; log. JV/r=97. 7993434.

NI =—0.0002735; N7'r =+0.0063000;

JV/ =—0.0001463 ; N7r =—0.0156907 ;

N7"=— 0.0027849 ; JV/' =+0.0006888 ;

JVTW=— 0.0093834 ; JV7 "'=+0.00007719.

18. If we now suppose the mass of Mars to be doubled, we shall have the pre-
liminary computations by making all the coefficients of equations (269-274) posi-
tive. We shall then obtain the following

Fundamental Equations for /'=+!; or for m"'=l-=-1340318.5

^=^+38.394808.^ +184.259392;
A=f+ 23.2280108.^+ 98.9712014;
^"=^+18.8430013^+ 73.7719022;
A=^+18.41 11446.0+ 60.3416523;
^2=(f +13.1958172.0+ 8.983726;
A,=02+26.3853904.0+ 9.939432.

(467)

D =02+44.9052274.0+596.989960 ;
D =02+52. 1576683.0+608.070284;
JD"=02+32.4642145.0+260.9092264 ;
D1=02+43.8990384.0+172. 465387 ;
Z)2=02+46.4946594.0+ 32.948855 ;
Z>,=02+ 3.4292114.0+ 1.69255259.

B= \ .7+32.74835 1 b ; B= ^+17.6069231 \ b •

(4GS)

C =— 10+22.553639| [9.4381 189]6';
C'=— 10+17.6669357|[9.1138076]6';
C"=+[6.3976676]6';
Crw=+[0.4411620]i'.

(469)

(470)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 137

E =— [9.9807377J6";
E> =— j^+22.3515875 j [8.9723624]Z>";
E"=— J0+17.5618825 j[9.7501125]6";
E'"=— [1.0016537J6".

F =+[8.4689676]&'";

F'=— [9.4463886J6'";

F"=— ^0+14.902331 |[0.7927855]Z/";

F"=— J0+34.490730 } [9.6907241]&'".

(471)

?1= {0+4.5202496 } 6i; £2={0+ 0.7124602

:;:}

Cl=— {0+4.1330154 j[9.5433087]£2;
C2=— {0+0.73073414 j [9.434971 l]6a ;
<7,=+[0.8644527]&a ;
^=+[0.8654649]^.

Ej=— [8.331 7448]63;
^2=— {0+39.7660231 } [8.74377 18]63;
EB=— { 0+ 0.64863718 } [0.7242832]&3 ;
Et=— [0.9647514]53.

^=+[7.7244692]&4;

J?i=— [0.3564628]64;

^=—10+ 2. 7805744 }[1.5514854]64;

Ft=— {0+45.7639253 }[9.4626364]Jt.

(472)

(473)

(474)

(475)

(476)

Sr*+47.9382786./+787.7033176./ ) _ , 47?,

+5108.828105.r/+11271.52356 j =

^4+29.5262630./+95.1517832.^2 ) ,

+51.352675.^+0.624611 )

The values of b, b', b", and b'" are given by equations (405), and the values of
b2, 63, and 64 are given by equations (406), by merely multiplying the coefficients
'

If we put equations (477) and (478) equal to nothing, they will give

g1=— 5".1830604, &=— 0".0124492,

* g2=— 6 .7024834, y5=— 0 .6632208,

03=— 17 .3261252, 9»=— 2.9170763,

04=— 18 .7266096, g,=— 25 .9335168.

18 March, 1872.

138

SECULAR VARIATIONS OF THE ELEMENTS OF

The solutions of equations (467-478) will now give the following values :

For the root g, we get,

ff =— 5M86718,
y =+0.11399082V
N" =+0.0802224JV
Nm =+0.01622926AT
N'r =— 0.0001963412V
NT =— 0.00025087662V
N" =+0.000213949 N
N r"=+0.000006300732V

For the root g^ we get,

0!=— 6".708872,
Ni =— 0.30098342V!
N? =— 0.23890962Vi
2VY" =— 0.050862162V!
Nt" =+0.00037137732Vt
2V/ =+0.00052924752Vi
Aiw =—0.0002649250^
Aiw=— 0.00001649956^

For the root

we get,

,=— 17*.327793,
=— 5.526427.ZV,
=+ 4.250740iV2

2V2""=— 0.0020564172V2
2V2F =— 0.016972172V2
2V2r'=+ 0.0016641 172V2
2V2r"=+ 0.0001913242Va

For the root gz, we get,

&=— 18".726623,

2V3' =— 6.2804372V,
2V8" =+7.1470222V3
2V3W =— 8.1177852V3
Nf =+0.00005835662V8
2V8r =+0.001772272V3
2V3" =— 0.00015168762V3
N3r"=— 0.00001 762792V3

For the root g» we get,

o4=0"; and 2V4=2Vr4'=2V4"=i

log. 9.0568698,

" 8.9042956,

" 8.2102986,

" 6.2930lllnf

" 6.3994600«,

" 6.3303095,

" 4.7993910.

log. 9.4785424n,

" 9.3782336n,

" 8.7063948»,

" 6.5698154,

" 6.7236588,

" 6.4231228«,

" 5.2174723w.

log. 0.7424444w,

« 0.6284646,

" 1.3255284,

" 7.3131111w,

" 8.2297373/z,

" 7.2211838,

" 6.2817698.

log. 0.7979898H,

" 0.8541251,

" 0.9094376??,

" 5.7660900,

" 7.2485300,

" 6.1809500/?,

" 5.2462000/,.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS.

139

IV

ir

ir

For the root g^ we get,

&=— 0".661664,

276 =+1.22936827,"
27,' =+1.12902577/'r
JV6" =+1.10586027/r
275'" ==+1.04915821?;

27,' =+0.9653170iV5
276r/=— 0.9382057JV5
275r/J=— 9.83088227/7

For the root #U5 we get,

#6=—2".9 16095,
N6 =+ 3.479802^V6/r
27a' =+ 2.020979^"
jVa" =^_ i. 810077 Na'v

N6r =+ 0.816421527,"
27," =—20.1157427,"

27.m=+ 2.15699527/7

log. 0.0896819,

" 0.0527036,

" 0.0436997,

" 0.0208410,

" 9.9846700,

" 9.9722980«,

" 0.9925925n.

log. 0.5415546,

" 0.3055617,

" 0.2576970,

" 0.1168248,

" 9.9119144,

" 1.3035360M,

" 0.3348492.

For the root r/7, we get,

^7=— 25".933517,
JV7 =— 0.0409446027
N,' =— 0.048154562f
Nj" =— 0.4073554JV7
N7'" =— 1.4730382V
N,r =— 2.490000JV

17
IV
IV
IV
IV

log. 8.6121966??,,

" 8.6826374»,

" 9.6099734«,

" 0.1682137%,

" 0.3961992n,

" 9.0385844,

" 8.0880201.

19. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities: —
For the root g=— 5". 1867 18, we get,
. 6.16813 v

x=-{ — ins— ^; y=-

15.530471 „ 134.20637 ._

-N; z= ^r, N2.

1015 1015

Whence /3=21° 39' 31".5 ; and log. JV=9.0952088.

N =+0.124511, #"=—0.000024449,

N' =+0.014194, Nr =—0.000031240,

N" =+0.0099895, NVI =+0.000026642,

N'" =+0.0020212, #m=+0.00000078455.

110 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root ffl=— 6".708S72, we get,

6.90464 . 8.259874 372.8292 2

Whence ^=140° 6' 26".4; and log. ^=8.4605329,

JVi =+0.028876, jy/F =+0.000010724,

JVi =—0.0086913, JV/ =+0.000015283,

Ni =—0.0068988, JVi" =—0.0000076500,

_ZVi"'=— 0.0014686, N1T"=— 0.00000047644.

For the root g,=— 17":327793, we get,

6.999264 3.944520 _ 4077.099

^2— To13 2' ^2 — ' 1013 Z2~ 1013

\Vhence /?2=299° 24' 14".3; and log. JV2=7.2945933.

JVi =+0.0019706, ^'"=—0.0000040523,

JV2' =—0.0108902, iV/ =—0.000033445,

J\r2" =+0.0083764, JV2" =+0.0000032793,

iV2"'=+0.0416986, Nt"= +0.00000037702.

For the root g3=— 18". 726623, we get,

6.863743 2.942093 _ 2202.062

X3 - ^13 ^3 - 13 •"«» Z3~ 13

JQ13 •«» 3~ 101

Whence /38=246° 47' 52".7 ; and log. ^,=7.5303588.

2f, =+0.0033912, JST3'T =+0.0000001979,

JV3' =—0.0212985, 1V3T =+0.0000060102,

JVs" ==+0.0242373, JV3r/ =—0.0000005144,

N™=— 0.0275294, NST"=— 0.00000005978.

For the root <74=0", we get,

0.7257311 0.21 12699 _ 27.24758

X*— JQ10 -"*» .V*— JQ10 •"«» Z4—

Whence |84=1060 13' 51".5; and log. ^=8.4431120, JV4=0.02774035.

For the root g&=— 0.6616640, we get,

OJ016924 0.2718731 „ .241.9923,

xi>— IO -"i > Vf> — +~ ~io -^V6 > Z6— IO — -"• •

Whence /?6=20° 307 42".3; and log. JV/r=7.0790118.

iVs =+0.0014747, N6'r =+0.001 19953,

JV6' =+0.0013543, jy/ =+0.00115794,

^"=+0.0013265, jV5r/=— 0.00112520,

jy,"=+0.0012685, N,Y"=— 0.01179220.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

For the root g,=— 2".9160954, we get,

0.3679185 0.3544211

xo -J~ 10,0 ^Vo ; yt- 1()

» y^~ ihio — -tva

581.1008

Whence $,=133° 55' 46".2; and log. ^'==6.9440507.

JV; =+0.0030592, JV7" =+0.00087912,

W =+0.00 17767, JV/ =+0.00071774,

N: =+0.0015913, JVaw =-0.01 76843,

.^=4*0.0011505, _#6rj/=+0.00190064.

For the root g-=— 25".936797, we get,

0.2995996 vrlr , 0.2201126

/V» . . - /\/ •* r - •«

7~ "To1"0 ^7

' TVT_^. 7T=+-

59.01481
10'°

Whence /37=306° 18' 15".6 ; and log. ^/F=7.7993080.

N, =—0.0002579, N7ir =+0.00629953,

Nj' =—0.00030335, N7T =—0.0156858,

JV7"=— 0.00256615, N," =+0.00068848,

#/"=— 0.0092794. ^ra

D =^3+44.6 177028.^7+590.564857 ;
17=,72+52.1872233.i7+610.731488 ;
JD"=/+32.3787870.(7+259.4006143;
Z)1=^2+44.0905279.«7+173.591533 ;
Z>2=<72+46.6785882..7+ 33.163967;
Z>3=<72 3.4403421. ' 1.70359455.

J3=jflr+32.52083}5; 5' = ^+17.7264928^; 1

B"= ^+13.180944 1 b. \

C =— {^+22.326122 } [9.4381 189]6';
C'=— j«7+17.7564991 } [9.1138076]^;
<7"=+[0.3976676]6';
(7"'=+[0.4411620]&'.

20. For an increment of +T£7, to the mass of the Jupiter, we shall obtain the
preliminary computations by merely making all the coefficients positive in equations
(280-293). We shall then obtain the following

Fundamental Equations for fj.ir=-\ ; or /orm/r=l-4-

^=^+38.155337.^ +182.786001;
A =#2+23.3356274.#+ 99.4308553 ;
^"=<72+18.7710411.(7+ 73.2781865;

60.7217122;

9/040174;

9.944072-

(479)

(480)

(481)
(482)

142

SECULAR VARIATIONS OF THE ELEMENTS OF

(483)

E =— [9.9807377]Z>";
£'=_{0+22.2915804j[8.9723624]&'';
E"=-\ (7+17.7039725 £[9.7501125]6";
E'"=— [1.00J6537]6",

.P =+[8.1679376]6'";

JJ" =— [9.1453586J6'";

F"=— ^+14.674814 1 [0.7927855]5'";

F'"=— 1«7+34.4307229 } [9.6907241J6"'.

(484)

52={flr+ 0.7142452
53=1^+19.0000972

} b, ;)
^!. (

[9.5433087]52;
(72=_^+0.73251914|[9.4349711]62;

(73=+[0.8644527]62;

Ez=
E3=

(4g6)

— [8.3317448]&s;

— ^+39.9481669 j [8.7437718]fe3;

— { (/+0.6504222 } [0.7242832]63 ;

— [0.9647514]6S.

_F1==-|-[7.7244692]&4;

jr2=_[0.3564628]64;

F3=— \ </+2.7899200 J [1 .55 14854]&4 ;

Ft=— J0+45.946069 } [9.4626364]&4.

(488)

^+47.7853999.^+782.14382516.^ 1 } (489)

+5044.303371.^ +11057.20317 j

0<+29.7164348./+96.0269583.tf' ) ) (490)

+51.793880.^ + 0.533011 j

The values of Jlt 62, &3, and J4 are given by equations (406); and the values of
i, fe', 6", and &'" are given by equations (405), by merely multiplying the coefficients
of tf'r by

If we now put equations (489) and (490) equal to nothing, they will give
g=— 5".1493604, g,=— 0".0104945,

9l=— 6 .6300170, &=— 0 .6646769,

^=—17.5198308, fft=— 2.9259522,

ga=— 18 .4861917, ^7=— 26 .1153112.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

The solutions of equations (479-490) will now give the following values:
For the root g, we get,

#=—5". 153064,
N =+0.1198583^

N'" =+0.01695429JV
NIT =—0.0002019713^
Nr =— 0.0002567140JV
N" =+0.0002241425 tf
JVm=+0.0000063200772V

143

For the root g{, we get,

gj== — 6".635874,
JW =—0.2841568^,
W =— 0.2283353 JVi
N,'" =— 0.04747286JVi
W =+0.000343187 JV,
NS =+0.0004843167^
^"=—0.0002492740^
N1T"=— 0.0000151962^

For the root gz, we get,

g,=— 17".520779,

W =— 5.627642JV2
JV2" =+ 4.727661^2
Nl" =+31.11766JV2
N2zr=— 0.001595785^2
NaT =— 0.01334776^
^2"=+ 0.001291801 N2
N™=+ 0.0001489017^

For the root g^ we get,

gs=— 18".486192,
JV3' =— 6.134120JV3
N;' =+ 6.742900^
N," =— 11.29640JV;
JV/r=+ 0.000000955724^
N3V =+ 0.0001854001^3
JV3rj=— 0.0000169937^3

0.000002026216^

log. 9.0786680,

" 8.9319690,

" 8.2292796,

" 6.3052894n,

" 6.4094494?i,

" 6.3505243,

" 4.8007224.

N3T"=—

log. 9.4535581«,

" 9.3585790n,

" 8.6764453n,

" 6.5355307,

" 6.6851294,

" 6.3966770n,

" 5.1817351n.

log. 0.7503265n,

" 0.6746463,

" 1.4930069,

" 7.2029742»,

" 8.1254084n,

" 7.1111957,

" 6.1728995.

log. 0.7877523»,

" 0.8288466,

" 1.0529402??,

" 93.9803324,

" 96.2681100,

" 95.2302878^,

« 94.3066957n.

For the root gt, we get,

g.=0", and iV4=i

'=iV4"=, &c.

144

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root gK we get,

gb=— 0".6633671,
N, =+1.231158#/r
N; =+1.130398#5'r
#5" =+1.107341#/r
N? =+1.049016#6/ir
^5r =+0.9653980#5
#"=— 0.9445346 #

#"'=— 9.899907#5/r

For the root #6, we get,

g6=— 2".9250S6,
N6 = 3.536719#6/F

^6" =+ 1.834513^"
2V6'" =-|- 1.310090JV6/r
^-6r =+ 0.8166500JV6Jr
^"=—20.25015^
Ntra=+ 2.169824^^

For the root /77, we get,

g,=— 26".117474,
N7 =—0.0421 1995^
N,' =— 0.04851416A^T/F
JV7" =— 0.4329533Ar/r
N7'".=— 1.4772167V/7
N,r =— 2.515433JV/"

log. 0.0903139,
0.0532315,
0.0442814,
0.0207820,
9.9847064,
9.975.2179«,

" 0.9956311H.

log. 0.5486005,

" 0.3111304,

" 0.2635208,

" 0.1173012,

" 9.9120360,

" 1.3064432/z,

" 0.3364244.

log. 8.6244879»,

" 8.6858685»,

« 9.6364410w,

" 0.1694440«,

" 0.4006128w,

" 9.0398312,

" 8.0897044.

21. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities : —

For the root g=— 5".153064, we get,

.

~~

0.616929

^ —

1.5676783

ioi4

13.836148

Whence /3=21° 28' 52".l, and log. #=9.0855076.

N =+0.121761, N'r=— 0.0000245922,

N1 =+0.014594, Nr =—0.0000312577,

N" =+0.0104106, #"=+0.0000272918,

=+0.0020644, #"'=+0.000000769538.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. U5

For the root #,=— 6".6358742, we get,

0.685590 _0.6767428 34.48222

U -«u y\— U" ivi> zi— 14 ~ 1%

Whence ^=134° 3T40".5; and log. #=8.4461821.

# =+0.027937, #'"=+0.00000958766,

#'=—0.0079385, #r =+0.0000135304,

#"=—0.0063791, #"=—0.0000069640,

#'"=—0.00132626, #r//=— 0.00000042454.

For the root #2=— 17".520779, we get,

69.06713 A7 , 26.31099 .. 4443.491 ,

*2= - — #; 2/2=H --- n— #; z2= - — #

Whence ft=290° 51' 15".3; and log. #=7.2209728.

# =+0.0016633, #'r=— 0.0000026543,

#' =—0.0093605, #r =—0.000022202,

#" =+0.0078636, #" =+0.0000021487,

#'"=+0.0517583, #m=+0.00000024767.

For the root <73=— 18".486192, we get,

67.51436 23.46891 20391.171

""""^ »' ^3~ H - ^3' Z3~

Whence /53=250° 49' 54".7 ; and log. #=7.5447249

# =+0.0035053, #/r =+0.00000000335,
#' =—0.0215019, #r =+0.0000006499,
#" =+0.0236359, #" =—0.00000005957,
#'"=—0.0395973, #"'=— 0.00000000710.

For the root #4=0", we get,

_ , 7.294416 2.119410 274.1191

x* — "r~ jQii -^ii y*— JQII #; Z4— JQII — #•

Whence /34=106° 12' 4".8; and #=#'=#"= &c.,
-+0.02771087, log. 8.4426502.

For the root g6=— 0".6633671, we get,

_ 1.016103 _. , 2.742753 . _245.2401

"« - I J^Qll 1V5 ) 5/5 - ~1 JQII r-"f > Z!> - JQ10 "I '

Whence /?5=20° 19' 41".0 ; and log. #/r=7.0765225

# =+0.0014684, #'r =+0.00119268,
#'=+0.0013482, #"=+0.00115141,
#"=+0.0013207, #"=—0.00112652,
#'"=+0.0012511, #m=— 0.0118074.

19 March, 1870.

146 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root */„=— 2".925086, we get,

0.3697416

0.3561999

v/.<^~ . "«> TIT/F. —

6=H inTo •"• ' 9V- K)10

588.7466

ITZ

IV, .

1010

Whence &=133° 55' 52".8; and log. JV/r=6.9405338.

N6 =+0.0030841, WT =+0.00087204,

Nt' =+0.0017851, N,T =+0.00071215,

JV6" =+0.0015998, JV6" =-0.0176594,

^'"=+0.0011424, ^"=+0.0018922.

For the root </7=— 26".117473, we get,

0 3026288 ,r/r , 0.2224873^ „ _60.04189

SBl= ~TOTo — 7 ; y"~ 1010 1010

Whence /?7=306° 19' 21".6; and log. iV/r=7.7962857.

N, =-0.0002635, ^'" = =+0.00625584,

JV7' =—0.0003035, ^7" : -0.0157362,

^7" =-0.0027085, NS' =,+0.00068567,

#7"=-0.0092412, W =+0.000076911.

22. For an increment of +4ij7/ir, to the assumed mass of Saturn, we shall obtain
the preliminary computations by merely making all the coefficients positive in
equations (307-312). We shall then obtain the following

Fundamental Equations for iur=+ir; or for ??ir=l-^3416.195.

^1=^+38.078720.^ +181.976176;
4'=02+23.1907166.<H- 98.4490593;
^"=^+18.7198881.^+ 72.8857520;
41=/+18.6278118.#+ 61.6540448;
J2=02+13.3828119.£+ 9.178963;

10:089397.

(491)

D =02+44.5181269.#+588.346190 ;
D =y2+52.0193531.^+605.574324 ;

D1=i/2+44.4628062^+176.258864 ;
/)2=^^_47.1787023^+ 33.673551 ;
Z>3=i/2+ 3.4692S99.-7+ 1.72977087.

(492)

£=^+32.45831

6; 1
. j

C =— j^+22.263602| [9.4381189J6';
C'=- 1<7+17.6256851 1 [9.1138076J&';
C"'=+[0.3976676]6';
C"*=+[0.4411620]&'.

(493)

(494)

THE ORBITS OF THE EIGHT. PRINCIPAL PLANETS. 147

E =— [9.9807377]i";
E> =— ^+22.2545242}[8.9723624]6";
E" =— j#+17.5731585 } [9.7501 125]6";
E"'=— [1.0016537J6".

F =+[8.1679376]&'";
F1 =— [9.1453586]6'";
^=—1^+14.6122941 [0.7927855]fc";;
F"'=— J0+34.3936667 } [9.6907241]i";.

^={^+4.5551100(6,; £2=j#+ 0.7176484}^; )

£,= {0+18.823159016!. J

C,=— j#+4.1678758 } [9.5433087]ia ;
C72= — {^+0.7359221 } [9.434971 1]62.
(73=+[0.8751766]62;
(74=+[0.8761888]&2.

— [8.3317448]S3;
—1/7+40.2949304 j [8. 7437718]&3;

g+ 0.6538252 }[0.7242832]S3;

— [0.96475 14]63.

^=

E3=

F2=— [0.3564628]64;

Ff=—\g+ 2.8154348 }[1.5514854]&4;

Fi=— j^+46.44278 } [9.4626364]64.

(495)

(496)

(497)

(498)

(499)

(500)

/+47.5409132./+774.11613432./) fil

+4966.568399.^+10830.80683 j "

«4+29.7477885./ +96.8557913.02 ) , ,

+52.S77272./ + 0.536670 } =(*" ^ *» ^

The values of 619 Z>2, &3, and 64 are given by equations (406); and the values of
b, V, b", and V" are given by equations (405), by merely multiplying the coefficients
of ^"by l+(*r= 1.025.

If we now put equations (501) and (502) equal to nothing, they will give,

g =— 5".1255241 ; #4=— 0".0104061 ;

gi=— 6 .5914740 ; ff&=— 0 .6688791 ;

#,=—17.4063959; g&=— 2.9523070;

ff3=— 18.4175192; ^7=— 26 .1161961.

143

SECULAR VARIATIONS OF THE ELEMENTS OF

Substituting these roots in succession in equations (491-502), we shall get the
following values: —
For the root ff, we get,

g=—5". 1293390,
N' =+0.1 224435 AT

N" =+0.08766774^
N'" =+0.0175016JV
N'r = — 0.000205739^
Nr =— 0.000261587JV
NTt =+0.000236310^
N r"=+0.0000062338tf

For the root gv we get,

</!=— 6".5971682,
AY =—0.277326^
JV/' =— 0.223546JV,
W/" =— 0.0468220 JV,
JV/r =+0.000334045^
NS =+0.000471677JV,
Air/ =—0.000250301^;
. Nlv"=— 0.0000148755JV,

For the root g%, we get,

^2=— 17".4073173,

Nj =— 5.57006^2

^2" =+ 4.

N2ir=— 0.00161379^3
NS =— 0.0137070^
^2F/=+ 0.00136814^2
#/"=+ 0.000156952^2

For the root ga, we get,

gz=— 18".4175195,
N3' =— 6.10257^,
^3" =+ 6.66495^3

N,ir=— 0.000017665^3

N3r =— 0.000117075^3

JV3r/ =+ 0.000()0985785JV3

^3^"=+ 0.00000108146JV3

log. 9.0879358,

" 8.9428398,

" 8.2430770,

" 6.3133160»,

" 6.4176164n,

" 6.3734826,

" 4.7947524.

log. 9.44299 lOw,

" 9.3493660H,

" 8.6704496«,

" 6.5238055,

" 6.6736443,

" 6.3984620n,

" 5.1724700n.

log. 0.7458600/z,

" 0.6609136,

" 1.5189348,

" 7.2078461n,

" 8.1369422/z,

" 7.1361306,

« 6.1957676.

log. 0.7855130n,

" 0.8237970,

" 1.0139370ra,

" 95.2471266»,

" 96.0684623«,

" 94.9937824,

« 94.0340085.

For the root <74, we get;

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. H9

For the 1'oot gM we get,

#5=— 0".6675803,

N5" =+1.109145tf,"

NJ =+0.965528W6/r
A77=— 0.925706JV,"

For the root

we get,
#6=— 2".95 14286,
N6 =+ 3.6381 6^V6'r
JVa' =+ 2.08409^"
Ne" =+ 1.86452^^
JV/' =+
N6r =+

log. 0.0914746;

" 0.0540246 ;

" 0.0449882;

" 0.0211454;

" 9.9847648;

" 9.9664731w;

" 0.9960956«.

log. 0.5608818;

" 0.3189167;

" 0.2705668 ;

" 0.1205640;

" 9.9122164;

" 1.3049538»;

" 0.3302494.

log. 8.6196418ra;

" 8.6896315n;

" 9.6301420%;

" 0.1584846«;

" 0.3855740n;

" 9.0359788;

" 8.0853954.

23. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities: —

For the root g=— 5M293390, we get,

For the root «/7, we get,

^=—26". 1183694,
JV7 =— 0.0416526iVT/
JV/ =— 0.0489363 JV/
NS =—0.426719^
JV7'" =— 1. 440405 JVr/r

, 606572

3!-— _L- • T/

1589168.6
1020

z==-

140213200

Whence (3=20° 53' 20".5 ; and log. JV=9.0839064.

N =+0.1213127; ^^=—0.00002496;

JV' =-[-0.0148551; ^"=—0.000031737;

N" =+0.0106362 ; N" =+0.000028671 ;

N'" =+0.00212364; 2VF7/=+0.0000007563.

150 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root ffi=— 6".5971682, we get,

702346 650773.9 _33379780

xi=H — i7y»~' 2/1 — HP ' Zl~ 1020 '

Whence /?,=132° 49' 3".9 ; log. AT1=8.4576530.

N! =+0.0286850; A7; /r= +0.000009582;

AY =—0.00795527; A7/ =+0.000013530;

AY =—0.00641252; ^" =—0.000007180;

A7," =—0.00134306; #/"=— 0.0000004267.

For the root gz=— 17".4073173, we get,

68779790 28706100 _ 4.840416

-I 2/2 — H TTwo > % — '

JQM •> 2— JQIO

Whence ^=292° 39' 13".8 ; log. ^=7.1874476.

NI =+0.0015397 ; JV2/r =—0.000002485 ;

jV2' ^—0.0085764 ; Ntr =—0.000021105 ;

^"=+0.0070528 ; NtTI =+0.0000021066 ;

^2m=+0.0508607 ; ^am= +0.00000024 17.

For the root ga=— 18".4175195, we get,

_ 67277160 22295350 _ 1.9362943

^ - JQ20 ' 2/3— 1Q20 » Z3— JQ10 »

Whence ^,=251° 39' 34".l ; log. ^3=97.5635'231.

JV3 =+0.0036604; N3'r =—0.00000006466 ;

Ar3' =—0.0223376 ; N3r =—0.00000042853 ;

.^"=+0.0243961 ; N3rl =+0.000000036083 ;

Nam=— 0.0377972 ; #3r//=+0.00000000039585.

For the root <74=0", we

_ , 0.7324877 0.2141555 _ 27.41415

x* — ~r JQIO > 2/4— JQIO 5 zi— J^QIO 5

Whence /34=106° 17' 49".9 ; log. ^=98.4446362.

For the root &=— 0".6675803, we get,

_ , 0.1042216 , 0.2727222 _245.6575

^S - H JQIO 5 3/6 - H IQ..I » ZS— JQ10

Whence ^s=200 55' 5".7 ; log. ^=97.0749906.

^6 =+0.00146716 ; N,'r =+0.00118851 ;

NJ =+0.001 3459; N,T =+°-00114755;

Nt" =+0.00131823; ^5r/=— 0.00110001 ;

^"=+0.00124781 ; N6r"=— 0.0117785.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 151

For the root 06=— 2".9514286, we get,

0.3712512

0.3561690

_

Z -

584.6008

Whence &=133° 48' 43".9 ; log. JV6/r=96.9445043.

N6 =+0.0032017
Nj =+0.0018341
#/ =+0.0016409
JV6'"=+0.0011616

Nf =+0.000880044
N* =+0.000718986
N™ =— 0.0177606 ;
JV6r"=+0.0018826.

For the root 07=— 26M183694, we get,

__0.2996395^ 0.2202906 _57.99673

HF 10l° "TO15 ;

Whence /37=306° 19' 22".0 ; log. ^"=97.8070262.

N7 =—0.00002671 ; N," =+0.00641248 ;

JVY =—0.00031380 ; N,r =—0.0155812 ;

Nj" =—0.0027363 ; N7" =+0.00069663 ;

#7=— 0.0092366 ; #T" '=+0.000078059.

24. For an increment of ^m", to the assumed mass of Uranus, we shall obtain
the preliminary computations by merely making all the coefficients positive in
equations (326-331). We shall then obtain the following

(503)

Fundamental Equations for ju"=+— ; or for m™=— — — .

£\j && 1 1 y

A =02+38.068970.0 +181.871338;
^'=02+23.1735927.0+ 98.3308977;
^"=//2+18.7132214.0+ 72.8344734 ;
^11=02+18.4995010.0+ 61.4402060;

A 2 I 1 O OAO^v^»OQ ^ I Q 1 .471 Oft •

^^=.n -|— io.^uyoo*o.^— |— y. i-r 1 1*0 ,
^3=02+26.3996602.0+ 10.055139.

D =^+44.5054401.0+588.063848 ;
£>'=02+51.9992925.0+604.964956 ;
Z)"=02+32.1623093.0+256.0208612;
Z>1=02+43.9809321.0+175.644437 ;

Z>2=0!+46.5212935.0+ 33.556164;
Z>3=02+ 3.4429483.0+ 1.72374142.

5=J0+32.45043|6; ^ = {0+17.5804200^; ) 5Q5.

^={0+13.139086^. j

C =—\g+ 22. 255718J [9.4381 189]tf;

C' =— {^+17.6104269 |[9.1138076]ft'; C5Q6)

Cf"=+[0.3976676]6';

C""=+[0.4411620]6'.

(504)

152

SECULAR VARIATIONS OF THE ELEMENTS OF

E =— [9.9807377]fe";
E' =—{0+22.2497227 } [8.9723624]6";
.£''=_ J£+17.557S996J[9.7501125]Z>";
E"=— [1.0016537]6".

p — +[8.1679376]6'";

f"=— [9.1453586]6'";

F"=— 1^+14.604409 } [0.7927855]Z/";

F"=— 10+34.3888650 } [9.6907241]//".

JB1= {^+4.6079205} 6,; Ba=\g+ 0.7255099(6,; 1

Baa=|(7-j-18.8315378}X I

\ [9.5433087]6, ;

C2=— ^+0. 7437838 j [9.434971 1]6S ;
Cs=+[0.8644527]62;
C4=-)-[0.8654649]ft2.

E,=— [8.3529341]&3 ;

^=—1 ^-|-39.7796076 }[8.7487718)^;

Et=— \g-\- 0.6616869 j [0.7242832]^;

F,=+[7.7244692]i4;

^=—[0.3564628]^;

Fs=— \g-\- 2.7812615 j [1.5514854]64;

FI=— j <7+45.77751 } [9.4626364]Z>4.

(507)

(508)

(509)

(510)

(511)

(512)

Sr*+47.5111025./+773.1351541.flrM ,

+4957.025771^+10802.91193 j"

» ^+29.5535550^ +95.6630308.^ ) ~~^. (514)

+52.197094.^ + 0.534229 j

The values of 6X, J2, 63, and 64 are given by equations (406) ; and the values of
b', 6", and I'" are given by equations (405), by merely multiplying the coefficients

If we now put the equations (513) and (514) equal to nothing, they will give

g=— 5".1224110; </4=— 0".()104337;

,;,=_ 6.5865615; 96=— 0.6748372;

«/,=— 17.3929254; «76=— 2.9245347;

^,=—18 .4092044 ; <77=— 25 .9437494.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 153

Substituting these roots in succession in equations (503-514), we shall get the
following values: —

For the root g, we get,

#=— 5".1262710;
N' =+0.12276772V
N" =+0.0879386^
N" =+0.0175667#
NIV=— 0.000208367JV
NT =—0.00026463^
NVI =+0.000232174^

N "'=+0.00000549425^

For the root g^ we get,

01=— 6".5922959;
Nf =—0.276494^,
JV," =—0.222958^
JVr =— 0.0467327tf,
JV," =+0.000336574^
NS =+0.000474690^
NS1 =— 0.000245839 JV,
Nlr"=— 0.0000143346JV1

For the root g2, we get,

«/2=— 17".3938608;

=+
=+
=— 0.00165952.V2

log. 9.0890840;
" 8.9441794;
" 8.2446904;

6.4226422w;

6.3658128;

4.7599088.

#/'=+ 0.00136629^
JV2r/'=+ 0.000156224^3

For the root g^ we get,

<73=— 18".4092048;

N3 =— 6.098831^8
N3" =+ 6.65618#3 '
N3'" =— 10.22645JV3
N3ZV=— 0.0000196421 jy3
N3T —— 0.000150200JV;
iV3r/=+ 0.0000125074^3
JV3r//=+ 0.000001 39286 N3

For the root g4, we get,

log. 9.4416856«;

" 9.3482238/i;

" 8.6696208»i;

" 6.5270807;

" 6.6764098;

" 6.3906513»;

" 5.1563868?z.

log. 0.7453350»;

" 0.6593380;

" 1.5216464;

" 7.2199837«;

" 8.1468598n;

" 7.1355420;

" 6.1937476.

log. 0.7852466%;

" 0.8232250;

" 1.0097248?i;

" 95.2931878n;

" 96.1766688n;

" 95.0971660;

" 94.1439073.

20 March, 1872.

154 SECULAR VARIATIONS OF THE ELEMENTS OF

ir

For the root gb, we get,

g&=— 0".6735122;
A& =+ 1.237 18A6/r
A",' =+1.133973^'
A6" =+1.110407A6/r
A6'" =+1.05043A5'r
N6r =+0.964612JV6'r
NbT' = — 0.937483iV6/r
A0r"=— 9.79928^'"

For the root g6, we get,

g6=— 2".9236175;
Ae =+ 3.57518^
JV6' =+ 2.06480JV67"
A;" =+ 1. 84969 JV6/F
JV8"' =_j_ 1.285565^6/r
JV/ =+ 0.816008^e/r
Ne"=— 1

N™=+ 2.16794iV6/F

For the root g^ we get,

<7,=— 25".9459i26;

AV =— 0.0466848W/
A"," =— 0.432476^/r
A," =— 1.466224A,"'
A"TF =— 2.491 785^'"
ATr/=+0.1093547JV/
ATF//=-i-0.0122059AV

log. 0.0924317;

" 0.0546030;

" 0.0454822;

" 0.0213676;

" 9.9843526;

" 9.9719633n;

" 0.991 1943».

log. 0.5532976;

" 0.3148787;

" 0.2670992;

" 0.1190940;

" 9.9116944;

" 1.2824678n;

" 0.3360471.

log. 8.6237552n;

" 8.6691754n;

" 9.6359617»;

" 0.1662004?i;

" 0.3965106»;

" 9.0388375;

" 8.0865696.

25. If we now substitute these values in equations (408) and (409), we shall
obtain the following quantities: —

For the root g=— 5".1262710, we get,

610397

y=+

1587285.2

14044917

10

10

Whence /3=21° 1' 11".9; log. A"=9.0833149.

A =+0.1211476;
N' =-(-0.0148742;
A"=-f0.0106545;
A"=+0.00212865;

N'r=— 0.000025246;
A" =—0.000032062;
Ar/ =+0.00002813;
Am= -1-0.000000697.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 155

For the root </!= — 6".5922959, we get,

696641 640902.9 _33247150

^ ~r 1020 ' 2/1— "Toso ' Zl~

Whence ^=132° 41' 2".3; log. ^=8.4538431.

N, =+0.0284342; N,'r =+0.000009570;

NJ =-0.0078621 ; N,T =+0.000013498 ;

JVi" =—0.0063398 ;. N^1 =—0.0000069904 ;

#T=— 0.0013288 ; N1r"=— 0.0000004076.

For the root ^=—17". 3938608, we get,

68615050 28882030 _ 0.4884728

^2— 1020 ' ^2 1020 ' Z*~ io10 '

Whence /32=292° 49' 39".4; log. JV3=97. 1830006.

Na =+0.0015241 ; N2'r =—0.0000025292 ;

Nj =—0.0084788 ; Nzr =—0.000021373 ;

NJ =+0.0069557 ; N,rz =+0.0000020823 ;

^"'=+0.0050658 ; ^2r//=+0.00000023809.

For the root gs=— 18".4092048, we get,

_ 0.006724489 0.00221 7478 _1.9257107

xs — ihio ~' 2/3 — iTjio -i 2s — ^TAIO •

Whence ^=251° 44' 58".0 ; log. JV3=97.5654837.

N3 =+0.0036769 ; N3IT =—0.00000007222 ;

JV3' =—0.0224249 ; N3r =—0.0000005523 ;

N3" =+0.0244742 ; N3n =+0.000000046 ;

^'"=—0.0376018 ; ^^=+0.00000000512.

For the root #4=0", we get,

_ , 0.7265241 0.2110814 27.31188

x* — ~r IO ~> 2/4 — —

Whence /?4=106° 12' 1".7; 10^^=98.4424954.

For the root #6=— 0".6735122, we get,

0.1020975 0.2702021 240.6404

5 2/5 -- H

- JQIO

Whence /3S=20° 42' 11".2 ; log. JV67F=97.0793126.

N, =+0.0014850 ; JV/r =+0.00120036 ;

^'=+0.0013612; N&r =+0.0011579 ;

^"=+0.0013329 ; ^"=—0.0011251 ;

JV6'"=+0.0012609 ; N,v"=— 0.0117624.

156

SECULAR VARIATIONS OF THE ELEMENTS OF

For the root 00=— 2".9236175, we get,

03679656 0.3547543 _555.2999

~; y*~~ io10 "io10"

Whence ft=133° 57' 9".9 ; log. #." =96.9640000.

#6 =+0.0032908 ; N" =+0.00092045 ;

#„' =+0.0019005 ; N6T =+0.00075109 ;

#6"=+0.0017026 ; W =—0.01763875 ;

#8'"=+0.0012109 ; #6™=+0.00199548.

For the root 07=— 25".9459126, we get,

0.3001025 0.2204575 _ 59.06886

JC7== |7)i6 » y~l~ IO10 ' Z-l~ IO10

Whence ft=306° 18' 4".6 ; log. #,-=97.7996217.

#7 ,= -0.0002651 ; NJr = =+0.00630408 ;

#/ =-0.0002943 ; N,T =-0.0157084 ;

#,"=-0.0027264 ; #/' =+0.00068938 ;

#,"=-0.0092432 ; #/"= +0.00007695.

26. For an increment of +Iynm, to the assumed mass of Neptune, we shall
obtain the preliminary computations by merely making all the coefficients positive
in equations (345-350). We shall then obtain the following

Fundamental Equations for 1ur"=+IV or for mm=lH-17072.73.
4=0^+38.068863.0 +181.870184;
A =/+23. 173407 1.0+ 98.3296126;

4"=/+18.7131480.0+ 72.8339084;

60.9399039 ;
42=02+13.2015603.0+ 9.072649 ;
43=-/+26.3912337.0+ 9.975760.

D =^+44.5053009.0+588.060747 ;

Z>'=02+51-9990747-iH-604-958351 ;
X>»=^4-32.1620579.0+256.0169344 ;
^=^+43.9488809.0+174.214515 .

Z>2=02+46.5094752.0+ 33.282984 ;
Ds=72+ 3.4725714.0+ 1.70970867.

£=|0+32.45034|6; ^=}0+l 7.5802549 (&; )

7J«= 10+13.139034^. )

tf =—{0+22.255630| [9.4381 189]Z>'; 1
m _ <„ i it cino«io»ro 1 1 oamrftW.

(515)

(516)

Cr"=+[0.3976676]&';
C"'=+[0.4411620]6'

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 157

(519)

(520)

(521)

(522)

E =— [9.9807377]fc";
E' =— {0+22.2496697 \ [8.9723624]i";
E* = — {0+17.5577346 1[9.7501125]6";
Em=— [1.0016537]6".

F =+[8.1679376]6'";

I"=— [9.1453586]Z>'";

F"=— {0+14.604322| [0.7927855]Z>'";

F'"—— J0+34.3888122 ^[9.6907241]Z>"'.

£,= {0+4.5635477^,; 52={0+ 0.7189043(6, ;)

B8= {04-18.8244977 K. J

Ci=— {0+4.1763135 j[9.5433087]62;
Cz=— {0+0.73900559 1[9.4349711]&2;
C3=+[0.8644527]62;
(74=+[0.8654649]62.

El=— [8.331 7448]63;
E2=— {0+39.7725674 |[8.7437718]63;
E3=—\g+ 0.64869894 *[0.7242832]&3;
EI=— [0.96475 14]63.

Jl=+[7.7658619]54;
^i=— [0.3978555]64;

^3=—{0+ 2.8238725 } [1.5514854]64;
FI=— {0+45.770470 j [9.4626364]&4. J

^+47.5107777./+773.1244605y 1
+4956.921649.0+10802.60734 j "

04+29.5753988.03+96.3534529.02 1 _( ,

f 52.082669.0 +0.529879 j =

The values of 515 &2, J3, and 54 are given by equations (406), and the values of
i, 5', b", and b'" are given by equations (405), by merely multiplying the coefficients

„ _ _ Yff i - — - _ _ —

of N b

(523)

(524)

If we now put equations (525) and (526) equal to nothing, they will give

01=_ 5".1223768; &=— 0".01037221 ;

<72=— 6 .5865075 ; g,=— 0 .66492499 ;

^=—17.3927801; flr,=— 2.96207581;

gl=— 18 .4091132 ; <77=— 25 .93802580.

158

SECULAR VARIATIONS OF THE ELEMENTS OP

Substituting these roots in succession in equations (515-526), we get the follow-
ing values : —

For the root g, we get,

r =—5". 1262327,
#' =+0.122771iV

N" =+0.0879417^
N'" =+0.01756792V
N'r =—0.000207983^
JV" =— 0.000264402IV
N" =+0.0002360322^
#""=+0.000006 1622

For the root #„ we get,

0,=— 6".5922343,
NJ =— 0.276482 JVi
N,' =—0.222950^!
N' =—0.0466245^1
N," =+0.00033610^
JV,r =+0.000474472iVi
Nin =— 0.000248245 Nj.
Nlv"=— 0.0000147153IVi

For the root ffz, we get,

gz=— 17".3937133,
NJ =— 5.56327iV2
N,' =+ 4.563742V2

N3'r=— 0.0016539^

J^r/=+ 0.001370242V2
^2r"=+ 0.00015714^

For the root ga, we get,

«78=— 18".4091136,

W =— 6.09878JV,
W =+ 6.656082V3

N3'r=— 0.00001 95968JVS
^a" =— 0.0001 50608 N3
^3^/=+ 0.0000125707iV3
N3"'=-\- 0.00000139394JV8

For the root gt, we get,

„ — 0". N AT' N" AT"'" — <Srr

ijl — J , JT^ IV^ JV^ ^V^ VXC.

log. 9.0890969,

" 8.9441950,

" 8.2447196,

« 6.3180273w,

" 6.4222642/j,

" 6.3729713,

" 4.7897356.

log. 9.4416674n,

" 9.3482076w,

" 8.6696144»,

" 6.5264690,

" 6.6762012,

" 6.3948800H,

« 5.1677685w.

log. 0.7453298«,

" 0.6593212,

« 1.5216780,

» 7.2185079»,

" 8.1470920«,

" 7.1367973,

" 6.1962851.

log. 0.7852438w,
" 0.8232187,
*' 1.0096800M,
" 95.2921852«,
" 96.1778487w,
95.0993576,
94.1442453.

"
»

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

159

For the root </5, we get,

</5=— 0".6634935,

N," =

^5r =+0.965270W/
NJ'=— 0.921756 Nbi

For the root g6, we get,

g6=— 2".9612185,
N6 =+ 3.6675(W6/r
N; =+ 2.09390JV/r

JV6" =+ 1. 87228 N61V
JV6'" =+ 1.32235^
Ntr =+ 0.813392JV/1
^"=—20.4503^

log. 0.0909496,

" 0.0537360,

" 0.0447544,

" 0.0210270,

" 9.9846490,

" 9.9646159^,

" 0.9515409w.

log. 0.5643695,

" 0.3209559,

" 0.2723698,

" 0.1213462,

" 9.9102996,

" 1.3106998/j,

" 0.3340172.

log. 8.6239109»,

" 8.6684721w,

" 9.6361780w,

" 0.1664891»,

" 0.3963997w,

" 9.0395107,

" 8.0882005.

27. If we now substitute these values in equations (408) and (409) we shall
obtain the following quantities : —

For the root #=—5". 1262327, we get,

. 611711 , 1586934.1 14045156

x=-\ — im>-; y=-\-

For the root </7, we get,

07=— 25".9401904,
JV7 =— 0.042064#/r
Nj' =— 0.0466092#/r
#7" = — 0.43269lJVT/ir
N? =—1.46720^

1020 1020

Whence (3=21° 4' 40".0; log. #=9.0831058.

N =+0.1210893; NIT =—0.00002519;

#'=+0.0148664; Nr =—0.00003202;

N" =+0.0106498; #"=+0.00002858;

N'" =+0.0021278; #"/=+0.0000007462.

160 SECULAR VARIATIONS OF THE ELEMENTS OF

For the root gl=— 6".5922343, we get,

693376 6J38974.9 _33245220

*!- 10.-o ; 2/1- -10» '• z»- 102o •

Whence /3,=132° 39' 44".3 ; log. ^=98.4527380.

2f[ =+0.0283621; 2^" = +0.000009533;

_#"/=— 0.0078418; JVir =+0.000013457;

N"—— 0.0063234; ^"=—0.0000070408;

.2V7"=— 0.0013254; ^,r"=— 0.00000041736.

For the root g>=— 17".3937133, we get,

__68626660> 28878680 _ _O4885253

1()» • 2/2— — JQM ; Z2— - 1()10 •

Whence &=292° 49' 18".4 ; log. JV2=97.1830089.

JV2 =+0.0015241; J\T2/r=— 0.000002521;

JT; =—0.0084789 ; N* =—0.000021385 ;

JV,' =+0.0069555 ; N™ =+0.0000020884 ;

^"=+0.0506625 ; JVr2r"=+0.0000002395.

For the root #,=—18". 4091 136, we get,

__67244580 _221 73520 _ 1.9255992

TO* ; y*~ io°" ; ZA~ "to10 "'

Whence /?3=251° 45' l".l ; log. ^3=97.5655046.

JV3 =+0.0036771 ; 2f," = -0.00000007206 ;

2ft =—0.0224258 ; JV^ =—0.0000005538 ;

JV3* =+0.0244750; ^3"=+0.00000004622;

N™=— 0.0375997 ; ^"^-fO.OOOOOOOOS^e.

_ 0.7310062 _0.2158651 _ 27.46919

x*~ '» ' y*~ io ' zt— IO

For the root gt=Q", we get,
_ 0.7310062

x*~ io'»

Whence /?4=106° 27' 6".5 ; log. JV4=98.4432303 ;

For the root f/6=— 0.6634935, we get,

_ 0^1014674 0.2722315 _ 222.4744

1Q10 '" 2/6— JQ,O ~~» Z8— JQ10 '

Whence /36=20° 26' 30".0 ; log. #6/r=97.1159056.

A\ ==+0.0016101; N&'r =+0.00130589;

JV8'= =+0.001 4779; Nbv =+0.00126053;

JV6"=+0.0014476; JV6"=— 0.00120371 ;

^"'=+0.0013738; N,r"=— 0.0116801.

THE OIIBITS OP THE EIGHT PRINCIPAL PLANETS 161

For the root </6= — 2".9612185, we get,

0.3722380^ 0.3654807 _600.4976

X6 T JQ10 » 2/6— ~JLO™ ' Z"~ IO10

Whence &=134° 28' 30".7; log. iV6/r=96.9388828.

N6 =+0.0031860 ; N6'T =+0.000868726 ;

N6' =+0.0018190 ; N6T =+0.00070661 ;

JV6" =+0.0016265; ^"=—0.0177657;

JV6'"=+0.0011488 ; iV/J/=+0.00187456.

For the root g^=— 25".9401904, we get,

0.2997138L 0^2203005^ _59.04657

' ^7 — ' 10 ' Z;~ ~

_

'l~ io10 ' 7 — ' io1

Whence /37=306° 19' 2".l ; log. JV/r=97.7993116.

N, =—0.00026499 ; NJr =+0.00629958 ;

^'=—0.00029362; JV/ =—0.0156932;

N,"=— 0.00272577; N^' =+0.00068996;

Nf=— 0.00924274 ; 2V//J=+0.000077181.

28. Having thus obtained the values of all the constants, corresponding to the
separate variation of the planetary masses, it now remains to find the coefficients
of these variations. This we shall do by taking the difference between the con-
stants corresponding to the assumed masses and those depending on the assumed
variation of mass for each planet, and dividing by the assumed variation. By this
means we shall obtain the following table — observing that the coefficients of ft, ^', ft",
&c., in the values of N, N', N", &c., are given in units of the seventh decimal place
of these coefficients.

<7=— 5".126112+0".207190^— 1".49906^— 0".88154^"— 0".060606,u"'— 2".6952^F

-0". 129080^— 0".00318^F/— 0".00121^m;

/2=21° 6' 26".8+6752".6p— 35940'X— 71 1 70y+1984".7/t/"+134530yr

+31452y—l 2596>r/— 1070V";
JV=+0.121076+276274a— 568000^'— 140640^"+3438V+688000lW/r

+95880/fr+29840"+1630J«ra;
Ar'=+0.014867+127798lu+276820/— 52360/— 6929^'"— 273100/F— 4800^'

^"=+0.010650+88422^+230840^'— 23860/— 6601/'— 239000^/r— 5360^

+ 19
^"'=+o.002128+17510/!i+51480iu'+7640/«"— 1066/'— 63400/r—

^FJ=+0.0000280+350.4iU+364/-74;«"-14.0iu"'-750«/F+252^F+36,«

21 March, 1872.

162 SECULAR VARIATIONS OF THE ELEMENTS OF

0,=— 6".592128— <)".314940«— 0".85282u'— 0".78390«"— 0". 116744,,'"— 4".3746lu'F

-0".20160pr— (r.OOSSfy"— (r.OOlOGp™7;
/31=1 32° 40' 57".8— 102434>— 490268^'— 205068y+26729y+ 700270'X F

+ 19444'> F+90>r/— 735>ra;
JV,=+0.0283520— 37374^— 3360Ju'+358300/u"+5235«'"—415(HyF+133120/uF

+ 16400/«F/+990iuF":
JV/=— 0.0078380— 8507/i+222860^'—75320/—8530/'—100200/F— 46800^"

-4760^"— Stop™;
A7=— 0.0063210— 9 157,u+ 167160^'— 68660/—5780u'"—58300;«/F—36680^F

^'"=—0.0013250— 2162^+32900^'— 2360/—1437/'—1400f!/r— 7280^

^/" =+0.00000952+5^—
JV/ = =+0,00001345+10^— 346/u'+218(u"+18^'"+80/t/r+32^r+10/uF/+0/Mr";
JV,r/ = -0.00000696+1^+194^'— 96/—
^""=-0.000000420-0^+10.8^-6.8^— O

<7S=17".393390— 0".028779/«— r.31820^'— 2".4080/+0"()Gr)597/'— 12".7389a/F

-0".55708//F— 0".00942^r/—

/?J=292° 49' 53".2+124l8>+333738V+270720y+23660y— 7"ll800>

-25592y— 276>"—
^=+0.0015240—1455^— 67740^'— 50820/+4466/u'"+139300^r+6280^

/F

Nt'=— 0.0084783+8450^+454660^+283020^"— 24120/'— 882200,^'

— 39240^— lO^u"7— (

^"=+0.0069546— 9445^— 392740^'— 348760/+14218,u'"+909000j(/F

+39280^+ 220^^+90 w;

^'"=+0.0506672— 13628^— 648600^'— 421340/+79686/'+1091100/F

+77400^ F— 180^F/— 47,ura;

N" -0.00000251+1.6^+66^+62^"— 15.4/'— 140/F+12/MV— 4^F/— l^^,-

N,T -. -0.00002140+10.1^+434^'+ 294/-120.4/u'"—800^F-120/uF+6/fFJ '

g3=— 18".408914— 0".087808/t— 5*. 146 16^'— 5".18060/— 0".317709/'— 7".7278

-0".3442V— 0".00582;«F/— 0".
/?,=251° 45' 8".6+4720>+157816y+123030/u''-17836y"-331390y

-12580>F— 212>F/— 7
N3= +0.0036775 + 1 277^+82380^+65 1 60/— 2863«'"— 1 72200^—6880^

JV,'=-0.0224278-7986^— 2904(%'-46.8080l«"+11283/u"'+925900/F

+36080//F+580/iF/+200/<m;

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 163

N,m=— 0.0375951 -J-27422/u+1075000/ti'4-732200/(t"+100657l«'"— 2002200M

-80840/i'F—

^3'=— 0.000000557— 11.32^—676.4^'— 525^+65. 7/T+1207/F+51.V+ V

+0.3^™';

JVr,r'=+0.000000046+1.0p+57.8{/+44.8/— 5.6^'"— 106/r— 4^r

.6^'+5.2/— 0.65^'"— 12^— 0.

= 106° 14' 18,".0— SS"^^—

3y— 2726y

+7685y";

-8140^ F/
+620/".

g&=— 0".661666+0".000020i;<+0".00004,u'+0".00002,ci"+0.000002/«'"— 0".

/^6=20° 31'24".6+24".3^+298y— 540y— 42".3^'"— 70360yF+56804>F

+12932>r/— 2946yz/;
^=-1-0.0014778+29^—620^—440^"— 31^'"— 9400/F—4240(Ur+l440^F/

r— 4360^+880^

+ 11850^m;

^"=—0.00112485—

=— 0.0117882— 55—

— 7500^"—

"'— 6650/"— 4328^ r+ 206^

— 6320^—4072^

—3.5^'"— 1670/F+9936^r—50^r/

— 7886,ur";

ge=— 2".916082+0".000028a+a".00014^'+0".00004^"— 0".000013/'— 0".9004/F

/— 0".45136/uF//;

R =-133° 56' 10".8— 175"»— 344y-)-310y— 24".6;u"'— 1800>/r— 17876>F

+ 1182y/+19400y";

164 SECULAR VARIATIONS OF THE ELEMENTS OF

^=+0.0018108+596/<+660,/-2820^-341/'-25700iu'r+9320/+17960iu^

JV6«=+0.0016228+378^+1220/t'-1540/-315iu"'-23000/''+7240iar

f-15960«"+370^r";
[,"_52ju'"—15300u/r+1560iu "+11040^"

^=+0.0008794-4^-80^-20^"— 3/u'"

#6"=— 0.01 76872+35^+1 100p'+280/+29//''+27800/r— 29360^+9680^'

— 7850/";
#."'=+0.0019010— 4p— 600^'— 40^"— 9/'—8800/r— 7360^+18900^'

— 2640/«fxr.

g,=— 25".934567— 0".000039^— 0".00196/t'— 0".00430/— 0".00223/'— 18*.2906/

-7".35208;ur— 0".22692u"'— 0".(
/?T=306019>21".2—2".4^—242'X+128y—65".6/«'"+40V"'+32"l«F— 1532V

VI

VII .
9

VI

VII

iv^ ;

V

JV7'=— 0.0002932+40^+5840^'+14100/— 102/'— 10300/7— 8240/^—220^

— 40;

JVT"=— 0.0027275— 93^—12780^'— 5500/+1614iu'"+19000/r—3520j«r+220l«

^"'=—0.0092499+12631^—2700^— 13300^"— 295l«"'+8700/M/I'+5280^

+ 1340^r/
^^=+0.00630053+0^— 80^'— 40^"— 10/'— 44700/u/r+44800iuF+720/«

N7r=— 0.0156928+3^+240/+200^"+70/'— 43400«/r+44640;«r— 3120^

-
^,"=+0.0006890— 1^—20^—20^—5^'— 3300/F+3040iu"+80/urr+100;«m;

VI

%"';

VI

We have thus obtained the system of constants and the coefficients of their
variations, corresponding to the ecliptic of 1850, as the plane of reference; and we
shall now inquire what modifications are necessary in order to refer the same
quantities to the invariable plane of the planetary system.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS.

165

CHAPTER III.

ON THE POSITIONS AND SECULAR VARIATIONS OF THE ORBITS WHEN REFERRED
TO THE INVARIABLE PLANE OF THE PLANETARY SYSTEM.

1. WE shall now refer the positions of the orbits to the invariable plane of the
planetary system, in order to discover whether there are any laws which control
their mutual positions, of a similar nature to those which we have shown to exist
relatively to the eccentricities and perihelia. For this purpose it is necessary to
first determine the position of the invariable plane with reference to the fixed
ecliptic of 1850; and we can then readily refer all the orbits to that plane. The
position of this plane is found by the principle, that the sum of the products,
formed by multiplying each planetary mass by the projection of the area described
by its radius vector, in a given time, is a maximum. If we put y for the inclination
of the invariable plane to the fixed ecliptic of 1850, and n for the longitude of its
ascending node on the same plane, we shall have (Mecanique Celeste [1162]),

(527)
But we have

c =

c tan y sin n=c"; c tan y cos n=c'.
— e2) cos <p+m'V(i'a' (I— e'2) cos <£'

+m"vVa"(l— e"2)cos $"+ &c.,
c'=ml/ [j.a (1 — e2) sin <£ cos 6-\-m'V (id (1 — e*2) sin <£>' cos 0'

+m"V(i"a"(l— e"2)sin $' cos 6"+ &c.,
c"=m'[/ pa (1 — /) sin <£> sin Q-\-m'\/ pa' (1 — eri) sin q> sin ff

-j-m'V//a"(l— e'^sin $>" sin 0"+ &c.

If we denote the sun's mass by unity, we shall have
p=l+w», f/=l+< /=l+m", &c.;
but we shall also have

TttWa"2, &c.

+mVa"2V/li::?2" sin ^>" cos 6"-\- &c.,
d'=mna?V f^Tsin ^ sin 6-\-m'ria?Vl— e* sin $' sin ff

(528)

Substituting these values in equations (528), they will become
c —mHO?]/ 1 — e2 cos $ -\-m'ria'2V 1 — e'a cos $'

+mWa"V l^e^cos $>"+ &c.,
d =m?ia2l/l^e2sin $ cos O+rn'ria^Vl— e^sin ^' cos tf

(529)

166

SECULAR VARIATIONS OF THE ELEMENTS OF

Substituting in these equations the values of m, n, a, e, ty and 0, given in §§ 5
and 17 of Chapter I, and § 6 of Chapter II, we shall get

c=+0.0035274157, c'=— 0.00002735230, e"=+0.00009393304. (530)

In finding these quantities n" has been supposed to equal unity, and the values
of n, n', n'", &c. have been multiplied by — - in order to preserve the same ratio.

?i

Substituting the values of c, c', and c", in equations (527), we shall obtain
n=106° 14' 6".00, and y=l° 35' 19".376. (531)

If we now denote by <£>0, ^>0', <£><,", &c., 00, 00', 00", &c., the respective inclinations
and longitudes of ascending nodes of the different planets, on the invariable plane;
the values of 00, 00', 00", &c- being reckoned from the descending node of the fixed
ecliptic of 1850, on the invariable plane; we shall have the following equations to
determine 00, 00', &c., <£><,» <fr>'> &c.

sin <2>0 sin 00=sin <£> sin (0 — II), )

sin $0 cos 00=cos y sin <£> cos (0 — n) — sin y cos $. j

These equations will give the following elements: —

(532)

Mercury,

fy =6°

20'

58".08,

00

=287°

54'

5". 12,

Venus,

, / o

11

13.57,

0o'

=307

14

8.

10,

The Earth,

<*>!" =1

35

19.376,

00

= 180

0

0.

00,

Mars,

<?>o"' =1

40

43.70,

00W

=248

56

21.

45,

Jupiter,

<?>o/F =0

19

59.674,

00

IV

=210

7

35.

44,

Saturn,

$>or =0

55

30.924,

00

r

= 16

34

26.

66,

Uranus,

, YT -I

1

45.27,

00

VI

=204

12

33.

78,

Neptune,

$>om=0

43

24.845,

00

"'=286

39

55.

10.

2. Now putting

tan $0 sin 60=p0,
tan <£>„ cos 00=<70,

we shall get the following values

pa =—0.1058879,
Pa =—0.0304057,
Po' = 0.
pf =—0.0273512,
p0'v = -0.00273067,
Por =+0.00460691,
p"=— 0.00736719,
Por"= +0.01 26079,

tan
tan

, sm 00 =
, cos Oa=q0

;=?}&c.'}

(533)

q0 =+0.0342038,
qj =+0.0231090,
q0" =-0.0277354,
q0" =-0.0105324,
q0'T =—0.00503058,
q0r =+0.0154792,
9or/=— 0.0163856,
?0m=+0.000734628.

If we substitute these values in equations (408) and (409), we shall obtain the
values of /3, /?j, /22, &c., N, N^ N2, &c., corresponding to the invariable plane. But
instead of performing this operation separately for each root, we shall proceed in
the following manner.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 167

3. If we neglect the squares of e, e', e", &c., d>, d>', d>", &c., y, we may put
cos y=l; sin y=tan y=y; sin d>=tan d>; sin $»'=tan d>', &c.;

then we shall have

p—sin $ sin 6, /=sin d>' sin 6', p"=sin d>" sin 6" &c., ) - „,.
5=sin d> cos 0, </=sin d>' cos 6', </'=sin d>" cos 0" &c. j

Substituting these values in equations (408) and (409), also putting ;=0, and
remembering that for the root #4=0, we have Nt=N^=Ntv, &c., we shall have

m
na

m

sn <> sn

na na

5. Jtf4sin/34;

^sin d>' sin 6'+ &c.= I
na {

sin d> cos 0+-™' sin d>' cos 6'+ &c.= I —^-—. 4-&c. 1 Nt cos #,.
Jza' I na ' «'a ' j

na • na I na ' n'a'

But if we neglect m2, m'2, &c., we shall have

(535)

in

i i m „ « m 0

mna~= , m'«'a-=-— , mnV= --- , &c.,

""

« a

and equations (527) will give, by substituting the values of c,c',c",

• A i wi' . . . „, , o \ m in' , n }

sin 04 — — sin d> sin 0 4-&c.== < — _L_&C. > y sin II:

' na [ na ' nrf ' J

i 04- mt , sin d>' cos 0'4- &c.— \ — — U-'~, 4-&c. i y cos II.
' na ( na ' na }

m •
— sin d> sm
na

-sm d> cos!

(536)

Comparing equations (535) and (536) we find n=/34, and y=N±. Now substi-
tuting n=/34, y=JV4, in equations (532), they will give

sin d>0 sin 00=sin <(> sin (0 — /34)=sin d> sin 0 cos j34 — sin <£> cos 0 sin /34

=p cos /34— ? sin /24=£>0 ;
sin ^0 cos 00=sin(|) cos (0 — /34) — y=:
sin d> cos 0 cos/?4 — sind) sin0sin/34 — y =qcos(3t — ^)sin/34 — N4=q0.

(537)

And since the relative values of N, N', N", &c., N^ Nlt N^", &c., are known we
may determine their actual values corresponding to the invariable plane, by the
analysis of Chapter II, § 5. We shall therefore suppose

?0 =a N cos (grt+^O+a! ^ cos (gr1<+/31(0))+a2 ^2 cos

9o'=a ^' cos (flr«+i8TO)+a1 ^' cos (9'1<+/?1(0)J+a2 ^' cos

&c.;

Po =a N sin (^-f/^^+a! ^ sin (gj+pW+a, N, sin (^<4-/8s(0>)+&c.,

p0'=a N1 sin (^+/30>))-[-a1 JV,' sin 071<+/31(0))+a2 JV2' sin

&c.,

(538)

a, HI, a2, &c., being the constant factors which are necessary in order to reduce
the numbers already calculated to the corresponding ones for the invariable plane ;
and /3(0), (3^\ 82(0\ &c., being the constants necessary to satisfy the equations for
the given epoch.

168 SECULAR VARIATIONS OF THE ELEMENTS OF

Equations (408) and (409) will also become

na

na

(539)

no,

«a

cos

If we substitute in these equations the values of
by equations (537), they will become

(540)

,po, &c., g0, g0', &c., given

• (541)

-a
(

na

w a

(542)

Now according to equations (410), the coefficient of Nt in this equation is equal
to nothing; and if we substitute the values of the coefficients of a cos /?4, and a sin /34,
which are given by equations (408) and (409), both members of equations (541)
and (542) will be divisible by the coefficients of a sin (gt-}-(3m), and a cos
and we shall find

Whence we get
Therefore

sn (#/-=a sn
cos (gt+p—pj=a cos

tan

>); >

>). J

—ft^tan («7<+13(0))

>=3-?, and «=1.

(543)

(544)

It therefore follows that in order to apply our numbers to the invariable plane,
we have only to diminish the constants /?, p^ /?2, &c., by the longitude of the
ascending node of that plane, on the fixed ecliptic of 1850, and neglect the con-
stant term.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 169
Therefore we shall have

'= A" sn

0'= JV' cos

sin
:)sin
sin
cos
cos
-\-N6 cos (s^+ft — ft

sin

sin
sin
sin
cos
cos
cos

sin
cos

—ft);

(545)

(546)

Substituting for p0 and q0 their values given by the first members of equations

(537), we shall easily find

sin (j!>0 sin (f>0 — gl — {3)=^ sin J (gl — g)t-\~Pi — P \ -\-N% sin j (g2 — g)t-\-@2 — /? ! 'J

-f-JVgsin j((/3 — g)t-\-(33 — ^l-l-JVjsin \(g6 — g)t-\-pt — (3\ > (547)
-\-N6 sin { (g6 — g)t-\-fis — P \ -\-Ni sin \ (g7 — g)t-\-(37 — (3 \ )

sin eft, cos (00 — gt — /3) 1

£$\ \ (548>

COS (gl-
COS 1 (ga-
COS t-

COS (gr.-
COS | (y._
COS -

From these equations it is easy to show that the mean motion of 00 is equal to
gt when N exceeds the sum of the coefficients of the cosines, all taken positively.
We shall also have

maxmum <po=
and minimum. =N—

T; )
^. j

4. If we now substitute in these equations the values given in Chapter II, § 7,
we shall obtain the following maxima, minima, and mean motions.

Inclination to invariable plane.

Mean motion of

T *

Maximum.

Minimum.

Julian year.

Mercury,

9° 10' 41"

4° 44' 27"

—5". 126076

Venus,

3 16 18

000

indeterminate

The Earth,

360

000

u

Mars,

5 56 2

000

KI

Jupiter,

0 28 56

0 14 23

— 25".934567

Saturn,

1 0 39

0 47 16

—25 .934567

Uranus,

1 7 10

0 54 25

— 2.916082

Neptune,

0 47 21

0 33 43

— 0.661666

22 March, 1872.

170 SECULAR VARIATIONS OF THE ELEMENTS OF

5. It thus appears that the mean motion of the nodes of Jupiter and Saturn, on
the invariable plane, are exactly the same. This indicates a relation of a perma-
nent character between the positions of the nodes of these two planets, the nature
of which we shall now examine.

If we divide equation (547) by (548) we shall obtain an equation similar to (147)
and (148). And if in this equation we substitute the numbers corresponding to
Jupiter and Saturn, we find that the mean places of the nodes, on the invariable
plane, will be the same if N,"' and JV/ have the same sign, and that they will dif-
fer by 180° if Nj'r and N7r have different signs. The computed numbers show that
the signs of these two quantities are different: it therefore follows that the mean
longitudes of the nodes of Jupiter and Saturn, on the invariable plane of the
planetary system, always differ by 180°. We shall find, by the analysis of Chapter
I, § 21, that the actual place of Jupiter's node may differ from its mean place to
the extent of 19° 38', while that of Saturn can deviate from its mean place only to
the extent of 7° 7'. It therefore follows that the longitudes of the nodes of these
two planets can differ from 180° to the extent of 26° 45'. Their nearest possible
approach is therefore 153° 15, while their present distance apart is 166° 27'.

We shall also find that the actual place of Mercury's node can differ from its
mean place to the extent of 18° 31'; while the nodes of Uranus and Neptune can
respectively deviate from their mean places to the extent of 6° 0' and 9° 40'.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 171

CHAPTER IV.

ON THE PRECESSION OF THE EQUINOXES AND THE OBLIQUITY OF THE

ECLIPTIC.

1. THE analytical formulae for the precession of the equinoxes and the obliquity
of the ecliptic to the equator, referred to a fixed and also to a movable plane, are
given by the formulae [3100, 3101, 3107, and 3110], Mecanique Celeste. In order
to reduce them to numbers we shall observe that the letter c in the notation of
the formulae corresponds to N", Nf, -iV2", &c. in this work. If we denote the mean
value of the precession in a Julian year by Z, and the mean obliquity of the ecliptic
by h, and also put

f=l+g, fi=l+gi, /2=H-<fo /»=*+#» ft=t+gi=1, /6=Z+ft. &c., we shall have
the following formula) for determining the precession and obliquity :

cot h— -t

n/t 1 sin

-f GJ | cot h—^tanh 1 sin
I /i i

-J-Ca < cot 7t— &* tan/t I sin
( h

Precession of the
Equinoxes on the

(550)
Fixed Ecliptic.

h
+c3 1 cot h—^-tanh 1 sin

+c4 { cot A— ^-tan/t 1 sin

I /4

+c5 1 cot h-totoah | sin (/6<+/?6)
+c6 1 cot h—^-tanh | sin (/8«+/36)

( /6

+c7 { cot h—^-tauh \ sin (/T«+/8T)
Ji

l=A— c cos (/<+/? )— cx cos (/iH-/?i)— c2 cos (/,<+iS2) I Obliquity of the
-c3cos(/3<+/33)-c4cos(/4<+/34)-c6cos(/6<+/35) }• (551)

)-c7 cos (/7<+A) g"*» t? ;he
J Fixed Ecliptic.

172

SECULAR VARIATIONS OF THE ELEMENTS OF

J,'== U-\-£ 9-N" \ cot h-\- -.-tan7i > sh

/I / J

-

— 9±W cot
/i

sin

_ •?» JV2" I cot A-|-^tan A V sin (/2<+/32)

^2 I /2

_9?-N3" { cot A-f }.-tan 7t 1 sin (/3<+/33)

/3 I /»

/6

cot 7t-

sn

_^JV6" cot ^+ tan 7t sin

/a /•

-

/7

-2-Jw cos

cot

sn

Precession of the
Equinoxes on the

(552)
Apparent Ecliptic.

cos

COS

-^iV3 cos

/3

^-J\K COS

/6

Apparent obliquity

(553)
of the Ecliptic.

In equations (550) and (552), £ is to be determined so that ^ and # shall be
equal to nothing when <=0. We may determine I and h as folio ws:-
If we take the differential of equation (552), we shall obtain

dt

—^2" {

cot

co

cot
cot

cos

tan 7i cos

cos

1

tan 7t 1 cos

I

cot fc+-tanA 1

-

cos

-flf^V.* | cot /i+ j tan h J c

— ^JV/ { cot A+-^tan A 1 cos (/7<+/37)

I /7

(554)

Now at the epoch of 1850, at which time we have supposed <=0, we have

e=23° 27 31."0; and — ^1=50".23572 according to the investigations of BESSEL.

dt

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 173

The first members of equations (553) and (554) are therefore known; and if we
substitute in them the values of g, </lt r/2, &c., N", Nf, Nf, &c., (3, /?„ 0,, &c., cor-
responding to the assumed masses, together with /,/„/,, &c., they will become
— the numbers in brackets being logarithms,

23° 21' 31" 0— h C8-7069903] [8.4509982] [8.6715146]

1+<J l+ffi 1+92

[9.1494996] [6.9157253] [7.5163249]

~f+ff7~

[8.6222025]

(555)

f PR TnfiQQO^I l~8 d^nQQROl 1

50".23572+0".05933222 coa=Z+ j I J-f L ^DU

[8.6715146] [9.1494996] [6.9157253]

[7.5163249] [8.6222025] \

~ 3

(556)

If we divide equation (556) by Z tan A, and add the quotient to equation (555),
we shall get

50".23572+0".05933222cotA 3l-.0=ft+cot h. (557)

Ztan/t
Whence we get

50".23572+0".05933222 cot Ti

l-J-(7i_ 84451".0)tan£

The direct determination of h and Z from equations (555) and (556) is trouble-
some, and it is better to solve them by approximation. A few trials will show
that 23° 17' 16"=83836" is a near approximation to the value of li. If we sub-
stitute 7i=23° 17' 16" in equation (558), we shall get Z=50".4382997; and if we
substitute this value of Z in equation (555) we shall find

7i=23° 27' 31".0— 10' 14".4265=23° 17' 16".5735.

Now, substituting this value of h in equation (558), we shall get

Z=50".4382387.

Having found h and Z, we must substitute them in equations (550-553), and we
shall obtain the expressions for the numerical values of the precession and obliquity
during all past and future ages.

Adding^, glt gz, &c. to Z, we shall get/,/,,/2, &c., as follows,

/=45".312168, ft=l =50".438239,

/=43. 846111, /6=49 .776573,

/2=33 .044849, /e=47 .522157,

/3=32 .029325, /7=24 .503672.

174

SECULAR VARIATIONS OF THE ELEMENTS OF

We also have

(j = 21° 6' 26".8

^=132 40 56.2

£2=292 49 53.2

0,=251 45 8.6 P»=

Equations (550-553) reduced to numbers become
^=50".438239<+8915".6+ 5800".35 sin (/<+/3 )

— 358V.52 sin '
-f- 5583".09 sin
-f-20438".28 sin
-j-13294".37 sin
4- 646".98 sin

+ 834". 76 sin (/6<+&)

— 3217".97sin

14' 18".()

pt= 20 31 24.6

#.= 133 56 10.H

19 21.2

Precession on
fixed Ecliptic

(559)
of 1850.

f.=23° IT 16'.57-2445".33 cos (/<+/? )+1499".78 cos
_2189".55 cos (/2<+/32)-7950».46 cos

— 5722".14 cos (/4<+&)— 277".79 cos (.<+/. i fixcd Ecliptic
_355".257cos(/6<+/36)+1158".01cos(/,<+/37) j of 1850.

^'=50".438239<+8915".6+696".462 sin (ft+P )

— 552".463 sin
-f 2250".29 sin
-|-8708".52 sin
-flO".0558 sin
-j-57". 102 si
— 1910".92 sin

Precession on
(561)

apparent Ecliptic.

f=23 IT 16".57— 248".520 cos (/<+/? )+ 196".017 cos
-755".057 cos (/2<-(-/32)-2901".753 cos

_ 3".644 cos

)- 20".539cos(/6<+/36)
-f 595".433 cos (/7<+/3T)

Obliquity of

(562)
apparent
Ecliptic,

If we take the differentials of equations (561) and (562), we shall get
'^'=50".438239+0".152998 cos (/<+£)— OM17438 cos
+0".360530 cos (/2<-f &)+l".352281 cos

+0".0024267 cos (/6<+/36)+0"-°13156 cos

— 0".227011cos(/7<+,37)

(563)

L==-f-0'.054595 sin

+0*.120965 sin
4-0".0008794 sin

— 0".041668 sin

. 004732 sin («76<+/36)
— 0".0707357sin(<77<+/37)

(564)

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 175

If we put <=0, in equations (563) and (564), we shall get the values of pre-
cession and variation of obliquity at the beginning of 1850. These values are

<Z7-=0".489682.

tit ~ dt

If we take the sum of the coefficients of the sines in the expression of 4' equation
(561), without regard to their signs, we shall get the maximum quantity by
which the true place of the equinox can diifer from its mean place. This sum is
14185".81=3° 56' 25".81 ; it therefore follows that, if we suppose the equinox to
have a uniform yearly motion equal to 50".438239, its place, when computed for
any epoch, will not differ from the true place by an amount exceeding 3° 56' 26".
This remark is of especial importance in regard to the computation of the elements
of terrestrial physics during past geological periods.

If we divide the number of seconds in the circumference of the circle by the
mean motion of the equinoxes, we shall get 1296000"-^-50".438239=25694.8=the
number of years required for the equinoxes to perform a complete revolution in
the heavens.

If we take the sum of the coefficients of the cosines in the expression of e
equation (562), without regard to their signs, we shall obtain the maximum
quantity by which the obliquity of the ecliptic can differ from its mean value.
This sum is 4720".96=1° IS 40".96. From this it follows that the obliquity of
the ecliptic is always confined within the limits 23° IT 16".57±1° 18' 40".96; or,
between 24° 35' 57".53 and 21° 58' 35".61. The amount of its oscillations cannot
therefore exceed 2° 37' 22".

If we take the sum of the coefficients of -^, equation (563), we shall get the

dt

maximum quantity by which the annual precession can differ from its mean value.
This sum is equal to 2".225841 ; whence it follows that the annual precession is
always confined within the limits

50".438239±2".225841.

The maximum value of precession in a Julian year is therefore equal to
52".664080, and the minimum value of precession during the same time is equal
to 48".212398. If we divide the difference of these two numbers by the time
required for the earth to describe one second of an arc, we shall get the maximum
variation of the tropical year, equal to

4".451682-=-(3548".1876)-^864009)=108s.40 seconds of time.

If we subtract the present value of the precession from its maximum value, we
shall get 2".42836 for the difference between them. Dividing this number by
the time required for the earth to describe one second of arc, we shall get the
amount of time by which the present tropical year exceeds the tropical year when
it is reduced to its minimum length. The number thus found is 59M3. In like
manner we shall find that the tropical year may exceed its present length by

H6 SECULAR VARIATIONS OF THE ELEMENTS OF

49'.27. The present length of the tropical year is 365d 5" 48m 478.26. Whence

we get

maximum length of tropical year =365" 5h 49m 369.53;
and minimum length of tropical year =365 5 47 48.13.

Lastly, if we take the sum of the coefficients of the sines in the formula for — - ,

we shall find that the annual variation of the obliquity is always confined within

the limits of

0" and ±0".744167.

THE ORBITS OF T 11 E EIGHT PRINCIPAL PLANETS. 177

CHAPTER V.

TABULAR VALUES OP THE ELEMENTS OF THE PLANETARY ORBITS.

1. IN order to complete the subject of the secular inequalities, and render the
preceding investigation of greater practical value, we have reduced all the astrono-
mical elements to tables. The tabulated values for the planets embrace a period
of 7200 years ; commencing 6400 years before 1850, and continuing till A. D. 2650.
These are given at intervals of a century during the entire period. The elements
of the earth's orbit, together with the precession of the equinoxes in both longitude
and right ascension, and the obliquity of the ecliptic to the equator, are given for
an interval of 16,000 years ; commencing 8000 years before, and ending 8000 years
after the year 1850. They are also given at intervals of a century during the first
half of that period, and at intervals of four centuries and eight centuries for the
remainder of the period. The tabulated values are computed from the constants
corresponding to a mass of the Earth which is equal to the assumed mass increased

by its tenth part, or for m"=— - of the sun's mass. This value of the Earth's

335172

mass corresponds to a solar parallax of about 8".775, which is but little less than
the recent determinations of that element; and it is remarkable that this value of
the Earth's mass is very nearly equal to that which permits the planetary orbits to
attain a greater eccentricity than any other mass moving at the same distance.
Slight changes in this value of the mass would therefore produce only very incon-
siderable changes in the variation of the elements of the planetary orbits.

2. The eccentricities and places of the perihelia have been computed from the
values of A, 7i', h", &c., ?, I', I", &c., given by equations (C) (page 32), by substituting
the values of N, N', N", &c., N» W, #,", &c., ft ft, ft, &c., g, ffi, gt, &c., given in
Art. 31 (pages 64 and 65), by means of the equations

tan &=h+-l, tancrWi'-f-r, &r. ; e=7t-=-sin ts=?-=-cos or, e'-=-7t'-4-cx'=Z'H-cos «/, &c.

In like manner the nodes and inclinations of the orbits, on the fixed ecliptic of
1850, have been computed from the values of q,p, q',p', &c., given by equations
(F) page 1 1 1, by substituting the values of N, N', N", &c., Nlt NJ, W, &c., ft ft, ft,
&c., g, g}, 02, &c., given in Art. 17, pages 134, &c., by means of the equations

tan.B=_p-r-q, tan 6'=//-H7', &c. ; tan $> =p -r-sin 9 =q -=-cos 0, 1

tan <2>'=y-=-sin ff=q'^-cos 0', &c. j

23 April, 1872.

178

SECULAR VARIATIONS OF THE ELEMENTS OF

3. The inclinations of the orbits of the planets and the longitudes of their
ascending nodes on the ecliptic, or variable orbit of the earth, are denoted by
$1, <fr'» $i'"> &c., 0i, Bi, 0r, &c-> and have been computed by the equations

tane^Cp-pO-T-te-s"); tan 0/= (/-/') -(</-</') ; &c.;

tan $, =(p — p")-=-sin Ol =(q —q")-^-cos Oi ; )
tan $>,'=(/— p")H-sin 01'=(<?'— ?")-^cos 0,', &c. j

The longitudes of the perihelia and nodes are in every case counted from the mean
equinox of 1850.0.

4. The precession of the equinoxes on the fixed ecliptic of 1850, and the incli-
nation of the equator to the same plane, are denoted by 4- and ^ ; and they are
determined by equations (550) and (551). Reducing them to numbers, and trans-
forming so as to dispense with the arbitrary constant quantities, they will become

^=^+[4.0317043] sin \ft cos |/*+[3.6828039] sin

—[3.5715890] sin2!/*

—[4.0645663] sin \fa cos.
+[4.6281028] sin2|/3<

—[3.0600136] sin J/af cos
—[3.0760078] sin2i/0«

cos

+[3.8013126]
+[3.5080433] sin J/a* cos J/.<
+ [3. 74642 12] sin2! fj.

— [3.8713230] sin \}\t cos ±f4t
-[4.4070533] sin2|/4^
+[3.0831687] sin \fbt cos | ftt

-[2.6561496] sin2|/5^

— [3.5903084] sin |/T/ cos
—[3.7238538] 8ins|/T<

(565)

6l=23° 27' 31".00+[3.1962358] sin %/tcos
+[3.656351 1]sin2i/i
-[3.4230485] sin |/^
+[3.3045398] sin2!/,*

:cos

i f3t COS

COS

—[3.3390526] sin
+[3. 1006747] sin
—[4.2160834] sin

-[3.6525469] sin
+[4.0408748] sin

-[3.5051445] siri
+[2.2889032] sin \fbt cos I
+ [2.7159223] s

+[2.704921 7] sin if6t cos j
—[2.6889275] sin2*/*
+[3.2799396] sin \f.i cos
-[3.1463942] sin2|/7<

(566)

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 179

The coefficients in these equations are logarithms of seconds of arc ; and/,/!,/2,
&c. have the following values : —

/=45".225870 Z=/4=50".439525

/1=43 .770258 /6=49 .777863

/2=32 .812Q39 /6=47 .523448

/3=31 .503029 /7=24 .504515

In all cases t denotes Julian years of 365J days.

5. The general precession of the equinoxes in longitude is very nearly the same
as the precession on the apparent ecliptic, which is denoted by $, and is given by
equation (552). But as the apparent ecliptic is continually shifting its position in
space, the motion of precession on such an assumed plane becomes the same as it
would be along a warped surface, and very imperfectly represents the general pre-
cession at times only a few hundred years from the epoch, although its maximum
deviation from the truth can never exceed one-fourth of a degree. But on account
of the importance of the subject we shall determine in a rigorous manner the gene-
ral precession of the equinoxes in both longitude and right ascension. For this
purpose we shall consider the spherical triangle formed by the fixed ecliptic of
1850, and the apparent ecliptic and equator of any time t. In this triangle there are
known the two angles and the included side ; namely, the angle of inclination of
the apparent ecliptic to the fixed ecliptic of 1850, which is denoted by <£>", and the
inclination of the equator to the same plane, which is denoted by c1} and the
included side which is equal to ^-\-6". The three remaining parts of the triangle
are the distances from the extremities of the known side of the triangle to the point
of intersection of the apparent ecliptic and equator, and the angle included by
these sides. We shall denote these quantities by ^'-\-Q", $•, and e. $ denotes the
general precession in longitude, 3 denotes the planetary precession, which is the
distance between the fixed and apparent ecliptics measured on the apparent equator,
and e denotes the apparent obliquity of the ecliptic. The fundamental equations
of spherical trigonometry will therefore furnish the following formulae for the
determination of $, 3, and e : —

sn s sn •= — sn ty sn

sin E cos 3=sin <?>" cos el cos (^-j-0") +cos q>" sin

sin e sin (4/+0")=sin EX sin

sin E cos (4/-j-0")=sin ca cos $" cos (4-|-0")-|-cos Sl sin

The negative sign is given to the first equation because a forward motion of the
equinox is a diminution of precession.

The equations (567) give the values of ^ and e ; and either one of the equations
(568) will give the value $ when e has been determined. Since $• is always a very
small arc, it is determinable with all desirable precision by means of its tangent ;
but this is not the case with e. This quantity cannot be determined with extreme
precision by means of its sine, without using logarithms to more than seven places

180 SECULAR VARIATIONS OF THE ELEMENTS OF

of decimals. But as e never differs greatly from f,, we may compute the difference
between e and ti by eliminating $ from equations (567), and we shall find,

sin (e — EI)=

sin2 ft" cos2 e l— sin3 ft" sin2 e, cos3 (^+^')+2 cos ¥ sm ft" cos ci sin c' cos C^+fl") (5(59)

sin (E+CI)

Altliough f appears in the denominator of this equation it is readily determinate
from its sine with sufficient precision to be used in finding sin (e — ej with accuracy.

Since $ never differs greatly from 4, we may readily transform equations (568)
so as to give the difference of these quantities, and we shall find

sin E sin (4> — 40=cos fi sm ft" sin C^+0") \ (•

-2 sin2 1 tf sin f , sin (4>+0") cos (^+6") j l

This equation determines 41 — 41' witn a^ desirable precision, E having been
previously determined.

6. We shall now consider the spherical triangle formed by the fixed ecliptic of
1850, the fixed equator of 1850, and the apparent equator of any time t. Since
the inclination of the equator to the fixed ecliptic of 1850 is given by equation
(566), we may suppose this quantity to be known for the given time. Then calling
fj the inclination of the fixed equator and ecliptic E/, the inclination of the apparent
equator to the fixed ecliptic, and ^ the total luni-solar precession during the time t;
the two angles and the included side of the proposed triangle will be known, and
the three remaining parts may readily be determined in the following manner:
Let the distances from the intersection of the fixed and apparent equators to
the fixed ecliptic be denoted by 90° — z and 90°-|-z', and the angle of intersection
of the two equators be denoted by 0, we shall have the following equations for
determining these last named quantities: —

sin 0 cos z =sin 4- sin EI

sin 0 sin z =sin BI cos e/ — cos 4/ sin e/ cos e 1

cos 0 =cos E! cos ^'-(-sin e, sin e/ cos ^ • (571)

sin 0 cos z^sin 4- sin ei
sin 0 sin z'=cos 4- sin sl cos e/ — cos el sin e,

These, equations determine z, z', and 0 rigorously ; but since e i is nearly equal
to e ,', they are under a very inconvenient form for accurate computation when 4- is
a small angle. They may, however, be readily put under the following form for
very accurate and convenient computation: —

sin 0 cos z =sin 4> sin e/

sin 0 sin z =2 sin2 \ 4/ sin f/ cos ^-j-sin (e, — e/)

sin2 \ 0 =sin2 \ (el — e^+sin^^sinfisinf/ }• (572)

sin 0 cos z'=sin ^ sin ei
sin 0 sin z'=2 sin2 \ 4- sin e, cos e/— sin (e,— E/]

The sum of the quantities z and z* might very properly be called the luni-solar
precession in right ascension ; and if to this we add the planetary precession we

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 181

shall have the general precession in right ascension at any time t, equal to z-f-z'-j-^;
all of which quantities being taken from the table with the argument t,

7. Tables I — VIII have been computed as explained in §§ 2 and 3. They show
the elements of the planetary orbits at the times given in the first column of each
table, and seem to require no explanation as to their uses. 4,, and £u in Tables
IX and X, have been computed from the formulae (565) and (566); $, $ and e have
been computed from equations (567), (570), and (569), by using the values of 6",
<£/', 4, and cj given in Tables VIII and IX; and lastly, z, z', and 0 have been com-
puted by means of equations (572), by using the values of 4, e, £\ given in Table IX.

8. Having explained the method of constructing the tables, we will now explain
the manner of using Tables IX and X in connection with 6" and $", which are
given in Table VIII. The quantities 6", $", and $ are useful in reducing the
longitudes of the celestial bodies from the mean equinox of 1850 to the mean
equinox of any other date, and vice versa. This transformation is effected by
means of the following equations, in which h and (3 denote the mean longitude
and latitude of a celestial body at the epoch of 1850, and X and (3' denote the same
co-ordinates referred to the mean equinox of any time t, before or after that epoch.

cos p cos (a/— 0"— 4/)=cos ^ Cos (a,— 6") ~|

cos p sin (X— 0"— 4/)=cos 0 Cos $" sin (X— 0")+sm (3 sin <?>" I ( 573)
sin p =sin (3 cos <£>" — cos (3 sin q>" sin (X — &') )

For reducing to the equinox of 1850 these equations take the following form: —

cos (3 cos (A— 0")=cos p cos (a,'_0"— «J/) ]

cos 13 sin (Ji— 0")=cos p cos <p sin (A'— 0"— 4/)— sin <$>" sin p I (574)
sin 3 =sin cos "+cos ^ sin $* sin (^— 0"— 40 J

It is sometimes desirable to find the difference in the longitude or latitude of a
star arising from the precession of the equinoxes. This difference may be found
by the following formulae, the employment of which is perhaps more laborious than
that of the preceding from which they were derived; but they may in ordinary
eases be managed by the use of five-figure logarithms, whereas equations (573) and
(574) require seven-figure logarithms to arrive at accurate results.

fare tan =

J tan ft sin ft" cos (a.-^ ff^-- gsm^ff^cos (3.— 0") sin (3,— 0") 1 T (575)
\ ~ l+tan/3sin^ sin (?L— 6*)— 2 sin2 £ $>" sin2 (^-6")

cos ,5 sin / sin (^-

'=3-2 arc sm

sin2

X=X'— ^'— ["arc tan =

f tan p sin <?>" cos (^'_0"-^')+2 sin^ ltf_caB&—ff'—M sin (^— y— ^*) 1 1 (57g)
t " ~T:_ tan p sin ^iii^^F-^-S sin2 J ^>" sin2 (^'-0"-^')

J cos p sin <?>" sin (a'-0"-1J/)-2 sin /?' sin2 1 ^," ) (577)
/3==/3'+2 arc sm j ^—^^-^

182 SECULAR VARIATIONS OF THE ELEMENTS OF

If, in these equations, we neglect quantities of the order sin2 <p", they assume a
very convenient form for computation, and at the same time possess sufficient
accuracy for computations extending over a period of several hundred years,
except for stars situated very near the pole, in which cases some of the preceding
equations must be employed if we wish to obtain accurate results.

a; =a, +^'+<?>" tan p cos (a,— 0") 1

P=p—^sm(^—er)', j

2, =X —y—f tan p cos (^'-0"-4') 1
P=p-WBm(X—ff'—V). }

In these equations <J>" is to be expressed in seconds of arc.

9. For the reduction of right ascensions and declinations all the necessary data
depending on the motion of the equinox are contained in Table X. The quantities
in the table are adapted to computation by the following formulae, in which a and 5
denote the mean right ascension and declination of a star at the epoch of 1850 ;
and a' and # denote the same co-ordinates referred to the mean equinox of any
time t before or after the epoch.

cos 3' sin (a' — z' — jy)=cos 5 sin (a+ z — &) "j

cos 5' cos (a'— sf— S')=cos 5 cos 0 cos (a+z— 3)— sin 5 sin 0 V (580)
sin 5' =cos <5 sin 0 cos (a+z — S)+sin 8 cos© J

For reducing to the mean equinox of 1850 these equations take the following
form : —

cos 5 sin (a+z — $)=cos 5' sin (a' — z' — &') ^

cos 5 cos (a-j-z— $)=cos # cos 0 cos (a — z1— $')+sin 0 sin 5' V f581)
— sin 8 =cos 6' sin © cos (a' — z' — &') — cos © sin 5' J

The first two of equations (580) will very readily give

tan =

( tan 5 sin 0 sin (a+z — &)+2 sin2 J 0 sin (a+z — S) cos (a+z — &) ) "j
t ~ 1— tan j sin 0 cos (a+z— £•)— 2 sin2 ^ 0 cos2 (a+z— £) J J

Here the term z+z'+^' — $• appears as the general precession in right ascension
common to, all the stars, and the last term of the equation gives the correction
depending on the place of each particular star.

In like manner the first two of equations (581) will give

a=a'— (z+z'+S'— S)- fare tan =

| tan y sin 0 sin (a'— z1— £')— 2 sin2 1 0 sin (a'—/— 3') cos (a'— z7— 3Q 1 "]/583\
( 1+tan «' sin 0 cos (a'— z1— 3')— 2 sin2 1 0 co?(a'— z'— 3') J J

For stars situated near the pole, equations (580) and (581) are preferable to
(582) and (583), because when 5 or i' is equal to 90° the terms depending on tan
5 or tan X become infinite, and the equations (582) and (583) assume an indeter-
minate form. But this is not the case with equations (580) and (581) ; for if 5=90°
equations (580) will give sin 6'=cos0, and then we shall find cos (a — z' — $')=
sin 0-j-cos f>'= — 1, whence a' — z' — 3'=18()°, from which a' is easily determined.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 183

Rigorous expressions for the difference of the declinations depending on the
precession are not readily deducible from equations (580) and (581) of sufficient
simplicity to possess any advantage in computation over the original formulae, since
0 is not necessarily a small angle.

10. Having given the necessary formulae for reducing the position of a star which
is given in longitude and latitude, or right ascension and declination, with refer-
ence to the equinox of 1850, to that of any other date, and the reverse, we shall
now give some examples by way of illustration.

Example I. The mean place of polaris, at the beginning of the year 1850, was,
a=16° 15' 22".815, 5=88° 30' 34".889; it is required to find its right ascension and
declination at the beginning of 1950, 2050, and 2150, and also the maximum decli-
nation of the star.

In 1850 the planetary precession was equal to nothing, therefore 3 disappears
from the second member of equations (580). For <=+100, Table X, gives
z=0° 38' 23".482, z'=0° 38' 36".527, 0=0° 33' 24".811, and $=— 0° 0' 12".433.
Whence a-|-z=160 53' 46".297 ; and the computation is as follows, using the first
and second of equations (580): —

a+z

8

a+z

0

0

sin 9.4633534

cos 8.4151046

cos 9.9808361

cos 9.9999795

sin 7.9876415

—sin 9.9998531rc

cos 8 cos (a+z) cos ©=+0.02488400
- sin 8 sin 0 =—0.009716 16

cos 8' cos (a'— z'— S')l=+0.01516784
cos 8' sin (a' — z' — 3') =cos 8 sin (a+z)
a'— z'— 3' =26° 29' 2V.70
38 24.10
a' =27° 7' 45".80

log. 8.3959202

cos 5' cos (a'— z'

=89° 1' 44".277

" 8.1809237
" 7.8784580
tan^.6975343

cos 9.9518314
log. 8.1809237
cos 8.229092T

Computing in the same way for (!=-)- 200 and <=-{-300, we shall find the values
of a! and <$' as in the following table : —

89°

89

89

1'

27
28

44". 277
20.654
13.594

1950 27° 7' 45".80

2050 56 51 57.27

2150 120 32 36.31

The declination evidently attains a maximum value some time between 2050
and 2150. If we compute its place for (1=250, we shall find z=l° 35' 59".93,
^=1° 36' 26".63, 0=1° 23' 29".80, and 3'=— 21".80.

184

SECULAR VARIATIONS OF THE ELEMENTS OF

Then we find a'=88° 13' 39".20 and #=89° 3? 32". 202.

Now having the declination for <=200, <=250, and <=300, we very readily
find that the maximum will take place when t=-\- 252.335, or a little before the
middle of the year 2102. Computing the place of the star for this last value of
t, we find

a'=89° 53' 2".78 and 6'=89° 32' 32".973.

The nearest approach of the pole to the star is therefore equal to 27 27".027.

Example II. Let it be required to find the declination of Polaris when t= — 8000.
For the value of t we get z=— 53° 0' 2".8, z'=— 54° 34' 37".2, B=— 39° 24' 51".5,
and ^'=+2° 43' 48".3. a+z=— 36° 44' 40".0.

Therefore the computation will stand as follows : —

a-\-z

5

a-j-z

0

0

5

cos 5 cos (a-f-z) cos ©=+0.0161008
-sin 3 sin 0=4-0-6347087

cos # cos (a'— z7— y) =-(-0.6508095
cos # sin (a'— z'— $')

a— z'— y = 358° 37' 49".7
=—.r>l 50 48.9

sin 9.7768805«
cos 8.4151046
cos 9.9038016
cos 9.8879407
sin 9.802721 3n
-sin 9.999*53 In

log. 8.2068-Ki!)
" ^.8025744

" T8134539
" 8.1919851re

a'= 306° 47
a'— sf— $•'
COS (V cos (a'— z7— 30

6' =49° 23' 0

tan 8.3785312»

cos 9.9998759
log. 9.81345:',!)
cos 9.81357SO

From this calculation it follows that the present polar star wasr 8000 years ago,
distant 40° 37 from the pole.

Example III. In 1850 the place of the star a Cephei was

a=318° 45', 3=+Ql° 56';
required its mean place when <=-)-5600 years.
In this example we find

z=+ 36° 55' *'+S'=4-370 35', 0=28° 44'.

Then we get a'— ^— $'=249° 44', whence «'=287° 19' and #=+87° 50'.
It therefore follows that the star a Cephei will be only about 2° distant from
the pole in 5600 years ; it will therefore constitute the pole star of that period.

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS, 185

The equations for reducing from longitude cand latitude to right ascension and
declination, and the reverse, are the following: —

cos 6 cos a= cos (3 cos A ^

cos 8 sin a= cos t cos (3 sin 2, — sin e sin (3 ^ (584)

sin <§= sin e cos (3 sin 2,-J-cos g sin (3 )

cos (3 cos X=cos 8 cos a ^

cos /? sin ?.=cos e cos £ sin a -[-sin e sin <$ V (585)

sin (3= — sin e cos 5 sin a-(-cos e sin 5 J

Example IV. The right ascension and declination of a Tauri (Aldebaran) in
1H50 was a=66° 49' 46".35, ,$=+16° 12' 11".(); required its longitude and latitude
for t=— 4900, or, for the beginning of the year B. C. 3050.

Since c=23° 27' 31".0, in 1850, equations (585) will give the longitude and
latitude for the same epoch, as follows : —

X=67° 41" 34'. 1, p=— 5° 28' 40".l.

And for t=— 4900. Tables VIII and IX will give

6"=5° 22' 51".7, $>"=0° 41' 22".45, and ^'=—67° 40' 32".2.

If these quantities be substituted in equations (573), we shall find

X— 6"— ^'=62° 16'45".5, sin /S'=— 0.106061;
whence we get

/L'=359° 59' 5".0, and /5'=— 6° 5' 17".9.

Therefore the star Aldebaran, at the beginning of the year B. C. 3050, was only
55".0 westward of the vernal equinox, measured on the ecliptic of that date, and
coincided with the equinox in the year 3049 B. C.

Example V. The right ascension and declination of Aldebaran being, as in the
preceding example, at the beginning of 1850, required its right ascension and
declination at the beginning of the year B. C. 3050. For t=— 4900, Table X
gives z=— 31° 43' 26".4, z'=— 32° 44' 31".8, 0=— 26° 14' 1".4, and $'=+1° 31'0"0.
If these quantities be substituted in equations (580), we shall find

a'— z'— 3'=33° 42' 2".8, and sin.«y=— 0.0969604.
Whence we get

a'=2° 28' 31".0, and #=— 5° 33" 51'.0.

We might have found these last quantities by means of equations (584) by sub-
stituting for Ji, j3, the values of X and /-.', found in example IV, and using the
value ofe corresponding to that epoch, which value is £=24° 3' 8*.2.

From this computation we see that, although the star Aldebaran was 55" westward
of the equinox when measured on the ecliptic, it was nearly 2| degrees eastward
of the equinox when measured on the equator, and instead of being in a northern
constellation then, as now, it was in reality in a southern constellation.

11. We will now determine the position of the pole of the equator. The longi-
tude of the pole on the fixed ecliptic of 1850 at any time t will evidently be equal

24 April, 1S72.

SECULAR VARIATIONS OF THE ELEMENTS OF

to 90° — ^; and the latitude of the pole will also be equal to 90°— ;••/, or to the
complement of the obliquity of the ecliptic of 1850, at the given time. If we then
put /t=90° — 4,^/3 =90° — si, in equations (584), the resulting a and 6 will evidently
be the right ascension and declination of the pole of the equator for the time /,
referred to the equinox and equator of 1850. Calling the right ascension of the
pole A, and the declination D, we shall evidently have

cos D cos J.=sin e^ sin o// )

cos D sin A= — sin ex cos e /-(-cos ^ sin e,' cos i|/ > (586)
sin Z)= cos e, cos e/-f-sin e i sin c/ cos 4- J

f, denoting the obliquity in 1850, and e/ denoting the obliquity of the fixed ecliptic
of 1850 at tlte time for which the computation is made. If we compare these
equations with equations (571), we find sin A= — sin z, and sin Z>=cos 0.

Therefore A=—z, and D=90°— 9.

Now since z and 6 are already tabulated we can enter Table X with the argu-
ment t, and take out the right ascension and declination of the pole by mere
inspection.

Example VI. What will be the right ascension and declination of pole 5600
years hence, when referred to the equinox and equator of 1850 ? Entering Table X
with the argument <=-f 5600, we find z=36° 55' 6".4, and 0=28° 44' 0".89.
Therefore .4=323° 4' 53".6, and Z>=61° 15' 59".ll. The mean place of a Cephei
in 1850 was a=318° 45', and 2=61° 56'. This star will therefore be the pole-
star of that period.

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 187

TABLE I. — Elements of the Orbit of Mercury.

t

e

•B7

0

<?

8,

*i

—6400

0.205301

65° 13' 40"

54° o' 37".5

7° '3' 4".8i

59° 31' 23".8

6° 36' 57". i

6300

0.205308

65 22 55

53 53 35-6

7 12 52.3«

59 19 30.0

6** •"
37 13-0

02OO

0.205314

65 32 II

53 46 33-8

7 12 39.81

59 7 35-5

6 37 48.9

6lOO

0.205321

65 41 26

53 39 32-1

7 12 27.32

58 55 40.3

6 38 14.6

6OOO

0.205328

65 50 42

53 32 3°-4

7 12 14.84

58 43 44.5

6 38 40. i

5900

0.205335

65 59 57

53 25 28.9

7 12 2.37

58 31 48.0

6 39 5-6

5800

0.205341

66 9'3

53 18 27.4

7 ii 49.90

58 19 51-1

6 39 3°-9

5700

0.205348

66 18 28

53 ii 26.3

7 H 37-44

58 7 53-5

6 39 56-i

5600

0.205354

66 27 44

53 4 25.2

7 i' 24.99

57 55 55-2

6 40 21. 1

5500

0.205361

66 37 o

52 57 24.7

7 n I2-54

57 43 56.4

6 40 46. i

5400

0.205367

66 46 16

52 50 24.1

7 ii o. 10

57 3i 57-0

6 41 10.9

5300

0.205373

66 55 32

S2 43 23.5

7 i° 47.67

57 19 57-o

6 41 35-5

5200

0.205379

67 4 48

52 36 22.9

7 i° 35-24

57 7 56-3

6 42 o.o

5100

0.205385

67 14 4

52 29 22.3

7 10 22.83

56 55 55-2

6 42 24.4

5000

0.205391

67 23 20

52 22 21.7

7 10 10.42

56 43 53-5

6 42 48.7

4900

0.205397

67 32 36

52 15 21. I

7 9 58.02

56 3' 51-2

6 43 12.8

4800

0.205402

67 41 52

52 8 20.5

7 9 45-64

56 19 48.4

6 43 36.8

4700

o. 205408

67 51 8

52 I 20.1

7 9 33-27

56 7. 45-1

6 44 0.7

4600

0.205414

68 o 24

51 54 19-8

7 9 20.92

55 55 4L2

6 44 24.4

4500

0.205420

68 9 39

51 47 19-4

7 9 8.58

55 43 36.8

6 44 48.0

4400

0.205425

68 18 55

51 40 19.1

7 8 56.25

55 3i 31-9

6 45 n.4

4300

0.205431

68 28 10

51 33 18.9

7 8 43.93

55 19 26.5

6 45 34.8

4200

0.205436

68 37 26

51 26 18.7

7 8 31.62

55 7 20.5

6 45 58.o

4100

0.205442

68 46 41

51 19 18.8

7 8 19.32

54 55 14-0

6 46 21.0

4000

0.205447

68 55 57

51 12 18.9

7 8 7.03

54 43 7.'

6 46 43.9

3900

0.205453

69 5 H

5l 5 19-2

7 7 54-75

54 3i 0.7

6 47 6.7

3800

0.205458

69 14 30

50 58 19.6

7 7 42-49

54 18 5'.8

6 47 29.4

3700

0.205463

69 23 46

50 51 20.1

7 7 30-24

54 6 43.4

6 47 5L9

3600

0.205468

69 33 3

50 44 20. 6

7 7 18.01

53 54 34.6

6 48 H.3

3500

0.205473

69 42 19

5° 37 21.3

7 7 5-79

53 42 25.3

6 48 36.6

3400

0.205478

69 Si 35

50 30 22. 0

7 6 53-57

53 30 15.6

6 48 58.7

3300

0.205483

70 o 51

50 23 22.7

7 6 41-37

53 '8 5-4

6 49 20.6

3200

0.205487

70 10 7

50 16 23.5

7 6 29.19

53 5 54.8

6 49 42.5

3100

0.205492

70 19 23

5° 9 24.3

7 6 17.02

52 53 43-8

6 50 4.2

3000

0.205497

70 28 40

50 2 25.2

7 6 4.87

52 41 32.4

6 50 25.8

2900

0.205501

70 37 56

49 55 26.1

7 5 52-74

52 29 20.5

6 50 47.2

2800

0.205506

70 47 '2

49 48 27.1

7 5 40-61

52 17 8.3

A 5I 8'I

2700

0.205510

70 56 28

49 41 28.1

7 5 28-49

52 4 55-7

6 51 29.6

2600

0.205515

7i 5 44

49 34 29.2

7 5 16.40

5i 52 42.7

6 51 50.6

2500

0.205520

71 15 o

49 27 30.3

7 5 4-32

5 1 40 29.3

6 52 ii. S

2400

0.205524

71 24 16

49 20 31.4

7 4 52-25

5i 28 15.5

6 52 32.2

2300

0.205529

?i 33 33

49 13 32.5

7 4 40.20

51 16 1.4

6 52 52.8

2200

0.205533

71 42 50

49 6 33.7

7 4 28.17

5 i 3 46.8

A 53 13.3

2100

0.205538

71 52 7

48 59 34-9

7 4 16.17

5o 51 3i-9

6 53 33.6

2000

0.205542

72 i 24

48 52 36.1

7 4 4-17

5o 39 i6-5

« 53 S3i

1900

0.205547

72 10 40

48 45 37-4

7 3 52-19

So 27 0.9

6 54 13.8

I800

0.205551

72 19 57

48 38 38.7

7 3 40-23

50 14 44-9

6 54 33-8

1700

0.205555

72 29 14

48 31 40.0

7 3 28.28

5o 2 28.5

6 54 53.5

l6oO

0.205559

r- 38 31

48 24 41.4

7 3 i6-35

49 50 n. 8

6 55 ,3.2

1500

0.205563

72 47 47

48 17 42.7

7 3 4-44

49 37 54-8

6 55 32-6

1400

0.205567

72 57 4

48 10 44. 1

7 2 52.56

49 25 37.5

6 55 52.0

1300

0.205571

73 6 21

48 3 45-5

7 2 40. 69

49 13 19-8

6 c6 ii. 2

6-*

1200

0.205575

73 15 38

47 56 46.9

7 2 28.84

49 i !-9

56 30.3

. I 100

0.205579

73 24 55 47 49 48.3

7 2 17.00

48 48 43-7

6 56 49.2

1000

0.205583

73 34 12 47 42 49-7

7 2 5.19

48 36 25.0

6 57 8.0

9OO

0.205586

73 43 29 47 35 51.1

7 53-40

48 24 6.0

6 57 26.7

. 800

0.205590

73 52 46

47 28 52.5

7 41-63

48 ii 46.7

6 57 45-2

7OO

0.205594

74 2 2

47 21 53.9

7 29.88

47 59 27.2

6 58 3.6

6oO

0.205597

74 ii 19

47 H 55-2

7 >8.i5

47 47 7-4

6 c8 21.8

6~* _

5OO

0.205601

74 20 36

47 7 56.6

7 6.43

47 34 47-7

58 39-9

400

0.205604

74 29 53

47 o 57.9

7 o 54-70

47 22 27.0

6 58 57-9

300

0.205608

74 39 10

46 53 59-3

7 o 43.07

47 10 6.5

6 59 15.7

200

0.205611

74 48 26

46 47 O.6

7 o 31.41

46 57 45-6

f 59 33-3

100

0.205615

74 57 43

46 40 1.9

7 ° !9-77

46 45 24.6

6 59 50.9

0

0.205618

75 7 o

46 33 3.2

7 o 8.16

46 33 3-2

7 o 8.2

+ 100

0.205621

75 16 16

46 26 4.5

6 59 56.57

46 20 42.0

7 o 25.5

200

0.205624

75 25 33

46 19 5-7

6 59 45.00

46 8 20.4

7 o 42.6

300

0.205627

75 34 So

46 12 7.0

6 59 33-45

45 55 58.6

7 o 59.6

400

0.205630

75 44 7

46 5 8.2

6 59 21.92

45 43 36-6

7 i 16.4

500

0.205633

75 53 24

45 58 9-4

6 59 10.42

45 3i M-4

7 i 33-2

6oO

0.205636

76 2 41

45 51 10.6

6 58 58.93

45 18 52.0

7 i 49-7

7OO

0.205639

76 ii 58

45 44 ». 7

6 58 47.47

45 29.4

7 2 6.1

+800

0.205642

76 21 15

45 37 I2-8

6 58 36.02

44 54 6.7

7 2 22.4

188

SECULAR VARIATIONS OF THE ELEMENTS OP

TABLE \\.-Elements of the Orbit of Venus.

t

e'

w'

6'

¥

V

?l'

— 6400

0.0104237

124° 30' 10"

92° 31' 48"

3° i8'33"-9

1 08° n' 5"

3° 19' 42". i

6300

0.0103630

124 42 13

92 16 15

3 18 45-6

'°7 39 55

3 '9 45-8

6200

0.0103023

124 50 12

92 o 41

3 >8 58.5

107 8 45

3 >9 49-4

6100

0.0102417

i 24 58 6

91 45 6

3 '9 '0-5

106 37 36

3 19 S3-'

6000

0.0101812

•25 5 54

91 29 30

3 19 22.2

106 6 27

3 19 56.8

5900

0.0101208

125 >3 37

91 >3 52

3 '9 33-7

105 35 20

3 20 0.5

5800

0.0100606

125 21 15

90 58 14

3 '9 45-0

105 4 '3

3 20 4. i

5700

0.0100005

125 28 47

90 42 34

3 '9 56-°

104 33 7

3 20 7.8

5600

0.0099405

125 36 14

90 26 53

3 20 6.7

IO4 2 I

3 20 11.5

5500

0.0098806

125 43 35

90 ii 10

3 20 17.2

103 3° 57

3 20 15.2

5400

0.0098209

125 50 51

89 55 27

3 20 27.4

102 59 53

3 20 18.9

53°°

0.0097613

125 58 2

89 39 43

3 20 37.5

102 28 50

3 20 22.6

5200

0.0097019

126 5 7

89 23 58

3 20 47.2

101 57 48

3 20 26.2

5100

0.0096426

126 12 6

89 8 ii

3 20 56.8

101 26 46

3 20 29.9

5000

0.0095834

126 19 o

88 52 24

3 21 6.0

ioo 55 45

3 20 33-6

4900

0.0095244

126 25 48

« 36 35

3 21 15.0

100 24 45

3 20 37.3

4800

0.0094654

126 32 30

88 20 46

3 21 23.8

99 53 46

3 20 41.0

4700

0.0094066

126 39 6

88 4 55

3 2.1 32-3

99 22 48

3 20 44.7

4600

0.0093480

126 45 36

87 49 3

3 21 40.6

98 51 50

3 20 48.4

4500

0.0092895

126 52 o

87 33 ii

3 21 48.6

98 20 53

3 20 52.1

4400

0.0092311

126 58 17

87 i? 17

3 21 56.4

97 49 57

3 20 55.8

4300

0.0091729

127 4 28

87 i 22

3 22 4.0

97 19 I

3 20 59.5

4200

0.0091149

127 10 33

86 45 26

3 22 II. 2

96 48 7

3 21 3.2

4100

0.0090571

127 16 32

86 29 29

3 22 18.3

96 17 13

3 21 6.8

4003

0.0089994

127 22 24

86 13 31

3 22 25.1

95 46 19

3 21 10.5

3903

0.0089419

127 28 9

85 .57 32

3 22 31.6

95 '5 27

3 21 14.2

3800

0.0088845

127 33 48

85 4' 32

3 22 37.9

94 44 35

3 21 17.9

3700

0.0088273

127 39 20

85 25 31

3 22 44-0

94 >3 44

3 21 21.6

3600

0.0687702

127 44 46

85 9 29

3 22 49.8

93 42 54

3 21 25.3

3503

0.0087133

'27 5° 5

84 53 26

3 22 55.3

93 "2 5

3 21 29.0

3400

0.0086566

«27 55 '7

84 37 22

3 23 0.6

92 41 16

3 21 32.6

33°3

o.oo8533O

128 0 21

84 21 16

3 23 5.7

92 10 28

3 21 36.3

3201

0.0085436

128 5 20

84 5 10

3 23 10.5

9i 39 4i

3 21 40.0

3100

0.0084873

128 10 ii

83 49 3

3 23 IS-'

9' • 54

3 2' 43- 6

3000

0.0084313

128 14 54

83 32 54

3 23 19-4

90 38 9

3 21 47.3

2903

0.0083755

128 19 29

83 16 45

3 23 23.4

90 7 24

3 21 50.9

2800

0.0083198

128 23 57

83 o 34

3 23 27.3

89 36 39

3 21 54.6

2700

0.0082643

128 28 17

82 44 23

3 23 30.8

89 5 56

3 21 58.2

2600

0.0382390

128 32 30

82 28 10

3 23 34.2

88 35 '3

3 22 1.9

2500

0.0081539

128 36 35

82 ii 57

3 23 37.3

88 4 31

3 22 5.5

2400

0.0080939

128 40 32

81 55 42

3 23 40. i

87 33 49

3 22 9. 1

2300

0.0380442

1 28 44 20

81 39 27

3 23 42.7

87 3 8

3 22 12.8

220O

0.0379896

128 48 o

81 23 10

3 23 45.0

86 32 28

3 22 16.4

2IOO

0.0079352

128 51 31

81 6 52

3 23 47.1

86 i 49

3 22 2O. O

2033

0.0078810

128 54 54

80 50 33

3 23 49.0

85 31 10

3 22 23.6

I9OO

0.0078271

128 58 8

80 34 14

3 23 50.6

85 o 32

3 22 27.2

I SOD

0.0077733

129 I 13

80 17 53

3 23 51.9

84 29 55

3 22 30.8

1700

0.0077197

129 4 9

80 i 31

3 23 53.0

83 59 18

3 22 34.4

1600

0.0076663

129 6 56

79 45 8

3 23 53.9

83 28 42

3 22 38.0

1500

0.0076132

"29 9 33

79 28 44

3 23 54.5

82 58 7

3 22 41.6

1403

0.0075602

129 12 I

79 12 19

3 23 54.9

82 27 33

3 22 45.2

1300 0.0075074

129 14 19

78 55 S3

3 2.) 55.0

81 56 59

3 22 48.7

I2O3

0.0074549

129 16 28

78 39 26

3 23 54.9

81 26 26

3 22 52.3

IIOO

0.0074027

129 18 26

78 22 58

3 23 54.6

80 55 54

3 22 55.8

IO33

0.0073506

129 20 14

78 6 30

3 23 53.9

80 25 22

3 22 59.4

900

0.0072987

129 21 52

77 49 59

3 23 53-i

79 54 5'

3 23 2.9

800

0.0072470

129 23 21

77 33 29

3 23 52.0

79 24 21

3 23 6.4

700

0.0071956

129 24 39

77 '6 57

3 23 50.6

78 53 52

3 23 10.0

600

0.0071444

129 25 46

77 o 24

3 23 49. i

78 23 23

3 23 13.0

500

0.0070934

129 26 44

76 43 5°

3 23 47.2

77 52 54

3 23 17.0

403

0.0070427

129 27 31

76 27 14

3 23 45.2

77 22 27

3 23 20.5

300

0.0069922

129 28 7

76 10 38

3 23 42.8

76 52 o

3 23 23.9

200

0.0069419

129 28 33

75 54 o

3 23 40:3

76 21 34

3 23 27.4

— 100

0.0068919

1 29 28 48

75 37 23

3 23 37.5

75 5" 8

3 23 30.8

o

0.0068420

129 28 52

75 20 4-5

3 23 34.4

75 20 42.9

3 23 34.4

+ 100

0.0067924

1 29 28 45

75 4 2

3 23 31.1

74 So 18

3 23 37.8

203

0.0067431

129 28 28

74 47 20

3 23 27.6

74 >9 55

3 23 41.2

303

0.0066940

129 28 o

74 30 38

3 23 23.8

73 49 32

3 23 44-6

400

0.0066452

129 27 22

74 13 54

3 23 19.7

73 '9 9

3 23 48.0

500

0.0065966

129 26 33

73 57 8

3 23 15.4

72 48 48

3 23 51.4

600

0.0065483

129 25 33

73 40 22

3 23 ii. o

72 18 26

3 23 54.8

700

0.0065002

129 24 22

73 23 34

3 23 6.2

71 48 6

3 23 58.1

+800

0.0064523

129 23 o

73 6 45

3 23 1.2

71 17 46

3 24 1.5

THE ORBITS OF THE E1GUT PRINCIPAL PLANETS. 189

TABLE III. — Elements cf the Orbit of Mars.

t

,111

or'"

8"'

tf»

8,'"

t,'"

—6400

0.0869446

304° 22' 34"

63° 53' 22"

2° 17' 9". 7

86° 36' 50"

0 54' 44".°

6300

0.0870448

3°4 5° 33

63 40 30

2 16 50.0

86 2 43

54 39-3

6200

0.0871449

305 I 8 30

63 27 35

2 1 6 30. 2

85 28 32

54 34-6

6100

0.0872449

305 46 26

63 14 37

2 l6 10.2

84 54 19

54 29.9

6000

0.0873449

306 14 19

63 i 36

2 15 50.1

84 20 2

54 25.3

5900

0.0874448

306 42 1 1

62 48 33

2 15 29.8

83 45 4'

54 20. 6

5800

0.0875447

307 10 I

62 35 27

2 15 9-4

83 ii 17

54 16.1

5700

0.0876445

3°7 37 49

62 22 18

2 14 48.8

82 36 50

54 II-5

5600

0.0877443

3°8 5 35

62 9 6

2 14 28.1

82 2 ig

54 7-0

55°o

0.0878440

308 33 20

61 55 52

2 14 7-3

81 27 45

54 2.4

5400

0.0879437

309 i 2

61 42 35

2 13 40-3

80 53 7

53 58.0

53°°

0.0880433

309 28 43

61 29 14

2 13 25.1

80 18 26

53 53-5

5200

0.0881429

309 56 23

61 15 51

2 13 3-8

79 43 42

53 49-1

5100

0.0882424

310 24 o

61 2 24

2 12 42.4

79 8 55

53 44-7

5000

0.0883418

3io 51 36

60 48 55

2 12 20.8

?8 34 4

53 40.3

4900

0.0884412

311 19 10

60 35 24

2 II 59.0

77 59 9

53 36-o

4800

0.0885405

3" 46 43

60 21 49

2 II 37-2

77 24 12

53 3i-7

4700

0.0886397

312 H 13

60 8 ii

2 II I5.I

76 49 n

53 27.4

4600

0.0887388

312 41 42

59 54 30

2 10 52.9

76 14 6

53 23.2

4500

0.0888378

3>3 9 10

59 40 47

2 10 30.6

75 38 59

53 i9-o

4400

0.0889367

313 36 35

59 27 o

2 10 8.1

75 3 48

53 14-9

4300

0.0890354

3H 3 59

59 13 I0

2 9 45-5

74 28 34

53 10.7

4200

0.0891341

3«4 31 21

S8 59 '7

2 9 22.7

73 53 '7

53 6.6

4100

0.0892327

314 58 42

58 45 21

2 8 59.8

73 17 56

53 2.6

4000

0.0893311

315 20 0

58 31 22

2 8 36.7

72 42 32

52 58.5

3900

0.0894294

315 53 «8

58 17 20

2 8 13.4

72 7 5

S2 54.6

3«oo

0.0895276

316 20 34

58 3 «5

2 7 5°-o

7i 31 35

52 50.6

3700

0.0896257

316 47 48

57 49 6

2 7 26.5

70 56 i

52 46.7

3600

0.0897236

3'7 15 °

57 34 54

2 7 2.7

70 20 25

52 43-9

35°o

0.0898214

317 42 ii

57 20 39

2 6 38.9

69 44 44

52 39-1

3400

0.0899191

318 9 20

57 6 21

2 6 14.9

69 9 i

S2 35-4

33oo

0.0900166

318 36 28

56 51 59

2 5 50-7

68 33 14

52 31-7

3200

0.0901140

3'9 3 34

56 37 34

2. 5 26.3

67 57 24

52 28.1

3100

0.0902113

319 3° 39

56 23 6

2 5 1.8

67 21 31

52 24.5

3000

0.0903084

3'9 57 42

56 8 34

2 4 37-2

66 45 35

52 21.0

2900

0.0904053

320 24 43

55 53 59

2 4 12.4

66 9 36

52 17-5

2800

0.0905021

320 51 43

55 39 21

2 3 47-4

65 33 33

52 14-0

2700

0.0905987

321 18 41

55 24 39

2 3 22.3

64 57 27

52 10.7

2600

0.0906952

321 45 38

55 9 53

2 2 57.0

64 21 18

52 7-3

2500

0.0907915

322 12 33

54 55 4

2 2 31.6

63 45 6

52 4.0

2400

0.0908876

322 39 26

54 40 12

2 2 6.0

63 8 51

52 0.8

2300

0.0909836

323 6 19

54 25 15

2 I 40.2

62 32 32

51 57-6

2200

0.0910793

323 33 10

54 10 15

2 I 14.3

61 56 ii

51 54-5

2100

0.0911748

323 59 59

53 55 12

2 0 48.2

61 19 46

51 5i-4

2000

0.0912702

324 26 47

53 40 4

2 0 21.9

60 43 18

51 48.4

igOO

0.0913654

324 53 33

53 24 53

59 55-5

60 6 47

5" 45-5

1800

0.0914603

325 20 18

53 9 38

59 28.9

59 30 13

51 42.6

1700

0.0915550

325 47 i

S2 54 19

59 2.2

58 53 35

51 39-8

1600

0.0916496

326 13 44

52 38 56

58 35-2

58 16 55

5i 37-1

1500

0.0917440

326 40 24

52 23 29

58 8.2

57 40 n

51 34-4

1400

0.0918381

327 7 4

52 7 58

57 40.9

57 3 25

5i 3]-8

1300

0.0919320

327 33 4i

5« 52 24

57 '3-5

56 26 36

51 29.2

1200

0.0920257

328 o 18

5' 36 46

56 46.0

55 49 44

51 26.7

IIOO

0.0921192

328 26 52

51 21 4

56 18.3

55 12 50

51 24.3

IOOO

0.0922125

328 53 26

5t 5 18

55 50.4

54 35 S2

51 22. 0

900

0.0923056

329 19 58

So 49 27

55 22.3

53 58 52

51 19.7

800

0.0923984

329 46 29

50 33 33

54 54- 1

53 21 48

51 "7-5

700

0.09-24910

330 12 59

50 i? 34

54 25.7

52 44 42

51 :5-3

600

0.0925834

330 39 27

50 i 31

53 57-2

52 7 33

51 13.2

500

0.09266:5

331 5 54

49 45 23

53 28.5 51 30 21

51 I 1.2

400

0.0927674

331 32 19

49 29 n

52 59-6

5° 53 o

Si 9-3

300

0.0928590

33' 58 43

49 12 55

52 30.5

5° J5 49

51 7-5

200

0.0929504

332 25 6

48 56 34

52 1.3

49 38 28

51 5-7

— IOO

0.0930415

332 51 28

48 40 8

5i 31-9

49 i 4

51 4.0

0

0.0931324

333 '7 47-8

48 23 36.8

5' 2.30

48 23 36.8

51 2.30

+ 100

0.0932230

333 44 7

48 7 i

50 32.6

47 46 6

51 °-7

20O

0-0933134

334 10 24

47 50 20

50 2.6

47 8 33

5° 59-2

300

0.0934035

334 36 4i

47 33 34

49 32.5

46 3° 57

5° 57-8

400

- 0.0934934

335 2 56

47 I" 43

49 2.2

45 53 18

50 56.4

500

0-0935830

335 29 10

46 59 46

48 31-7

45 15 36

5° 55- '

600

0.0936723

335 55 23

46 42 44

48 i.i

44 37 5°

5° 53-9

-_. P/> Q

700

0-0937613

336 2t 34

46 25 36

47 30-3

44 o 2

50 52.8

+800

0.0938502

336 47 44

46 8 22

46 59.2

43 22 n

5° 5'-7

190

SECULAR VARIATIONS OF THE ELEMENTS OF

TABLE IV.— Elements of the Orbit of Jupiter.

t

ff

."

fl/f

*"

V

1,"

— 6400
6300
6200

0.0393563
0.0394984
0.0396406

3° 10' 38"
3 '5 47

3 21 2

92° 20' 14"
92 21 55
92 23 45

° 32' I4".9
3' 59-o

31 43- 1

124° 29' o"
124 5 5
123 41 9

0 41' I9".4
40 58.0
40 36-7

6100

0-0397830

3 26 25

92 25 42

31 27.2

'-3 J7 13

40 '5-4

6000

0.0399255

3 3' 55

92 27 48

31 'i-4

1^2 53 17

39 54- 1

5900

0.0400681

3 37 32

92 3° 2

3.o 55 -6

122 29 22

39 32-7

5800

0.0402109

3 43 l6

92 32 24

30 39-9

122 5 26

39 ii-4

5700
5600

0.0403538
o. 0404967

3 49 6
3 55 3

92 34 54
92 37 34

3° 24.3
30 8.7

121 41 31
121 17 36

38 50.0
38 28.7

5500

0.0406397

4 i 7

92 40 21

29 53-i

120 53 40

38 7-3

5400

0.0407827

4 7 18

92 43 !7

29 37-6

1 2O 29 45

37 46-0

5300

0.0409258

4 '3 35

92 46 22

29 22.2

120 5 50

37 24.6

5200

0.0410689

4 '9 58

92 49 36

29 6.9

"9 41 55

37 3.3

5100
5000

0.0412120
0.0413550

4 26 29
4 33 5

92 S2 57
92 56 28

28 51.6
28 36.4

119 18 o
"8 54 5

36 41.9
36 20.6

4930
4800

0.0414980
0.0416410

4 39 48
4 46 36

93 o 7
93 3 55

28 21-3

28 6.3

118 30 io
118 6 15

35 59-2
35 37-8

4700

0.0417839

4 53 3'

93 7 52

2? 51-3

117 42 20

35 16.5

4600

0.0419267

5 o 32

93 ii 57

27 3^-5

117 18 24

34 55.1

4500

0.0420694

5 7 39

93 16 12

27 21.7

116 54 29

34 33-8

4400

0.0422121

5 H 52

93 20 34

27 7.0

116 30 34

34 12.4

43°o

0.0423546

5 22 IO

93 25 6

26 52.4

116 6 39

33 5'.°

4200

0.0424969

5 29 34

93 29 47

26 37.9

115 42 43

33 29.7

4100

0.0426391

5 37 4

93 34 36

26 23.S

115 18 48

33 8.3

4000

0.0427812

5 44 40

93 39 34

26 9.3

»4 54 S2

32 47.0

3900

0.0429231

5 S2 21

93 44 41

25 55-i

114 30 56

32 25.6

3800

0.0430648

608

. 93 49 57

25 41.0

»4 7 o

32 4.2

3700

0.0432064

680

93 55 22

25 27.1

"3 43 5

3i 42.9

3600

0.0433478

6 15 58

94 o 55

25 13.2

"3 19 9

31 21.6

35°°

0.0434890

6 24 I

94 6 38

24 59-5

112 55 12

31 0.2

0.0436300

6 32 io

94 12 29

24 46.0

112 31 16

30 38.9

33OO

0.0437708

6 40 23

94 18 30

24 32-5

112 7 20

30 17.6

3200

0.0439113

6 48 42

94 24 39

24 19.2

"I 43 23

29 56.3

3100

0.0440516

6 57 6

94 30 57

24 6.0

in 19 26

29 35-0

3000

0.0441917

7 5 35

94 37 23

23 S3-0

no 55 29

29 13.7

2900

0.0443315

7 '4 9

94 43 59

23 40.0

no 31 32

28 52.4

2800

0.0444711

7 22 48

94 50 43

23 27.3

no 7 34

28 31.1

2700

0.0446104

7 3' 3i

94 57 36

23 14.6

109 43 37

28 9.9

2600

0.0447494

7 40 20

95 4 37

23 2.2

109 19 39

27 48.6

2500

0.0448882

7 49 13

95 " 47

22 49-9

108 55 41

27 27.3

2400

0.0450265

7 58 H

95 '9 6

22 37.7

108 31 42

27 6.1

2300

0.0451645

8 7 '4

95 26 34

22 25.7

108 7 43

26 44.8

2 2OO

0.0453023

8 16 22

95 34 9

22 13.8

107 43 44

26 23.6

2100

0.0454398

8 25 34

95 4i 54

22 2.2

107 19 45

26 2.4

2000

0.0455769

8 34 50

95 49 46

21 50.7

106 55 46

25 4L3

I9OO

0-0457137

8 44 n

95 57 47

21 39-3

106 31 46

25 20.0

I800

0.0458501

8 53 37

96 5 56

21 28.2

106 7 46

24 58.8

1700

0.0459862

936

96 14 13

21 17.2

105 43 45

24 37-7

l6oo

0.0461220

9 12 41

96 22 38

21 6.4

105 19 44

24 16.5

I50O

0.0462574

9 22 19

96 31 12

20 55-7

i°4 55 43

23 55-4

I4OO

0.0463924

9 32 2

96 39 53

20 45-3

104 31 41

23 34.3

1300

0.0465271

9 41 48

96 48 42

20 35.1

104 7 38

23 13.2

I2OO

0.0466613

9 51 39

96 57 38

2O 25.0

103 43 35

22 52.1

I IOO

0.0467952

io I 34

97 6 43

2O 15.2

103 19 32

22 3I.O

IOOO

0.6469286

io n 32

97 IS 54

20 5.5

102 55 28

22 IO.O

900

0.0470^16

io 21 35

97 25 14

19 56.1

102 32 23

21 48.9

800

0.0471942

io 31 42

97 34 40

19 46.8

102 7 18

21 27.9

700

0.0473264

io 41 53

97 44 H

•9 37-8

ioi 43 13

21 6.9

600

0.0474581

io 52 7

97 53 54

19 28.9

101 19 7

20 45.9

500

0.0475894

II 2 25

98 3 42

19 20.3

loo 55 i

20 24.9

400

0.0477202

II 12 48

98 13 36

19 ii. 8

too 30 54

20 3.9

300

0.0478505

II 23 I4

98 23 38

19 3-6

loo 6 46

19 43-0

200

0.0479804

'i 33 43

98 33 46

18 55.6

99 42 38

19 22.1

IOO

0.0481098

ii 43 16

98 44 o

18 47.8

99 18 30

19 1.2

O

0.0482388

" 54 53-i

98 54 20.5

1 8 40.30

98 54 20.5

18 40.3

+ 100

0.0483672

12 5 34

99 4 47

18 33.0

98 30 io

18 19.4

no

0.0484952

12 16 17

99 15 20

18 25.9

98 6 o

17 58.6

300

0.0486227

12 27 4

99 25 58

18 19.0

97 41 48

"7 37-8

400

0.0487496

12 37 55

99 36 42

18 12.4

97 17 36

17 17.0

0.0488760

12 48 49

99 47 31

18 6.0

96 53 23

16 56.2

600

0.0490020

12 59 46

99 58 25

17 59-8

96 29 9

"6 35-4

700

0.0491274

>3 io 47

loo 9 25

'7 53-9

96 4 55

16 14.7

+800

0.0492523

13 21 51

loo 20 29

17 48.2

95 40 39

'5 54-0

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 191

TABLE V. — Elements of the Orbit of Saturn.

t

f

my

6r

<?r

•/

»IF

—6400

0.0708682

62° 8' 48"

127° 38' 42"

2° 8' 28".6

144° 49' 21"

2° 41' 3i".7

6300

0.0706762

62 34 9

127 25 42

2 8 57.i

144 19 58

2 41 23.4

6200

0.0704829

62 59 31

127 12 37

2 9 25.4

"43 50 34

2 41 15.0

6100

0.0702882

63 24 55

126 59 29

2 9 53.4

143 21 8

2 41 6.4

6000

O.070092I

63 50 19 | 126 46 17

2 10 21. I

142 51 41

2 40 57.8

5900

0.0698947

64 15 45

126 33 2

2 10 48.6

142 22 II

2 40 49. 1

5800

0.0696958

64 4! 12

126 19 43

2 II 15.8

141 52 40

2 40 40.2

5700

0.0694955

65 6 40

126 6 21

2 II 42.7

141 23 7

2 40 3L3

5600

0.0692939

65 32 9

125 52 55

2 12 9.4

HO 53 31

2 40 22.3

SSoo

0.0690909

65 57 39

125 39 27

2 12 35.8

140 23 54

2 40 I3.I

5400

0.0688866

66 23 ii

125 25 55

2 13 1.9

139 54 '4

2 40 3.9

53°°

O.06868IO

66 48 44

125 12 20

2 13 27.8

'39 24 33

2 39 54-5

5200

0.0684740

67 14 19

124 58 4I

2 '3 53-4

138 54 50

2 39 45-'

5100

0.0682657

6? 39 55

124 45 o

2 14 l8'7

138 25 5

2 39 35-5

5000

0.0680560

68 5 32

124 31 16

2 14 43.8

137 55 18

2 39 25.8

4900

0.0678450

68 31 ii

124 17 28

2 15 8.6

137 25 29

2 39 16. i

4800

0.0676327

68 56 51

124 3 38

2 15 33-0

"36 55 37

2 39 6.2

4700

0.067419!

69 22 32

123 49 45

2 15 57-3

136 25 44

2 38 56.2

4600

0.067204!

69 48 15

123 35 49

2 l6 21.2

J35 55 49

2 38 46.2

4500

0.0669878

70 14 o

123 21 51

2 16 44.9

135 25 51

2 38 36.0

4400

0.0667702

70 39 46

123 7 49

2 I? 8.3

134 55 52

2 38 25.7

4300

0.0665513

7i 5 34

122 53 46

2 I? 31.4

134 25 50

2 38 15-4

4200

0.0663310

7i 3« 23

122 39 40

2 I? 54.2

133 55 47

2 38 4-9

4100

o 0661095

7i 57 H

122 25 31

2 18 16.8

133 25 41

2 37 54-3

4000

0.0658868

72 23 6

122 II 2O

2 18 39.0

132 55 33

2 37 43-6

3900

0.0656628

72 49 o

121 57 7

2 ig I.O

132 25 24

2 37 32.8

3800

0.0654375

73 '4 56

121 42 51

2 ig 22.6

131 55 12

2 37 22.0

3700

0.0652110

73 40 53

121 28 33

2 19 44.O

131 24 58

2 37 ii.o

3600

0.0649832

74 6 52

121 14 13

2 2O 5.1

130 54 42

2 37 o.o

3500

0.0647542

74 32 53

120 59 50

2 20 25.9

130 24 23

2 36 48.8

3400

0.0645239

74 58 55

120 45 26

2 2O 46.4

129 54 3

2 36 37.6

33°°

0.0642924

75 25 °

I2O 31 O

2 21 6.6

129 23 40

2 36 26.2

3200

0.0640597

75 5' 6

120 16 31

2 21 26.5

128 53 16

2 36 14.8

3100

0.0638257

76 17 14

1 2O 2 I

2 21 46.1

•128 22 48

2 36 3-2

3000

0.0635905

76 43 24

"9 47 29

2 22 5.4

127 52 19

2 35 51-6

2900

0.063354*

77 9 36

119 32 54

2 22 24.4

127 21 48

2 35 39-9

2800

0.0631165

77 35 5°

119 18 18

2 22 43.2

126 51 14

2 35 28.1

2700

0.0628776

78 2 6

119 3 39

2 23 1.6

126 20 37

2 35 16.2

2600

0.0626375

78 28 24

118 48 59

2 23 19.7

125 49 59

2 35 4-2

2500

0.0623962

78 54 44

118 34 17

2 23 37.5

125 19 18

2 34 52-2

2400

0.0621537

79 21 6

118 19 33

2 23 55.1

124 48 35

2 34 40.0

2300

0.0619100

79 47 31

118 4 48

2 24 12.3

124 17 49

2 34 27.7

2200

0.0616652

80 13 57

117 50 2

2 24 29.2

123 47 2

2 34 '5-4

2100

0.0614192

80 40 26

"7 35 '4

2 24 45.8

123 16 12

2 34 3-°

2OOO

0.0611720

81 6 57

117 20 24

2 25 2.1

122 45 19

2 33 5°-4

1900

0.0609237

8 1 33 3°

"7 5 33

2 25 18. i

122 14 24

2 33 37-8

I800

0.0606743

82 o 5

116 50 41

2 25 33.8

121 43 27

2 33 25.1

1700

0.0604237

82 26 43

"6 35 47

2 25 49.1

121 12 28

2 33 12.3

I&OO

0.0601720 82 53 24

116 20 52

2 26 4.2

I2O 4! 26

2 32 59-5

1500

0.0599192 83 20 7

"6 5 55

2 26 18.9

I2O IO 22

2 32 46.5

1400

0-0596652 ; 83 46 52

»5 5° 57

2 26 33.4

119 39 16

2 32 33-5

1300

0.0594101 84 13 40

»5 35 57

2 26 47.5

119 8 7

2 32 20.3

I2OO

0.0591540 84 40 30

115 20 56

2 27 1.3

118 36 56

2 32 7-i

MOO

0.0588968 | 85 7 23

"5 5 54

2 27 14.8

118 5 42

2 3' S3-8

IOOO

0.0586384

85 34 18

114 50 51

2 27 28.0

117 34 26

2 3' 40-5

900

0.0583789

86 i 16

114 35 46

2 27 40.8

"7 3 7

2 31 27.0

800

0.0581183

86 28 17

114 20 40

2 27 53.4

116 31 45

2 3' "3-5

700

0.0578567

86 55 21

H4 5 34

2 28 5.6

116 o 21

2 30 59.8

6OO

0.0575940

87 22 28

113 50 27

2 28 17.5

115 28 55

2 30 46.2

500

0.0573302

87 49 38

"3 35 '8

2 28 29.2

114 57 26

2 3° 32-4

4OO

0.0570653

88 16 50

113 20 8

2 28 40.4

"4 25 54

2 30 18.5

300

0.0567994, 88 44 6

"3 4 57

2 28 51.4

"3 54 19

2 30 4.6

200

0.0565325 89 ii 25

112 49 46

2 29 2.O

113 22 42

2 29 50.6

IOO

0.0562646 i 89 38 46

iiz 34 34

2 29 12.4

112 51 3

2 29 36.5

0

°.°559956

90 6 12. o

112 19 20.6

2 29 22.4

112 19 2O.6

2 29 22.4

+ 100

0.0557257

90 33 40

112 4 7

2 29 32.1

in 47 36

2 29 8.2

20O

0.0554547

91 I 12

III 48 52

2 29 41.4

in 15 48

2 28 53.9

300

0.055:827

91 28 47

in 33 36

2 29 50.5

i 10 43 58

2 28 39.5

4OO

0.0549097

91 56 25

in 18 19

2 29 59.2

no 12 5

2 28 25.0

500

0.0546358

92 24 7

III 3 2

2 30 7.6

109 40 9

2 28 IO.5

6dO

0.0543609

92 51 52

1 10 47 44

2 30 IS.?

109 8 10

2 27 55-9

700
+800

0.0540850
0.0538082

93 «9 4i
93 47 34

HO 32 25

no 17 6

2 30 23.4
2 30 30.9

108 36 8
108 4 4

2 27 41.2
2 27 26.S

192

SECULAR VARIATIONS OF THE ELEMENTS OF

TABLE VI. — Elements of the Orl'it of Uranus.

t

<r"

«"

a"

»"

8,F/

*"

—6400

0.0480857

165° i' 47"

70° 3' 6"

o° 53' 14". 98

130° 34' 2"

o° 54' 3o".4O

6300

0.0480544

165 7 18

70 4 45

o 53 8.46

129 46 15

o 54 13.82

6200

0.0480232

165 12 48

70 6 27

o 53 i-94

128 58 12

o 53 57-49

6100

0.0479920

165 18 18

70 8 10

o 52 55-41

128 9 52

o 53 41-42

6000

0.0479608

165 23 47

70 9 S6

o 52 48.89

127 21 17

o 53 25.58

5900

0.0479297

165 29 15

70 ii 43

o 52 42.36

126 32 26

o 53 10.00

5800

0.0478987

165 34 43

7° 13 33

o 52 35.84

>25 43 '9

o 52 54.66

5700

0.0478677

165 40 10

70 15 25

o 52 29.31

'24 53 57

° 52 39-58

5600

0.0478368

165 45 36

70 17 20

o 52 22.79

124 4 19

o 52 24.75

55°°

0.0478059

165 51 2

70 19 16

o 52 16.27

123 14 26

o 52 10.18

5400

0.0477751

165 56 27

70 21 15

o 52 9.75

122 24 l8

0 5' 55-87

53°°

0.0477443

166 I 52

70 23 17

o 52 3.23

121 33 54

o 51 41.83

5200

0.0477136

166 7 16

7O 25 2O

o 51 56.71

120 43 15

o 51 28.05

5100

0.0476829

166 12 39

70 27 26

o 51 50.19

IIO 52 21

o 51 14.54

5000

0.0476523

166 18 2

70 29 35

o 51 43.68

119 I 12

o 51 1.29

4900

0.0476217

166 23 24

70 31 46

o 51 37.18

i 18 9 49

o 50 48.30

4800

0.0475912

I 66 28 46

70 34 o

o 51 30.68

117 18 ii

o 50 35-59

4700

0.0475607

166 34 7

70 36 16

o 51 24.19

116 26 19

o 50 23.16

4600

0-0475303

166 39 27

70 38 34

o 51 17.70

US 34 13

o 50 ii. oo

4500

0.0474999

1 66 44 46

70 40 56

0 51 II. 21

114 41 54

o 49 59.14

4400

0.0474696

166 50 5

70 43 19

0 5« 4-73

113 49 21

o 49 47.56

4300

0-0474394

166 55 24

70 45 46

o 50 58.24

I'2 56 35

o 49 36.28

4200

0.0474092

167 o 42

70 48 15

o 50 51.75

112 3 36

o 49 25.29

4100

0-047379'

"67 5 59

70 50 46

o 50 45.27

in 10 24

o 49 14.60

4000

0.0473490

167 ii 15

70 53 20

o 50 38.80

no 17 o

o 49 4.21

3933

0.0473190

167 16 31

70 55 57

o 50 32.34

109 23 24

o 48 54.12

3803

0.0472891

167 21 47

70 58 36

o 50 25.88

108 29 37

o 48 44-34

3703

0.0472592

167 27 I

71 i 18

o 50 14.43

107 35 38

o 48 34.87

3600

0.0472294

167 32 15

7' 4 3

o 50 13.00

106 41 28

o 48 25.71

35°o

0.0471997

167 37 29

71 6 51

o 50 6.59

i°5 47 7

o 48 16.86

3400

0.0471700

167 42 42

71 9 41

o 50 o. 18

104 52 36

o 48 8.33

33°°

0.0471405

167 47 54

7' 12 34

o 49 53-78

i°3 57 56

o 48 o. 1 1

3200

0.0471110

'6? 53 5

7' «5 30

0 49 47-39

103 3 5

o 47 52.20

3100

0.0470816

167 58 16

71 18 29

o 49 41.01

102 8 6

o 47 44.61

3000

0.0470523

168 3 27

71 21 30

0 49 34-63

101 12 58

o 47 37-33

2900

0.0470231

168 8 37

71 24 34

o 49 28. 26

100 17 42

o 47 30.38

2800

0.0469939

168 13 46

71 27 41

o 49 21.90

99 22 19

o 47 23.74

2700

0.0469648

168 18 55

71 30 51

o 49 15.56

98 26 48

o 47 17-43

2600

0.0469358

168 24 3

7' 34 3

o 49 9.23

97 31 10

o 47 11.44

2500

0.0469069

168 29 10

71 37 '8

o 49 2.91

96 35 27

0 47 5-77

2400

0.0468780

168 34 17

71 40 36

o 48 56.60

95 39 37

o 47 0.43

2300

0.0468493

I 68 39 23

7i 43 57

o 48 50.31

94 43 42

o 46 55.42

22O3

0.0468206

168 44 29

71 47 21

o 48 44.03

93 47 43

o 46 50.72

•too

0.0467921

I 68 49 34

7' So 47

o 48 37-77

92 51 39

o 46 46.35

MOO

0.0467636

'68 54 39

7' 54 16

o 48 31.52

9' 55 32

o 46 42.32

1900

0.0467352

i 68 59 43

7« 57 48

o 48 25.28

90 59 22

o 46 38.62

1800

0.0467069

169 4 47

72 i 24

o 48 19.05

90 3 9

o 46 35.24

1703

0.0466787

169 9 50

72 5 i

o 48 12.83

89 6 53

o 46 32. 19

1600

0.0466506

169 14 52

72 8 42

o 48 6.63

88 10 36

o 46 29.47

1500

0.0466227

169 19 54

72 12 26

o 48 0.45

87 14 18

o 46 27.08

1400

0.0465948

169 24 55

72 16 12

o 47 54.28

86 18 o

o 46 25.01

1300

0.0465670

169 29 56

72 20 2

o 47 48.13

85 21 41

o 46 23.27

1 200

0.0465393

169 34 56

72 23 54

o 47 41.99

84 25 25

o 46 21.84

1 100

0.0465117

169 39 56

72 27 50

o 47 35-88

83 29 10

o 46 20.75

IOOO

0.0464842

I69 44 55

72 31 48

o 47 29.79

82 32 56

o 46 19.99

900

0.0464568

169 49 54

72 35 49

o 47 23.72

81 36 45

o 46 19.55

800

0.0464295

169 54 52

72 39 54

o 47 17.66

80 40 36

o 46 19.44

700

0.0464023

169 59 49

72 44 I

o 47 11.63

79 44 29

o 46 19.65

600

0.0463752

170 4 46

72 48 II

o 47 5.61

78 48 28

o 46 20.17

500

0.0463482

«7o 9 43

72 52 24

o 46 59.61

77 52 32

O 46 21. OI

400

0.0463213

170 14 39

72 56 40

0 46 53-63

76 56 40

o 46 22.17

300

0.0462945

'7° '9 34

73 o 59

o 46 47.67

76 o 55

o 46 23.64

2OO

0.0462679

i 70 24 29

73 5 2'

o 46 41.74

75 5 '4'

o 46 25.4?

— IOO

0.0462413

170 29 24

73 9 46

o 46 35-83

74 9 4i

o 46 27.53

o

-f-IOO

0.0462149
0.0461886

'70 34 17-7
170 39 ii

73 H 13-4
73 '8 44

o 46 29.93
o 46 24.06

7j 14 '3-4
72 18 54

o 46 29.93
o 46 32.64

no

0.0461624

170 44 4

73 23 18

o 46 18.21

71 23 42

o 46 35.65

300

0.0461363

170 48 56

73 27 54

o 46 12.38

70 28 39

o 46 38.96

400

0.0461103

170 53 48

73 32 33

o 46 6.57

69 33 46

o 46 42.57

500

0.0460844

1 70 58 40

73 37 «5

o 46 0.79

68 39 i

o 46 46.47

600

0.0460587

>7> 3 3'

73 42 o

o 45 55-03

67 44 26

o 46 50.67

, l°°

0.0460330

171 8 21

73 46 48

o 45 49.30

66 50 o

o 46 55.16

4.800

0.0460074

171 13 ii

73 5« 39

0 45 43-58

65 55 44

0 46 59-93

THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 19,3

TABLE Nil.— Elements of the Orbit of Neptune.

t

eva

»"'

jra

f"J

!,*»

*,m

—6400

0.0088883

49° 26' l"

130° 29' 8"

0 46' 14". 21

149° 3'' 7"

2° 22' I2//.02

6300

0.0088924

49 26 35

130 28 47

46 14.90

149 12 59

2 21 41.70

6200

0.0088963

49 27 10

130 28 27

46 15-59

148 54 52

2 21 11.29

6100

0.0089006

49 27 45

130 28 6

46 16.28

148 36 45

2 20 40. 78

6000

0.0089047

49 28 21

130 27 45

46 16.98

148 i 8 38

2 2O 10.17

5900

0.0089088

49 28 57

13° 27 25

46 17.67

148 o 30

2 19 39-47

5800

0.0089130

49 29 33

130 27 4

46 18.36

147 42 23

2 19 8.66

57°°

0.0089172

49 30 10

130 26 44

46 19.05

147 24 16

2 18 37.76

5600

0.0089214

49 30 48

130 26 24

46 19.74

147 6 8

2 18 6.77

5500

0.0089256

49 3' 25

130 26 3

46 20.43

146 48 i

2 17 35-69

5400

0.0089299

49 32 4

13° 25 43

46 21.12

146 29 54

2 i? 4-5°

53°°

0.0089342

49 32 42

130 25 22

46 21., Si

146 ii 46

2 16 33.22

5200

0.0089385

49 33 21

I3O 25 2

46 22.51

H5 53 39

2 16 1.85

5100

0.0089428

49 34 I

130 24 42

46 23.21

145 35 31

2 15 3°-39

5000

0.0089472

49 34 41

130 24 21

46 23.91

145 17 24

2 14 58.84

4900

0.0089515

49 35 2'

130 24 I

46 24.61

144 59 16

2 14 27.2O

4800

0.0089559

49 36 2

130 23 4I

46 25.31

144 41 9

2 13 55-47

4700

0.0089603

49 36 44

I3O 23 20

46 26.01

H4 23 i

2 13 23.65

4600

0.0089647

49 37 25

130 23 o

46 26.72

144 4 53

2 12 51.75

4500

0.008969!

49 38 8

130 22 40

46 27.43

143 46 46

2 12 19.76

4400

0.0089736

49 38 50

130 22 2O

46 28.14

143 28 38

2 II 47.69

4300

0.0089780

49 39 33

I3O 22 0

46 28.85

143 10 30

2 II 15-53

4200

0.0089824

49 40 17

130 21 39

46 29.57

142 52 22

2 IO 43.30

4100

0.0089868

49 41 I

130 21 19

46 30.28

142 34 14

2 IO IO.99

4000

0.0089913

49 41 45

130 20 59

46 31.00

142 16 6

2 9 38.60

3900

0.0089957

49 42 30

130 20 39

46 31.72

141 57 58

2 9 6.12

3800

0.0090002

49 43 15

I3O 2O ig

46 32.44

14' 39 49

2 8 33.56

3700

0.0090046'

49 44 i

130 19 59

46 33.16

14! 21 40

2 8 0.92

3600

0.0090091

49 44 47

130 19 39

46 33-89

Hi 3 32

2 7 28.20

35°°

O.OO9OI35

49 45 33

130 19 19

46 34.62

140 45 23

2 6 55.40

3400

0.0090180

49 46 20

130 18 59

46 35-35

140 27 14

2 6 22.51

33°o

0.0090225

49 47 7

130 18 39

46 36.08

14° 9 5

2 5 49-54

3200

O.OO9O27O

49 47- 54

130 18 19

46 36.81

139 5° 56

2 5 16.49

3100

0.0090315

49 48 42

•3° '7 59

46 37-54

139 32 46

2 4 43-37

3000

0.0090361

49 49 3°

'3° 17 39

46 38.28

139 H 3°

2 4 10.17

2900

O.OO9O4O6

49 50 18

130 17 19

46 39.01

138 56 27

2 3 36-89

2800

0.0090452

49 5i 6

13° '7 °

46 39-75

138 38 17

2 3 3-53

2700

0.0090497

49 51 56

130 16 40

46 40.49

138 20 6

2 2 30. 10

2600

0.0090542

49 52 45

130 16 20

46 41.23

138 I 56

2 I 56.59

2500

0.0090587

49 53 35

130 16 o

46 41.97

137 43 45

2 I 23.01

2400

0.0090633

49 54 26

130 15 40

46 42.71

137 25 34

2 O 49.36

2300

0.0090678

49 55 l6

130 15 21

46 43-45

137 7 23

2 ' 0 15.63

22OO

O.OO9O724

49 56 7

130 15 i

46 44.20

136 49 12

59 41-83

2IOO

0.0090769

49 56 59

130 14 41

46 44-95

136 31 I

59 7-96

2000

O.OO9O8l5

49 57 51

130 14 21

46 45.70

136 12 49

58 34-01

1900

0.009o86l

49 58 43

130 14 I

46 46.45

135 54 37

57 59-99

1800

0.0090907

49 59 36

130 13 42

46 47.20

135 36 25

57 25.90

1700

0.0090953

5° o 3°

130 13 22

46 47-95

135 18 13

56 51-74

1600

0.0090999

50 I 23

130 13 2

46 48.71

135 o o

56 I7-51

I5OO

O.OO9IO45

50 2 17

I3O 12 42

46 49-47

134 41 47

55 43-20

1400

O.OOglOgi

50 3 12

130 12 22

46 50.23

134 23 33

55 8.84

1300

0.009II37

5° 4 7

130 12 3

46 50.99

134 5 20

54 34-4°

I2OO

0.0091184

5° 5 2

13° I' 43

46 51-75

133 47 5

53 59-91

I 100

O.OO9I23O

50 5 58

130 II 23

46 52-51

133 28 51

53 25.35

IOOO

O.OO9I276

50 6 54

13° ii 3

46 53.28

133 10 36

52 50-72

900

O.OO9I322

5° 7 5°

130 10 43

46 54.04

132 52 20

52 16.04

800

0.0091369

50 8 47

130 10 24

46 54.81

132 34 5

5i 4i-29

700

0.0091415

5° 9 45

130 10 4

46 55.5f

132 15 48

51 6.48

600

O.009I46I

5° I0 43

130 9 44

46 56.36

131 57 32

50 31.60

500

O.OO9I5O7

50 ii 41

130 9 24

46 57.13

131 39 15

49 56.66

400

0.0091554

5O 12 40

13° 9 4

46 57.91

131 20 58

49 2 i . 66

300

0.0091600

5° 13 39

130 8 45

46 58.69

131 2 41

48 46.60

200

0.0091646

5° H 38

130 8 25

46 59-47

130 44 23

48 11.47

100

0.0091692

5° '5 38

13° 8 5

47 0.25

130 26 4

47 36-29

0

0.0091739

50 16 38.6

13° 7 45-3

47 1.04

13° 7 45-3

47 l-°4

-f-ioo

0.0091785

5° i? 39

130 7 25

47 1.82

129 49 26

46 25.73

2OO

0.0091832

50 18 40

130 7 6

47 2.61

129 31 6

45 50.37

300

0.0091879

50 19 42

130 6 46

47 3-40

129 12 46

45 H-95

400

0.0091926

50 20 44

130 6 26

47 4-19

128 54 25

44 39-47

500

0.0091972

50 21 46

130 6 6

47 4.98

128 36 3

44 3-94

600

O.OO92OI9

50 22 49

130 5 46

47 5-77

128 17 41

43 28.35

7OO

0.0092066

50 23 52

130 5 27

47 6.56

127 59 18

42 52-7I

-J-8OO ; O.OO92II3

5° 24 55

13° 5 7

47 7-36

127 40 55

42 17.01

25 May, 1872.

194

SECULAR V A K 1 A T I O N S OF THE ELEMENTS OF

-

TABLE VIII. — Elements of the Orbit of the Earth.

t

e"

-

6"

f

— 8000

0.0192055

75° 23' 45"-8

13° 12' 44.3

0 8' 9. 1 14

7900

0.0191870

75 4i 40.1

12 57 32-4

7 17-243

7800

0.0191682

75 59 35-6

12 42 20. 6

6 25.364

7700

0.0191489

76 17 32-4

12 27 9.0

5 33-48o

7600

0.0191294

76 35 30-6

12 n 57-5

4 4I-59I

75oo

0.0191096

76 53 30.1

ii 56 46.2

3 49.698

7400

0.0190894

77 ii 30.9

n 4i 35-'

2 57.802

73oo

0.0190689

77 29 32.9

ii 26 24.2

2 5.904

7200

0.0190482

77 47 36.2

ii n 13-4

i 14.004

7100

0.0190272

78 5 40.7

10 56 2.8

o 22.105

7000

0.0190058

78 23 46.4

10 40 52.3

o 59 30.207

6900

0.0189841

78 41 53-3

10 25 42.1

o 58 38.312

6800

0.0189620

79 o 1.4

10 10 32.0

o 57 46.420

6700

0.0189396

79 18 10.7

9 55 22.0

o 56 54-533

6600

0.0189170

79 36 21. i

9 40 12.3

o 56 2.650

6500
6400

0.0188941
0.0188708

79 54 32.7
80 12 45.5

9 25 2.7
9 9 53-3

o 55 10-773
o 54 18.904

6300

0.0188473

80 30 58.6

8 54 44-0

o 53 27.044

6200

0.0188234

80 49 13.1

8 39 34-9

o 52 35- '94

6100

0.0187992

81 7 28.8

8 24 25.9

o 51 43-354

6000

0.0187747

81 25 45.7

8 9 17-2

o 50 51.524

5900

0.0187499

81 44 3-9

7 54 8.6

o 49 59.707

5800

0.0187248

82 2 23.3

7 39 o.i

o 49 7.902

5700

0.0186994

82 20 44.0

7 23 1-9

o 48 16.120

5600

0.0186737

82 39 5-9

7 8 43.8

o 47 24.347

55oo

0.0186477

82 57 29.1

6 53 35-8

o 46 32.590

54oo

0.0186214

8.1 15 53-7

6 38 28.0

o 45 40.852

53oo

0.0185948

83 34 19.6

6 23 20.4

o 44 49-I32

5200

0.0185678

83 52 46.7

6 8 13.0

o 43 57-431

5100
5000

0.0185405
0.0185130

84 n 15.0
84 29 44.6

5 53 5-7
5 37 58.6

o 43 5-750
o 42 14.090

4900

0.0184852

84 48 15.4

5 22 51.7

o 41 22.453

4800

0.0184570

85 6 47-5

5 7 44-9

o 40 30. 840

4700
4600

0.0184285
0.0183998

85 25 20.8
85 43 55-2

4 S2 38-3
4 37 3'-8

o 39 39-253
o 38 47.692

4500

0.0183708

86 2 30.8

4 22 25.5

o 37 56-158

4400

0.0183415

86 21 7.6

4 7 19-4

o 37 4.652

43°°

0.0183119

86 39 45.6

3 52 13-4

0 36 I3-I75

4200

0.0182820

86 58 24.8

3 37 7-6

o 35 21.729

4100

0.0182518

87 17 5-i

3 22 1.9

o 34 30.315

4000

0.0182213

87 35 46.7

3 6 56.4

o 33 38-93'

3900

0.0181905

87 54 29.6

2 5' 5'.'

0 32 47-577

3800
3700
3600

0.0181595
0.0181282
0.0180965

88 13 13.7
88 31 59.0
88 50 45.6

2 36 46.0
2 21 4I.O
2 6 36.2

0 3' 56-254
o 31 4.962
o 30 13.706

3500

0.0180645

89 9 33-4

I 51 31.6

o 29 22.487

3400

0.0180323

89 28 22.4 •

I 36 27.1

o 28 31.301

33°o

0.0179998

89 47 12.6

I 21 22.8

o 27 40.153

3200

0.0179670

90 6 4.0

I 6 18.6

o 26 49.047

3100

0-0179339

90 24 56.8

o 51 14.6

o 25 57.981

— 3000

0.0179005

90 43 51.0

o 36 10.8

o 25 6.958

THE ORBITS OF. THE EIGHT PRINCIPAL PLANETS. 195

TABLE VIII. — Elements of the Orbit of the Earth — continued.

/

e"

•".''

6"

*"

— 3000

0.0179005

90° 43' 5i".o

o° 36' io".8

o° 25' 6^.958

2900

0.0178668

91 2 46.5

0 21 7.1

o 24 15.977

2800

0.0178329

91 21 43.4

o 6 3.6

o 23 25.040

2700

0.0177987

91 40 41.6

359 51 0.2

0 22 34.149

2600

0.0177642

91 59 41.1

359 35 57-o

0 21 43-304

2500

0.0177294

92 18 42.0

359 20 53.9

0 20 52.504

2400

0.0176943

92 37 44-2

359 5 5i-o

0 20 1.748

2300

0.0176589

92 56 48.0

358 5° 48.3

o 19 11.038

22OO

0.0176233

93 15 53-3

358 35 45-7

o 18 20.378

2100

0-0175875

93 35 0.0

358 20 43.2

0 17 29.769

2000

0.0175513

93 54 8.3

358 5 41.0

o 16 39.210

1900

0.0175148

94 13 18.0

357 5° 38-9

o 15 48.703

I800

0.0174781

94 32 29.3

357 35 36-9

o 14 58.248

I7OO

0.0174411

94 51 42.0

357 20 35.1

o 14 7.844

l6oO

0.0174038

95 10 56-3

357 5 33-5

o 13 17.494

1500

0.0173662

95 30 1 1. 6

356 50 32.0

0 12 27.200

1400

0.0173284

95 49 28. i

356 35 3°-7

o ii 36.961

1300

0.0172903

96 8 45.6

356 20 29.6

o 10 46.779

1200

0.0172520

96 28 4. 1

356 5 28.6

o 9 56.657

1 1OO

0.0172134

96 47 23.9

355 5° 27-8

o 9 6.592

IOOO

0.0171745

97 6 44.9

355 35 27.2

o 8 16.586

900

0-0171353

97 26 7.6

355 20 26.7

o 7 26.642

800

0.0170959

97 45 3'-7

355 5 26.4

o 6 36.759

700

0.0170562

98 4 57-5

354 50 25.7

0 5 46.937

000

0.0170163

98 24 24.8

354 35 25-3

o 4 57.178

500

0.0169762

98 43 S3-6

354 20 25.4

o 4 7-484

400

0.0169367

99 3 24.0

354 5 25.7

o 3 I7-854

300

0.0168949

99 22 55.9

353 5° 26.1

O 2 28.290

200

0.0168539

99 42 29.4

353 35 26.6

o i 38.792

100

0.0168127

100 2 4.4

353 20 27.4

o o 49.362

oo

0.0167712

100 21 41.0

353 5 28.0

o o o.ooo

+ 100

0.0167295

loo 41 18.9

172 50 28.6

o o 49.293

2OO

0.0166875

ioi o 58.5

172 35 29.9

o i 38.516

300

0.0166452

ror 20 39.5

172 20 31.2

O 2 27.667

4OO

0.0166027

101 40 22.0

172 5 32-6

o 3 '6-747

500

0.0165599

IO2 O 6. 1

171 50 34.1

o 4 5-754

600

0.0165169

IO2 19 51.8

171 35 35-8

o 4 54-688

800

0.0164302

102 59 27.8

171 5 39-8

o 6 32.333

1200

0.0162535

104 19 0.7

170 5 48.4

o 9 46.700

1600
2OOO

0.0160731
0.0158888

105 39 0.3
106 59 24.9

169 5 59.8
168 6 13.6

o 12 59-799
o 16 11.576

2400
32OO

0.0157009
0.0153141

108 20 18.9
i" 3 35-3

167 6 29.3
165 7 6.9

o 19 21.978
o 25 38.45

4000
4800

0.0149133
0-0144993

113 49 3.6
116 36 49.1

163 7 51.8
161 8 46.0

o 31 48-79
o 37 52.70

5600

0.0140723

119 26 59.1

159 9 46.5

o 43 49.70

6400

0.0136340

122 2O I.O

157 I0 53-9

o 49 39-49

72OO
+8000

0.0131843
0.0127243

125 16 6.1
128 15 28.5

155 '2 7-3
153 13 26.4

o 55 21.67
* o 55.94

196

S E C U L A R V A II I A T I 0 N S OF THE ELEMENTS OF

TABLE IX.— Precession of the Equinoxes and Obliquity of the Ecliptic.

t

4

'J

+'

I

—8000

—112° 24' 57".5

24° 24' 5 ' "-3

—109° 55' 42.2"

24° 15' 24".8

7900

III 2 3-0

24 23 '3-°

108 34 12.4

24 15 n. 8

7800

109 39 3-5

24 2' 35-7

107 12 41.6

24 14 57-9

7700

108 15 59.0

24 '9 59-4

IO5 50 9.8

24 H 43-3

7600

106 52 49.6

24 18 24.3

104 29 37.0

24 14 28.0

7500

i°5 29 35-4

24 16 50.1

'°3 8 3.3

24 14 ii. 8

7400

104 6 16.6

24 15 17.2

101 46 28.5

24 >3 55.°

73°°

102 42 53.1

24 '3 45-5

100 24 52.6

24 13 37-4

7200

101 19 25.3

24 12 15.1

99 3 '5-7

24 13 19.0

7100

99 55 53-'

24 10 45.8

97 41 37.6

24 12 59.9

7000

98 32 16.8

24 9 18.0

96 19 58.3

24 12 4O. I

6900

97 8 36.2

24 7 5«-4

94 S8 17-9

24 12 19.6

6800

95 44 5'-7

24 6 26.3

93 36 36.5

24 II 58.4

6700

94 21 3-2

24 5 2.5

92 14 54.0

24 II 36.4

6600

92 57 n. o

24 3 40-2

90 53 10.2

24 II 13.8

6500

9« 33 IS-"

24 2 19.3

89 3' 25.3

24 10 50.4

6400

90 9 15.8

24 i o.o

88 9 39.2

24 10 26.4

6300

88 45 12.9

23 59 42.2

86 47 51.7

24 10 1.7

6200

87 21 6.9

23 58 25.9

85 26 3.2

24 9 36.4

6100

85 56 57.6

23 57 »-2

84 4 13.4

24 9 10-3

6000

»4 32 45-3

23 55 58-i

82 42 22.3

24 8 43- 6

5900

83 8 30.1

23 54 46.5

8 1 20 '30. i

24 8 10.2

5800

Si 44 12.1

23 53 36.5

79 58 36.5

24 7 48.2

5700

80 19 51.3

23 52 28.0

78 36 41.6

24 7 "9-5

5600

78 55 28.1

23 51 21.2

77 "4 45-3

24 6 50.2

5500

77 3» 2-3

23 50 16.1

75 52 47-7

24 6 20.3

5400

76 6 34.3

23 49 «2-7

74 3° 48.7

24 5 49.8

S300

74 42 4-'

23 48 10.9

73 8 48.3

24 5 "8-7

5200

73 »7 3i-9

23 47 10.8

71 46 46.5

24 4 47-0

5100

7i S2 57-8

23 46 12.4

70 24 43.3

24 4 14.6

5000

70 28 21.9

23 45 15.8

69 2 38.5

24 3 41-7

4900

69 3 44-2

23 44 20.7

67 40 32.2

24 3 8.2

4800

67 39 5-'

23 43 27.4

66 18 24.5

24 2 34.2

4700

66 14 24.3

23 42 35-6

64 56 15.0

24 i S9-6

4600

64 49 42.4

23 4' 45-6

63 34 4-2

24 i 24.5

4500

63 24 59.3

23 40 57.2

62 II 51.9

24 o 48.8

4400

62 o 15.2

23 40 10.6

60 49 38.0

24 o 12.6

43°°

60 35 30.1

«3 39 25.5

59 27 22.6

23 59 35-8

4200

59 10 44.2

23 38 42.1

58 5 5'4

23 58 58.6

4100

57 45 57-5

23 38 0.2

56 42 46.6

23 58 20.8

4000

56 21 IO.3

23 37 20.0

55 20 26.0

23 57 42.6

3900

54 56 22.5

23 36 41-3

53 58 3-7

23 57 3-8

3800

53 31 34-4

23 36 4-3

52 35 39-6

23 56 24.6

3700

52 6 46.0

23 35 29.8

51 12 13.9

23 55 44-9

3600

50 41 57.6

23 34 55-0

49 50 46.4

23 55 4-8

35°°

49 >7 9-1

23 34 22.6

48 28 17.2

23 54 24.2

34°o

47 S2 20.7

23 33 S1-8

47 5 46.1

23 53 43-2

33°°

46 27 32.4

23 33 22.4

45 43 "3-2

23 53 «-8

3200

45 2 44-5

23 32 54-5

44 20 38.4

23 52 2O.O

3100

43 37 56-8

23 32 28.0

42 58 1-5

23 5' 37-7

— 3000

— 42 13 9-7

23 32 2.9

— 41 35 22.8

23 5° 55-'

•

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 197

-

TABLE IX. — Precession of the Equinoxes and Obliquity of the Ecliptic — continued.

i

*

V

i

—3000

—42° 13' 9.7

23° 32 2.9

—41° 35 22.8

23° So 55-i

2900

40 48 23.2

23 31 39-2

40 12 42.3

23 50 12. 1

2800

39 23 37.5

23 31 16.9

38 49 59.8

23 49 28.7

2700

37 58 52.4

23 30 55-9

37 27 15.3

23 48 45.0

2600

36 34 8.3

23 30 36.1

36 4 29.0

23 48 0.9

2500

35 9 25.1

23 30 17.6

34 4i 40.7

23 47 16.5

2400

33 44 42.95

23 30 0.4

33 18 50.41

23 46 3«-7

2300

32 20 1.86

23 29 44.3

3i 55 57-95

23 45 46.5

2200

30 55 22.04

23 29 29.4

30 33 3.47

23 45 1-4

2100

29 3° 43-5°

23 29 15.6

29 10 6.89

23 44 15-5

2000

28 6 6.29

23 29 2.86

27 47 8.21

23 43 29.56

1900

26 41 30-59

23 28 51.20

26 24 7.52

23 42 43.32

1800

25 16 56.32

23 28 40.52

25 i 4.64

23 41 56.83

1700

23 52 23.62

23 28 30.80

23 37 59.64

23 41 10.09

1600

22 27 52.57

23 28 22.01

22 14 52.52

23 40 23.23

1500

21 3 23.21

23 28 14.08

20 51 43.23

23 39 36-05

1400

19 38 55.62

23 28 7.00

19 28 31.79

23 38 48.69

1300

18 14 2g.g7

23 28 0.72

18 5 18.21

23 38 1.17

1200

16 50 6.00

23 27 55-23

16 42 2.47

23 37 13-46

1 100

15 25 44.02

23 27 50-47

15 18 44.50

23 36 25.56

IOOO

14 i 24.01

23 27 46.26

13 55 24.33

23 35 37.46

90O

12 37 6.01

23 27 42.72

12 32 1-95

23 34 49-26

800

ii 12 50.06

23 27 39-75

ii 8 37.37

23 34 0.98

700

9 48 36.20

23 27 37-31

9 45 10-55

23 33 12.55

6oO

8 24 24.452

23 27 35.352

8 21 41.494

23 32 23.960

500

7 o 14.858

23 27 33-823

6 58 10.204

23 3' 35.302

4OO

5 36 7-442

23 27 32.680

5 34 36-670

23 30 46.562

300

4 12 2.229

23 27 31.874

4 ii 0.886

23 29 57.750

2OO
100

2 47 59.240
— i 23 58.492

23 27 3«-357
23 27 31-082

2 47 22.850
—i 23 42.555

33 29 8.877
33 28 19.956

O

o o o.ooo

23 27 31.000

o o oo.ooo

33 27 31.000

+ 100

+ I 23 56.225

23 27 31.063

+ i 23 44.818

33 26 42.020

2OO

2 47 50-175

23 27 31-223

2 47 31.903

23 25 53.027

300

4 ii 41.846

23 27 31.432

4 Ii 21.255

23 25 4.034

400

5 35 31-238

23 27 31.640

5 35 >2.875

23 24 15.054

500

6 59 18.351

23 27 31.800

6 59 6.765

23 23 26.097

600

8 23 3.192

23 27 31.863

8 23 2.924

23 23 37.175

800

II IO 26.IO

23 27 31.51

ii ii 2.05

23 20 59.49

1200

16 44 45.24

23 27 27.53

1 6 47 27.49

23 17 44.78

1600

22 18 30.18

23 27 16.70

22 24 29.04

23 14 32.43

2000

27 51 43.22

23 26 56.1

28 2 6.45

23 II 22.40

240O

33 24 27.47

23 26 23.0

33 40 19-39

23 8 15.7

3200

44 28 45.5

23 24 29. i

44 58 29.9

23 2 14.7

4000

55 32 2.9

23 21 17.2

56 18 55.9

22 56 34.9

4800
5600

66 35 7-2
77 38 52.4

23 16 33-7
23 10 10.4

67 4' 31-2
79 6 6.5

22 51 21.2
22 46 38.8

6400

88 44 16.7

23 2 4.8

90 32 30.1

22 42 32.0

720O
+ 8000

99 52 19.0
+ IH 3 56.2

22 52 2O.6

22 41 7-7

IO2 0 25.8

+ "3 29 33-3

22 39 4.8

22 36 21.0

-

198

SECULAR VARIATIONS OF THE ELEMENTS OF

•

TABLE X. — For Precession in Right Ascension and Declination.

•

i

z'

z+z'

e

3

— 8000

—S30 o/ 2". 8

—54° 34' 37"- 2

—107° 34' 40».o

—39° 24' 5 ' "-5

+2° 43' 48".27

7900

52 >7 37-9

53 5« 58-'

106 9 36.0

39 3 3°-2

2 42 13.99

7800

5' 35 «7-9

53 9 «9-7

104 44 37-6

38 41 50.0

2 40 35-30

7700

5° 53 3-°

52 26 42.0

103 19 45-o

38 19 51-0

2 38 52-32

7600

50 10 53.0

5' 44 5-'

101 54 58.1

37 57 33-4

2 37 5-'7

7500

49 28 48.0

51 i 29.0

loo 30 17.0

37 34 57-4

2 35 '3-97

7400

48 46 48.0

50 18 53.9

99 5 41-9

37 "2 3-4

2 33 18.85

73°°

48 4 53.0

49 36 20. o

97 4i '3-°

36 48 51.6

2 3: 19-92

7200

47 23 3-1

48 53 47-4

96 16 50.5

36 25 22.2

2 29 17.35

7100

46 41 18.3

48 II 16.0

94 52 34-3

36 ' 35-6

2 27 11.25

7000

45 59 38-6

47 28 46. i

93 28 24.7

35 37 3>-8

2 25 1.74

6900

45 '» 3-8

46 46 17.6

92 4 21.4

35 '3 >'-2

2 22 48.98

6800

44 36 34 '

46 3 50.9

90 40 25.0

34 48 34-i

2 20 33-11

6700

43 55 9-3

45 21 25.8

89 16 35.1

34 23 40.6

2 18 14.29

6600

43 '3 49-5

44 39 2.5

87 52 52-0

33 58 3'-°

2 15 52-63

6500

42 32 34-8

43 56 41- '

86 29 15.9

33 33 5-7

2 13 28.32

6400

41 51 25.0

43 '4 21-9

85 5 46.9

33 7 24.9

2 II 1.44

6300

41 10 20.0

42 32 4.7

83 42 24.7

32 41 28.8

2 8 32.18

6200

40 29 19.9

41 49 49.7

82 19 9.6

32 15 17.7

2 6 0.77

6100

39 48 24.7

41 7 37-°

80 56 1.7

3' 48 5"-7

2 3 27.23

6000

39 7 34-3

40 25 26-7

79 33 i-o

31 22 II.3

2 0 51.77

5900

38 26 48.9

39 43 "8-8

78 J° 7-7

3° 55 l6-6

58 I4-56

5800

37 46 8.2

39 i '3-4

76 47 21.6

3° 28 7.9

55 35-72

5700

37 5 32-2

38 19 10-4

75 24 42.6

30 o 45.4

52 55-44

5600

36 25 0.8

37 37 'o-3

74 2 ii. i

29 33 9-4

5° 13-84

55°°

35 44 34-o

36 ss 12-8

72 39 46.8

29 5 20. o

47 3'-'3

5400

35 4 n-7

36 13 18-3

71 17 30.0

28 37 17.8

44 47-44

S300

34 23 539

35 3» 26-8

69 55 20.7

28 9 2.9

42 2.93

5200

33 43 40-5

34 49 38-4

68 33 18.9

27 40 35-6

39 17-77

5100

33 3 3i-5

34 7 52-9

67 ii 24.4

27 ii 56.0

36 32.12

5000

32 23 26.8

33 26 io-7

65 49 37-5

26 43 4-6

33 46-13

4900

3« 43 26.4

32 44 3i-8

64 27 58.2

26 14 14

30 59.98

4800

3« 3 3°- '

32 2 56.0

63 6 26. i

25 44 46.9

28 13.80

4700

30 23 37.9

31 21 23.6

61 45 1-5

25 15 21. 1

25 27.78

4600

29 43 49-7

3° 39 54-5

60 23 44.2

24 45 44-4

22 42.05

4500

29 4 5.6

29 58 29. i

59 2 34.7

24 >5 57-0

19 56.77

4400

28 24 25.2

29 17 6.9

57 4« 32."

23 45 59-3

17 12. II

43°°

27 44 48.8

28 35 48.5

56 20 37.3

23 >5 5'-4

14 28.22

4200

27 5 16.1

27 54 33-6

54 59 49-7

22 45 33-7

II 45.26

4100

26 25 47.2

27 13 22.4

53 39 9-6

22 15 6.2

9 3.36

4000

25 46 21.8

26 32 14.8

52 18 36.6

21 44 29.4

6 22.69

3900

25 7 °-i

25 51 10.7

50 58 10.8

21 13 43-5

3 43-39

3800

24 27 41.6

25 10 10.7

49 37 52.3

20 42 48. 7

i 5.62

37°°

23 48 26.2

24 29 14.7

48 17 40.9

20 II 45.3

o 58 29.51

3600

23 9 14.0

23 48 22.5

46 57 36.5

'9 4° 33-5

o 55 55-21

35«>

22 30 5.2

23 7 34-'

45 37 39-3

19 9 13.6

o 53 22.87

3400

21 50 59.3

22 26 49.7

44 '7 49-0

'8 37 45-9

o 50 52.63

33«>

21 II 56.5

21 46 g.I

42 58 5-6

18 6 10.5

o 48 24.62

3*°°

20 32 56.4

21 5 32-5

41 38 28.9

17 34 27.8

o 45 58.99

3100

19 53 59- 1

20 24 59.8

40 18 58.9

17 2 38.0

o 43 35-87

—3000

—19 15 4.4

—19 44 31.1

—38 59 35-5

— 16 30 41.3

+o 41 >5-38

THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 199

TABLE X. — For Precession in Right Ascension and Declination — continued.

t

z

2'

•+*'

e

&

— 3000

—19° IS 4-4

—19° 44 31.1

"I

-38° 59' 35"- 5

— 16° 30' 4i".3

4-0° 4i' 15. "38

2900

18 36 12.5

«9 4 6-5

37 40 19.0

15 58 38.1

o 38 57.66

2800

17 57 22.9

18 23 45.9

36 21 8.8

15 26 28.5

o 36 42.82

2700

17 18 35.8

17 43 29.4

35 2 5.2

14 54 12.8

o 34 30-99

2600

16 39 50.8

17 3 16.8

33 43 7-6

14 21 51.3

o 32 22.29

2500

16 i 8.2

16 23 8.2

32 24 16.4

13 49 24.2

o 30 16.84

2400

15 22 27.5

15 43 3-7

31 5 3'-2

13 16 51.7

o 28 14.77

2300

14 43 49.0

'5 3 3-o

29 46 52.0

12 44 14.2

o 26 16.14

2200

14 5 12.2

14 23 6.4

28 28 18.6

12 II 31.8

o 24 21.04

2IOO

13 26 37.1

13 43 "3-9

27 9 5i-o

II 38 44.8

0 22 29.62

2000

12 48 3.7

'3 3 25.3

25 51 29.0

11 5 53-5

o 20 42.00

igOO

12 9 31.9

12 23 40.6

24 33 12.5

10 32 58.1

o 18 58.24

I800

II 31 1.5

ii 43 59.8

23 15 «-3

9 59 58.8

o 17 18.45

1700

10 52 32.5

II 4 22.9

21 56 55.4

9 26 56.0

o 15 42.70

I60O

10 14 4.6

10 24 49.7

20 38 54.3

8 53 49-8

o 14 ii. 08

I5OO

9 35 38-2

9 45 20-3

19 20 58.5

8 20 40.5

o 12 43.68

1400

8 57 12.6

9 5 54-6

18 3 7-2

7 47 28.4

o ii 20.55

I30O

8 18 47.8

8 26 33.0

16 45 20.5

7 14 13-7

o 10 1.77

I2OO

7 40 23.7

7 47 '4-9

15 27 38.6

6 40 56.6

o 8 47-43

I 100

7 2 0.5

7 8 0.4

14 10 0.9

6 7 37-5

o 7 37-59

IOOO

6 23 37.8

6 28 49.4

12 52 27.2

5 34 l6-5

o 6 32.30

900

5 45 iS-8

5 49 41. 9

" 34 57-7

5 o 53.9

o 5 31.61

800

5 6 54-0

5 10 37-8

10 17 31.8

4 27 3°-°

o 4 35.58

700

4 28 32.5

4 3i 37-3

9 o 9.8

3 54 4-9

o 3 44.26

600

3 50 11.032

3 52 39-776

7 42 50.808

3 20 39.064

0 2 57.696

S00

3 «« 49-738

3 *3 45-575

6 25 35.313

2 47 12.593

O 2 15.920

40O

2 33 28.307

2 34 54-553

5 8 22.860

2 13 45.771

o i 38.971

1 3°0

i 55 6-727

i 56 6.585

3 5« J3-3'2

i 40 18.850

o I 6.879

200

i i 6 44.874

i 17 21.582

2 34 6.456

i 6 52.081

o o 39.673

100

— o 38 22.631

—o 38 39-455

— i 17 2.086

— o 33 25.715

+0 o 17.373

0

o o o.ooo

0 0 0.000

o o o.ooo

O 0 O.OOO

o o o.ooo

4-100

+o 38 23.482

+0 38 36.527

4- i 17 0.009

+o 33 24.811

—o o 12.433

200

i 16 47.594

I 17 10.547

2 33 58-141

i 6 48.471

— o o 19.917

300

i 55 12.512

I 55 42-112

3 50 54.624

I 40 10.724

— o o 22.444

400

2 33 38-37°

2 34 "-277

5 7 49-647

2 13 31.327

— o o 20.013

500

3 I2 5-23

3 12 38-1°

6 24 43.33

2 46 50.04

— o o 12.626

600

3 5° 33-13

3 5i 2.64

7 4i 35-77

3 20 6.62

— o o 0.292

800

5 7 33-36

5 7 46.46

10 15 19.82

4 26 32.15

4-0 o 39.178

I20O

7 4' 52-11

7 40 52.87

15 22 44.99

6 38 46.37

0 2 56.756

1600

10 16 41.14

10 13 38-94

20 30 20.08

8 49 57-76

o 6 30.865

2000

12 52 7.6

12 46 14.2

25 38 21.8

10 59 50.14

o ii 18.674

2400

15 28 17.5

15 18 48.0

30 47 5-5

13 8 7.12

o 17 16.358

3200

20 43 12.7

20 24 34.2

41 7 46.9

17 18 49.10

o 32 21.726

4000

26 2 13.6

25 32 26.2

51 34 39-7

21 19 51.28

o 50 59-337

4800

31 26 0.6

30 43 56.8

62 9 57.4

25 9 o.oo

I 12 10.125

5600

36 55 6-4

36 o 38.9

72 55 45-3

28 44 0.89

I 34 44-92

6400

42 29 53.8

41 24 i.o

83 53 54-8

32 2 40.02

i 57 27.37

7200

+8000

48 10 33.3
53 57 1-6

46 55 21.8
52 35 46.6

95 5 55-'
4-106 32 48.1

35 2 45-43
+37 42 10.22

2 18 55.87
4-2 37 46.67

UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY

Return to desk from which borrowed.
Thb book is DUE on the last date stamped below.

7.JuI'52JPA
JUN2 8 "•

4Jan'56R£Y

ro LD

APR 5 1961

p»=:c'D v
MAY 1 1 1962 i c

LD 21-95«-ll,'50(2877sl6)

UP SE

MAY 6 1

LD

DEC 1 2 1967

RECEIVED

DEC 5 '67

L.OA/N

w?

U C BERKELEY LIBRARIES

THE UNIVERSITY OF CALIFORNIA LIBRARY,-,