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MECANIQUE CELESTE.
MECANIQUE CELESTE.
BY THE
MARQUIS DE LA PLACE,
PEER OF FRANCE; GRAND CROSS OF THE LEGION OF HONOR ; MEMBER OF THE FRENCH ACADEMY, OF THE ACADEMY
OF SCIENCES OF PARIS, OF THE BOARD OF LONGITUDE OF FRANCE, OF THE ROYAL SOCIETIES OF
LONOON AND GOTTINGEN, OF THE ACADEMIES OF SCIENCES OF RUSSIA, DENMARK.
SWEDEN, PRUSSIA, HOLLAND, AND ITALY; MEMBER OF THE
AMERICAN ACADEMY OF ARTS AND SCIENCES; ETC.
TRANSLATED, WITH A COMMENTARY,
NATHANIEL BOWDITCH, LL. D.
TELLOW OF THE ROYAL SOCIETIES OF LONDON, EDINBURGH, AND DUBLIN; OF THE ASTRONOMICAL SOCIETY
OF LONDON j OF THE PHILOSOPHICAL SOCIETY HELD AT PHILADELPHIA; OF THE
AMERICAN ACADEMY OF ARTS AND SCIENCES J ETC.
VOLUME III.
BOSTON :
FROM THE PRESS OF ISAAC R. BUTTS ;
MILLIARD, GRAY, LITTLE, AND WILKINS, PUBLISHERS.
M DCCC XXXIV.
/i^i
/
fâb"^
Entered, according to Act of Congress, in the year 1829,
By Nathaniel Bowditch,
in the Clerk's Office of the District Court of Massachusetts.
TO
BONAPARTE
MEMBER OF THE NATIONAL INSTITUTE.
Citizen First Consul,
You have permitted me to dedicate this work to you.
It is gratifying and honorable to me to present it to the Hero, the
Pacificator of Europe,* to whom France owes her prosperity, her greatness,
and the most brilliant epoch of her glory ; to the enlightened Protector
of the Sciences, who, himself distinguished in them, perceives, in their
cultivation, the source of the most noble enjoyment, and, in their
progress, the perfection of all useful arts and social institutions.
May this work, consecrated to the most sublime of the natural sciences,
be a durable monument of the gratitude inspired in those who cultivate
them, by your kindness, and by the rewards of the government.
Of all the truths which this work contains, the expression of this
sentiment will ever be the most precious to me.
Salutation and Respect,
LA PLACE.
[* This volume was published, by La Place, in 1802, soon after the peace of Amiens.]
VOL. III. B
ADVERTISEMENT.
This volume contains the numerical values of the secular and periodical
inequalities of the motions of the planets and moon ; the numbers, given
in the original work, having been reduced from centesimal to sexagesimal
seconds, to render them more convenient for reference. The Appendix
contains many important formulas and tables, which are useful to
astronomers in computing the motions of the planets and comets. Some of
these tables are new, and the others have been varied in their forms, to
render them more simple in their uses and applications : none of them have
heretofore been published in this country. Several of the formulas have
been introduced into the calculations of modern astronomy, since the
commencement of the first part of the original work. The portrait of
the author, accompanying this volume, was obtained in France, and is an
impression from the original plate, which was engraved under his direction,
for the Système du Monde. The fourth volume of the work will be put
to press in the course of a few weeks.
PREFACE.
We have given, in the first part of this work, the general principles of
the equilibrium and motion of bodies. The application of these principles
to the motions of the heavenly bodies, has conducted us, by geometrical
reasoning, without any hypothesis, to the law of universal attraction ; the
action of gravity, and the motions of projectiles on the surface of the earth,
being particular cases of this law. We have then taken into consideration,
a system of bodies subjected to this great law of nature ; and have obtained,
by a singular analysis, the general expressions of their motions, of their
figures, and of the oscillations of the fluids which cover them. From these
expressions, we have deduced all the known phenomena of the flow and ebb
of the tide ; the variations of the degrees, and of the force of gravity at the
surface of the earth ; the precession of the equinoxes ; the libration of the
moon ; and the figure and rotation of Saturn's Rings. -We have also pointed
out the cause, why these rings remain, permanently, in the plane of the
equator of Saturn. Moreover, we have deduced, from the same theory of
gravity, the principal equations of the motions of the planets ; particularly
those of Jupiter and Saturn, whose great inequalities have a period of above
nine hundred years. The inequalities in the motions of Jupiter and Saturn,
presented, at first, to astronomers, nothing but anomalies, whose laws and
causes were unknown; and, for a long time, these irregularities appeared to
be inconsistent with the theory of gravity ; but a more thorough examination
has shown, that they can be deduced from it ; and now, these motions are
VOL. III. c
PREFACE.
one of the most striking proofs of the truth of this theory. We have
developed the secular variations of the elements of the planetary system,
which do not return to the same state till after the lapse of many centuries.
In the midst of all these changes we have discovered the constancy of the
mean motions, and of the mean distances of the bodies of this system ;
which nature seems to have arranged, at its origin, for an eternal duration,
upon the same principles as those which prevail, so admirably, upon the
earth, for the preservation of individuals, and for the perpetuity of the
species. From the single circumstance, that the motions are all in the
same direction, and in planes but little inclined to each other, it follows,
that the orbits of the planets and satellites must always be nearly circular,
and but little inclined to each other. Thus, the variations of the obliquity
of the ecliptic, which are always included within narrow limits, will never
produce an eternal spring upon the earth. We have proved that the attraction
of the terrestrial spheroid, by incessantly drawing towards its centre
the hemisphere of the moon, which is directed towards the earth, transfers
to the rotatory motion of this satellite, the great secular variations of its
motion of revolution ; and, by this means, keeps always from our view, the
other hemisphere. Lastly, we have demonstrated, in the motions of the
three first satellites of Jupiter, the following remarkable law, namely,
that, in consequence of their mutual attractions, the mean longitude of the
first satellite, seen from the centre of Jupiter, minus three times that of the
second satellite, plus twice that of the third satellite, is alivays exactly equal
to two right angles ; so that they cannot all be eclipsed at the same time.
It remains now to consider particularly the perturbations of the motions of
the planets and comets about the sun ; of the moon about the earth ; and
of the satellites about their primary planets. This is the object of the
second part of this work, which is particularly devoted to the improvement
of astronomical tables.
PREFACE. xi
The tables have followed the progress of the science, which serves as
their basis ; and this progress was, at first, extremely slow. During a very
long time, the apparent motions only of the planets were observed. This
interval, which commenced in the most remote antiquity, may be considered
as the infancy of Astronomy. It comprises the labors of Hipparchus and
Ptolemy ; also, those of the Indians, the Arabs, and the Persians. The
system of Ptolemy, which they successively adopted, is, in fact, nothing
more than a method of representing the apparent motions ; and, on this
account, it was useful to science. Such is the weakness of the human
mind, that it often requires the aid of a theory, to connect together
a series of observations. If we restrict the theory to this use, and
take care not to attribute to it a reality which it does not possess, and
afterwards frequently rectify it, by new observations, we may finally discover
the true cause, or, at least, the laws of the phenomena. The history of
Philosophy affords us more than one example, of the advantages which may
be derived from an assumed theory ; and, of the errors to Avhich we are
exposed, in considering it to be the true representation of nature. About
the middle of the sixteenth century, Copernicus discovered, that the
apparent motions of the heavenly bodies indicated a real motion of the
earth about the sun, with a rotatory motion about its own axis : by this
means, he showed to us the universe in a new point of view, and completely
changed the face of Astronomy. A remarkable concurrence of discoveries
will forever render memorable, in the history of science, the century
immediately following this discovery ; a period which is also illustrious, by
many master-pieces of literature and the fine arts. Kepler discovered the
laws of the elliptical motion of the planets ; the telescope, which was
invented by the most fortunate accident, and was immediately improved
by Galileo, enabled him to see, in the heavens, new inequalities and new
worlds. The application of the pendulum to clocks, by Huygens, and that
xii PREFACE.
of telescopes to the astronomical quadrant, gave more accurate measures
of angles and times, and thus rendered sensible the least inequalities in the
celestial motions. At the same time that observations presented to the
human mind new phenomena, it created, to explain them, and to submit
them to calculation, new instruments of thought. Napier invented
logarithms : the analysis of curves, and the science of dynamics, were
formed I)y the hands of Descartes and Galileo : Newton discovered the
differential calculus, decomposed a ray of light, and penetrated into the
general principle of gravity. In the century which has just passed, the
successors of this great man have finished the superstructure, of which he
laid the foundation. They have improved the analysis of infinitely small
quantities, and have invented the calculus of partial differences, both infinitely
small and finite : and have reduced the whole science of mechanics to
formulas. In applying these discoveries to the law of gravity, they have
deduced from it all the celestial phenomena ; and have given to the theories
and to astronomical tables an unexpected degree of accuracy; which is to
be attributed, in a great measure, to the labors of French mathematicians,
and to the prizes proposed by the Academy of Sciences. To these
discoveries in the last century, we must add those of Bradley, on
the aberration of the stars, and on the nutation of the earth's axis : the
numerous measures of the degrees of the meridian, and of the lengths of
the pendulum ; of which operations, the first example was given by France,
in sending academicians to the north, to the equator, and to the southern
hemisphere, to observe the lengths of these degrees, and the intensity of
gravity : the measure of the arc of the meridian, comprised between
Dunkirk and Barcelona ; which has been determined by very accurate
observation, and is used as the basis of the most simple and natural
system of measures : the numerous voyages of discovery, undertaken to
explore the different parts of the globe, and to observe the transits of
PREFACE. xiii
Venus over the sun's disc ; by which means, the exact determination of
the dimensions of the sokir system has been obtained, as the fruit of
these voyages : the discoveries, by Herschel, of the planet Uranus, its
satellites, and two new satellites of Saturn : finally, if we add to all these
discoveries, the admirable invention of the instrument of reflexion, so useful
at sea ; that of the achromatic telescope ; also the repeating circle, and
chronometer ; we must be satisfied, that the last century, considered
with respect to the progress of the human mind, is worthy of that
which preceded it. The century we have now entered upon, commenced
under the most favorable auspices for Astronomy. Its first day was
remarkable, by the discovery of the planet Ceres ; followed, almost
immediately afterwards, by that of the planet Pallas, having nearly the
same mean distance from the sun. The proximity of Jupiter to these two
extremely small bodies ; the greatness of the excentricities and of the
inclinations of their mutually intersecting orbits, must produce, in their
motions, considerable inequalities, which will throw new light on the
theory of the celestial attractions, and must give rise to farther improvements
in Astronomy.
It is chiefly in the application of analysis to the system of the world,
that we perceive the power of this wonderful instrument ; without which,
it would have been impossible to have discovered a mechanism which is
so complicated in its effects, while it is so simple in its cause. The
mathematician now includes in his formulas, the whole of the planetary
system, and its successive variations ; he looks back, in imagination, to the
several states, which the system has passed through, in the most remote
ages ; and foretells what time will hereafter make known to observers.
He sees this sublime spectacle, whose period includes several millions of
years, repeated in a few centuries, in the system of the satellites of
VOL. HI. D
XIV PREFACE.
Jupiter, by means of the rapidity of their revolutions ; which produce
remarkable phenomena, similar to those which had been suspected, by
astronomers, in the planetary motions ; but had not been determined,
because they were either too complex, or too slow, for an accurate
determination of their laws. The tlieory of gravity, which, by so many
applications, has become a means of discovery, as certain as by observation
itself, has made known to him several new inequalities, in the motions of the
heavenly bodies, and enabled him to predict the return of the comet of 1 759,
whose revolutions are rendered very unequal, by the attractions of Jupiter
and Saturn. He has been enabled, by this means, to deduce, from
observation, as from a rich mine, a great number of important and delicate
elements, which, without the aid of analysis, would have been forever
hidden from his view: such as the relative values of the masses of the
sun, the planets and satellites, determined by the revolutions of these bodies,
and by the development of their periodical and secular inequalities :
the velocity of light, and the ellipticity of Jupiter ; which are given,
by the eclipses of its satellites, with greater accuracy, than by direct
observation: the rotation and oblateness of Uranus and Saturn; deduced
from the consideration, that the different bodies which revolve about
those two planets, are in the same plane, respectively : the parallaxes
of the sun and moon : and, also, the figure of the earth, deduced from
some lunar inequalities : for, we shall see hereafter, that the moon, by
its motion, discloses to modern astronomy, the small ellipticity of the
terrestrial spheroid, whose roundness was made known to the first observers
by the eclipses of that luminary. Lastly, by a fortunate combination of
analysis with observation, that body, which seems to have been given to
the earth, to enlighten it, during the night, becomes also the most sure
guide of the navigator ; who is protected by it from the dangers, to
\vhich he was for a long time exposed, by the errors of his reckoning.
PREFACE. XV
The perfection of the theory, and of the lunar tables, to which he is
indebted for this important object, and for that of determining, with
accuracy, the position of the places he falls in with, is the fruit of the
labors of mathematicians and astronomers, during the last fifty years:
it unites all that can give value to a discovery ; the importance and
usefulness of the object, its various applications, and the merit of the
dififlculty which is overcome. It is thus, that the most abstract theories,
diffused by numerous applications to nature and to the arts, have become
inexhaustible sources of comfort and enjoyment, even to those who are
wholly ignorant of the nature of these theories.
CONTENTS OF THE THIRD VOLUME.
PARTICULAR THEORIES OP THE MOTIONS OF THE HEAVENLY BODIES.
SIXTH BOOK.
THEORY OF THE PLANETARY MOTIONS.
Object of this theory 1
CHAPTER I. FORMULAS FOR THE INEaUALITIES OF THE MOTIONS OP THE PLANETS, WHICH
DEPEND ON THE SaUARES AND HIGHER POWERS OF THE EXCENTRICITIES AND INCLINATIONS OF
THE ORBITS 4
ON TUE INEaUALITIES WHICH DEPEND UPON THE SaUARES AND PRODUCTS OP THE
EXCENTRICITIES AND INCLINATIONS 4
Form of the terms which produce them [.3703,3704]. Influence of the ratio of the mean
motions upon these terms, by reason of the small divisors, which are introduced by the
integrations [3712]. Preparations of the diflerential equations for the different cases of these
inequalities which occur in the solar system §L2
Considerations, by which we may distinguish the most important of these inequalities
[3732-3735] §3
Development of the terms, which result in the expressions of the radius vector, of the longitude,
and of the latitude of the disturbed planet [3736— 3800] §4,5,0
OX THE INEaUALITIES DEPENDING ON THE CUBES AND PRODUCTS OF THREE DIMENSIONS OF THE
EXCENTRICITIES AND INCLINATIONS OF THE ORBITS, AND ON THEIE HIGHER POWERS 45
Form of the terms which produce them [3807—3807'] § 7
Examination of the cases where they become sensible. They depend on the circumstance, that
the ratios of the mean motions are nearly commensurable. Application of these principles to
the theory of Jupiter and Saturn, in terras of the third degree [3828, &c.] § 8
Inequalities depending on terms of the fifth degree [38.56']. They are sensible in the theory of
Jupiter and Saturn. Calculation of them for these planets [3860, «Sic] § 9
VOL. MI. E
XX CONTENTS OF THE THIRD VOLUME.
longitude and the radius vector of the earth [4300', 4304]. The planets which produce them,
are Venus, Mars, Jupiter and Saturn.
Inequalities which are independent of the excentricities [4305,4306].
Inequalities depending on the first power of the excentricities [4307, 4308].
Inequalities depending on tlie second dimension of the excentricities and inclinations of the
orbits [4309].
Inequalities depending on the third dimension of the same quantities [4311].
Inequalities of the motion of the earth in latitude [4312]. They are produced by the action of
Venus and Jupiter ^ 2<j
Inequalities of the motion of the Earth, produced by the action of the Moon [4324,4326]. §30
On the secular variations of the earth's orbit, of the equator, and of the length of the year
[4329", &c.]. The action of the sun and moon has a considerable influence on these values.
Determination of the epoch, when the greater axis of the earth's orbit coincided with the line
of the equinoxes [4363"], and when these two lines were perpendicular to each other
[4367'"] §31
CHAPTER XI. THEORY OF MAES 26S
Examination of the limit to which the approximations must be carried, in the valuation of the
radius vector [4371, &c.]. Numerical values of the sensible inequalities which affect the
longitude and radius vector. The planets which produce them, are Venus, the Earth, Jupiter
and Saturn.
Inequalities which are independent of the excentricities [4.373, 4374].
Inequalities depending on the first power of the excentricities [4375, 4376].
Inequalities depending on the second dimension of the excentricities and inclinations of the
orbits [4377—4380].
The inequalities in latitude are hardly sensible [4384]. The greatest of them arises from the
action of Jupiter §32
CHAPTER Xll. THEORY OF JUPITER 275
Examination of the limit to which the approximations must be carried, in the valuation of the
radius vector [4385, &c.]. Numerical values of the sensible inequalities afiecting the
longitude and the radius vector. The planets which produce these inequalities, are the Earth,
Saturn, and Uranus, but chiefly Saturn.
Inequalities which are independent of the e.xcentricities [4388,4389].
Inequalities depending on the first power of the excentricities [4392,4393]. They are so large
as to render it necessary to notice the variation of their coefiîcients.
Inequalities depending on the squares and products of the excentricities and inclinations
[4.394 — 4.397], They are produced only by the action of Saturn.
CONTENTS OF THE THIRD VOLUME. Xxi
Inequalities depending on the third and fifth dimensions of the excentricities and inclinations ;
and also on the square of the disturbing force [4401, «fee.]. These last terms, which depend
on the inequalities of a very long period, have considerable influence on the secular variations
of the elliptical elements.
Great inequality of the mean motions [4434]. It is produced by the action of Saturn. . §33
Inequalities in latitude [4457]. They are produced by the action of Saturn §34
CHAPTER XIII. THEOUV OF SATURi\ 299
Examination of the degree to which the approximations must be carried in the valuation of the
radius vector [4460, &c.]. Numerical valuesof the sensible inequalities affecting the longitude
and radius vector. The planets which produce them are Jupiter and Uranus.
Inequalities which are independent of the excentricities [4463,446].
Inequalities depending on the first power of the excentricities [4466, 4467].
Inequalities depending on the squares and products of the excentricities and inclinations
[4463—4471].
Inequalities depending on the third and fifth dimensions of the excentricities and inclinations,
and also on the square of the disturbing force [4472', &c]. Great inequality of Saturn. It is
the reaction of that of Jupiter § 35
Inequalities in latitude [4511]. They are produced by the action of Jupiter and Uranus. . §-36
CHAPTER XIV. THEORY OF UEANUS 314
Examination of the degree to which the approximations must be carried, in the valuation of the
radius vector [4521, &c.]. Numerical values of the sensible inequalities affecting the
longitude and radius vector. They are produced by the action of Jupiter and Saturn.
Inequalities which are independent of the excentricities [4523, 4524].
Inequalities depending on the first power of the excentricities [4525, 4526].
Inequalities depending on the second dimension of the excentricities and inclinations
[4.527—4529].
Inequalities depending on the third dimension of the excentricities and inclinations [4530].
There is only one of them produced by the action of Saturn § 37
Inequalities in latitude [4531]. They are produced by the action of Jupiter and Saturn. § 38
CHAPTER XV. O.N SOME EaUATIONS OF CONDITION, BETWEEN THE INEaUALlTlES OF THE PLANETS,
WHICH MAY BE USED IN VEEIFn.NG THEIR NUMERICAL VALCES §39—43 318
CHAPTER XVI. ON THE MASSES OF THE PLANETS AND MOON 333
VOL. III. F
XXII
CONTENTS OF THE THIRD VOLUME.
Reflections on the values given to those masses in § 21. New determination of those of Venus
and Mars [4G05, 4608]. Discussion of that of the Moon, by the comparison of several
phenomena which can determine it [4619 — 4637], such as the observation of the tides, the
lunar equation in the tables of the Sun, the nutation of the Earth's axis, and the Moon's
parallax. From these examinations, it appears, that this mass is rather less than that which is
deduced from the tides observed at Brest [4037] §44
*
CHAPTER XVII. ON THE FORMATION OF ASTRONOMICAL TABLES, AND ON THE INVARIABLE PLANE
OF THE PLANETARY SVSTEM §45,46 341
CHAPTER XVIII. ON THE ACTION OF THE FIXED STARS UPON THF PLANETARY SYSTEM. . . . 343
The great distance of these bodies renders their action insensible [4673]. Reflections on the
comparison of the preceding formulas with observations [4687, &c.] §47
SEVENTH BOOK.
THEORY OF THE MOON.
Explanation of this theory ; its particular difficulties [4692, &c.]. Considerations that must influence
us in the approximations. How we may deduce from this theory, several important elements
of the system of the world [4702, &c.], and among others, the oblateness of the Earlh [4709],
which is thus obtained with greater accuracy than by direct observations 356
CHAPTER I. INTEGRATION OF THE DIFFERENTIAL EaUATIONS OF THE MOON'S MOTION 2QQ
Difierential equations of this motion given in § 15 of the second book [4753 — 4756]. Method
of noticing in the calculation, the non-sphericity of the Moon and Earth [4773]. ... § 1
Development of the quantities which occur in the differential equations, supposing these two
bodies to be spherical [4780, &c.] §2
The ecliptic, in its secular motion, carries with it the moon's orbit, so that the mean inclination of
this orbit to the ecliptic, remains always the same [4803]. This circumstance, indicated by
analysis, simplifies the calculations, because it permits us to take the ecliptic for the fixed plane
of projection [4804] §3
Investigation of the elliptical part of the motions of the Moon and Earth [4826, 4828, 4837,4838]. § 4
Principles relative to the degrees of smallness of the quantities which occur in the expressions
of the co-ordinates of the moon [4841]. Examination of the influence of the successive
integrations upon the different terms of these co-ordinates [4847, &c.]. Indication of the
terms of the radius vector, wliich produce the evection [4850], and annual equation [4851]. §5
CONTENTS OF THE THIRD VOLUME. Xxiii
Use to be made of these considerations. Development of the differential equation which produces
the radius vector ; noticing only the first power of the disturbing force [4858 — 4903]. § 6, 7
Investigation of the terms of the order of the square and the higher powers of the disturbing
masses, which acquire a sensible influence by integration [4904, &c.]. It is necessary to
notice the perturbations of the Earth by the Moon [4909', 4948, &c.] §8
Connection of these terms with the preceding. Complete development of the differential
equation which produces the radius vector [4961] § 9
Integration of tliis equation [4904, &c.]. Inequalities resulting from it. Expression of the
motion of the lunar perigee [4982, &c.].
The variableness of the excentricity of the Earth's orbit produces a secular inequality in the
constant term of the Moon's parallax ; but this inequality is insensible [4970].
The same cause produces a secular inequality in the motion of the Moon's perigee, which is
conformable to observation. Analytical expression of this inequality [4985].
The excentricity of the Moon's orbit is subjected to a secular variation, which is analogous to
that of the parallax, and like it, insensible [4987] § 10
Development of the differential equation which gives the latitude [501 8, &c.], noticing, in the
first place, only the simple power of the disturbing forces §11
Investigation of tlie terms of the order of the square of those forces which acquire a sensible
influence in the expression of the latitude [5039,&c.] §12
Connection of these terms with the preceding, and the complete development of the difierential
equation which gives the latitude [5049] §13
Integration of this equation [5050, &c.]. Inequalities resulting from it. Expression of the
retrograde motion of the nodes [5059].
The variableness of the excentricity of the Earth's orbit, produces in this motion a secular
inequality. Analytical expression of this inequality [5059]. Its ratio to that of the perigee
[5060].
The inclination of the lunar orbit to the true ecliptic, is likewise variable by means of the same
cause ; but this variation is insensible [50G1] §14
Development of the differential equation which gives the time or the mean longitude in terras of
the true longitude [5081, &c.] Integration of this equation. Inequalities which result from
it [5095, &c.]
The mean longitude also suffers a secular change, resulting from the variableness of the excentricity
of the Earth's orbit; expression of this inequality. Analytical relations of the secular equations
of the mean motions of the Moon, its perigee and nodes [5089, «Sic] § 15
Numerical determination of the several coefficients, occurring in the preceding formulas [51 17,&c.]
and the numerical développent of the expression of the mean longitude [5220]. The
perturbations of the Earth's orbit by the Moon, are reflected to the Moon by means of the Sun
and are weakened by the transmission [.5225, 5226]. Numerical value of tlie motion of the
perigee [5231], and of its secular equation [5232]. This equation has a contrary sign to that
of the mean motion [.5232']. Numerical expression of the motion of the node [5233], and of
Xxiv CONTENTS OF THE THIRD VOLUME.
its secular equation [5234]. This equation has also a contrary sign to that of the mean motion
[5234'] ; hence it follows, that the motions of the nodes and perigee decrease, while that of the
Moon increases. Numerical ratios of these three secular equations [5235]. Secular equation
of the mean anomaly [5238] § 16
The most sensible inequalities of the fourth order, which occur in the expression of tlie mean
longitude [5240— 5305] §17
Numerical expression of the latitude [5308] §18
Numerical expression of the Moon's parallax [5331] § 19
CHAPTER II. ON THE LUÎJAR INEQUALITIES ARISING FROM THE OBLATENESS OF THE EARTH A.\D
MOON 585
The oblateness of the Earth produces in the latitude of the Moon but one single inequality. We
may represent this effect, by supposing that the orbit of the Moon, instead of moving on the
plane of the ecliptic, with a constant inclination, to move with the same condition, upon a
plane which always passes through the equinoxes between the ecliptic and equator [5352].
This inequality can be used for the determination of the oblateness of the Earth [.5358]. It
is the reaction of the nutation of the Earth's axis upon the lunar spheroid [5398], and there
would be an equilibrium about the centre of gravity of the Earth by means of the forces
producing these two inequalities, if all the particles of the Earth and Moon were firmly
connected with each other, the Moon compensating for the sraallness of the forces acting on it,
by the length of the lever to which it is attached [5424].
The oblatenes of the Earth has no sensible influence on the radius vector of the Moon [.53G6] ;
but it produces in the Moon's longitude one sensible inequality. The motions of the perigee
and node are but very little augmented by it [5396, &c.] § 20
The non-sphericity of the Moon produces in its motion only insensible inequalities
[5445, 5451, &c.] §21
CHAPTER III. ON THE INEaUALITIES OP TUE MOON DEPENDING ON THE ACTION OF THE PLANETS. G17
These inequalities are of two kinds, the first depends on the direct action of the planets on the
motion of the Moon [5479, 5481] ; the second arises from the perturbations in the Earth's
radius vector produced by the planets [5490]. These perturbations are reflected to the Moon
by means of the Sun, and are augmented by the integrations which gives them small divisors.
Determination of these inequalities for Venus, Mars, and Jupiter [5491, &c.]. The variableness
of the excentricities of the orbits of the planets, introduces, in the mean longitude of the
Moon, secular equations, analogous to that produced by the variation of the excentricity of the
Earth's orbit, reflected to the Moon by means of the Sun ; but they are wholly insensible in
comparison with this last. Thus the indirect action of the planets on the Moon, transmitted by
means of the Sun, considerably exceeds their direct action, relative to this inequality [5539]. §22
CHAPTER IV. COMPARISON OF THE PRECEDING THEORY WITH OBSERVATION 642
Numerical values of the secular inequality of the mean motion of the Moon [5542, &c.], and those
of the mean motions of the perigee and node of the Moon's orbit. Considerations which
confirm their accuracy [5544, &c.] § 23
CONTENTS OF THE THIRD VOLUME.
Periodical inequalities of the Moon's motion in longitude [5551, &c.]. Agreement of the
coefficients given by the theory, with those of the lunar tables of Mason and Burg [5575, &c.].
One of these inequalities depends on the Sun's parallax [5581]. If we determine its
coefficient by observation, we may deduce from it the same value of the Sun's paralla.ï,as that
which is obtained by the transits of Venus [5589']. Another of these inequalities depends on
the oblateness of the Earth [5590]. The value of its coefficient determined by the tables of
Mason and Burg, indicates that the Earth is less flattened than in the hypothesis of homogeneity,
and that the oblatenes is ^^-g. [5593] , S 34
Inequalities of the Moon's motion in latitude [5595, &c.]. Agreement of the coefficients given
by the theory with those of the tables of Mason and Burg [5596]. One of these inequalities
depends on the oblateness of the Earth [5598]. Its coefficient, determined by observation,
gives tlie same oblateness [5602], as the inequality in longitude depending on the same element.
So that these two results agree in proving, that the Earth is less flattened than in the
hypothesis of homogeneity ^25
Numerical expression of the Moon's horizontal parallax [5C03]. Its agreement with the tables
of Mason and Burg [5605] §26
CHAPTER V. ON AN INEaUALITV OF A LONG PERIOD, WHICH APPEARS TO EXIST IN THE MOON'S
MOTION QQQ
The action of the Sun on the Moon, produces in the motion of that satellite an inequality, whose
argument is double the longitude of the node of the Moon's orbit, plus the longitude of its
perigee, mimis three times the longitude of the Sun's perigee [5641, &c.]. The consideration
of the non-spherical form of the Earth, may also introduce into the motion of the Moon, two
other inequalities [5633, 5638'], with nearly the same period as that which we have just
mentioned ; and in the present situation of the Sun's perigee, they are all three nearly
confounded together. The coefficients of these three inequalities are very difficult to compute
from the theory ; it appears that the two last must be wholly insensible [5637', 5639']. . § 27
The first is evidently indicated by observations. Determination of its coefficient [5665]. [This
result was afterwards found to be incorrect, as is observed in the note, page 666, &c.]. § 28
CHAPTER VI. ON THE SECULAR VARIATIONS OF THE MOTIONS OF THE MOON AND EARTH, WHICH
CAN BE PRODUCED BY THE RESISTANCE OF AN ETHEREAL FLUID SURROUNDING THE SUN. . . g-^g
The resistance of the ether produces a secular equation in the Moon's mean motion [5715] ;
but it does not produce any sensible inequality in the motions of the perigee and nodes
[5713,5717] §29
The secular equation of the Earth's mean motion, produced by the resistance of the ether, is about
one hundredth part of the corresponding equation of the Moon's mean motion [5740]. §30
VOL. III. ^
XXV
Xxvi CONTENTS OF THE THIRD VOLUME.
APPENDIX BY THE AUTHOR.
The chief object of this appendix is to demonstrate a theorem, discovered by Mr. Poisson, that the
mean motions of the planets are invariable, when we notice only the terms depending on the first and
second powers of the disturbing forces [5744, &c.] This is done by giving new forms to some of the
differential expressions of the elements of the orbits, as is observed in [.5743, &c]. Forms of these
differentials, including all the terms depending upon the first power of the disturbing masses
[5786 — 5791]. Expressions of the mean motion [5794] ; of the periodical inequalities in the
elements [5873 — 5879] ; and of the secular inequalities of the elements [5882 — 58SS].
Investigation of the mutual action of two planets upon each other, referring their inequalities to
an intermediate invariable plane [5905, &c].
New method of computing the lunar inequalities, depending upon the oblateness of the earth
[5937—5973].
On the two great inequalities of Jupiter and Saturn ; correcting for the mistake in the signs of the
functions JVC), JV(}) Sic. [5974—5981].
IN THE COMMENTARY
Among the subjects treated of in the JVotes, we may mention the following :
Correction to be made in the formula mfàR-{-m'rdR' = 0, [1202], in some of the terms of the
order of the square of the disturbing masses [4004c, &c]. The necessity of this correction was first
made known by Mr. Plana [400Gw, &c.]. Results of the discussion upon this subject, by Messrs.
Plana, Pontecoulant, Poisson and La Place [40056'— 4008î]. New formula by La Place, relative to
some of these terms [4008x]. This formula has been called " the last gift of La Place to Astronomy,"
being the last work he ever published.
On the values of the constant quantities f,,f',g, &c. ; introduced into the integral expressions of
or, ÔV, OS, by La Place [4058c, &c.] ; which were objected to by Mr. Plana. The results of La Place's
calculation proved to be correct by him, and by Mr. Poisson, in [4058c — 40G0/i].
Corrected values of the masses of the planets, finally adopted by the author [40Gld].
Elements of the newly discovered planets Vesta, Juno, Pallas and Ceres ; corresponding to the 23d
July, 1831, as given by Enckc [4079i].
Elements of the orbits of the comets of Halley, Olbers, Encke and Biela [4079»i].
Inequalities in the motions of Venus and the Earth, having a period of 2.39 years, and depending
on terms of the fifth order of the excentricities and inclinations ; discovered and computed by Professor
Airy [4296 a — q, 4310 c — /].
Mr. Ponteooulant's table of the part of the great inequality of the motion of Jupiter, depending on
the square of the disturbing force [4431/]. Similar table for the inequalities of the motion of Saturn
[4489c].
Results of the calculations of Professor Hansen [4489 n — p].
CONTENTS OF THE THIRD VOLUME. XXvii
The action of the fixed stars affects the accuracy of the equation ta. m. \/ n -(- c's. ?)i'. y/a' -f- &c. = 0
[46S5g-].
Results of the calculations of several authors relative to the sun's parallax, hy means of the parallactic
inequality in the moon's longitude, and by the transits of Venus over the sun's disc [5589 a — m].
Inequality in the moon's longitude, whose period is about 179 years. It is found to be insensible
[5611 a — g]; instead of being 15V39 at its maximum, as the author supposes in [5GG5].
The planets and comets move in a resisting medium, according to the observations of Encke's
comet [5067 a — c].
Notice of the papers published by La Grange and Poisson, relative to the invaiiableness of the mean
motions of the planets, which is treated of in the appendix to this volume [5741a — I].
It appears from the calculations of Nicolai, Encke and Airy, that the estimated value of the mass of
Jupiter, adopted by La Place from Bouvard's calculations of its action on Saturn and Uranus, must be
increased, to satisfy the observed perturbations of the planets Juno and "Vesta ; as well as those of
Encke's comet, [5980 i — p].
APPENDIX BY THE TRANSLATOR.
Formulas for the motion of a body in an elliptical orbit [.5985(1—19)] ; with their demonstrations
[5984(3-25)].
Formulas for the motion of a body in a parabolic orbit [5986] ; with their demonstrations [5987].
Determination of the symbol log. A: = 8,2355814 .. . which is used in these calculations [5987(8)].
Formulas for the motions of a body in a hyperbolic orbit [5988] ; with their demonstrations [5989].
Kepler's problem for computing the true anomaly from the time, or the contrary, in an elliptic orbit.
Indirect solution of this problem, according to Kepler's method, but arranged in formulas
by Gauss [.5990].
Simpson's method for determining the true anomaly, in an ellipsis or hyperbola, where e is
very nearly equal to unity, noticing only the first power of 1 — t, or e — J [5991(1— 12)].
Bessel's improved method for computing the terms depending on the second power of
1 -e or e-1 [.5991(1-40)]
Gauss's method, in a very cxcentric ellipsis, noticing all the powers of e — 1 [5992].
Gauss's method of solution in a hyperbolic orbit, in which e — 1 is very small, noticing
all tlie powers of this quantity [5993].
Olbers's method of computing the orbit of a comet [.5994, &c.].
Table of formulas which are used in this calculation [5994(.9— 45)].
Geometrical investigation of this method of calculation [5994(46—130")].
Remarks on the manner of determining the approximate values of the curtate distance p
of the comet from the earth [5994(132—172)]
Examples for illustrating these calculations [5994(173—242) ], using tables I, II, III.
Remarks on the calculation of p by means of the equations (C), (/)) [.5994(136—103, 242', 242")].
Forms of the fundamental equations, adopted by Gauss for the determination of the curtate distance,
or its equivalent expression it, by means of logarithms [5994(244, &c.)].
Solution of two examples, reduced to the form of Gauss [5994(247 — 250)].
Analytical investigation of the method of computing the orbit of a comet, [5994(263-403)].
Great advantage in having the intervals of times between the observations nearly equal to each
other [5994(349)].
XXVÏn CONTENTS OF THE THIRD VOLUME.
The method usually employed in this calculation requires some modification, when M appears
under the form of a fraction, in which the numerators and denominators are both very small
[5994(257)]. These methods are explained in [5994(387—392)].
Mr. Lubbock's method of computing the orbit of a comet [5994(405—458)].
Method of computing the elements of the orbit of a heavenly body; there being given the two
radii r,r', the intermediate angle v' — v = 2f, and the time f — t of describing the angle 2/
[5995].
Collection of formulas for solving this problem, in an elliptical orbit [5995(4—67)] ; with their
demonstrations [5995(08—174)]. Examples of the uses of these formulas [5995(175—193)].
Collection of formulas for solving this problem in a parabolic orbit [5996(2— :i;5)] ; with their
demonstrations [.5996(26—50)] ; illustrated by an example in [5996(51-53)].
Collection of formulas for solving this problem in a hyperbolic orbit [5997(1 — 59)] ; with their
demonstrations [5997(60—172)]. Example of the uses of these formulas [5997(173—183)].
Gauss's method of correcting for the efiect of the parallax and aberration of any newly discovered
planet or comet, in computing its orbit by means of throe geocentric observations, with the intervals
of time between them [5998].
Corrections in the places of the earth, on account of the planet's parallax [5998(47 — 50)].
Method of calculating the longitude and latitude of the zenith [5998(67 — 71)&,c.]; also the
longitude and latitude of the planet from its right ascension and declination [5998 (97— 107) ], with
examples.
Method of correcting for the aberration of the planet [5998(108 — 117)].
Example for illustrating the calculations relative to the parallax and aberration [5998(118 — 126)].
Gauss's method of computing the orbit of a planet or comet, by means of three geocentric longitudes
and latitudes, together with the times of observation [5999.]
Table of the symbols and formulas which are used in this method [5999(9 — 54)].
Demonstrations of these formulas [5999(58, &c.)].
Example, containing the whole calculation of the elements of the orbit of Juno, from three observa-
tions of Maskelyne [5999(274—650)].
CATALOGUE OF THE TABLES IN THE APPENDIX.
Table i. Contains the square roots of the numbers from 0,001 to 10,1 ; to be used in Olbers's
method of computing the orbit of a comet ; in finding r, r", c ; from j-9, r"% c^ ;
which are given by three fundamental equations of this method [5994(31, 32, 33)].
Table II. To find the time T of describing a parabolic arc, by a comet ; there being given the sum
of the radii r-\-r", and the chord c, connecting the two extreme parts of the
arc. This table is computed by Lambert's formula [750], namely,
7 = 9"'''^', 688724. j (,- + r" + c)^ — (r + r" — c)* ( ;
and the numbers are given to the nearest unit in the third decimal place, expressed in
days and parts of a day. This degree of accuracy being abundantly sufficient for the
purpose of computing the orbit of a comet, by Dr Olbers's method ; and the table serves
to facilitate this part of the calculation.
CONTENTS OF THE THIRD VOLUME. XXIX
Table III. To find the anomaly U, corresponding to the time t' from the perihelion, expressed in
days, in a parabolic orbit; where the perilielion distance is the same as the mean distance
of the earth from the sun. The arguments of this table, as they were first arranged by
days days
Burckliardt, are the values of r, from I'^O ,0 to t' = G ,0; and the logarithm
of t' from log.<'^0,700 to log. <'^5,00; the corresponding anomalies being given
from 17=0'' to [/= 172''32"'09',9. We have also given Carlini's table for the
first six days of the value of t'. This last table has for its argument log. of t' days ;
and the corresponding numbers represent log. U in minutes, minus log. t' in days. 9S7
Table IV. To find the true anomaly v, in a very excentric ellipsis or hyperbola, from the
corresponding anomaly U in a parabola; according to the method of Simpson,
improved by Bessel. This table contains the coefficients of Simpson's correction,
corresponding to the first power of (1 — e); and those of Bessel's correction,
corresponding to the second power of (1 — c) ; for every degree of anomaly from
0** to 180''; as they were computed by Bessel 996
Table V. This table was computed by Gauss, for the purpose of finding the true anomaly v,
corresponding to the time t from the perihelion, in a very excentric ellipsis, noticing
all the powers of 1 — e 999
Table VI. This table is similar to Table V, and was computed by Gauss for finding the true
anomaly r, corresponding to the time t from the perihelion, in a hyperbolic orbit,
which approaches very nearly to the form of a parabola; noticing all the powers
of (e— 1) 1002
Table VII. This was computed by Burckhardt, for the purpose of finding the time t, of describing
an arc of a parabolic orbit ; there being given the radii r,r', and the described arc
v'—v = 2f. 1005
Table VIII. This table was computed by Gauss, and is used with Table IX or Table X, in finding
the elements of the orbit of a planet or comet, when there are given the two radii r, r',
the included heliocentric arc v' — x) = 2/; and the time t' — t, of describing this
arc, expressed in days. . 1006
Table IX. This table is used with Table VIII, in the computation of an elliptical orbit, by means of
r,r',v' — V and t' — t 1012
Table X. This table is used with Table VIII, in the computation of a hyperbolic orbit, by means
of r,r', v' — V, and t' — t 1013
Table XI. To convert centesimal degrees, minutes and seconds, into sexagesimals. ..... 1014
Table XII. To convert centesimal seconds into sexagesimals, and the contrary 1016
The Tables V — X, include all those which Gauss published in his Theoria Molus, etc. We have
altered, in some respects, the arrangement and forms of these tables, to render them more convenient
for use ; and upon comparison it will be found, that this appendix contains the most important of the
methods which are given in that great work, as well as in that of Dr Olbers. The methods of Gauss
being somewhat simplified, by reducing many of the processes to the common operations of spherical
trigonometry, instead of using a great number of unusual auxiliary formulas, expressed in an analytical
manner; and Olbers's calculations are abridged by the use of Tables I, II.
VOL. III. ^
ERRATA.
CORRECTIONS AND ADDITIONS
IN VOLUME I.
PagR. Line.
119 6 bot. For dw read dia.
120 13, 19, 21 J'or (zdx—xdy) read (zdx—xdz).
125 12 For dZ-j-dy,, read dZ-\-dz,.
7 \)o\.. For — l.m.Sn.ds, read —Xj.m.Sv.ds.
7 bot. For — y'ddx', read — yddx'.
4 bot. For Y read y.
3 bot. Insert dm in the last term.
7 Insert ( after xK
9 bot. For |. read 4..
4 bot. For .2 read ^.
9 bot. For axis of z, read axis of x.
16 For dy read Sy.
10 bot. For {dp) read (J-p).
3 bet. For dr' read dr.
4 bot. For ag read a,g.
8 bot. For 0-.U read au'.
7 Change tlie accents in the denominator of I'.
1 bot. For /2a, read /2-2.
7 bot. For 2, read r2.
12 For shi.mt, read sin. ?nn<.
13 For cos.mt, read cos.mnt.
3 For sin.2»i<, read 2.sin.2ji(.
11 For [6S8a], read [66Su].
1 For sin.4.(t),— 6), read sin.4.(i',— 6).
2 2
10 bot. For — , read — .
r * r
3 For 0",5, read 0«,5.
1,2 bot. For logarithm, read logarithmic.
8 bot. For tang.(/g"— ;■), tang. {IS"'—j) ; read
sin.(/3''— /), sin (/3"'— /).
7 bot. For éy, read d'y'.
6 For c, read c'.
4 For y',y', fcc, read y,y', &c.
IS i^or iy',)/', &c., read y,y',&.c.
5 bot. For 6' reaa j'.
6 bot. For c=V', read c'=V'.
4 liot. For .111, read Jl^D.
8 bot. For [1034(;], read [1069a].
1 bot. For the exponent — è, read 4.
134
147
147
147
159
182
183
209
215
220
230
234
235
280
281
301
371
371
378
378
381
398
413
455
464
475
478
480
487
495
499
542
581
585
Page. Line.
593 5 bot. for [109Sa], read [10976'].
608 Id For B, read B„.
618 15 For spherical angle, read spherical triangle.
679 5 bot. For m'p, read m'p' ; and/br m'q, read m'q'.
693 4 bot. For m, read m'.
715 15 bot. For andt, read an, in both formulas.
IN VOLUME II.
370 16 For [1581a], read [1851a].
510 11 bot. For >. read e.
L' , L-
780 4 bot. For — , read — .
r'3 T.'-i
781 5 bot. For
read —- .
IN VOLUME III.
The same measures have been used for .correcting the
mistakes ol the press in Volume III, as in printing the
preceding volumes. The reader will also omit the third
line from the bottom in page 501, which is unnecessarily
repeated ; and at the end of the paragraph, page 556, line
16, will make the following addition of a paragraph which
was accidentally omitted. " The function [5082s] contains
also the terms depending on 120m2..4(8), 120m2../î(9)
[5261c, e, line 1], which are derived from the part
— Ja. funct. [4931;)] contained in [50825]. For by combining
the term A&> ee' .e.os.{cio-\-e'mv) in [493 J^, col. 1] with
— |.e.sin.(2i) — 2nit!— cr), in col. 2, we get the first of these
terms; and by combining the term .4(9).e£'.cos.(CB — c'mtj),
with — Je. sin. (2b — 2niii — ev), in col. 2, we get the second
of these terms." Lastly, in page 458, line 3, we may add,
that the function [4957] must be multiplied by the chief
term of [4S90], or
[4961 or 4960e].
to obtain the corresponding terms of
SECOND PART.
PARTICULAR THEORIES OP THE MOTIONS OF THE HEAVENLY BODIES.
SIXTH BOOK.
THEORY OF THE PLANETARY MOTIONS.
The motions of the planets are sensibly disturbed by their mutual
attractions, and it is important to determine accurately the inequalities which
result from this cause ; for the purposes of verifying the law of universal
gravitation, improving the accuracy of astronomical tables, and discovering
whether any cause, foreign from the planetary system, produces a change in
its constitution or its motions. The object of this book is to apply to the
bodies of this system, the methods and general formula given in the first part
of this work. We have developed in the second book, only those inequalities
which are independent of the excentricities or inclinations of the orbits, and
those which depend upon the first power of these quantities. But it is often
indispensable to extend the approximation to the square and to the higher
powers of these elements ; and sometimes it is also necessary to consider the
terms depending on the square of the disturbmg force. We shall first give
the formulas relative to these inequalities ; and shall then substitute in these
formulas, and in those of the second book, the numbers or values of the
elements corresponding to each planet. By this means we shall obtain
the numerical expressions of the radius vector, and the motions of the ^jlanet in
longitude and in latitude. Bouvard has willingly undertaken the calculation
of these substitutions, and the zeal with which he has prosecuted this
laborious work, deserves the acknowledgment of all astronomers. Several
mathematicians had previously calculated the greater part of the planetary
inequalities ; and their results have been useful in verifying those of Bouvard ;
for when any difference has been found, he has examined into the source of
VOL. HI. 1
PARTICULAR THEORIES OF THE
the error, in order to satisfy himself of the accuracy of his own calculation.
Lastly, he has reviewed with particular care, the calculation of those
inequalities which had not been before computed ; and by means of several
equations of condition, which obtain between these inequalities, I have been
enabled to verify many of them. Notwithstanding all these precautions,
there may possibly be found in the following results, some errors, which
almost inevitably occur in such long calculations ; but there is reason to
believe that they amount only to insensible quantities, and that they cannot
be detrimental to the general accuracy of the tables founded upon them.
These results, on account of their importance in the planetary astronomy,
of which they are the basis, deserve to be verified with the same care that
has been taken in the calculation of the tables of logarithms and of sines.
The theories of Mercury, Venus, the Earth, and Mars, produce only
periodical equations of small moment ; they are, however, very sensible, by
modern observations, with which they agree in a remarkable manner. The
development of the secular equations of the planets and of the moon will
make known accurately the masses of these bodies, which is the only part
of their theory that remains yet somewhat imperfect. It is chiefly in the
motions of Jupiter and Saturn, the tAvo greatest bodies of the planetary
system, that the mutual attraction of the planets is sensible. Their mean
motions are nearly commensurable ; so that five times that of Saturn is
nearly equal to twice that of Jupiter, and the great inequalities in the motions
of these two bodies arise from this circumstance. When the laws and
causes of these motions were unknown, they seemed, for a long time, to
form an exception to the law of universal gravitation, and now they are one
of the most striking proofs of its correctness. It is extremely curious to see
with what precision the two principal equations of the motions of these
planets, whose period includes more than nine hundred] years, satisfy ancient
and modern observations. The development of these equations in future
ages, will more and more prove this agreement of the theory and observation.
To facilitate the comparison with distant observations, we have carried on
the approximation to terms depending on the square of the disturbing force,
and it is hoped that the values here assigned to these equations will vary but
very little from those found by a long series of observations continued during
an entire period. These equations have a great influence upon the secular
variations of the orbits of Jupiter and Saturn, and we have developed the
analytical and numerical expressions arising from this source. Lastly, the
MOTIONS OF THE HEAVENLY BODIES. 3
planet Uranus is subjected to sensible inequalities, which we have determined,
and which have been confirmed by observation.
The first day of this century is remarkable for the discovery of a new
planet, situated between the orbits of Jupiter and Mars,* and to which the
name of Ceres has been given. It appears as a star of the eighth or ninth
magnitude ; its excessive smallness renders its action insensible on the
planetary system ; but it must suffer considerable perturbation from the
attractions of the other planets, particularly Jupiter and Saturn, which ought
to be ascertained. It is what we propose to do in the course of this work,
after the elements of the orbit have been determined by observation to a
sufficient degree of accuracy.
It is hardly three centuries since Copernicus first introduced into
astronomical tables the motion of the planets about the sun. A century
afterwards, Kepler made known the laws of the elliptical motion, which he
had discovered by observation ; and from these laws, Newton was led to the
discovery of universal gravitation. Since these three memorable epochs in
the history of the sciences, the progress of the infinitesimal analysis has
enabled us to submit to calculation the numerous inequalities of the planets
depending upon their reciprocal action ; and by this means the tables have
acquired an unexpected degree of accuracy. It is believed that the following
results will give to them a much greater degree of precision.
* (2341) This volume was published by the author shortly after the discovery of Ceres,
January 1, ISOl ; and before the discovery of the planets Pallas, Juno, and Vesta. He did [3698a]
not compute the numerical values of the perturbations of their motions as he had intended.
4 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
rèr.
Fi ret form.
Radius.
CHAPTER I.
FORMULAS FOR THE INEQUALITIES OF THE MOTIONS OF THE PLANETS WHICH
DEPEND UPON THE SQUARES AND HIGHER POWERS OF THE EXCENTRICITIES AND
INCLINATIONS OF THE ORBITS.
ON THE INEQUALITIES WHICH DEPEND UPON THE SQUARES AND PRODUCTS OF THE
EXCENTRICITIES AND INCLINATIONS.
Differen ^* '^^ determine these inequalities, we shall resume the formula [926],^
tial equa-
[3699] 0 = -jj^ + -^ + 2fdR + r.(j-y
We have, as in [605', 669], f
[3700] T = W^
[3701] r = a.{l +|e^ — e . cos. (nt-\-s — zi) — i el cos. 2. (n ï + s — w)} :
hence the preceding differential equation becomes,}
Differeu- tj^ rS
tial eijua- -^ a . / u
;;^:,ir" 0=-^ + n\r6r+3rî'a.ôr.{e.cos.(nt+i—z,) + eKcos.2.(nt+e—z=)l
[3702] "^ ^
sroo^nd ^2fdR + r.C^
form. -^ \ch-
[3699a] * (2342) Substituting, in [926], the value of r R [928'], it becomes as in [3699].
r3700a] ^ {2343) The equation [3700] is easily deduced from [605'] ; and the value of r [3701]
is the same as that in [669], neglecting tenns of the order e^.
f (2344) If we use, for brevity, the same symbols as in [1018a], namely,
[3702o] T=nt~nt-Jf-s'—s, W=nt+s—zi, b = ie^—e.cos.W—ié^.cos.2W,
[3703i] we shall have r=a.(l+è) [3701]; hence r-^=a-^.{l-\-b)-3=a-^.{l—3b-\-6b^);
neglecting 6' and the higher powers of b ; or, in other words, neglecting e^ e*, 8ic. Now, by
VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', 7. 6
Now all the terms of the expression of R, depending on the squares and [3702]
products of the excentricities and inclinations of the orbits, may be reduced .j,^^^^^,.
to the one or the other of these two forms,* dopSdin-
on anglea
R = M. cos. { i . (n' t — nt + B' — s)-{-2nt-\-K\; (Pi-t form.j [3703]
of two
Rz=N. cos. { i . {11! t — n i + s' — 0 + -^1 ; f®'"™'' '■"'"■i '^^'"■*]
different
forms.
in which i includes all integral numbers, positive or negative, comprehending
also i = 0 [954"]. JVe shall, in the first place, consider the term [3703]. [3704']
It produces, in 2fàR-\-r.(^—\ the function f [3^04"]
\ .^•^^~^);" . M-^a.(y^\ \ . cos. {i.(n't—nt+s'—{)+2nt+K\.
\tn' -\-{2 — t).n ' \da J ), * ^ ' yi 1 4
[3705]
[3702c]
retaining ternis of the order e^, we get, successively, 66^=6e^.cos.^fF=3e^-|-3e^.cos.2 W;
hence \ — ^h-{-Qb^=l + ie^-\-2e. cos. W+ f 1 œs. 2 W. Substituting this in r'^
[.37026], and then muhiplying by i^.rSr, we get [3Î02(?] ; which is easily reduced to
the form [3702e], by the substitution of n^ [3700] and r^a. (1 — e.cos.^ [3701] in the
last temi of the second member. Now we have — 3e^.cos.^ ?f = — f e^ — fe^.cos.2 fV;
hence [3702e] becomes as in [3702/],
'~^ = a3''^ ^ '''^a^' '^ ^ ^ ' ^^ "^ '^ "^ ^ ' ^°^' ^^+|fi^-C0S.2 TVl [3702a;]
= 71^. rSr-\-n-.a5r.l<^e^+3e. cos. W-\-^e^. cos. 2W\.\\ — e. cos. W\ [3702e]
=^n\r5r~\-n^.aSr .\Qe. COS. fF+ 3 e^. cos. 2 W\. [3702/]
Substituting this in [3699], we get [3702].
* (2345) This will be e^^dent by the substitution of «,, v,, &,c. [1009,669] m [957].
It also appears from [957"", &c.] ; for in [3703], the coefficients of n' t, —nt, are i, i-2, [3704o]
respectively ; their difference 2 expresses the order of the coefficient k [957'''', &c.], or
that of M [3703] ; which must therefore be of the order 2 or e^. In like manner, the
coefficients of ?i' t, — n t [3704] being both equal to i ; the coefficient JV may contain
terms of the orders 0, 2, 4, he. [957"'S &;c.], which include those of the order e^ ; and [3704i]
a very little attention to the remarks in [957^, &tc.] will show, that these are the only forms
of this kind containing e^.
t (2346) Substituting the expression r .(——j=a.( — J [962], in the function [3705a]
[3704"], we get 2/dfl + 7- . (^^) =:2/di?+ a. ('^V In finding d R, we must [37056]
suppose, as in [916'], the ordinaies of the body m to he the only variable quantities; or, in other
words, we must consider nt as variable, and n't cojwton^, as is done in finding d/î [1012a — c].
Now in taking for R the form [3703] , R—M. cos. { i . (w' t — nt-}-s' — i)J^2nt-\-Kl, [3705cl
VOL. HI. 2
6 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
We have seen, in the second book [1016], that the parts of — depending
on the angles i.(n't — nt + e' — e) and i.(n't — nt-\-s — 5) + ni + £,
°'^"" ° are of the following forms,
[3706] — =F.cos.i.(n't — nt+e' — e)-^eG.cos.{i.(n't — 7it-\-s — s)-{-nt+e — ^ \
depending "
o?Z'fiT.. -te'H.cos.{i.(n't—7it+i'—^) + nt+B—z>'\;
hence the function
[3707] 3n^.a&r.\e. cos. (7it + e — ra) + e\ cos. (2 n i + 2 s — 2 w) }
will produce, in [3702], the following terms,*
C{F+G).e\cos.{i.(7i't — nt + s'—i) + 2nt + 2s—2^] }
[3708] |»-«-.^ -^H.€e'.cos.{i.(7i't—nt + e'—B) + 2nt + 2^—^—^'\l'
Therefore, if we notice only the terms depending on the angle
i.(n't — 7it + s—s) + 27it,
[3709] and put (x = 1 ; which is equivalent to the siipposition that the sim^s mass is
[3709'] equal to tmity, 7ieglecting the mass of the planet ; f 7ve shall have n^ «^ = 1 ;
[3705d] we obtain d R = — {2 — i) .n. M .sm. {i. {n't — 7it -Jf- I'—s) + 2 7it -{- Kl-ctL Integrating
this, and multiplying by 2, we get
[3705e] 2fàR= .^,f~'':" .M.cos.{i . {n't-nt + ^-i) +2n t ^K\.
The partial differential of R [3705c], relative to a, being multiplied by a, gives
[3W] a.Ç^) = a.{'^).cos.ii.in't-r,t + s-s)+2nt + K}.
Adding this to the expression [3705e], we get 2/d R-\-a . (j^j r as in [3705].
* (2347) The forms of the temis of -, assumed in [3706], are the same as those
computed in [lOlG] ; the constant part corresponding to i = 0; and the secular
[3708a] teiTOS being made to disappear, as in [1036", &c.]. Substituting these in [3707], and
reducmg by formula [20] Int., retaining only the teims dependmg on the angle
i . („' t — nt + s'—s)-\-2nt + K [3703], we get [3708].
t (2348) M being the mass of the sun, and m that of the planet, we have JU+m^M-
[3709a] [914']. If we put Jkf=l, and neglect m on account of its smalhiess, we shall have fA=l ;
and then fiom [3700], we shall get [3709'].
VI. i. §1.] TER]\IS OF THE SECOND ORDER IN e, e', y.
and then the differential equation [3702] will become*
^ rf2.(,6,) ^i{F+G).c\cos.\i.{n'l-nt^s'-z)+2nt + 2s-2v>\\
</«2 - ^ -\-H.ee'.co5.\i.{r^t—nt^^—s)-\-2nt^1i-zs-vi'\)
(i)i+(2 — i).n \(/o/)
Hence we get, by integration,!
^ ( (F+G).Ê2.cos.^ù(n'< — w< + £'— 5)+2ji< + 2s — 2-nf
A «". < > I Values of
( +H.ee'.cos. fi.(ji'i — n<+E' — s) + 2?i <+2 s — «—• n'} )> f ^(îr
V J depending
[3710]
rSr
(in'-\- (2 — i) .n \da J '^
on angles
of the first
form.
[3711]
a2 {i.n'+(3 — i).Mf .{!'n'+(l — «).n|
TOT
If this expression of be considerable, and one of its divisors
i n' + (3 — i) .n, i n' -\- (\ — i) . n, be very small, as is the case in the
tlieory of Jupiter, disturbed by Saturn, when we suppose i = 5 ; 2n being [3712]
nearly equal to 5n' ;% the variableness of the elements of the orbit will
* (2349) Substituting, in [3702], the value of its third and fourth terms [3708], also
the values of the fifth and sixth terms [3705], multiplied by n^ a^ = 1 , for the sake of [3710a]
homogeneity; it becomes as in [3710].
[3711a]
t (2350) If we put, in [865, 870'], y=rSr, a = n, a Q = 2 .a/f.™- (m,^ + 6,),
the letters m, s bemg accented to prevent confusion in the notation, and 2 denoting the sign
of finite integrals; we shall have the differential equation [3711&], whose integral [871]
is as in [3711c],
r 5 r = 2 . -4^, . f^- (m t + s)= "- ^ . [3711c]
m^a_„a cos. ^ ' i i' m,^—n^ ^
Comparing the coefficient of < in the expressions [3710, 37116], we get OT,=i. (w'— M) + 2n; [3711d]
hence m,^ — n-z={m,-{-n) .{m^ — ?i)= ^fn'_[-(3 — i).n\ .\in'-\-{l — i).n\; substituting [3711e]
this in [3711c], and then dividmg by a^, we get [3711].
t (2351) We have, in [4077], for Saturn n'=43997''; and for Jupiter m=109256' r37n/-n
nearly; hence 5?i' — 2 n = 1473^; which is quite small in comparison with n or n',
being only y\ part of n.
8 PERTURBATIONS OF THE PLANETS.
have a sensible influence on this expression ; it is important, therefore, to
notice this circumstance. For this purpose we shall put the differential
equation [37 1 0] under the following form,*
0= ^^^^-\-n-.rôr-\-7i-a^P.cos.{i.(n't — nt-\-s'—i)-{-2nt + 2s\
[3713] dt^ ' ^ / I I 3
+ n'a^F. s'm.\i. (n't — nt + b'—b) J^2nt + 2s\.
Integrating this, and neglecting the terms depending on the second and
higher differentials of P, P', we shall obtain f
r 7i"
^aTulof a^ {i7i'~\- (3 — i) . 71] . [ill' -\-{l — i) . ii]
noticing / C _ \ d P'
the secu- «/-.-. . . „
lar varia-
tion of the _
elements. S / ' \in'-\-[^-i).nl . iin'-\-{l—i).n\
[3714] x< ^ > i i -ry > ) . v.(B)
+ \ p- 2.^KK-„)+2n|.— / ^.^ ^iÇn't-nt+s'-s)+2nt+2-:} ]
[371%] * (2352) If we put, for brevîty, T,= i . {n't — nt -\- i' — s) -\-2n t -\-2s, the
term depending on J*', in [3710], mil become
[3711^] 3 „3 «2 2^ e^. COS. ( T; — 2 in) = f «2 «a Fe^. {cos. T, . cos. 2 « + sin. T, . sin. 2 ra | ;
[3711i] if^veput |Fe2.cos.2zj=P; |Fe8,sb.2a=P', it becomes mV. {P. cos. T+P'. sin. TJ,
as in [3713]. In like manner, the terms of [3710], depending on G, H, M, maybe
reduced to the forms [371 li] ; P, P' being functions of the variable elements e, ra, Sic,
■■ ^ and r, T' mdependent of these variable elements ; observing, that n, a, s [1045', 1044"]
are considered as constant, as well as the similar elements of the planet m'.
■f (2353) Using the abridged symbols m,. T, [3711(Z, g], and substitutmg, in [37116],
r3714o] tlie flmction [3711/], instead of the temis under the sign 2, this differential equation
becomes of the form [37146], and the integral [3711c], taken in the hypothesis that P, P'
are constant, becomes as in [3714 c],
[3714i] 0 = ^l^Sdll j^n^,rôr + rv" é. {P. cos. T,+ P'. sin. T,\ ;
712 a2. 1 p. COS. r, +P'. sin, rj
[3714c] r 5 r = .
«1,2 — rfi
We shall suppose r & r, to be increased by the quantity [r (5 r] , in consequence of the
secular variation of P, F, so that mstead of [3714c], we shall have, generally,
[3714d] ràr= ^^^^^^^ +['•5'-].
VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 9
The formula [931] becomes, by putting (^ = 1,*
— — î> • S i \ Values of
a^ndt - ^ j^H.ee'.\i.{n't — nt-Jfi'—s)-^2nt-Jr2e—a—zi'l) [ Sv
1[^) depending
on angles
of the first
è V = ; [3715]
V/l — e2
and by giving to i all positive and negative values, including zero [3704'], [3715']
we shall obtain all the inequalities, in which the coefficient of n t differs
from that of n't by two.
Now as the value of r 5 r [37 1 4 c] satisfies the equation [3714J], supposing P,P' to be constant,
and by hypothesis the value [3714(Z] satisfies the same equation [37146], when P, P' aie [37Ud']
variable by reason of the secular inequalities, we may substitute [3714d!] in [37 14 J], and
then, from the resulting expression subtract the equation [37146], and we shall obtain an
equation of the form [3714/], observing, that we must retain only the terms depending on the
first and second differentials of P, F, namely, dP, dP', d^P, d^P', to the exclusion of P, F, [3714e]
^ d^.{ràr] o r t n I "a ^^.f p.cos. r, + P'.sin.rJ
0=-^^ + n-. ir&r-]+n~a^ (,„.-n^).rf<a ^^714/]
Now we have, generally, d^.{P.co%.T)^d^P,cos. T,+ 2dP.d.{cos.T,) + Pd^.{cos.Ty, [37Ug]
in wliich the term containing P is to be rejected [3714e] ; and if we neglect the term
depending on d^P, on account of its smallness, we shall obtain
d^. {P . COS. T,)=2dP.d. cos. r,= — 2dP.m^dt. sin. T, [371 1<Z, g]. [^W]
In like manner we have
d^. (F. sin. T,)=2d F. d . sin. T,= 2dF.m,dt. cos. T, ; [3714A]
hence [3714/] becomes
0 = ^^^ + n^[r<5r] + -^--^.j2m,.— .cos.T,-2m,.-.sm.r,|. [3714.]
This is similar to the equation [37116], changing rSr into [riîr], representing by aQ
the tenns depending on d P', d P. These ternis being divided by m^ — n^, give,
as in [3711c], the following value of [?• 5 r] ;
r-j,.-! S ^"^ iZ cos T- ^"'- — sb t\ [37144]
Substituting this in [3714f/], connecting together the terms depending on cos. T^, also
those depending on sin. T,, then substituting the value of m^ — n^ [3711c], and dividing
by a^, we get [3714].
* (2354) We have 2 r . d5r -{- dr . ar = 2 d .{r^r) — dr . 5 r, as is easily [3715a]
proved by developing the first temi of the second member, and reducing. Substitutmg
VOL. III. 3
10 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
If the coefficient in'-\-(2 — i).n be very small, and this inequality
be very sensible, as is the case in the theory of Uranus, disturbed
[3715"] by Saturn [4527]; we must put the part of R depending on the
this and [3705a] in [931], we obtain
^ -\- I \ 3 a In at . dli 4-2 an at . a . I — }
[37156] <J w = ■
/(l-e^)
The differential of [3701], being; multiplied by :; — — , becomes
■- a-ndt
[3715c] — ^î^ "= ~ TT • ^^ • ^'"' (« ^ +^ — «) + «■• S'n-2 . (n < +-= — a)^
This is to be reduced, as in [370Sn], by substituting the value of — [3706], using tlie
fonnula [IS] Int., and retaining only the terms depending on the angle T, [371l£-] ;
hence we get
[3715rf] -J^ = -^(F+G).e^.sin.(T-2«)-iifee'.sin.(T-.-.').
[3715e] Again, if we put, for brevity. To = i . {n't— n t + /— e) + 2 n( -\-K, tlie term of R [3703]
will become R = M.cos.T^; hence the differential à R, found as in [916'], upon the
[3715/] supposition that nt is the variable quantity, is dR = — (2 — i) .n d t . M .sm. T^i-
Multiplying this by 3 a.ndt, integrating and using m, [37 lid], we get
oafndt.dR = - . a M.cos. T„ .
[3715g]
To this we must add 2andt .a. (-j—j = 2andt .a . (-j—j • cos. T!, ; and then, by
integrating the sum, we obtain
[3715/.] f[saJndt.dR + 2andt.a.(^-^)]=\^^^^.aM+^Z^l^^
Substituting this and [o715£Z], in [3715i], we get [371.5].
[3715i]
In the great inequalities of Jupiter and Saturn, the most important parts of Sv, Hv'
[37 15 J, &c.] are those depending on the double integration of AR, d!RI, which
introduces the divisor (5 m' — 2n)2. These paits are to be applied to the mean motions
[3715fc] of the planets, as is shown in [1066", 1070"]. As we must frequently refer to these
parts 6v, 5v', of the mean motions <^, ^' of the planets m, m', we shall here give their
values, deduced from [1183, 1204, 3709a], or from the appendix [5794], representing
the chief parts of & v, 5v' [37156, Sic.] ;
[3715Z] Sv deduced from ^ = 3a n .fd t .fd R ;
[3n5m] (5 v' deduced from ^' = 3 a' n'.fd i ./d' R.
VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 11
angle i.(n't — nt-\-s — s ) + 2 ?i ^ + 2 s [3703] , under the following form,*
R= Q.cos. {i . (n' t — nt + s' — s) + 2nt + 2sl
+ Q'. sin. [i . (n' t — nt + s' — s) -^27it + 2sy^
and we shall have,t
[3716]
(6— 3i).«2a { ^--rr } , ^ ,^
la.ffndt.àR^- ji- I q + . ,, ''^ > .sin. \i.{>i't-nt+s-s)J^2nt+2
•'-' \in'+{2-i).n\^ I ^ in'-\-{2-î).n)
[3717]
(6-3»).n a_^ ) Q'___HL_ } .cos.{;.(M7-n<+s'-£)4-2n<+2s}.
{in'+(2— i).nP / ni'+(2-
* (-2355) Using, for brevity, K,= K — 2 s, and T, [.3711^], the expression of [3716a]
R [3703] becomes iî = JIf .cos. (T,+Ar)=JM.cos./i:;.cos.2'— ^f . sin. Z;.sin.T, ; and
by putting ^I.cos.Z,=::Q, — J/.sin.Z,= Q'; it changes into il=.Q.cos.r,4-Q'.sin.T, [37166]
as in [3716] ; Q, Q' bemg like P, P' [3711À;], functions of the variable elements of the
orbits, and T, independent of them. Now we have, in [4077], for Uranus n = 15425*';
for Saturn n'=: 43997'' nearly; hence 3m — ?i'=2278»'; which is much smaller than n [3716c]
or n'; and by putting i^^ — 1, in the divisor in' -{-{2 — i).n, it becomes 3m — n'; [3716(/]
therefore this small divisor must occur in computing the perturbations of Uranus by Saturn,
as is observed in [3715''].
t (23.56) The difierential d R, deduced from [.3716i], considering nt as the variable
quantity, as in [3715/], is
dR = — {2 — i).7idt.q. sin. T, + (2 — i) .ndt.q'. cos. T, ; [3717a]
hence we have
3 a .ffn dt.d R=ffa n^.di^.l{—6 + 3i).q.sm. T,-{- (6—3 î).q. cos. TJ. [37174]
If the integral of the second member of this expression be taken, supposing Q, Q' to be
constant, it ^\^ll produce the terms independent of d(^, c? Q' in [3717]. The terms
depending on d(^, d(^ may be estimated by means of the general formula [1209è],
which, by changing ^, B into Q, A, respectively, and neglecting d^Q, rf^Q, Sic, becomes
ffAqdt^=qffAdt^-2.'^.fffAdt\ [3717c]
From this formula, it appears, that the term depending on —, is easily deduced from
that depending on Q , by changing Q into — 2 .—^.dt, and then integrating relatively
to t, supposing — to be constant. In this way we easily deduce the term depending [3717*^]
on dq [.3717] from that of q-, and in like manner we get the term depending on rfQ'
from that of q.
12 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
Hence the formula [37156] will give*
2d.{r5r) UF-\-G).e^.smAi.(n't-7it4-s'-s)4-2nt + 2s-2ôil )
a^ndt - ( -\-H.ee'.sm.li.{n't—nt-\-e'—=)~{-2iii-\-2s—ss—zi'])
bein;_
variable.
[3719]
^ fin'-\-{2-i).n\^' [_" ^'' in'-\-(2-î).nJ in'-lf-[2—i).n
-^{rr—, ~\ «QH — + }.sm.ii.{n't-nt+s'~i)+2nt + 0g\
Another
form of tins
value of
ÔV,
[3718]
[3718'] For greater accuracy, we have neglected the divisor \/i — e^ in this
expression of (5ï? ; because it does not affect the part of this expression
which has the square of mi' +(2 — i) . n for a divisor, as we have seen
in [1197]; and in the present case, this part is much greater than the
others. Moreover, we must, as in [1197"", 1066", 1070"], apply this part
[3719'] of &v to the mean motion of mf ; and as it is very nearly equal to the
* (2357) Using the value of R [3716], or rather [37166,3711^]; taking its partial
differential, relatively to «, which will aflect only Q, (^' ; multiplying by 2a^.ndt, and
then integrating, we get
[d/lBaj 2 and t . a. I —— = -r- ■ sin. 1 , . - — . cos. i ;
■^ \da / m, \da J ' m, \da J '
m^ being, as in [3711f/]. Substituting this in [3715J], also the values of the terms
[3717, 3715rf], it becomes as m [3718] ; except that the divisor \/(l — c^) is neglected,
[37186] for the reason mentioned in [3718'], namely, that the chief part of Sv or ^ [1195 or 1197]
does not contain this divisor ; and as the other terms are very small, it may also be
neglected in them.
t (2353) The tenns of Sv [3718], having for divisor the square of j'w'-f- (2 — i) ■ n,
[3719a] are those depending on 3 affndt .dR, computed in [3717]; and it is evident, that
this part of S v much exceeds the other parts depending on F, G, H, he. Now,
by [1066", 1070"], or by [1197>'"], the parts depending on 3affndt.dR, must be
applied to the mean motion, and as the other parts, depending on the same angle, are much
[37196] smaller, we may suppose that the whole of this equation is to be applied to the mean
motion, as in [3720]. We may remark incidentally, that the expression of r [1066], as
well as that of v [1070], contains the double integral ffndt.dR; hence, at the first
view, it would seem that if v contain terms depending on this double integral with the small
divisor [in' -{-{2 — ?') . «P, as in [3718], ?• would contain similar terms of the same
order. But we must observe, that these terms of r, v [1066, 1070] are multiplied,
[3719c] respectively, by ( — — - ) , f — — ) , or by their equivalent values a e . sin. (n i -}- s — a),
l-t-2 e.cos. (n< -|- s — «) [669]. Hence these terms of v will be multiplied by I,
VI. i. §1.] TERMS OF THE SECOND ORDER IN e, e', y. 13
whole term depending on the angle i.{n't — nt-\-i' — t) -\-2nt -^-2^,
ive may apply this tvhole inequality to the mean motion of m. [3720]
TIT 1. 11 1, • 1 1 r dP iJP' (JO dO'
We shall obtani the values of — — , — — , — -^, —-i^, by takmg the
(It (It ^ dt dt •' ^
differentials of the expressions P, P', Q, Q', relative to the excentricities
and inclinations of the orbits, the positions of their perihelia and nodes, and
then substituting the values of the differentials of these quantities. But we
may obtain these values of — — , &c. more simply in the following manner. [3721]
Find the value of P, for an epoch which is distant by two hundred years
from the epoch taken for the origin of the time t ; then putting P^ for this
value, and T for the interval of two hundred years, we shall have*
Formula
, ^ for the
yj^ d ± J-. j-> détermina-
1 . j-^i'—f. [3723]
" ' tion of
dP, dP',
T . £ J 1 1 c dP" do dq &c.
In the same manner, we may nnd the values oi — — , — -^, — -^.
•^ dt dt dt
or TÔT
To deduce the expression of — from that of — j-, we shall denote
by -^, the part of — depending on the angle i.{n't—nt-\-i — 6)+2ni+2s, [3724]
and we shall havef
ràr r S^ — \-F.cos.i.{n't — nt-\-s — s)-{-Ge.cos.\i.{nt—nt-\-^—s)-\-nt-\-s — ra\ (
5" — „ ' S '^ C ' [3/25]
[3722]
+ i/e'. COS. ^i . {in—nt-\- s'— ;) -\-nt-\- s — jj'J
S
and those of r by the small quantity e, which will make it of a less order ; it will also be of [3719rf]
a different form from those contained in this article, by reason of the factor sin.(n<-|-£— ra).
dP (PP
* (2359) From Taylor's theorem [617], we have P,=P + T.-— +| T^. — + &c.;
and if we neglect the square and higher powers of T, on account of the smallness of the [3723a]
terms, it becomes as in [3723].
t (2360) Adding — to the part of — [3706], we shall obtain all the terms of
a 'a
depending upon the angles i . [n' t — nt-\-i' — i), i.{n't — nt-\-i' — s) -^nt-\-i, [3725o]
i.{n't — nt-\- s' — s) -{- 2 n t + 2 !. Multiplying this by - , we get [3725].
VOL. III. 4
14
PERTURBATIONS OF THE PLANETS.
[Méc. Cél.
Value of
f), r,
[3736]
for the
angles of
the first
form.
[3726']
[3727]
Computa-
tion for
angles of
the second
form.
Hence we deduce*
-^ = ^+i.(i^+2G).e-.cos.{?:.(n'i — »/ + /— 0 + 2/1^ + 2;— 2^J >
+ lH.ee'.cos.li.(n't—nt-Jrs—i) + 2nt + 2i — z: — -/\ )
2. JVe shall compute, in the same manner, the terms depending on the
angle i.Çn't — nt-\-s' — e) ; and shall suppose, that, by carrying on the
approximation to the first power only of the excentricities, we shall havef
— = F.cos.i.(7i't—nt-\-s—i)-\-Ge.cos.\i.(7i't — nt + s — !) + nt + s — ^J
+ G'e .cos.j — i.(7i't — nt^s' — £) + nf-(-£ — to j
+ iïe'.cos.| i.(n't—nt+s'—s)-}-nt + s—^'\
-{'H'e'.cos.\—i.(n't—nt + s—s)-J^nt + i—^'\ ;
[3726a]
[37261]
[3726c]
[3726d]
[3726e]
[3726/]
[3726g]
[3726;i]
* (2361) Using the symbols [3702«], namely, T=n't — nt-\-e'—s, TF=znt-\-s—a,
W':= n't-{-s' — zs', the expressions [3725] will give, by transposing the terms depending
on F, G, H ; f,..n,^ ca,,-~w^-^/yiXr+ V--UT'
.F.cos.iT—-.Ge.cos.{iT+W) — -. He', cos. (i T-j- W/) ;
r (5, )■
a ' a
rSr
and from [3701] we get - . :^ 1 -)- 4 «^ — ^ • cos. W — | e^. cos. 2 TV; which is to be
substituted in [372Gè]. In making this substitution, we have, by hypothesis, only to notice
terms of the order c^, ee', e'^, &,c. [3702', &c.], and of the same form as [3703]. Now
(W* . /*
the term -^ [3724] being already of the second order, we may substitute for the factor - ,
by which it is multiplied, the first term of its value [3726c], namely 1 ; in the coefficient
of -F, we may use the term — i e^. cos. 2 TV ; and in the coefficients of G, H, the
term — c . cos. TF ; by this means it will become as in [3726^]. Reducing this
expression by means of [20] Int., and retaining only terms of the form [3703], it becomes
as in [3726A], which is of the same form as in [3726].
S^^rjr^ e^ cos. 2 TV) . F. cos. iT-\-(e. cos. TV) .Ge. cos. (i T+TV)
rSr
+ {e.cos.TV).He'.cos.{iT~\- TV,')
= -^+ IFe^.cos. {i T+2?F) + JGe^.cos. {i r+2fr) -(-iffee'.cos. {i T+TV+T¥/).
■j- (2362) The expression of — [3727] is the same as [3706], making the alteration
required by the supposition, that i is positive [3727']. If we use, for brevity, the
symbols [3726a], this formula will become
[3727a]
[37276] — =F. cos.i T-\-Ge. cos.(i T-\-lV)+G'e-cos.(-i T+W)+He'. cos. (i T+ W/)+H'e'. cos.(- i T-\- W/).
5r
The case of i^O, is separately considered in [3755'", Sjc.].
VI. i. §2.] TERMS OF THE SECOND ORDER IN e, e', y. 15
i being positive [3727rt, 6]. We shall then get* [3727]
r ( G'+ G') . e2. COS. i . {n' t — nt + B' — e) ^
+ He c' . COS. { i . {n't — nt^ s' — s) + -.
(E)
[3728]
r 0 r
< -\- jn ee . cos. ^i.yn i — ni -^ s — jj -j- a — -n 5 >
( +H'ee'.cos.^;.(«'<-n? + 6'— £) — a + a'^) , , ^_^^_^^^^
* (2363) In finding the pai-t of rSr dependmg on the angle i.{n't — nt-{-s'—i), or iT,
by means of the fomiula [3702], it is necessary to compute the part of 2/di? + r.( ) [3728a]
depending upon the same angle, or upon R^^ JV . cos. {i T-\-L) [3704]. This [37286]
gives for dR, similarly to [3705^7], the expression àR=nJ\r .i . sin. [i T-\-L) .dt •
2»
hence 2/d R = ^— . JV . cos. {i T -\- L) ; also from [3705a], we obtain [3728c]
Multiplying the sum of these two expressions by 1 = ji^ a^ [3709'], we get
2/dil+..(^) = «^«^5«^.(^)-^-^^.aA-^.cos.(.-T+Z). [3728.]
Again, if we multiply [37276] by 3 n^ a^. {e . cos. PV-}- e^. cos. 2 W], we shall obtam [.3728/]
the terms of [3702], which are multiphed by 3n^a.Sr; and as we have to notice
only the terms depending on angles of the fonn i T [3726'], we may neglect the second rg^gg -,
tenir of this factor e^.cos.2W [3728/], and then it will become 3 74^ a- . e . cos. W.
In multiplying [3727è], by this last factor, and reducing by [20] Int., the term jP
produces no term of the required form, and each of the other terms G, G', H H'
produces one ; hence we finally obtain
3 n» a . 5 r. J e . cos. W-{- e^. cos. 2 ?F|:= f « V. { ( 6+ G') . e^ cos. i T+He e'. cos. {i T-\- W,'— TV)
+11' eé. cos . {i T— W;+ W) ^^^^^''^
= |«V.{(G+G').e2.cQs.i T+Hee'.cos. (iT+a— «') ^^,„^.,
(3728*1
+ifée'.cos.(îT— a + tj').
The sum of the second members of the expressions [3728e, z], being represented by a Q
rfs irSr) [3728ft]
for brevity, the differential equation [3702] becomes 0 = -^^^ — -\-n^.r()r-\-a.Q^; and
we find by mspection, that a Q is equal to the numerator of the second member of [3728], [3728i]
miltiplied by a^. This equation, being solved as in [37116, c], gives rôr = -^-^,
usi^g^, [371k/]; hence we get '^ = „Tg;ê^ = „,(,„_" ^(^^^„) > as in [3728]. [3728.]
16 PERTURBATIONS OF THE PLANETS. [Méc. Cèl.
ç{G—G').e\sm. i.{nt — nt + ;'—s) ■\
Value of
6v.
— Hee.sm.\L{n't — 7it-Jrs' — s) — zs-\-zi'l') } .* (F)
. , ■ .a^A-r-] — r-, — r-;; ■aJS >.sm.\i.{nt — nt-\-^—£)-{-L\
[in'— in \da/ [tn'—my ) / > >
[3729] ^ V =
^1=1
[3730]
r fi r*
If we put -^ for the part of — , Avhich depends on angles of the
rt
[3729,
form i,{n'l — n^ + s' — s),t and is also of the order of the square of
* (2364) The value of 5 d [3729] is easily deduced from [37155]; since the
denominator ^(1 — e^) is the same in both, also the first tenn of the numerator; and
the other terms may be obtained by a calculation similar to that in [3728« — »]. For if we
multiply the expression [3728c] by §andt, and [3728fZ] by 2 a n d t, and take
the sum of the products, we shall get
«] 3aj7idtAR + 2a.ncItui.(~)=\2n.a^('!^) — ^-.a:N'l.cos.{iT-{'L).dt.
•' ' \da J i \da J n—n > \ ■ /
Integrating this we get the two last terms of [37156], which are the same as the two last
terms of the numerator of [3729], or those depending on JV, dJY. The only remaining
term of [37156] is the second, which is found by multiplying the differential of ?• [3701]
[3729i] by ^ ; whence we get — ^^^ = ^ . 5 e . sin. Jr+ e^ _ si„_ 2 Wi.
■' a-ndt a~n dt a
Sr
Substituting — [3727], we may neglect the term c^.sin. 2AF, and the term F, as
in [-3728^, kc] ; the other terms being reduced as in [18, 19] Int., retaining only angles
of the form i T; we get, in like manner, as in [3728A, Sic] ;
—^±^==—-x.smW=l\{G-G'U^.smJT+Hc(^.s\n.{lT+W;-m-Hee\sm.(IT--l^^+W)\
a^n dt a
[^''^^'l =h-\{G-G').e^sm.{T+Hce'.sm.(iT+:s - z/)-Hee'.s\n.(iT -in+^n')] ;
[3729d]
being the same as the terms depending on G, G', H, H\ [3729]. We may remark,
that from the formulas [3728, 3729], we may deduce others similar to [3714, 3718], in
which the secular variations of the elements c, -m, &,c. are noticed.
I (2365) The second member of [3727] being denoted by F', it will include all the
or . ■ ■ ■
r3731ol terms of — , depending on the angle i T, as far as the first power of the excentricities
a
(5 r
[3726']. Adding to this the expression —, depending on the same angle, and on terms
[37316] of the order e-, ee', Sic, we get — =ij"-f— , for the expression of —, containing
VI. i. §3.] TERMS OF THE SECOND ORDER IN e, e', y. 17
the excentricities or inclinations, we shall have
— = ^j^ + i .lGi-G'—F\.c~. COS. i.(n't — nt+ s' — 0 V'''""''
+ i. ATee'.cos.^î. (n'f — «^ + s' — s) + ^— ^'J [3731]
for onglcB
of the
second
form.
+ I . He e'. COS. \ i . (n' t — nt-\-^' — s) — ^ + ra' | .
In these three expressions i must be supposed positive [3727']. [3731'
3. The great number of inecjualities depending on the squares of the
excentricities, and of the inclinations, makes it troublesome to compute all
of them ; and we must be guided in the selection of those which are of a
sensible magnitude, by the following considerations. First. If the quantity
in'-\-(2 — i).n differ but little from ±?i; then the one or the other
of the divisors in' +(3 — i) . n, in'-\-(\ — i) . n, in the formula [3711], ^^^^^^
will be quite small, and by this means the expression may acquire a sensible
value. Second. If the quantity in' -{-(2 — i).n be small, those terms of [3733]
the formula [3715], having this quantity for a divisor, may become sensible, slîecîîtg^
Third. If the quantity i . (n' — n) differ but little from rh n, the one imp^tLt
or the other of the divisors in' — (i-\-l).n, in' — (i — l).n, of the [3734]
formula [3728], will be small, consequently this expression may acquire a
sensible value. Fourth. If the quantity i . {n — n') be small, the terms
terms as far as the order e^, ce', &ic. inclusively. Multiplying this by —, we
r i,r rSr '' 7-1/ t i /- 1 c 1 • *■
get -. — ==; — g- . f . In the first member of this expression, we may put -=1, [37316']
as in [3726(Z], and in the factor of F', we may use the value [3726c] ; hence we shall get
^=:'^ + F'.{ — 1— ic^+e.cos. rr+ i e"". COS. 2 JV I [3731c]
= ^— 5e^--F.cos.ir+F'.e.cos. W; [373W]
the second of these expressions being easily deduced from the first, by observing, tliat of
the four terms comprising the factor of F' [3731c], the first teim, — 1, produces nothing [3731e]
of the order e^, when the value of F' [3727] is substituted ; the second tenn, — i e^,
produces the term depending on F in [373 Ir/] ; the third produces the term depending
on jP' [3731c/] ; and the fourth term, ^ t^. cos. 2 fV, produces nothing of the proposed
form and order. Now substituting, in the term F'.e.cos.JV [313ld], the value of F', [3731/]
or the second member of [3727], reducing the products by [20] Int., and retaining only
angles of the form i T, it becomes as in [3731].
VOL. III. 5
18
PERTURBATIONS OF THE PLANETS.
[Méc. Cél.
[3735]
•General
value of
R.
[3736]
First form.
[3737]
[3738]
[3739]
Values of
^■> y. 2,
[3740]
x', y', z'.
[3740']
[3736a]
of the formula [3729], which have this divisor, may become sensible.
We must therefore estimate carefully all the inequalities subjected to either
of these four conditions.
4. The quantities F, G, G', H, H', are determined by the
approximative methods in the second book [1016, &c., 372T]. We shall
now determine M, N ; and for this purpose we shall resume the value
of R [913, &c.];*
m'.[xx'-^yy'-^ zz')
R^
m
r' being the radius vector of m'. We shall take., for the fixed plane, the
primitive orbit of m, and for the axis of x, the line of nodes of the orbit
of m' upon this plane. If we put v for the angle formed by the radius r
and the axis x ; v' for the angle formed by the same axis and by / ;
also 7 for the tangent of the inclination of the two orbits to each other,
we shall havef
y = r . sin. v, 2^0;
a; = r . cos. v
X =^r . cos. V ;
y-
r . sm. V
^^
f .y . sin.w
(2366) As there are only two bodies m, m', the value of R, X [913, 914] become
n'.[xx'-\-yy-\-~z') X X
R:=
m'
\ >
M
.■■'\
[37366] and by using r'^ = a'^ + if -{- z!^ [914'], we get [3736].
f (2367) In the annexed figure 72, C is the origin of the co-ordinates, or centre of
the sun; C X, C Y, C Z, the
axes of X, Y, Z, respectively ; M
the place of the body to, supposing
it to be situated nearly upon the
[3740o] plane of xy [3737] ; M' the place
of the body in. The co-ordinates
of TO ai'e CA=x, AM^y, z=0 \
nearly ; those of to' are CA' = x',
A'B'=y', B'M'=z'. Moreover
angle.MC.^=«[924^],.lfC.^W,
Then in the rectangular triangle CAM, we have C A — C M .cos. A C M,
AM=C AI. sm. A CM, or in symbols, a: = r.cos. «, y = r .sin. i) [3740]. In the
VI. i. §4.] TERMS OF THE SECOND ORDER IN e, é, 7. 19
Hence we get, by neglecting the fourth powers of 7,* [3741]
R = -j^ . COS. {V I') J- . 3-3 . i COS. (V V) COS. (v^v)\
m r , , , m.y' r
— - .COS. (V I') . — . !COS. (V' V) COS. (V'-\-V)\ Second
.,.'2 V / ^^2i V -' V'/l fo,„ „f
R.
<i!.y~ rr'.{cos.(t)' — 1;) — cos.{v'-\-v)\ [3742]
|r2— 2r;-'.cos.(D'— t>) + r'2}4 4 ' jr2_Orr'.cos.(D'— t') 4-r'2}2-
We shall suppose, as m [954, 956],
-^.cos.(n'i— n^ + s'— f) — Ja-— 2aa'.cos.(n'^— n^ + s'— s) + a'^î~^ [3743]
= I x.A^'K COS. i.(n't — nt-\-s — 0 ^t-',B<o.
!«-— 2«a'.cos.(n'^— ni + s'— .^) + rt'2|-f=i2.jB»cos.ù(n7— n^ + £'— 0; [3^44]
rectangular triangle CAM', we have C^'=CJ/'.cos.^'CJ»f', ^'Jf ^^CJU'.sin.^'CJ»/;
or in symbols, x'=r'.cos.v' [.3740'], .4'J'/'=r'. sin. j;'. In the rectangular triangle A'B'M',
we have, A' E=A'M'.cos.B'A' M', B' M'=A'M'.5m.B'A' M' ; substituting in these [3740c]
1 r
the preceding value of ./2' ./»/', also cos.B' A! M' ^=-—-, , sm.B'A'M= ,
/(1 + 7-) v/(l + 7^)
we get y', z' [3740'].
* (2368) If we neglect 7^ as in [3741], we shall have (1 + 73)-*= l_ | y2 .
hence we obtain from [3740'], y' = i-' . sm. v' — J 7^. r*. sin. u'; z'^='y^ .r'^.sm.^v' • [3742a]
substitutmg these and the other values [3740, 3740'], in the first member of [37426], and
then reducmg by [24, 17] Int., we get [3742c] ;
^ '^'+ i/ /+- ~'= '■'■'• (cos. u'. cos. ij-f-sin.u.sm.t;') — J 72.7-/. sin. jj. sin. o' [3742i]
^.irr'.cos.^!;'— î;) — i72.rr'.jcos.(t)'-i')— cos.(«'-fî))}. [3742c]
Substituting this last expression in the first tenn of R [3736], we get the two first
terms of [3742]. Again, if we develop the first member of [3742e], and substitute
r^=x"-+y^+z^ r^=x^+y'^+^'^ [3740,3740'], also the expression [3742c], we get ^
(^-^)'+ {y'-yf+ (^-~)'=(-^"+2/'+~-)-2. (^^'+yy+zz')+ (x'==+2/'2+^'2) [3742e]
= ^-2-2;y.cos.(«'-v)+r'2}-|-i72.rr'.{cos.(î,'-i,)-cos.(z>'+^)}. [3742/]
Invc^lving this to the power — ^, we get
\{^'-xf+{y'-yfJr{^'-zf\-i=\r'-'2rr^.cos.{v'-v)-^r'^]-^
_3 [3742^:]
^ Y^.rr. \coz.{v'—v)—co%.{v'\v) \ . {r^— 2r/.cos.(/— t;)-!-/^ j 2.
substituting this in the last term of [3736], we get the two last terras of [3742].
2» PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3744'] and shall represent R = M.cos.\i. (n't —nt-i- s'—s) + 2nti-K\ [3703] , by
the followmg function ;
K™f]f ^= M'-'Ke'' .cos.\i.(n't—nt + s'—:) + 2nt + 2s—2z.\
[3^45'] -\-M^'Kee'.cos.\i.(n't—nt+s — s) + 2nt-\-2s—^ — ^'\
of the first
'?^^^^«-, + M^~\ e'^cos. I i.(ii't—nt + i—s) + 2nt+2i—2z,'\
[3745'"] + M^'K y^ .cos.\i.(n't—nt+s'—s) + 2nt + 2s—2n \ ;
n. n being the longitude of the ascending node of the orbit of m' tipon that
[3746] fyj- j^^ counted from the line which is taken for the origin of the angle ni + f.
We have, as in [669],
[3747] -= 1 + ^e- — e. COS. (ni+£ — -a) — i e". cos. 2 . (nt-\-£ — ^i) :
[3748] v = nt^i — n + 2e. sin. (n i + s — ^) + | ^'' sin. 2 . (n t + e — a).
From these we get the values of —, v', by marking with one accent, the
quantities n, e, s, &c. Then we have, as in [955], the product of
2 . A (". cos. {i . (7i't — nt + .='— s)|,
by the sine or cosine of any angle ft-\-I\ which is equal to
[3749] 2 • A'' ■ "^i li-(n't-n t + /— =) +ft +I\.
Hence we easily obtain*
[3750] M-^= f.J.-.(4^-5)..^(') + 2.(2.-l).«.('^) + «^('^)5;
Values
pondins 4^^-^ ^ ^ ^ \ da J \ da J \ dada / ^
to tlie first
^^, M-.= |.^,;_.,.(4;-3).^--..(.,--a,...(^')+...C^)^,
[3750'"] M '31=— — . « a'. £''-'1 .
8
[3750a] * (2369) In [952, 953] we have r = o . ( 1 + wj ; v = vt -}-s — U-\- v,; the terni n
being added to conform to the present notation. Comparing these witli [3747, .?748],
[3750t] we get the following values of u^ , v^ , also the similar ones of w/, w/, using the abridged
symbols [3726a] ;
[3750c] «,= — e.cos.^F + 4 e^ — J e^. cos.2 ^F ; t), =2e .sin. ^F + f . e^siii.2 TF ;
[3750rf] «; = — c'. COS. W'-\-^e"^—l e'- . cos. 2 W ; v;=2e'. sin. ?f '+ f . e'^ . sin. 2 W;
M. ï. §4.] TERMS OF THE SECOND ORDER IN e, e', y. 21
and in the case of i=\ [3150y, ?/'], we have
4 a'* 8
Finding the squares and products of these quantities, then reducing them by [17 — 20] Int.,
retaining merely the temis of the second degree in e, e', y, which are the only terms now
under consideration [3702'], we obtain the following system of equations. In these
expressions we have substituted for JV its value W =^ T -\- IV -\- zs — -a' [3726a], [3750c]
in order that the quantity n't-\- e' may not appear in the tenns of R, except in connexion
with i, as in the assumed form of these terms of R, given in [3745, &c., 957]. The
numbers prefixed to the formulas [3750/] express the order of the terms in the
value of R [957].
= Je'2 — Je'2.cos.2.(r+ ?F+t3 — ^);
= f Ê'2.sin.2.(r+ ?r+« — î/);
= fe^.sin. 2^;
= ie^-f le'-i.cos. 2 7F;
'== è ee' . cos. (r+ w — ^') + A ce', cos. (7+ 2 W-\--a—Ta') ;
= Je'2 4-Je'2.cos.2.(r + ?F+« — ^');
= — fe'.sin. (T^-^ — ^)— ee'.sin. {T +2 TV+zi—z,');
= — e^.sin. 2fF;
' = _ e'^.sm.2.{T-i-W+z, — z/);
,= ce'.sin. (T+w — •n') — ee'.sin. (r+2 W -}- ■a—zy') ;
= 2e'2 — 2e'2.cos. 2.(T+ ÏV -^ zi — zi') ;
= 2ee'.cos. (T+1^ — to')— 2e'2.cos. (T+2 ^F+a— to');
= 2e^ — 2e^cos.2W.
Substituting these in [957], we shall obtain the terms of R depending upon M'-''\ M'-'^\ M'^',
[3745, kc.]. The ternis of the fonn iVT^', arising from the terms of z, a/, in the two [3750g]
lower lines of the value of R [957], will be considered hereafter in [3750m, &;c.]. In
making these substitutions, we must use the following formulas, which are the same as those
in [954f, 955a, 955/], changing TV into TV^, to prevent confusion in the notation.
COS. JV^.iX.A <'> . cos. i T == i 2 . ^« . COS. ( J T + TV,) ; [3750^]
sin. TV, . J 2 . iA^'l sin. {T= — ^2.iA « . cos. (i T + TV,) ; [3750i]
COS. fT .is.i^^ra. COS. i r=i 2 . i^^w.cos. {i T+ W,). [3750A]
VOL. III. 6
2
"/
3
"/
4
Î'/
5
^',
6
<-
7
v,u;
8
nr-
9
u,v;
10
uv.
11
w>;
12
ujv.
13
v',^
14
v,v,'
15
r,2
[3750/]
22 PERTURBATIONS OF THE PLANETS. [Méc. Cèl.
We shall represent R = N. cos. \i . (n' t — 7it + I'—s) + L\ [3704]
We shall consider the terms depending on each of the factors M'-''\ iV/'", M'--'' [3745, &;c.]
separately; and in the first place, shall take the tenns of the form M'^Ke^-cos. {iT-\-2W).
These are evidently produced by the factors sin. 2 IV, cos. 2 TV, which occur in the terms
of [3750/], marked 2, 5, 6, 10, 15; reducing the products by the formulas [3750A— A:].
These five terms, marked in the order in which they occur, without reduction, supposing them
all to have the common factor ^ • c^- cos. {i T-(-2jF), and omitting 2 for brevity, are
[37502]
This expression is easily reduced to the form of the coefficient of —, in the value
of M'-"'' [3750]. Proceeding iii the same manner with the parts of the terms 7, 9, 12, 14
[3750/], depending on the angle T+2?F-f-w — zs', we find that they produce
in R [957] terms of the form iV/^'^e e'. cos. {{i + 1) . T+ 2 JV -{- zs — zs" } , which may
[3750n] be represented by — ^ . e c'. cos. {(i + 1) . T+ 2 JF+ is — î/^, multiplied by the
4
following expression, which includes the terms as they occur, without any reduction ;
[3750O] -««'(7a^) + 2*«-(^)-2-'(77-) + 4^'-^"-
We may change in this i into i — 1 [.3715'], and then we get for the coefficient
[3750;)] of — j.ce'.cos.(ir+2yF+ro— ra'), or — j . e e'. cos. {i T~{-2nt -{-2 s—zi— a'),
an expression which is the same as the coefficient of — — , in the value of il/"' [37.50'].
Again, the terms 3, 4, S, 11, 13 [3750/], depending on the angle 2. (r+ (F+ra— n'),
produce in R [957], terms of the form M^-\ e'^ cos. { [i + 2) . r+ 2 fF+ 2 ro — 2^} ;
[3750?] which may be expressed by ''^ .c'^.co5.\{i-\-2).T -{-2W -^2is — 2«'|, muhiplied by
the following function, which includes all these terms as they occur, without reduction ;
or, as it may be written,
[3750,-'] i.(4i + 5).^'''-2.(2i+l).«'.('^) + a'^.(^).
We may change in this i into %■ — 2 [3715'], and then we have for the coefficient
[37505] of -.c'2.cos.(iT+2?F+2^— 2j:'), or ^. cos. (» T+2ni + 2s — 2^'), the
8 o
m'
[3750«] same quantity as the coefficient oî —, in the value of M^-' [3750"].
VI. i. §4.] TERMS OF THE SECOND ORDER IN e, e\ 7. 23
by the following terms ; * Terms of
^, -.-.-,«. . . - » depentlinff
R= ^W, COS. I. (n't — 71 t+s'—e) [3752]
on angles
+ N^'K ee'. cos. [i . (n't — n t + e'— s) + w — ^'| "'[ih^-]
second
+ iV'^' . e e'. COS. ^ z . (n7 — ?i ^ + e' — s) — ra + ra' | ; 'farsa"]
We shall now notice the terms depending on z, z , which were neglected in [3750^] ;
these are the same as those depending on 7^, in the value of jR [3742]. As we neglect
terms of a higher order than 7-, we may substitute, in these terms, the values r = a;
r' = a'; v=^nt -{-s — Ii ; v =^n' t-\- ^ — U ; v' — v = n't-—nt'\-s — s=T; [3750u]
v'-\-v = n't-[-nt-{-^-{-e — 2ll=T-{-2nt-{-2s — <iin; hence this part of
R [374-2] becomes
R = —'l^ .-^.{cos. T— cos.(jr+2n< + 2e — 2n)j
,^ , ■ [3750i>]
* \a? — Saa.cos. T-|-« P
Substituting, in the last term, the value of the denominator [3744], namely \ ^.B'^'K cos. iT,
and reducing by means of the formula [3750^], it becomes
m'ya f _-^.cos. T+4"-cos. (T+2n<4-2s— 2n) )
^=~T~-< a- a~ V [3750k'1
(+|aa'.2.J5('\cos.(J+l).r-iaff'.2.5w.cos.{(i+l).r+2»U + 2s-2nn
The last temi of this expression, changing i into i — 1 [3715'], becomes
X.aa'. 2. B''-i'. COS. (ir+ 2 îii + 2£ — 2n); [3750i]
which is of the same form as [3745'"], and is equal toit by putting M'-^'^ = . a a' . S . B^''~'\
8
as in [3750'"]. In the case of i = l, the term [3750z] becomes
— ^.^a a'. S^»^ COS. ( T + 2 n ^ + 2 £ — 2 n) ; [3750t/]
connecting this with the second term of [3750f<'], namely,
^.^.cos.(T+2n< + 2j— 2n); [3750)/']
and putting the whole equal to this value [3745'"], we get, for this case, the same value
of JW ">, as in [3751 ].
* (2370) By proceeding as in the last note, we shall find, that the substitution of the
values [.3750/] in iî [957], produces terms depending on the angle i T, iT+zs — ra', [3752a]
ïî"— « + «', as in [3752—3752"], without W, which occurs in the fomis [3745—3745'"].
24 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
and we shall have
CoefB-
;3753] JV(»)=-^.5(e«+e'=).r4i''.^«>-2a.f^')-a».f^)]-$.aa'.[B('-') + 5«^»]^;
depending ^C. L \ da / \ da- /J 2 )
on angles
, 4f^' ^ ' \ da / ^ ' \ da J ' \dada JS
form.
m.., ^».= -.^,.,+„,^..„+,.(..+,,..(^')+,.,+,).,.(^>)+„,(^)|.
We shall calculate these terms separately, commencing with the angle i T, which is
[37525] produced in R [957], by the substitution of the terms ^e^, Je'^, occurring in the
terms of [3750/], marked 2, 3, 6, 8, 13, 15. These quantities produce in R, the
expression — .cos. i T, multiplied by the following terms, WTitten down in the order
in which they appear, without reduction, and omitting 2 for brevity ;
Now if we multiply the first of the equations [1003] by — 1, and the third of these
equations by — | ; the sum of their products will give
, /dA^i)\ . ,„ /ddA<ii\ fd.»o\ AW^(.)\
substituting this in [3752f], we find, that the coefficient of e'^ is the same as that of c',
and the whole expression becomes
[3.5.,, _^.(..+ ..)4.i,^»_„.(l^)-i,...(^-^);.eo..-r.
To this we must add the third term of [37502^], depending on cos. {i -\- I) . T, which,
[3752/] by changing i into i—l, as in [3750c), becomes — . J aa'. 2. jB^'-i'. cos. i T. The
expression [375'2e] is the same for —i, as for +i; because A'-'^ = A'''^ [954"].
Moreover, the term [3752/], by the same change of i, using J?(-i-i) == ^c.+D [956'],
[3752g-] becomes '^ . i a a'. 2 . J5''+" . cos. i T. Hence, if we use only positive values of i, we
must double the fonction [3752e], and add to it the two expressions [3752/ g] ; the
sum of these three ftmctions, being put equal to N'-^'' . cos. i T [3752], gives the same
value of iV^*", as in [3753]. In the case of i = \, this sum must be increased by the
[3752/il first term of [3750w] ; by which means iV^°^ is increased by the quantity given in [3754].
The case of i=0, which is separately considered in [3755'''], produces, in R, the
following expression, which is deduced from [3752c,/], by putting i=^0;
VI. i. §4.] TERMS OF THE SECOND ORDER IN c, e', y. 25
In these three last expressiotis i is supposed to he positive and greater than [3753'"]
zero. Incase i = \, we must add to iV"" the term — ^. ^^ [3752/i]. [3754]
It is more convenient, for numerical calculations, to have the differentials
relative to only one of the two quantities «, a', in these formulas.*
Proceeding m the same manner with the angle iT-^-zi — w' [3752'], we find, that
terms of this form are produced in R [957], by the substitution of the parts of the terms
of [3750/] depending on the angle T-f-ra — i^, and marked 7, 9, 12, 14; reducing
them by means of the formulas [954c, 955a,/]. Hence this part of R becomes equal
to ~.ee'.cos.\{i -\-\) .T -\--a — -m'], multiplied by the following expression, retaining [37524]
the terms according to the order of the numbers, without any reduction ;
aa'.f -— — ) — 2ia. ( — - ) — 2ia'. — — ) + 4 i^. ./î*'^. [3752q
\dada / \ da / \da /
Changing i into i — 1, in [3752À:, Z], we find, that this part of R maybe represented
by . ee'. JV^''.cos. {i T+ra — ra') [3753'] ; observing, that this change in the value [3752m]
of i, reduces the expression [3752Z] to the same form as the factor of —, in the
value of iV"' [3753']. We must retain only the positive values of i in [3752', 3753'] ; for
if we ciiange the sign of Î, the expression cos. (i T-j-w— ■zs'), becomes cos.( — iT-\-vs—-a') [3752)1]
or COS. {i T — ■n-j-'sj'), which is of tlie same form as [3752"]. Hence it appears, that we
may deduce JV'-' [3752"] fi-om iV*'> [3752'], by changuig the sign of i. Performing [3752o]
this operation on [3753'], we get [3753"], using ^^-'-i> = ^('+i> [3752/']. Finally, the
case of i = 0, is found by putting i = 0 in [3752ot], or in the similar terms depending [37520"]
on JV^2i [.3752o]; observing, that when i = 0, the expressions JV'^', JV'2> [3753', 3753"]
become equal to each other ; and this part of R becomes
f . ... [ 4^...+ .„ . (1^) +.».. (•-■) +,y. (1^') I . e„. („-.,. ,,.„
* (2371) In making the reduction of M<" from [3750'] to [3755], it will be convenient to
use the abridged symbols a™ . (-^^j = ^ï ; a"" . (Ç^') = .//'<::' ; and as the [3755o]
index n is the same for all the terms depending on M''^\ we may neglect it, and put simply
' [37556]
VOL. III. 7
26 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
This is obtained by means of [1003], from which we get
[3755] JtfU,__^.^(2,-_2).(2,-_i).^(-«+2.(2;-l).«.(!^) + «^(î!^')^;
[3755'] Jlf(^)= |\^(4i^_7i + 2).^^-> + 2.(2z-l).«.('-fp) + «^(''';;;^^^
Reduced
values.
[3755"] ./V-= ^.^(2._2).(2.--l).^<--2«.('^)-«^(^)5;
[3755'"] JV^^>== -'.^(2^• + 2).(2^• + l).^u4-n_2«.(^'^^_„..(^''f^^^.
[3755i>] 5, The case of i ^ 0 deserves particular attention. We shall
resume the expression [923], and shall consider, in the first place, the
and the same symbols may be used in the reduction of Jlf' -' , .A'"'" , JV'-' . Then the
coefficient of — ^m', in the value of M'-^'' [3750'], will become, by the substitution of
the first and second formulas [1003],
[3755c] = 2. (i — 1) . {2 . (i — 1) + 1 } . ^„+ ^4 i — 2^ . ^, + A.,
^{2i — 2).(2i — l).A,-^2.{2i — l).A, + A,;
which is the same as the coefficient of — J m' in [3755]. In hive manner, the coefficient
of — , in [3750"], becomes, by using the first and third of the formulas [1003],
8
(;_2).(4i— 3).^o + 2.(2i — 3). lA, + A,l + {2Ao + 4.A, + A,\
f^^^^'^J =\{i-2).{ii-3)+4i — 4l.^, + 2.{2i-\).A,+A,;
which is easily reduced to the form of the factor of —, in M^~'> [3755']. Again, the
factor of i m, in the value of iV'" [3753'] becomes, by the substitution of the values
in the first and second formulas [1003] ;
A.{i — \f.A,—2.{l—\).A, + 2.{i — \).\Jl^+A,\ + \ — 2A,—A^\
^2.{i—\).\2.{i—\) + \\.A, — 2Ay — A^;
which is the same as the coefficient of \m, in tiie value of jV"' [3755"]. From this
we may easily obtain A*'-', by merely changing the sign of i, as in [3752o].
* (2372) The terms of R depending on i = 0, are given in [3752», 3752p] ; they
are independent of n t, n' t, and produce in ^ d a secular equation [3773] ; and on this
account, they are carefully computed, though it is finally found, in [4446, 4505], that
[3755e]
VI. i. §5.] TERMS OF THE SECOND ORDER IN e, e', 7. 27
term
— 5^ ^ ^, of the expression of d5v, given by this [3755"]
On the
secular
'■dv
formula. We have, as in [1037], by noticing only the terms affected with pi" of
the arc of a circle n t*
r
a
S r
a
— I m'. (h C + h' D).nt. cos. {n t + e)-
^ 1 _ /i . sin. (« t-\-i) — l. COS. (n t + i); [375G]
= \m'.{lC + l' D).nt . sin. {n t + s) [3756']
they are insensible. To reduce these terms of R to the form [3764], we may use the
following symbols, given in [1022, 1033] ;
A:=c.sm.«; Z = e.cos.is; A'= e'. sm. w' ; r = e'.cos. ra' ; [3756a]
e2=A2+/2. e'2 = A'2+Z'2; [37566]
y.sin.n^y— p; y .cos.n = q' — q ; 7^=(p'—p)-+{q'—lf- t^^'^^'^l
Now substituting, in [3752t], the values of c^, c'^, y^ [3756e, c], they will produce,
respectively, the first, second, and fourth lines of the expression of R or ô R [3764] ;
observing, that, by using the sign S, as in [917'], these terms of R may be represented [3756rf]
by &R. The term [3752j7] produces the thii-d line of the same value of 5R; for
we have, by using [3756a],
e e'. cos. (a — -n') = e e'. (sin. ^s . sin. a' + cos. -m . cos. z^) = A h' + / Z' ; [3756e]
substituting this in [3752p], it produces this term of &R [3764], having the factor hh'-\-ll'.
This value of oR is to be used in the formula [923], to compute the part of 5v, which |-.,„„
is independent of the angles n t, n't; and of the second degree in A, A', /, V , &.C.
* (2373) The object of the present computation is merely to ascertain the part of à v,
mentioned in [.3756/], by means of the expression of dàv [923]. This may be reduced
to the form [-3757^], by observing, that r i?' = r . ("^ W « . (^) [928', 962], [3757o]
and that we have, identically, 2 r . 5 R' -{- R' . 6 r ^2 & . {r R!) — R' . 5 r. From the first [37574]
of these equations, we see that R' is of the same order as R, or of the order m ; and
by rejecting tenns of the order m'^, as in [-3768'], we may neglect the term — R'.Sr,
and then this expression [37576], by the substitution of r R' [3757a], becomes
'd'>R\ [3757e]
2r.5R'-\-R'.ôr=2S.{rR') = 2a.( ^) .
Substituting this in [923], also the value of r^dv [.3759], we get
'dûR
d.{2r.d5r+drJr) + dt^.\^3fôàR+^a.(^)] ^^^^^^^
1 r
a'J.nrf<.v/(l— e9)
28 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
These give, by noticing only the terms depending on the squares and
[3757] p^fjdy^cis qJ- ji^ i^ ]i'^ i'^ independent of the sines and cosines of nt + e,
and its multiples *
[3758] d.(2r.d5r + dr.&r) = —''^^^^^^.\(h'+n.C + (hh'+in.D\.
In this we must substitute SR [3764], and those terms of dr, S r, which produce
quantities of the form and order mentioned in [3756/]. Now these quantities will be
obtained by selecting, from the general value [1037], the three terms contained in the
r Ô V
[3757e] second member of [3756], for -; and the terms in the second member of [3756'], for — .
It is unnecessary to use any other tenns of a higher order in h, 1, &c. ; for if we retain,
in -, any teim of the order h^, hi, 1% connected with sin.2.(?i^4-6) or cos.2. (m^ + s),
a
it must also be connected, in [3757rf], with terms of —, or of its differential, of the same
[3757/] forms and order, producing terms of the fourth order in h, 1, and independent of the
angles n t, n' t, which are neglected in this article. The same remarks will apply to
other terms of - , depending on higher multiples of the angle nt-{-s. Having adopted
[3757g-] this form of -, it will be unnecessary to retain any terms of — [1023, 1037], except
a o
Sr
those in the second member of [3756] ; for, though other terms in — [1023], of the
[3757/i] forms P, P'. sin. ()i / -(- s), P". cos. {nt -\- s), might produce, in 2r .d è r-\- dr .&r,
quantities independent of the sine or cosine of the angle nt -\-e, or its multiples ; yet
if we notice only terms of the order m', they will vanish in its differential, which occurs
in [3757d, 3760] ; and this does not happen with the arcs of a circle retained in [3756'],
as is shown in [3760].
* (2374) In finding the terms of 2 r . tZ i5 r -|- <7r . 5 r, of the order m', it is only
[3758o] necessary to notice quantities of the form Q ■ « t.dt, containing the arc of a circle n i,
Q being constant ; for if the function contain any constant term, or elements of tlie planet's
orbit, it will either vanish from its differential ' [3760] or become of the order 7«'^, &;c. ;
and terms depending on the sine and cosine of nt-\-s, ai-e neglected [3757]. Substituting
r [3756], and its differential, in the first member of the following expression, we get
[37585]
2 r . d 5 r -{- d r . 5 r = \2 a — 2 ah . sin. [nt -\- e) — 2al. cos. [ni -\-e)\ .d S r
4- I — ah . cos. {nt-\-s) -{- al . sin. [nt -\-e)] . n dt . or ;
in which we must substitute the values of S r , d Sr. Now if, for a moment, we
[3758c] put im'.a.{lC-Ji-rD) = L, im'. a .{h C-{-h'D) = H, we shall get, fiom [3756']
VI. i. ^5.] TERMS OF THE SECOND ORDER IN e, e', y. 29
We then have r" dv = a" n d t . \/ï^^ [1057]; hence we shall obtain [3759]
d.{Or.dSr+dr.Sr) m'.ndt <(^j^,_^pyc+(hh' + ll').D\. [3760]
r~ dv 4
We have, in [1071],
(0, 1) = — 1 m' nC; [^^^rn'riD; [3761]
therefore *
d.{2r.dSr-{-dr.Sr) _,_^^,^Q^^y ^,^,j^p^ _ ^^ _ (hh'+ll')\. [3762]
r^ dv
We shall now consider the term — ^-^, , of the same formula [923]
r-' dv
[3758d]
and from its difterentlal, the following expressions, retaining only the tenns which contain
the ai-c of a circle, as in [3755'] ;
S r =L .n t . sin. {n t -{- s) — H .nt . cos. (» t-\-s);
dSr=L.7i^. tdt .COS. {71 t-j-s) -{- H . nP. t d t .sm. (jii + e).
Substituting these values of &r, dSr, in the first members of the equations [3758e],
reducing by [17 — 20] Int., retaining only the terms containing the arc of a circle,
independent of the sine or cosine of nt-^ e, we get
'2a.dSr = 0;
— 2 a h. s'm. {nt -{- e) . dSr = — ahH .rfitdt ;
— 2al.cos.{nt+s).d6r = — alL.n^tdt; [3758e]
— ah . cos. (nt-\-s).ndt.5r^iahH.7i^tdt;
-[- al . sin. [nt -{- s) .nd t .5r = ^ al L . n^t dt .
The sum of the tenns in the first members of [3758e] is equal to the second member
of [37586] ; consequently the first member of [3758J] is equal to the sum of the second
members of [375Se] ; hence we get
2r.dSr + dr.5r= — iahH.n^tdt — ialL.n^tdt. [3758/]
The differential of this expression becomes, by resubstituting [3758c],
d.{2r.dSr + dr.Sr} = — in^a.dt^.{hH-{-lL)
Dividing this by the expression of r'^dv [.3759], neglecting the divisor \/{l — e^), which
only produces terms of the fourth degree in h, h', e, &c., it becomes as in [3760].
* (2-375) Substituting the values [3761] in [3760], we get [3762]. [3762a]
VOL. III. 8
[3758g]
30 PERTURBATIONS OF THE PLANETS. [Méc. Cèl.
or [3757 d'\. If we notice only the secular quantities depending on the
[3763] squares and products of the excentricities and inclinations of the orbits, we
shall have, by the analysis of the preceding article [37ô6d—f],
Pan of V \ ^ \
6R,
..™- +f(Mwj4.... + .„.('-)+2...(M^)+„..(J^),
ponding to
t=0.
[3765]
[3766]
m
p, p'l q, ([, denoting the same quantities as in [1032]. Hence we easily
obtain, from Book II, ^ 55, 59,*
aw.<5i2 = — 1.(0, l).{/r + r + /t'2 + Z'2| + [^].j/t/t'+n'i
which gives f
an.àùR = dh.\ — {Q,\).h+\^.h'\—dl. \(0,\).l—[^.l'\
-(0,1). dp. (p'-p)-(0,l)-dq.(q'-q).
* (2376) If we multiply [3764] by an, we shall get the value of aii.ôR, which
may be easily reduced to the form [3765] by the following considerations. The coefficient
[3765a] of h^-^P is equal to '— [1073], and the coefficient of h'^-^-l'^ is of the same
value ; as evidently appears by the substitution of the expression [3752rf]. The coefficient
of {p'—p)^+ir/—fjf, in this product, is ^ m' ?i . «^ a'. B"'== J . (0, 1) [1130].
Lastly, the coefficient of h h'-\-Jl' in this product, is evidently equal to | m n, multiplied
[37656] by the expression of D [1013], and this is shown iii [1071] to be equal lo [ôTÎ],
as in [3765].
t (2377) In taking the differential of [3765], relatively to the characteristic d [37056],
we must consider h, I, p, q as the variable quantities, and h', I', p, g' as constant ;
and then we shall get
an .do R = — {0,1) . {h d h -j-ld I) + [^] . {h'd h -^r d I)
[3766a]
^ (0,1). \-{p'-p).dp-(q'-q).dri\;
being the same as in [3766], with a slight alteration in the arrangement of the terms.
VI. i. <^5.] TERMS OF THE SECOND ORDER IN e, e', y. 31
The second member of this equation becomes nothing, in virtue of the
equations [1089, 1132] ; therefore we have*
a ?t . d (5 K = 0 ;
[3767]
hence we deduce, by observing that n" a^ = \ [3709'] ,t [37671
3dt.fdt.d&R_ 3m'.gdt
m''g being the arbitrary constant quantity added to the integral fdôR [1012'].
It now remams to consider the function „ ,— 1 , which
r- d V
occurs in the expression of d&v [923]. If we neglect the square of [3768']
2 S . (r R') d t^
the disturbing force, this function will be reduced to — „ , — ,
r^ dv
*
(2378) Taking into consideration only two bodies, m, m', we get, as in [1072],
^=(0,l).?-[irr]./'; ^ = -(0,l).A+[ôZ].A'. [3767a]
Multiplying the first of these equations by — R/, the second by dh, and adding the
products, we find, that the sum of the terms of the first member vanishes ; consequently [37676]
the sum of the terms in the second member, being the same as the terms depending
on dh, dl, in [3766], must also vanish. Again, we have, in [11.31],
^=(0,1). (<?'-<?); ^'__(o,l).(/__p); [3767.]
multiplying these, respectively, by — dq, dp, and taking the sum of the products; the
first member becomes identically nothing, and the second member is the same as the terms [3767rf]
depending on dp, dq [3766], which are therefore equal to nothing, as in [3767].
We may incidentally remark, that the quantities (0, 1), [Ôj]], &c. [3761] ; also dh, dl, &ic.
[1102, 1102ff], are of the order m' ; consequently the second member of [3766] is
of the order m'^ ; but its integration, in [3768], introduces divisors of the order
g> gi> ,?î' ^c. [1102, 1102«], which are of the order m' [1097t] ; by this means, the [3767e]
integral fdt.àèR [3768], is reduced to terms of the order m, like the other terms
computed in this article.
t (2.379) The integral of [-3767], using the constant g [1012'], is an.fàSR = an.m'g;
lultiplying this
we get [3768].
multiplying this by , and then dividing by r^ d v z^ a^ n d t . \/ {I — è') [3159}, [3768a]
32 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
^ „ /dSR\ ,,
[3769] or by [928, 962], to V^ .* This quantity produces,
m.ndt.a-'.i— — I
[3769'] in the first place, the term .^-—^ " ,t which is to be added
?iin' p'lJ t . , .3 m'. as: ndt
[3769"] to , '; [3768], or to the equivalent expression /-;— ^ ,
deduced from ?r «^ = 1 [3767']; and the sum vanishes by the
substitution of g = — -^a.l— — j [1017].
Resuming the expression of ôR [3764], we shall observe, that the function
[3771] I' .aa'.B^'K{(p'-py+ (ç'-ç)^} + &c.î
* (2380) We have, in [3757&, c], by neglecting tlie square of the disturbing
[37690] force, 2 r 5 iî' + R ' 5 r = 2 5 . (r iî ' ) = 2 a . (jj^) • Multiplying this by d t"- and
by l = n^tt^ [376T], and then dividing by r^ d v:^ a~ nd t .</{\—e^) [3759], we
2 a^. I —— ).ndt
[37695] get ^ °" \ [3769], for the corresponding terra of d>]v.
t (2381) The value of R [957], or rather [1011], gives, for the case of { = 0,
[3770a] and for terms mdependent of nt, n't, S R = I i7i' . A'-''\ Substituting this in the term
/djKOA
of dSv [37696], it becomes as in [3769']. Now if we substitute g-=— Ja.(-^j
[37706] . m'.ndt.a^ /rf.^(oi\ , ,. .
[1017], in the term of dêv, [-3769"], it becomes — ";,^_^o, ■\d^)' '^
destroyed by the equal and opposite term obtained in [3769'] ; so that this sum becomes
[3770c] nothing, as in [3770]. The calculation [3767—3770] is in some respects a repetition of
that in [1016", &ic.] ; and we see that the value of g, assumed in [1017], suffices even
when we notice the parts of R contained in [3764].
% (2382) Taking into consideration only two bodies m, m, the differential of [3771]
[37710] '^'" ''^ im'.aa.B''\\{p'—p).{dp'—dp) + {f/—q).{d(/—dq)l; observing
that B^^^ [956] is a function of the constant quantities a, a' [1044"]. Now the first
and second of the equations [1132] become as in [3767c], and the third and fourth of those
[37716] equations give ^ =— (1,0). (q'—q) ; -^ =(1,0) .(/ — p). Hence the differential
expression [3771a] becomes
[3771c] ^ ^/ . „ „'.5(n . (p'—p) . (ç'_ î) . {— (1, 0) — (0, 1) + (1, 0) + (0, 1)( . ^ < ;
VI. i. >§5.] TERMS OF THE SECOND ORDER IN e, e', y. SS
is equal to a constant quantity independent of the time t, because its
differential becomes nothing, in virtue of the equations [1132]; and if we
consider only two planets, m, m', as we shall hereafter do, (p'—pf+{q'—qT [^TTr]
will l)e a quantity independent of the time, in consequence of the same
equations. Therefore the preceding function [3771] can produce in
^ n d t . a .{ )
'\'i^/ [3769], only a quantity independent of tdt, kc, which [3771]
. v/l— e^
may therefore be neglected, since it may be supposed to be included in the
value of ndt. Hence we shall have, by eliminating the partial differentials [3771"]
of A^°^ and A^'\ relatively to a', by means of their values [1003],*
[3772]
in which the tenns between the braces mutually destroy each other, and render this
quantity equal to nothing ; therefore the expression [3771] must be constant, and may be
represented by G, and it will introduce into 5 R [3764] the constant quantity G. Now
as this quantity, considered as a function of a, produces in [3771"], only a term wliich
may be inckided in the expression of ndt, we may neglect it, and reject the tenn '■ '
depending on jB'* in [3764].
* (2-383) It appears from [3752(/], that the coefficients of }m'.{P-\-P), ^m'.{h'^+l'^),
are equal in the value o{ S R [3764] ; these terms may therefore be connected together,
as in [377*26]. Now if we put the two expressions of JV-^' [3753", 3755'"] equal to
each other, then divide by | m', we shall have, for the case of i = 0,
- ^,,, , ^ /dJim\ /(/.4(i)\ , , /dd.m)\ ^ ^,,, ^ /dA(.^i\ „ fddJiw\
4^'-+2«.(-^) + 2a'.(— ) + «a'.(^-^,)=2^«>-2a.(-^)--«^(^; [3772«]
substituting this in the coefficient of lm'.{hh'-\- II') [3764], it becomes as in [3772J] ;
hence we get
Taking the partial differential of this expression, relatively to a, and multiplying it
by 2ndt .a^, we get [3772].
TOL. III. 9
[37726]
34 PERTURBATIONS OF THE PLANETS. [Méc. Ccl.
Now if we collect together all these terms, we shall obtain,*
irof" ,- vi'.ndt C /dAin\ _ fdd.m\ fdKm\~}
epend-
Expres-
sion of
[3773]
[377:%]
[3774] In this expression we may neglect the terms independent of the time t [3773e].
Hence it is easy to deduce the expression of (/ h v', by changing what relates
to m into the corresponding terms of m' and the contrary ; and observing,
[3775] that, though the value of J^'' [997], relative to the action of m' upon m,
is different from its value relative to the action of m upon m', yet we may
[3775] use, in the preceding expression, either of these values at pleasure.! But
* (2384) The value of dSv [3773] is found, by adding together the several parts of
the expression [3757(/], computed in this article ; and as tlie terms [3768' — 3771"] destroy
[3773a] gj^jjjj Qfjjgj.^ there will remain only the terms [3762, 3772], to be connected together.
The expi-ession [3762], by the substitution of the values of (0, 1), [""Til [1073] becomes
and as the factors without the braces are the same as in [3772], the sum of the two
expressions [3772, 37736] is easily found to be as in [3773] ; which is a function of the
[3773c] elements of the orbits similar to that mentioned in [1345'"']. If all the terms of this
function were constant, they might be included in the expression of the mean motion ndt.
But e^ = h^ + P, e^ = h'^-^r^, he. [1108, 1109], are composed of con«to«^ quantities,
and of others depending on the secular periodical variations of c, e, Stc. ; and it is evident,
that the constant quantities produce in d 5 v terms of the same form as the mean motion ;
they may therefore be neglected, as in [3771'", 3774].
[377.3rf]
[3773e]
t (2385) Substituting [964] in [963'], and then putting s:=i, we get
[3775a] (a2_2 „ «'. cos. è + a'^)-i=a'-K\i bf + i'|>. cos. ê + if . cos. 2é -f- &c.^
Now the first member of this equation is symmetrical in a, a' ; tlierefore its second member
must also be symmetrical ; so that we shall have, generally, a'~'.è'f equal to a synmretrical
flinction of a, a'; and if we refer to the formulas [996, 997], we shall see, that for all
^ ' values of i, except i=l, the function ^<'' is likewise symmetrical. In the case of ? = 1,
VI. i. §5.] TERMS OF THE SECOND ORDER IN e, e', y. 35
we may obtain dov' more easily by the following considerations. If we [3775"]
add the value of d^iv, multiplied by w\/â, to the value of d!iv', multiplied
by in'\/'a , Ave shall have, by substituting the partial differentials of A^^\ A'^\
relative to «, instead of those relative to a',*
m\/a.(ISv-{-7nya'.cJov:^ ^ .lh^-{-PJf-h --{-I ^ . j^a . \^-j^ j + i a . {^-j^ J ^
[3776]
corrresponding to the action of?»' upon m, we have ,/2''" = -- ;.& [997]; and in
J " " [3775c]
the action of m upon m', it becomes ./2'"=— — ^—-^i 5 hut we may neglect the
parts —, — :^, because they produce nothing in dSv, dôv'. To prove this, we shall [3775rf]
a
observe, that by noticmg only the part .4''*= — , we shall get
/rf./3Ui\ 1 /■ddAO)\ /ddJim\
1-77-; = ^' [-1^)=^' l^I^j=^' [3775.]
substituting these in [3773], the terms mutually destroy each other ; so that we may
neglect this part of ./2'^*, and for similar reasons we may neglect the part ^'''=--, in [3775/-]
computing the action of m upon m', and then the two expressions [3775c] become
symmetrical in a, a', as in [3775'].
* (2.386) Multiplying [3773] by tn^a, and dividing the second member by na^a=^l
[3709'], we get, by reducing the factors without the braces to a symmetrical form,
+.,„,....,.(*-+n. i^..(r)+^«'-c-^')+"'-r-^) \
Changing the elements m, a, v, h, I, he. into m', a, v', h', /', &c. and the contrary ;
which does not, in the present case, alter the values of A^°^ or ^'" [3775], we obtain [3776t]
the expression of m' \/a' .dSv'. The factors between the braces corresponding to the
first, second, and third Imes of [3776a], become, respectively, as in the first members
of [3776rf,/, h], and by means of the expressions [1003], they may be reduced to the
36 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3776] If we consider only two planets, m and m',* the differential of the second
f„__„ , forms [3776e, ^, i]. In making these reductions, we may use the abridged symbols
Ad, Ai, A.2, A3 [3755i], observing, that the index of A'^"'' or .4<" remains unchanged;
^^''^^ a'.(— ) + J«-.(^ + «'3.(-^):=|-.4o-^J + M2^„+4^,+^,i
•\.\—QAu—\SA, — 9A^—A.:,\
[3776e] = — 5 .^j — ^A.2 — A^;
l^nn ,,.(^)+,,'.{'^) + ,P. (^)=2.i-A-^.!+4.i2A+4^,+^,!
+ l — 6Ao—l8Ai—9A.,—A3\
[377%] =—4A^ — 5Aç,— As;
/(/./3(iA /rf9.4<')\ /rf3./4''A
[3776/.] -.^<"+«'.(— )-^V^.(^-2a'=.(^ = -.4„+|-.4„-^,|---.|2^,+4^,+^,|
— 2 . 1 — 6.^0—18^1—9^2—^31
[3776i] = — 5.2o + 5A + -y-. ^2+9^^3 •
Now substituting the values corresponding to [31~6c, g, i] in the value of m'^a. dSv,
deduced from [3776«], by the change of the elements [37766], we get
„V...«= i„„..„.(..+r=).|-.„.('/^)-,v...(^-^)-.=.('-^
+i„..,.(".-+"').S-5^-+-.C^')+V.'.(^)+..'.('5S')S-
Adding together the two expressions [3776a, t], we obtain [3776], observing, that in
this sum, the coefficient of h^ -\- P is found to be the same as that of A'^-j-/'^. We
[3776i] may remark, that the factor ^ — , in the second line of [3776], is erroneously
printed '■ in the original work. If we multiply the second member of [3776]
by ?ia*^l [3709'], and substitute the expressions (0,1), [ôT\ [1073], we shall get,
bstead of [3776], the following equation ;
[3776m] m^a.d&v+mya'.dôv'= im\/a . dt . (0,1) .{h^ -\-P + h'^ + 1'^)
— Zm\/a.dt. [KI] • {h li -\-ll').
* (2387) The differential of the equation [3776m], may be put under the following form;
d.\m^a.dàv\-m!s/a'Mv']= Zm^a.dt.\{Q,\).{hdh.-\-ldl) — [^.{h'dh + rdl)\
^^''^''"^ -\-Zms/a.dt.\{0,\).{Kdh:+l'dl') — [-^].{hdK^ldV)\.
VI. i. ^5.] TERMS OF THE SECOND ORDER IN e, e', y. 37
member of this equation will be nothing, in virtue of the equations [1089] ;
therefore %oe have, by noticing only secular periodical quantities, "for "■
0 = m s/a.dàv-]r m' \/7i ■ d <> v' ; [3777]
which immediately gives d tS v\ when d o v is knoivn.
The value of div is relative to the angle formed by the iivo radii veclores
r and r + dr. To obtain its value relative to a fixed plane, we shall [3778]
observe, that if we put dv^ for the projection of dv upon this plane, and
neglect the fourth power of the inclination of the orbit, we shall find,
as in [925],*
dv=dv.\\^-ls''—\/^À. [3770]
We have, as in [1051],
s = q.sm.{nt + s) — J9.C0S. (n/ + j) + &C. ; [3780]
which gives f
rf5 = ^j2_^yji.cos.(7t^ + 0 + (np + ^Yf/i.sin.(«^-f3) + Sic. ;
[3781]
Substituting, in the first line of the second member of this expression, the values d h, d I
[3767a], it vanishes, because the terms mutually destroy each other. The second line
of the second member becomes, by the substitution of the formulas [1093, 1094], equal
to ^m^n'.dt.\{\,Q).{li'dh'-{-rdl') — ['ûV^.{hdh' + ldV)\, which vanishes also by [37776]
the substitution of </A'= {(1,0) . /'— [To] J} . J<, dV=.\—{\,Q).h'-^[']^].h\.dt, [3777c]
deduced from the third and fourth of the equations [1089]. This is also evident from the
consideration, that the expression [3777i] may be derived from the first line of [3777a],
by changing the elements relative to m into those corresponding to m', and the contrary ; [3777rf]
and as that line is found to vanish by the substitution of the values of dh, dl [3767 «],
the other will in like manner vanish by the substitution of the values of d A', d I
[3777c]. Now the second member of [3777a] being equal to nothing, we have, by
integration, m \/a.d&v-\- m' s/à .dhv'=Gdt \ G being a constant quantity independent
of the secular periodical equations. This quantity Gdt may be supposed, as in [3771'"], [3777c]
to be connected with ndt, ri'd t ; so that by noticing only the secular periodical equations,
we may put tlie first member of the preceding equation equal to nothing, as in [3777].
• (2388) The equation [925] maybe put under the form di\=dv\/\l-\-s'^ — (T+TIilTrâv
Developing this, and neglecting terms of the fourth degree in s or ds, we get [3779].
t (2389) The differential of .9 [.3780], considering p, q, t as variable, becomes as
in [3781]. The squares of these expressions, which enter into the function [3779], are
VOL. III. 10
38 PERTURBATIOiNS OF THE PLANETS. [Méc. Cél.
[37811 hence we shall find, by neglecting the periodical quantities depending on n t ,
and observing that d v ■-= n d t, very nearly,
[3782] dv^^dv-\-}2 .(q dp — p d q) ;
therefore to obtain the value of d5v^, we must add the quantity
[3783] \. (qdp — p dq) to the preceding value of dàv [3773].
If we only consider two planets m, m', we shall ha\r, by means
of [1132, 1130],*
[3784] {q('p—pdq) ■ m\/â+{f/dp'-p'dq').my^= — imm'.dt.aa'.B''\\{p'—p)^-lr{q'—qfl ;
[3779a] of the order of the terms computed in this article [3702'], and by neglecting terms of a
d s~
higher order, we may omit, in — [3779], the terms of dv [3748] depending on e,
[37796] and put dvz^ndt, by which means we shall get d v^t=:z dv.\l-\-i s^ — i-ir — (»
in which we must substitute s, ds [3780, 3781]. In making these substitutions, and
noticing the terms independent of the sine and cosine of nt or its multiples, as is done in
this article, where the secular periodical terms only are retained, we may, as in [3651a], put
[3779c] sin.2 (n < + e) = è , cos.^ (n ^ + e) = i , sin. (« /■ + s) . cos. (n < -f s) =0 ;
then the squares of [3780, 3781] will give, by neglecting dq^, dp^, which are of the
order of the square of the disturbing forces,
[3779rf] i,a_i.(ç2+_p2);
Substituting these in [3779], we get [3782].
* (2390) Substituting the values of dp, dq, dp, dq' [.3767c, 3771 i] in the first
members of [3784«, i>], and reducing the second expression by means of [109.3, 1094],
we get the second members [3784o, c] ;
[3784a] my'a.{qdp — pdq)^ms/a.{Q, \) .dt . \q ■ [q' — q) -^ P -(p' —p)\ ;
[37846] w'v/«'- {q'fip'—p'dq') = m'y/a. {1,0) . d t .\— q'. {q'— q) —p'. {p' — l})\
[3784c] ==^m^a.{(),l).dt.\— q' . (7'— q) — p' . {p' — p)\-
The sum of the two equations [3784f(, c] gives the value of [3784] under tlie form
[3784c/] — m^a. [0,1). dt.\{q' — qf-j-{p'—p)^]; substituting (0,1) [11.30], and dividing
3
by na-==l [3709'], it becomes as in the second member of [3784].
VI. i. §6.]
TERMS OF THE SECOND ORDER IN e, e', y.
39
and the second member of this equation is equal to dt, multiplied by
a constant quantity ; * therefore by noticing only the secular periodical
quantities, we shall have
0 = m y'â .d&v^-^m' sjd . d^v^ ;
& V and (5 v^ being relative to the fixed plane.
6. We shall now consider the inequalities in the motion in latitude,
dependinrr on the products of the excentricities and inclinations of the orbits.
For this purpose we shall resume the third of the equations [915] ;
We shall take for the fixed plane the primitive orbit of m, in consequence of
lohich we may put 2 = 0 in the expression of (-i— )• We shall have,
by [3736—3741], observing that z' = r's',\
dR
rf7
[3784']
Tho Bame
formula for
reduced to
the fixed
plane.
[3785]
r"" 1^2— 2r/.cos.(î)' — i;)+r'2}*'
[3785']
Differ-
ential
equation
for the
latitude.
[3786]
[3786']
[3787]
[3788]
* (2391) The differential of the second member of [3784], being divided by — 'imdt,
becomes as in |'3771rt], and is therefore equal to nothing, as is shown in [3771c] ; hence [3785a]
we find, as in [377 ItZ], that the first member of [3784] is equal to dt, multiphed by a
constant quantity G, wliich may be neglected as in [377 le] ; so that by noticing only the
secular periodical equations, we shall have {qdp—j)dq).m\/a-\-{q'dp' — p'dq').nJ\/a'^zO. [37854]
Now we have found, in [3782], that by reducing « to a fixed plane, the value of dv or dàv
must be augmented by ^.{qdp — pdq); and in like manner, the quantity d^v' must
be increased by i.{q'dp' — pdq). Multiplying these by m\/a, m'\/a', respectively, [3785c]
and adding the products, we get the increment of the function [3777], or the quantity to be
added to it, to obtain the value of m \/a . dSv^-^-m' \/c! . d 5 v/. Now this increment
vanishes by means of the equation [37856] ; consequently the function [3777], varied in
this manner, becomes as in [3785].
t (2392) The latitude of the body ot', neglecting terms of the third order, being ^3737^-]
represented by s , and the radius vector by r', we shall have, by the principles of
orthographic projection, 2' = //, as in [3787]. Now / [37366] being independent
of z, the partial differential of H [3736], relative to z, becomes
(àR\ m' 2'
m'.(z'-
l(x'-
^f+iy'-yf+i^'-'-)]^ '
[3787a']
40 PERTURBATIONS OF THE PLANETS. [Méc. Cèl.
the differential equation in z, will by this means become*
[3789]
0 = ^ + n=2.{l + 3e.cos. (n^+£ — ^)|
+
m'. n^ a^. s
\ r^ — 2 r r. cos. («' — ^)-\-'i'"~\~
We shall now putf
■clR
[3790] ('l:^)^M.sin.{?:.(n7— n« + /— s)+2ni+^i+iV.sin.^'i'.(n7-nï+/— i)+L|,
for the jiart of [3788]
ffJR
[3791] I77
r- — 2r r. cos. {v' — v)-{- r' -} ] '~
depending on the angles i.(n't — 7it-\-s' — s)-\-2nt and i.(7i't — nt-\-;' — s) ;*
[-3792] and shall suppose, that by noticing only the inequalities of z, depending on
and if we neglect quantities of the order s'^, we may reject tenns of the order z'~ or 7*
in the denominator; then, as in [3742^], we shall have
[3787i] {x'— xf 4- (i/' — i/f + {z' — zf = /-a _ 2 r r'. cos. {v — v) + r' 2 ;
substituting this and z := 0, z'==r's' [3786', 3787] in [3787a'], we get [3788].
We may here remark, that the method used in this article, in finding the motion in
[3787c] latitude, depending on terms of tlie order of the product of the excentricity hy the inchnatlon
of the orbit, is difTerent from tliat proposed in [948], and used in [1025, &c.] in finding
the terms independent of the excentricity. This last method may, however, be applied
without any difficulty to terms depending on the excentricity, and we shall obtain the same
[3787rf] result as in [3795 — 3797] ; as has been shown by Mr. Plana, in Vol. XII, page 449, &c.
of Zach's Correspondance Astronomique, he.
* (2393) We have, by means of [37026, c, 3700],
[.3789o] (A ?--3= ,a a'^. { 1 + 3 e . cos. (n t + s—zi) -f- &c. | = 7i^. ^1+36. cos. (n t + 1— «) + &;c.| .
Substituting this in [3786], also the expression [3788], multiphed by n'a^=l [3709'],
we get [3789].
t (2394) The reasons for assuming these forms are evident from [3704a — 6], observing
[3790«] that the object proposed at the commencement of this book, is to notice merely the terms
depending on the squares and products of the excenlricities and inclinations.
VI. i. §6] TERMS OF THE SECOND ORDER IN e, e', y. 41
the first power of the inclination of the orbits, the part of z, relative to
the angle i . {n't — n i + s' — e) + nt, will be *
z=^yaF. sin. \ i . («' t — n t + s' — s) + n t + s — n\. [3793]
We then have, by retaining only the terms depending on the products of the
excentricities and inclinations,!
0=^--— -\-nrz + %rf.€y.aFA ,.'.,, , , ^ , , [
+ n''a\M.ûn.\i.{n't — nt-\-i—;)+27it + K\
+ n'a\N.ûa.\i.(ri!t—nt + B'—c) + L\ ;
[3794]
* (2395) Putting, for brevity,
we shall have, for the terms of s [1034] depending on iî''~", the expression
F. \ (?' — q) . sin. Tg — (y — ^) . COS. Tal ; [3792i]
substituting in this the values p' — p ^= y . sin. n, g' — q = y • cos. 11 [1033], it becomes [37926']
Fy . {sin. T3 . COS. n — cos. T3 . sin. n j = Fy . sin. ( T3 — n)
= F7. sin. {i . {n' t—n t -J[- ^— s) -\- n t -{-E — n\. [3792e]
Multiplying this by r, we get the corresponding part of z=^rs [3787,3796], to be [3792(/]
substituted in the term 3 n^ e z . cos. {at -{- s — zs) [3789]. Now this term is of the [3792«]
second order, or of the same order as the terms now under consideration [3702'] ; and by
neglecting tliose of a higher order, we may substitute a for r, in the expression of z [3792(7],
and we shall have z=a«; hence the term of s, computed in [3792c], produces in z [3792/"]
the quantity [3T93]. Substituting this in [3792e], and reducing by means of [18] Int.,
we get the tenus depending on F in [3794]. In computing the value of the term [3792e],
and neglecting quantities of the order m'^ or e^, it is not necessary to notice any other
terms of s [1034], except those depending on B'-^'^^ or F, which we have used above. [3792g-]
For the terms depending on the arc of a circle nt, in the second and third lines of [1034],
vanish, as in [1051], in consequence of the secular variations of p, q. Again, having
taken the primitive orbit of m for the fixed plane, we have z = 0 or s = 0 [3786'], at
the commencement of the motion, corresponding to p = 0, q^O [1034, 1032] ; so that
these terms may be neglected in computing [3792e]. Lastly, the terms of s depending
on sin. {n't -]- e), cos. («' C -(- s'), in the fourth line of s [1034], may be considered as
included in the term of 7*3 or of F [3792a], depending on i = I ; consequently the [3792i]
function [3792é] is rightly expressed by the terms depending on F in [3794] ; the
quantity F being of the order m' [3792a], as well as M, JV [3790, 3791]. [3792*]
t (2396) The equation [3794] is easily deduced from [3789] ; for the two first terms
are identically the same in each ; the third term depending on e, reduced as in [3792/, Sic],
VOL. III. 11
[3792A]
[3795]
4*2 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
hence we get, by integration,*
iin^.ey.aF.sm.\i.(n't — nt + s'—s)-\-2nt-ir2î—7z—n\ ^
._ ( +n''a\M.sm.\i.(n't—nt + i—s)-\-2nt-^K\ \
\ in'—(i— l).n\.\ in'—(i—3) . n \
i%n?.ey.aF.sm. [i.{n't — nt-\-^' — 0 + '^ — n| )
\ +n?a^.N .ûn.\i.{n't—nt-]-s'—i)-\-L\ \
\in' — (i + \).n\.\in'—{i—\).7i\
We have the latitude s, by observing thatf
s ^=~ =:--{-- .e . COS. (nt -\-i — ■a) :
r a a ^ ^ '
therefore s may be obtained by dividing the preceding expression of z by a,
and adding to it the quantity %
ley.F.?.m.{i.{n't—nt-\'S—i) + 2nt+2t—^ — T\]
+ \ey.F.%m. \i.(n't — nt-\-s — £) + « — n|.
[3796]
[3797]
[3794o] produces the terms depending on F [3794] ; the two remaining terms, comprised in the
second line of the second member of [3789], are represented by the fonction [3791], or by
the equivalent expression [3790] multiplied by 7t^a''=l, as in the two last lines of [3794].
* (2397) The equation [3794] is of the same forai as [865a], putting y = z, « = w;
[3795o] then any term of [3794] depending on F, M, or JV, being represented by a.K.s\n. (m^t-\-s),
[379561 the corresponding term of z will be represented by , ; , '"^ ', 'as in [871"] ; the
' X u i. -^ (m,-(-n).(m, — n)
letters m,, e,, being accented to distinguish them from the similar letters of the present
[3795c] article. Now putting m^:=zi.{^nl — n)-\-2n in the first and third of ^Aese ^erm^ of [3794],
and m.i = i.{n' — n) in the second and fourth, we get, successively, the terms of z [3795] ;
[3795d] all of which are of the order ?«' [3792Ar].
f (2398) We get, in like manner as in [3787], r5 = c; dividing this by r, or its
[3796a] equivalent expression a.\\ — e. cos . {nt -\- e — ra)} [3701], we get the two values
of s [3796], neglecting, in the last of them, the terms of the third order in e and z.
J (2399) Substituting, in — .e.cos. (?i<-|-s — ra) [3796], the term of z of the first order y,
assumed in [3793], and reducing the product by means of [18] Int., we obtain the
corresponding values [3797]. Adding these to the term of - [3796], deduced from
[3795], we get the terms of s. of the proposed forms and order. These terms are neglected
VI. i. §6.]
TERMS OF THE SECOND ORDER IN e, e', y.
45
Nothing more is required but to ascertain the values of M and iV; which
may be easily found by the analysis in § 4. We shall, however, dispense
with this calculation, because the inequalities of this order in latitude are
insensible except in the orbits of Jupiter and Saturn, whose mean motions
are nearly commensurable, and we shall give, in [3884 — 3888], a very
simple method for the determination of these inequalities.
If we refer the motion of m to a fixed plane, which is but very slightly
inclined to that of its primitive orbit, putting tp for the inclination of the
orbit to this plane, and a for the longitude of its ascending node ; we shall
have the reduction of the motion in the orbit to this plane, by the method
explained in Book II, ^22 [675, &c.],*
— J- . tang."(p . sin. (2 v^ — 2 é) — tang. <?) . J 5 . cos. {v^ — ^) ;
» being the motion v referred to the fixed plane. Hence the motion in
latitude produces in the motion in longitude, upon the ecliptic, inequalities
depending on the squares and higher powers of the excentricities and
[3797']
[3798]
[3799]
[3800]
[3600']
by the autlior in [3797'] on account of their smallness. The most important terms of the
perturbation in latitude, of the second order, computed in [3885, 3886], are reduced to numbers [37976]
in [4458, 4513], for Jupiter and Saturn, in whose orbits these terms have a sensible value.
* (2400) In the annexed figure 73, AB \s tlie primitive orbit of the planet rn, A G the
fixed plane, D the place of the planet, B D=^&s the perturbation in latitude now
under consideration, which is perpendicular to A B ; lastly, the arcs B G, D EF are
perpendicular to AG, and BE perpendicular
to DF. Then by using the notation [669"],
we have AB:=^v — 13, AG^=v^ — ê,
BAG^ip; and in [676'], by neglecting tf^,
^B=^G' + tang.2iç,.sin.(2i;,— 2^) ; but
on account of the smallness of cp, we may
put tang.3 J 9 = ( ^ tang. <p )^ = J- tang.^ «j ;
so that to reduce A B \o A G, we must apply the correction — ^tang.^ip.sin. (2t), — 26),
as in the first term of [3800]. Again, since B D is perpendicular to AB, and BE
perpendinijar to DF or B G, we have nearly, the angle ABG = angle D B E ;
moreover, in the spherical triangle A B G, we have cos. ABG = sin. BAG . cos. A G
[1345*-], or in symbols, cos. D5 jB== sin. ip . cos. («^ — d). Now in the right-angled
triangle Bfil>, we have, very nearly, BE = BD.cos.DBE=iàs .sm.(p.cos.{v;—ê);
and on account of the smallness of p, we may change sin.cp into tang.ç, also BE into FG;
hence F f? = 5 s . tang. (?. cos. (j;,— Ô). Subtracting this from AG, we get AF; and
in this way we obtain the second term of [3800].
i;-«
[3800a]
[38006]
[3800c]
[3800(/]
[aSOOe]
[3800/]
[3800^]
44 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3800"] inclinations of the orbits ; but these inequalities are insensible except for
Jupiter and Saturn.
If we notice only the secular quantities, and put, as in [1032],
[3801] tang. <p . sin. Ô =p ; tang, ip . cos. â = ç ;
we shall have*
[3802] is = t.~. sin. (n i + s) — t-'j^ • cos. (n t + s).
[3803] The term — tang. (? . 5 5 . cos. (v^ — ê) produces the following expression,
[3804] ^•^9'P~~P l! . so that we shall havef
[3805] V-V + t. —^ ,
which agrees with what we have found in the preceding article [3782]. Î
* (2401) If we suppose s to be a function of t, which becomes S, when t = 0, we
[3801a] shall have, by the theorem [607, &c.], s = S + t . (^ — j + — . (^— j + &ic. If we
neglect t- and the higlier powers of t, and notice only the secular inequalities, we shall
get s — S=t.(-—-]. Now s — S, being the variation of s in the time t, is what is
represented above by &s ; hence S s ^^ t . I j ; and by noticing only the secular
inequalities depending on dp, d q, in [3781], we obtain
/dS\ dq . , , dp , , >
[3801c] [-j^) = -^ ■ sm. (« t + s) — ~. cos. (n t + s) ;
consequently &s becomes as in [3802].
t (2402) Developing cos. (v— è) by [24] Int., and then substituting the values [3801], we get
[3804a] — tang, (p . cos. (d, — ^)^ — tang. (p . \ cos. â . cos. i\-\- sin. Ô . sin . d J = — q .cos. i\ — p . sin. d,
[38046] = — q. COS. {nt-\-s)— p. sm.{n(-\-s) ;
observing, that as this quantity is of the order j}, g, and is to be multiplied by 5s, in [3800],
which is also of the same order [3802, 3767c], we may put v^=^nt-\-s, neglecting, as usual,
the terms of a higher order in p, q. Multiplying together the expressions [3802, 3804è],
[3804c] and retaining only the quantities independent of the periodical angle 2nt-{-2s, we may
use the values [3779c], and we shall get, for — tang, (p . i5 «. cos. (r, — è), the same
[3804rf] expression as in [3304]. This represents the secular change of v, arising from the last
term of [3800] ; and by adding it to v, it gives n,, as in [3805]. We may observe, that
[3804e] the first term of [3800] produces no secular terms, or such as are independent of 2t', — 2^,
and it is therefore neglected in this estimate of v, [3805].
[3805a] t (2403) Neglecting terms of the order t^ or m'^, we may suppose i.{qdp — pdq)
VI. i. §7.] TERMS OF THE THIRD ORDER IN e, c', 7. 46
[3806]
[3806']
ON THE INEaUALITIES DEPENDING ON THE CUBES AND PRODUCTS OF THREE DIMENSIONS OF THE
EXCENTRICITIES AND INCLINATION'S OF THE ORBITS AND THEIR HIGHER POWERS.
7. The inequalities depending on the cubes and products of three
dimensions of the excentricities and inclinations of the orbits, are susceptible .p„of„,„,
r ^ r u. of /i of
01 two lorms, the third
order.
R = M. sin. {i . (n' t — nt-\- «' — s) + 3 n ^ + ^| ; [First form.] [3807]
R = N. sin. {i . {n't — n / + s' — s) + nt + L]. [second fum,.] [3807']
We may determine them by the analysis employed in the preceding articles ;
but as they become sensible only when they increase very slowly, we can
make use of this circumstance to simplify the calculation. We shall resume
2d . (rS 7-)
the expression [37156], and shall neglect the term — 3' , . > which is [3808]
ci III (Xi z
then insensible, t because of the smallness of the coefficient of t, in the
inequalities now under consideration. Then this formula becomes
àv= — '^^^^ + Sa.ffndt.AR+'2fndt..a\('^^^.X 13809]
to be equal to Cdt, C being a constant quantity ; then [3782] becomes dvp=dv-\-Cdt, [3805J]
whose integral is v^=v-\-C t, as in [3805].
* (2404) The reason for assuming these forms is evident from the principles used
in [3704a — 6], For the coefficients of n't, — nt, in [3807], are i, i — 3, respectively; [3807o]
their difference 3 expresses the order of the coefficient k [957"'", &c.], or that of M [3807],
which must therefore be of the order e^. Again, the coefficients of n' t, — nt [3807']
are i, i — 1 ; their difference is I , consequently N may contain terms of the order [3807fc]
1, 3, 5, fee. [957'", fiic.] ; which include those of the order (? ; and it is evident from
[957", &c.], that these forms embrace all these terms of the third order.
t (2405) This remark applies exclusively to terms of the form [3807], like those
in the three first lines of the second member of [3819], depending on the angles
i.{r^ t—nt-\-s' — i) -\-Znt, whose differential introduces the very small factor i.(n' — n)+3n
[3818(/]. But this small factor is not produced in the differential of the terms of the
form [3807'], contained in the last line of the second member [3819] ; and then the
term [3808] is not neglected, but is computed in [3822c].
X (2406) In the terms treated of in §7, and depending on the cubes of the excentricities,
no quantities are finally retained except those which have the small divisor i.{n' — n)-\-3n,
or its powers; and as the expression of 5v [37156] contains the function 2 d . {r5r),
divided by a^. ndt ; we must examine whether this function contains the small divisor we
have just mentioned. Now by the inspection of the value of rôr, or rather of Sr [1016],
VOL. III. 12
[3808a]
[38086]
[3809a]
[3809i]
46 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
The divisor y'l — e^ [37156] must be neglected for greater accuracy, as
in Book II, ^54 or [3718']. We must also, by the same article, ayply these
inequalities to the mean motion of the planet m, in computing its elliptical
motion [3720]. This being premised, if we suppose
R zzz: m! P. sin. \i . (n!t — nt -\-s'—^) + 3nf + 3-=!
[3810]
+ m' F. cos. {i .{n't — nt-{-z'—i)-\-Snt + Si\;
which comprises all the terms of R, where the coefficient of nt is greater
[3811] or less than that of n't by the number 3 ; we shall get, as in [1209],*
3.(3 — i).m'n^.a
[3809rf]
[38126]
3a.ffndt.aR-.
|i.(ra'— ?^)-j-3n}2
;„, , 2(/P 3ddP' ).,.,, , , V , „ , „ ,
r»ûioi I -j r* + (•■ , , , I „ > ., — TT-r-, , , „ )^ , „>.Bm.]i\nt—nt-\-s—s)-\rmt-\-3i\
\_mii\ 1 ^ \i.(n'—n)-\-3n\.dt \i.(n—n)->ç-3n\^.dt^S
in 2dP' 3rfrfP ) ,. , , , , ,,„ ,„,
[ {%.[n'—n)-{-Qn\.dt {i.(n'—n)-\-Qn\Kdt^S ■
we shall not find, in the preceding function, any term depending on the first power of e,
[3809c] and having the divisor i.[n — »t)-j-3n. In quantities of the second order in e, c',
given in [3711, 3714], we find such terms having the first power of that divisor ; but these
terms depend upon angles of the form i . {n't — n t -\- ^ — s)-\-2nt , which are different
from those under consideration in this article [3806' — 3807'] ; so that they may be
neglected. To investigate the similar terms of the order e', which depend on the angle
i.{n't — n t -\- e' — s)-\-3 nt, we may go through a calculation similar to that in
[3703—3714], changing, however, the angle i . [n' t — nt-\-s' — t)-\-2nt into
i.(n't — nt-\-s' — s)-\-3n't; which is the same as to increase the integral ntimher 2 — ;",
connected with nt by unity; by which means the divisors in-{-{\ — i) .n, in'-\-{2 — i).n,
in' -{-{3 — i).n, which occur in [3705,3710,3711,3714], are changed, respectively,
into î»' + (2 — i).n, in'-\-{3 — i).n, iw'-j-(4 — i).n. Hence the quantity r^r,
[3809e] similar to [3711], will contain a term of the order t^, depending on the form [3807], and
having for divisor the first potver of the small quantity in'-\- {3 — i) .7i , as is hereafter
found in [3819]; but this divisor will vanish from the dilTerential d.[r6r); therefore it
may be neglected, as in [3809rt] ; and then the formula [37156] becomes as in [3809] ;
omitting the divisor, \/{l — e^), for the reasons given in [3718'].
* (2407) Substituting, in the first member of [1209], the assumed value of
[:3812o] k.sm.{i'n't — int-\-^) [1208^'], it becomes
ffan^.dt^.\q.sm.{i'n't — int-'ri's'—ie)-\-q.cos.{i'n't — int-{-i's'—is)\ =
riia.ain.(i'n't — int+i's'—i£) (j ^ , 2dQ' , 3rf3Q Ad^q ^ ^
^) ^_o
(i'n' — inf 'i ^~'~ {i'n'—in).dt~'' (i'n'—inf.dt^ [i'n'—inf.dfi
i.(i'n't — int + i'e'—{e) ^_ _ 2rfQ 3rf2Q- jd^q
(i'n'—inf 'I ^ {i'n'—in).dt'{i'n'—inf.dVi'^(i'n'—inf.dt3~
VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e', y. 47
Then we shall have *
2m'n
^ , „ /dR\ 2m
2.rndt.a-.('—) = — ^--. — , ,„ -.
«2. f'^JLYcos.{i.{n't-nt+e-s)-^3nt-\-3sl
-a". ('^^)-sin. {L{n't-nt+^-s)-\-3nt-\-3i]
Lastly, we shall suppose, that by noticing only the angle
i . (n't — nt + s — '.)-{-2nt + 2B, '
[3813]
we havef
r S r
= H. COS. {i . (n't — nt + B'—s)-}-2nt + 2B-\-Al; [3814]
Now if we take the difierential of [3810], relatively to d, then multiply it by 3 a .7idt,
and prefix the double sign of integration, we shall get, by using for brevity,
T=n't — nt + £'— 1 [3702a],
[3812e]
C — 3.{3 — {).m'.P'.sm.(iT+3nt-\-3i)) ^„^,„,,
■'■' •'•' ^+ 3. (3 — i) .m'.P.cos.(îT+3n!: + 3£)S
The second member of this expression is of the same form as the first member of [38126],
as is easily perceived by changing, in [3812J], i' into i, and i into i — 3; also
putting Q = — 3 . ( 3 — z ) . m'. P', Q' = 3 . ( 3 — i) m'. P ; then making the same [3812<]
changes in the second member of [3812è], we obtain for 3a .ffn dt . àR, the same
expression as in [3812]. We may observe, as in [3714cZ'], that the secular variations of [3812/]
the elements are noticed by the introduction of the differentials dP, dP', ddP, ddP',
which are computed in [4415, &c., 4484, &c.].
* (2408) The partial differential of R [3810], taken relatively to a, being multiplied, [3813a]
by 2/1 dt . a^, and then integrated, gives [3813].
t (2409) The expression [3814] is equivalent to that in [3711]; if being taken for
the coefficient of any one of the terms of this formula, and A representing that one of the [3814a]
quantities — 2 in, — « — ra', K — 2 s, which is connected with this coefficient H ; ^001411
observing that H is of the second dimension in e, e'. The differential of [3701], is
dr=^ae.ndt.sm.{nt-^E — «) -|- &:c. ; multiplying this by [3814], and neglecting [3814c]
terms of the fourth order, we get, by using T [3812c],
^-^ .dr^ Hae .ndt .cos.{i T-\-2nt -\-2 s -{- A) .sin. {nt-{-B — ts)
= lHae.ndt.sin.{iT-\-3nt-{-3s—-a-{-A) [3814d]
— i Ha e.nd t .s'm. {iT-}- nt-{- s -\- zs -\- A).
As this is of the third order [38146], we may, in the first member, put r= a, and then
dividing by — andt, we get
'^'•^' -|ffe.sin.(iT+3n<+3E-^ + ^) [3814,]
(findt
+ iiîe.sin. (iT-f ««-(- t-^-a + A).
48 PERTURBATIONS OF THE PLANETS. [Méc Ctl.
[3814] ^ ^*'^"S determined as in [3814a], and having the very small divisor
r. (n'— w) +3n ; then the first term of ôv [3809] gives the following
expression ;
[3815] __^Ll_! ^_i/fe.sin.|z. (ra'i — «f + s' — f) + 3»i + 3s — a + Jj.
Hence we shall find, by noticing only terms which have the divisor
[3816] i . {n' — w) + 3w,*
SC _, Sa.rfP 3a.ddP' ■) . C{.{n't-nt+i'-s)')
i {t.{n'-,i)i-3n].dt \i.(n'-n)-+-3n]-2.dPS l+3nt~\-3s S
5 p 2a. dP' 3a. ddP } C{.{n't-nt+;'-s)-)
'—UP-
•COS.
Ter mi of
\i.{n'-n)-l-3n].dt \i.{n'-n)-\-3nl'i.df^S (+3n«+3e
<5« ( a^. (~). COS. U. (n't — 7it + s' — s) + Snt-\-3s\
heihird 2m'n y \da/ ' ^ ' ' ' ' *
i.(n' — n)-|-3n ] /dP'\
/ — a'^.f^j.sin. \i.{n't — nt-\-s' — E) + 3n<+3£}
— 3 Ue.sin. \i.{n't—nt^^—s)-\-Znt-{-Ze—vi-[-A\.
The differential equation [3699]
[3818]
o = ^' + '-^ + Vaie+..Q.t
The first term of the second member is the same as in [3815] ; the second term is noticed
r'î8l4/'l *" [3822rf]. We may observe, that it is not necessaiy to notice terms of the order t^
in dr [3814c], because they depend on the elliptical motion, and have no divisor of the
form i . (n' — ?i) -j- 3 ?! ; moreover they must be multiplied by terms of the order e,
[3814^] which occur in — [1023], to produce terms of the third order now under consideration ;
and these terms of [1023] do not contain the small divisor just mentioned.
* (2410) Substituting, in the expression of i5i; [3809], the values of the terms in its
[3816a] ^^^^^^^ member, given in [3815, 3812, 3813], we get [3817].
f (2411) The expression [3818] is the same as [3699], from which we have deduced
[3702], and by using [3705a], it becomes
[3818a] 0==^^^' + n^r5?-4-^3«2a.5r.[e.cos.(n<+j-rt)+e2.cos.2.(«)'4-E-^)]+2rd/î+a.('^y^.
This is solved as in [3711&, c], and if any term of the expression between the braces be
[3818o'] represented, as in [37116], by aif. sin. (?»,< -}- «,)> or o.K .cos. {mt -\- s^), the
corresponding terms of rhr [3711c] will contain the divisor m^ — n^, or rather the two divisors
(m.-\-n), {m^ — n). To find the values of m^ producing the divisor i.{n' — n)-|-3n [3818'],
VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e\ y. 49
gives, by noticing only the terms which have tlie divisor i.(n' — n)-{-3ii, [3818']
2.{{~3).mn ( aP .sm. \i.{n't—nt i-s' — i)^3nt-\-Ss\ ■)
J- (5 )•
a
i . («'—«) +3 11 ' I -JraP'.cos.\i.{nt — nt-\-!'—s)-\-3nt + 3;l ^
— ^He. COS. \i. (n't— nt + s—s)-^3nt + 3s—^ + A}
+ iHe.cos.{i.(n't—7it + s—s) + nt + s + ^ + Al.
[3819]
Terms of
ri)r.
we shall put it successively equal to ???, + n and m, — n ; and we shall get
m,= 2.((i' — «)-)-2n, 'm^=i.[i\! — n)-\-An', but we may neglect the last, because
the coefficients of n, n dilfer by 4, and the terms depending on it must be of the fourth
dimension in c, e' [3704n', &lc.], which are here neglected. Therefore, in finding r5r,
we need notice only the following terms. First. Where m.^ = i . [n' — n) -(- 2 n .
Secoii'l. Where the quantity R, or rather fdR, contains the divisor i . [n' — ?i) -j- 3 w
[3813']. Hence it is evident, that we may neglect a . (— — j, which produces no such
terms. The part of R, given in [3810], produces in 2/d R, the following terms,
-2.(i — 3).m'.?i (
77(7i'— n) + 3V ■ ^
[3818fc]
[3818c]
and [3818rfl
P.sm.{i.{nt — nt-\-s'—e)-{-3nt-\-3s] )
+ P'.cos.f/.(n'<— ni + s'— s)4-3 7i< + 3£| V
These come under the second form [38186], in which o-K has the divisor i.{n' — n)-\-3n.
The part of rSr [3818a"], depending on these terms, is found by dividing them by
jn/ — )i^ ; ?H, being in this case equal to i . {ii — n) -j- 3 ?i ; and by hypothesis it is very
small in comparison with n. Thus, for Jupiter and Saturn, where i=5, it becomes
m^=i.(n' — «) -l-3?( = .5 w' — 2n^=j\n [3711/]; so that m,^ is less than ~7^,
for the divisor m^ — n^, we may write simply — 71^=^ — a~^ [3709']. Therefore, by
multiplying [3818c], by — a^, we get the part of rSr corresponding to these terms
of 2/d R ; and then dividing this result by a^, we obtain the corresponding terms of
The terms thus computed agree with those in [3819], depending on P, P'.
necessary to notice the terms of 2/d R, like those depending on [3703, 3704], because
terms depending on different angles from those proposed
in [3807, 3807'], or else such as have not the small divisor mentioned in [3818'].
The next term of ajBT [3818a'J, which we shall notice, is that depending on the quantity
Sn^a.ôr .t^. cos. 2 . (?i i -f s — ra) [3818rt] ; and as we retain merely the terms of the [3818/]
third dimension in e, e', &:c., it will only be necessary to notice terms of the first dimension
in 6r. Now if we examine [1023], we shall find, that none of its terms, of that order,
have the small divisor [3818']; therefore we may neglect this part, and then the only
remaining quantity in [3818a], producing terms of a ^, is Sn^a.Ar .e .cos. {nt -\- e — ts).
As this contains the factor e, we may notice in o r only terms of the second dimension, in [3818^]
VOL. III. 13
rSr
It is not
[3818f]
they will produce in —
50 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
Adding this expression to that in [3814],
[3620] '-Jl^ff^cos.{i.(n't — nt + i'—^) + 2nt + 2s + A],
or
we obtain*
To^r^sof sr^ H .cos.li.(n't — nt-^e'—s)-i-2nt + 2^ + Al
[3821]
— He.cos.{i.(n't—nt + i'—^) + Snt + 3s—^ + Al
-irHe.cos.{i.(7i't—nt + s—B)-{-nti-s + z: + A\
2.(1— S). m n ( aP.sm.\i.{n't — nt+s'—i)-\-3nt-\-3sl
'^ i.{7i'—n)-i-3n' i + aP'. COS. {i. {n't — iit-^-s'—sJ-^-Snt-^Qs]
order to procure those of the third dimension, which are the only ones investigated in this
article. The terms of the second dimension, which can produce the angles proposed
in [.3807, 3807'], are evidently included in the form [3814] or [3820] ; multiplying this
by 3n^o^. e . cos. (?i< + ^ — «)> and reducing by [20] Int., it becomes
r „ „ „Ç cosAi.{n't—nt-{-s'—s)-\-3nt-{-3s-zs-\-A\}
[3818A] 3n^a.ôr.e.cos.{nt + e-^) X a=^^''-''''-l + cos.li.{n't-nt + s'-s) + nt+s+^+A\ V
Now He [3814t] is of the third dimension in c, e', he, and by neglecting higher
dimensions, we may put - = 1 [3701], and then we shall have for the remaining terms
of o.K.cos.{m,t + e,) [3818n],
&He.7i^a'^.cos.ii.(nt—nt-\-s — s)-\-3nt-\-3e—Ts+Al
[3818i] z (
-Jf-îHe.n^a^.cos.{i.{n't—7it + ^—c) + nti-B-J^zi-\-.^.
Dividing this by m^ — n^ [3Sl8a"], we get the corresponding terms of rSr. Now for the
first of these angles i.{n't—nt-{-s — £)^3nt-lrSs—zi-\-A, we have OT,=i.(n'— n) + 3M,
and as this is very small [3818rf], it may be neglected ; and then the divisor becomes — n^-
[3818A:] In the second angle [3S18i], the value of 7«, is i.{n' — n)+n or \i.{n' — ?(.)+3n} — 2ra,
which is nearly equal to — 2n ; hence m^ — n^ is nearly Sn^; consequently this divisor
is nearly equal to 3n^- Therefore if we divide these terms of [3818i] by — Ji^ and 3n^,
respectively, we shall obtain the corresponding terms of r S r ; lastly, dividing these result»
by a^, we get the terms of —5- depending on He, as in [3819].
[38180
* (2412) None of the terms of ^ or — , of the order m'e, contain the small
a- a
divisor [3818'], as is evident from the inspection of the formula [1016] ; so that the terms
of — , containing this divisor, and which must be noticed, are Included in the functions of
the second members of [3819, 3320]. Adding these quantities together, and multiplying
VI. i. §7.] TERMS OF THE THIRD ORDER IN e, e', y. 51
This value of — produces in 2 v, an inequality depending on the angle [3822]
i . (7i' t — nt-\-;' — e) + n f + f, which has i.(n' — n)-\-Sn for a divisor. [3822^
To determine it, we shall resume the expression of (5 v, given by the
formula [931].* The part — \ , ' — of this expression produces [3822"]
It m 1i Or V
in (5 V the term
6v = ^ He . s'm.[i . (n't — nt-\- s' — e) + ^ î + ^ +« + -<4 } ; [3823]
which is the only one of this kind having the divisor i . (n' — n)-}-3n.
The inequality of i5î) depending on the angle i.(n't — nt-\-s' — B)~\-2nt-\-2s, [3824]
noticing only the terms having the divisor i . (n' — «)-}-3n, is, by
[3715, 3814], very nearly equal to
2H.s\n.{i.(n't—nt + B'—B)+2nt + 2e + Al. [3825]
their sum by -, which, by [3701], is equal to l-\- e .cos. {nt -\- e—zs) -j-kc., we [38216]
Of
get the coiTesponding temis of — . The quantities produced by this multiplication are
equal to the sum of the terms [3819, 3820], with the additional term produced by
multiplying the Rinction [3820J by e . cos. {nt-\- s — «), and this term is
He. COS. {nt-{-e — w) .cos. \i.{n' t — nt-{-s' — s) -f 2 n i + 2 e -f .4| , [3821c]
which, by [23] Int., becomes
iHe. cos. \i .{lit — nt-^s' — s) + 3 n i + 3 s — ts + A]
-\-^ He. COS. {i.{nt — nt-\-s'—2) -{- 7i t -^ s -{- vs -}- jl] .
Connecting this with the other terms [3819, 3820], we obtain, by reduction, the
function — [3821].
[3821d]
*
(2413) This formula, by the substitution of [3715a, 3705a], becomes as in [37156],
the part mentioned in [3822"] being represented by — '— . Now the last [3822o]
a^. ndt <fi. ndt
term of the second member of [3819] depends on the angle i T -\- n i -{- s -\- is -{- A
[3702a], mentioned in [3822'], and if we substitute it in the first term of the preceding
2d.{rûr) . , ,
expression „ -~— , it produces the tenn
* a-. ndt ^
— \i. {n — n)+7i].-^. sin. \i T-|- n < + e + w + ^ f ; [382261
and as we have, very nearly, — \i.{n' — n)-\-n] =2n [3818Ar] ; it becomes
2He . sin. \i T -\-nt -\- -a -\- A). Again, the second term of [3822a] has already been
computed in [.3814e], and contains the quantity i He .sm.{i T-\~nl-\-s-{-zi -\-A) ; [3822rf]
connecting this with the preceding [3822c], the sum becomes as in [3823].
52 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
Therefore, if we denote this inequality by
[3826] K.sm.{i.(n't—nt + i'—s)-\-27it+2B-{-B}*
Terras of weshallhave, in 6v, the following expression,
ÔV.
[3827] i<v = ^Jï^e.sia.{i.(n't—nt-JrB'—s)-j-nt + B-Jr^ + B\.
8. // is chiefly in the theory of Jupiter and Saturn that these different
inequalities are sensible. If we suppose i =^ 5, the function
[3828] i . («'— ?i) + 3 n = 5 n' — 2 n ,
becomes very small [381 8rf], in consequence of the nearly commensurable
ratio which obtains between the mean motions of these planets ; and from
this cause the corresponding terms of or, ov acquire great values. To
determine them, we shall resume the expression of R [3742]. The partf
[3829] — .cos.(y— f) -j-\cos.{v'—v) — cos.{v'+v)\+-^.— ^ ^,
^ 4 r^ 4 J,-2-2r)-'.cos.(y'-«;)-fr'2p
* (2414) The parts of R [957, 1011], represented by M, JV [3703, 3704], do not
contain the small divisor i . {71 — n)-\-3n, as is evident from inspection. Moreover,
[3826a] F, G, H [3706], being the parts of — [1016], depending on terms of the first degree
in e, e, do not contain this divisor, as appears by the inspection of [1016]. Therefore no part
2rf.(?"(5r)
of ÔV [3715], except the first term —^^ — 7—, contains this divisor ; and if we substitute
a^.ndt
in this term the value of r (5 ?• [3814], we shall obtain, in ô v, tlie terra
2
[38266] .li.{n'—7i)-{-2nl.H.sm.li.{n'i—7it-\-e'—s)-J[-2nt-{'S-j-A} ;
substituting — \i.{7i' — ?))-(- 2 «}=?i [3S22c], it becomes as in [3825]. If we now
compare the expressions [3825, 3823], we find, that [3823] may be derived from [3825]
[3826c] by multiplying its coefficient by |e, and decreasing the argument by nt-^B — sj.
The same process of derivation being used upon the assumed form [3826], produces the
expression [3827] ; which is computed in [4439] for Jupiter, by tliis very simple process.
t (2415) We shall suppose, as in [1009, 956c, 963'', 1018a], for tlie sake of brevity,
[3829a] r =a(l + Mj; r' =a'.(l+j«;); v =^nt + s-\- v, ; v'=n't + s'-^- v,' ;
[-38296] a^ = a m, ; a' = a' w/ ; a"=: v,' — r, ; a. ^ - ;
[38296'] T ^n't-nt-\-^—e; dT^{n'—n).dt;
[3829c) W=nt-\-B — -ro ; W'=^n't-\-B' — -a;
[38S9e'] M,, «/ v' — » are of the order of the excentrlcities, and a is changed into a^, to
VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 63
produces no term of the third order of the excentricities and inclinations, [3830]
distinguish it from a [963'^]. If we represent the function [3829] by m, and suppose U to
be the part of this value independent of u,, «/, f,, vj, we shall have U as in [3829/]; [3829d]
observing that the last term of [3829J becomes in this case, by using [3744, 3749],
im'.'j^.aci'.cos.T.la^-2ad.cos.T+a'^l~^ = ^^m'.f. ad. cos.T .is . B''\cos.iT
= im'.f.aa'.iS.B"Kcos.{i-\-l).T
= 1 m'. y^. a a'. -2. B^'-il. cos. i T ; t3829e]
[3829/]
[3829g:]
U=L — ^g- . cos. T — T ™'' 7*- "7^ ■ cos. T -j- i '«'• y®- -^ • cos. [n t -\- n t -\- ^ -{- 1)
+ I m. f. aa'.S.. B^-^\ cos. i T ;
i being as in [3715']. The development of u, as far as the second powers of aj, a', a"
being found as in [957e], is
«= -■+-(^) +-• O +^"- © +H.'. (^)+^..^'. G^.)
the tenns of the third order, obtained in the same manner, are
+i-^"'G-S^)+--^"-G-i^)H-*-'-(^)-
We have given this full development of îi, because it will hereafter be of use in the notes on
this article ; and for the same purpose, we shall also insert the following expressions, deduced [3829t]
from the comparison of the values of ao, a', a" [3829J, a] with [659, 668, 669] ;
ao = rt .|i e^ — (e — f e') . cos. fV — i e^.cos. 2W — i e^. cos. 3Wl=au/, [3829*]
a'=a'.|Je'2— (e'— tc'3).cos. fP— ^ e'2.cos.2 JF' — § e'^. cos. 3 ^'} = «'m/ ; [3829i]
"■~^-(2e-ie3).sin. fF-Je2.sin.2?r-ife3.sin.3^FS^"''~'''' ^^^^^
From these values it appears, by a slight exammation, that none of the terms of U [3829/]
produce quantities of the third order, depending on the angle 5 7i'i — 2nt, now under
consideration. For the terms of [3829/], multiplied by y% of the second order, depend [3829«]
on the angles T, n' < -f n < + s' -f s , i T ; and when we combine these with terms of the
VOL. III. 14
64
PERTURBATIONS OF THE PLANETS.
[Méc. Cél.
Value of
R
for tliis
case.
[3831]
[3832]
depending on the angle 5 n't — 2nt; such terms can therefore only arise
from the remaining part*
m'.y^ rr'.cos.{v'-\-v)
R
^j.2 — ,2;-/. COS. (j;' v) -\-
JSfS
I r^ — 2 r r'. cos. (ti' — v)-\-r'^l-
and then the expressions of P and P' [3810] will be the same, whether
we consider the action of m' on m, or that of m on m'. We shall now
investigate these values of P, P'.
[3829o]
[3831a]
[3831o']
[38315]
[38316']
[3S31c]
[3831d]
[3831e]
[3831e']
[3831/]
^w< order in a,,, a', a" [38297i: — ?»], they will not produce the angle bn't — '^nt. The
only remaining term of C/ [3329/] is the fii'st, depending on cos. T or cos.(?i'i — nt-\-^ — s);
and if this were multiplied by a term depending on the angle An't—nt, it would produce a
quantity of the required form ; but none of the powers and products of cio , a', a" [3829Ar — ?»];
retamed in [3829^, K] contain terms of the third order depending on this angle ; therefore
we may also reject this temi, as in [3830].
* (2416) If we reject the terras of R [3742], mentioned in [3829], which we have
proved, in the last note, not to contain terms of the required form and order, we shall obtain
for R the âmction [3831]. This expression is not altered by changing r, v into r', «',
respectively, and the contrary ; so that it w'Ul be of the same form, whether we compute the
action of ?«' upon m, or that of m upon to' ; but in the first case it will be multiplied
by in, in the second by m. Supposing, as in [3829fZ], that the general value of the
fonction R [3831] is represented by u, and that it becomes equal to U, by putting
r = «, ?•'=(/ v^^nt-\-s, v'^n't-{-e', v — » = ?t'i — n ^ + s' — s=T,
we shall get the first of the following expressions of U [3831c]. The second expression
[3831<?] is deduced fi-om the first by the substitution of the values [.3743, 3744], neglecting,
however, the first term of [3743], which makes an exception in the value of A'-''', in the
case of i = 1 ; because this term produces no effect in the present calculation, as we
have seen in [3829o] ;
lJ=—m'. \(v^-2 a a. cos. T+a'^-^—{m'.y^.aa'.cos. {n't+nt-\-s'+2) .{(?- 2 aa'. cos. T-\-a'^\-i
= im'.S.A^'\cos.iT—im'.f.aa'.cos.{n't-{-7it-\-s'-j-s).S.B'-'\cos.{T
= i m. 2 . A^'\ COS. i T— I iri. y^. a a'. 2 . B'^ ". cos. [i T-\- 'int + 2 s— 2 n) .
We may remark, that, in reducing [3831(/] to the form [3831e], we obtain, in the
first place, from [3749],
cos.{n't-\-nt + s'-Jrs).:s.B^'\cos.iT=X.B^'\cos.{iT-\-n't-j-nt-{-s'-\-s)
= 2.B''\cos. {(i+l).T+2«^ + 2£} ;
and by changing i into i — 1, it becomes X. B'-'~^\ cos. \iT-\- 2 n t -\- 2 b\ ; but as this
quantity is to be multiplied by y^, we must change 2n<-|-2s into 2nt-{-2s — 2n, as
in [.3745'" — 3748], and then the value of U becomes as in [383 le].
VI. i.^,^ 8.]
TERNIS OF THE THIRD ORDER IN e, C, 7.
55
We have, in Book II, ^22, by carrying on the approximation to terms
of the third order of the excentricities [659, 668, 669],*
— ^e\cos.(3nt+3B—3^) ;
v = ni4-e+(2e — ie=').sin. (w^ + s— tï)+ f e-.sin. (2nï + 2s— 2x^)
+11 e^sin. (3w< + 3s— 3a).
Values of
r, 1'.
[3834]
* (2417) We shall now commence the investigation of the part of R depending upon
the first term of [3S31e], namely, U^=^ m'. 2 . ^''\ cos.i T; the other terms depending
on B''"", being computed in [3840a, Sic.]. Substituting this value of U, in the
terms [3829^, A], we get the following value of R,
[3834o]
1
R =
2, 3
4
5, 6
7, 8
9, 10
11, 12
13, 14
15, 16
17, 18
19, 20
im'.2.^<''.cos.^T
+
+ * m' . ag . 2 . f — — \ . cos. I r+ J m' . a'. 2 . ( -7-^ j -cos. i T ,
— im'. 0.". Si. A^'\ sin. iT
, „ /ddA(i)\ .^, 1 , , /dd.m\ .^
+ im'.ao2. 2 . -r-7- ) . cos. i T+Jm'.ao a' . 2 . ( -— — ) .cos.i T
\ da^ J " \dada J
+ l+i m'. a' ^ . 2 . i^-^) .cos.iT-i m'. ^a". 2 i . (1^) . sin. i T
—im'.oJa/'.Xi. (-j^) ■ sin. i T— ^rn'.a''^. 2 Î^A^'K cos. i T
+ A-'.S^2.('-^).cos.iT+.™'.ao^a'.2.(^).cos.ir^
+ ^-'-'^-(£^.)---^+^V..'.a'3.2.(^^).cos..-T
+ < ->'.a„V'.2;.('i^) .sin.zr-i».'.aoa"^.2P.(^).cos.tr
-im'.a'2a".2i. (^) . sb. i T-im'.a'a"^. 2^^. (^^Vcos.i T
— àm'.aoa'a".2i/^^,') . sin. i r+^ . a"^. 2 i'. ^». sm.i T
\dada/ ' 12
Terras of
R
depend-
ing on
[38346]
We must substitute, in this expression, the values of a^, a', a" [3829A: — ni], and retain
only the terms of the third dimension, and of the form 5n't—2nt [3834"], in which the
coefficients of n't, nt differ by 3. Now as these coefficients are equal in the angle i T,
which occurs m [38346], this difference in the coefficients of ri't, nt must arise from the
[3834c]
56
PERTURBATIONS OF THE PLANETS.
[Méc. Cél.
[3834'] This being premised, if we develop R [3831] according to the order of the
powers and products of a^, a', a" ; and it is evident, from [957^'", Sic], that such terms
[3834d] must have for a factor, some one of the four quantities e'*, e'^e, e'e^, c^. If we take
the powers and products of tlie quantities a,,, a', a" [3929/!: — to], of the tliird dimension,
and reduce them by means of [17 — 20] Int., we shall find, that the greatest angles connected
[.3834e] with these factors e'\ e'^e, c' <?, ê, are, respectively, 3 7P, 2W'^W, W'-{-2Jr, 3fV;
it is not necessary to notice the smaller angles TV, W , 2 W — W, Sic, because they
do not produce terms of the form bn't — 2n t [3834c] ; substituting ?F'= T-\-nt-\-B—-a',
W= nt-{-s — -m [3829c] ; they become, respectively.
[3834e']
[38.34/]
[3834êr]
[3834^]
[3834i]
[3834ft]
3 T+3Mi + 3£ — 3«'; 2T+3n<-f-3e — 2i3'— ra;
r_|_3,j;_j-3£_,3'— 2i3 ; 3m; + 3£ — 3îï.
Now we perceive, by inspection, that the cosine of any one of these angles is multiplied,
in [38346], by a tenn of the form ^/''. cos. z T ; and its sine by a term of the form
^/''. sin. I T; the products reduced by the formula [3749], are found to depend,
respectively, upon the angles
(i + 3).T-}-3n< + 3£ — 3«'; (« + 2). r+ 3 n < + 3 e — 2^^' — ts ;
(i_|_l). T-\-Znt-{-3s — z^—2-a; i r+ 3 n < + 3 s — 3«.
In order to reduce all the angles to the form i T, we must change, in the first, i into i—3;
in the second, i into i — 2 ; in the third, i into i — 1 ; and make the same changes in
the index of ^/'' ; by this means the terms in question become of the forms
e'3. 2 . ^i''-=>. COS. (i T+ 3 Ji < + 3 £ — 3 ^) ;
e'=e . 2 .A^^'-^K COS. (i T + 3 n « + 3 s — 2 73'— i^) ;
e 62.2.^''-». COS. (i T+ 3 B ^ + 3 £ — TO — 2 13 ) ;
e» . 2 . ^ w. COS. {i T+ 3 ÎI i + 3 £ — 3 w) .
Putting i=:5, as in [3828], these expressions become of the same forms as the four first
terms of R [3835], depending on M'°^ M'", Jf'^', M<^\ respectively. The two
remaining terms M''^\ M^^\ depend on JS""", which was neglected in [3834a], and
will be computed in [3840a, &c.]. We may remark, that the exponent of e, in any one
of the terms [3834A], being increased by i — 3, gives the corresponding index of ^, ,
and when i = 5, we have for this increment i — 3 :^ 2 .
We shall now proceed to the computation of the values of the powers and products
of a,, a', a", which occur in the expression of R [3834J], retaining only the tenns
[38342] depending on e''
e^, which are wanted in finding the values of M'-°\
VI. i.§8.]
TERMS OF THE THIRD ORDER IN e, e', 7.
67
terms depending on the angle 5 n't — 2nt, we shall obtain an expression [3834"]
of the following form,
J\P^\ M'--\ M'-^K These quantities are arranged in the following table, in the order in
wliich they occur in [38346], noticing only the greatest angles mentioned in [3834e] ;
[3834m]
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0-0
a'
a"
tt,'
= — frt.c». cos. 3 TV;
= J-l e' 3. sin. 3 W— if e^. sin. 3 W ;
0
= i 0^ f3. COS. 3 W ;
aott
a' 2
i a' « . e'2 e . cos. (2 W'^ W) -\- ^a' a .e' ê. cos. ( ?F'+ 2 7F) ;
irt'2.e'3.cos.3fF';
a^a" = fa.e3.sb.3»^— |«e'2e.sin.(2?F+?F) — Jrt.e'e2.sin.(JF'+2ff)
a' a" = — f a', e' 3. sin. 3 ?F'+ 1 a', e'e^. sin. ( ?F'+2 ?r) -f i a', e'^ e . sin. (2 W'^ W)
i c\ C0S.3 W+ye^cosX W'-{-2 7F)+f e'2e.cos.(2 fF'^- W^)— f e'3.cos.3 W
= ftf3
ao' = — i fl^. e'. cos. 3 W ;
a/ a' = — i a' a", e' f?. cos. ( ?F'+ 2 ?F) ;
a.oa'2 == — ia'2a.e"2e.cos. (2?F'+ W');
a'3 = — |«'3.e'3.cos.3l'F';
a;-ia" = — \ «2. c^. sin. 3 W + * «^ g' ^a, gin. ( /F'+ 2 W) ;
aoa"2 = a.e3.cos.3?F— 2a.e'e2.cos.(?F'+2/^) + «.e'2e.cos.(2^'+rr) ;
a'^a" = i«'2.e'3.sin.3 JF'— ia'2.e'2e.sin. (2R^'+ ?F);
a'a"2 = a'.e'3.cos.3 TF'— 2a'. e'^e .cos.(2ff' +?F)4-«'.e'e2.cos. (?F'+2(F) ;
ao a' a" = — 1 a a', e e^. sin. ( W'-\- 2 W') + J a a', e' ^^ e . sin. (2 fF'+ W) ;
a"3 = 2e3.sin.3fF— 6e'e2.sin.(fF'+2?'F)4-6e'2e.sin.(2?F'+fF)-2e'3.sin.3W'' .
We shall use these expressions in the following notes, in computing Jlf", JV/<", fiic. ; and
we shall also make use of the following formulas, which are deduced from [95.5e — A], by
taking the differentials relative to T, and dividing by àzdT, changing also W into ?F^ ,
as in [3T50A, &tc.] ;
sin. W^.is. P. A^'-> . sin. i T= — ^ 2 . P. ^^'l cos. {i T-\-TV);
COS. fF, . 1 2 . P. A'^\ sin. i T=: i^.P. A^'\ sin. {i T+ IV,) ;
sin. W,.\-s..P. A''\ cos.i T^ ^ 2 . P. A^'\ sin. (/ T-\- W,) ;
cos. fF, . I 2 . i3. ^'0. cos.i T= X V . î3. ^(0. cos.(i r+ fFJ,
VOL. III. 15
[3835a]
[38356]
[3835c]
[3835(f]
[3835el
58 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
General
form of
for termaof
the third
order-
R= M^''\e'\cos.(5 7i't — 2nt + 5s' — 2s — 3z^')
~R +ai^'Ke'-e.cos. (5n'i — 2 n i + 5 s' — 2 s — 2«' — ^)
+ M(^>. e'e-. COS. (5 h' i — 2 jU + 5 s' — 2 e — ^' — 2 ^)
[3835] +M(^'.e^cos. (5 n' t — 2 n i -^ 5 e' — 2 s — 3 ^)
+ M'^' . e'f. COS. (5n't — 2nt + ôs' — 2s — zi'-~2n)
+ M'-'Key-. COS. (5 n' t — 2n t + 5 e' — 2 s — z^ — 2 n) ;
and we shall find, after all the reductions,*
!(2) (3) (3)
389 6';' + 201 a . ^-^+ 27 a^ ^ + a3. Ç^
•4 a a da-' rf a^
[3836a]
* (2418) The pait of R [3835], depending on e'*, may be put under the form
M'-°K e'\ COS. {iT-\- 3 W) or iH'»\ e'^. cos. (2T+3?F'), using T, ÏV, &lc. [38296', c] ;
the coefficient of T being i=2. Terms of this kind are produced in i?, by multiplying
the quantities which are connected with e'^ in [3835aJ, by the corresponding terms with
which they are combined in [38346], and then reducing the products by means of the
formulas [955, 955a — h, 33356]. The terms depending on ^® and its differentials, are
[38366] giygjj ;,^ ^jjg value of Jli"" [3S36c/], in the order in which they occur, without any reduction,
and omitting 2 for brevity ; so that the terms of [3835a], marked 4, 10, 20, are connected
/dA'''i\ /rf2^(i)\ /rfS^Wx
With ^«; 3, 9, 18 with (^); 7, 17 with (-^-^j ; 14 with (—-) .
Substituting i^^2 [3S36o] in this first value of M^°\ we get the second value of [3836e] ;
[3836c] and this, by using the values [1003], becomes as in [3836/], or by reduction, as in [3836^].
Lastly, substituting in this the values [996 — 1001], we get [3836/j], which is easily
reduce d to the form [3836] ;
[3836d]
/(/./?(3)\ /d-A^~^\ „ /rf3./3f3)\
[3836e] =W-^^--W-'.«^('-;^)+il-«-.(^)-.V-'--•(^
+ -.^6^'^'+18a.(— j+9a^(-^) + a^.(^(
[3836g] =^s^W.^«,^.^^^^«.(^__j + |^^^«3.^__j + _.«3.(-^-^j
/- (3) (2) (21 ■
7,1' S 12) dbk c. d^bi , d3Ji (
[3836/.] =:i^,.)— 389 6,— 201a. -^ — 27 a2.^--a3.—f ,
VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 59
16 / * a a, dix.^ ria-* >
* (2419) Proceeding as in the last note, we find, that the part of R [3835] depending
on e'=e, may be put under the form M'-'\ t'^ e .cQs.{iT -^'2 W'-\- W) [3829è', c], [383ra]
in which the coefficient of T is i = 3. Substituting the values [3835a] in [3834&],
we obtain the first of the following values of JV/'" ; observing, that the terms of [3835a]
depending on e'^e, marked 10, 20, are connected with ^<'^ ; the terms 8, 16, with i— — j ;
the terms 9, 18, vnû\ ( , , ) ; the terms 6, 19, with ( , ', ,) ; the term 17
\ da' J \dadaj
with C^-^^); and the term 13 with (^^^^^r^A- Substituting i = 3 in [3837(-], [38376]
we get [3837f?] ; and this, by using the values [1003], becomes as in [3837e], or by
reduction, as in [3837/]. Lastly, substituting in this the values [996 — 1001], we
get [3837 0-], which is equivalent to [3837] ;
[3837c]
, , /dd^\ ,,,,., ; , ,a /'d--A(0\ ^ , ,2 /rf3^U)>,
= - \\«- m'. A ^3>- 1^ m'. a . (^-j^j + f |- m'. a'. (^ j
[3837rf]
, ,, , , /ddJ10)\ ^ , ,„ /rf2^(3K /d3jia)\
60 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
( W (4) (4) ■)
[3838] „'iJf(.= _^.- 3966';+ 184a. ^ + 25a^^ + a^^;*
16 ( ^ da. do.'' da.-^ )
* (2420) We may compute [3838, 3839] as in the two last notes, but it is rather less
laborious to derive them from M'-''\ M'-'-\ by changing the symbols as below, namely,
[3838o] For i, n't, nt, e', i, zî , •n, e', e, «', a; a', aj, T;
[38386] Write — i, nt, n't, i, s', zi , -a', e, e' , a, a; a„, a', — T.
The changes in these three last values of a', o.^ , T, evidently follow from those proposed
in the other symbols, using [3829À:, /]. The value a" [3829m] is not altered, except in its
sign, because e.sin. W^ changes into e'. sin.?F', and e'.sin. ?F' into e.sin. fF, &ic. ;
moreover, A^'^ is not altered, because we have A'-~''' ^^^ A^''' [954"]; we also have, as
[3838c] j^ [3831c, rf], —\a^— 2a a', cos. T-\- a'^l-i=i X .A^^\co5.iT ; and as the first
member is symmetrical in a, a', the second, or A''\ must also be symmetrical, and wll
[3838rf] not be varied by putting a, a' for a', a, respectively; lastly, the expression of iî [38346]
is not altered by making these changes ; observing, that the quantities i a", i T remain
unchanged. Now the part of R [3835] depending on c'e^, may be put under the form
[3S38e] J»f (2). e' e^. cos. {i T+ 2 W-{- W), in which the coefficient of T is i = 4 . Comparing
this with [3837a], we find, that by making the changes [3838a, 6], the expression [3837a],
corresponding to i = — 4, will become like [3838e], and M'^^ will change into M'--^ ;
we may therefore obtain the values of M^"'^ [3838/1], by changing a, a', i into «', a, — i,
respectively ; then putting i = 4, we get [3838A']. This value may be reduced to the
form [3838J], by the substitution of the values [1003], and also the partial differential of
the second of this system of equations, taken relatively to a, which gives
[3838/]
Reducing the expression [.3S38i], we get [3838^] ; and by the substitution of the values
[996—1001], it becomes as in [3838?], being the same as [3838] ;
M'-''-> = m'.A<~^{—^i^^^i^+iii:.n'.(^^yy^i-li^]+m'.a.(^^^y\ —
[3838/i']
[3838i]
dada' J '
/f/.-î'-DX /(/.^HA /ddJl'^^\
=W»'.-«»-tt..'.»'.(-^)+W-«'...(^)-iS.«'..«'.(,-^,)
VI. i. §8.]
TERMS OF THE THIRD ORDER IN e, e', 7.
61
a' M'^' =
in
48
!(5) (5) (5) -\
2 ri a d iS? do? )
C (3) ^
a'ilf W _ _ ^ . ^ 10 63 +a . lï^ > ;t
Id / ^ f/a >
[3839]
[3840]
f (J) (-1) '41 -J
= 7-r-, . J — 396 0 i — 184 a . — 25 a-', -—^r a'. — — - ( .
lb a f - a a da~ aa'^ )
* (2421) The part of R [3835] depending on e^, may be put under the form
M^^''.e^.cos.{iT-\-3 TF), in which the coefficient of T is x=5. Comparing this with
[3836a], we find, that by making the changes a, a', i, &ic. into a, a, — i, &c., respectively,
as in [3838a, 6], the expression [3836(/] will become as in [3839i]. This represents the
value of Jf^"", or the coefficient of c^ in [3835]; and by putting i^5, it becomes as
in [3839i'] ; which, by means of [996—1001], is easily reduced to the form [3839] ;
48a C * ' do. ' rfa2 ' do. .
t (2422) The values of iV/C", M*^) [3840, 3841] depend on the second term of [3831e] ;
and by retaining only this term, we shall have JJ ^ — | m'. 7^. aa'.'S. . B^'~^K cos. T, ,
supposing, for a moment, that T^z=i . (n' t — nt -\- s' — i) -\-2nt -{-2s — 2 n .
As this expression is multiplied by 7^, of the second order, we need only notice terms
of the first order in ao, a', a", in the development of u or R, and we shall get
for this part of jR, the following expression [3829^],
«--C^)+'^'-(7
""U.-.f^
ft' /
dT,
obser\ing, that we notice in this article only terms of the third dimension. The values
of aQ, a', to be substituted in this expression, are the same as in [3829A:, Z] ; and
by retaining terms of the first order, w-e have ao = — ae . cos. fV, a == — a'e'. cos. M'".
The angle T, represents the mean value of i . {v — v) -\-2v ; its increment, depending
VOL. III. 16
[3838^]
[3838Z]
[3839a]
[38394]
[38396']
[3839c]
[3840o]
[3840a']
[3840A]
[.3840c]
[3840(/]
62 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3841] «'M<^' = ^.<7 6l + a.^^.
r3840rf1 °" ^'" ^'' [^^~^"]' '•'' o-"=i.{vJ — ■dJ -j- 2 !;,= «!)/ — {i — 2).i;,, and by substituting
v;=2e'.sm.TV', v,= 2e.sin.fV [669], we get a", and then [38406] becomes
R^-e'.[a'.oos.W'.(^)-2i.sin.W'.(-^^\
[3840e]
_e.^«.cos.^F.(^)+C2.--4).sin.rF.(j^)|;
and by substituting the partial differentials of U [3840a], we obtain, without any reduction,
R= ^- m'. c' f. COS. TV. J a' a . 2 . B^'-^K cos. T^ + a' ^ « . 2 . C^^^) ■ cos. T, I
+ i m'. c' 72. sin. W .a'a.Xi. £('-i>. sin. T.
[3840/-]
+ J m' . c 72 . COS. TF. \a'a.-s:.B <'-». cos. T^ + a^ «'. 2 . (^-^^) . cos. T^ I
— i m' . c 7^ . sin. W .a'a.X. (2 i — 4) . S''-», sin. T^ .
The terms of this expression, depending on c' 7^, contain the factors cos. tV'.cos. T^,
[3840g-] and sin.fF'.sin.T^, both of which, as in [17, 20] Int., produce the terms icos. (T^-^W),
which, by putting i = 4, becomes icos.{5n'( — 2nt-\-5s' — 2s — •ra' — 2n) [3840»'].
Comparing this with the term depending on Jfef '''' in [3835], we get the first of the
following expressions, omitting 2 for brevity, and then by successive reductions, using
[963''', 1006—1008], we finally obtain [3840/], which is easily reduced to the form [3840] ;
[3840/1] M'-^^=-i^m'. i a' a .B^'-^^-{- a'^a. (^fj^) \ — ^ m'. a'a.i. S^'-"
[3840i] = J^ m'. a' a . \— 1 B'^^ + a'. (^-^^ j= J, m'. a'n . J- 7 5«' + [-3B'^>-a . (^')] I
(3)
,:««, =,,.,..4-,OB™-..(i^')|=,,„..„.„.)_i£,4'_^..^ I
[3840i]
ICa C 2 ' rfa
In like manner, the terms of [3840/"], depending on e 7^, contain the factors
COS. ^F. cos. T4, sin. ?r. sin. T4, producing the term | cos. (T4+ ^F), which,
[:3840»i] by putting i = 5, becomes ^cos.(5n't — 27it-{-5s — 26 — « — 2n) [3840o'].
Comparing this with the term depending on M^^^ [3835], we get the first of the following
VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', y. 63
Hence we deduce*
General
+ a'M(^). e'e~. sin. (^+ 2 w) + a'M^^'. el sin. 3 « [3842]
+ a'M<^'. e'7'. sin. (2 n + ^') + a'M'^'. e y". sin. (2 n + ..).
and of
We shall get in', a' P', by changing the sines into cosines, in this expression °^^,'
of in!, a' P ; and it will be easy to deduce the values of a P, a P, by [3843]
expressions, in which we must put i = 5, and then, by reducing as above, it becomes as
in [3840p] ; whence we easily deduce [3841],
M^^^=j'^m'. I «'rt.B<'-" + a2rt'Y!^^") I -j-j\m'.a'a.{2i—4).B^'-'^ [3840n]
[3840o]
(4)-
16 o' C t "T d a )
* (2423) In the case of i = 5, if we use, for a moment, the abridged symbol [3842a]
T5=5n't — 2nt-\-5e' — 2s, the value of R [3810] becomes
R = m'. P. sin. Tg -f 7n'. P. cos. T^ . [3842o']
Now each tenn of R [3835] may be easily reduced to the form [3842»'] ; since, if we
take, for example, the fii-st ^<'". e'lcos. (T^ — 3ra'), and develop it by [24] Int., it [38426]
becomes J/"'>.e'3.sin.3w'.sin.T5+^/<®.e'3.cos.'n'.cos.r5. Comparing this witli [3842a'],
we get for the parts of m'.P, m'.P', the following expressions,
m'.P = Jlf(°i.e'3.sin. 3^3', m'. F=M^°\ e'^ cos-Sz/, [38426']
as in [3842, 3843]. In like manner, we obtain the other terms of [3842] from [3835].
The values of P, P', deduced from [3842, 3843], may be put under the following
fonns, which will be of use hereafter. Expres-
sions of
P=S.M'.e"'. e". f\ sin. (6' îi'-f 6 ts -f 2 c n), ^' P-
[3842c]
P'= 1 . M'. e"'. e\ f. COS. (6' î3'+ 6 rt + 2 c n) ;
2 being the characteristic of finite intégrais, and h\ b, c, integral numbers, including zero,
satisfying the equation è' + 6 + 2c=:3.
64 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3843'] multiplying a' P, a' F, by ~ or a. We shall then find, by putting
i — 5, in the expressions of ôv and — [3817,3827,3821],*
Ex près -
sioiiofthe /' ( ^n HP ?t« rltfP' "^
\ ^-P'+T^^Vt.-,-/ o P ^,ol -sin. {5n't-2nt+5s'-2s)
ÔV —6m'. n^ ) i. (5n—2n).dt {5n'—'2nf.dt^) ^ ' '
of the °"
third
order.
[3844]
(5n'_2„)9 \ r 'ia.dP' Sa.ddP
( „ 2a. dP' Sa.ddP ) , ,
— ^ ciF— , ,, — ,g , o V, j.o ^ •cos.(5?i'/-2n<+5s'— 2s)
C (5n'— 2>i).d< (5n'—2nf.dt~ ) ^ ' '
2 / \ — „ cc'v n.. V r r J o cA
5 n' — 2 n'
' — a^f—ysm.{rjn't—2nti-5s'—2s)
— ^He.s'm. {5n't — 2nt-l-5E'—2s—zj-\-A)
-]-^Ke.s\n.{5n't—4nt-J^5s'—4s-\-7S-\-B) ;
Exprès- ^ J,
terTo.-'" — = H .cos.(5nV— 3/U + 5s'— 3e+^) — iîe.cos.(5n'^— 2ra« + 5£'— 2s— «+^)
of.the _|_ He.cos.{bn't — Ant-\-bs' — As-\--ui-\-A)
order.
[3845] +^^^.\aP.sm.{bn't — 2nt-^bs'—2i)-YaP'.co5.{^n't — 2nt + bt'—2s)\.
[=3845'] If we suppose i = — 2,t and change the elements of m into
[3844a]
* (2424) Adding the terms of .5^ [3817,3827], and putting i=b, we get [3844].
Putting i = 5, in [3821], we obtain [3845].
t (2425) By restricting ourselves to terms of the first order of the masses, and of the
[3846a] third dimension In e, e', y, the expression of — [3831] becomes symmetrical in the
elements of m, m', so that these elements may be Interchanged without altering this value
R R
of — [3831 «, «']. The same symmetry obtains in the expression of — [3810] ; for
[384(36] if we put, for a moment, T^ = 5n' t — 2jit ^5s'—2s, T,.= 5nt — 2 ?f'C+ 5 s — 2e',
and retain, in [3810], only the two terms arising from the successive substitution of the
values » = 5, i = — 2, It becomes
[3846c] ^=P- sin. T, + P'. cos. T, + P^ . sin. Tg + P'o- cos. T, ;
Py, P'o, Tq, being, respectively, the values of P, P', T^, when the elements a, n, e, &c.
are changed into a', n', e, Sic, and the contrary, this being necessary to preserve the
[3846(/'] symmetry [3846»]. In computing the action of tn upon m, it Is not necessary to notice
VI. i. §8.] TERMS OF THE THIRD ORDER IN e, c, y. &b
the conesponding ones, relative to Ht', and the contrary, we shall obtain
C ,„, , '2a'. (IP Sa'.ddP' ) . ,, , ^ , ^ , r^ ^ \
I '{5n'—2n).dt {57i'— inf. dt^ y ^ ' ^f
( ,„ 2«'.rfP' Sa'.ddP } /r '. f> V 1 C ' O \ ( Expros-
I (5n'-2n).rfi i5n'-2nf.dt^ S ^ 'J tcrmsof
_ . 15m.)i'-
he
terms of
iv'
of the
third
2m. n' ) vaa / • ■ - r order.
«'-.r^j.cos.(5?i'^— 2n^ + 5e'— 2s)
5n'— 2n ) .„ /dP
— a'^.(—\.sm.{on't—2nt-\-b^—2s)
[3846]
— IH'e'.sin. (57!'< — 2)U + 5s'— Qe— t3'+^')
+ îi:V.sin. (3n7 — 2;U + 3£'— 2ê-[-w'+B') ;
t / Exprès-
^= if'.cos.(47i'<— 2n< + 4£'— 26+^')— iîV.cos.(5n'!!— 2n!;-f5£'— 2e— w'+^O ,';°"„^"ffi-
+ i/V.cos.(3«V— 2n< + 3£'— 25 + w' + ^') «nhe
order.
^°™'"' |«'P.sin.(5n'< — 2ni + 5E'— 2£)+a'P'.cos.(5n7 — 2n< + 5£'— 2£)} ; [3847]
5n'— 2n
if', cos. (4 n' i — 2 n ^ + 4 e' — 2 £ + J' ) being the part of — r^- depending [3848]
onthe angle An't — 2nt* and ^'. sin. (4n'i — 2n i + 4=' — 2£+-B')
the angle T^, because it does not produce terms having the small divisor 5 n' — 2n. [.3846rf"]
In making the change of the elements of m into those of m', according to the directions
[3845'], the value of — , corresponding to the action of m upon 7n', becomes
- = Po . sin. T. + P'o . COS. To + P. sin. T, + P'. cos. T, . [3846e]
m
The second members of [3846c, e], are evidently identical ; but in this last expression
the terms depending on the angle Tg, are derived from those of [3846c], which depend r„o,^^^
on i^ — 2 ; by changing the elements ?«, «, e, Sic. into those of m', a, e', &c., as in [3845'].
Lastly, we may observe, that the quantities P, P', connected, respectively, with
sin. Tj, cos. Tj, are the same in [.3846c, e]. Hence we may derive ôv from Sv, by
taking the sum of the two parts of 5 » [3817,3827], putting i = — 2, then changing
m, a, n, e, H, K, k.c. into m, a', n', e, H', K', &.C., respectively ; by which means
we get [3846]. In like manner, we may derive [3847] from [3821].
* (2426) These terms correspond to [3814, 3826], putting i= — 2, and changing
the elements as in [3845'].
A'OL. III. 17
66 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
being the part of 6 v' relative to the same angle. In these various
inequalities, we shall, for greater simplicity, refer the origin of the angles to
[3840] the common intersection of the orbits of Jupiter and Saturn ; as Ave have
already done in the development of the expression of R [3736 — 3738], and
shall continue to do in the following article. For the sake of symmetry,
we shall retain the angle n, which must be supposed equal to nothing.
lioZf * We shall determine the differentials - — , , ' , - — , — — , in
,P,dir, iW dt^ ' dt ' dt^ '
^^- the following manner. We shall compute, for the two epochs of
[3849] 1750 and 1950, which embrace an interval of 200 Julian years, the
, - (7e (/« de' d-ul d y du i i n
values or --, —, —, —— , —, -— ; and shall represent these
[3850]
dt' dt ' dt' dt ' dt' dt
quantities, at the second of these epochs, by — ', — -^, -~-, &c. ;
we shall then have, by supposing t to be expressed in Julian years,*
[3851] ^'^^ + 200.^%
dt dt dt''
in which the differentials de, dde, in the second member, correspond to
the epoch 1750. The value of e,t for any time t, neglecting the cube
* (2427) We have, as in [607, &c.],
[3850a] ,^f7+,.(^) + .,..(^J_^) + &e.,
[38506]
d € (i€
u beins; a function of ^, which becomes U, when t=0. Now puttine m=— -', C7:= — ,
,- T , ••11/' r '^^, ^^ 1 dde
as in [3850], we get, by retaining only the farst power of r, 77 =" 77 + ^ • jTs > which,
by putting ^=200, the interval mentioned in [3849'], becomes as in [3851]. From this
roo,-/. -, ddi 1 \de, de} de de, .
[38.50c] we get ^^ =§00 ' Jdl " rfIS ' ^^^^ ^^"'"'^ °^ rf"i ' ^ ' being computed, as m
[4238, he, 4330a, &c.], for the epochs 1750, 1950 ; we obtain, by substitution, in [3850c],
dde
the value of -t^j corresponding to the epoch 1750.
t (2428) Putting U=e, M=e,, in [3850«], we get
[•3852a] e=e + t/^^+it^.'^ [3852];
in which we must substitute the values of e, ^^ , ^ [3850, 3850c], for the epoch 1750^
VI. i. §8.] TERMS OF THE THIRD ORDER IN e, e', 7. 67
of t and its higher powers, is
de (1 d I
clt' dt-'
being supposed to correspond to the year 1 750 ; this
expression maij be used for ten or tioelve centuries before or after that epoch* [3853]
In like manner, we may determine the values of ^, e', ^', 7, and n ;
Til [OOOO J
thence we may compute the values of P, corresponding to the three
epochs 1750, 2250, and 2750. If we represent these values by P, P,, P„,
and the general expression of P byf
P4./ 'll^'l ^i^. [3854]
^+^•77 + 2 ' dt^ ' ^
we shall have, by putting successively, t = 500, t = 1000,
dP. 950000 1 — ^
— + 250000.2- ^^,
P =P+ 500.^+ 250000 . 1 . -r;^ ; [3855]
Values of
dP, ddP.
[3856]
p^ = p + 1000 . ^ + 1000000 . è • ^ ; ^^855']
Ct Z (t I
hence we obtain!
dP 4P — .3P — P, ddP P„—2P, + P
d t ÏÔÔÔ ' ~dl^ 250000
* (2429) To give some idea of the rapidity with which the terms of the series [3852]
decrease, we may take the value of e'" [4407] for the case of < = 1000, and we shall
find t .-^=329% — i<2 «=8^; so that the second is about ^V P^rt of the [3853a]
(13 e
fii-st ; and with the same rate of decrease, the third tenu it^-:r^ will be insensible; [38534]
similar remarks may be made relative to tlie other terms of [4407, Stc.].
t (2430) Tlie expression [3854] is similar to [3850a], and by putting, successively,
< = 500, f=1000, we get p, P„ [3855,3855'].
Î (2431) Multiplying [3855] by 4, [3855'] by —1, adding the products, and then
dividmg by 1000, we get — [3856]. Again, multiplying [3855] by — 2, adding [38560]
the product to [3855'], and then dividing by 250000, we get jjy [3856].
68 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
9. The terms depending on the ffth powers of the excentriciiies may have
[3856'! " sensible influence on the great inequalities of Jupiter and Saturn ; but the
calculation is very troublesome on account of its excessive length. The
importance of the subject has, however, induced that very skilful astronomer
Burckhardt, to undertake the computation. He has discussed, with scrupulous
.ggg„ attention, all the terms of this order depending on the angle bn't — 2n^,
neglecting merely those terms which depend on the products of the
excentricities by the fourth power of the mutual inclinations of the orbits ;
which produce only insensible quantities. The expression of R [3742]
[3857'] corresponds to the action of in' upon m ; and the part of the expression
which has the most influence on this inequality, is the product of m' by
the following factor,*
„ J ~.ri-'.\cos.{v'—v) — cos.{v'-\-v)\
[3858] -= — -—==== + 3-.
m Vr'^—'2rr'.cos.{v'—v)+r'~ [,■''— 2rr'.cos.{v'—v)^r'^^
[3858'] This factor is the same for both planets ;\ by developing it, and noticing
* (2432) If we proceed by a method similar to that used in [3d29«, &.C.], we may
prove, as in [3829?i, &c.], that the second and third terms of R [3742], namely,
[3858o] J- . — -{cos. (d' — v) — cos.{v-\-v)],
do not have any influence in producing terms of the order now under consideration, depending
on the angle bnt — 2nt, and by neglecting them, and also the first term of [3742],
which is noticed in [3S61, 3868], we obtain the value of -r [3858].
t (2433) As 7 enters into R [3858] only in the even powers, and the quantities
[3859a] multiplied by y^ are neglected [3857], the terms of R of the fifth order, must contain
factors of the following forms,
[38596] e'^ c'^e, e'^e^, t'^c\ e' e^ e" ; y^e'^, y^e'^e, y'^ e' e\ y^ e^ ;
of which the six first terms compose all the combinations of e, t', of the fifth dimension,
and the remaining terms all the combinations of e, e, of the third dimension, multiplied
by 7^ of the second dimension. Now we see, as in [957"", 957''^], that if R contain a
series of terms of the form ?;*'. Ar. cos. (5?i'/ — 2nt-\-A), the first term of the series
[3859c] will be of the order i' — i = 5 — 2^3, or of the third order ; the second term will be
of the order i' — i-\-2, or of the ffth order ; and by noticing only terms of the fifth
order, the angles will become, respectively, of the forms [3859]. For in the elliptical
[.3859d] motion the angle nf-\-s is always connected with — w, 7i't-\-^ with — «' [669, 957'^'] ;
VI. i. §9.]
TERMS OF THE FIFTH ORDER IN e, e', 7.
69
only the products of the excentricities and inclinations corresponding to the
angle 5 n't — 2 w ^, we shall have a function of this form,
R
VI
~= N "". COS. (5 «' i — 2 ?U + 5 s' — 2 £ — 4 ^' + -.:)
+ iV (' ) . COS. (5 n' i — 2 n Ï + 5 a' — 2 £ — 3 ^')
-{-N^''\cos.{5n't—2nt + ôs' — 2s — 2^' — ^)
+ N'-^K COS. (5 n't — 2nt + 5 =' — 2 s — ^' — 2^)
+ N '^'. COS. (5nt — 2nt + 5^' — 2s—3z^)
+ iV(^>. COS. (5 7i' t — 2n t + Ô s' — 2 s + z^' — 4>^)
+ TV (**'. COS. (5 71' t — 2n t + 5 s' — 2s — 2^' + ^ — 2u)
+ N '-'K COS. (5 n' t — 2 n t + 5 B —2 s —^' —2n)
+ iV(^'. COS. (5 n' t — 2 71 1 + 5 s' — 2 s — ^ — 2 n)
+ ^<^cos. (5n'i — 27if + 5s' — 2£ + ^' — 2^ — 2n).
and we find*
[3858"
Forms
of the
terms in
R
uf the fifth
dimen-
sion in
(0) [3859]
and in tlie terms depending on 7^, the angle 2n't~\-2s' is connected with — 2n;
so that if the coefficients of w, -n', n, be represented by g, g", g", respectively, we
shall always have, by noticing the signs g -\- g' -\- g" :^ — 3; which is similar to [959], [3859e]
changing the signs of the coefficients. Moreover, the sum of the coefficients g, g', g",
considering them all as positive, must not exceed 5 [957'"], because the present calculation
is restricted to terms of the fifth order. Thus, for example, a term depending on the
angle 5 n't — 2nt-\-5^ — 2 s — 5to'+2«, must be rejected, because the sum of [3859/]
the coefficients of -n', «, taking them positively, is 7, corresponding to terms of the seventh
order. Now a slight examination will show, that the values of g, g, g" , which satisfy the
equation g -^ g^ -\- ^' ^ — 3 [3S59e], with the prescribed condition, are as in the [3859g-]
following table ; the corresponding numbers being placed in the same vertical lines.
These numbers agree with [3859] ;
Values of g', _4, — 3, — 2, — 1, 0, 1;
Values of g, 1, 0,-1, —2, —.3, —4;
Values of g", 0, 0, 0, 0, 0, 0 ;
■2, —1, 0, 1;
1, 0, —1, —2;
■2, —2, —2, —2.
[3859;i]
* (24.34) The signs of ah these values of a' N^'>\ a! N^'^', &c. [.3860— .3860'"], have
been changed from the original so as to correct the error mentioned by the author
in [5974, Sic.]. Before the discover)' of this mistake, he had computed and used these [3860a]
erroneous values in ascertaining the inequalities of Jupiter and Saturn [4431, 4487] ;
hence it becomes necessary to apply the corrections of the mean longitudes, given in
[5976, 5977, &ic.]. We have given [3860—3860''] as they were printed by the author,
VOL. III. 18
70
PERTURBATIONS OF THE PLANETS.
[Méc. Cél.
[3860] a'iV('" = —
768"
(') (1) (1)
3138 b , — 13 a. — -^ — 1556 o?. —— — 438 a». 1
à
— 38a^
(1)
cW
d a.
f/as
do.^
Terms of
the fînh
dimen-
siou in
r, e', y.
[3860']
«'iV(»=_
+
e'3y2
384""
(2)
— (20267 e' 2+ 24896 ê) .h''— (7223 e'=+ 8 1 44 e^) . a .
(2)
(2)
f/2i,
(2)
,3
z^ . < + ( 1 094 €'-+ 3692 e^) . a=. Vf + (482 e'^+ 1 436 e^) . a^ '-^^
' "° ) a a"' ^ c/ a'
(2) (2)
+ (41 e'"~+ 140 e^) . a\ ^ + (e'2 + 4e^) .«-^^
(3) s
590a.(6^ + 6j + 255a^(^l^ + ^)
0) (3), , (1) (3)
2~ + w.,2 / ~i "" • V~T7ir I
f/a2
f/ tt' d a'
[38606]
[3660c]
[:3860d]
[36(!0f]
[3860/]
correcting the signs as above ; but without pretending to verify more tlian one or two terms
of each of the coefficients. Tiie calculations of Burckliardt, on this subject, are given in
the Mémoires de FInstttnt, T. IX, 1808, p. 59, supp., but generally with wrong signs.
From what has been said in the preceding notes [3809a — 38.56rt], concerning the terms
of the third order, we may form some idea of the great labor of computing and reducing the
terms of the fifdi order [3860-3860''^]. The series [3829^—?», 38346] must be very
much increased by the introduction of terms of the fourth and fifth orders ; a table similar
to [38350] must be formed, containing terms of the fifth order, depending on the proposed
angles and on the powers and products of a^ , a', a", as far as the fifth order inclusively.
Then we obtain, as in [38.36f/, 3837c, &ic.], values of iY'»>, JV"', &c., depending on ^'''
and its differentials relatively to a, n' ; which may be reduced to the differentials relative to a
only, by extending the table [1003] to differentials of the fifth order; finally, by the
substitution of the values ./2'*', B'-'\ and then- differentials, in terms of ftj^, èj, and their
differentials [996—1003], we get the required values of JV"", JV"', &ic. This short
sketch of the method of computing the terms of the fifth and higher orders, must suffice ;
more minuteness would be inconsistent with the prescribed limits to the notes on this work ;
in which we have proposed to point out and illustrate the methods of computing the various
inequalities, by occasional examples, without attempting to verify the immense number of
numerical calculations with which the work abounds.
VI. i. ^9.] TERMS OF THE FIFTH ORDER IN e, e', y. 71
(3)
—(109392e'2+53064e=).&'"— (42368 e'^+23436.e^).a.^
'''^''~ '76^*\ +(1064e'^+2088e^).a^Ç^+(1572e'^+1710e^).a='.'^
(3) (3)
+ (152 e-+192e==) .a^ i^ + (4e-+6e^).a^^
e'^e^s
/ (2) (4)\ „ fdb^ dbi.
(2) (4)
128 , (2) (4). / (2) (4)
1
da? ^ da? y ' \ (/a»
(4)
—(42912c'^+199848e'-2).6'"— (21728 e2+82032e'^).a.'"
è da.
(4) (4)
+ (116 e'-+210 e'^) . a^ ^ + (4 e^+ 6 e'^) . a^. ^
(3) (5)
580a. (63 + 63) +234a^(^+^
\ z s ' \ da. da.
/ (3) (5)v / (3) (5),
I d'^hs. d^h^\ fd^h^ d^b"\
(51
— (11840e=+152000e'^).6®— (6560e-+65168e'2).a.^
4 </a
g3 ) „ (5) (5)
"'^'' ~ 768 • \ — (592 e^+ 4720 e'^) . a^ ^ + ( 1 52 e^ 920 e'^) . a^. ^-
(5) (5)
+ (26 e" + 128 e'2) . a^ Ç^ + (e"- + 4 e'^) . a^. ^
554a.(6:+6")+222a^(^ + ^-^)
' / (4) (6)v , (4) (6),
384 '
[3860"
(4) ,^,«^« o , r^^/^oo «X db^
Terms of
, g 1 (4) (4) I the fifth
a'iV(^)=__./ _(640e2+2970e'=).a=. ^ + (864e^+1854e'2).al 1_^
sioo in
c, e', 7.
[3860"
[38601-
72 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
(6) „ (O
[3860^] «'A^W^ 1_^
768
41448. 6/+ 18392 a.. ^ + 1780 a=^. , ,
4 do. «a^
(6) (6) (fi)
156 a^ î-^ — 29 a^ lif — a^ ^
C ,o, (2) (2) (3)
[3860V.] a'iV<^)= ^^. < — 85 a . 6 3 + 85a^^+ 21 a^. ^ + a^ 1-1^
1-28 ( ^ a a «/a-' rfa^
(3)
(3) f] 7, 3
3
2
Terms of
the fiflh
dimen-
sion in
1
e, e', r. \ (56 e^ + 842 e'=) . a . 6 3 + (4 e^ + 87 e'^) . a^ -ip
[3860vii] d' ]\fO)^ <^^ ■
128 1 (3) (3)
-(16e^+20.a3.^-(2e^+e-).a^Çi|
(4)
_ _ (174 e^ 196 e'^) . a. 6*^' + (50 e-+ 180 e'~) . a=, '^
[3860vi.i] ^('^^(S)— IZ!.
128 \ W „ W
1
.3îÇ^ + (2e-+e^).a^îÇM
+ (14 e'^— e^) . «.3. — f + (2 e'^ + e"^) . a^
e'e^ yS ^ (6) (5) (r.) ■\
[3860U] a'iV<«)=:l^.<580a.èr+86a^^— 8a3.1!ij_a^^i •.
( ? eta da- «/a-*)
When we consider the action of m! upon m, we must augment «W"' [3860],
by increasing h with the term r? or — a [3743], which increases
5 (t
3125 a. c'*e
a'N'-"'' by —^ .* When we consider the action of m upon m',
768 '■
[3861]
* (2435) In [996], we have, generally, — . è ' = — ^''' ; but in the particular case
[3861a] of 1=1, this becomes, as in [997], -.Z."' — -^ = — A^^K The part -^^ being
introduced by the tenn -^^ .cos. {n't — nt -\- s' — s) [954], which does not occur in the
terms noticed in the value of R [3858], so that wherever the quantity — , •^'', occurs,
[38616] we ought to add ; or in other words, b ought to be increased by the term , ,
fit * u Ct
or — a . To notice this circumstance, we must apply a correction to the vakie
VI. i. §9.] TERMS OF THE FIFTH ORDER IN e, e,y. 13
, , jti) , 1 , . , . , ,T,n^ 1 SOOe'^e
we must add to b, the term = ; which increases a N'-"' by -;.^ „ • [3862]
•i a-* 763 a-^
This behig premised, we shall multiply the preceding values of a' N'-°\
a'iV''', &c. by m', and shall reduce each of the cosines by which
they are multiplied in the function [3859], into sines and cosines of [38(32']
5 n't — 2nt-{-5s — 2s; Avhich gives to this function the following form,*
Value of
..'/?= m'.a'P,.sm.(ôn't — 2nt + 5s'—2s) «•
[Action of m' on ml. [38631
+ m'. a' P;. cos. (5n't — 2nt + 5 s'— 2 e).
We shall likewise multiply by m the values of «'iV<% rt'iV*'', &c.
relative to the action of m upon m' ; and shall reduce the sines and cosines
of a' N'-'^'' [3860], which may be computed by supposing II = — a, whicli
(11 U)
(Z 6 1 (id h jL
gives "^ =^ — 1) "d'^'^^' ^'^' Substituting these in [3860], It becomes
_ '^ . |_3138 a + 13 a? = ^J^˱iîll , [3861c]
/Uo /Do
as in [3361]. When we are computing the action of m on m', the fonnula [3861a] becomes
« * a- a' I i cfiS a' I i S'
SO that the correction of è^j' is — a-^, and the correction of a'./V"'* for this case, will
be found by putting &j = — a.-- in the expression [3860]. Now this value of 6 j gives
,,(1) (1) (1) (1,
substituting these in that expression of a' N ^^\ it becomes
~ïml^-^~^^^^ — ^ X 13 + 6 X1556 — 24 X 438 + 120 X 38 — 720}='|^,
as in [3862].
* (2436) The reduction here used is the same as that in [3842J, &c.], by which
tiie fonction [3835] is reduced to the form of [3842n'], and were it not for the terms
[3861,3862], the values of P,, P/ [3863] would be identical with P„, P,,' [3865],
respectively ; for the factor [3358] is the same for both planets ; and the reasoning made [3864a]
use of in [3846a— ^] will serve to prove, in [3863, 3865], that P,, P/ will be respectively
equal to P„, P/, if we neglect the ternis [3361,3862], and we shall show, in [3866i], [38646]
that these ternis do not affect the result.
VOL. III. 19
74 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3864] of the function [3859] to sines and cosines of brilt — 2/1^ + 5=' — 2;;
which will give to it the following form,
Value of
R.
'R = m . «' P^, . sin. (ôn't — 2nt + ôs'—2s)
[38651 [Action of «I on m'].
^ +m.a'PJ.cos.(5n't — 2nt + ôs'—2s).
We shall then substitute these values successively, in the expressions
of 6v, 6v', of the preceding article [3844,3846], neglecting their second
[3865'] differences, because of the smallness of these quantities ; and in this
way we shall obtain the parts of the inequalities of Jupiter and Saturn,
corresponding to the angle 5 n't — 2nt, and depending on the powers and
products of the excentricities and inclinations of the orbits of the fifth order.
We may here observe, that in consequence of the ratio which obtains
[3806] between the mean motions of Jupiter and Saturn, we have 3125 a^^ 500;*
ji' 2 n' ** 4
[3867] for a^==— and on' is very nearly equal to 2n; consequently ■- ^ = — .
71" 11" /Co
Hence it follows, that the value of a' N *°^ is the same, ivhether ice consider
the action of m' upon in, or that of m upon m'. Hence we may deduce
the preceding part of 6 v' from the corresponding part of 6 v, by multiplying
[3868] the latter by — ^J'~ . -.f
[3806a]
[38666]
[3868a]
[38686]
* (24.37) We have nearly l=7v^a^ = n"' a' ^ [3109']; hence iL.=-^^=:a3 [,38296] ;
n' 2 /n'\2 4
but by [3318f/], we have nearly 5n' — 271^=0, or - = -; therefore a^=(-)=— ,
31 O \7l / fit)
as in [3867], and 3125a^ = 500, or 3125 a = '—j- ; substitutmg this in the increment
of a'JV^"' [3861], correspondmg to the action of rn! upon m, it changes into the
expression [3862], representing the increment of «'JV'"' in the action of m upon m',
as we have remarked in [38646].
■j- (2438) If we multiply the factor — '— — - , connected with the chief term
[an' — 2 rap
of ^t; [-3844], by tlie quantity — ^ ' „ . - [3868], the product becomes
im.n- a ^
\5m.n"^ a' 15m.n'~ 1
(5n'— 2>!)3 ■ a ~ (5n'— 2n)3 ' a '
j the same as the corresponding fac
the other part, -, being multiplied into the terms aP, aP', adP, adP', kc. [-3844],
in which the part — — is the same as the corresponding: factor of the terms of i5 y' [3846] ;
(on — 2n)3 X o
VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 75
10. In the theory of Mercury disturbed by the Earth, we must notice the
ine([uulity depending on the angle nt — 4 n7 ; because the mean motion [3869]
of Mercury is very nearly four times that of the Earth [4077a]. Supposing inequaiuv
m to be Mercury and m' the Earth, we shall obtain the proposed inequality Siércuô"
by putting i = 4, in the expression oi àv [3817]. Considering the [3870]
extreme minuteness of this inequality, we may neglect all the terms
dP (IP'
depending 071 -r-, ^— , and retain only those having the divisor (n— 4?i'/. [3871]
Hence we shall get*
iv = , "'•"" .iaP'.sin.fa^— 4rt'^ + s— 40+«P-cos.r?z^— 4n'^ + -— 40i- [3872]
(« — Any ' ^ ^ ' '
We can easily determine P and P' in the following manner. We may
T S T
calculate, by formula [3711], the value of — g-, corresponding to the
angle -I n't — 2nt, by substituting in it i := 4. Hence we obtain a [3873]
value of —5- of the form,t
«-
^-: L . e^cos. (4n'i — 2nf + 4s' — 2e — 2w)
+ L"'.ee'. cos. (4n'« — 2w^ + 4s' — 2s — ^ — ^')
+ U^K e'K cos. (4 n't — 2 n ^ + 4 s' — 2 s— 2 ^')
+ U^\ y"~ . cos. (4 n't — 2n ^ + 4s' — 2 £ — 2 n).
We shall then observe, that this value of ^ results from the variations
(r
of the excentricity and perihelion, depending on nt — 4<n't, in the elliptical
[3874]
produces the corresponding expressions a P, a P, a' d P, a d P', he. [3846] ; the
values P, P' of S v', having been proved in the two last notes to be respectively equal
to those of P, P', in S v.
[3868c]
* (2439) Neglecting dP, dP', ddP, ddP, and H, in [3817], and putting z'=4, [3872a]
we obtain the expression [3872].
t (2440) The two first of the angles [3874], connected with e^, e e', are explicitly
contained in [3711] ; the others, as well as these two, are included in the form
cos. \i .{n't — nt-[- s— s)'\-2nt-\- K\, [3873o]
which occurs in [3711], and is developed in [3745 — 3745'"].
76 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3875] expression of ^.— . This expression contains the term — c.cos. (n/+;— cî),
whose variation is*
T Ô T
[3876] — ^ = — èe . cos. (n Ï + s — ra) — eôzs . sin. (n t -\- s — ûî) ;
6e and (5w being the variations of e and ^3, depending on 7i t — i'li't.
[3876c]
* (2441) If we square the value of r [3701], and substitute
cos.^ {7it -\- e — ■ui)=zi -\-^ COS. '2.{nt -\-s — k),
we shall get
r3=a2.{l + f c^— 2c.cos.(ji?+s— to) — ie-.cos.a.Cnf + e — ro) + &;c.|.
[3876rt]
In the troubled orbit the elements r, a, e, s, ss, n t, are increased by the variations
[38766] 'J''; <5 «) Se, Sis, Sv, respectively; and if we neglect the squares and products of these
variations, the increment of the preceding expression will be found by taking its differential
relatively to the characteristic S ; hence we get
2riir='iaôa.\l-lr§ c^— &,c. \
-j- a-. \3 c 5 c — 2(Se. cos. («<-)-£ — ûj) — 2 c ô a . sin. {n t -\- s — zs) — &c. } .
Dividing this by 9 a^, it becomes of the form
r 0 1'
[3876rf] — 2 = — Se. cos. {nt-\- s — to) — e f5 to . sin. {n t -{- s — to) -(- X ;
representing, for brevity, by the symbol X, all the terms of the second member, excepting
the two parts explicitly retained by the author in [38T6]. If we neglect X, and substitute
'-■ '*''•' in the remaining terms the values of Se, e o a [3877, 3878], we sbal] get the expression
of — [3879], which the author supposes to be identical with [3874], and thence by
integration obtains Sv [3882]. In the Memoirs of the Astronomical Society of London,
Vol. II, page 358, Sic, Mr. Plana has pointed out some defects in this method, and ha?
shown, that the terms depending on X materially alter the result. To prove this, he has
computed directly the terms of Sv depending on the divisor [n — 4 n')^, using formulas
similar to those in [3335 — 3841] ; which we shall give in [3881r — w'] ; after going over
[.3876?] the calculation by the method of the author. From the comparison made in [3883w, y],
it appears, that this method of La Place cannot be considered, in an analytical point
of view, as a very near approximation to the truth ; though he seems rather unwilling
[3876/t] to admit the fact, in a note he published on the subject in the Connaisance des Terns,
for 1829, page 249.
[3876/]
VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 77
We shall have, by [1288, 1297],*
ôe= __ ,■< i j^j.sm. {4nt—nt -{-As — i)-\-l--- ). COS. {in' t — nt-\-4£— s)Ç; [3877]
y COS. (4 n't — nt+4 s'— s) + ('^^ . sin . {4n't — ni -{-é^—s)l; [3878]
yn'.an C /d P
e 0 a = j— , . < — I -^
n — 4 H ^ \ a e
hence the variation of — e.cos. (ni + ' — ^) becomesf
r^r m'.a?i Ç/dP
:£;^,.j('^).sin.(2n<-4n'i+2£-4£'-«) — ('^Vcos.(3n<-4n'<-f2s-4£'-î:r)l. [3879]
This function is identical with the preceding expression of -y [3874] ;
therefore if we change, in both of them, 2nt-\-2s into » ^ + s + ^ + - , [3880]
V being the semi-circumference, we shall obtain J
T-;- ^ -T- ).cos.rwi — 4w'^+£ — 4/)+ ^— ) . sm. (îi^ — 4n'i + £ — 4s')^
M — 4w i\de/ ^ \de J ^ ')
= L . e^sin. (4w'i — wi + 4E' — 5 — 3ot)
+ L" '. e e'. sin. (4 m' i — n i + 4 a' — s — î^' — 2 ^) [3881]
+ L(=>.e'^sin. (4?j'^ — n^+4a' — s — 2a' — w)
+ L'^>.7-.sin. (4n7— >i^ + 4s' — E — 73 — 2n).
[3877o]
[3879o]
* (2442) The expression of R [12S7] is the same as in [.3810] ; so that P, P' have
the same values in both formulas. Now putting t' = 4, » = 1, (j.= 1 [-3709], in the
expression of à -a [1297], and then multiplying it by e, we get the value of e5a [3878].
The variation i5e [1288] becomes, by similar substitutions, of the same form as in [3S77].
t (2443) Putting, for a moment, Ati! t — nt -\- A^ — s^rT^, nt-\-e — -^s^zW;
then multiplying [387'i] by — cos. ?r, also [3878] by — sin. ?r, and adding the
J* 6 J*
products, we get for the second member of [3876], or the value of —^ , the expression
[3879e] ; reducing this by means of [22, 24] Int., it becomes as in [3879c], which
is equivalent to [3879] ;
'■^='^^,-\(^)-i-sm.T,.cos.W+cos.T,.sm.JV}-(^\{^^^^ [38796]
t (2444) We have two expressions of — [3874, 3879], depending upon the
angle 2nt — Aii't, and it is evident, that if it were not for the terms produced by the
VOL. Ill, 20
78 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[38S1'] If we integrate this equation relatively to e,* and then multiply it by
11 — An
we shall obtain
I\L . eKsin.(4^n't — nt-\-4,£' — s — 3^)
+ 1LW. e^e'. sin. (4n'ï — ni + 4s' — ; — ^' — 2^)
+ L(^'. ee'~. sin. (4 n'^ — Ji ^ + 4 s' — s — 2 ^' — ^)
+ L<^'. ey-.sin. (4n'/ — wi +4£' — s — ^ — 2 n)
function X [3876e], they would be identical ; therefore they will still be equal to each other,
if we change the angle 2nt-\-2e into 7it-{-e-\--s-\-^ir. Now if we make this change
in [38741, we shall find, that a term of the form cos.(4n't—2?ii-\-'ie' — 2s4-A), becomes
[38806] L J'
cos.{An't—nt-\-4£' — i-{-A — zi — i •r) =^ sin. (4 7i' t — n t -\- 4 (' — s-\-A — ■si);
and the second member of the expression [3874] changes into the second member of [3881].
In like manner, sin. (2 ?i ( — 4 n't -{-2 s — 4 s' — w) becomes
[.3880c] sin. {nt — 4 n' t -\- s —■ 4 s' -\- ^v) — cos. {nt — 47i't-{-i — 4s);
and COS. {2nt — 4n't -\-2s — 4 s' — «) becomes
[3880d] COS. {n t—4n't-\-s — 4 s'+ è *) = —sin. {nt — 4 n't -\-s — 4 s') ;
hence the second member of [3879] becomes as in the first member of [3881].
* (2445) Multiplying the equation [3881] by de, and then integrating it relatively
to e, in order to obtain the values of P, P', we get
J!!Î:^Ap.co5.(nt — 47i't-\-B — 4s')-\-P'.sm.{nt — 47i't^s — 4s')\
n— 4n' i >
= iL . e^ .sm.{4n't — nt-{-4s — s — 3-a)
[38816] + i L^^'. e^ e'. sin. {4n't — nt-\-4 s'— s — u'— 2 ra)
+ L<2'. ee'2. sin. (4 7i't — n < + 4 e'— £ — 2^— tn)
-}- L'-^1.ey^.sm.{4n't — nt-\-4e — s — -us — 2n).
3n
The first member of this expression being multiplied by _ , , becomes equal to the
value 0Î Sv [3872] ; therefore 5 v will be obtained by multiplying the second member
[3881c] of [38816] by — — ; and in this way we obtain [3882]. In the integration relative
to e [3881a, 6], we may add terms depending on e'^, and e 'f, which are considered
as constant in the integrations ; but the excentricity of the Earth's orbit e', being only
[3881rf] about -rV of e [4080], the term depending on e'^, must be much smaller tlian the
VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 79
In this integration, we neglect the terms of P and P' depending on [3882]
[3881e]
[3881/]
others ; and the same remark will apply to the term depending on e' 7^. The author
has neglected these terms, because they are so much less than those which are included
in the expression [3882].
Having followed the author in this indirect method of computing the value of <^v [3882],
we shall now proceed to the direct investigation of the same inequality. For this purpose
we must have an expression of R, similar to [3835], depending on the angle 4 ti! t — nt.
This expression is evidently of the following form,
R = M«» . e" . COS. (4 71 1 — 71 1 + 4 e'— s — 3 z>')
+ Jtf "' . c'^e . COS. (4 ?i'< — ?i < -f- 4 e'— s — 2 ra'— -55)
+ M'-> . e' c2 . COS. {An't — 71 1-{- 4 s'— s _ ra'— 2 ts)
+ JU"'. e» .COS. (4?i'< — ?8/ + 4£'— £ — 3«)
+ JJf («. e' f. COS. (4 n'< — ?U + 4 e'— s — a'— 2 n)
-\-M^^\ey^. COS. {A7i't — 7it-\-4^—s — zs—2n);
but the factors JW«>, M'-^\ he. are different from those in [3836, Sic] ; we shall give
their values in [3S8lr — !«']. If we suppose, for a moment, the preceding expression
of R to be put under the form R= 2 M .cos. {4 n't— Tit -{- K), we shall have
d JÎ = )! 2 M . sin. {4 n't — nt + K) [916']. Substituting this in the expression of the ^^
mean longitude ^ [3715/], we shall get the corresponding term,
Sv = 3rrandt.àR= — -^^„.:sM.sm.(4n't — nt-i-K); [388U]
•''' [4n'—nf
therefore the value of 5v may be easily derived from R [38S1/], by multiplying it
by ; , and changing the cosi7ies into sines. The terms of jR may be very [3881il
easily obtained from the values of Jkf'"', M'-^\ he, computed in [3836(7— 3840o], by
merely decreasing the value of i by unity ; so as to change the angle 5 n't — 2 7it
into 4 n't — nt. In this way of computing M'-°\ we must use the decreased value
i=\ [3836a], and then [3836(Z] becomes as in [3881r]. In computing .M"' from [3881/]
[3837c], we have the decreased value i^2 [3837a]; hence we get [3881s]. [.3881jn]
From [3838e, A], we get the decreased value i=3, and JF-' [38810- From [3839«, b], [3881n]
we get the decreased value i = 4, and M'^' [3881m]. These expressions are reduced, in rgggj ^
the fii-st place, by means of the formulas [1003], and then by [996—1001] ; so that we finally
obtain the values [3881r', s', t', u']. Observing, that in computing JIf ' [3881/], we must
(1)
notice the increments of 6, and , represented by — a and — 1, respectively, [3881o]
' da.
as in [38616 — c], by which means we shall obtain the first term, — —,.\—256o.l,
[S88U]
80 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
[3883] e'^ and e' y- \ but as the excentricity of the orbit of the Earth is quite small,
in the expression [388 b'J, which is omitted by Mr. Plana by mistake. In like manner,
[3881o'] from [3840Â] and the decreased value z — 3 [3840^], we obtain Jl/™ [.3881 «] ; also from
[3881;>] [3840rt] and the decreased value i=A [3840m], we obtain M'^) [388 1 w] ; which, by
similar substitutions [1008, Sic], are reduced to the forms [3831t)', w']. In making these
[3881?] successive reductions, we have used the abridged expression [3755a], ./2 '"=«'"
'3
rfa'3
[3881.] JIf .0,^ ^ . 564^<"-48«'. C-f^) + 12«-. f'-^Ua'
48 ^ \da' J ^ V f'a'- /
^^, C64^'i> + 48.[^'» + ^/')] + 12.[2^(" + 4^/» + ^3W]
"^^'1 + [6.^W+18^i(')+9.^a<"+^3("]
= J . ^ 14-2 ^'"+114^/"+ 21 ^3<'> +^3(1' I
,0) (1) (IV
[3881,-'] = -^ 5_256a+142è!' + 114a.'^ + 21a^ '"' ^ ^^-^ '''' '
104^^-' + 26a.(^)-40«'.('if!^Ul0a'«.r^^
\ da J \da / \dadaj
104 ^'2) _^ 26 ^^(2) _j_ 40 . |-^(2) _|. ^^(3)-j _^ 10 . [2 ^/2) ^ ^ ® ■]
+ 4 . [2^«>+ 4 ^p + ^2^2,-] _|. |-6 ^^(2)+ 6^3(2)+ ^3«
152^«)_)_ 108 ^ ®_|_ 20 ^3'2i_^ _^^c3)^
[3881«] J/(i> = — -.
16
ÏG
(S) <2) .„,(2) __(2)
[3881(] 3f(2)^ ^
[3881,'] = J!-,.jl526r+108a.liL + 20a2.^+a3.^ii
'-•«"'--«■•C^')+-«-(^')--«-G^)
C 126^"^+ 21 . [^(3)+ ^ ffl-| ^ 60 ^/='+ 10 . [2 ^/3' + ^2i3)j
(+6^3^=1+ [3 ^2"'+^3'3>]
I 147 ^'■»+ 101 ^ o)_|. 19 ^^(3) _^ _^^C3,|
ÏG
m'
16
, C „v (3) (3) (3)
[3881f ] = _ -^, . j 147 6, + 101 a . !iil- + 19 a=2. ^ + a^. ^^ ;
VI. i. §10.] TERMS OF THE THIRD ORDER IN MERCURY. 81
in comparison with that of Mercury, and the inequality in question is very [388!?]
f (4) (4) (4) -J
..».= |....^_5B».+..(i^)^='^...,.^-5««+[-3i,...-„(';£:>)]
IG t ' \ da J) W o' C f ' da. ^'
[3881t)]
[3881w']
JIf (5)= -^. a'a . ^ 5 B® + « . ( — ) (■ [3881i<;]
I 5 6 3 4- a . -—^ C • L'^ooiw) j
16 " a' ■ C ^ '/■^
(1)
rfSfe.
If we substitute in these the numerical vahies [409.5 — 4102'!, also ^ = 5,340815,
da?
(2)
a.-^-^^ 1,96112, given by Mr. Plana, in Vol. II, page 366, of the Memoirs of the
Astronomical Society of London, we shall obtain, by supposing «'=1,
a' JH'0'= — m'. 0,-3411; a'JW*'>= m'.3,3192; a' M'2'= _m'. 1,4808 ;
o'J»f(3i== m'.0,2181; a'JJf ^4' = — >«'. 0,1921 ; a'^®^ m'. 0,0690.
[3883a]
[38836]
The last four of these numbers agree nearly with those given by Mr. Plana; but he
finds «']»/<«' = — m'. 2,40567, a'.y»f(» = m'. 2,94.30 ; so that he makes .M'"' seven [3883c]
times too great, and .Af <" about a seventh part too small. The first of these mistakes
arises from the omission of the term — 256 a [388 lo] ; the second is an error in the
numerical calculations. We must observe, that the indices of M in La Place's
notation, namely, 0, 1, 2, 3, 4, 5, correspond, respectively, to 3, 2, 1, 0, 5, 4, in [3883d]
the notation used by Mr. Plana. In computing the value of 5 v, Mr. Plana uses the
elements coiTesponding to the year 1800, namely,
e'= 0,0163.5.32; e = 0,2056163; 7 = tang. 7' 0" 6' ; w' = 99'^ 30™ 5' ;
[3883e]
«=74'' 21™ 47^; n = 4.5''.57"'3P ; «'=1; « = 0,.38709; and ?i', w [4077] ;
.329630 t^^^^3 '"^ 35^36'
he also reduces the mass m' from .JoqV^ [4061] to ^, which makes [3883/]
VOL. Ill, 21
82 PERTURBATIONS OF THE PLANETS. [Méc. Céî.
[3883"] small, we may neglect these terms without any sensible error [388 Id].
H."=— 0,0713 [42.30']; then by the method [3881 î], he finally obtains
[3883g-] 6v = 0^5596 . sin. {4:n't — nt-\-4s — i—l6'' 59" 20").
If we correct the errors mentioned in [3883c] ; also another error, in his substitution of the
value of 2 n, which is taken too small by 40'', in [3881/] ; it will become
[3883^] Sv = 0',61 . sin. {An't — nt-^-A s — s — 21'' 19"=).
This differs but very little from the computation of La Place in [4283], namely,
ôv = (1 +|x") . 0',69 . sin. {4n"t—nt-\-4^'—s~19^2'^ 1.3')
[3883t]
= 0-',64 . sin. (4 n" < — 71 f + 4 e"— s — Id'' 2'" 13') [3883/].
Notwithstanding this near agreement in the numerical results, the method of La Place is
essentially defective, as may be seen by comparing the term depending on e^ in the
expression [3881i,/], namely,
[3883i] Sv = —~~ . JJf (31. c». sin. (4 n't — nt-}-4s'—s — 3 o),
with that given by La Place in [3882],
t-'^'^^^'] ê V == ". , . L . c^.sin. (4n't — n t + 41'— s — 3z:).
[38S3n To compute the value of L, we may observe, that L.ë'.cos.{4n't—2nt-{-4e'—2B — 2zs)
is the term of ~, depending on e~, in [38741. Now the term of — [3711],
[3883m] ai ^ ^ a'^ <- ■>
corresponding to i=4, and having the divisor 4n — n, is
[388.3n] 4'l?3^-«-^^+«-(77'
-r- . a Jli 4- a^.[—-)
-rpL , \?±J . n^. COS. (4 n'< — 2m < + 4 s'— 2 5 — 2 s)
(4n — î!.).(4n— 3n) ^ ' ' ■
and as we retain here only the terms depending on c^, we may put M^=M^^'^ e^
[.3703,3745]; moreover, we have, in the present case, very nearly 4 ?i' — 2n = — n,
4n' — 3n = — 2n [3869] ; hence this term of ^ becomes
\4aM-^+a^.('-^\\
[38830] _(__ \ da /),nc'.cos.{4 7H — 2jit + 4i'—2s — 2^).
2.(4J^'— n) ^ ' '
Now we may obtain the expression of M'"' [3S8.3p], by putting i=^4 [3883m], in [3750],
The partial differential, relative to a, is as in [38837]. Substituting these two values
VI. i. §11.] TERMS OF THE SECOND ORDER IN THE LATITUDE. 83
11. It follows, from [1337'— 1342], that the two terms of R [3835],
represented by
R= M^*K e' r. COS. (5 n't — 2n t + 5 s' — 2s — zs'—2n)
[3884]
+ M^'\e f- cos.(5 n't — 2ni + 5 s' — 2s — z: — 2n),
in the first member of [3883;-], and making the same reductions as in [999, &c.]. we
get [3S83«], by putting «'=1,
r (4) (4) (4) \
= — .)176è +n4a.-— 4-20a^. -j-^+a^.-— --( . [3883*]
Substituting this in [3883o], and putting the resuk equal to
L . e^ COS. (4 n' i; — 2 » ^ -f- 4 £'— 2 c- — 2 tn) [3883Z'],
vre get
r (4) (4) (4) -s
L^^:^ 176è';+114a.^ + 20a^Çf + a3.^% ; [3883^
16.(4)1 — n) C 2^ da. ' dofi da-' ) "■ J
consequently the part of 5v [3883?], computed by La Place, is
C (4^ (4) W 5
16. (-In'— nf ( 5- ' da. ' daS ^ rfaS V
whereas the real value, obtained by the direct method [3881i, m'], is
3, = _"•"";•-" „.^1.366l' + 93a.^+18a^.^+a3.^^ [3883.]
, , ,. , r5 . ) 1.36 6 1 + 93 a . -^ + 18 a^. -7-^- + «-'• -7—0
10.(4,i'— n)2 ( 4 ' da ' da^ ' rf a^
If we substitute in these expressions the values given in [4095, &c.], we shall find, that
the coefficient of — — ^ '■ , in the first is 12, 54, and in the second 10, 50; so [3883jc]
16. (4 n' — n)2
that La Place's method makes this term too great by about one fifth part ; and the same [.3883i]
discrepancy occurs in the coefficients of most of the terms of these two formulas.
84 PERTURBATIONS OF THE PLANETS. [Méc. Cél.
produce in the value of s, or in the motion of m in latitude, the inequality,*
[3885] 6 5 = — — -; — - • < > •
5n — 2?i ^ _j_ 3j(5) _ f ^ _ 5;„_ (5 ,^/^ _3nt + 5s'—3s — -a—n) )
Moreover the same terms produce in the value of s', or in the motion of m'
in latitude, the inequality f
2a'»' m S ^''^•e'7-sin.(4n'<-2«^ + 4a'_2s-^'-n)^
[3886] i s = ■^, — . - . < > ;
5n'— 2?J m ( j^ jyj^^K ey . sm. {47i' t—2n t -\- 4 s'— 2 s — zs — n) )
There is a small inequality in the motion of the Earth, depending on the same angle
nt — 4.n"t, given by the author in [4311]. He seems to have computed it from
[3883y] the term for Mercury [4283], hy means of the formula [1208], ôv" = — 5v.~^,,
using (5« = — 0',690412 [4283], and the other elements [4061, 4079]. This method
will answer, as the inequality is extremely small.
* (2446) Putting;, in the term of iî [1337"], tang. 9/ = 7, it becomes
[.3885o]
R = m k . y^. cos. [i' n' t — int -\- Jl — g ()/) ;
[3885a'] comparing this with [3884], we get 5" =2, d/ = n, j':=5, i = 2; also in the
first term, m'k = M^'^'^ . e', ^ =5 s' — 2 £ — s/ ; and in the second term, mk = M^^'^. e,
[388561
•' i^ = 5 s' — 2 s — ro. Substituting these in [1342], which is obtained from the integrals
[1341«, 1341], we obtain in s, from the first term, the quantity
[3885c] _-l^.J»fW).e'y.sin.(5rt'/-2«^-« + 5 3'-2.-^'-n);
and from the second term, the quantity
[3885d] --^^^^.M^'\cy.^m.{bnt-2nt-v-\-b^'-2^-^-U);
observing, that (j,=:l [3709]. Putting, in these, for v, its mean value nt-\-e [3834],
and connecting the two preceding terms, they become as in [3885].
t (2447) The terms of R [3884], used in computing s [3885], are deduced from
the fonction [3831], which is multiplied by the factor or mass m. In computing the
[3886o] value of s' , corresponding to the planet m', and to the same angles, we must use the
factor m, instead of in ; therefore the value of R to be used in computing s', is equal
to the function [3884], multiplied by — ; which amounts to the same thing as to change
[38866] M^^\ M^^\ into — ,.JI/^", and —,.M^^\ respectively.
VI. i. §11.] TERMS OF THE SECOND ORDER IN THE LATITUDE. 85
n being, as in the preceding inequality of s, the longitude of the ascending rgggg,,
node of the orbit of in! upon that of m. These are the only sensible
inequalities in latitude, in the planetary system, depending on the product
of the excentricities and inclinations of the orbits.
We have seen, in [3800], that the value of 5 s produces in the motion
of m, reduced to the fixed plane, the term — tang. ? . 6 s . cos. {i\ — '') ; [3887]
by substituting the preceding inequality of s [3885] in this term, we shall
obtain a term depending on bn't — 2 n ^ , which must be added to the
If we now compare the value of s [-3885] with the vakie of R [3884], we shall find,
that s may be derived from R, by multiplying it by '■ — ; then integrating relatively [388(;c]
to t, as in [.38S5è, &c., 1341»], and after integration, decreasing the angles by the quantity
i' — n [3885c], or by its mean value nt-\-s — 11. In like manner, we may derive s' [3886rf]
from R [3884], after multiplying it by the factor ^ [38866]. This value of ^,.il
is to be multiplied by — '- — , to correspond with [3886c], and it will become
„ ( JW<«.e'y.cos. (.5?i'^ — 2?U + 5£'— 2 e — ra' — 2n))
-2a'ri'.f/^.-,.^ >; [3886e]
-(- JIf ^5)_ e y _ COS. (5 ?t'^ — 2 Ji < + 5 e'— 2 £ — « — 2 n) )
m
and then by integration, we get
2a' n' m <» iV/«' . e'"/ . sin. (5 n'^ — 2 71 i! + 5 s'— 2 £ — ^' — 2 n)
5ji' — 2îi ' m' '
-\- iW^s). g y . sin. (5 „'^ _ 3 „ i _|_ 5 £'_ 2 c _ js — 2 n)
[388(;c'l
The angles bnt — 2nt + 5s' — ^2£ — is — 2n, Sic, must now be decreased by
v — n'=nt-\-s' — n', corresponding to the planet m', as in [.3886c/]; the angle n'
being the longitude of the ascending node of the orbit of 7n upon that of m' ; in the same rsgsry]
manner as n [3746] is the ascendirig node of m' upon that of m ; and it is evident,
that n' = 180'' -f n ; hence v'—U'=n't-{-s'—n— 180"'. Subtractmg this from
the angles which occur in [3886e'], it becomes
9«'„' ™ C M^'^\e'y.sm. (4n't — 2 nt-{- As — 2 s — zj'—n + ISO"))
5«'_2n m' ^ _j_ J/(5)_ g ^ , ^j,,, (4 ^'^ _ 2 „ ^ + 4 a' _ 2 £_ « _ n + 180")
which is easily reduced to the form [3886].
VOL. III. 22
86 PERTURBATIONS OF THE PLANETS, [Méc. Ce).
g great inequality of the motion of m; but this term is insensible for
Jupiter and Saturn.*
* (2448) The functions 5 s, ôs' [3885,3886], reduced to numbers in [4458,4513],
[3887o] j^j.g Qf jjjg Qj.£jej. 3i or 9' ; these are muhiphed by tang. 9 in [3887], and as this tangent
is very small [4082], these terms may be neglected.
VI. ii. sU2.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 87
CHAPTER II.
INEaUALITIES DEPENDING ON THE SQUARE OF THE DISTURBING FORCE.
12. The great inequalities which toe have just investigated, prochice other
sensible ones, depending on the square of the disturbing force. We have
given the analytical expressions in [1213, 1214, 1306 — 1309] ; and it
follows, from [1197, 1213], that if we put
the great inequality of Jupiter ^= H. sin. (5 n't — 2nt + ÔB — 2s+ Â), [38891
we shall have e^^juaiu!";
of Jupiter.
^^^_^.(^'»V«;+^"'V«).sin.2.(5u'^-2n^ + 5a'-2s + J), [3890]
8 m ya
for the corresponding inequality of Jupiter, depending on the square of the
disturbing force* This inequality, like that from ivhich it is derived, is to [3890]
be added to the mean motion of Jupiter.
In like manner, if we put
the great inequality of Saturn ^'= — ïï'. sin. (5 n't — 2nt-\-ôs' — 2s+Z')» [3891]
we shall have ^Sties
of Saturn.
6v'=--. ^- !^-—L )LJ . sm. 2.(5n't — 2nt-i-5s'—2;+A), [3891']
8 m y a ^
* (2449) The great inequality of Jupiter is found, by substituting, in ^ [1197],
,A=1 [.3709], also i = 2, i'=5; and if we put
6m'.an^k
^ = 5.^— 2e + :4, T,= 5n't — 2nt + 5^ — 2s, ^=— (5n'-2n)g' ^^^^^''^
we get ^ = H.sln. (jr5 + ;i), as in [3889]. Making the same substitutions in the [3890c]
terms of the second order [121.3J, it becomes as in [3890].
88
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
[3891"] for the correspondmg inequality of Saturn,* which must be added to the
mean motion of Saturn.
The variations of the excentricities and perihelion may introduce similar
inequalities in the mean motions of the tivo planets. To determine them,
[3891"'] we shall ohserve, that if we notice only the cubes and products of three
dimensions, of the excentricities and inclinations of the orbits, we shall havef
[3892] 3a.ffndt.dR = — Ga m'.ff^i- dt".
P.cos.{5n't—2nt + 5s'—2s)
— P'. sin. (5n'<~9n t + 5s'—2 s)
[3891a]
[3891&]
[3891d]
[3891e]
[3892ol
[38925]
[3892c]
* (2150) Substitutmg ^ [3S90c] in [1208], we get
the great inequality of Saturn ^' =
mi/a — . ,„ , _,
-,.H.sm.{T.^-JrJl);
m' j/a'
putting tliis equal to the assumed value [3891], we obtain
— , mv/o _ _
H =-—-,. H, and jI = J1l.
m\/a
Now by comparing the two formulas [1213, 1214], we find, that the part of the great
inequality of Saturn, depending on the square of the disturliing force, is equal to the
m\/a
[3891c] corresponding part of the great inequality of Jupiter, multiplied by — , , ,
using the expression of this inequality of Jupiter [3S90], that of Saturn becomes
and by
_g TOv/a (2m'\/a'-\-5ms/a)
m' y/a!
m't/a'
sin.2.(7;+:5)^i^'-.'^-'^;^-^°Uin.2.(7; + :^);
the second of these formulas being deduced from the first, by the substitution of H [38916].
This last expression agrees with that in [3891'], except that  is changed into JÎ', so as
to make both the expressions [3S91, 3891'] depend on the same argument; observing,
that these quantities are very nearly equal to each other, since, in the year 17.50, we have
^ = 4''22'"2P [44.34], and .1'= 4'' 21'" 20" [4492].
f (2451) The part of R depending on the angle Zn't — 2nt, and terms of the
third degree in e, c', y, h.c., is given in [3842«']. Its differential, relatively to the
characteristic d [916'], is
àR = — 2nJ.ndt.\P. cos. % — P' . sin. %].
Multiplying this by Za.ndi, and prefixing the double sign of integration, we get [3892],
which represents the part of 5v [3715i], depending on diî, the divisor \/(l — e®) being
neglected, as in [3718']. The quantities P, P', which occur in this expression, are, given
in [3842, 3843], in terms of the elements of the orbits of m, m'.
VI. il. §1-2.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 89
which gives, in 3 a .ffn d t . d R, the quantity*
àe-\(j-)-C03.[57it—2nt+5s'—2B)—{'—-\sm.{5n'l-2ntj-5s'—'is)l
+ 0 75. J (-—^ . cos.(Mt—2nt-^5£'—2s)-r—-Ymn.(5n'l-Qnt+5s'—2 s)l
+ de'. }('~ycos.{Wt-'2nti-5e'-2s)~Çj^\sm.{5n't--'2nt+5s'—2s).
-Qam'.ffn\W.( ); [3893]
y+6ra'. j ('y-Vcos.lSn'f— 2jie+5c-'— 2e)— f— Vsin.(5n'<— 2?U+5s'— 25); /
-^5 y. j r_-yco3.(5)i'<— 2;i<+5£'— 2=)— ^j- Vsin.(5ra'«— 2(!<-(-5s'— 2;)^
^+5n. \{j^ • '=°^- (ô«'«-2n/+5s'-2;)-(^^) .sin.(5n'<-2n<+5='-2.=)|^
6e, 6 a, 6e', 6^', 6), 6U, being the parts of e, ra, e', ra', 7, n, respectively,
depending upon the angle 5n't — 2nt. We have, by means of [3842c], f
/dP\ fdP'\ /dP'\ /'dP\
[j^)-'-[j7)' W;=-^-(rfrj' [3894]
/dP\ , /dP'\ /dP'\ , fdP\
* (2452) We have already noticed the effect of the secular variations of P, P', in the
terms of 3a.ffndt.dR [3812,3812/], depending on sin.Tj, cos.Tj; using, for brevity,
T5 [38906]. The object of the present investigation is to ascertain whether the periodical
variations of e, e', to, -as', r, n, depending on the angle T5, which are computed in [3893a]
[1288, 1297, Sic], produce, in the function 3 a .ffndt . dR, any secular or periodical
inequaUties. Now if we suppose the elements e, e', w, -ra', /, II, to be increased by the
variations 5e, &e', S-a, ôts, Sy, Su, respectively, the corresponding increments of P, P',
will be obtained, by means of [607 — 612], in the following forms,
these parts of the general values of P, P', being substituted in [3892], produce the
expression [3893].
t (2453) The equations [3894 — 3894"], are easily deduced from the general values
VOL. III. 23
90 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
Moreover we have, as in [1297, 1288],*
[3895] (— Vco3.(5«'i-2n^ + 5£'-26)-('^Vsin.(5n'^-2n/+5£'-26)= '^"^]~^^"lerî^;
[3895'] (^).cos.(5«V-2?ii + 5s'~2£)+('^Vsin.(5n7-2ni+5£'-2e) = -'^^7^.<5e;
we likewise havef
[3896] (~\.cos.{^n't~2nt + bi-2s) — (^\.Bm.(bn't-2nt+b^-^i)= ''^"'^"le'cî^^:
\dc' J ' \de: J ^ ' ^ m.a'n' '
[3896'] (^).cos.(5n't-2nt + 5s'-2s)4-('^).s[n.(57i't-2}it4-5B'—2e)=-^^^^^^^^
\"6/ ' \de / ^ ' m.an
of P, P' [.3S42c], which give
[38940] (^\ = 2 è . JJi'. e'". e*. y^'. cos. (6V+ & ra + 2 c n) ;
[38946] (^— 'j=:2&.iVf'.e"''.e''-^y'^^cos.(i'îï' + 6w4-2cn).
These expressions satisfy the first of the equations [3894] ; and in hke manner, we may prove
the others to be accurate, by the substitution of the partial differentials of P, P' [3842c].
* (2454) The value of R [3842a'], is the same as that assumed in [1287],
[3S95o] supposing (*=!, i' ■=b, i = 2, as in [3890«]. Making the same substitutions in
ÔC, i5a [1288, 1297], we get, by using the abridged symbols [3846^,(1], the following
expressions, which are easily reduced to the forms [3895', 3895] ;
[3895i]
6e = — -r-, — — . i -r- • COS. i 5 + ( -7— . sm. ^5 >
[3B95C] s^= .^!!l^5(l£yeos.T,-f^).sin.rJ.
f (2455) The values Se', e'S'm', depending on the angle Tr,, noticing only terms of
the third order in e, e', 7 [3891'"], are easily deduced from those of ôe, cSts [3895,3895'],
by a process similar to that employed in [3846« — gl ; using also the same abridged symbols
[.3895e] Ts, Tg, Po, Pq, kc. For if we substitute, in [1288], the values i'= — 2, i = —5,
we get the following term of Se, which may be added to [38956], to obtain a symmetrical
form of Se, similar to [3S46i, Sic],
This last temi may, however, be neglected in computing the value of Se ; because it has
not the small divisor 5 71' — 2n. Now changing the elements m, a, n, e, &ic. into
[SSgSg-] ^,^ ^^1^ ^^^/^ ^t^ ^ç,_^ ^j^j jj^g contrary, as in [3846a, d\, we find, that the part of S e', arising
VI. ii. §12.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 91
To obtain the values of 5 y and Sn, we shall observe, that the latitude of m, [3696"]
above the primitive orhit of m', is 5= — y . s'la. (v — n),* which gives [3897]
as = — 5y . sin. (v — n) + 7 . a n , cos. (v — n), [3898]
Now we have, in [1342],t
( ('iPl.cos.(ôn't — 2nt + 52'—2s — v + n)
7)i.a?i ) \dy/ "^ ' ^
''•'-^^^ZJ^rS ,.p,, . _ . „ ^ [3899]
dP'
rf7
sin. (5 n't — 2nt + ôs' — 2 s — v + n)
[3895h]
[38975]
from [3895i], has the divisor 5n — 2n', which is large ; therefore this part is small and
may be neglected. The other part, derived from [3895/], becomes
„ , m.a'n' i/dP\ ^ , fdP\ . ^ ■)
5n—Zn i\de / \de'/ ^^
whlcli is easily reduced to the form [3896']. In the same manner, we may derive S ■a'
[3896] from Sz, [3895c]. ^^^^^'1
* (2456) It may not be amiss to remark, that the object of the calculation in
[3896"— 3902], is to ascertain the parts of ôy, 7 5 n [3900, 3901], arising from the [3897a]
perturbation of m in latitude, by the action of m' ; supposing the fixed ■plane io le the
primitive orbit of m! [3897]; these parts are denoted by 5^/, y5„n, respectively,
in [3899']. In like manner, the action of m upon m' affects the values of ôy, y Su
by terms which are represented by «, 7, 7 5^ n, respectively, [3904]. The sum of these l35J7c]
two paits of i5 7 gives the complete value of 5 7, as in the first equation [3905] ; and the
sum of the two parts of <5 n gives the complete value of S n, as in the second of the
equations [.3905]. Having made these preliminary observations, we shall now remark, that L^*'-''']
the expression [3897] is similar to [679], changing v, into v, tang. 9 into 7 [669", 3739] ;
and à into n+180'' [669", 3746] ; observing, that as n [3746 or -3902] is the longitude [3897«]
of the ascending node of m' upon the orbit of m, we shall have n 4- 180'', for that of the
[3897/^1
ascending node of m upon the orbit of m, taken for the fixed plane [3896"]. Hence
[679] becomes 5 = 7.sin.(i' — n— 180'')=:— 7.sin.(j; — n), as in [.3897]. Supposing [3897^-]
now 7, n to vaiy ; the corresponding variation of 4- will be as in [3898].
t (24.57) Using the values [.3895«], also ^ = 2, tang.ç); = 7, é;=n [3902, 1.3-37'] ;
also, for brevity
Ts=^5n't — 2nt-\-5s'—2e, Ts = 5n' i — 2n t + A — 2n; [3899a]
the expressions of R [13.37"], and « or us [1342], become
Ii = m k.y-. COS. T^, ûs=—^~y~^^.y.sm.[Ts — v-{-n). [38996]
[3900]
92 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
Comparing this expression with the preceding [3898], we shall obtain,
[3899'] for the parts of 6y, y an, depending upon the action of m! upon m, which
ô„. we shall represent bj s^^y, 7^,,^^
6,7==— "';"" ■5fl^Vsin.(5n'^-2ftt+5£'-2s) + ('^Vcos.(5n7-2n<4-5s'-2c-) j;
"' 5?i'— 2ra i\dy / ^ ' \dy J ^ ' ^3
[3901] yS, n= -4^-5f^Vcos.(5«7-2w<+5s'-2.0-('^Vsin.(5?i'^-2u<+5a'-2;)?;
' " 5?l — 271 i\dy J \<iy/ )
7' n- in which y is the mutual inclination of the tivo orbits to each other, and n the
longitude of the ascending node of in' upon the orbit of m [37461. These
[3903] ^ . . *^, , f . ^ ' , , T
g_ quantities also vary by the action oi m upon m ; so that it we put these
[3904] last variations equal to 6^ y, 5^ n ; the whole variations being 6y, iu ;
we shall have*
[3905] 5y = S,y +ô^^y ■ 6 H = ^, H + 6„ H ;
m.a'n' ms/a m.a'n' m\/a
[3906] ^,7='^, • '5,/ '/ = -TT^ • «5,, /- ; ^,^=^, • '5„n = -^-— .6 n.
•■ J ' m!. a n ' m\/a m.aii " m \/a "
If we compare this value oî 5 s with that of R, we shall find, that &$ = —-, — ^;r •{ ~, — ),
[3899c] 5,1—2/1 \dyj'
provided we increase the angle 5>i'/ — 2nt by the quantity 90'' — v-\-'n., by which
[3899</] means cos. Tg will change into cos. ( Ts + 90'^ — v-\-Jl) = — sin. ( Tg — v -\-n);
and if we use R [3842a'], the expression of 8s [3899c], becomes as in [3899/,^, or 3399].
[3899c] Now if we put, for brevity, v — n = v, , and develop the terms of [3899^], by means
of [22, 24] Int., it becomes, as in [3899A],
[3899/] 5s=g-^^^.»/.^Q.sin.(T,+90^-r + n) + (^).cos.(r5 + 90"-. + n)^
^'""'^^ =5-£k--'K^)-'^°^-(^^-^')-(i^)-^*"-^^'-^')^
^--^ -^B-n ■ { [(?).sln.T.+Q.cos.Tj.si„.,+ [Q.cos.T,-(^).sin.T.].cos.^.
Comparing this with 5«^ — (5y. sin.D,-)-7i5n.cos. t), [3898], and putting the coefficients
of sin. v^, cos. V, , separately equal to each other in both expressions, we get [3900, 3901].
If we compare the value of 7<5„n [3901] with that of R [3842a'], we easily perceive
[3899A;] that it may be put under the form 7(S,,n = — an .fdt .i-—\; and having found 7(5„n
[3899/] by this formula, we get from it the value of S^y, by changing the angle T^ into T^-\-9(y^,
as is evident by comparing the two expressions [3901, 3900].
* (24.58) From the expression of 7i5„n [3899A:], we may obtain the value of y <î; n,
[3906a] corresponding to the action of m upon m' ; by observing that the values of P, P' , which
VI. ii. ^^ 12.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 93
This beins; premised, if we substitute these different quantities in the c,,,mi'ityof
■- tlie mean
function [3893], Ave shall find that it vanishes.* Therefore the variations ^,''11^1'
of the excentricities, of the perihelia, of the nodes and of the inclinations of [3906']
the orbits, corresponding to the two great inequalities of Jupiter and Saturn, téms here
do not introduce into the mean motion of Jupiter, or into the greater axis of its l^^^^fj
vanishes.
occur in R [3842(}'], are the same in both cases, as is remarked in [3832 or 3846/', &c.] ;
so that it is only necessary to cliange R [3831] into —-R, and an into a'nf, to
obtain from [3899^-], the expression yô^n = — —,. a' n'.fd t .U-—\. Dividing this [390C6]
by 7Ô„n [3899t], we get the first form of '5,n [3906]; and by applying the principle [3906c]
of derivation [3899Z] to this value of y^'^, we obtain that of 5/ [3906]. The second
forms [3906] are derived from the first, by putting an=a-^, a'n'=a-i [3709'].
Substituting the values [3906] in [3905], we get
m'.an-\~7n.a'n' , . m'.anA-m.a'n' .
Sy= T .5„y; y^n= -f~- .yS,,n; 3906e
in which we must substitute for 5,,y, 7 5„ n, their values [3900, 3901]. Therefore,
to obtain the complete values of 5y, y 5n, we must change the factor m'. a n into [3906/"]
m'. an -\- m . a n', in the formulas [3900, 3901].
* (2459) If we substitute the values [3S94— 3901] in [3393], we shall find, that the
terms of this expression mutually destroy each other. In proving this, we shall neglect
the factor — 6 am'.ffn^dt^, which affects all the terms; and shall use the symbol
Tj [38906], also, for brevity,
5n^^ nn'-^n^ 5n'-2n ^3907»]
m .an m.an m .an--\-m.a n
Then the expressions [3895, 3895'] may be put under the following forms [39076] ; the
similar values [3896, 3896'] become as in [3907c] ; and if we change, in [3900, 3901],
the factor m'.an into m'. an -\-m . a'n', in order to obtain the complete values of
Sy, yon [3906/], they will become as in [3907(/] ;
(^-£-)-^os.T, + (^).sln.T,=-M.Je; (^) . cos.r,-(^^) .sin.T,=./lf,.e..; [3907.]
{'£)-^os.T, + ('l^yn.T,^^M.,.Se', (If ) . cos.T.-('iÇ).sin.r,=M,.e'.-.'; [3907,
(^).cos.7,+ (^).sin.T, = _J>/3.57; (^) . cos.r,-(^').sin.r,=./If3.7<Sn. [3907d]
VOL. III. 24
94
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
[3907]
[3907^-]
[3907A]
[3907i]
[3907fc]
[S9071]
[3907m]
[3907n]
[3907o]
[3907p]
[3907j]
oi'bit, considered as a variable ellipsis, any sensible inequality, depending on
the square of the disturbing force ; and it is evident, that the same result holds
good in the mean ^notion of Saturn and in the greater axis of its orbit.
Substituting tliese values in the first members of the following equations [3907e — ff], then
reducing, by the neglect of the terms which mutually destroy each other and putting
sin.^ 7^5 -(- cos.^ Tr,= I, we get
[3907f] —Mi.5e.cos.T,—M,.e5v5.sm.T,= (^^y, M,.5e.sm.T,—M,.eô,z.cos.T,= —(^);
[3907/] —M^.Se'.cos.T,—J\1^.e'&^'.s[n.T,= (j£^ ; M,.S e'.sm.T,—M^.e'&^.cos.T,=—(^^') ;
—J\l^.&'y.cos.T,—M;.y&n.s\n.T,= (~^^ ; M,.Sy .sm.T,—M,.ySu.cos.T,= -('^) ;
Now the first line of [3893] becomes, by the substitution of M^ . e f5 « [3907//] equal
to Se. {Ml . e <5 w) ^ Mi . e 5 e . (5 ra . The second line of [3893] becomes, by the
substitution of [3894], equal to e 5 ra . j (-i~) ■ cos. T^ -f ( — ) • sin. T^ I , and by
using — ^j.^e [3907i], it is reduced to C(5«.( — Mi.&c) = — Mi-eS e.S-a ; adding
this to the first line [3907 /i], the sum becomes zero. In lilce manner, the third line of [3893],
by the substitution of M.^ . e' 5 ra' [3907c], is equal to S e'. (Jk/^ . e'Szs') ^M^.e'S e'. d -n' ;
and the fourth line, by the successive substitutions of [3894'] and — M^.Se' [3907c],
is e'&z/.^ — Mg . 5 e') ^ — 31^. e S e .6zs' ; the sum of these two lines is therefore
equal to zero. Substituting M^-yàlî [3907<^] in the fifth line of [3893], it becomes
S y . {M3 .yôJl)Tz= M3 .yôy.SH; and by successively using the equations [3894"],
also the value of — M^Sy [3907rf], we shall find, that the sixth line of [3893] is
ySu.( — ^3.157)= — M3.ySy.oll; therefore the sum of the fifth and sixth lines
is equal to zero. Hence we see that all the terms of [3893], included between the braces,
mutually destroy each other, as is observed in [3906'] ; consequently the values of
èe, ÔZS, 5e', Sz/, 5 y, Su [3895—3901], do not produce in 3a.ffndt.AR
[3892 or 3715i] any term of the order of the square of the disturbing forces. The
function 3 a .ffndt .dR, represents the mean motion of the planet m [1183]; therefore
the variation of the mean motion, arising from these values of i5 e, S -a, S e', &c. is nothing.
Again, from [3709'], we have 2a = 2n '" , and as the mean motion nt or n, is
not affected by these values of Se, ^ ra, Sic, it follows, that the transverse axis of the
ellipsis 2 a is not affected by the variations Se, S -a, &c. now under consideration, as is
observed in [3906"]. The same result holds good when we notice the variations of the
motions of the body m', disturbed by m, as in [3907].
VI. il. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 95
13. TVe shall now consider the variations of the excentricities and of the
perihelia. We have given, in [1287 — 1309], the expressions of the
^ (1 C (/ CT d c' d Ts' ,
increments ot —-, —, —, 77"' dependnig on the two great [3i)08]
inequalities of Jupiter and Saturn, and we have observed, in [1309", &c.],
that the variations of e, ro, e', ra', relative to the angle 5 n' t — 2>t^,t
* (-2460) Tlie expression de [1284], is integrated in [1286], and put under another
form in [1283]. Now as this last expression is used in this article, we shall take its
differential relatively to t, and then change the angles n't, nt into ^', ^, respectively, [39080!
as in [1194'"] ; for the purpose of noticing the inequalities of the mean motion. If we
put fA=l, i'^o, i^2, as in [.389.5rt], we shall get from [1288] the following value
of de; and in like manner, from [1297], we get d-m [.3908f/] ;
de=-m.andt.^ (^^).cos.(.5^'-2^+5s'-2s)_ (^^).sin.(5^'-2^+5s'-2a)^; [390Sc]
rf^=-,«'.««rf<.^^.(lÇ).sin.(5>'-2^+5s'-2£)+i.('^').cos.(.5f-2^+5s'_23)^. [3908d]
t (2461) If we put the values of §, ^', under the forms ^=nt-}-N, ^'^n't-{-JV', [.3909o]
we shall find, by comparison with [1304, 1305], and using the symbols [3890a, b],
^= (IÉS^-^^-^°^-^^-^'-^'"-ï^5l ; [39095]
,,, 6m'. a n^ m^a ,„ „
^ = -W^;ji:2^^-^a' ■ ^^--- ^^-^'- ^■'"- î'^l- [39095']
Substituting the values [3909«] in the first member of the following expression, we get
5 ^'— 2 ^ + 5 £'— 2 E = 5 «7— 2 M < + 5 e'— 2 £ + (5 JV'— 2 JV) ^T^ + (5iV'— 2iV), [3909c]
and by neglecting the square and higher powers of 5N'—2JV, using also [60,61] Int.,
we obtain
sin. (5 ^— 2 ^ + 5 E-— 2 0 = sin. T, + (5 N'— 2 N) . cos. T, ;
cos. (5 ^'— 2 I + 5 £'— 2 0 = COS. T, — {bN'— 2 N) . sin. T^. ^^^^^'^^
Substituting these in the value of de [3908c], or, as it may be called, dàe, we get
dhe= m!.andt.\-(;^-£).cos.T, + (^-£).^n.T,\ [3909e]
+ rr^.andt.{b N'- 2 iV) . ^ (^^-^) . cos. T, + (—") . sin. tJ .
96 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
may introduce in these expressions some variations similar to those produced
The part of this expression, depending on the factor 5 JV' — 2 JV, is of the order itJ^ ;
and as the other terms are of the order m', we must notice, in them, the variations
of {- — 1, (— — ), arising from the variations of Se, (5 ra, &tc. The additional
[3909/]
terms of the vahies of f— — j, {— — j, from this source, may be found by changing
P, P' into [ - — ) , ( -T— ) , respectively, in [3393c, rf] ; and as the former quantity
is multiplied by — m'.and t .cos-T^, in [3909e], and the latter by m'. andt .sin.T^,
the complete expression of doe will be
d5e^= rn'.andt .< — (~T~) • cos. Ts + ( -;— j . sin. Tg ^
+ m'. andt.{5 JY'— 2JV)A (^^ . cos. T5 + (^) ■ sin. tJ
( + ('^).Se + (^).S.+ (:^).Se'
\'\de^ J \dedTSj ' \dedc'J
[3909;»] — m'.andt. cos. Ts . <
\.^\ded^'J ^\dedy) '^\dednj
\~\de^J ^\ded-a) '\dede'J f
-)- m'. audi . sin. Î5 . <^ > •
\'\ded-!a'J ^\dedyj '^~\deduj )
Now if we take the partial differentials of [3894—3894"], relatively to e, we get
/ddP\ (dP'\, f''''P!\ (i±^'\ — _('!JL\_, fill-']
\d7d^) = \d7)'^''-\ de-i J' \ded^)— \de J -Kdc^)
^^^^■3 {d^)='\-d^)' \:d7d^j=-'\iûd7)'
\d7dn)~~'^''\dedy)'' \dedn ) '''\dedy)'
Substituting these in [3909/t], and retaining only the terms of the order m!^ ; or in other
words, neglecting those terms of the first line of [3909A], which are independent
VI. ii. § 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 97
by the two great inequalities. If we apply this method to the elements of
of the factor 5 A'' — 2 JV and the second differentials ddP, ddP', we get
dÔ6= m'. a ndt. (5 JV'— 2 A") . | (^\ . cos. T, + Cj^) ■ sin- T, I
ddP\ /dP'\ , fddP'\ fddP\,,
, /ddP'\ ,, ,, /ddP\ , , /ddP'\
(ddp\ fddP\ ^ ,-
— ?«'. a lid t. COS. T^.
. de^ J \ de
-\-m' . and t .sin. T^
ddP'\ , / dP
oe —
[.3909fr]
/ddP\ ,,,,/'ddP'\ , /ddP\ ,^
[-d7d7)-'^^+[-dId^)-^^^-[d7di)-^^^
We must substitute in this the vahies [3895 — 3896', 3906/], and tlien by integration, we
shall obtain 6 e [3910], as will appear by the following calculations, using the abridged symbols
to denote the factors of the three difterent groups of terms which occur in [3910]. If ^ve
compare these expressions with those in [3907a], we shall obtain the following values of
m'.an, which will be used hereafter ; these equations are easily proved to be identical, by the
substitution of [3907a, 3909/] and reducing. m'.an=:MiK:,=zJ\'LJV2—M^.{J^^ + J^:,). [3909ml
First. We have, by means of [3909&, U],
m'.andt .{i)N'—2JV)=:^— —-:-—— . ^ — ~^, , ' .{P. cos. Tr,— P'. sm.T^l.d t
[3909rt]
= — 2 A*! .f P. cos. T^ — P'. sin. Ti\.dt.
Multiplying this by (-v— ) • cos. Tj -f f— — j .sin.Tj, we obtain the value of the first line
of [3909Â-], as in the first member of the following expression, which, by means of
[1, 6, 31] Int., is reduced to the form [3909o] ;
-<iN,.dt.\P.cos.T,-P. sin.Ts^.J (^) .cos.Ts + ^^Vsin.Ts^
> • [3909ol
Its integral gives the terms of i5e [3910], depending on the factor (5mv/a + 3m'v/«')-
VOL. III. 25
98 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[3909] the orbits of Jupiter and Saturn, and put àe, «î ts, for the variations arising
Second. The term of [3909A"], connected with the factor ( — ~j . dt, is as in the
first member of [3909p] ; wliich, by the successive substitutions of [3909»!, 3907e],
becomes as in [39095], whose integral gives the corresponding term in the fourth hne
of S e [3910] ;
[3909p] m'.an . I— Se .cos.T^—eûzi .sm.Ts] = MijX^. { — Ô e .COS. T^— e Ù 7Ô . sm.T^l
[3909g] =JV,J—Mi.Se.cos.7\—J\I^.e5z:.s\n.T,l=Mj'^\.
ITiird. The term of [3909A:], connected with [-j-tt )-dt, is as in [3909r], and
by reduction, using [3909m, 3907e], it becomes as in [3909s] ; whose integral gives the
corresponding term of the fourth line of [3910] ;
[3909r] ni'.an.{Se.sin.T^ — eS-a.cos.T5l=^M^J\'^.\5e .s'm. T^ — e^^.cos.Tgl
[3909«] =zJV2.{Mi.ôe.sm.T5—M^.eôzs.cos.Ts\ = —M.{^\
Fourth. We may proceed in the same manner with tlie terms of [3909Ar] . connected
with the factors ( . , , ] .dt, { - — — r r"; which will be found to be represented,
\dede / \dede / ^
|3909«1 respectively, by the first members of [3909/j, r], accenting the symbols e, 8e, S-!^.
If we also put 7n'. anz^MjJV^ [3909m], and reduce the formulas as in [3909c, «] by
[3909m] using the expressions [3907/], they will become, respectively, JV^.i — ], ~"^3-(^j-
Multiplying these by the factors [3909;:], and integrating relatively to t, they become as in
the last line of the expression [3910].
/ddP\ ,
Fifth. In like manner, the terms of [3909t], connected with the factors , , ).dt,
[m9v] , \dedyj
( -—].dt, will be represented by the first members of [3909p, j-], changing e, êe, ôzs,
XdedyJ
into 7, (5 7, 5n, respectively. Then substituting in', an = M^. {Nç,-{- JV^) [3909m],
and reducing the formulas, as in [3909(7, *]> using [3907^-], they become respectively,
[.3909«>] (A'a + Ns) . (^) , — (JVo + A',) . (''^^ . Multiplying these by the factors [3909y],
and integrating relatively to t, we get the corresponding terms of oe [3910] ; the terms
depending on JVo being in the fourth line, and those on JV^ in the last line of [3910].
Sixth. The two remaining terms of [3909A:] are as in the first member of [3909x] ;
which is reduced to the form in the second member, by the substitution of m'. an^ M^ JV^
[3909m], and M^.Su [39076]. Reducing the products by means of [31, 32] Int.,
VI.ii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 99
from the square of the disturbing force, we shall find
[ \de) \de
{5n'—2nf mV«' \ 2.(5ji'— 2n
^ \ de J '\de
Inequality
of the ex-
centricitv
dP'\ ^ /dP\y \ ofJupitei.
2.(5)i'— 2îi)
. C0S.2 . (5 n't—2nt-\-5;'—'2s)
rfF\ f<idP\ JdP\ {ddP'\ . /dP'\ fi^\f'[f\ fddP'\y •
I^J-yde"- ) \de)\dei J'^Kdy j\dedyj \dYj\ded y )ç^
[3910]
WiP\~_fdP'Yl
m'g.aans ) , iMlZ_AfiZ_l.cos.2.(5n'«-2n<.+5E'_2 5)
"5n'— 27i*\ '^ 4.(5n'— 2n).e
- lliZ_AlijL.sin.2.(5ji'f— 2n<+5e'— 2j)
2.(5n'— 2îi).e
,m'.aa'.nn'.t WdP'\ /ddP\/dP\ /ddP'\ /dP'\ /ddP\ /dP\ /ddP'\]
5n'—^n 'i\d^ )\dede' ) \de')'\dede')'~\ dj )\dedy) \dy)'\dedy)\
.*
5
[3909x]
it becomes as in [3909(/] ; then integrating relatively to t, it produces the terms depending
on COS. 2 T, sin. 2 T, in the fifth or sixth lines of [3910] ;
m'.andtA — \~T~) •5'"- ^s — \~r~) '^°^' "^ A ' ^'^
- •^.^'.l-©.s-.n.n-('^),cos.r,|. J(^^).».n-(-).sin.r.|
* (2462) If we compare the expressions of de, dis [3908c, r/], we shall find, that
dvi may be derived from de, by subtracting 90"^ from the angle 5^'— 2 ^ + 5 e'— 2e, [3910a]
and connectbg the factor - with each of the quantities (-7- )) (-7—) 5 '^y th's means [39104]
the angle Ts is also changed mto T5 — 90'', in all the terms of [3909e, h, A;], in which
100 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
a e
[3911]
Inequality 3m'2.aan3 (5mv/a+2mVa') j i '- Vrfe / \d<"/> . „ ,£. „ o ^ic- o<
perigee of (5n — 2ref.e jnVa' \ 2.(5n' 2 ni
Jupiter. 1 ^ '
+ i ^ 1 ^^ i-^.co3.2. 5;i'<— 2n(+5;'-2s)
2.(5re'— 2ji)
dP\ /ddP\ /'dP'\ /ddP'\ /dP-\ f'l±^\,/<l£\ f'^'^^'M
d7)\de^ )'^\de )\de^ J^Uyj\dedj)'^\d^)\did:^)y^
M'^.a^n^ j+^V^^"^^^ >..in.2(5„'/-2n< + 5.c'-2.)
^(5?i'— 2ra).e ' "j 2 . (5 ?i' — 2 n) . e
+ \£L-L-h£— L. COS. 2. [5 n't — in t-j- 5 s'— 2 s)
(5ra' — 2 re), e
I?! m
"(5
re'.gg'nn'.< 5/'ijP\ fddP\ /dP'\ /ddP'\ /dP\ /ddP\ /dP'\ /ddP'\
n'—2n).e'l\d7)\dede')'\de')\dede')'^\(r^)\dedy)'\dy)\dedyl
[3910c] T5 explicitly occms ; observing that no change must be made m the factor 5 JV' — 2N.
Hence it appears, that if we change in [.3909A] the angle T5 into T5 — 90"^, without
ahering 5 JV' — 2 JV, and then muhiply tlie resuUing expression by - , we shall obtain
[3910rf] all the terms of d Szi, except those arising from the variation of the factor - , connected
with the quantities (— — J, (—. — ) [39106]. These last quantities depend upon the
two following terms of dôzs, namely,
[3910e] ,«'. an dt.ll -("£). sin. T.- ('i^') . cos. T, ] ,
corresponding to the two first terms of [.3909e] ; and as the variation of - is
13910/] _^_f^_l_.^(l^).sin.T,+ (^).cos.r,} [39076];
also m'.an=M^JVn [3909m], this part of dozs will be represented by
JVo , ( /dP\ . „ /dP"'
[3910g]
VI. ii. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 101
The parts of these expressions, proportional to the time t, give the secular
variations of the exceutricity and of the perihelion, depending on the
square of the disturbine; forces. To obtain the periodical terms of v depending [3912]
on this sqxiare, we shall consider the term 2e.sin. (n^ + s — -in) [3748], [3012']
in the elliptical expression of the true longitude. If we put 6 e, (5 ra, for
the variations of e, ■^, depending upon the angle bn't — ^.nt-^-bi' — 2;,
Tills is to be connected with the terms mentioned In [3910(ZJ, to obtain the complete value
of do -a; and then by integration, we shall get 5-ui [.3911], as will appear by the following
investigation, taking the terms in the same order as in the preceding note [3909« — y].
[3910A]
[3910î]
In the first place, the terms depending on .5 A'"' — 2 JV, are multiplied by the factor
{— — j . cos. Tj-j-f —— j . sin.Tg, in the expression of d5e [3909A], which becomes
-.(— — J.sln.Ts . ( — — j .C0S.T5, in dô-a [3910f/]. Now it is evident, by inspection,
that this last expression may be derived from the first, by changing ( — -- 1 into - . ( — ) ,
/dP'\ . 1 /dP\ « \ e/ [39ioi]
and (-7-] iiito .( — j, without varying the angle T5, or the factor 5 JV'— SA";
therefore v. e may use the same process of derivation in obtaining the part of dozs, depending on
oN' — 2.¥, from the similar part of due [3909A:] ; or in other words, the part of us [3911],
connected with the factor bm,y'a-\-'im s/a', from the similar part of &e, [3910].
We shall now apply the principle of derivation mentioned in [3910f/], to the terms
[3909p — «■], and we shall find, that the factor of -.{— — ].dt, in do-,, deduced
e Kde'i J ' [3910mJ
from [3909^], is N^.\—M,.5e.ûn. Ts-f- itfj . e 5 « . cos. Ts | = A', . (^\ [3907e] ,
producing the term — • ("3~) • (-7~r) • ''^- '" dàzi, whose integral is as in the first [3910ml
term of the fourth line of 5 a [3911]. The term [3909s], by similar reductions,
A% /dP'\ /ddP'\
gives — • ( 7^] • ( "TT" ) • ^ ' ^'^^ *^™^ [3909^] give [3910n]
A3 /rfPN (ddP_^ A-3 (dJF\ (ddP'\ [3910,]
e \dt)-\dtde')-'' e \de' J ' [dede' J ' ^ '
the terms [3909ic] give
as in the fourth and seventh lines of 5 a [3911].
VOL. III. 26
102 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
and upon the first power of the disturbing force,* also 6' e, 6' w, for the
[3913] preceding variations of e, w, depending upon the double of this angle ; f
[3913'] moreover, if we denote by à s the sum of the two inequalities of m, the
Lastly, the terms of dSzi, deduced from those of doc, in the first member of [3909r],
by the principle of derivation [3910f7], are
[3910p] m'.ajirf<.| — -.(^— j.sin.T5+-.(^— J.C0S.T5J .Sis;
which, by the substitution of 7n'.a7i = M^JV„ [3909m], and 5 a [3907i], becomes
[39105]
Adding these terms to those in [3910o-], and putting cos.^ Tj — sin.- T3 = cos. 2 Ts,
2 sin. Tj . COS. Tg = sin. 2 T^ , we get
and by integration, it produces the terms of i5 -si, depending on sin. 2 T5 , cos. 2 Tj ,
in the fifth and sixth lines of [3911].
[3912a] * (2463) These values of 5 c, 5^, are given by the formulas [3907i].
t (2464) The formulas [3910—3912'] give, by using T5 [38906],
3m'2.a3n3 (5»n/a-t-amVa') A L V f<e / \dej_i
3.(5n'— 2n)3" my a' '\ r^,/dP'\ ^/dP\
[3913a]
[39136]
3m'3. f)3»,3 (^m^a-\-^mVa') 1 L \de/ \dt/A
VLii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 103
one depending on the angle 5n't — '2.nt-{-Bi — 2s, the other upon
the double of this angle,* the term 2e . sin. (nt^s — ^), will become [3913"]
(2 e + 2 5 e + 2 <5' e) . sin. (« ^ + = -f- d s — ^ — 5 =i — 5' ^) . [39i4i
If we neglect the cube of the disturbing force, the preceding expression may [3i)U'|
be developed in the following form,t
2 e . sin. (nt-\-s~\- 6 s — ra)
+ 2 (5 e . sin. (n t 4- s — ra) — 2e6is . cos. (nt -\-s — ra)
^ [3915]
+ \2ô' e + 265^ .5s — e. Ç6v>)-\. sin. (n t + s — w)
— [2ed'îJ + 2oe.da — 26s .ôe}. cos. (nt + s — zi).
The term 2 e . s'm. (nt -\- s -{- 5; — i^) is that obtained by increasing the [3915]
* (2465) The great inequalities [1197, 1213, Sic.], are to be applied to the mean
motion of the planet [1070"]. If we notice only the chief terms of â s, having the divisor [3914a]
(5?i'— 2n)^ they will become, by putting i^^5 in [3817], and using Tj [3S90è] ;
^ ' = (sl'-atp • ^ ^- ^"'- ^= - ^'- ''"• ^= ' • P^^^''^
We may remark, tliat the terms of v [3748], depending on e^, e^, &c., are here neglected [3914c]
by the author, on account of their smallness ; they are, however, noticed by him in the
fourth volume [9062, &c.].
I (2466) Putting a = nt + i-[-5s — TS, b^ô-^^ôt^', in [22] Int., we get [3915a]
the second member of [3915&], which is successively reduced to the form [3915c],
by usmg [43, 44] Int., neglecting terms of the order m'^, and finally putting [3915a']
cos. o = cos. [nt-\- S' — to) — us. sin. (w t -}- s — ra) in the term multiplied by 5 w ;
sin.(n<-|-£-|-5£— ^— 5i3— ô'î3) = sin.«.cos.((533-{-5'xn) — cos. a.sin.(r5x3-|-i5'-5j) [39156]
= {1 — i . (Sts)^] .sin.w — (6z!-\-S'zi) .cos. a
=sin.a — J.((Jûj)~.sin.(7i^-f"^~^)~("''+'^'®) •cos.(w^-|-£— ro)
-\- o-us. 5 1. sin. {7it-\-s — ra).
Multiplying this by 2e4-2i5e-r2i5'e, and neglecting terms of the order m' ^, it
becomes as in [3915] ; observing, that in the term multiplied by 2 i5 e, we may put
sin. a = sin. (n < -}" ^ — w) + i5 s . cos. (lit -\- i — w). [3915rf]
104 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[3915 "] mean motion n t, by & j, in the elliptical part, according to the directions
in [1070"]. The two terms
[3916] 2ie . sin. (nt -\-s — n) — 2 e (î ^ . cos. {nt-\-s — cr) ,
form the inequality depending on the angle Qnt — Sn't^Ss — 5e', given
by the formula [3718].* If we then substitute in the other terms, the
* (2467) If we put ?' = 5 in [3814,3825], where only the terms having the divisor
5 ft'— 2 m are retained [3818', 3824], we get
[391Go] '-^==i/.cos.(5n'i— 3n« + 5£'— 3s + .^); (5j; = 2H.sin.(5n'i— 3«« + 5e'— 3e+^) ;
and we may observe, that this value of ^ v is easily obtained from that of r(ir, by means
[3916!>1 of the formula [3718]; retaining only its first term 5v=^ ^ , which contains the
small divisor 5 n' — 2;i [3814, &ic.]. If we substitute, in this last expression of (5 v, the
value of r (5 r [3876f/], neglecting the small terms depending on X, it becomes
[3916e] (St) = 2 ue. sin. (n C + s — w) — 2 e (5« . cos. [nt -\-s---ui).
Comparing these two values of 5d [3916a, c], we find, that the two terms in the second
fine of [3915], depend on the angle bn't — Qnt-\-bi' — 3e, or ^nt — bn't-\-'è3 — 5s',
as in [3916']. The same result may be obtained by the substitution of the values
of <3 e, c 5 w [39076] in [3916], and using the symbols Tg == 5 w'ï! — 2 « < + 5 s'— 2 e,
W=nt -\-s — Î3 ; since it becomes, by successive reductions, as in [3916^]; being
of the form mentioned in [3916'] ;
25e .sin.?^— 2e5«.cos.?f =— ^ 'KS) ' ''°'' ^^ "^ (Ï) ' "'"" ^' \ ' ™' ^
[3916e] '
= — ^ ■ (^) • Icos. Ts . cos. TV-\- sin. T, . sin. W\
Ml \de/ '
[3916V]
-I- J- . (^\ . |sin. Ts'. COS. fV— cos.Ts . sin. ?F}
[3016/1 =--. {-) • COS. in - TV)+^. f^') . sin. (T.-W)
=—■—. (--] .cos. (5 7i' t — 3 n t 4- 5 t'—3 s + zs)
[3016g] 2
+ — . f — ] .sin. (5n'< — 3?ii-f-5s'— 3e + -5î).
VI. ii. «^^ 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 105
values of he, ôt^ [39076], and for ô' e, b' -., their preceding values
[391 3f/, 6] ; the sum will give, by neglecting terms depending on the sine
and cosine of «ï + 5, because they are comprised in the equation [3917]
of the centre,"
(on— 2»)3' mVa' f \(/e/ \de/S
_ 3m-s.«^r.3 [5mM±ivW^) ^j,,(dJ-\_^_ f^V..sin. (5. ^-10,/^ + 5.-10.-'-^). [3918']
{5n'— 2n)3 m'y/»' C \de J \dejS ^
ir we put, in [3916-], (^) = — 7»/jIJ.sin.(^ — ^), (^-£^ = M,H.co5.{A — ^), [3916A]
and reduce the result by means of [21] Int., it becomes equal to
2 H. sin. (5 Ji7 — 3 n t + b^ — 3^+Jl). [3916il
This is of the same form as [382.5], which represents the most important term of this form
and order, having the small divisor 5?i'— 2 k [3824]. The factor ZJ' is of the second .gg^g-,!
fdP\ fdP'\
dimension in c, e' [3314i], being of the same order as the quantities (77)' ITT")'
For the values of P, P [12S7], which correspond to the angle T5, are of the third
dimension in e, e', &,c. [957'"', &c.], and their differential coefficients, which occur [3916t]
in [391fiir], are of a lower order by one degree.
* (2468) The first and second lines of the expression [3915] are accounted for in
[3915", 3916] ; the remaining terms become, by using the abridged symbols W, T^ [3916rf],
\2à'e-{-2e5cz.6s~e.{S^f\.sm.Tr-\-\ — 2c.o'is — 2Se.ii^-{-2ôe.Se\.cosJV; [3917a]
in which we must substitute the values of Se, 5 -a [39076], (V e, 5' ■a [3913a, b],
Si [39146]. In making these substitutions, the terms Szi.Ss, [S-a)^, Se. 5 -a, Ss.Se,
will produce factors of the forms ^.cos.^Tj, ^'. sin.^Tg, »4". sin. Tj . cos. T, ,
or à^ + ^^.cos.2r5, i.(]'—hA'.cos.2T^, ^ ^ . sin. 2 T., [1,6,31] Int.
Substituting these in [3917«], we find that the parts ^ A, ^Jl', independent of 2 T^ ,
produce terms depending on sin. W, cos. W, of the form a . sin. W -\-h .sm. (V ;
which, by putting a=k .sir., p, h^Tc . cos. p, and reducing by [21] Int., becomes
fc . sin. (/f'-j- (3) = Ar. sin. (n ^ + ; — « + (3)- This maybe connected with the equation [3917c']
of the centre [3915'], as is observed in [3917] ; therefore these terms may be neglected,
and we may substitute in [3917«] the following values,
C0S.3 Tj = 1 COS. 2 T5 ; sin.^ T,= — ic^.2T^; sin. T5 . cos. T, = | sin. 2 1\ . [3917d]
Substituting these in the square of Svs, multiplied by — e, deduced from [39076, a], we get
VOL. III. 27
[39176]
[3917c]
106
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
This inequality may be put under the form [3921] ; for if we represent by
[3919] 6v = K. sin. (5 7i' t — 3 ni + 5 s' — 3 e + B),
This term is destroyed by the corresponding terms of 2à' e, deduced from the third
hne of [3913a], so that the sum becomes
[3917/] oye_e.Mn)2==— ,--
3 m''^. a^n^ (5 m \/a -j- 2 m V«')
:^-©+^-©]---^'
15»'— 2n)3
m'^/af
+
:''■■©-^•(^^)]
.2T,
Multiplying the value of e&zs [3901 b, a], by as [3914a], and reducing the product
by means of the expressions [3917(7], we get, by putting the factor 6, in this last
1 1 /• r. 2m'i/i'
expression, under the lorm 3 . , , , ,
mya
[39l7g]
2eSzs,(is:
3»i'2.a2n3 2mVa'
(5n' — 2n)3' my a'
[39177i] Adding this to [3917/], and putting, for brevity, ^£, = -
3m'3.aan3 (5mv/fl + 4 m'y/a')
(5n'-2n)3'
711' y^a'
we get
[39l7i] 25'e + 2e5«.⣠— e.(^î3)2=
Again, multiplying together the two equations [39076], and dividing by ^M^^.e [3907a],
we get, by substitutmg the values [3917»^/],
[3917fe]
— 25c J« =
m'a.a3 7i2
de J \de
(5n'— 2n)2.e J r/dP\ /dP'
Adding this to the expression à' tz [3913?»], multiplied by — 2e, it is destroyed by the
terra depending on the third line of [3913i], and the sum becomes
3m'2.a2n3 (5mv/a + 2?nVa')
\o»l.{l\ (5„' — 2n)3 m'\/a
vJ+[m^)+-q:
.cos.2Ts
-[-m-^m--
VI. ii. §13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 107
the inequality of m, depending on 3 7it — 5n't-\-3i — 5 s';* and as
in [3889],
the great inequality ^ = H. sin. (5 n'i — 2 « ï + 5 e' — 2 s + Â), [392oj
Multiplying —My.Se [3907i] by — jj^, also by <5 e [3914J], and then reducing
by means of [3907a, 3917(/], we get
+[--f^)+'--(^:
.cos.Srs
The sum of [3917/, m], using JI:/4 [3917A], is
5 — Seô'îs— 25e.5t3 + 2i56,5e}=<( >. [3917n]
Multiplying [3917i] by sin. W^, and [39177i] by cos. fF, then adding the products,
we find that the first member is equal to the expression [3917a] ; and the second
member, by the substitution of sin. 2 Ts . sin. W -f cos. 2 T5 . cos. ?F = cos. (W—2T5),
cos. 2 T, . sin. TV— sin. 2 T5 . cos. fV= sin. ( W— 2 T5 ), becomes [39iro]
and by resubstituting the values of M^ , T5 , ?F [3917A, 3916J], it becomes as
in [3918,3918'].
* (2469) The expression [3919] is of the same form as that assumed in [3826], or
that computed in [3916^], assuming 1 = 5; moreover [3920] is the same as [3889]. [3920o]
Hence if we put, for brevity, T5 = 5 ?i' < — 2 71 < + 5 e' — 2 s, fV^ = nt -\- s, and [3920o']
then make the two expressions [3919,3916^] equal to each other; also [3920, 39096, a],
«sing M [3907a] ; we shall obtain the two following equations ;
K.ûn.{n-W,+B)=^^^^^.\-(^i^ycos.{T,-W,+^)+(^-^ysm^^^^ [392061
H.sin.CTs-f- 1) =(5 Jlyjja •{P-c0s.r5-F.sb.T5}. [3930c]
108 PEUTUllEATiOiVS OF THE PLANETS, [Môc. Cél.
the preceding inequality will be, by ^69, of the second book,*
[3921] ^^_ (5>V«+4>»V«')^ff^-,3i^^.(5,,,_10^,,^5,_10,_^_^j,
m\/a ^ '
In like manner, we shall find, by noticing only the secular variations,!
* (2470) Multiplying together the equations [39206, c], and reducing the products
by [17 — 20] Int., we find that the first member becomes equal to
[3931a] I ÛK . COS. {W^J^Â — B) — lTl K. COS. ( fF» — 2 T-.— B — A) ;
and the product, in the second member, depends on similar angles JV», W^ — 2T5.
Now as these expressions must be equal to each other, whatever be the value of t, we
may put the terms depending on the angle /Fg — 2 T5 in both members, separately equal
to each other, and v/e shall get
_ _ 6m'2.(i2n3 ) L \de. J ' \de/_\
[.•39216] — è HK. cos. { W..—'iTi,—B-A) = — ,r„>_g„a . <
^D7l — ^raj-' \ |- //p/v /,!P\-\
This equation being identical, we may change W., — 2T5, into !-V„ — 2 T5 -{- OO"* ;
by which means, the expressions cos. (TF. — 2 75 — B — 7]), sm. {IV.2 — 2 T5 — ûj),
[3921c] COS. (H'., — 2T5— ®), become, respectively, —ejn. (FF3 — 2 '/'s — iî— J),
cos. ( JV^ — 2 Ts — tn), — sin. ( TFg — 2 Tj — -) 5 substituting these in [3921^-], and
multiplymg the result by '^,—^ , t'le first member of the product becomes as
in the second member of [3921] ; and the second member of this product includes the terms
[392W] [391S, 3918'] ; observing, that fF.j — 2 1\=: but — 10 «' < + .5 ; — 10 s' [3920rt'] ;
therefore the inequality [3921] is equal to the sum of the two expressions [3918, 391S'].
t (2471) Using the abridged symbols P„, P^, T,,, Sic. [38466— f7] ; also
[3922a] Z = b?J — 22,-\-b^ — 2s. Z^=b2, — 2^' + 5£ — 2s'; we find, that the expression
of de [3908c] may be rendered symmetrical by the introduction of the two terms
depending on the angle Z^ , or T^ , in the value of R [3S46c] ; so that we may put
, <.fdP\ „ fdP'\ . _, /dP^\ _ fdP'n\ . „?
[3Î326] de = —m'.andt.j^[-jjycos.Z—i^-j-ysm.Z+^~ycos.Z, — l^^j^ysm.Z,^.
In computing S e from this expression, it is not necessary to notice the angle Zg , because
[S922C] it does not produce terms which are so essentially increased by the small divisor 5n' — 2n,
as has been already observed in [3846f/"]. From this expression of de. we may derive
VI.ii.§13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE,
depending on the square of the disturbing force,
ôe'=-
3 irfi. a3 n3. t (ôm\/a-\-2m'\/a')
[5n — 2n)-.a'' m\/a
5)i'
a'g n'3 ■ t ^/(/P'N /ddP\ /dP\ fddj^
,i'—2n 'i\de' )\ilc"i J\d7)\d7^
dP
77
dP'
dy
+
mm'.aa'.nn'.t (.^dP'\ /ddP\ A/j
5n' — 2?^ l\de. Jydedc'J \d
dP
e
ddP'
dede'J xdy
ddP
de'dj
ddP_
de'dy
dP\ /ddP'\
J^,}\de'dy)
■dP\ /ddP'
dy J \dt'dy^
109
Secular
inenuality
of ilie ex-
centricity
of Saturn.
[3922]
iliat of de, by changing the elements of tlie body m into those of ?«.', and tlie contrary ;
by which means P changes into Pq [3846(/, &c.], P' into P'o , Z into Zo, « into «',
e into c', &:c. ; hence we have
(/e':=: — m.a'n' (lt.\ [ -r-ri -cos. Z,
U'/e'/'
,sIn.Zo + (^).
cos
Z —
rfP'
77
.sin.Z^
Neglecting the terms of this expression depending on the angle Z,,, because they do
not produce by integration the small divisor bn' — 2k; then substituting the values
of sin. Z, cos.Z [.3909fZ, .3922rt], we get the following value of dc, or as it may be
written d&e', being similar to [3909e],
d <i c == m .a n d t .< — — — . cos. i s + — r-r • sm. 1 r, >
I \de J ^ ' \de' J '' )
+ m . a' n' d t . {5 JV'— 2 A") . ^ (''|^') . cos. T, +Çj^) ■ si"- T, I .
The part of this expression depending on 5 JV' — 2 JV, is easil)- deduced from that in the
tirst line of [3909fc], or from its development in [3909o] ; by multiplymg it by —, ,
and changing the partial differentials of P, P', relative to e, into those relative to e.
Hence we obtain the following expression of the part of d5e', depending on the
factor (5JV'— 2JV) [3922/],
'■r^)---©+[''-©+-'-(S)
-., m. an ,
— J\ , . — .d t.
Now by successive reductions, using an = a [3709'], an'=a' we get
[3922d]
[3922e]
[39226']
[3922/]
[3922^]
[3922A]
m.a ' a
hence from [3909/], we obtain
m'.an
VOL. in.
3 m'~. «2 }i3 (.5 m \/a -\- 2 j^V"') "^ • "
(on'— 2n)2
28
3mS. a3?i3 (.5nn/a-f-2mVa')
{5îi' — 2nf.a'' m^a
[3922^1']
[3922i-]
110 PERTURBATIONS OF THE PLANETS, [Méc. Cùl.
of the
perigeo of
Saturn.
[3922/]
[3922?»]
[3923c]
{5n'—2nf.a'e'' m\/a 'X \de'J~ '\de'J^
dP\ /ddP\ /dP'\ fddP'\ fdP\ fddP\ fdP'\ /ddP'\)
nAa"-inKt (,/dP\ /ddP\ , fdP'\ fddP'\ , fdP\ fddP\ ,
yr\dy )\U'dyJS
mm
■(5^
.aa'.7in.t WdP\ /ddP\ /dP'\ /ddP'^. /dP\ /dJP\ ^ /dP"\ / d d P' \) ^.
—2n).e''i\dej\dede')'\de )\dede')^\dy )'\dedj) ' \d^ )\d7I^) ^ '
Substituting this in [3S22A], and integrating, we find, that the terms multiphed by i,
become as in tlie first line of [3922] ; the otlier part depending on 2 T5 , produce
in (5 e' tlie terms
[39224] ~2.(3n'-2n)W" '^Wa j r- /.^p^x /,;p
. sin. 2 T,
. cos. 2 75
If we compare the terms of d rS e, whicli are independent of (5 .V — 2 7V) [SOODe],
with those of dSe [3922/*], we find, that the latter maybe derived from the former
by changing the elements m, a, n, e, w, &,c. into m', a', n', c', ■cr', &ic., respectively,
without altering P, P', T5 ; and as the divisor 5// — 2« is introduced merely by
the integration of terms depending on the sine or cosine of the angle T^ and its multiples,
this divisor xvill also be unchanged. Now making these changes in the secular terms, in
the fourth and seventh lines of 5 e [3910], we obtain the similar terms in the second and
third lines of S e [3922] ; moreover the periodical terms, depending on 2 T5 , in the fifth
and sixth lines of oe [3910], produce the following terms of oc',
t3^22n] u-E^;r^^,- \ K77) -[-17) J-^°^-'^^^-^- W) • (77) -^'"-^^^^ 3 •
[3922o] The sum of ihn expressions [3922Àr, rî\ may be represented by o'e', to conform to the
notation in [3913], the characteristic &' being used to include the terms depending on
[3922p] the angle 2T5. These terms are used in [3924c].
* (2472) In the same manner as we have deduced the expressions [3922'^, e,/]
from [3903c], we may obtain the following expressions of d -,, d 3', d & t^' from [3903(Z] ;
[3923«1 ^.=-.'.««^..51.(^).sm.Z+^.('^).cos.Z+J.(^).sin.Z,+ ^(^).cos.Z„>;
[3923.] d.'=-m.a'n'dt^^,.(^ysn.Z,+l('^^^^^
d5zi'=. m.a'n'dt}—\. f^") . sin.7',— -,. f'-^Vcos. T, \
( e \de / e \de' J )
+ m.aVrf^(5JV'-2A').^-i.(^).cos.T5 + ^,.(^).sin.T,|.
VI. ii. § 13.] DEPEiXDING ON THE SQUARE OF THE DISTURBING FORCE. ] ] I
We also find, that the motion of wi' in longitude, is affected with
the inc(|na]itj*
3iiAa3n3 (3 mv/a-|-2niVV)
(an'— 2n)3.a''
m^a
.cos.(4 nt—0 n't+i 5—!) e'— tj')
+
■ , /dP'\ „ /dP\l
[3924]
This last expression being developed, as in [392r2o-, &,c.], and integrated, gives this part
of ^33'. It may also be derived from die' [39:2:2/], in the following manner. We
perceive, by inspection, that the part of [3923c], depending on the factor 5 JY' — 2 JV,
can be derived from the corresponding terms of doe [3922/"], by changing
same
/dP\ . 1 /dP'\ , /dP'\ . 1 /dP\ ,^ , ,
( — 1 into -, . -r-r , and -rr mto î-\~r~, )■ Ii we make the
\de'/ e' Kde'J' \de'J e \de' J
changes in the first line of S e' [3922], which was derived from the factor 5 JV' — 2 JV,
[3922 j, &.C.], we get the first line of the expression of ù'bj' [3923] ; and the periodical
terms of e'tJ^', corresponding to [3922A:], become equal to the following function, which
is used in [3924n] ;
3m-.a?n^ {5m^a-{-^m'^a)
rfc'
2.(5»i'— 2)1)3. a'
m\/a
-[-(f)--(S)]
. cos. 2 Ts
1. 2 7;
[;3933f/]
[392:iÉ]
[3923/1
[3923g-]
The part of (/ 0" to' [3923c], which is independent of 5 JV' — 2 N, may be derived
from the corresponding part of (ZtSts [3908f/, 3910a — e, or 3911 J, by the principle
of derivation mentioned in [3922/, &c.] ; that is, by changing m, a, n, e, to, &z;c.
into to', a', n', c', to', S:c., i-esjjectively, ivithout (dtering P, P', T~,, or the divisor
on' — 2?i. In this way, we find that the fourth and seventh lines of [3911] give the
second and third lines of [3923] ; and the periodical terms, corresponding to the fifth and
sixth lines of [3911], produce in c' d to' the following terms,
7n2.o'2n'2 Cr/£/P\2 fdP'V^^ . ^™ , ^ /dP\ /dP'\ ^)
The sum of the expressions [392.3/, h] depending on the angle 2 T5, represents the value
of c' 0' to', [3913] ; which is used in the next note. [3923i]
* (2473) The expression [3924] represents, for the planet m', the terms similar to
those in [3918, 3918'], which correspond to the planet tn, and are derived from the
function [3917a]. The similar function, relative to the planet to', using the symbols
T, = 5n't — 2nt-\-5^—2s, JV'=n't -\- ^—-^^ is
|2 Ô' e' + 2 e' 6 to' . 5 / — e'. (f3 to')2| . sin. TV — \2 e 0' zi' -{- 2 5 e'. à to'— 2 .5 s'. S e'\ . cos. W.
[3924«]
[39246]
112
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
If we denote the inequality of m', depending on the angle 2nt — ■in't-{-2; — 4s',
[3925] 6 v' = K'. sin. (4 n'^ — 2 M i + 4 s' — 2 E + 5'),
[3924c]
By the inspection of [39076, c, a], we perceive, that o c, 5zi, become equal to <) ë , '3ra',
respectively, by changing the elements m, a, e, he. into m', a', ë, he, iviihout
altering P, P', T5, or the divisor 5n' — 2n ; upon the principles of derivation
used in [3923^]. By this method of derivation, we may obtain — ë.{5zi')^ from [3917c],
and we find, that it is equal to, and of an opposite sign to the part of 2 5' c' [3922n] ; so
that these terms destroy each other, in the value of 2 0' e' — c'. (0 ro')^ ; and then the
other part of 2'5'e' [3922/:;], spoken of in [3922o], produces the following expression ;
[3924rf]
-e'.(<5ra')^
3m3.a3n3 (5m/a + 2HtVa')
(5)i' — 2n)3. a'' m\/'a
+
dp
de'
.cos.2T^
Now if we represent, as in [3913'], by 5 s', the part of 5 v' [3846, &c.], depending on
the angles Tg, 2T^, and notice, as in [3914a, Sic], only the chief terms of 5s' depending
on Tj, we shall get the following value, which is similar to [39146],
[3924e]
6e'-.
15 m . a'n'^
S-P.cos.T, + P'.sin.T,}.
■(5n'— 2îî)a
Multiplying this by 2 e' 5 to' [3907c, n], and substituting the values [3917c?], we get
/dP'-^
[3924/]
[3924e']
[3924/1]
[3994i]
[3924*]
dP
t'
' à ûî . f) s := Ti — . <
+ [
(5?i'— 2îi)3
■^■■(^
dP
de'
2T,
.C0S.2T,
We have very nearly 5?i'=2w [38 18c/], and n^(P^n~c? [3709']; multiplying
these two ecjuations together, and the product by 3 m^, we get 15 m^. a'^ k'^= 6 vr. a^ n^ ;
substituting this in the first factor of the second member of [3924/], it becomes
15to2. a'2?i'3
3 m9. a3 n3 2 711 \/a
{5n'— 2/1)3 {5n'— 2îi)3.a' m /a '
and then the sum of [3924c/,/] becomes, by \vi'iting, for brevity,
M^
3 nfi. (t3 »3 {3mv/a + 2mVV)_
(5ji' — 2n)3.a'
my' a
i.2ï;
2 <5' e' + 2 e' <5 i;i'. <S£' — e'. ((5 îs' )2=
+^.[P'.©-p.O].cos..r.
Vf. ii. § 13.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 113
and the great inequality of m' [3891] by*
^'= — H'. sin. (5 n't — 27it + 5s' — 2s + Â'), [3926]
Again, if we multiply together the two equations [-SOOTc], and divide the product
by iJ\I\.e' [3907n], we shall get an expression of — 2ôe'.ôzi', similar to [3917À;],
»k', a, n, e, being changed into m, a', ii! , e', respectively, without altering the divisor
5n' — 2n. Adding this to the part of — '^t'h'-ui, deduced from [3923A], we [3924?]
find that the sum becomes nothing ; and the term of e' i5 ' ra' [3923/] produces the
following expression,
^p.(^^^+P^(l^^1.eos.2T,
— 2e'ô'a'— 2(5e'.5w'= — — T-;— — .' -—^, ^^^< >. [3924m
-M-£)--m--^'
2
Multiplying —M-.-oe' [3907c] by —j^, and by (5 s' [3924e], and reducing, using
[3907a, 3917(/], we get
rp.(^)+P'.(^)1.cos.2r,
^^^•^'=ron'-2nf ■< - ..^.. ..- - >; [3924n]
H-m-^-m---^
in which we must substitute the factor [3924/t] ; then the resulting expression being added
to [3924w], using M^ [3924*], the sum becomes
^'•■['■■('S)+^'-(S):
. cos. 2 7\
-|2e'a"V + 25e'.ow' — 2(5s'.5e'}=<( \. [39240
-'=■[-■■©--• (^:
.sin.STs
Multiplying the equation [3924^'] by sin. W, and [3924o] by cos. W, then adding [3924p]
the products, we find that the first member of the sum is equal to the flmction [39246] •
the second member, reduced by formulas similar to [3917o], is
which, by resubstituting the values [3924i, «], becomes as in [3924].
* (2474) If we interchange the elements of the bodies m, m', in [3826], and suppose
B to become B', and i = — 2, we shall obtain an inequality of the body m', of the
form [392.5]. Substituting % = bnt—2nt-\-bz -2e, ?F3=n7 + £', W'=nt-\-î—u/, t^^^*^
we find that the expressions [3925, 3926] become, respectively,
àv'=K'.5m.{Ts—W^-{-B'); ^'= — H'.sin. (T5 + J'). [3926t]
VOL. III. 29
114 PERTURBATIONS OF THE PLANETS, [Méc. Ce}.
we shall find, that the inequality of m', depending on the angle
^nt — 9n'i + 4s — 9/, is represented by
[3927] 6v'=\S^''''^''^^"^^'''\h' K'.ûxi.ant — 9n't + ^B—9^'—B' — 7i').
m\/a ^
These may be reduced to forms similar to [39206, c], respectively, by observing, tliat
the term 2e'.sm.{n' t-^-s — zi), in the motion of m' , similar to that of m [3913''], may
be developed as in [3915], and will contain the terms 2 (5 c'. sin. W — 2 e' ^ w' . cos. W,
which may be reduced, as in [3916/"], to the form
n , and by the usual process, as in [3916A, i], it may be reduced to the form K'.^m.{T^—W'-\-B^).
Now if we put B^=B'—zi', and W=W3 — -^' [3926a], it becomes, as in [39266],
iT'. sin. (Tg — fFg-j- J5') ; so that by substituting the value of Al^ [3907a], we shall
have identically, in like manner as in [39206],
[39-26e] K\àn.{T,-W,+B') = ^^l;^^^^^.^-(^^^
Putting the two expressions of the chief terms of the great inequality [3924e, 39266]
equal to each other, we get, by changing the signs,
[3926/] E'. sin. {T, + .5') = '^^:^,-\P- cos. T,- P'. sin. T,^
The identical equations [3926e,/] are similar to [39206, c], and may be derived from them
[.3926g-] by changing m!, a, n, e, to, J3, B, K, H, fV^, into m, a', n, e, -a', A', B', K', H', W^,
respectively; also multiplying the second member of [3920c] by if-, without altering the
angle T^ , or the divisor ( 5 n' — 2 n ). Making the same changes in the product of these
two equations, and in [39216], we get from this last the following equation ;
15m9.o'3n3 3 L \de J \de
[3926.] -àH'jr'.cos.(^3-22'.-5'-..';=-^^;^;^^.^_ ^
sin.(^3— 275— ra') ,
This equation being identical, we may change fV^ — 2T5, into IV^ — 2 T5 -\- 90'' ;
then multiplyine; by 7p~, , we find, that the second member of the product
^ •' ° -^ 2 ni v/o ^
becomes equal to the expression [3924] ; and the first member becomes equal to [3927] ;
[.3926t] observing that W^-2T^ = 'int-9n' t'j-4e-9s' and 15^^. a'^M'^^Gm^ a^n^ [3924§-] ;
therefore the expression [3927] is equivalent to [3924].
VI. ii. § 14.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 115
14. The nodes and inclinations of the orbits of Jupiter and Saturn are
subjected to variations analogous to the preceding. To determine them, toe
shall observe, that c, ç', being the inclinations of the orbits to a fixed plane,
and ^ 6' the longitude of their ascending nodes, we shall have, as in [1338], [3928]
bj reason of the smallness of <p, 9',*
7 . sin. n = (?'. sin. è' — ç . sin. è ; [3029]
'/ , COS. n = (p. cos. 6' — ip . cos. é. [3929']
Moreover, from [3906], we havef
m'\/d
in'\/a
i . (?'. sin. 0') = — 5:^ .5.(9. sin. 0 ; ^3930]
à . ((?'. COS. ù') = — '^^^^ .5.((p. COS. Ô). [3930']
The subject of the small inequalities, treated of in this article, is resumed by the author [3926A]
in the fourth volume [9062, Sic] ; where he notices terms of the order m'^. e^, &c.,
which are omitted in [.3914c]. His object in using the indirect methods, adopted in this [392(5q
article, is to avoid the great labor of a direct calculation ; assuming as a principle, that these
very small inequalities may be determined in this manner to a sufficient degree of exactness,
for all the purposes of practical astronomy ; as will appear from the minute examination f3926»i]
of the terms of this kind in [9041 — 9114].
*i
(2475) Comparing the notation in [1337', 3902], we get â/ = n; tang.i|)/=:tang.7=y [3929a]
nearly; hence the equations [1.338] become p' — p=y.sm.n, q — q = y .cq?,.TI. [39296]
Now on account of the smallness of 9, we have very nearly |? = 9 . sin. â, q= o . cos.
[1334]; and in like manner, for the orbit of m', p' = 9' . sin. â', q' = ip' . cos. 6
Substituting these in [3929J], we get [3929, 3929'].
[3929c]
t (2476) The variation of the second member of [3929], arising from the action
of the body m' upon m, is represented by — (5 . ( 9 . sin. ê ) , because ç', é', do not [3030a]
vary by the action of m'. The variation of the first member of the same equation,
usbg the characteristics &,, <S„, as in [-3399', -3904], is <5,,. (7 . sin. n) ; hence by
development, we have
— S.{((>. sin. 0 = ^7- sin. n + 7 . J,, n . cos. n. [39306]
In like manner, the variation of the second member of [3929], relative to the action of the
body m, which does not affect 9, ê, is ô . {tp. sin. ^) ; and that of the first member is
116 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
From these four equations, we deduce the following,*
in! \/a' . ^ . • ^ X )
[3931] &,==-^^^_ç-^^^,.{6y.cos.(n-é)-y.6n.sm.(u-ê)];
[3931'] cp5è :z= jn^ a c, _ gjj^_ ^^ — o) + y.^n . COS. (n — ê)\ ;
[3932']
[3930c]
0) h^ = — . "', , , ,.Uy . sin. (n — â') + 7.6n.cos. (n — ô')L
m \/a -\- m \/a ' ^ / ■ ' v ^ >
5^.(7sin. n); hence we get [.3930fZ]. Substituting successively in this the values
[3906, 39306], we finally obtain [3930/], as in [3930],
[3930rf] (5 . ( ip' . sin. â' ) z= 5, y . sin. n + 7 • <5, n . cos. n
[.3930e] = £;^.|<5„7.sln.n + 7.<S,,n.cos.n^ [3906];
[3930/] =-£^.5.(9.sin.â) [3930è].
[3930iir] In the same way, we may deduce [3930'] from [3929'].
[3931ff] * (2477) We shall put, for brevity, M= — ,'"^", , ,, Jf, = — 7 ^'', -, ;
then taking the variation of [3929], relative to the characteristic 5, we get, by the
substitution of [3930], the following equation,
5.(7. sin. n ) ^ <5 • ( <?'• sin. Ô') — (5 . ( p . sin. Ô )
[.39316]
= — ^^.5.(?.sin.é)— 5.(ç,.sin.â) = — ITT • 5 . (9 -sin. â),
(«V» •'"7
or
[39316'] 5 . ( (p . sin. () ) = — M7 . 5 . ( y . sin. n ) .
[39316"] In like manner, from [3929', 3930'], we get 5 . ( <p . cos. ^ ) = — Jlf7 . 0 . ( 7 . cos. n ) .
Developing these two equations, we obtain
[3931c] i5 (p . sin. Ô -f (p (5 Ô . cos. & == — M~ . (^ 7 • sin. n + 7 . 5 n . cos. n ) ;
[3931d] ^ (p . cos. â — (p (5 â . sin. 0 = — .M- . (^ 7 . cos. 11 — 7 . 5 n . sin. n ) .
Multiplying [3931c, t/] by sin. Ô, cos. ^, respectively; adding the products, and
substituting sin.- ^ -f cos.^ â ^ 1 , sin. n . sm. â + cos. n . cos. â = cos. ( n — Ô ) ,
[3931e]
cos
. n.sin. é — sin. n .cos. ô = — sin. (n — é), we get [3931]. Again, multiplying
[3931c, rf] by cos. d, — sin..", respectively; adding the products, and making similar
substitutions, we get [3931'].
t (2478) We may compute the equations [3932, 3932'] from [3929—3930'], in like
[3932a] manner as in the last note ; or more simply by derivation, in the following manner.
VI. ii. >^ 14.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 117
Therefore the variations of ç, è, <?', o', depend on the variations of 7 and n.
We have, by ^12,*
( (—].cos.(57i't—2nt + ôs'—2B)
,.,,.. , (—).sm.(57i't—2nt + 5e'—20
(/ r m' \/a'
+ (^ycos.(ôn't—2nt + 5^'—2s)
If we change m, a, (p, ê, 7, into ?»', «', ç»', «', — 7, and the contrary respectively, in [39326]
the equations [.3929 — 3930'], they will remain unaltered, as will be evident by changing the
signs of the two first of these equations, and multiplying those which are derived from the
two last by the factor —'^- Making the changes [3932&] in [3931,3931'], which [39.32c]
are deduced from [3929—3930'], we get [3932, 3932'].
* (2479) Substituting the values <5„ 7, 3„ n [3900, 3901], in [3906e], and using,
for brevity, the symbols T5 [3890i], also an = a~^, a n' :i=^ a' ~ ^ ,
M, = "''■ ° "+"^ • "' "' = (!!LV^+J!^Vg:) ^ ^j M _,„,,„ „_(i!H^+_Z!^vV).„,-.„„^ [3933«]
m'.an m' \/a' m ^/a
we get
The divisor 5n' — 2n is introduced in 5 s, &c. [1342,3899 — 3901], by the integration
relative to t, spoken of in [1341a. &c.], in finding p, q, s [1341, 1342] ; where the
angle T5 is considered as the only variable quantity ; the very small terms, of a different [.39.33c']
form or order, depending on the variations of the elements, which enter into the second
members of [1342, he, 39336, c], being neglected. If we again resume the differentials
of the expressions [.3933è, c], upon the same principles, we shall get
d
[.3933rf]
^^-M3..'.an.^(4^).cos.T,-Q.sin.T,^;
^) . sin. T, + (^) . cos. n^ . [393:3.]
[3933/]
rf.{6n) ,, m'.an ( /d P
— ; — = — Mg. .
dt 7
These equations are equivalent to [3933, 3933'], omitting the characteristic S, which
merely signifies, that the calculation is restricted to terms depending on the angle Ts [3893'].
VOL. III. 30
118 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
r3933"l Hence we deduce, by neglecting periodical quantities* ivhose effect is
insensible, and observing thatf
* (2479a) If we compare the expressions [3842, 4401] with the numerical vahies
[SPSSg'] of e'% e", y, or tang. 7 [4080, 4409], we shall easily perceive, that the terms of P [3842],
depending on 7, are not a thirtieth part so great as some of the tenns depending on e", e" ;
therefore the periodical inequalities depending on the variation of 7, will evidently be
much less than those arising from the variations of c'", e". Now from the computation
made in [4438, 4496], it appears, that these last inequalities are nearly 4' and 9' ; hence
it is evident, that we may neglect the periodical quantities spoken of in [3933"].
13933/t
[3933i]
[39346]
[3934c]
[3935a]
[39355]
t (2480) Dividing [3842] by a', and taking the partial differentials relatively to 7, we get
[3934a] ^'- (^) = 2 M^'\ e' 7 . sin. ( 2 n + «') + 2 M^^\ e 7 . sin. ( 2 n + ^ ) ;
m'. ('^] = 2 M^». c'. sin. (2 n + ^j') + 2.¥'5). e . sin. (2 n + -^ï).
\0 7^ /
Multiplying the second of these equations by 7, it becomes equal to the first ; hence we
get, by dividing by m', 7./-— -j = (- — j. In like manner, from the values of
m'.ct'P' [3843], we obtain y .( ., j:^(- — j; dividing the first of these
expressions by the second, we get an equation, which is easily reduced to the form [3934].
t (2481) To obtain the effect of the variations of P, P', ,?, ^', in dy [393-3], we
may proceed in the same manner as we have done in notes 2461, 2462 [3909a, &ic.], in
finding the variations of de, d^. In the first place, we must substitute, as in [3908a],
^, 1^' for nt, n't, in [3933], and use the symbols [3933»]; hence we get
rf7 = _.;^f8.m'.anrf^^(^).cos.(.5^'-2^^-5s'-2.)-(î^).sin.(5^'-2^+5.'-2£)^.
Substituting in this the values [3909f?], we get the following expression, which is nearly
'" similar to [3909e], changing e into 7, &:c., and writing, as usual, dSy for Sy,
Ç /dP\ /dP'\ ')
d5yz= Ma ■ m'. andt . < — ( -— ) . cos. 7^ + ( —r~ ) . sin. T5 f
C \dyj \dy J >
[3935c]
+ M^.m'.andt.{?>^'—2]V) Âi^-^Vcos. r5+('~) .sin. 7^^ .
The variation of this expression, arising from i5 e, '5 w, (5 e', (5 13', 5y, 5 n, in the two
first terms, may be found as in [3909e — fc] ; or more simply by derivation, in the following
VI. il. § 11.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 119
3 »i'2. gg ?i3 (my/a-f wtVa') (5my'a+2mVa') ( /'<'^'\_p' {iJL\ \
'{ân'—^nf' mVa' ' ?nV«' ' ( Xd-y) Kdj/S
<5r = -
+
+
5 n' — 2 )i m V"'
mm'.aa'.nn' [mi/a-\-m'\^a')
'Sn'— 2^"
f/rfP'N /ddP\ {dP\ /rfrfP'N)
+ mVa') ^ f''P'\ f'ldP \_fdP\ (ddP'\ } _
^-V^^ "^^ i Vrf«^/ 'Ue'rf"// \de')'\de'dy)\'
Inefiuiilily
ill tlin
inclinutioii
of the
orbits of
Jupiter mid
Siituni.
[3935]
manner. If we change, in d <) c [.3909e], e into y, zs into n, and the contrary ; also [3935;;]
m' into Ms-m', without aUering the values of P, P\ JY, JV', T5, e, z/, he;
we shall find, that this expression of d5 e becomes equal to that of do y [3935c] ; and
by making the same changes in the other expressions of d5e [3909A, /c], we shall get the
similar values of day. After making these changes in [3909/c], and putting, for brevity)
M^=^Mg.m'.an [-39330], we may alter the arrangement of the quantities, so that the [-39.356]
ternis depending on the same differential coefficient may be connected together, and
we shall get
dh= ^/9.rf^(5JV'-2JV).[('^^).cos.r,+ (^).sin.T5j
+ JI/g.^/^f^Y(-^e.cos.T5-cfe.sin.T,)+J»f9.(/^(^^y(r5e.sin.7;-efc.cos.r5)
+ ^ig.rft.['^).(-5e^cos.^5-cW.sin.^3)+^/o.r/^('^^V(^e'.sin.T5-e'^^'.cos.^5)
de'd
dedy
\dedy
[3935/]
-^M,.dt. (^^).(-57.cos.2'5_76n.sin.r5)+M9.(/^-(^).('5r.sin.T5-y5n.cos.r5)
_JI/.../^5n.^(^i^).sin.T.+ ('^).cos.T4.
We may neglect tlie fourth and fifth lines of this expression,
values [3907O-] in the fourth line, it becomes equal to — ,
For if we substitute the
multiplied by the terms in
[3935e-]
the first member of [3934], and is therefore equal to nothing. IMoreover, by using the
value of ^ n [3907 fZ], we find that the lower line of the expression [3935/] becomes of a
similar form to tiiat in the second member of [3909a;] ; the partial difl^erentials of P, P'
being taken relative to y, instead of e. Hence we find, as in [3909y], that this line
of [3935/"] depends upon xhe periodical quantities sin. 2 T5 , cos. 2T5, which are
neglected in the present calculation [3933"] . The three remaining lines of the expression
[3935/] being reduced, and integrated relatively to t, produce respectively the three lines
of the expresson of ^7 [3935]. For if we compare the first line of [3909/t], multiplied
Mg=-^ [3933«], with the first line of [3935/], we shall find that they become
[-393.5A]
by
[3935i]
identical, by changing the partial difierentials relative to e into those relative to y ; hence
120
PERTURBATIONS OF THE PLANETS,
[Méc. Cél
Inequality
in the
place of
the node.
[393G]
<5n:
3m'2.a2ji3 [nn/a-^m'y/a') [5m\/a-\-2m'^/a']
(5n' — 2n)2.y 7n'\/a' m'^a'
+
73 «2
{nn/a~\-m'^a')
(5n'— 2n).y m'^a'
mm' aa' nn' {m\/a-\-in'y^a')
(5n' — 2n).y m'\/a'
\de J' \dedy ) ~^ \d7 ) ' \d7d
edy J
/ddP
\d e
dP'
7
ddP'
7
ddP
7
,/dP\ 'ddP\ , fdP'\ /ddP'\
/dP\ fddP\ fdP'\ (ddP
\dt' )' \de'dy) ' \d7) ' \de'd
' \de'dyj
' . (dP\ /ddP\ /dP'\ /ddP'\
[3935it]
[3935/]
[3935j?i]
[3936a]
[39366]
[3936c]
[3936rf]
[3936e]
we obtain the coefficient of t, in the term of 5 y, depending on the first line of [3935/"], by
multiplying the first line of [3910J, which is derived from the first of [3909A:], by M^ [3935i],
and changing the differential divisor de into dy, as in the first line of [3935]. Again,
substituting the values [3907e] in the second line of [3935/], and using
Mg m' 2. «2 jtS {m\/a-\- my a')
[3933a, 3907o],
we get the second line of [3935].
Mg
m' \/a!
Lastly, substituting [3907/], and
mm. aa tin
[m\/a-\-m!\/a!^
m' \/a'
[3933a, 3907a],
Mç^ 5 n'— 2 II
in the third line of [3935/], we get the third line of [3935].
* (2482) We may compute (5n from [3933e], in the same manner as we have
found h y [3935] from [3933f/] in the last note ; or we may use the principle of derivation;
observing that the expressions of dy, ydïl [3933(Z, e] have a relation to each other,
which is similar to that of de, ed-m [3908c, (/]. Moreover the former values may
be derived from the latter, by changing e, «, &tc., into y, n, Sic, respectively, as
in [3935f/] ; therefore we may derive the expression of (5 n from that of 5 y, in the same
manner as we have derived «îw from 8 e, in note 2462 [3910cf, &ic.]. Proceeding now
as in that note, we shall find, by changing e into 7, &:c. in the terms [3910p, q], and
reducing as in [3910?-], that these terms depend on the periodical quantities sin. STs,
cos. 2r5, which are neglected in [3933"] and in [3935/(]. In the terms depending on
the factor 5JV' — 2 JV, we find, by proceeding as in [39107i:], that we must change
\dy)
nito - .
7
— — ) ; and by making these
(dP'\ , fdP'\ . 1 /dP'
changes in the first line oî S y [3935], we get the corresponding terms of ^n in the first
line of [3936]. The remaining terms corresponding to those which are computed in
[3910m — 0], depend on the second differentials ddP, ddP', and maybe computed
from the second, third, and fourth lines of [3935/]; changing T5 into T5 — 90'', as
VI. ii. {s 15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 121
15. If we wish to determine, for any time whatever, the elements of the
planetiuy orbits, we must integrate the differential equations [1089, 1132],
by the method explained in [1096, Sic.] ; but in our present ignorance
of the exact values of the masses of several of the planets, this calculation '■ " ''
would be of no practical use in astronomy ; and it becomes indis])ensable to
notice the secular variations, depending on the square of the disturbing
force, which we have just determined ; since they are very sensible in the
orbits of Jupiter and Saturn. These variations increase the values of
, , -=— , -f-, — — , &c., relative to these two planets, by the
at at at at d t
. . ^ h'\Se"' I'^iTu'" l''\Se'" h".ôu" p^.&ca'" , q''\Sè'"' „
quantities* __+-^; ~; ii___ + i__; &c., [3938]
in [3910a — d], and substituting the values [3907e — gl ; by this means we shall obtain
the corresponding terms, which are to be multiplied by — in d 5u ; or by —
in Su, namely,
A/P\ /ddP\ , /dP'\ /ddP'\-) , Ma (,/'dP\ /ddP\ , /dP'\ (ddP
[393G/]
\ /ddP\ /dP^\ /ddP'\~) Ms_ WdP\ /ddP\
•Ml ■ l\de J' \dedy ) ^~ \d7 ) ' \dedy )\~^ M2 i\de')' \de'dy) ' \de' J ' \de'dy
Mg WdP\ /ddP\ /dP'\ /ddP'\
+ .¥3 ■ l\dy ) ■ yi^J + Wr / ' \ dy^ )
Substituting in this the values [3935/, m], also
17-= 5 K ■ o + '~^> — S — C • T-r, [3933a, 3907oJ, [3936g-]
we get, by a slight reduction, the second and third lines of [3936].
* (2483) The equations [1022], corresponding to Jupiter and Saturn, are
Ai" = e'\ sin. ra'" ; I" = é\ zo%. v>" ; A"=e\ sin. to" ; /"= e". cos. ■b". [3938o]
Taking the variations of these quantities, relatively to the characteristic i5, used as in [3938'],
and then substituting the values of sin. ra'", cos. ra'", &ic., deduced from [3938^], we get
<5 Ai'= 5 e'". sin. ra'" + e'". <5 w'\ cos. to'"= rîe'" . -|^ + e'\ cS to-". -^ ; [.39386]
J /*" = 5 e'". cos. TO — t" . 5 to'", sin. to'" = ô e'" . ^ — e'". h to'". — , &c. [3938c]
e'" e'"
The secular part of any one of the quantities (5e'", (Îto'", he, 5ra" [3910, 3911, 3922, 3923],
may be put under the form ht'^^At; A being a function of the elements of the orbits,
of the order m'^. Its differential, divided by at, gives —— =^A== — ; observing, [39,'3S(/]
that the variations of A may be neglected, because they are of the order m', and are
VOL. III. 31
122 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
considering only in 6 é" , 6 n'% the quantities proportional to the time t,
^'"' J determined in the preceding articles. We must substitute, in these last
quantities, the values of e", sin. ~", cos. ra'', &c., expressed in terms
of h", /'", &c.* The diiferential equations [1089] will then cease to
[.?939] be linear ; but it will be easj to integrate them by known methods of
approximation, when, after the lapse of many centuries, the exact values
of the planetary masses shall be known. In the present state of astronomy,
it is sufficiently accurate to have the secular variations of the elements of the
orbits, expressed in a series ascending according to the powers of the time,
carrying on the approximation no farther than to include the second poAver.
We have seen, in [1114", 1139'"], that the state of the planetary system
is stable, or in other words, that the excentricities of the orbits are small,
and their planes but little inclined to each other. We have deduced this
important result of the system of the world from the equation [1153],t
[3940]
[3940']
[3941] constant = (e- + o"") . m \/a + (e'' + <?'-) . m' \/a + &c. ;
for the second member of this equation being small in the present state
of the system, it must always remain so ; consequently the excentricities
r394'>l ^^^ inclinations of the orbits Avill always be quite small. J We shall now
prove that the differential of the preceding equation [3941],
[3943] (cde + ^dv) .m^a-\- (e' d e' + <f>' d v') . m' \/a' + &c. = 0 ,
multiplied by He'", which is of the order m'^, producing terms of the order m'^. For a
similar reason, we may nesrlect the variations of ^ , — , Sic. in findine the differentials
of [39386, Sic.]. Hence the differential of the last expression in [3938e], divided by dt, is
,.^^t,o -, ^^f^'" doe'" h'" , . rf(5wv Jiv ^jiv 7jiv iT^iv liy r„„„„-, . .
[3938e] --— ^ — — . — + e" — -— .- = —-.-_ -f e'' . -, as m [3938], omitting the
dt dt e" ' di e'" t v ^ t e'" l J' t>
characteristic 6 in the first member. In a similar way, we may obtain the other values
~dT' 'dJ
[3938/] [3938] from [3938c, fee] ; also the variations of "^ , '^ , &c. from [1132,1032].
* (2484) The equations [3938a] give e'^' = ^{h'^^ + l'^^), e" =^^{Jc'' + l"""), as
[3939a] in [1108]; which are to be substituted in [3938]; and when the resulting quantities are
added, respectively, to the second members of [1089, 1132], they cease to be linear
in A"', l", he, as is observed in [3939].
t (2485) Neglecting terms of the order (p*, we may put tang.^<p=fp^ and then [1153]
r3940a] becomes as in [3941].
[.3941a] t (2486) This must be understood with the restrictions mentioned in note 762 [1 1 14a, &c.].
VI. ii. v^l5.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 123
[3943'1
[3943"]
obtains even when we notice the secular variations of the elements of the orbits
determined in the preceding articles [3910, 3922, 3935, &lc.]. Hence it
will follow, that these variations do not affect the stability of the planetary
system. To render this evident, it is only necessary to prove, that if we
represent the mass of Jupiter by m, that of Saturn by ??j', and put
ie, âe', Ô!?, à if, respectively, for the secular variations of e, é, <p, ip',
which were found by the preceding calculations, we shall have
(e 6 e + ? â ?) . m \/« + (e' 6 e' + 9' ô ?') . m' /a' = 0 . [3944]
If we substitute, in the function (pa© . m^/a + 'P'^'P'- w*'v/^<') the values
of «, 6ç, 9', 6ip', given in the preceding article, it becomes*
m m' y/a a
[3944']
yày ; [3945]
[394Gn
m s/ a -j- m' y/n'
which changes the equation [3944] into
, , , , / / , , mm'\/aa' .
eôe.mi/a-Jreôe.mi/a'-\ ; — , , . , . y ôy = 0. [3946]
^ m \/a -\- m \/a ■'
We shall now commence with the consideration of the first line of the
expression of ie [3910], which becomes, by the substitution of a^n^=\
[3709'] ,t
-5 6=- r^, ; ,7 , ; \nt.]P. (---]— P'. (--)[. [3947]
* (2487) Multiplying [3931,3932] by <f.7n^a, (p'.mYa', respectively, and adding
the products, we get
/>** ■(
, , , ,//'' mmVaa' ^ -^r • 1 — P • cos. ( n — ^ + o'. cos. ( n — é')L
^/..<p^^ + q)^9= _^ .<^ V. [3944a]
m^a + nWa' ( ^ y S H . { ^ . sin. ( H — â)- 9'. sin. (
n-é')\ )
Now multiplying [3929, 3929'] by sin. n, cos. 11, respectively, adding the products, and
putting sin.^n + cos.^n^l, sin. n.sin.â'-f cos n.cos.â' = cos.( 11 — Ô'), 8ic. [24] Int., [39446]
we get [3944c]. In like manner, multiplying [3929] by — cos. n, and [3929'] by
sin. n, and reducing the sum of the products, it becomes as in [3944f?] ;
(?'. COS. ( n — è') — 9 . COS. ( n ^ é ) = y ; [3944c]
(?'. sin. {U — é') — cp. sin. ( n — â ) = 0 . [3944(i]
Substituting these in [3944a], it becomes as in [3945] ; and by this means [3944]
changes into [3946].
t (2488) Substituting a^ n^ = - [3946'] in the first line of 5 e [3910], it becomes
as in [3947]. Again, substituting a^ n^ = n [3946'], in the first hne of oe' [3922], [394eo]
we get [3943] ; in like manner, the first line of [3935] becomes as in [3949].
124 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
In the second place, we shall consider the first line of the expression
of àé [3922],
[3948] , V _ .3m.(5,V«+2mV«') C /rf^N /dP\ )
Lastly, we shall notice the first line of the expression of 5 y [3935],
[3949] 3m'.(5mv/« + 2mV«') (>V« + mV»') ^^ ^ (^ /^^^ p' /l^\?
(5n'— 2h)2. «^/f/ ■ mV«' ' ( ' \'h J 'vhj\'
If we notice only these terms, we shall find*
, , , , / , / , mm'\/na'
e à e . m\/a + e à e . m u a -\ — — — , , , , .y&y
* * m \/a ~\- m \/a'
[3950]
3mm'.(5m\/f> + 2m'\/a') \ ' L '\d7 J '^ ^ ' \de' ) ~^"'" \J^ )j
[3950] Now P, F', being homogeneous functions of e, e', y, of the third
dimension, we shall havef
therefore the equation [3950] will become
,,,,,,, mm'\/aa' ^
[3952] ede .m\/a + e àe.m i/a -i ; — . , , , .y6y = 0.
* (2489) Substituting the terms of 5e, 6c, 6 y [3947,3948,3949], in the first member
of the expression [3946], it becomes as in the second member of [3950].
f (2490) The expressions of P, P' [3842, 3843], are evidently homogeneous in
e, e', y, and of the third dimension. Now the theorem in homogeneous functions
[3950a] [1001a], by putting n = 3, a = e, a'^e', a"=y, A"^=P, becomes as
in [3951]; and if we put ^''':= P', we get [3951']. Substituting these in [3950],
we get [3952].
VI. ii. §15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 125
We shall, in tlie next place, consider the followhig terms in the fourth line
of 6e [3910],*
^ {5n'-'in).a'l\de )\de'i ) \de ) \ dt"~ )^\dy )'\dedj ) [d y ) '\d e dyJS ' ^^
and the terms in the third line of 6e' [3922],
,, mm'.t ^/rfP'\ /ddP\ fdP\ /ddP'\ fdP'\ / ddP \ /dP\/ddP'\-i
"^ — [5n'—%n)Vaa''l\d c )\jid^j~\J7 J\J777)^\1^ jXdTd^r^JTy )'\^^ ^1
also the terms in the second line of & y [3935],
. m'2.< (mt/a+)«Va') i/dP'\ /ddP\ (dP\ (ddl
['yii—'in).a' m'^a' C\de J ' \dedyj \de J '\ded
m'^.mt j \ de J
fdP\ C /ddP'-^
ddP'\ , (ddP'\ tddP'\~i
dti )+^- Vrferfe'J+^" \dtdy)\
(bn!—2n).\/a'\ , /dP'\ C /ddP\ , , /ddP\ , /ddP\-)
I -^[-djji'idûyJ+'-UVdVj+^i-d^n
fdP\ i /ddP'\ , , fddP'\ , /ddP'\}
-w}-r\iTj^)'^'-Wd^)+^-[ih^)s
[3955]
we shall have, by noticing these terms only, and observing that we have,
as in [3934],
/'dP'\ /ddP\ /dP\ /ddP'\
/ , , , , / , , m m'\/a a'
eôe .m\/a + e 6e. m i/a -j ; — ; , , , .y6y
'!^^ 5, (^-\+c- r^^^Uy i'^-^\l\
de )'l \de^ )^^-\dede')^''-\dedy)S \
;t [3957]
* (-2491) The part of (5 e in the fourth line of [3910], by tlie substitution of
«2n2^- [,3946'], becomes as in [.395.3]. Again, we have an = ^ , a'ti=— [.3946'],
a a', n «'= — — - ; substituting this in the tiiird h'ne à e' [3922], it becomes as in [3954].
Lastly, substituting a^ n^ = - [3746'], in the second line of f5 y [3935], it becomes [39526]
as in [3955].
t (2492) Adding the two terms [3956] to the two terms hettveen the braces, in tlie
last factor of the expression of '5 y [3955] ; it becomes of a symmetrical form with the [3957o]
values of 5e, a e' [3953,3954]. Substituting these values of & e, he', &y, in the first
member of [3957], and connecting togetlier the terms depending on the same factors of the [3957i]
first order, it becomes as in the second member of [3957].
VOL. III. 32
126 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[3957'] ( ) and ( ) are homogeneous in e, e', 7, and of the second
\d e J \de J
dimension ; hence we have*
<^
dP_
in
fcJdP\ . , fddP\ . fddP\
/ddP'\ , , /ddP'\ , /ddP'\ _ /dP'\
[3958"] Moreover ( - — ■ ) , ( ^ ) are homogeneous in e, c', 7, of the second
\«7/ \'h /
dimension ; therefore we have
hence we find, by noticing these terms only,!
[3960] eôe.m^a + e'ô e'. m' k/o! + -^p''^y ,.76-^ = 0.
^ ^ m \/a -f- m \/a
Lastly, we shall consider the following terms of f>e,X included in
[3958a]
[39586]
* (2493) It evidently appears from tlie values of P, P' [3842, 3843], that
/dP\ fdP'\ /dP\ /dP'\ , ...,-,
( -— j , ( — — j , ( 7~ ) ) ( ";; — ) îire homogeneous tunc.tions m e, e , 7, of the second
degree, corresponding to the formula [1001a,], supposing « = e, n'=e, a" = y, m=2.
If we put, in this formula, ^(''=^-— j, we get [3958]; and ^"i=r-_j gives
[3958']. In like manner, by putting successively, ^"i = ( — j, ^'■' = f-—j [1001a],
we get [3959, 3959']. ^
t (2494) Substituting the values [3958, 3958'] in the first and second lines of the
[3960a] second member of [3957], we find that these terms mutually destroy each other. In like
[.30(106] manner, the terms in the third and fourth lines of [3957], are destroyed by the substitution
of [3959, 3959'] ; and the whole expression becomes as in [3960].
t (2495) Substituting aa'.nn'= [3952«], in the last lines of the values
[3061a] of Sc, 5 y [3910,3935], we get [3961,3963], respectively. Putting a'^n'~=-,
[3952o], in the second line of & t [3922], we get [3962].
VI. il. § 15.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 127
the seventh line of [3910],
m m'. I
WdP'\ fddP
/dP\ ('i'^P'\_,(dP'\ fddP \ /dJl\fddP'\)
{r}n'—'în]Vaa''l\,le'J\dede'J \de' j\de de' )'T'\d y )\,ledy j\,ïi )\77d^) )'
and the terms of 6e', in the second line of [3922], namely,
g .^ »A< W'^P"\ fddP\ /dP\ /ddP'\ /dP^\ /ddP\ /dP\ /ddP'\-)
also those terms of o;. , in the third line of [3935],
5y =
(my/a + mVn') W'^fl\ (ddP\ (dP\ I'ddP'
(.5?i' — 2n).^/aa' m'\/a' ' ' ' \\dt' J ' \de' dy
Hence we shall have, by noticing these terms only,*
dP\ /ddP'\ -)
de' J\de'dyJ ) '
, , , , , , , , mm' Wa a'
e ie . m i/a 4- e ôe. m wa -\ —- — , . -. 5 7 = 0 .
Therefore the equations [3946, 3941] hold good, even ichen ive notice the terms
depending on the square of the disturbing force [3910, 3922, 3935].
* (2496) Substituting the values of à e, Sc', 5 y [.3961—3963], in tlie first member
of [3964], and reducing, as in the preceding notes, by means of formulas similar to
[3958 — 3959'], we shall find, that the terms mutually destroy each other. But without
taking the trouble of writing down these formulas at full length, we may abridge the
calculation, by the principle of derivation, in the following manner. If we multiply
the values of 6e, 5e', 5 y [-3953, .3954, 3955], by the factor
m\/a
III' y/a'
and in the terms
whicli are connected with the two differential coefiîcients ( — — ) , ( -—- ] , change the
partial differentials of P, P', of tlie first order relative to de, into those relative
to de; and in the differentials of the «ccowfZ on/cr, d e^ into de de', de de' into de'^,
d e d y into d e' d y, the other differentials being unchanged ; we shall obtain the three
expressions [3961, 3962, 3963], respectively. The same changes in the partial differentials
may be made in [3958 3958']; as is evident by putting, in [1001«], a = e, a'=^t', a"^y;
and then .7'" ^ f — j , to obtain the equation corresponding to [39.58] ; also ./2®==(-T-r j ,
to obtain the equation corresponding to [3958']. To render the expression [3963]
symmetrical, we may, as in [3957a], add the two terms [3956] to those between the
braces in [3963]. Hence it is evident, tliat if we substitute these values of oe, 5e', Sy
[3961, 3962, 3963, 3964/], in the first member of [3957], liie result will be equal to
the second member of [3957], multiplied by the factor [3964i], changing also the partial
f39(;il
[.3002]
Tlie sta-
hility of
the orbit of
a planet 19
not (lis-
lurbed by
[3904]
lerrns of
the order
of the
[30tJ4']
ptjuare of
the dis-
turbing
forcen.
[39<34a]
[39046]
[.39G4c]
[39(>4(/]
[3964e]
[:39C4/]
[3964g:]
1-28 PERTURBATIONS OF THE PLANETS, [Mtc. Cél.
The determination of the invariable plane, given in ^62, Book II, is
founded on the three equations,*
[3965] c =m \/a.{l-t^) • COS. (p + m' ^f77(l^^'2) . cos. y' + &c. ;
[3965'] c' = m i/a ."(1— e^) • sin. (? . COS. â -j- ?/«' y/«'.(i — e'"-^) . sin. <?'. cos. o' + &c. ;
[3963"] c"= 7rt \/aT{l^^) . sin. (? . sin. ô + m' \/«'.(l— e''-^) . sin. a', sin. ;)' + &c. ;
« and a' being constant, having regard even to the terms [3906' — 3907],
[3965'"] depending on the square of the disturbing force. The first of these
equations gives, by neglecting the products of four dimensions in e, e', &c.,
W, ({>', &c.,t
[3966] constant = ( c" + if" ) . hj \/rt + ( e' - + o' - ) . m' \/a' + &c. ;
and we have just seen, in [3964'], that the terms depending on the square
[3966] of the disturbing force, do not- affect the accuracy of this equation. The
[3964/i] differentials, as in [3964f]. Now the third and fourth lines of the terms between the
braces, in the second member of [3957], remain unchanged [3964(/] ; they must therefore
vanish, as in [39605], by the substitution of the expressions [3959, 3959'J. In hke
[3964il manner, the first and second lines vanish, as in [3960a], by the substitution of the two
equations found in [3964e], corresponding to [3958, 3958']. Hence the second member
wholly vanishes, and the result becomes as in [3964]. We may remark, that this
[3964/t] demonstration is restricted to terms having the small divisor (5n' — 2rt); but it is
extended to other terms in [5935, Sic.].
* (2497) Substituting ( 1 -f- tang.-.p)~' =cos. 9 ; ( 1 -)- la'ig-^ <?')~*=^ cos. p', &c.
[3965a.] j^ [1151], it becomes as in [3965]. Making the same substitutions in d , d' [1158,1159],
and putting also, as in [1156],
?; . cos.ffl = sin. 9 . sin. ^ ; (jr. cos. (p= sin. 9. cos. ^ ; y. cos. 9':^ sin. 9'. sin. d', &jc.,
we get [3965', 3965"] It may be remarked, that the quantities c', c", are in the original
work called c", c', respectively ; tliey are here altered so as to conform to the notation
in [1158, 1159].
t (2498) If we neglect terms of the order t"*, ©'', we shall have
[3966a] /a.(i_e2) = (l — ie2)./«, cos.(p=l — Iv^ [44] Int. ;
hence m \/a . (1— e^j . cos. o == m \/a — J . («^ -(- 9-) . ?« \/o ; substituting this and the
similar terms of a', c', 9', Sic, in [3965], it becomes
[39666] c = m /« -f- ?«'/«' + &.C. — I .\{t^-\- (f) .m\/a-\- (c'^ + 9'^) . m' \/a' -f- &:c.|.
Multiplying this by — 2, and transposing the constant terms — 2m\/a, — 2in'\/a' — &«;.
to the first member, we get [3966].
[39656]
[3965c]
VI ii. À 16.j DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 129
equation [3965"] gives, by neglecting the products of three dimensions
in c, e', &;c., (.?, ?', &.c.,*
6.(3. sin. (1) . m v/fl + 6 . (.;'. sin. c') . m\/a' + &c. = 0. [3967]
Now if we notice only the terms depending on the square of the disturbing [3967']
force [3931 — 3936], f this equation will hold good ; therefore the expression
f"= 7)1 ^/^(i^.2) . sin. V sin. <-' + m' \/a'.{[^^^) . sin. ç. sin. l,' + &e. [3961:']
[3965"], ^vill not be affected by these terms. In like manner, we find, [3968]
that a similar result is obtained from the equation [3965'],
c'= m y/a7(l— 6^) . sin. o . cos. J + m'/a'.(l— e'^) . sin. ©'. cos. è' -f &c. [3969]
Hence the invariable plane, determined in § 62, of the second book [3969']
[1162, 1162'], remains unchanged, even when toe notice these terms
depending on the square of the disturbing force.
16. The terms depending on the square of the disturbing force, have a
sensible influence on the two great inequalities of Jupiter and Saturn ; t we [3969"]
* (2499) Neglecting terms of tlie order y^ ^'^ we may put siii.(p=(p; sin.o'=:(p', &c.
[4.3] Int. If we also neglect terms of tlie order e^?, e'^ç)', &c., the equation [3965"]
may be put under the form c"=: ( 9 . sin. 6) . m s/ a -\- ( ç'. sin. a') . ni' \/a' -\- &c. ; and [3967a]
if we take the variation relatively to the characteristic (3, it becomes as in [3967].
t (2500) The terms here referred to, are those mentioned in [3943'], and computed
for two planets in [3929— 393-3']. The equations [3930, 3930'] may be put under the [3968a]
following forms,
0.(9. sin. é) .m s/ a -f- ^ • ( ç'. sin. ^ ) . m' i/(/'^ 0 ;
[39686]
0 .((p .cos. é) .m\/a -f- 0" . (ç'. cos. H) .m'^a=^0.
In the same manner, other planets produce similar expressions, and the sum of all the
equations, corresponding to the first, forms the equation [3967] ; a similar equation may [3968c]
also be obtained from the sum of the equations of the second form.
% (2501) Substituting the expressions [37566, c, e], in SR [3764], it becomes as
in [3970J ; observing, that the coefficients of h^ + P, h'^+l'^ [3764], are equal to [3969a]
each other, as appears by multiplying [3752i] by — 4.
VOL. III. S3
130 PERTURBATIONS OF THE PLANETS, [Méc. Ctl.
shall proceed to determine the most considerable of these terms. We have
seen, in [3764], that the expression of R or iR contains the function
<«= -^.(.H.-).i2„.(^)+„..(''-:)(
,™, +.'.....os.(.-.).|4.».+.„.(:^) + 2..('->„..(^)
[3970] ^f ^^'*^ increase the quantities e, fi', w, ra', r, in this expression, by their
variations, depending on the angle b n t — 2nt,* we shall obtain in R
some terms depending on the same angle ; and it would seem, on account
of the divisor on' — 2n, connected with these variations, that these
terms mioht become sensible. But we must observe, that this divisor
[3970"] disappears in d R, because the differential characteristic d, refers only
to the co-ordinates of m, or to the variations of e, ^ [916'] ; so that it
introduces the multiplicator on' — 2 n. Now we have seen, that the great
[3970'"] inequality of m depends chiefly on the term 3 affn dt . dR [1070"].
The inecjualities of the radius vector and the longitude, Avhich depend on
the variations of the exccntricities and perihelion, relative to the angle
[3971] 5 n't — 27it, have therefore very little influence on the two great inequalities
of Jupiter and Saturn.
We shall see hereafter [4392, &c., 4466, &c.], that the most sensible
inequalities of these two planets, depending on the simple exccntricities
* (2502) The variation of c, «', ■ro, 8ic., here referred to, are tliose represented
ro970a] '^y ^ ^' '^*'' ''^' ^''" [3907 J, c, c/] ; all of which have the divisor 5 «' — 2 îi [3907«] ;
but the divisor is destroyed in finding their differentials (/ e, d -a, Stc, as is evident from
[3908c, &ic.]. Hence it follows, that the differential of the expression [3970] gives,
[39704] in d H R or d R, terms depending on ede, e e' d w, &c., wliich do not contain
this divisor ; and if we substitute them in the chief term of the great inequality [3970'"],
they will produce terms which are of the order ?h'^. or of the order m', in comparison
[3970c] with the chief terms computed in [3844, 4418, 4474] ; but as these terms of the order w'^,
[3970(i] have the same divisor (5 n — 2 ?i)^, a* the chief terra, it seems proper to examine
carefully into their exact values, instead of neglecting them, as the author has done. We
shall also see, in [4006^, &ic., 4431/"], that several terms, omitted by the author, similar
[39/ Oc] ■- . . -11 -1
to those treated of in this article, are quite as important as those which he has retauied.
VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 131
of the orbits, are relative to the angle nt — 2 n't. We shall put*
-= F. COS. (7ii — 2n't + s — 2i' + A), [397^]
-5,
a
5r
for the term of — , depending on this angle ; and
6v :=E.sm. (nt — 2n't-\-s — 2s'-JrB), [3973]
lor the term of 6 v , depending on the same angle ; also for the
correspondmg terms oi -7- and & v ,
^ = F'. COS. (nt — 2tt't + s — 2s' + A'); [3974]
ôv' = E'.sm.(nt—2n't + s — 2s'-^B'). [3^4']
If we suppose that R corresponds to Saturn, disturbed bij Jupiter, and [3974"]
then develop it relatively to the squares and products of the excentricities
and inclinations of the orbits, noticing only the angle Sn' t — n t, we shall [3074'"]
obtain, as in [3745, Stc], a function of this form,t
R = il/(»'. é~ . cos. {Qn't — nt + 3^' — i — 2 z-I)
+ il/fi' . e e'. cos. (3n'i — M i + 3 i' — s — ^ — ^')
+ M^-Ke'. cos. (3m'< — n ^ + 3;' — 5 — 2«)
+ M'^'.7=. cos. {Sn't — nt + Qi'—s — 2n).
* (2503) The terms of 5« [4392], depending on the angle nt — 2 n't, or rather
on 2»i''i — 71'^/, are of the order 136' or 56', and may be reduced to the form [3973]; [.3973a]
those of 0 v' [4466] are of the order 182% 417% and may be reduced to the form [3974'] ;
they are the largest terms of the expressions [4392, 4666]. In like manner, tlie parts
of —, 4 [4393, 4467], may be reduced to the forms [3972, 3974]; the last of [.3973i]
" CI
which is the greatest term of [4467] .
t (2504) This value of R is similar to that assumed in [3745 — 3745'"], changing
reciprocally the elements of m' into those of m ; also M'--'> into M'-°\ M^"^ into M^-'' ; [3975a]
and afterwards putting i = — 1. This form of the angles in the value of R, is selected
because it produces, in connexion with the variations [3972—3974'], terms in dR, d' R, [39756]
of the order m^, depending on the same angle 5 n't — 2 n t, as the great inequality, as is
seen, in [3979, 3982, 3985, 3989, 3991]. We may remark incidentally, that in this article
[3975]
132
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
[3976] The quantity 3I^''\ e'". cos. [3 n't — ?i< + 3s' — s — 2y>') arises from the
development of the term of R, denoted by yi'''. cos. («' — v) ;* in which
we must increase ?• by S r, r' by i r', v by 6v, v' by i v'. This
is the same as to increase, in the development of this term, a by or.
a by 6 /■', and n t — nt by &v' — -5 1; ; by which means it produces the
following expression,!
iJ = _ 3/(»). e'=. {iv' — i v) . sin. (3 m' ^ — n i + 3 / — J — 2 ^')
[3976']
[39'
[3978]
+ a. (—,--) . e'-. ~ . COS. (3n't—7it + 3^:'—s — 2zi')
d a
a
+ a'. (^^^ .e'-.~. COS. (3n/t — nt+3/—s — 2 ^').
\ da J a
[3975c]
[3975rf]
[3975e]
[3976a]
[3976t]
the values B, i?, , difler from those in other parts of the work ; since B, B, [3974", 4005']
take the place of jf?', B [1199'], respectively; m being the mass of Jupiter, »«' that
of Saturn. The object of the author, in making this change in the value of B, is to
obtain express formulas for the direct computation of the inequalities of Saturn, which are
much larger than those of Jupiter ; and then to deduce the corresponding smaller ones
of Jupiter, by means of the formula [1208] ; it being evident, that this method of
deduction, in the cases where it can be applied, must be more accurate in finding the small
inequalities of Jupiter from the large ones of Saturn, than in an inverse process.
* (2505) The part of B, independent of j'^, corresponding to the action of Jupiter
iipon Saturn, is found by changing, in [3742], ?»', r, r', v, v', into m, r . r, v', v,
respectively ; and if we suppose, that when a, a, nt -\- s, ii t -\- s', are changed
into r, r, v', v, respectively, the quantity .4''' [3743] becomes .^/'', we shall get,
from [3742, 3743], for this part of B, the following expression,
:- . 5; . .^/''. cos. ?" . («' — i;).
[3976c] jR = — .cos. («' — v) — ■ ,, o — h — ; 7~~i rT~^^w''
'■ ,.a \ '' v/i' — 2rr.cos.[v — «) + ? jj
Substituting in this the values of r, r', v, v' [952, 953], we obtain an expression of B,
[.3976(/] of the same form as [957], and possessing the properties mentioned in [957 — 963] ;
moreover, the term multiplied by the factor e'^, being represented by
[3976c] M^''\e'^.cos. \i . {71' t — nt -^ s' — i) + 2n' t -{- 2s' — 2 z:'} [9.57—959'],
becomes of the form [3976], by putting i=l ; then the corresponding term of B [3976r]
is of the same form as in [3976'].
t (2506) The term Jf '"'. c'^. cos. (3 ?i'/— ?i < + 3 s'— £— 2 to') [3975], is produced
in the function B, by a development similar to that which is used in [957], that is, by
[3977a] the substitution of the cV/p^icoZ values of u^, v,, &c., without noticing the perturbations
[3972 — 3974']. If we wish also to notice these terms, we may suppose a, a', v, v', to be
VI.ii.^16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. \S3
This produces in R, the terms*
R^—i, il/"". E'. e'-. COS. ( 5 »' / — 2 n ^ -f 5 /— 2 .■ — 2 ^'— B')
+ h M"". E . e'-. COS. (5n't — 2nf + 5s'—2i — 2^' — B)
[3979]
+ 1 a' . (^^]. F'. e\ COS. (5 I,.' t-~2nt + o^'—2i-^2^'—A')
\ a a J
+ 1 a . ('^'*^) . F. e'\ COS. (5 n' ï — 2 « « + 5 .-'— 2 -= — 2 ^' — ^ ).
increased, respectively, by Sr, (ir, 5v, 5 v' ; liy which means A'Kcos.i.{v' — r) [SQ^ya']
will be augmented by the three terms in the second member of the following expression, in
which we have retained the factor i=l, for the purpose of more easy derivation hereafter;
[39776]
/>.\A''\ cos. i.{v' — v)] = — A''\ i .{Sv'—5v). sin. i . ( v'~v)
and in the same manner as we have derived from .^''^ cos. i .{v' — v) the term
.¥<">. f'^ cos. \i . {n t — nt -\' s — s) -i^ 2 n' t -\- 2 s'—2tz'] [3976e], [.3977f]
we may derive the three terms [.3978] from those in [39776]. Thus the first term of the
second member of [39776] is the variation of ^'\cos.i.{v' — v) or of J">.cos.(t)' — v), [3977rf]
supposing the angle i . {v — u) to increase by i.((5j)' — àv); in like manner, the
first line of [3978] is the variation of the term
iH'"'. e'2. COS. \i . {n't — nt + i' — e) -f 2 u'< + 2 e'— 2 zi'], [3977e]
supposing the angle i .{n't — nt -\- s — s ) , corresponding to i . {v' — v), to increase
by the same quantity 6 v' — 5 v . The second line of [3978] is deduced from the second [.3977e']
term in the second member of [39776], by supposing a to be increased by S r in ^'"
and .W". Lastly, the third line of [3978] is derived from the third term of the second [3977/]
member of [39776], by supposing a to be increased by 5 r , in ^"' and JW'"'.
* (2507) The expression [3979] is deduced from [3978] by the substitution of
[3972 — 3974], and reducing by [17 — 20] Int., retaining only the angles which are similar
to that of the great inequality, depending on
bn't — 2nt = {Zn't—nt) — {nt — '2n't) ; [39796]
or the difference between the angles contained in [3978] and those in [3972 — 3974'].
VOL. in. 34
134 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
^^- We shall pxit à' R for the differential of R, supposing the co-ordinates
of m' to be the only variable quantities. In the terms multiplied by E'
[3981] and F', the part 5 7i't — ni, of the angle 5 n'i — 2n/,* is relative to
these co-ordinates. In the terms multiplied by E and F, the part 3 n t^
of the same angle 5 n't — 2nt, is relative to the same co-ordinates;
therefore we shall have, by noticing only the preceding terms of R [3979],
rt'd'i?z= i.{5n' — n).di.a'M(°KE'.e'^.sm.{5n't — 2nt-{-às' — 2; — 2m' — B')
[3982]
[3983]
— i.(5?i' — ?0-'/^.«'^-(^^)-F'.e'2.sin.(5n'<— 2»(« + 5.='— 2.-— 2^'— ./î')
— ^.n'dt.a'J\I'°\E.e'^.sm.{5n't~2nt-{-5s' — 2e — 2zi' — B)
— ^ .n'tl ( .aa'. (■^~-\ . F.e'^.sm.{5 n' t — 2nt + 5 ! —2s —2-:^' — A).
The term ilf ". e e'. cos. (3 ?i'ï — n ï + 3 /— .= — ^ — ^') [3975],
results from the development of A'^-\ cos. 2 . (v' — v), in the expression
* (2508) The difterential relative to d' [-3980], does not affect nt in the angle
[.3989«] 3ii'f — nt, which occurs explicitly in [3975], so that d'.{3n't — nl) = 3 7i'cl t ; but
6 v'
[39836] this cliaracterlstic d' affects the w/io/e of the values of —, i5d' [3974,3974'], connected
with F', E', consequently the whole of the angle nt — 2 7i't, which occurs in these
values, must be considered as variable, and its differential is n (t i — 2n'dt. The
difference of these two expressions gives
[393'25']
[3982f] à'.{ron't — 2nl)=à'.[3n'l—n() — à'.{nt — 2n'i) = {5n'—n).dt;
which must be taken for the differential of tlie angle b n' t — 2nt [3979J], depending
on E', F', in the first and third lines of [3979] ; hence we obtain the first and second
(3982(/] lines of [3932]. In like manner, the differential relative to d' does not affect the
[398ae] expressions of —, Sv [3972, 3973], connected with the factors F, E ; or in other
words, the differential of the angle nt — 2n' t, connected with these factors, must vanish :
and we shall have A'.{nt — 2?i'<)=0; subtracting this from [3932a], we get, in
this case, for the differential of [3979i],
[398%] d'. (5 7!'< — 2h<) =d'. (3?i'<— M^— d'. (?i t — 2^^ t)=3 n' dt .
Substituting this in the differential of the second and fourtli lines of [39791, we get,
[3983/!] ^ L J' 6 '
respectively, the third and fourth lines of [3982].
[39841
[3985]
VI. ii. ^ 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 135
of R* Therefore we must vary, in this term, a by ir, a' by ir', ^^
also 2 n't — 2nt by 26v' — 2iv; and by this means we obtain the
following terms of R,
R = —2 M^'\ e e'. (d v' — 6v). sin. (3n' t — ti t + 3e —s — z^ — zi')
+ a. ) . ee. — . cos. (3n r — nt + 3e' — s — ra — ra')
\ d a J a
+ a'. f^^!L_) .ee'.^.cos. (3 n'/ — n r + 3e' — s — ^ — ^').
\ an y a
Hence the part of a'd'R, relative to this expression, is
a'à'R= {5n'—n).dt.a'J\P^\E'.cc'.sm.{5n't—2nt-lr5s—2s — 7Z—zi' — B')
— i.{57i' — n).dt.a'-^.('^-^^\F'.ee'.sm.{5n't — '2nt + bs-22—z^-zs'-A')
— 37idt .a' M^'lE .ce'.sm.{5n' t — 2nt-j-5i' — 2i — ^—^' —B)
— in'dt.aa'.(^-^^\F.ee'.sm.{57i't — 2nt + 5s' — 2s—z!—'u/—A).
The term M<^>. e". cos. (3 n' t — nt + 3^— ^ — 2^) [3975], arises [.3986]
from the development of J'^'. cos. (3 y' — 3i'), in the expression
* (2509) Proceeding witli the term depending on M^^K [3975], in the same manner
as we have done with that multiphed by AI"^\ in tlie tliree preceding notes, we find, that
it may be put under the form
M^'Kee.cos.\i. {n' t — n t -^ s — s) -J^- ,1' t -j- n t -j- s" -{- e — -ui' — tz], [3984a]
supposing i = 2 ; by whicli means it becomes as in the second Une of [3975], and tlie
corresponding term of [39~6c], is of the form
à?ft.^/*^. cos. t . {v'— v)~A'''''.cos.2 . {v'—v). [.39844]
The variations of this term, depending on or, fir', ou, f5 «', are as in [3977è], supposing
i = 2; and from these we may deduce the functions [3984, 3985], by a computation
similar to that used in finding [3978, 3982]. We may, however, obtain the former
by derivation in a more simple manner; for if we change M''^\ c'^, — 2 -a', into rr^,.u.
M'", ee, — « — to', respectively, we shall find, that the first term of [-3975] becomes
like the second ; and the doubling the values of '5 v', & v, in [.397761, on account of the
^ ' ' L J' [3984rfl
factor r = 2, make it necessary that we should double the values of E, E' [3973,3974'].
Making these changes in [3978, 3982], they become, respectively, as in [3984, 3985).
136 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
of R* Therefore we must vary, in this term, a by 6 r, a' by 6 r', and
13987] 3n't — 3nt by Sôv' — 3iv ; hence we get the following terms of B.
R = — 3 M''~\ e\ (ôv' — iv) . sïn. (3 n' t — n t + 3 s'—s — 2zs)
[3988] + a . f'^^\ . e". - . COS. (3n't — nt + 3s' — s — 2^)
\ d a J a
+ «'. y^^) • è\ ~ . COS. (3n'i — n ï + 3 -=' — s — 2^).
Therefore the part of a' d'R, relative to this expression, is
a'à'R= §.{5n'—ii).cIt.aM^-^''.E'.e^.sm.{5n't—2nt + 5s' — 2e—2vs — B')
— i.{5n'— n). dt.a"^.(-^^^].F'. e^.sm. {5 n't — 2nt + 5 s'— 2 s — 2zi — A')
L3989] —^.n'dt.a'J\'r-\E.e''.sm.{rj7i't — 2nt + 5s'—2e—2z^—B)
— %.n'dt.aa'.f-^\F.e''.sm.(5n't — 2nt-{-5s'—2i—2z^—A).
[3989'] Lastly, the term M'=' . -/. cos. (3n' t — 7it + 3 s' — s — 2n) [3976],
[3989"] arises from the term multiplied by -/.cos. (3r' — v), in the expression of i2;t
* (2510) Proceeding as in the last note, we may put the term [.3975], depending
on M'-'^-, under the form
^3988o] M<^\ e^. cos. \i . {n' < — n Ï + s'— s) + 2 n < + 2 e— 2 ^f ,
supposing i = 3 ; and then the corresponding term of [3976f] is of the form
pjjjggj, i to'. A}'K cos. i . {v — v) = A^^'. cos. 3 .{v'—v).
The variations of this term are as in [3977 J], supposing i = 3; from which we may
get [3988, 3989], in the same manner as [3978, 3982] were found. The same result
may be obtained more easily by derivation, as in the last note ; by changing, in [3975, &,c.],
[3988c] M^"', e'^, A'-'\ 2zi', into M'-'', e^, A'-^\ 2 s, respectively; by which means the first
term of [3975], changes into the third; and tlie trebling of the values of ôv', ôv, in
[3988(n [3977i], on account of the factor /:=:3, makes it necessary to change E, E'
[3973,3974'] into 3E, 3 E', respectively. Making these changes in [3978,3982],
they become as in [3988, 3989], respectively.
f (2511) We must now compute the terms arising from the introduction of the increments
[3990a] i5 r, Si-', &v, 5v', in the expressions of J-, r', v, v , connected with the factor 7®, in
the value of R [3742] ; which were neglected in [3976«]. These terms of R may be
VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 137
we must therefore vary a by 6r, a' by àr', S n't by Siv', and
n t bv à V ; hence we obtain the following terms,
R =. — M<''. y^.(3ôv'—&v). sin. (Sti't — nt-\-3/ — s — 2u)
+ a . ('^_:^\ . "-, iZ'. COS. (3n't — n t + 3 /— e — 2 n)
\ da J a
+ «'. ( ^/V)./'. 4-cos.('3n'« — n< + 3/ — E — 2n).
\ (I a J a ^
[3990]
deduced from those depending on y^, in [3742], by changing the elements as in [3976r(].
These four terms of R [3742] are ah'eady muUiphed by the factor y^, of the second
dimension, and as none of a higher order are noticed in [397.5], we may substitute in
these terms, r=a, r' = a' , v ^=nt -{- b — n, v'=n(-\-s' — IT; and retain only
angles of the form 3n'< — nt, assumed in [3975]. Now it is evident, that the two
first of these terms of R [3742], depending on the angles cos. (i;' — v), cos. {v' -\- v),
produce the angles n't — nt, n't-\-nt, which are not included in the proposed form.
The third of these terms [3742] contains v' — v in its numerator and denominator,
and when the denominator is developed, as in [3744], the whole term will depend on
quantities of the form cos. ?*.(«' — v) or cos. i.(n'^ — nt), which are not comprised
in the form ^n!t — nt, now under consideration ; so that we need only retain the last
term of [3742], which, by making the changes indicated in [3976a], may be put under
Ttt 'V T T COS I y' ~l~ v ^
the form R = {- . '- ' j. Now if in the formula [3744],
4 {,-2 — 2 rr'. COS. (w'—î))+»-'2 1 ^
we change a, a', nt-\-s, n't-\-s', B'-'\ into r, r, v, v', J5/", we shall get
[3990i]
[3!)90c]
[3990</]
[3990(/']
\r^— 2rr'. cos. {v'— v ) -f r'~l ^ = ^S. 5». cos. i . ( v'—v).
Substituting this in R [3990e], and reducing by means of formula [3749], it becomes
iî = — :i m . f. r r. * 2 . B'p. cos. \i . (v'—v) -f «' + jj j.
If we change î into i — 1, and put — |- ?« . r r*. i?/'-" = JW''', we get
R = f.S.M'-'^.cos. {i.{v'—v)'}-2v];
which in the case of i^3, produces a term of the form R = M'-^\'^^. cos. {3 v' — v).
Taking the variations of this term, as in [3977a', &c.], we get the following expression,
similar to [3977è],
&.{M'^\f.cQs.{Sv'-v) \ = —.¥"1.^2. {3Sv'— S «) .sin. (3 v'—v)
[3990/]
[3990/']
[3990g-]
[3f>90A]
[3990i]
Substituting in this the values [3990è], we obtain [3990].
VOL. III. 35
[3991]
138 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
Hence we obtain in a' d' R, the following terms,*
aà'R= ^.{5n~7i).dt.a'M'-^\E'.f.sm.{5n't — 27^-^-5;'— 2! — ^n — B')
— i.{5n'—n).dt.a'~. [~j^) F'. f. sin. (5 n't — 2n t + 5 s'— 2 s — 2 n— ^')
— ^n'dt. a'M^^'lE. f. sin. (5 wV — 2 ?i ^ + 5 s' — 2 s — 2 n—B)
— ^n'di.aa'.l -j^j ■ F.f. sm. {5n't — 2 nt -{-5i'— 2c — 2 n—^).
The most sensible inequalities, arising from the squares and products of the
[3991] excentricities and inclinations of the orbits, which neither have 5 n' — 2 nf
for a divisor, nor depend upon the variations of the elements relative io the
* (2512) Tlie expression [3991] is deduced from [3990], in the same manner as
[3982] is from [3978] ; or more easily by tlie principle of derivation. For if we cliange
[3991a] M'^°\ e'2, 5v', — 2 «', into Jl/"\ y^, Si'iv', — 2n, respectively, the function
[3978] will become as in [3990] ; consequently E' [3974'] must be changed, as in [3984f/],
[.39916] jj^^^^ g^,_ Making the same changes in [3982], which was deduced from [3978],
we get [3991].
t (2.513) The divisors in [3714, 3715], which may be small, in the theory of the
perturbations of Jupiter and Saturn, are i>i'-\-{3 — i)-n, in' -{-{I — î).n, în'-{-[2 — i).n;
"•^ and since n'^fn nearly [38 18fZ], they become (3 — %i)-n, (1 — f?).H, (2 — |-î).«.
If we put / = 5, the first divisor becomes 0, the others being large. If i = 4, the
• last divisor becomes — f ?i, and the others are larger. If / = 3, the last divisor
becomes ^ n, and the others are greater then this quantity ; and it is evident, that next
to i^5, this value of i gives the least value to the divisors [3992a] ; therefore the terms
of 7-0 r, ÔV [3714,3715], of the second order, relative to the quantities e, e', y, and
depending on the angle 3?*'/ — nt, maybe increased by this divisor, so as to become
greater than other terms of the same order, relative to e, e', y, which have not a small
divisor. This reasoning is confirmed a posteriori by the inspection of the numerical values
of 5r"', Sr", Hv", û v" [4397,4470,4394,4468], in which the terms depending on
the angle 3 n't — n t, are generally greater than any of those that are noticed in [3991'],
[.3992^] excepting 4n't—2nt. This last angle is here neglected, because the terms or, ôv,hc.,
depending upon it, do not produce in [3995], functions of the form [3998], depending
on the angle 5 n' t — 2 n t, which are the only ones under consideration at the present
moment. Now if we notice only the temis depending on the angle 3 7i't — 7it, in
[39926]
[3992c]
or
[3992c] [3714, 3715], we shall obtain for —, Sv, quantities of the forms [3992, 3993],
6r'
and in like manner, in —, Sv', terms of the forms [3994, 3994'].
VI. il. §16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 139
angle bn't — Int, are those corresponding to the angle 5 n't — nt.
We shall put
— = G . COS. (3 n'i — n < + 3 e'— Î + C) , [3999]
(5 r
for the part of -, depending on this angle ; also
ÔV = H. sin. (3 n't — n t + 3 s' — s + D), [3993]
for the part of ô v, depending on the same angle ; in like manner,
^-4 =^ G'. COS. (3 71' t — nt + 3 i' — z + C') , [3994]
a ^
Ô v = H'. sin. (3n't — nt-ir 3s'— s+D'), [3994']
5 r'
for the parts of -7-, èv', depending on the same angle. The expression
of R, developed relative to the first power of the excentricities, contains
the two following terms,*
R= iV'O'.e.cos. (nt — 2n't + s — 2i'-\-zs)
[3995]
+ N^'Ke'. cos. (nt — 2n't + e—2s' + ^').
* (2514) In the same manner as we have deduced, from R [3976c], the three
terms [3916e, 3984», 3988a], of the second order in e, e', we may obtain two of the [3995a]
first order in e, e', of the following forms,
R= :^-i^\e.cos.\i.{nt-nt^e'-s)-{-7it-i-s-z,\ ^^^^^^
+ JV('>.e'.cos. li.(«'<— ?i<4-s'— 6) + ?i7 + e'-ra'}.
If we put i = 2, in the first of these terms, it becomes of the same form as the first [3995c]
term of [3995] ; and by proceeding in like manner as in note 2506, we easily perceive [3995(/]
that this term arises from the development of A'-'^^.cos.i . {v' — v), supposing i = 2, [3995e]
as in [3995c]. Moreover the second term of R [3995è], becomes of the same form as
the second term of [3995], by putting i=l; and then the term Jl'^'\cos. i . {v'—v), [3995/]
upon which it depends, becomes .a'", cos. [v — v), as in [3998'].
We have already computed, in the case of i = 2, the effect of the substitution of the
variations 5r, or', ôv, Sv', in the development of .^^-'.cos. 2. ( y' — v) [3984i], and [3995g-]
we have found that this substitution, in [3984i(»], produces the function [3984]. A similar
method may be followed with the first line of R [39956] ; but it is more simple to derive ^ '
it from [3984a, 3984]. This is done by changing, in [3984a], the factor M^^Kee'
into JVC", e, and decreasing the angle, which is contained under the sign cos., by the [3995i]
[3996]
[3997]
140 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[3995'] The first of these terms arises from the development of J'^.cos. (2«' — 2v),
in the expression of R ; and in this development we must increase a hy 6r,
[3995"] a' by or', 2 n't — 2nt by 2ôv' — 26v; from which we obtain the
following expression,
R= 2 iV'"'. e . (6 v'— &v). sin. (^nt — 2n!t + s — 2i'-\--:z)
+ a. (~ — ) .e.— .COS. (îit — 2n't + s — 2s' i-^)
\ a a y a ^ ^
/,7JV(0)\ xj
+ «'. -7^).e.— .cos. (71/ — 2n7 + £ — 2£'+^).
\ da J d ^ ^
Hence we get in iî, the following terms,*
/?= iV'^i/'.e.cos. (5n'/ — 2n/ + 5E'_2£ — ^ + Z)')
— m\H.e.(io%.{b'n:t — 2nt^bi—2i — ^ + D)
+ ia'.f-^j.G'. e.cos. (5w'/ — 2 7i/ + 5f'— 2.^ — ^ + C")
+ i«. (— — j .G.e.cos. (5 7t'/ — 2ni + 5/— 2s — ra + C).
To obtain the corresponding part of d'7?, we must vary the angle
[3997'] 2jj/^ — jj^^ jj^ j^l^g terms multiplied by H' and G';t but in the terms
[39954] quantity ?i'i-|-£' — ra' ; by which means it becomes as in the first line of [39956];
then putting ! = 2, it becomes as in the first term of [3995]. The same changes being
made in [3984], which was derived from [3984n], it becomes as in [3996] ; observing
that when the angle 3 n' i — nt-\-Z^ — i — « — ■s/ [3984] is decreased by the quantity
nV -\- s' — ra' [3995fc], its sine becomes
[3995i] sin. (2 w' ^ — n < + 2 s' — s— ^ ) = — sin. (n < - 2 «'< + s - 2 £'+ ra) ,
as in the first line of [3996], and its cosine is as in the second and third lines of the
same expression.
* (2515) Substituting, in [3996], the values of hr, hv, 5/, <5 y' [3992-3994'],
[3997a] reducing the products by [17—20] Int., and retaining only the terms depending on the
angle 5 ii! t — 2?i<, it becomes as in [3997].
t (2516) The characteristic d' [3980] affects only the angle 2 m'!', in [3995], so
[3998o] that in these terms we shall have à'.^nt — 'i.n t) = — '2,n dt ; but d' aflects the
the whole values of ~7 , ^ f', consequently also the whole of the angle 3 n t — nt.
VI. il. §16.] DEPENDING ON THE SCJUARE OF THE DISTURBING FORCE. 141
multiplied by H and G, we must only vary 2 n't; hence we obtain [3997"]
a'd'B= — (ôn—n).clt.a'j\'^'>\H'.e.sm.{:in't~2nt^5s—2s — r.~\-D')
_j.(5„'_„).rf^„'aY''^VG'.e.sin.(5«7 — 2H^ + 5a'-2£— ra+C)
4--2n'tlt.a'JY^'>\H.e.sm.(5n'i — 2nt4-5.' — 2s — a^D)
[3998]
-~-^].G.e.sm.(5n't — 2nt-{-5e' — 2E—zi-\-C).
The term N^'K e'. cos. Çti t — 2 n' t + s — 2 s' + z,') , arises from the
r3998'l
development of the term of iî, represented by ^''. cos. («' — v)* [3d95f'\ ; '
which occurs in the terms [-3994, .3994'], which are multiplied hy G', II'; so that in
these terms we shall have d'.[3n't — n l) =^3 n' c1 1 — ndt. Subtracting [3998a] [39986]
from this, we get
d'. {5n't—2nt) = d'.(3ji't—n <) — d'. {nt — 2n' t) = {5 n'—n) . d t, [3998e]
for the dlTerential of the angle 5 n' t — 2?!.^, which occurs in the terms of R [3997],
depending on G', 11' ; it being evident, that the angle 5 ft' t — 2 n t is produced in these
^ terms by combining the angles 3 n' t — nt, ni — 2 n't, as in [3998c]. Substituting [3998ci]
this in the differential of the first and third lines of [3997], taken relatively to d', we get
the first and second lines of [3998], containing the flictors G', H', as in [3997'].
or
Again, the characteristic d' [3930] does not affect —, îi v, so that in their values
[3992, .3993], which contain the factors G, H, we have d'.{3n't — nt) = 0;
subtracting from tliis the expression [.3998n], we get
[3998e]
d'. ( .5 7i't — 2ni) = d'.{3n't—nt) — d'. {nt — 2n't)=2 n' d i ; [3998e']
which is to be substituted in the differenlial of tlie second and fourth lines of [3997],
taken relatively to d', to obtain the third and fourth lines of [3998], containing tlie
factors G, H, as in [3997"]. The whole value of d'^ is to be mukiplied by a', to '"^^^^•^^
obtain ddR [3998].
* (2517) We have seen, in [3995/], that the second term of [3995],
./V'". e'. COS. ( /i ^ — 2 n't-\-s—2 i' + to'), [3999a]
is derived from a term of i?, of the form .,4^". cos (i;' — v), corresponding to i=\;
being of the same form as [3977(/]. Now tlie effect of the substitution of the variations
of or, (5/, Ô (', dv', in tlie development of this quantity, having been computed in [3978],
we may deduce from it the terms of R [3999], corresponding to the present case, by a
similar method of derivation to that made use of in [3995/i— /]. Thus, instead of the ^^^^^^^
VOL. III. 36
142 PERTURBATIONS OF THE PLANETS, [Mtc. Cél.
r3998"l ^^ must therefore vary, in this term, a hy 6 r, a' by ô r', n' t — nt by
i v — 6v, and we get the following expression,
R=. N^'K e'. (6 v' — 6v) . sin. {n t — 2n' t + s — 2s' + v>')
[3999] + a. ( -— — ) . e'.— . COS. (n t — 2 n'?; +s— 2 e' + jj')
(/ a
a
[4000]
+ «'. [~T-r .e'. — .COS. (n^— 2n'^ + f — 2s' + ï5').
\ ail J a
Therefore the part of a'd'B, relative to these terms, is*
a'à'R = — i.{5n'—n). lit. a JV-^'.H'.e'. sm.(5n't—2ni + 5^—2 s — z^'-]-D')
///jV(i)\
— i.(5?i'— nj.f/i-.a's. f-— -j.G'.e'.sin. (5M'i — 2m< + 5s'— 2s— to'+C)
+ n'(Z<.aW'.H.c'.sin.(5n'^ — 2n< + .5='— 2s— ûj' + D)
— ?i'rf^.o«'. ^-^\G.e'.sin.(5?).'/ — 2Ki'-)-5s'— 2j— îj'+C).
The values of M<% iV/'^', M'*', M' =>, are determined in the formulas
[4000'] Qf ^^^ jjy changing the quantities relative to m into those relative to m',
and the contrary [3975a, 6].t The values of A'^"* and N'-^^ are determined
operations mentioned in [3995?], we must, in the present case, change the factor M'-"''. e'"
r3977el into A'"', e' ; and decrease the ani^le which is contained under the sign cos.,
[3999cl •- J ' o c- J
'■ by n't-\-e' — to'; by which means [3977e] becomes as in the second line of [39956],
[3999(i] or tlie second line of [3995], supposing ?'=1. Now making the same changes in [3978],
which is derived from [3977e], it becomes as in [3999] ; observing that when the
angle 3n'i — nt-{-3^—s—2-a' [3978], is decreased by n't-\-i — ia' [3999c],
it becomes 2 )/< — n <-(- 2 s'— s — to'= — ( ji < — 2«'< + e — 2 s'+ra') .
* (2518) The function [4000] may be deduced from [3999], by the method we have
used in computing [3997] from [3996]. It may, however, be deduced more easily from
[3999/] [-3995^ 3997J . by changing JV*»', e, ra, 6v, ôv', into .V<'>, e', to', iSv, i&v', respectively.
For by this means, [3996] changes into [3999]; and H, H' [3993, 3994'] become
L ^■' i H, I H', respectively. These changes being made in [3998], it becomes as in [4000].
f (2519) If we put i = — 1, in the terms of R [1011], depending on e, e', and
[4000a] retain only these two terms, putting also .4'~'> = .4<'' [954"], we get, for this part of R,
relative to the action of Saturn on Jupiter,
R=^ — -~.]a.[- — — 2^<')^.e.cos. {2nt — ntA-2s — s — zi)
i ( \ da / S ^ ' '
[40005]
— ^ ■\<''- ('-TT^W4^^='^.e'. cos. (2nt — v!t-\-2i — s — z^').
VI. il. § 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 143
by the equations,
a'A'^(»)=— 2m.rtVi'-'— im. ««'. (-^) ; [400i]
a' iV<" = m . «' . J('> — i m . a'\ C^^ • [4001']
Connecting together all these partial expressions of a'A'R, we obtain
a term of this form,*
a'à'R = m n'. I.dt . sin. {5n' t — 2nt + 5^' — '2,^ — 0) . [4002]
Hence the term 3 a ffn d t . d' R, of the expression of <5 v', givesf [4002]
iv'= — ,^"'''''''\.sm. (5n't — 2nt + ôs'—2s — 0). [4003]
[5 II — 2 II)- ^
This is the most sensible term of the great inequality of Saturn, depending
on the square of the disturbing force.
[4000c]
Changing, reciprocally, ibe elements of m' into those of m, we get the corresponding part
of -R, relative to the action of Jupiter on Saturn. Comparing this with the assumed
form [-S^QS], after having changed the signs of all the terms contained under the sign cos.,
in [3995], we get the expressions of JV'», ^'" [4001, 4001'].
* (2520) Adding together the parts of a d' R [3982, 3985, 3989, 3991, 3998, 4000],
and putting, for brevity, T^ = 5nt — 27it-{-5s' — 2e, we get a series of terms [4002a]
of the first form [4002f] ; /' being used for brevity, for the coefficients, and O' for the
quantity connected with Tj. Developing this by [23] Int., we get the second form
[4002c or 4002fi?J ; in which we may substitute
2./'.cos. 0'=mM'. /.cos. O, 2./'. sin. 0'= — mn'. /. sin. O, [4002J]
and we obtain the first form [4002e], which by means of [22] Int., becomes as in the
second form of [4002e], agreeing with [4002],
a'd'R = dt.-Z.r. sin. (Ts -\- 0')=^dt .S. . F. {sin. T^ . cos. O'+cos. Tj . sin. 0'\ [4002e]
= (/ ^ . sin. Tj . 2 . /'. cos. O'-^d t . cos. T^ . 2 . /'. sin. O' [4002(f|
= mn'.I.dt.\sm. T, . cos. O — cos. T5 . sin 0\ = mn'.l. dt .sm. (T. — O). [4002e]
t (2521) Multiplying [4002] by S n' d i , and then integrating it twice, relatively
to t, we get, for 3 a'ffn'd t . à'R, the expression [4003] : and this quantity is evidently [4003o]
the most important one in the value of u v, depending on the term now under consideration,
included in the expression [3715m].
144
PERTURBATIONS OF THE PLANETS,
[Méc. Céî.
[4003']
[4004]
[4005]
If the expression of R, divided by the disturbing mass, be the same
for Jupiter and Saturn, we shall have, as in [1208], the coiresjjonding
inequality of Jupiter 6 v, by substituting the preceding value &v' [4003]
in the formula
m' \/tt'
6V =
m\/a
.6V\
but the value of ^4'" [3775c] is not the same for the two planets,
therefore the terms*
ilf C). e'\ COS. (3 m' ^ — /U + 3 .-'— s — 2 ^J) ;
iV">. e'. COS. (nt — 2n't + s — 2 e + ^') ;
divided by the disturbing mass, are different for each of them. But it
follows, from [1202], that by noticing only the terms having the divisor
(5 n' — 2 n)", we shall have in this case,t
m.fdR^+in'.fd'R^O ;
[4004a]
[4004ft]
[4004c]
[4004i]
* (2522) The terms mentioned in [4004] are derived from «3'^\ cos. (îj' — v), as
it appears in [3976', .3998'] ; but the value of A'-''' is not the same, in computing the action
of m upon m' ; as it is in computing the action of m' upon m [377.5c]. Now we have
already remarked, in Vol. I, page 651, that the method of finding the inequality of Jupiter
from that of Saturn, by means of the formula [1208 or 4003'], is not applicable, without
some restriction, to the computation of terms of the order of the square of the disturbing
force. This is evident from the consideration, tliat in the equation
ni.fdR^m'.fd'R' = 0 [1 202] ,
from which the formula [1208] is derived, terms of the third order in m, m' are neglected,
which is equivalent to the neglect of terms of the second order in R, R' ; being of the
same order as the terms computed in [3982 — 4002].
t (2523) This formula is corrected for a typographical mistake in the original work,
[4005a] and is the same as in [4004c] ; terms of the third order in m, m being neglected.
We have already spoken of the different meanings of the symbol R, and it may not be
amiss again to repeat, that ?» is the mass of Jupiter, w' that of Saturn ; also in formula
[40056] [4004c], the value of R corresponds to the action of rd on m [913], and R' to the
action of m on ?«' [1199']. These are changed in the present article to R^ [4005']
and R [3974''], respectively.
VI. il. §16] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 146
R^ being tvhat R becoiries relatively to the action of Saturn on Jupiter, and ,Af.,.r,^
the differential characteristic d referring to the co-ordinates of Jupiter.*
* (2524) Substituting &v' [4003] in the formula [4003'], we get the corresponding
inequality of &v [4006]. This method of deriving 5v from <)v', would be sufficiently
accurate, were it not for the terms of the third order in m, m', omitted in [4004c, 4003']. These
neglected terms make it necessary either to correct the result obtained in [4006], or to compute,
tn a direct manner, the value of 5v from the formula ôv^Saffndt .dR [3715Z]. Thus,
for the terms of R,, which are similar to those of R [3978, 3984, 3988, 3990, 3996, 3999],
we must compute the corresponding values of adR^, similar to [3982, 3985, &c. — 4000],
and by combining all of them together, we get the value of adR^, corresponding to [4002].
This is to be substituted in [4005f], to obtain the required inequality 5v, which is to be
used instead of [4006]. It will not, however, be necessary to repeat the whole of these
calculations, since we shall soon show that the terms of R, of the form and order in the
development [3742], combined with those of a similar development of R^, satisfy the
equation [4005], when we except the terms depending on A'-^\ and notice only such
quantities as have been under consideration in this article, namely, those which are of the
order of the square of the disturbing force, and depend on the angle 5 n' t — 2nt.
For if we put
A = cos. ( v' — v) — ^7®. cos. {v' — I' ) ~t~ 4 7^' ^^^- ( '^'~H * ) >
X
B =^ — {r^ — 2 r r'. cos. {v' — i' ) 4~ '"' '^ ^ ^
3. .
4" ^7^-\cos.{v' — v) — COS. ( !)'-[- 1' ) } • \r^ — 2rr'. cos. (d' — v)-{-r'^\ ^ '
we shall obtain the value of R [4005/], corresponding, as in [3974''], to the disturbing
force of Jupiter upon Saturn ; the expression is derived from [3742], by changing m, r, v
into m', /, v', and the contrary. Moreover R^ [4005/', 4005'] corresponds to the action
of Saturn upon Jupiter, being the same as in [3742],
R=m.^ .--if-mB;
[Action of Jupiter on Saturn.]
R/=^ m'A . \-m B ; [Action of Saturn on Jupiter.]
respectively, in [3975—3991]; also JV'^ JV»>, into -.A*'»', - .JV'" [3995— 4001'] ;
or in other words, we may compute the parts of R^ , depending on B, by multiplying the
VOL. III. 37
[4005i']
[40056"]
[4005c]
[4005rf]
[4005e]
[4005/]
[4005^-]
[4005/i]
[4005i]
[4005*]
[4005/]
[4005/']
If we neglect, for a moment, the term A, we shall have R^mB, R, = m' B ;
I
whence R,^ — .R; so that the terms of R^, corresponding to R [3975], maybe [4005m]
found by changing M^'>\ M^'\ M'^\ M^^\ into -.J/«\ - .M^", -'.J/<2', -.JfO),
[4005»!]
146 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
Hence it follows, that the inequality of Jupiter, corresponding to the
corresponding terms of it [3978, 3984, &jc.] by — . In finding tlie differentials relative
to d, we shall proceed in the same order as we have done in finding those relative to d'
[4005o] [39S2ff, 8ic.], observing that d does not affect Sti't, in the angle 3 n't — nt, which
[4005;/] occurs explicitly in [3975]. Hence we shall have d . {3 n't — ni)^^ — n dt, similar
6r'
to [3982rt] ; moreover, as the sign d does not affect the values of —, Sv', the differential
of the angle nt — ■2n't, which occurs in these values, or in the terms connected with
[4005î] £', i^' [3974', 3974], is d . {nt~2n'i) = 0. The difference of these two expressions,
corresponding to the equation [3982c], is
[4005r] d . {5 n' t — 2 n t) = d . {3 n t — 71 1) — d . {71 1 — 2 n' t) = — )i d t ;
[4005r'] now we have very nearly 5 71' — 2 ?i= 0 [3818rf] ; and the inequalities S v, iv', under
consideration, are very small, as we shall see in [4431/] ; therefore we may put
— J! = — ( 5 71' — n), and the preceding expression becomes
[4005s] d.{57it—2nt)^ — ( 5 n' — 71) . d t ;
which is equal to that of d'. ( 5 m' t — 2 7it) [3982c], but has a different sign. Hence,
by noticing only the part of R, depending on B, and connected with the factors E', F',
we have d/? = — d'iî ; substituting this in the differential of R^ [4005»j], taken
relatively to d, we get dR=~.dR^ .A'R; which is easily reduced to the
[4005u] fo'™ ;« .diî, -[-?«'. d'/{ = 0 [4005]. In like manner, the differential d affects the whole
of the values —, &v [3972, 3973], depending on the factors E, F ; so that the
differential d, of the angle 71 1 — 2iH, connected with these terms, is
[4005i'] d .{7it — 2 7i't) =: ndt — 2n'd t .
Subtracting this from [4005j:>], we get
d.{5n't — 27it) = d.{3n't — ni) — d.{7it — 27i'i)=^27i'dt — 27idt:
and by substituting 2 m' — 2 }i = — 3 ?i' [4005/-'], it becomes
d.{5 7it — 2 71 1)= — 3 71' dt = — d'. {5 n't— 2 nt) [3982^] ;
r4005rl hence, for these terms, we also get, as in [4005^], dR^ — d'R and ?«.di?,-}-m'.d'-R = 0.
The same result holds good when the terms of R, instead of depending on the angle
[4005yJ 3 n't — 71 1 [3975], have other forms, as for example, nt — 2 7i' t [3995] ; which are to
be combined with the corresponding terms of S 7-, ôv, (5 /, 6 v', so as to produce the angle
5 7i'i — 2 7it. Thus, if instead of the particular values of R, — [3975, 3974], we
assume the following general values,
[4005î" R = M.cos.{ i\ n't — i^nt + Jli), ~ = F'. cos. ( it n t — i'., n' t -\- A^^) ;
[400.5«P
[4005w'i
VI. ii. ^S 16.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 147
preceding expression [4003], is
^ 3m. «ay- ^ ^■^^^.Q^,^_2nt-\-5s—2e — 0). [4006]
(5ji' — 2n)2 ^
in which i\ -j- i'^ = 5 ; {^ -(- i, = 2 ; we shall find that the products of these two [4005i']
expressions, contained in a function similar to [3978], will produce a term depending on
the angle 5n't—2nt, as in [-3979]. In this case, the equations [3982c, 4005r]
become, respectively, by suhstituting i\-{-i'„ = 5 [400.5-'], [400(Ja]
d'.(5ii't — 2nt)^zà'.{i'i n' t — iiiit) — à'.{^nt — i'^ n' t )
[4006i]
= i\ n (It — {^i^ndt — i'o n' d () = 5 n d t — i.^ ndt ;
à..{bn't — 2nt)=^à.{i\nt — i\?U) — d. (îoni — i'.^n't)^^ — i^ndt. [4006<;]
The sum of these two equations, substituting iy -|- to= 2 ; 5 ?j' — 2 ?« = 0 [400.5cr', /], is
ù'.{bnt—2nt)-i^A.{biît — 'int)^bn'dt—2ndt = Q, or à'R^àR^Q, [4006rf]
as in [400.5^] ; and from this we get, generally, as in [400.5x, 4005] , m.àR^-\-m' .à'R=zO. [4006e]
Hence it follows, that if we put àvy, i5z).,, for the parts of èv, of this form and order,
dependuig on Jl, B, respectively; also &v\, ôv'ç^, for the similar parts of ôv', we shall have
5 D =: 5 Di + 5 1>2 ; ôv' = Sv\-{-Sv'„; [4006/]
and the formula [4006e] gives, as in [1202, Sic], the following expression, similar to [4003'],
[4006e']
Sv^= — 5v'2."^. [meg]
my a
From this formula we may compute 5t)o, after having found or'j, by a direct process
similar to that used in [3975 — 4003].
In computing the terms of avy, àv\, depending on A [4005A], we may neglect the
two terms containing y^, for the same reasons as in [3990ff— c]. Then we shall have
simply ^ = cos. (î;' — v) ; hence the corresponding parts of R, R, [400.5/,/'], become [4006/i]
R = m. ^ .cos.{v'—v); R^=m'.-^^.cos. .{v'—v). [400fo-]
These quantities evidently depend on the term connected with the coefficient A <'', in the
development of — [954, 957], as is evident by the substitution of the values [952, 953].
Hence we have, by noticing only this part of A'-^\
A'^'> = m • -J ; in computing êv\, arising from the action of Jupiter on Saturn ; [4006^]
^'i>=m'. — ; in computing Sv, arising from the action of Saturn on Jupiter. [4006/]
Now A^^' occurs only in the development of the term .^'". cos. ( r' — v); and it is [4006m]
148
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
17. In the inequalities of Jupiter and Saturn, in which the coefficient
[4006'] of t is neither 5n'-
nor differs from it by the quantity n, in
[4006n] therefore found in JJf*"' [3976,3976'], also in JV'(" [4001'] ; but not in M^'\ M'--\ M"',
[400(3o] JV'"' [3983, 3986, 3989", 3995'] ; so that in these last terms we shall have (5 Uj = 0,
[400G/)] 5^'j = 0, à'Vç,=^iv, (5 î)'g = (5 1)' ; consequently the value of 5v may be correctly
obtained from i5 v', in these cases, by means of the formula [4003']. A different process
[4006?] must be used with the terms depending on M'-^^, JV*'\ which contain A^^\ For we must
compute (5 ti'j in a direct manner, by means of the value of ^'" [4006Zr] ; also dv-^, from
[4006r] [4006Z] ; by a process similar to that used in computing &v' or &v'^, in [3982,4002'].
[400C«] Having thus obtained i5 Dj , ùv\, iv'.^, we get àv^, by means of the formula [4006^],
and then by substitution in [4006/"], we obtain the values of 5v, Sv', corresponding to
r.««^ „ these terms. These remarks are not restricted to the two forms of R, treated of by the
[4006s ]
author in [3975, .3995], but apply generally to others of a similar nature, contained in the
general table, which we shall give in [4006zt].
In addition to the terms of R, depending on the angles 3 n't — ni, ni — 2n'i ;
[4006<] treated of by the author in [3975, 3995] ; there is an infinite number of a similar nature ;
some of which are deserving of peculiar notice, on account of their magnitudes ; and one
of them is of nearly the same order as those we have already noticed. The 20 forms of
R, S 7-, 5v, êr, ôv', Sic, producing the angle 5 n't — 2 n i , are contained in the
annexed table. Thus the form which is marked with the number 6, includes the terms
of R, depending on the angle 3 n't — nt, as in
; the first form assumed by the author in [3975] ; and
when this is combined with 6r, 5v, &:c., of the form
2n't—nt, it produces terms depending on 5n'i — 2nt,
as in [3979]. We may also take these angles in an
inverse order, corresponding to the accented numbers,
supposing, as in the number 6', that R depends on the
angle 2n i — n t , corresponding to the second form
of the author, in [3995], and ér, 5v, he. depend on
the angle 3 n't — nt . The numerical values of these
terms of ^i', 5v', are given inaccurately in [4432,4488];
as was first observed by Mr. Plana, in the second
volume of the Memoirs of the Astronomical Society of
London ; in which he has given the calculations of the
[4006d] separate terms at full length ; and has also noticed the terms of R, of the forms 5', 3, 4 ;
observing, however, that they have hardly any sensible effect in the complete values
of &v, 5 v'. The final values of ôv, ô v', computed by Mr. Plana, by a direct process,
and independently of each other, did not satisfy the equation [400-3'] ; and this numerical
result, he considered as a demonstration a posteriori, that this formula could not be applied
[4006!^] ^^ ^j^ ^1^^^^ jg^.^^^^ ^j. jj^g ^^,jg|. ^f ^jjg square of the disturbing masses. In consequence
[4006«]
No.
Coefficienl3 of ( in
the terms of
R.
Coefficients of t in
tlie terms of
Sr, év, or', iv'.
V
2!
3'
4'
5'
6'
1
2
3
4
5
6
0
n'
2ji'
3n'
n' — n
3 n' — n
5 n'- 2 n
An'—2n
3n' — 2n
2n'— 2n
An' — n
2n' — n
v=n'—n;
{ = any positive integer.
7
8
9
10
5n'—2n-\-i\i
5n' — 3 ?i -|-iv
5n' — 4?i-|-iv
5n' — 5n-\-iy
I V
t V — n
iv — 2n
iv — ■ 3 n
r
8'
9'
10'
No.
Coefficients of £ in
tlie terms i>f
Sr, iv, 6r', 6v'.
Coefficients of ( in
the terms of
R.
VI. ii. §17.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 149
Jupiter, or n' in Saturn; we must increase nt and n't by their great r^^Qg,,,
inequalities depending on bn't — Int. For we have seen [1070"],
of tliese remarks, La Place resumed the subject in a memoir published in the Connaissance
des Terns for the year 1829 ; in which he tacitly admits the inaccuracy of the application of
the formula [4003'] to all these terms of the order of the square of the disturbing forces ; and
gives a new formula [400Si], expressing the relation between the complete values of the
terms of 5v, 5 v', like tJiose computed in this article, and others of a similar form and order,
calculated by Mr. Plana [4006v]. This new formula has been called the last gift of La Place
to astronomy. Upon applying the numerical values of ôv, 5 v', given by Mr. Plana, to this
formula, it was not satisfied ; whence La Place inferred, that these numerical calculations
of Mr. Plana were incomplete or inaccurate. Some strictures having been made on this
formula by Mr. Plana, in the Memorie dclla Reale Accademia delle Scienze di Torino,
Tom. XXXI ; it was followed by two other demonstrations of this new formula ; the first
by Mr. Poisson in a memoir published in the Connaissance des Terns for 1831 ; the second by
Mr. Pontécoulant, in the same work, for 1833. In the memoir of Mr. Poisson, he notices
the term of the form 1, in the table [4006m], and shows, that it is of sufficient importance
to be introduced into the calculation. Under these circumstances, he recommends a
revision of the whole calculation, by taking into consideration all the forms comprised in
the table [4006it], which produce terms of i5 v, Sv'. of any sensible magnitude. This
extremely laborious task has been performed by Mr. Pontécoulant, who has given the
abridged results of his investigation in the Connaissance des Terns for the year 1833, from
which we shall make some extracts, in the notes upon the twelfth and thirteenth chapters
of this book, in treating of the orbits of Jupiter and Saturn. These results, so far as they
relate to terms of the forms 6, 6' [4006?;], computed in this article, differ but very little
from those of La Place [4432, 4488], except in the signs ; and upon referring to the
original manuscript, in which these last calculations were made, a mistake in the signs
was discovered. Finally, Mr. Pontécoulant suggested to Mr. Plana, some corrections
which were necessary in his work ; and upon the revision of his calculation, it was found,
that the results were almost identical with those of Mr. Pontécoulant ; these corrected
values, combined with the other terms of this kind computed by Mr. Pontécoulant, are
found to satisfy very nearly the new formula of La Place [4008x]. We shall now give
the demonstration of this formula.
For this purpose, we shall use the same notation as in [1198], in which M represents
the sun's mass, m the mass of Jupiter, in' the mass of Saturn ; x, ij, z, the rectangular
co-ordinates of Jupiter, referred to the sun's centre ; r its radius vector, &c. ; and the same
letters accented correspond to the orbit of Saturn. Then putting, for brevity.
x^'+yy'+-'^'
VOL. III.
w
xx'+yy'+-'
^lf^x'-xf+{y'-yf+{z'~zfl
[4006x]
[4006y]
[4006z]
[4007a]
[40076]
[4007c]
[4007 (i]
[4007e]
[4007/]
[4007e-]
38
150 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[4006'"] that these great inequalities must be added to the mean motion, in the
[4007/1] we get, as in [949,1200], by observing that r^^x^-V-y^-^-z^ r'^=x'^-^y"^-^z'~ [ÇiW],
[4007il R = in • (iv' -\- X) ; [For tho action of Saturn upon Jupiter.]
[4007/c] R':= m . (it) -\- X) ; [For the action of Jupiter upon Saturn.]
Now if we multiply the formula [1198] by M-\-m-{-m', it will become of the form
[4007o] ; for the two first terms of the second member of the product, or those in the
first line of [1198], may be put under the form,
[4007i]
^{dx^ + dy^ + dz"~) , ,3 {dx'^+dy-!i+dz"-
dt^ ' dt^ '
of which the first line is the same as in the first line of [4007o]. Connecting the terms in
the second line of [4007?] with those produced by the second line of [1198], namely,
{mdx4-m,'dx')^ {mdy-j-m'dy')- (mrfz-j- m'rfz'p
f*°°'"l dt^ dT^ ■ dV^ '
it produces the second line of [4007o] ; observing, that
??i^ d x^ -{- m' - dx'~ — ( m d x -\-m' dx'Y = — 2 m m'. dxd x', he.
The first and second terms of the third line of [1198] produce, without any reduction, the
[4007;i] third line of [4007o], and the last term of [1198] gives the last of [4007o], using
X [4007^] ; hence we have
constant = ( M+ m!).m. ■ ^ -* J^ ' -\- {M-\-m) .m'. ^ ^ ^
[4007o]
[4007p]
_ , Crfxrfx' , dvdy' , dzdz'")
-2mm'.^-^^ + -^ + ^
+ 2 . ( JIf + m + 7n') . m m'. X.
Tr 1-11 1 r dx'i + dif- + dz^ rfa:'2 + rfy2 + (/i'2 , ^^ ^ .
If we multiply the values of /^^ — , f,o [1199,1200], by
[M -\- m') . m, {M-{'m).m', respectively; and add the products, we shall get, for
the first line of the second member of [4007o], the following expression,
(^i+^')..,.^^J^,2/diï^ + (J^f+>»)•m^f•^•7^'"'^-2/d'i^|
VI. ii. ^^ IT.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 151
formulas of the elliptical motion ; they must therefore be added to the same
If we substitute this in [4007o], we shall find, that the term having the divisor r, is
2 m
— . \ {M + m') .{M+m) — {M+m+ m') . 31], [4007p']
which, by reduction, is ; and in like manner, the term depending on r , is — ;; — ;
so that if after this substitution is made, we divide the whole expression by 2, and transpose
the tenns depending on d -R, d'R', we shall obtain the following equation, in which
nothing is omitted, the constant quantity being included in the signs /,
( M + m') .m..fàR + {M-\- m ) . m'.fd' R'= m m. 0 + ^)
, /dxdx'-\-dydy'-\-dzdz'\
- '" '" • V dt' — ) ^^^^^^^
-{- {M + m -\- m') .mm'.y-..
We must now consider the terms of this equation affected with the small divisor 5n' — 2n,
and ha\Tng 5 n't — 2nt for the argument ; these temis being the only ones which can
acquire the di\Tsor (5»' — 2n)^ by another integration in J'fdR, ffd'R', or in [4007)-]
the expression of the longitudes of the two planets [3715/, »*] ; and in making this
investigation, we shall reject all terms of the order in'*. In the first place, we shall
observe, that the expression in the second line of the second member of [40075'] ^'^^^ ^'^^
contain such tenns of the order ??i^, as is evident from the reasoning in note 819 [1201'], [4007«]
where it is sho^^Ti, that these terms of the order ?«^, arise fi'om the substitution of the
elhptical values of x, x', y, ij , &c. ; and to obtam terms of the order »i', we must augment
these elhptical values of x, x, Sic. by the terms depending on the perturbations. These
terms may be easily obtained by considering the orbits as variable ellipses, in which we may
suppose X, x', to be of the forms,
x = ^1 -f- 5i . cos. (n / + Ci) + &:c. ; [4007<]
x' =: Ay + B.2 . COS. ( n't -)- Co) -j- &c. ; [4007k]
Ai, B^, Ci, &c., c/^2, Bo, Cj, &.C. being functions of the elements of the orbits.
These elements for the planet Jupiter are ; the mean longitude of this planet nt -\- e;
E the mean longitude of the epoch ; a the semi-transverse axis of the ellipsis ; e the
excentricity ; « the longitude of the perihelion ; y the inclination of the ellipsis to a fixed
plane ; and è the longitude of the ascending node. The same letters being accented, [4007u"]
represent the corresponding elements of the orbit of Saturn. In the values of all these
elements, the secular inequalities are supposed to be included. The differential of the
expression [4007/, u], bemg found as in [1168'], become
dx = — B,.ndt. sin. (n / + C^) — &c. ; [4007t.]
dx'= — Br,.n'dt. sin. {nt-{- Co) — &c. [4007w]
[4007u']
152
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
quantities in the development of R. Let
[4007] R^H. COS. (i' n' t — int + A),
[4007a-]
[40086]
[4008i']
[4008c]
[4008rf]
[4008e]
[4008/]
Tlie product dx dx', will therefore contain only periodical quantities of the form,
H . cos. {in't — int-\-E);
H, E, being functions of the elements of the orbits ; and i', i, integral numbers, positive
or negative ; moreover n't, nt, in the planetary system, are incommensurable quantities
[1197"]. Now if we consider the elements as variable, their variations, corresponding to the
great inequalities of Jupiter and Saturn, will have the same argument as these inequalities,
[i007y] namely, 5 n t — 2nt, and they have 5 ?i' — 2 n for a divisor, as is evident from what
we have seen in [1197, 1286, 1294, 1341, 1345'], or more completely in the appendix to
this volume [5872 — 5879]. Substituting these variations in [4007x], and reducing by
[17 — 20] Int., we shall obtain terms having this divisor; but it is evident, that they will
[4007z] not have the same argument, except z' = 10 and i = 4; in which case /J" will be of the
order e^ [957^''', &,c.], which is neglected, because we notice only terms of the third order
relative to the excentricities e, e', and of the same order relative to the masses in, mf.
[4008a] The same remarks may be made with regard to the products d y dy', d z d z' ; hence we
conclude, that the fonction included in the second line of [4007^] does not contain terms
of the order n? or it?, which has for its argument 5h7 — 2)i<, and for divisor 5/i' — 2?i;
so that we may substitute, in [40075], ^^^ following expression.
■mm.
dxdx'-\-dy dy'-\-dz dz'
0.
In the fonction comprised in the third line of [4007 (^], namely, (./li-f-w + ?»') . mm'. X,
we may change the factor M -{- m-\-7n' into ./If ; it being evident, that the neglected
quantities do not comprise terms of the order m^, having the argument 5 n' t — 2nt
and the divisor 5n' — 2n. Then substituting, in X [4007^], the elliptical values of x, x'
[4007<, u], and the similar values of y, y', z, z' ; it becomes, by development, of the form,
■k = A-\-K.cos. {5n't — 2nt-J[- I) + Q.,
in which A represents the part depending on the argument zero, and Q all the terms
depending on angles of the form i'n't-\-int, i', i, being integral numbers, positive or
negative, excluding those pi-oducing the argument 5 n't — 2nt, which is connected
with K, and the argument zero connected with A ; hence we have
(.W + w + m) .mm! .\ — M . m-rri .\A-\- K . CQ's,.{'ô'){ t —2nt ^ I) A^ q}.
The quantity mm'.— [4007<7], is of the third order in 7n, in, and as the value of r
[4008g-] contains no term having the divisor 5 ?i' — 2n, except it be of the order ??/, we may
neglect this term, because it produces nothing except of the order m"" ; and the same is to
VI. ii. §1~] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 163
be any term of this development ; and
6v = L. sin. (i'n't—int-\-B), [4008]
VI
be observed relatively to m m'.—. Substituting these and [40086,/] in [4007^], we get
M. { mfdR + m'fd'R' } + m m'. \fdR +/d'jR' \r:^M. m m'. \ A+K. cos. (5 7t7 — 2 n ï +/ ) + Q ^ [4008A]
We shall represent by (R), {R'), the parts of R, R', respectively, of the order m ; [4008t]
then using the characteristic (S of variations, we shall put àR, ôR', for the remaining parts
of the same quantities of the order mP, &lc., and we shall have
R={R)-\-5R, R'={R')~\-6R'. [4008i]
If we also put [(.2) + (^) .cos.(5?i'<-2n^-|-/)] for the part of .^+Z:cos.(5n'i;— 2n<+/), [4008J]
which is independent of m, m ; and prefix the sign <5 before the same quantity, to denote
the remaining part, we shall have
^ + Z.cos.(5 7i'< — 2«ï' + /)+ Q=[(.^) + (^).cos.(.5îi'^ — 2n<+/)]
-\-&.\A-^K.cos.{biït — ^nt + I]+q.
Substituting [4008fc, m] in [4008/(], and neglecting the terms mtn'.fd5R, nim'.fd' 5R',
which are of the order m^ ; also the terms M.mm. Q, because the integration does
not introduce the divisor 5 ?i' — 2 7i, we get
M.\mfd{R)-^m'.fd'{R')\-i-mm'.\fd{R)-^fd'{R')]-JrM.\mfd5R-Jrm'fd'ôR'l
=M.mm'.[{A) + {K).cos.{5n't—2nt+I)]-JrM.mm'.S.{A-\-K.cos.{5n't—2nt-^l)].
Now equating separately the parts of this equation, which are of the order m^, and those
of the order m?; putting also M=^l [-3709], in terms of the order m^ we get
M. \m. f d{R) +m'.f d' {R)\ = M. mm'. [{A) -{-{K). COS. {57i't — 2nt-\-l)']; [4008p]
mm'.{fd{R) +/d' (R) ] + m ./d 6 R + m'./d' ÔR'=mm'.5.\A -^K. cos. (.5 n' ( — 2 n t+I) | . [4008?]
14008m]
[4008?»]
[4006o]
[4008r]
If we neglect the terms of the second member of [400S»/], or in other words, the terms
of the elliptical value of X, depending on the two arguments zero and bn't — 2nt, we
shall have the following expression [4008s], which includes all the arguments except these
two ; and is accurate both as it regards terms of the third order of the masses m, m', and
of the third order relative to the excentricities and inclinations,
m m'.\fd {R) +/d' {R) \ + m.fd8R-{- m'.fd'ôR r= 0. [4008s]
Substituting M=^ 1 [4008o] in the product of [4008p], by the quantity m', we get, by
neglecting terms of the two forms 0 and 5n't — 2nt [4008;-], mm'fdR-^m'-.fd'R'^^O.
Subtracting this from [4008s], we obtain
m.fdiR-{- m'./d' Ô R' -{- {m — m') . m'./d' R' = 0. [4008«]
VOL. III. 39
[4008<]
154 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
the corresponding inequality of Jupiter.* If we increase 7it, n't, by their
great inequalities in the expression [4007], there will result in ii a
term of the form,t
[4009] R = ±qH. COS. {i' n't — i7it±{5 n't — 2nt) + A±E).
and since a'~ n = a' '~ n' = 1 [3866»] , neglecting terms of the order m, this may be
put under the following form, terms of the order in* being neglected,
[4008t;] m «* n.fdSR^ m'. a' ^ n'.fd' 5 i?' + ( m — m') . m', a' ^ n'.fd'R' = 0 .
Now if we put ^, ^', for the great inequalities of Jupiter and Saturn; S^^, S^^', for the
[4008t)'] parts of i^, ^', depending on dôR, d'SR'; or in other words, those which depend on the
combinations [4006m], excluding the angles zero and 5 n't — 2nt, we sliall have,
as in [.371 5Z, m],
[i008w] S^^ = 3an.ffdt.d&R; S^ ^'=3 a' n'.ffd t .d'5 R' ; ?,'^3 a' n'.ffdt .d'R
lastfoT*^" Now multiplying [4008?;], by 3dt, integrating and substituting [4008 w], we get
mula,
which
[4008.r] m /a • 5, ? + '«' /«'• 'I ■? ' + ( »* — '»') • ™'- /«'• ■? ' ^ 0 5
inctudoa
terms of wMch IS the last formula of La Place, proposed to be demonstrated in [4007^^ ; and the
the order
trfi. complete values of (S, ^ , (5^ ^ ought to satisfy it ; so that if one of these quantities be
rifioR 1 accurately computed, the other may be deduced from it ; but the usefulness of the theorem
is restricted by the circumstance, that it can only be applied to the results obtained from all
[4008z] the sensible terms of this kind, taken collectively; or to all the terms corresponding ic
each of the six factors e', e^ «', e e'^, e'*, ey^, e' y^.
* (2525) The relation between R and 5v is expressed by the equation [.37155].
A particular case of this formula is considered in [3703, 3715], in wliich
[4009a] R = M. cos. ( m,t + K) [3703, 371 Irf] ;
[40095]
and we find, by mere inspection, that the third and fourth terms of uv [37155] have, as
in [3715A], the divisors m^, m^ ; also by comparing [3702, 371 If], we find, that the
terms of hv [37155], depending on hr, have the divisor mf — ?t^, or ?;?, ±h- It is
[4009c] easy to generalize this result, as in [4010], where lUi^i' n' — in.
t (2526) If we increase n't by the great inequality of Saturn [3891], and nt by that
of Jupiter [3889], the angle i'n!t — int, which occurs in [4007, 4008], will be
increased by a quantity, which we shall represent by p ; then putting, for brevity,
[4012a]
Ts=bn!i — 2nt-\-b e'—2i; — i'H'. cos.A'—iU. cos.^= 2f/ .cos.c ;
— i'H'.sin.J'— 2'il.sin.^=2^.sin. c; 5 e' — 2 s -f c = £.
[4011]
[4012]
[4012']
VI. H. § 17.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 155
Now tlic series of operations, which connects H and L, gives to the parts
of // the divisors (/'n' — inf, i' n' — in, i' n' — in±n [40096, c] ; [4010]
and the same series of operations gives to the inequalities corresponding
to the parts of R [4009], the divisors* [i' n' — in±:(5n' — 2n)}-,
i'n'—inzt(5)i' — 2n), i' n'—in± {bn' — 2n) ±n. If i'n'—i7i
or i' n' — inztn be not small quantities of the order 5n' — 2n, we
may neglect 5 n' — 2n in these divisors,! and then the inequality,
corresponding to
R= ±qH. cos. [i' n' t — int±(5n't — 2nt) + A^E}, [4013]
will be
ÔV = ±qL . sin. { i' n' t — i n t ± (5 n' t — 2 7it) -^ B ±El ; [4014]
we get, successively,
p = —i'H'. sin. ( n + ^') — i H- sin. {T, + ^) [40126]
= — i' H'.\sm. Tj . COS. J'+ COS. T^ . sm.Â'1—i H. |sin. T^ . cos. 7l + cos. Tg . sln.^|
= 2«j'.{sin.T5.cos.c-[-cos.T5.sin.c^^25'.sin.(T5-j-<^) = 2q.sm.[bn't — 'ilnt-\-E). [4012c]
If we increase the angle i' n' t — int-{-Jl [4007] by the quantity p ; then develop the
expression by means of [61] Int., we shall obtain an additional term of the order p, and
represented by — p H .s\n. {i' 7i't — int-\-A). Substituting in this the value of [4012«i]
p [4012c], and then reducing by [17] Int., it becomes, as in [4009],
qH.cos.{in't — int-{-{57i't—2nt)-j-A-\-E]—qH.cos.\i'n't—int — {5n't—27ii)-\-A—E\. [4012e]
* (2.527) The coefficient of t, in [4007], is i' 7i' — in, and from this arise the
divisors [4010] ; but in the term [4009], this coefficient is augmented by the quantity
±(5 7i' — 2)t); which requires a corresponding increase in the resulting divisors [4010]; [4014o]
by this means the divisors [4010], depending upon the term [4007], change into those
given in [4012]. If we suppose 5 7i' — 2 ?i to be very small, in comparison with [40146]
i' 7i' — t?t or i' 7i' — in ±71, we may neglect it ; and then the chain of operations
connecting H, L [4007,4003], will have the same divisors as that connecting q H, q L [4014c]
[4013, 4014]. Now [4007] is changed into [4013], by multiplying by ± ?, and
augmenting the angle i'n't — int by ±{57i't — 2 7it)zizE. Applying the same [4014rf]
process of derivation to [4008], we get the corresponding inequality of Jupiter, as
in [4014].
t (2528) In restricting the formula [4014] to the terms mentioned in [4006'], we
5,j' 271
may consider the part which is neglected in [4012'], as of an order , or j\ of
that retained [3818fr] ; so that the error of the terms ôv [4014] is of the order ^^qL;
156 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
which is the same as to increase nt, n't, by the great inequalities in the
term of àv [4008].*
We must also increase, in the terms depending on the first power of the
excentricities, the quantities e, e', -us, ra', by their variations, depending
[4016] upon the angle bii!t — 2?i i ; but it is evident, that this will not produce
any sensible inequalities.!
18. The coefficients of the inequalities of the planets vary on account of the
Manner of ^ ^ n i * t ' • i • • i
ihflffirA secular variations of the elements of their orbits : we may notice this in the
secular followinq manner. We must first put the inequality relative to any angle
variations *■
^lemans. *' *^' t— i^t, undcr the form t
[4017] P. sin. (i' n't — int + i't — is)-\-P'. cos. (i' 7i' t — in t + i' s' — is).
and as rj is of the order ^p [4012c], it becomes of the order ^l^p L. Now the great
[40156] inequalities of Jupiter and Saturn being nearly 1265', — 2957', [44.34, 4474], the quantity
2) [4012ff] becomes — 5 X 2957'— -3 X 1265' = — 18580% or about y^- of the radius ;
r4015c] consequently the quantity -j-^^pL is less than tïs ^ tV -^' °'' ^^^^ than y J^ij L ; and
the error of this computation of i5 y [4014], arising from this source, will generally be less
than ■j^jjjy of the inequality [4008], which is under consideration.
* (2528«) If we increase n'i, nt, by the great inequalities, using j; [4012J], the
expression 6 v [4008] will become S v z= L . sin. [i' n't — i ni -{-B ~\- p). Developing
[40]5(/] this as in [60] Int., we get ôv = L.sm.{{'n't — i nt ~{- B) -j-jiL. cos. [i' n't — int-j-B).
Substituting j} [4012c], and reducing by [19] Int., it becomes equal to the sum of the
two expressions [4008, 4014].
t (2529) The smallness of these terms may be seen, by a rough examination of the
increment of the value of R [1011], arising from the introduction of the part of c oi ô e
[4016a] [1286], when we put ?:'==5, z = 2, a=l, " := 74 [3818f/], m'=^-J^^,
e = 0,05 [4061rf, 4080] ; observing that as i' — i==: 3, ^- [1281'], may be considered as
of the order e^ and (~) of the order e^ ; so that 5e [1286] may be considered as of
[40106]
[4017a]
the order 74 m'. e^. cos. (5 n'< — 2nt-\-A}, or ^i^ c . cos. (5 ?i'i — 2nt-\-J) nearly.
Consequently this increment of e produces terms of the order y^i^, in comparison with
those depending on e, in [4392], none of which amount to 200'; hence it is evidenti
that these terms are insensible.
X (2530) The form assumed in [4017] has been frequently used, as, for example,
in [371 li].
Vl.ii.-^IS.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 157
We must determine the values of P, P\ for the epoch 1750, and then put
tang. A = ^ ■■, L = ^/W+pr^ ; [4018]
the sign of sin. A is the same as that of P', and its cosine is the same [4018]
sign as that of P [401 9f?] ; then the proposed inequality will be*
L . sin. {i'n' t — int^ i' /— i s + J) . [40i9]
We must determine the values of P, P', for 1950, noticing the secular
variations of the elements of the orbits ; and we shall have for this
inequality, in 1950,
(L + <5 L ) . sin. (i'n't—i nt + i'e — is-\-A + 6A). [4020]
If we denote by t the number of Julian years elapsed since 1750, the
preceding inequality relative to the time t will assume the following form,t
Çl + ^-^^ . sin. U'n't — int + i' s'— i t -\-A +
200
[4021]
Under this form it may be used for several centuries before and after 1 750.
But this calculation is not necessary except with those inequalities which
are quite large.
In the two great inequalities of Jupiter and Saturn, it will be useful to
continue the approximation as far as the square of the time, in the part
[4021']
* (2531) Using, for brevity, i'71't — int-\-i's' — ie=^Tg; then developing [4019] r^Q^g^-,
by means of [21] Int., and putting the expressions [4017, 4019] equal to each other,
we get, identically,
P. sm. Tg + P'. COS. Tg = L. sin. {Tg-^A)=L. cos. A . sin. Tg + L. sm.A. cos. Tg. [40196]
Comparing the coefficients of sin. Tg, cos. Tg, separately, in both members, we get
P = L.cos.Jl, P'=Z,. sin. ^. Dividing the second by the first, also taking the sum [4019c]
of their squares, we get [4018]. The quantity L being considered as positive, we [4019(f)
get, from [4019c], the signs of sin. A, cos. A, as in [4018].
t (2532) If 5L, &A, represent the variations of L, A, in 200 years, between
i.bl. t.SA
1750 and 1950; then their variations in t years will be represented by "^^j '2ÔÔ ' '■ ^^^
respectively. Substituting these in [4020], it becomes as in [4021].
VOL. III. 40
158
PERTURBATIONS OF THE PLANETS,
[Méc. Cél.
[4022]
[4022']
Great in-
equality
of Jupiter,
reduced to
a tabular
form.
which has the divisor {bn! — 2nf. This part of the expression of àv
is as in [3844],
\aP'-
2a. dP
Sa.ddP'
ûv =
6 m'. n~
[5n'-2n).dt (5n'— 2n)2.rf(2
'.sm.{5>i't—2ntJ^5i — 2s)
{5n'—27if
— ^aP-
(OJI'
2a. dP' 3a.ddP }
:—2n).dt (an—2n)-.dl^^ ^ '
the values of P, P', and of their differentials, being relative to any time
whatever /. By developing them in series, ascending according to the
powers of the time, and retaining only the second power, and the first and
second differentials of P, P', the preceding quantity will become*
[4023] 5i, = -
6 ml, rP'
(5n'-2n)2
2a. dP
Sa.ddP'
(5n'—2n).dt {5n'—2nf.dt^
( dP' , Oa.ddP 7 , , , aap,>.^in.{5n't-2nt+5^-2s)
I dt ^ (5n'
2a. dP'
;-2H).dri\'-
i).dl-i)
Sa.ddP
{5n'—27i).dt (5n'—2n)KdtZ
, C dP 2a.ddP' ) , , ,
' I dt {5n'~2n).dt^^~ -
dfi
ddP\
dt^
>.cos.(5re'<— 2n<+5s'— 2s)
* (2533) The values of P, P', and their difFerentials [4022], must be computed for
the particular time t, for which the value o( 5v is wanted ; but this is an inconvenient
method; therefore the functions by which sin.Tj, cos.Ts [3842a], are multipHed in [4022],
[4022al ^''^ developed in [4023] in series, ascending according to the powers of t. This is done
by means of the formula [oS50«], neglecting i^, and the higher powers of t. Thus,
if we put the factor of sin. Tj, included between the braces in the first line of [4022],
equal to u, and take its first and second differentials, neglecting the differentials of the
third and higher orders ; we shall get the following values of U, and its differentials ; in
which the terms in the second members correspond to the epoch < = 0 ;
[40326]
[4022c]
[4022d]
U = aP'
2a. dP
Sa.ddP'
/dt
\d
dU
T
(5n'-2n).dt (5n'-2n)2.rf<a '
dP' 2a.ddP /ddU\ a.ddP'
dt
' (5n'-2 7i).dV2'
/ddU\
dfi
Substituting these in [3850a], we get for u, the same expression as the factor of sin. Tj,
in the first and second lines of [4023]. In the same manner, the factor of cos. T^, in
the second line of [4022], produces the corresponding factor, in the third and fourth
lines of [4023].
VI. il. § IS.] DEPENDING ON THE SQUARE OF THE DISTURBING FORCE. 159
The values of P, P', and their differaitials, correspond to the epoch
of 1750, and are determined by the method in [3850, &c.] ; the other parts
of the great inequality of m being rather small, it will be sufficient, by
what has aheady been shown, to notice the first power of the time. This
great inequality will then have the folloAving form,
[4024]
ôv= (A +5 t + Ct^). sin. (5n't — 2nt + 5 s'— 20
+ (A'+B't + C't") . cos. (5n't — 2nt + 5 /— 2 0 .
We may also put the great inequality of m' under the same form, by which
means it will be easy to reduce these inequalities into tables.
If ice wish to reduce the preceding inequality to one term, loe must calculate
it for the three epochs 1750, 2250, 2750. Let
f3 . sin. (5n't — 2nt + 5 b'— 2s + a) [4025]
be this inequality in the year 1750; and |3^, a,; (3,, a„, the values of p, a [4025]
at the epochs 2250, 2750 ; then the inequality corresponding to any fquaiuyof
time whatever t, will be* reduced
/ ds , -, „ ddP;\ . ^_ , _ , ,~ , ^ . . dA , , „ ddA )
the differentials p and a correspond to the epoch in 1750; and we shall
have, by [3854— 3856], f
to one
term.
[4026]
d^ 4 3,— .3(3 — p,,^
dt 1000 '
dd^ p„— 2(3,+ 3_
dt^ 250000 '
[4027]
d\ 4 a,— 3 a — A„ ,
dt ~ 1000
dd\ A„ — 2a, + a
dt^ ~ 250000
[4027']
* (2534) p and A being functions of t,
we shall have, as in [3850«],
e4-t '^^^^t^ '^'^^
and
A _i_y ^^ J^ii2 '^'^^
[4025a]
for their values ; using for p, A, and their differentials, the values corresponding to the
epoch in 1750. Substituting these in [4025], it becomes as in [4026].
t (2535) If in the general formulas [3854—3856], we change P, P,, P„, into
|8, 3,, p„, the expression [3854] will become like the first of the functions [4025a] ; [4027o]
J , , . , d 13 ddp
and by making the same changes in [3856], we shall get the values of — , —
160 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
In conformity to the remark we have made in [3720], these two great
[4027"] inequalities of Jupiter and Saturn must be applied respectively to their
mean motions.
[4027]. In like manner, by changing, in [3854—3856], P, P,, P„, into A, A,, A„,
the formula [3854] will become as in the second of the functions [4025a], and [3856]
[4027c] will give the values of —, — - [4027'].
VI.iii.§18'] DEPENDING ON THE OBLATENESS OF THE SUN.
161
CHAPTER III.
PERTURBATIONS DEPENDING ON THE ELLIPTICITY OF THE SUN.
18'. Since the sun is endowed with a rotatory motion, its figure will
not be perfectly spherical. We shall now investigate the effect of its
ellipticity on the motions of the planets ; putting
p = the ellipticity of the sun, expressed in parts of its radius ;
q = the ratio of the centrifugal force to the gravity at the sun's equator ;
(X = the sine of the planet's declination relative to the sun's equator ;
D = the sun's semi-diameter ;
1 = the sun's mass, usually called M ;
R = {?-\fi)-^'i^'-\)'
Symbols.
[4028]
then it will follow, from [1812], that the sun's ellipticity adds to the vaiuoof
function R [913], the quantity* dependine
on the
ellipticity.
[4029]
* (2536) We shall suppose m', m", ??i"', &c. to represent the particles of the sun's
mass ; considering it as being composed of concentrical elliptical strata of variable densities,
symmetrically arranged about its centre of gravity, taken as the origin of the co-ordinates
of these particles x', y' , £ ; x", y"
&ic. The co-ordinates of the attracted planet m
being represented by x, y, z, and its distance from the sun 7-=\/(.r^-j- )/^-f-c^). In
this case, the expression of R [91.3] will be reduced to its last temi 7?= — — ;
{xx'+yy'+zz')
any term of the form
because
depending on the particle m', whose co-ordinates
are x', y', s^, is destroyed by a similar term, depending on an equal particle m', whose
co-ordinates are — x', — y', — 2'. Substituting, in [4029è], the value of X [914],
VOL. III. 41
[4029a]
[40296]
[4029c]
162 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[4030] If we notice only this part of R, and put fdR = g-^R; g being a
constant quantity ; we shall find, that the differential equation in r 6r
[926, 928'] becomes, by neglecting* the square of tJ.,
[4031] 0=-^^^ + -^^— + 2^+^ ^ .t
neglecting terms of the order m' m", and using the sign / to represent the sum of the
[4029dl terms depending on all the particles, we get iî = — C'TTr', ,„ , , , ,„ , , , —.
This expression of R corresponds to that of — V in [1385"', 1386], m' being the attracting
particle, and \/ \ {x' — a,)^ -\- {y — y)^ -{- {z' — z)^ } its distance from the attracted planet ;
hence iî = — V; and by substituting the value of V [1812], we get
[4029e] -« — —,.— ra •JU.[iJ. s).
The last term being multiplied by D^, to render it homogeneous with the first, because
in [1812, 1795"], the semi-diameter of the body M is put equal to unity, and here it is
[4029/] supposed to be D. Again, by comparing [1670', 4028], we get a.(p = q; also by
comparing [1801, &lc., 4028], we get o.h = p. Substituting these in [40296], we obtam
[4029^] -^7 r3 -M-ii^—s).
Now if the sun were of a spherical form, with no rotatory motion, we should have
M
[4029A] P = 0, 7^0, and then J? = — — [4209^]. Subtracting this from the general value
of R [4029^'-], we get the part of it depending on the sun's ellipticity, namely,
[4029i] R^_^Illl^t^.M.ii^^—^),
and by putting, as in [4028], the sun's mass Jlf = 1, it becomes as in [4029].
* (2537) The inclination of the sun's equator to the ecliptic is less than 8'^, and its sine
[4030a] j^ ^g^^jy ^^ g^ ^j^^j ^2 „-j„st be less than {if, or ^3-; which may be neglected in
[4030t] comparison with ^ 5 and then [4029] becomes R = — ^.(P — s?)-"^-
t (2538) Substituting, in [926], the value of rPt'=r. (— ) [928'], also i^=n^a^
,-100], we get
d2.(,v5r) , ,fia?.rSr , „^,„ t /dR\
[40316] 0=-±^ + ^^~ + 2fàR + r.(jy).
[4031a]
^ ^ [3700], we get
Now the value of R [4030&], depending on the sun's ellipticity, gives
[4031c] fdR^-i.{?-hq)-D'-fà.'^- = -h{p-ii)-^+g; '••C^)=(''-*?)-Tr^
VI.iii.§lS'.] DEPENDING ON THE OBLATENESS OF THE SUN. 163
To determine the constant quantity g, we shall observe, that the formula
[931] gives, in àv, the quantity*
3a.ngt + {^ — h(]) ' — .lit; [4032]
a'
n t denoting the mean motion of the planet ; this quantity must be equal
to zero ; therefore we have
ST = ^ ^ .
^ 3 «3
Hence the differential equation in r&r becomes, by neglecting the square
of c, and observing that n-a'^^=\ [3709'] ,t
+ ^l^Mï , n~. Z)-. { 1 + 3 e . COS. (n t + i — ^)].
but from [4031c], we get
a?'
3a/diî + 2ar.(^)==3«i^ + (p-Aî).^ = 3«^ + (p-H)'
[4032']
[4033]
[4033']
[4034]
substituting tliese in [40316J, we get [4031]. We may observe, that the symbol (J- [4031a]
is entirely different from that in [4028].
* (2539) The constant quantity g is to he found, as in note 699, Vol. I, page 550, by
putting the terms of [931], multiplied by t, or rather by ^oZT^)' equal to nothing.
These terms are evidently produced by the two last terms of [931],
3 afn dt.fdR + 2afndt.r. 0^^ ; [4032a]
[40326]
noticing merely the term a of the value of r, which is evidently the only part which affects
the coefficient of t, now under consideration. Multiplying this last expression by ndt,
and integrating, it becomes as in [4032], which represents the part of ô v, connected with [4032c]
the factor t. Putting this equal to nothing, we get [4033].
t (2540) We have r = a.\l—e. cos. ( n < + s — w ) } [3747], neglecting e^ ; ^^^g^^,^
hence we get, by using [4033'],
i = 1 .n + 3 e . cos. (ni4-s — zi)\=^nm + 3e. cos. (n t + t—z,)]; [4034i]
substituting this, and g [4033], in [4031], we get [4034].
164 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
This gives, by integration,*
[4035] ^ = i.(p — 19).^.^ — 3e.«^.sin.(«< + f — «)}.
The elliptical part of - is 1 — 2 e . cos. (nt + s — zi) [3876a] ; and
if we suppose w to vary by 6^^, we shall have [3876f/],t
[4036] -g- =r — eô'a . sin. (nt-{- s — w).
* (2541) This integration is made as in [865 — 871"], putting rSr^^y'; hence [4034]
becomes, by connecting together the terms depending on e,
[4035a] o = ^+n^.y'—i.{p—hq)-n^-D^+\n^y'-\-i.{p—iq).7v'-D^.3e.cos.{nt-\-e—:;!).
[40356] Putting y'^y-]-^.(p — ^q).D'^, and neglecting the term of the order ye, or e^, we get
[4035c] 0 = -j^l^ n^ y + 2 . {f ~i q) . 71^. D^. e .COS. {n t -\- s — zi);
[4035rf] which is of the same form as [865a, 870', 871'], changing a or m into n, s into s — w,
a -fir into 2.(p — ^q)n^. D~.e, and then [871"] becomes
y = — ^ — .sin.(?t < + £ — «) = — (p — iq) -711 .D^.e .sin. Çnt-\-e — to) ;
substituting this in y' or rSr [4035i], we get
[4035e] r5r = i.(p — i<?).D2 — (p — iy).ni.I32.e.sin. (n^-|-£ — to);
dividing this by «^, we obtain [4035]. We may remark, that the term of the form
air. cos. {nf-{-i — to) [871'J is included in the elliptical motion, and it is not necessary
to notice this term in the present calculation.
V Ô r
■f (2542) Comparing together the expressions of — j- [3876^, 4035], we find, that
if the coefficients of sin. (7it-\-s — to) be put equal to each other, we shall get
D-
[4036a] — e ^5 TO = i . ( p— i Ç ) . — . ( — 3e .7it);
whence we obtain 'îis, as in the first equation [4037]. The second expression [4037] is
deduced from the first by the substitution of n = a ^ [3709']. Again, since the
formula [4035] does not contain a term depending on n t . cos. {71 i -\- e — to), and
[4036c] in [3876] this cosine is connected with the factor ôe, we shall have (îe = 0. The
VI.iii.§18'.] DEPENDING ON THE OBLATENESS OF THE SUN. 166
If we compare this expression of %- with the preceding, we shall obtain ^^^ ,^^
"" of the
perihelion f
rjo T)- f arising
6^=.(p_i9).^.nï = (p-è7).-^ [4036«,6]; [4037]
W ^ from the
Ct oblateneai
of Ihe
3UU, is
therefore the most sensible effect of the ellipticity of the sun, upon the motion '"«nsibie.
of a planet in its orbit, is a direct motion in its perihelion ; but this motion [4037']
being in the inverse ratio of the square root of the seventh power of the
greater axis of the planetary ellipsis, îve see that it cannot be sensible except [4038]
in Mercury [4036/],
To find the effect of the sun^s ellipticity upon the position of the orbit,
we shall resume the third of the equations [915]. This equation may be
put under the following form,*
d,lz n^a^.z , f(lR\
^^dr-^-^ + yiû)' f4039]
2:2
We shall take the solar equator for the fixed plane, which gives n^= — ^ [4039']
[4040fl] ; then by observing that r = x^ + ^/^+z', we shall havef
— j = 3.(p — i9).-^^5— .3; [4040]
constant part of —3- , which is nearly equal to that of — , is represented in the present
case by the first term of the second member of [4035] ; so that we shall have
'i^i.{9-iq).^, [4036^]
as in [4042]. Now we shall see, in [4262 — 4265'], that if the sun be homogeneous,
we shall have, for the orbit of the planet Mercury, 5j3 = (p — ^q) . — .<=0',012.? nearly [4036c]
[4265] ; and this expression is much smaller for the other planets, on account of 4he divisor a^ ;
so that it produces only 12°" in a thousand years for Mercury, and is much less for the other [4036/"]
planets. The quantity 5 r [4036'/, 4260 — 4263] is evidendy insensible.
* (2543) Substituting i>.^n^ a^ [3700] in the third equation [915], it becomes [4039a]
as in [4039].
t (2544) In [4028], (a is put for the sine of the planet's declination above the plane [403951
of the sun's equator, its perpendicular distance above this plane being z, and its distance
VOL. III. 42
166 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[4041]
hence the preceding differential equation becomes*
dch
now by what precedes [4036f/], we have
[4042] ^=i.(p-ly).^^;
hence we obtain
[4043] 0 = ^" + n^..|l+2.(p-ic).^'|.
This gives, by integration, f
[4044] z = ip .sin. <nt . (1 + (p — I q) • ~^ ) — 4?
[4045] - being the inclination of the orbit to the solar equator,! and d an arbitrary
z
[4040o] from the sun's centre r; hence we evidently have (a^-; also r = y/(x^ -j- y^ 4" ~^)
[914']. Substituting this value of fA in [4029], we get
[4040i] ij_(p_iç).D2.^E:__L^.
/ d r\ z
Taking its partial differential relatively to z, neglecting z^, and observing that i—]=-,
we get
[4040c] (^) = (p-i<?).i3^.^?| + i^ = 3.(p-i<?).f.z.
1 1 _ 7l2
î"5 «5 o2
1 1 n~
Retaining only the constant part of r, we may put - = — = — - [3709'], and then the
preceding expression [4040c] becomes as in [4040].
* (2545) Noticing only the terms of r, depending on the sun's ellipticity, we may put,
[4041a] as in [4036c/] , r=za-j-5r, whence -==-.n '^j. Substituting this and [4040]
in [4039], we get [4041] ; and if we use [4036(^], it becomes as in [4043].
t (2546) Comparing [865', 4043], we get y = z, a = n .U -j- {p — ^q) . — ^,
" by neglecting (p — hlT- Substituting these in the first value of y [864a]; changing
also b into <p, and (p into — ê, we get [4044].
X (2547) The sine of the declination is equal to - [4040a], and its greatest value
[4045a] is equal to - [4044] or - nearly ; which evidently represents the sine of the
inclination of the orbit to the solar equator.
VI. iii. § 18'.] DEPENDING ON THE OBLATENESS OF THE SUN. 167
[4045']
[4046]
constant quantity. Tims the nodes of the orbit on this equator have a
retrograde motion equal to the direct motion of the perihelion, and which
cannot therefore be sensible, except in the orbit of Mercury* At the same
time ive see that the sun'' s ellipticity has no influence on the excentricity of the
planeCs orbit [4046f ], or on the inclination of this orbit to the solar equator ;
it cannot therefore alter the stability of the planetary system.
* (2548) It is evident from the form of the angle, which occurs in [4044], that the
D-
retrograde motion of the node in tJie time t is represented by nt . {^ — J?)--t7) [4046a]
because the body is in the node wjien c = 0, and it completes its revolution, to the
same node, while the angle nt -{-nt .{if — è <p) . -5- increases by 360''; the mean [40466]
periodical revolution being performed in the time t, which makes nt = 360'' [4032'].
Hence it is evident, that the retrograde motion of the node in the time t is nearly equal to
the difference of these quantities, as in [4046a], being the same as the direct motion of the
perihelion [4037]. As (5e = 0 [4036c], the excentricity is not affected by the sun's [4046c]
ellipticity, neither does it affect the inclination - of the planet's orbit to the sun's equator
[4045a], which is constant, because ç is one of the constant quantities obtained by integration.
The results found in this chapter agree with those found by Mr. Plana in the Memoirs
of the Royal Society of London, Vol. II, page 344, &c., noticing the term neglected by [4046rfl
La Place in [4030] ; makmg also the computation directly from the formulas [5788—5791],
and carrying on the approximation to a rather greater degree of accuracy.
168 PERTURBATIONS OF THE PLANETS, [Méc. Cél.
[4047]
CHAPTER IV.
PERTURBATIONS OF THE MOTIONS OF THE PLANETS, ARISING FROM THE ACTION OF THEIR SATELLITES.
19. The theorems of ^10, Book II [442", &c.], afford a simple and
accurate method of ascertaining the perturbations of the planets from tlie action
of their satellites. We have seen, in [451', &;c.], that the common centre
of gravity of the planet and its satellites, describes very nearly an elliptical
orbit about the sun. If we consider this common orbit as the ellipsis of the
planet ; the relative position of the satellites, compared with each other and
with the sun, will give the position of the planet, relative to this common
centre of gravity, consequently also the perturbations which the planet suffers
from its satellites. Let
M^ the mass of the planet ;
Symbol.. Ji :^ ^j^g radlus vector of the common orbit, or the orbit of the centre of gravity
of the planet and satellites, the origin being the sun's centre ;
V == the angle formed by the radius R, and the invariable line, taken in
the comm07i orbit, as the origin of the longitudes ;
m, ml, &c. the masses of the satellites ;
[4048] r, /■', &c. the radii vectores of the satellites, the origin being the common
centre of gravity of the planet and its satellites ;
V, v', &c. the longitudes of the satellites, referred to this common centre ;
s, s', &c. the latitudes of the satellites above the common orbit, and
viewed from the common centre ;
X, Y, Z the rectangular co-ordinates of the planet ; taking the common
centre of gravity of the planet and its satellites for their origin ;
the radius R for the axis of X ; and for the axis of Z the line
perpendicular to the plane of the common orbit.
VI.iv.§19.] ARISING FROM THE ACTION OF THEIR SATELLITES. 169
We shall have very nearly, from the properties of the centre of gravity,
and by observing that the masses of the satellites are very small, in
comparison, with that of the planet,*
0 = MX + mr. cos. ( v — f/) + m' r'. cos. (v' —U) + &c. ;
0=^MY-i-mr. sin. ( i' — C/) + m' r'. sin. {v'—U) + hc.\ [4050]
0 = M Z + m . r s + m', r s + &c.
The perturbation of the radius vector is nearly equal to X; consequently
it is equal to Perturba-
tions.
.r. cos. Ct; — U) .r'.cosJv' — U) — &c.=: Perturbation of radius vector. [4051]
The perturbation of the motion of the planet in longitude, as seen from the
r
R
— ^•-B-sin,(t; — U) — - . — .sin.(v' — U) — &c. = Perturbation in longitude. [4052]
wU -cCr Jim. JAj
Y
sun, is very nearly — ; therefore it is equal to
m r
* (2549) If we let fall from the points where the bodies M, in, m', &IC. are situated,
perpendiculars upon the axes of X, Y, Z, the distances of these perpendiculars from [4050a]
the common centre of gravity of the planet and its satellites, taken as the origin, will be,
respectively, as follows;
On the axis of X ; X; r . cos. {v — U) ; ?•'. cos. {v' — U), &,c. ; [40506]
On the axis of F; Y ; r . sin. {v— U) ; r' . sin. {v — U), &c. ; [4050c]
On the axis of Z ; Z ; r s ; r's',hc. nearly. [4050d]
Multiplying the distances [4050/^] by the masses M, m, m, &c. ; and taking the sum of [4050c]
these products, it will become equal to nothing, by means of the first of the equations [124] ;
hence we get the first of the equations [4050]. In like manner, by multiplying the
distances, measured on the axis of Y, by M, m, m!, &ic., respectively, and putting the sum [4050/"]
of the products equal to nothing, we get the second of the equations [4050]. The third of
these equations is formed by a similar sum, corresponding to the axis of Z. From
Y Z
these three equations, we may find the values of X, —, —, as in [4051,4052, 4053];
and as the radius R, or axis X, passes through the place of the common centre of gravity,
Y Z
it is evident that these quantities X, —, — will represent, respectively, the perturbations [4050^]
of the radius vector, of the longitude and of the latitude, conformably to what is said above.
VOL. III. 43
170 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Lastly, the perturbation of the motion of the planet in latitude, as seen from
the sun, is very nearly -^ ; hence it is nearly equal to
m rs m! r' s' c -r» i • • i • i
[4053] ■ — IT; • p^ M ' ~R "^ Perturbation m latitude.
These different perturbations are sensible only in the earth, disturbed by the
moon. The masses of Jujnter^s satellites are very small in comparison with
that of the planet, and their elongations, seen from the sun, are so very
[4054] small, that these perturbations of Jupiter are insensible. There is every
reason to believe that this is also the case for Saturn and Uranus.
VI. v.§-20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 171
CHAPTER V.
CONSIDERATIONS ON THE ELLIPTICAI. PART OF THE RADIUS VECTOR, AND ON THE MOTION OF A PLANET.
20. We have determined, in [1017, &c.], the arbitrary constant
quantities, so that the mean motion and the equation of the centre may
not be changed by the mutual action of the planets. Now we have, in
the elliptical hypothesis,* — y— == **"5 ^^^^ ''^('-^^ of the sun being put equal [4055]
to unity. Hence we obtain
2.
a = n ' . (1 + X»i) ; [4056]
for the semi-transverse axis, which must be used in the elliptical part
of the radius vector.
If we suppose, in conformity to the principles assumed in [4078 —
4079, &c.], that
«==n~*; a'=n'~'\ &c. ; [4057]
we must increase a, a', &c. in the calculation of the elliptical part of the
• (2550) This is the same as [3700], putting, as in [3709a], iJ. = M-\-m, and
M^l, as in [4055]. From this we get
a = 7i'~^.(l-f m)^ = n~"^.(l + im — T^V^^+^O; [4056o]
which, by neglectmg terms of the order m^, becomes as in [4056].
172
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
[4058] radius vector by the quantities ^m a,
\ m a.
Stc. respectively ; but this
Increment
of the
radius
augmentation is only sensible in the orbits of Jupiter and Saturn.
* (2551) The values of a", ce', for Jupiter and Saturn [4079], are respectively
augmented by the correction [4058], in the expressions [4451, 4510]. The similar
augmentation, corresponding to the other great planet Uranus, is |^m"o", which, by using
[4058a]
[40586]
?»'' [4061], becomes
If this quantity were an arc of the planet's orbit,
[4058c]
[4058rf]
58512
perpe7i(Ucu1ar to the radius vector, it would subtend only an angle of 3'~,6, when viewed
from the sun ; but being in the direction of the radius vector, it produces no change in the
longitude, seen from the sun ; or from the earth, when the planet is in conjunction or in
opposition. The most favorable situation for augmenting the effect of this correction, in
the geocentric longitude of the planet, is when the earth is nearly at its greatest angle
of elongation from the sun, as seen from the planet. This angle for the planet Uranus
is quite small, its sine being represented by — ; = j^ nearly [4079] ; and as the above
correction 3°",6 is to be diminished in the same ratio, it produces only 0'',2 for the greatest
possible effect of this augmentation of the radius, in changing the place of the planet Uranus,
as seen from the earth ; consequently this correction is wholly insensible.
[4058e]
We have already observed in the commentary in Vol. I, page 561, that Mr. Plana
makes some objections to the introduction of the constant quantity^, in the integral [1012'],
and he has also urged similar remarks against the use of the constant quantities _/), f^
[1015'], in finding the integral i5m [1015] ; but a little consideration will show, that these
objections do not apply to the accuracy of the results, or to the astronomical tables founded
upon them ; but merely to the most convenient way of ascertaining, as a mere matter of
curiosity, the orbit a body would describe if it were not acted upon by the disturbing force,
or of computing the whole effect of the disturbing force in a given time. This subject has
been discussed very ably by Mr. Poisson, in the Connaissance des Terns for the year 1831,
[4058/"] pag. 23 — 33 ; and we shall, in the remaining part of this note, avail ourselves of his remarks.
The complete integrals of the three differential equations [545], which determine the
co-ordinates x, y, z, of the planet referred to the sun's centre as their origin, contain
six arbitrary constant quantities [571«], which we shall denote by a, h, c, Sec. ; and the
same is true in using the polar co-ordinates r, v, s; as we have already seen, in [602"],
in the Jirst ajrproximation, where the disturbing forces are neglected, and the simple elliptical
motion obtained. In a second approximation, in which we notice only the first power
of the disturbing forces, we may put &r, Sv, 5 s for the increments of r, v, s ; and then
the integrations being made, as in [1015, &:c., 1021, 1030], will introduce six new arbitrary
constant quantities, a', b', d, &;c. ; these accented letters being taken for symmetry, instead
0Î g, fi, fl, &.C., used by La Place. A third approximation includes terms of the second
order of the disturbing forces, and by similar integrations, produces six other constant
quantities o", h" , c", Sic, and so on successively. If ive restrict ourselves to the second
[4058g
[4058;i]
[4058i]
VI.v.s^20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 173
We must then apply to the radius vector the corrections given by the
approximation, neglecting terms of the order of the square of the disturbing forces, tlie [40584]
polar co-ordinates will be r-\-6r, v-\-5v, s-\-&s, containing the twelve constant quantities
a, h, c, &,c. ; «', 6', d, &ic., which must, by the nature of the question, be reduced to six
only, or to six distinct functions Jl, B, C, D, E, F, of these twelve quantities. The [4058i]
values of A, B, C, &;c. may be determined by the position, velocity, and direction of the
planet at a given moment ; or by the comparison of the values of r-\-ô r, v-\-Sv, s -\- ôs,
with those deduced from observation ; in each case the result will be fixed and determined.
On the contrary, we may assume at pleasure any values of a, b', c', &c. ; and the values [4058m]
thus assigned to these terms, will determine absolutely the quantities a, b, c, fee, which
differ but little from A, B, C, he. on account of the smallness of the disturbing forces.
If we wish that or, Sv, &s should express the effects produced by the disturbing forces
011 the radius vector, the longitude and the latitude of the disturbed planet ; we must
determine a, b, c, &.c. so that the elliptical co-ordinates r, v, s, and their differential
coefficients —, —, — , may represent the position, the velocity, and the direction of
dt dt dt
the planet at the commencement of this interval of time ; and afterwards determine
a, I', (,•', &ic., so that we may have at the same epoch
0,
&v = 0, (is=^0;
d.ir
~dt
= 0,
lit
= 0,
d.Ss
~di
= 0.
At the end of the time t, counted from the same epoch, r will be the distance of the planet
from the sun, wliich will obtain, if the disturbing force cease to act from the commencement,
and 0 r will be the augmentation of distance produced by this force. Similar remarks
may be made relative to the quantities v, Sv ; or s, Ss. If we determine a', b', d by other
conditions, the perturbations of the troubled orbit will no longer be loholly expressed by the
quantities 5 r, S v, S s ; because the elliptical parts r, v, s, are also affected by means
of the constant quantities a, b. c, Sic, tvhich partake of the disturbing forces, and are
different from what they would be if these forces were suppressed. But this is not attended
with any inconvenience ; since it does not prevent these complete values of r -[- 5 r,
v-\-Sv, s -{-5 s, from representing, at every instant, the true position of the planet, wliich
is the object of the tables of its motion, into which tliese values are finally reduced.
Instead of considering directly the increments ô r, 6 v, ô s, of the elliptical orbit, we
may use the method depending on the variation of the arbitrary constant quantities ;
supposing Sa, Sb, S c, &c. to be the increments of the constant quantities a, b, c, he.,
contained in r, v, s. These six variable quantities S a, Sb, S c, &c. will be given by
direct integration of formulas similar to [1177], or like those collected together in the
appendix [5786 — 5791], supposing that we neglect the second and higher powers of the
disturbing forces. These values will then be of the forms,
(5a=^o, + a; Sb = b^-\- fi; Sc^c,-{-y, he.
VOL. III. 44
[4058n]
[4058o]
[4058p]
[40589]
[4058r]
[4058*]
[4058*']
[4058f|
174 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
formulas of Book II, ^50 [1020, &c.], and bj the preceding articles
[4058u]
rt, , b^, c, being new arbitrary constant quantities, and a, p, y, &c. functions of t, and
of a, h, c, &.C. Substituting a-\-ôa, b-{-Sb, c-\-Sc, &.C. for a, b, c, he. in the values
of r, V, s, we shall obtain for the co-ordinates of the disturbed planet, expressions which
are equivalent to the preceding values of r -]- 5 r, v -\-ôv, s -{- 8 s. The constant
[4058t)] quantities «^, f>^, c^, &c., as well as a, p, y, Sic, are of the order of the disturbing
forces; therefore, by neglecting terms of the second order, as in [4053s'J, we may put,
in the values of a, p, y, &,c.; a + », for a, ^ + 6, for p, c+c, for c, Sic;
by which means a-f-a,! ^-f"^,j c+c,, he. will be the six arbitrary constant
quantities, which occur in the values of r-\-5r, v -\- S v, s-\-Ss. This shows how the
arbitrary constant quantities, contained in the co-ordinates of the disturbed planet, as found
by the two first approximations, are reduced to the number corresponding to the system
of dilTerential equations upon which they depend.
[4058u»]
If ive ivish to chtermint the total effect of the disturbing forces upon each of
the elliptical elements, during a given time, we must find, as above, the constant quantities
[4058r] a, b, c, &c.; by means of the position, the velocity, and the direction of the planet
at the commencement of this interval of time; and tlien the constant quantities a^, b^, c, ,
by means of the equations
[i058y] a -fa =3 0, 6^-|-p = 0, c^-{.y=zO, kc,
corresponding to the same instant. The effect of the disturbing force at the end of any
proposed time t, will be expressed by means of the quantities 5 a, Sb, So, &c., which will
then contain nothing arbitrary. This is practised in the theory of comets, In wiilch the
[40582] values of Sa, 5b, 5 c, he. are calculated, by quadratures, for the interval of time between
the two successive appearances of a comet.
These general considerations agree with the method used by La Place in the second
book of this work. In the abovementioned paper of the Connaissance des Terns for the
year 1831, page 29, he., Mr. Poisson has applied these principles to the investigation
of the effect of the whole disturbing force of a planet m', upon another planet m, moving
f4059al in the same plane. The radius vector and the longitude of the planet m being affected by
this action, but not its latitude, because the bodies m, m' move in the same plane. In this
case, the six arbitrary constant quantities mentioned in [4058/], are reduced to four.
If we neglect terms of the order e^ in the elliptical motion of the body rn, the expressions
of the radius vector and longitude [669, 605'], become
[40596]
[4059c] r=^a — ae .cos. {n t -\- s — -n) ;
[4059d] t) = n < + s + 2 e . sin. {ni-\-e — ra ) ;
[4059c] n^a^ = M^m = !x.
If we suppose the body m to begin to disturb the motion of m at the commencement
VI. v.§20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 175
[3706 — 4058]. The expression of àr [1020] contains these two terms,
ir = — m' a .fe . cos. (7it~\- s — ra) — ni' a ./' e'. cos. (nt-{- ; — ra') ; [4059]
of the time t, we may determine the effect of the perturbation of the radius vector by
means of the value of Sr [1016], in whicii the arbitrary constant quantities are retained. [4059/"]
The expression o{ &v [I0-21] would give the perturbations in longitude, if particular values
had not been assigned to the arbitrary constant quantities g, f, f. To obviate this
objection, we must retain these arbitrary quantities as they are found in the functions
\\0-2\b, c, f/, e], whose sum is assumed in the first line of the note in page 556, Vol. I [4059g-J
[1021e — -/], for the value of S v. In order to simplify this calculation, it will be convenient
to change the form of the terms depending on /, /' ; by developing the sines and cosines
of the angles nt-\-s — «, 7it-^s — ra', into terms depending on sin. ?i ^, cos. jj f,
by the method used in [1023((] ; and changing the values of the arbitrary constant
quantities /, /', so that the part of the expression of — [1016], depending upon them, [4059A]
may be put under the form /. cos. n t +./ '• sin. n t. The corresponding terms of the value [40.59t]
of 5 r may be found by multiplying this expression by 2, and changing the angle n t into
n ( -j- 90'' ; as is evident, by comparing the terms of — [1016], depending on f, f,
with those of ay [1021i]; hence these terms of àv become — 2f .s\n. nt-\-2f'. cos. nt. [405941
We may also add an arbitrary constant quantity h, to the part of <S v, computed in either
of the integrations [1021 rf, f], and retain the terms
m'.ant.j3g-\-a. f — — j ^ [1021t?, e], [4059J]
which were put equal to nothing in [102iy]. Making these changes in the expressions
or
of —, Sv [1016, 1021] ; neglecting the other terms of the order c or e', because this
degree of accuracy is sufficient in our present calculation, which is only designed for the
purpose of illustration ; and supposing also, for brevity, as in [1018a],
[4059m]
v = n — n'; T=7i't — nt + s' — s; G = a\ (^^) ~{- ^ . a jî%
we get
— = — 2m'.ag—im'. a^. ( — — ^ + J m'. n^. 2 . — . cos. i T4-f. cos.nt4- f. sin. n t ; [4059n]
o \ da J f^v-^ — n^
&v = h — 'if.%m.nt-\-2f'.cos.nt->rm'.nt.'X3ag + a^.(^^\i
Cna .,., 2n3. G ? . . _,
which are substantially the same as the equations (5), (6), of Mr. Poisson, in the paper
[4059o]
176
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
f and f being determined by the two following equations, given in [1018],
[4059p]
[4059c]
[4059;]
abovementioned ; observing, that i includes all integral numbers, positive and negative,
except i= 0 [1012'] ; whereas he only uses the positive values of i. Now if we use
the expression of g [1017], the terms depending on nt will vanish from &v, and then
& r [1020] will contain the constant part | m . a''
da
but this is not the whole
[40595]
[4059<]
effect of the disturbing force upon the radius vector ; because a part of this perturbation is
introduced in the value of 7!, which is affected by the value of g, assumed in [1017],
and n is connected with a by means of the equation [4059c].
We shall, for greater simphcity, take, as the epoch, the instant of the mean conjunction
of the planets m, m' ; so that we shall then have < = 0, r=0 ; also s' = s. We shall
also suppose that the body m', at that instant, commences its action upon the radius vector,
and upon the longitude of the body m. Now we may find, from the tables of the planet's
motion, the numerical values of r, v, —, —, when t = Q; and these are to be put
equal to the values deduced from [4059c, f/]. These four equations, being combined
with [4059e], determine the constant quantities n, a, e, s, w ; and then the formulas
[4059c, t1] determine the elliptical motion, which obtains, if the disturbing force cease to
act at the epoch ^ = 0. This being premised, we must put t^O, T=^0 [4059r],
in the four equations [4058o],
d.èr „ d.ôv
dt
[4059m]
[4059t)]
[4059«i]
iSr^O; , 6v^0; -^=0; ^=0;
and by substituting in them the values [4059», o], we may obtain the values of the four
arbitrary constant quantities g, f, f, h, introduced by the second approximation.
If we substitute these values of g,f,f', h, in 5r, Sv [4059n,o], they will express, at the
end of the time t, the effect of the disturbing force during that time. Now the differential
of 5r [4059?i], relative to t, being found, and substituted in the third equation [4059<],
gives /'=0, when t = 0, T^O [4059/-]. With this value of /', and those of
5v [4059o, «], together with < = 0, T= 0, we get A = 0. Substituting these values
d.6v
of t, h, f, in the equations ^j- = 0,
we obtain the follow ng equations,
■d.û<-0)
dt
= 0 [4059<], using also the values [4059n, o],
0 = — 2 m'. a g — J m', ft^.
da
+ i7}i'.n^.S.
!v9-
-/;
0== — 2/n + ?«'.«. ^Saj+fl^.r^^^^^—Jm'. 2.^^. «^o-
2 «3. G
[4059:c]
Multiplying the equation [4059i;] by 2 n, and adding the product to [4059io] we find
that the terms depending on /, G, ( —-: J , vanish from the sum, which becomes
0 = — m. nag — J m'. 2 . — . a A^''' ;
VI.v.§20.] ELLIPTICAL PART OF THE RADIUS VECTOR. 177
[4060]
[m9y]
whence s; = — — . 2 . .^ '\ Substituting this in [4059y], we get
/=- — • 2 . A-'^ + i m'. «2. ( — — — I m'. n^ 2 . :^^ — - .
By means of the values of /', /(, g, f [4059it, y], the expressions [4059m, o] become
— = .2.^^''.(1 — COS.ÎtO irn.rt^. .(1 COS.?in
a V \ d a /
f. _ [40592]
+ i m. 11^. 2 .— —; ; • (cos. i T — cos. n 0 ;
C San .,., , „ /d.mx}
2\
[4059z']
[40600]
[40606]
If we retain merely the non-periodical parts of r, v, 5r, Sv [4059f, d, z, s'], and
resubstitute the value of v [4059m], we shall get
, ^ , m'.a^n .... . , „ fd.m\
' ' »i— n \da /
v4-5v = nt-\-i-^m'.nt.]^- . 2. ^'^-fa^. (— — )C ;
' ' ' t 2.()i— 7i') \ da / !) '
for the expressions of the mean distance and mean longitude of the planet m-
The expressions of the same mean distance and mean longitude, according to La Place's
calculation [1020, 1021], are
r-\-àr = a-\-\m:.a^.(^-^\, v + àv = nt. [4060c]
The differences between these values, and those in [4060a, &], are merely apparent, and
arise from using different values of n, a, in [4060c] from those in [4060a, i]. To
render this evident, we shall suppose, for a moment, that n, t represents the mean motion
of the planet m, derived from observation ; then, by putting the coefficient of t, in the
equation [40606], equal to n^t, we shall have
, / <^ 3an ^ am i 2 fdA^^ ^ [4060d]
^ i 2.(n— 7i') ' \ da J <)
VOL. III. 45
178 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
The preceding part of the radius vector [4059] may be united in the same
table with the elliptical part of the radius.*
Let a, be the value of a, deduced from the equation a = fA^.?i '•' [4059f], when
'• *■' n, is substituted for ?t ; so that this equation holds good for a, n, and also for «^ , n, ;
we shall have successively, by development, neglecting the square of n, — n,
Substituting in this the value of n^ — n [4060cZ], we get, by transposition,
[4000^] «=-«'+«»•«• 1-2-:i;;=;ô-'--^+"-V-^) I •
This value of a being substituted in [4060«], we find, that the parts depending on A'-'''
destroy each other, and we have
/ d A'-O'i \
[4060/i] '• + (5 ?• = a, + ^ ?«'. a^. \~J^) •
Now as we neglect terms of the oi'der m ^, we may change a into «, , in the part depending
on ^"" ; and then the expression [4060/t] becomes of the same form as in [4060f] ; being
equivalent to that found by La Place. This calculation serves to illustrate and confirm his
[40(j0i] method of calculation ; and shows, at the same time, how we can dispose of the additional
arbitrary constant quantities, which are introduced by the integrations of 6 ?•, Sv; so as to
conform to the actual situations and motions of the attracting bodies ; and to investigate the
part of the effect of the disturbing forces, that we have particularl}' considered in this note.
* (2552) We have here omitted a clause, in which the author directs, that the sign
[4060ft] of the term of f, depending on cldA^^'', should be changed; because we have previously
corrected the mistake, and given the accurate expression of /' in [1021g-], which agrees
with that in [4060].
VI. vi.<^21.]
NUMERICAL ELEMENTS.
179
CHAPTER VI.
NUMERICAL VALUES OF THE DIFFERENT QUANTITIES WHICH ENTER INTO THE EXPRESSIONS OF Till;
PLANETARY INEaUALITIES.
21. To reduce to numbers, the formulas contained in the second book
and in the preceding chapters, we shall use the following data ;
Masses of the Sun and Planets*
Sun, M = 1 ;
Mercury, m = ^^ ; log. m = 93,6934013 ;
Venus, m' =: j^ ; log. m' = 94,4166538 ;
The Earth, m" = j^^ ; log. m" = 94,4819733 ;
Mars, m"'== \^~ \ log. w"'= 93,7337490 ;
Jupiter, ^"=1^^ ; log- m" = 96,9717990 ;
1067,09 °
Saturn, vf = ^±^ ; log. m" = 96,4737383 ;
Uranus, m"= "^^ ; log- m"= 95,7098763.
* (2553) The factors l+fx, 1+(ji', Uc. in the values of m, m', &c. [4061], are
not inserted in the original work ; but as they are introduced in [4230'], and frequently
Masses
of the
planets,
the masi
of the
sun beiiii?
unity.
[4061 J
[4061o|
180
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
Of all these masses, that of Jupiter is the most accurately determined ;
it is obtained by means of the formula [709]. If we put T for the time
[40615]
[4061c]
Masses
finally
adopieil
by the
author.
[406 W]
used in computing the perturbations of the motions of the planets, it was thought best,
for the sake of convenient reference, to insert them in this place. When the author printed
this part of the work, he supposed, in conformity with the best observations, which could
then be procured, that the masses of the planets were as in the table [4061], putting each
of the quantities (x, (a', &ic. equal to zero. Since tliat time, he has been induced, by other
observations, to make successive corrections in these masses, as in [4605, 4608, 9161, &c.].
In his last edition of the Système du Monde, he adopts the following
Corrected Masses of the Planets.
^ (A =0; log. ??i =93,6934013;
^' =z — 0,0.56030 ; log. m = 94,3916120 ;
(a" = — 0,0T1297 ; log. m" = 94,4498499 ;
(;/"= — 0,275000 ; log. m'" = 93,5940870 ;
|xi-= — 0,003186 ; log. m'':= 96,9704133 ;
f;.^ = — 0,043451 ; log. m' == 96,4544455 ;
ij:'= 0,088514 ; log. m"'' = 95,7467105.
Saturn, nf =
Uranus, m'' =
The alterations here made in tlie values of »«', ?»"', are in conformity with the results of the
calculations of Burckhardt, in his late solar tables, by comparing the observed perturbations
[4061e] of the earth's orbit with the theory. The change in the value of m", arises from the
supposition, that the sun's horizontal parallax is nearly equal to 8',6 [5589], instead of 8^,8,
assumed in [4073]. Lastly, the values of nt", m'', m'", are obtained, by Mr. Bouvard,
from the observations used in constructing his new tables of Jupiter, Saturn, and Uranus, by
comparing the theory with the actual perturbations depending upon their mutual attractions.
[4061/] Putting the values in [4061] equal to those in [4061fZ], respectively, we get the
corresponding values of (a, f.'/, he. [4061f/]. Lindeneau, in his tables of Mercury, printed
r4061ffl ill 1813, supposes that the mass of Venus ought to be increased to a-jaVioJ making
|j,'= 0,09643 nearly; to satisfy the perturbations of Mercury, by the action of Venus.
Encke, in his Astronomisches Jahrbuch for 1831, states, that the mass of Jupiter tû5 j.fls^ >
deduced by Nicolai, from the perturbations of Juno, agrees better with the observations
[40G1/I.] of Pallas and Vesta, than the mass adopted by La Place [4061, 4065], and that it probably
VI. vi.^21.] NUMERICAL ELEMENTS. 181
of the sidéral revolution of the planet m' ; T for that of one of its satellites ;
q for the sine of the greatest angle, under which the mean radius of the
orbit of this satellite appears, when viewed from the centre of the sun, [40G2]
at the mean distance of the planet from that centre ; then the mass of the
sun being taken for unity, that of the planet will be expressed by *
T
.,r!
\-q\
7=— i; = mass of the planet. [4063]
T
[4061&]
agrees also better for Vesta. Comparing this with [4061], we get (a''' =0,012492. When [406lt]
we take into consideration that \he first value of fi''==0 [4061, 4065] is obtained from the
observed elongations of the sateUites of Jupiter; the secondvdXue, (a'= — 0,003186 [4061«/],
from the perturbations of Saturn and Uranus ; the third value, (^'=0,012492 [4061z],
from the perturbations of the newly discovered planets ; we shall not be surprised in finding
these small diflerences in the results of methods, which are so wholly independent of each
other. Nothing is known relatively to the masses of these new planets or the masses of the [4061ot]
comets, except that they are all very small ; so that their action on the other bodies of
the system is wholly insensible.
* (2554) This is deduced from [709], —^^ — .i—\, in which we must write [4062a]
I* for M, as is evident from [706'] ; and as m' represents the mass of the planet, in the
present notation, we have n = M + ?»'. Moreover p is the mass of the satellite [Î07'],
and M that of the sun [706'] ; h the mean distance of the satellite from the planet ;
a the mean distance of the planet from the sun ; so that — represents the quantity
we
[40626]
[4062c]
q [4062] ; hence the preceding equation [4062a] becomes - J^ , = ^^ / \ if
/ T \ 2 1
neglect p in comparison with m', and put JW= 1 ; also, for brevity, cf. (—\ =~ , we
a 1
get, as in [4063], m'=^ =- — - . If we put r, p" for the mean densities of the [4062rf]
1_- ft-i
bodies m'', m"; also R'% R" for the radii ; we shall have nearly, as in [2106],
?«>'■= 4 * . piv. (/3iv^3 . ^v ^ I ^ ^ pv_ ^ji-y^ [4062e]
Hence we easily obtain the relative densities of these two bodies, ^~ = — .(-—] • [4062/1
pv m^ yR'" /
This may be used for ascertaining the densities of all the bodies, whose masses are known,
and whose apparent diameters have been well observed.
VOL. III. 46
182 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
We have, relatively to the fourth satellite,*
q = sin. 1530",38 = sin. 495',84 ;
[4064] T = 4332'")%602208 = 433244* 27"- 10',8 ;
r= 16'^'",6890 = 16"16''32"'09-,6.
From [4063, 4064], we obtain
[4065] m"
[4064a]
1067,09
The mass of Saturn is found by the same method ; supposing the sidéral
revolution of its sixth satellite to be 15"'''%9453 = 15"22"41'" 13',9, and
the greatest angle, under which the mean radius of the orbit of this satellite
appears, when viewed from the sun, in the mean distances of Saturn,
[4066] 552'',47=179'. The mass of Uranus has, in like manner, been obtained, by
supposing, conformably to the observations of Herschel, that the duration of
the sidéral revolution of its fourth satellite, is 13'''ys4559 = 13'10''56'"29',8;
[4067] jji^j ^]^Q mean radius of the orbit of tliis satellite, viewed from the sun,
at the mean distance of Uranus, 13S",512 = 44',23. But the greatest
elongations of the satellites of Saturn and Uranus have not been so
accurately ascertained as that of the fourth satellite of Jupiter. Observations
of these elongations deserve the careful attention of astronomers.
The mass of the earth is found in the following manner. If we take
the mean distance of the earth from the sun for unity, the arc described
by the earth, in a centesimal second of time, will be obtained by dividing
the circumference of a circle, whose radius is unity, by the number of
[4068] seconds in a sidéral year, 36525638"''-,4. Dividing the square of this arc
[4068] by the diameter, we obtain its versed sine = — r;^»^ jf which is the space
the earth falls towards t!ie sun in a centesimal second, by means of its
relative motion about the sun. On the parallel of latitude, whose sine is
* (2555) The values of c/, T [4064], are nearly the same as those used in the theory
of this satellite [6781,6785] ; the value of T corresponds to the mean motion n'" [4077].
t (2556) The radius of the orbit being 1, its circumference is 6,28.318 nearly; if we
[4068a] divide this by 36525638,4, and take half the square of the product, we get the expression
of the versed sine, corresponding to this arc, as in [4068'].
VI. vi. }21.]
NUMERICAL ELEMENTS.
183
equal to \/}, the attraction of the earth causes a body to fall through
3""", 66553* ill one centesimal second. To deduce from tiiis the earth's
attraction at the mean distance of the earth from the sun, we must multiply
it by the square of the sine of the sun's parallax, and divide the product by
the number of metres contained in that distance. Now the earth's radius
on tlie proposed parallel, isf 6369374'™'-; therefore, by dividing this
number by the sine of the sun's parallax, supposing it to be 2T',2 — S%8,
we obtain the mean radius of the earth's orbit, expressed in metres. Hence
it follows, that the effect of the attraction of the earth, at a distance equal
to that of the mean distance of the earth from the sun, is equal to the
product of ihe fraction i'^'^^., by the cube of the sine of 27",2 ;
^ bSby-i 1 4
consequently it is equal to J
10-
Subtractins this fraction from
1479565
10-33-'
we obtain —
1479560,5
10-
for the effect of the attraction of the sun,
[4069]
[4069']
[4070]
[4071]
[4071']
* (2557) This computation varies a little from that in [388"] or in [3SSf/] ; probably
owing to a small difference in the ellipticity, used in reducing the observations.
t (25.58) LT^sing the polar and equatorial semi-axes of the earth, 6356677™'-,
6375709"'"'' [2035i], whose difference is 19032""^'-, we find the radius corresponding to
the latitude, whose sine is /-L, to be 6375709""='- — i X l9032'"'='-= 6369365""^'-,
agreeing nearly with [4069'].
J (2559) Gravity decreases, in proceeding from the earth's surface, inversely, as the
square of the distance of the attracted point ; or as the square of the sine of the horizontal
parallax of that point nearly. Hence the earth's attraction, at the distance of the sun,
will cause a body to fall through a space represented by 3"""-,66553 X (sin. O's par.)"^,
in one centesimal second of time. To reduce this from metres to parts of the mean
distance of the earth from the sun, we must divide it by that distance, which is evidently equal
earth's radius 6-369374 '"o'- r n i i •
so that the space lallen through m a second, becomes
to
sin.27",2
\3.
sin. Os' par.
^ô^\~ ■ ■ (sin- Q's par.)^ = , as in [4071'!. Now in [4063'], we have found, that
Doo9.!}/4 10-0 ■- -" ■- -•
the earth falls towards the sun, in the same time, by the combined action of the sua and
1479565 — 4,488.5 1479560,5
earth
1479565
10^0 '
nearly ; and as that of the earth is
hence the effect of the sun alone is
4,4885
low '
4,4885 1479560,5 „^„„, , ■ . r.-,
to — ,^.,„ , or 1 to 329630 nearly, as in [4072].
1020 1020
the mass of the earth is to that of tiie sun
[4069a]
[4070a]
[4071a]
[40716]
[4071c]
[4071rf]
lOio
10-0
184 PERTURBATIONS OF THE PLACETS; [Méc. Cél.
at the same distance. Hence the masses of the sun and earth are in the
ratio of the numbers 1479560,5 to 4,4885; consequentlj the mass of
[4072] the earth is . If the sun's parallax differ a little from the quantity
we have assumed in [4070], the value of the earth's mass will varv as
[4073] the cube of that parallax, compared with the cube of 21", 2 = 8",8 [4071c].
We have computed the mass of Venus from the formulas [4251, 4332, &c.],
which express the secular diminution of the obliquity of the ecliptic to the
[4074] equator; supposing it, by observation, to be 154',30^50'. This diminution
is obtained from those observations which appear the most to be relied upon.*
With respect to the masses of Mercury and Mars, we have supposed, according
to observation, that the mean diameters of Mercury, Mars, and Jupiter,
viewed at the mean distance of the earth from the sun, are, respectively,
[4075] 21",60 = 7-; 35",19 = 11%4; 626",04 = 202-,84. Now Jupiter's mass
being ascertained, we could, by means of these diameters, obtain the masses
of Mercury and Mars, if the relative densities of these three planets were
known. It we compare the masses of the Earth, Jupiter, and Saturn,
with their magnitudes, respectively, we find, that the densities of these
planets are very nearly in the inverse ratio of their mean distances from the
* (2560) K we change 7, A [310-2f] into 0", «", respectively, to confonn to the
[4074a] notation used in [4082. 4083] ; we shall find, that the arc F G^y . cos. A [3109c],
which represents the difference between the inclinations of the equator to the fixed echptic
of 1750 and to the variable ecliptic of 1750 -|- ^j is equal to o". cos. é", or q" [4249].
[40746] The value of q" is found by integrating the second equation [4251]. In this expression
of q", the coefficients of fi, fif", (1% fi", are small, and the value of i^'^ [4061 J] is small
and tolerably well ascertained ; therefore we need only retain /. so that the intesn^
[4074c] becomes q" = — ( 0". 500955 -p 0',309951 . ,u.') .t. If we suppose ,a' =^ 0, the annual
[4074dl decrement becomes 0*..500955, being nearly as in [4074]. The action of the planet Venus
has more effect in producing this change of obliquity, than that of all the other planets
taken together; as is evident fcom the inspection of the value of d q'' [4251]; in which
[4074e] we find, that the coefficient of ,a' exceeds the sum of the coefficients of the other quantities,
ji, (i'", 11'", (Ji\ fi". We have already remarked, in [3380/! — q], that the author increased
the annual variation to 0'\521154 [4613] ; on the other hand, Mr. Poisson uses 0 45692
[4074/1 [33Sqp], and Mr. Bessel 0-',48368 [3380j] ; each of them varying the values of ,a, ^', &c.,
so as to conform to their assumed decrements.
VI. vi.§21] NUMERICAL ELEMENTS. 186
1 / O \3 a" ,„ 1 /D"'\3 a'"
' "' ine-r nn ' \ 7)iv ^ • „w J
1067,09 V-D'V « 1067,09 V^
and by substituting the values [4076c, 4079], we get, for m, m", rather greater values
than those in [4061]. These diflerences probably arise from having used different values
of D, D", D\ which cannot be obtained, by observation, to a great degree of accuracy.
In some of the subsequent calculations, it will be sufficiently accurate to use the values
of n, n, Sic. to the nearest degree; and for convenience of reference we have here
inserted these approximate values ;
71=1661°; 7i'=650°, n"=400^, n"'=212=',7, w» = 330,7,
n- = 13^,6, ■nr' = 4P,Q.
VOL. III. 47
[4076]
sun ;* we shall therefore adopt the same hypothesis, relatively to the three
planets IMercury, Mars, and Jupiter ; whence we obtain the preceding
values of the masses of IVIercury and Mars [4061]. The irradiation
and the other difficulties attending the measures of the diameters of the
planets, taken in connexion with the uncertainty of the hypothesis adopted
on the law of their densities, render these estimated values somewhat
doubtful, and this uncertainty seems to be increased from the circumstance,
that the hypothesis is not correct relative to the masses of Venus and
Uranus. Fortunately, Mercury and Mars have only a very small [4076]
influence on the planetary system ; and it will be easy to correct the
following results, so far as they are affected by this cause, whenever
the development of the secular inequalities shall make known exactly the
values of these masses.
* (2561) The densities of the Earth, Jupiter, and Saturn, given by the author in the
Système du Monde, are 3,93 ; 0,99 ; 0,55 ; respectively, being found as in [4062/, Sic.]. [4076a]
These densities of Jupiter and Saturn are nearly in the inverse ratio of the distances
a", a" [4079] ; but the density of the earth differs considerably from this rule. If we
suppose this ratio of the densities to hold good for the three planets Mercury, Mars, Jupiter, and
represent their apparent diameters [4075], by D=21",60, I>"'=35",19, -D'^=:626",04; [4076c]
[40766]
#13 n"'3 71' ^3
the corresponding masses will be m = b . — ; ?»'"= b . —^ ; m''= J . — — ; i being [4076(f]
a constant quantity, to be found by means of the value of m'" [4061] ; which gives
[4076/]
[4076^]
[4076A]
186
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Mean
motions
of the
planets.
[4077]
[4078]
The
time t 18
expressed
in Julian
years.
Mean
distances
of the
planets
from the
flun.
[4079]
22. Mean sidéral motions of the Planets in a Julian year of 365| days,
or the values of n, n', &c.
Sexagesimals.
Mercury, .n = 16608076",50 = 5381016%786 ; log. n =6,7308643
Venus, . . . n' = 6501980",00 = 210664r,520 ; log. n' =6,3235906
The Earth, n" = 3999930",09 = 1295977%349 ; log. n" = 6,1 125974
Mars, n"'= 2126701",00= 689051%124 ; log. /i"' = 5,8382514
71'"= 337210",78= 109256-,293; log. ?i'^ = 5,0384465
n^ = 135792",34= 43996%718; log. n^ = 4,6434203
n''= 47606",62= 15424^545, log. n" = 4, 1882124
Jupiter,
Saturn,
Uranus,
If we use these values of n, n', &c., the time t ivill be represented in
Julian years ; hence if we put the mean distance of the earth from the sun
equal to xmity, we shall obtain, from Kepler's law [385'"], the following
mean distances of the planets from the sun.
Mean distances of the Planets from the Sun, or the semi-major axes
of their orbits.*
Mercury, a = 0,38709812
Venus, «' = 0,72333230
The Earth, a" = 1,00000000
Mars, «'"= 1,52369352
Jupiter, a'^ = 5,20116636
Saturn, a" = 9,53787090
Uranus, a"
19,18330500
log. a
log. a'
9,5878211
9,8593379
log. a" = 0,0000000
log. «'"=0,1828976
log. rr = 0,7161007
log. a" = 0,9794514
log. «^' = 1,2829234.
* (2562) These values of «, a', &C. are deduced from [4077], by putting them,
[4079a] respectively, equal to I — V, ( — r, (— :F, &c.
The elements of the orbits of the newly discovered planets, Ceres, Pallas, Vesta, and
Juno, were first computed by Gauss, and have since been repeatedly corrected by him,
VI.vi.§â-2.]
NUMERICAL ELEMENTS.
187
The mutual action of the planets alters a little their mean distances ; we
shall, in [4451, 4510], determine these alterations.
aiul by other astronomers ; taking notice of the most important perturbations, from the [40794]
action of the nearest phinets ; so that we can now compute the places of these bodies
with a considerable degree of accuracy. The usual methods of finding the perturbations
can be applied to these small planets ; but the great excentricities and inclinations of some [4079eJ
of their orbits, will make it necessary, when great accuracy is required, to notice the terms
depending on the powers and products of these two elements, of a higher order than is
generally used with the other planets. The laborious task of ascertaining all the inequalities
of these four planets, was not performed by the author of this work ; and it will probably be [4079<i]
a long while before it can be done completely, on account of the small imperfections in the
present estimated values of the elements, which have not yet been determined with perfect
accuracy in the short period since the bodies have been observed. It is evident, also, that [4079e]
until these elements have been found very nearly, it will not be of much use to compute
several of the very small inequalities, with tiie extreme minuteness which is used relatively
to the other planets.
In computing the Jahrluch, it has been found most convenient by Encke to apply the
corrections directly to the elements of the orbit, rather than to the elliptical places of [4070/']
the bodies ; in a manner similar to that which is used in finding the elements of a comet, in
two successive returns. He finds, when the elements are thus adjusted to any particular [407yg']
moment of time, that they will give, tolerably well, the places of the planet for a considerable
period, on each side of this epoch. The elements of the orbits obtained by him, for these
four planets, about the time of the opposition of Pallas, in the year 1831, are as in the [4079A]
following table ; which will serve to give an idea of the relative positions of the orbits
at that time ; remarking, that these elements must not be confounded with the memi values.
Epoch 1831, July 23d, 0'', mean time at Berlin.
I Vesta.
Mean longitude, 84'' 47" 03'
Mean anomaly, 195 35 26
Longitude of the perihelion, .... 249 11 37
Longitude of the ascending node, . 103 20 28
Inclination, 7 07 57
Excentricity, 0,0885601
Mean daily sidéral motion, 97775540
Semi-major axis, 2,.361484
Periodic revolution corresponding, . 1325,5 days
74''39"'44'
20 22 31
54 17 13
170 52 34
13 02 10
0,2555592
813',525.33
2,669464
1593,1 days
290'' . 38"' 12»-
169 33 11
121 05 01
172 38 30
34 35 49
0,2419986
768%54421
2,772631
1686,3 days
Cereg.
307'' 03'" 26'-
159 22 02
147 41 23
SO 53 50
10 .36 56
0,0767379
769-26059
2,770907
1684,7 days
Elements
of Veatu,
Juno,
Pallas,
and Ceres.
[4079i]
188
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
Eicen-
tricities of
the orbits
of the
planets.
[4080]
Ratios of the excentricities to the mean distances, or the values of e, e', ^c.
for the year 1750.
Mercury, e = 0,20551320
Venus, e' = 0,00688405
The Earth, e" =0,01681395
Mars, e"= 0,09308767
Jupiter, e'' = 0,04807670
Saturn, e'' = 0,05622460
Uranus, e" = 0,04669950
log. e =9,3128397;
log. e' = 7,8378440 ;
log. e" = 8,2256698 ;
log. e"'= 8,9688922;
log. e"= 8,6819346;
log. &■ = 8,7499264 ;
log. e''= 8,6693122.
[407M]
[4079Z]
Elements
of the
orbits of
the four
known
periodical
comets.
[4079m]
[4079n]
The distances of tlie planets Pallas and Ceres from the sun, are so nearly equal to each
other, that it may sometimes happen, in finding the apparent orbits, in the precedins;
manner, that the order of the bodies will be inverted, relative their distances from the sun,
by means of the perturbations.
Besides these planets, there are four comets, whose periodical revolutions have been
discovered by Halley, Gibers, Encke, and Biela. They have been usually called by the
names of the discoverers i-espectively. That of Olbers has been observed only once, at
the time of its return to the perihelion in 1815 ; the others have been observed in several
successive revolutions.
Periodic revolution,
Time of perihelion,
Longitude of perihelion on the orbit,
Longitude of the ascending node,
Inclination,
Excentricity,
Semi-major axis,
Of the seven periodical bodies, which have been made known to astronomers since the
commencement of the present century, three were discovered by Dr. Olbers of Bremen ;
namely, Vesta, Pallas, and the comet of 1S15. His great success in the discovery of
these remarkable bodies, which had silently performed their revolutions in the heavens
for ages, unperceived by astronomers, induced an eminent German writer to style him»
the fortunate Columbus of the planetary ivorld.
Halley's.
Olbeis's.
Encke's.
Biela's.
7G years
74 years
1204 days
6,7 years
Nov. 7, 1835
April 26,1815
Jan. 10, 1829
Nov. 27,1832
304' 31 "'43'
149"^ 2-"
157'^18'"35'
109'' 56™ 45'
55 ,30
83 29
.334 24 15
248 12 24
17 44 24
44 30
13 22 34
13 13 13
0,9675212
0,9313
0,8446862
0,751748
17,98705
17,7
2,224346
3,-53683
VI.vi.§22.]
NUMERICAL ELEMENTS.
189
Longitudes of the perihelia in the year 1 750, or the values of ^, ts', ^c.
Mercury, « = 8P,7401 = 13'33^5S'
Venus, • ^' = 142°,1241 = 127 54 42
The Earth, ^" -= 109^,5790 = 98 37 16
Mars, ^"' = 368°,3037 = 331 28 24
Jupiter, ^'"^ 11°,5012= 10 21 04
Saturn, zy" = 97°,9466 = 88 09 07
Uranus, ^"= 185°,1262 = 166 36 49.
Inclinations of the orbits to the ecliptic in the year 1750, or the values
of f, <p', ^c.
Loagitudes
of the
perihelia
in 1750.
[4081]
Mercury,
Venus,
The Earth, cp" =
Mars, ^'"^
Jupiter, tp"' =
Saturn, <?' = 2°,7762
Uranus, <p" = 0^,8596
9 = 7°,7778= 7''00™00';
9' = 3°,7701 = 3 23 35 ;
?" = 0° ;
2°,0556 = 1 51 00
1°,4636= 1 19 02
2 29 55
0 46 25
Inclina-
tions of
the orbits
to the fixed
ecliptic of
1750.
[4082]
Longitudes of the ascending nodes on the ecliptic of the year 1750, or
the values of ô, 6', ^c.
Mercury, . .
Venus, . . . .
The Earth,
Mars, . . . .
= 50^,3836= 45''20™43^;
== 82°,7093= 74 26 18 ;
as in [4249—4251];
'=: 52°,9376= 47 38 38
Jupiter, «'"= 108°,7846 = 97 54 22
Saturn, ô' = 123°,8960 = 111 30 23
Uranus, r = 80^,7015 = 72 37 53
VOL. III. 48
Longitudo;^
of tlie
ascending
nodes of
the orbits
on the fixed
ecliptic of
1750.
[4083]
190
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Epoch. All these longitudes are counted from the mean vernal equinox, at the epoch
[4084] of December olst, 1749, mid-daij, mean time at Paris. ^Ve may here
Lpnguude observe, that hij the longitude of the perihelion, is to be understood, the
rlôJiT <^'^fc>^c^ of the perihelion from the ascending node, counted on the orbit,
increased by the longitude of that node.
23. We have obtained the following results, by the formulas of §49,
Book II.
MERCURY AND VENUS,
[4085]
[4086]
hence we deduce
Then we obtain*
0. = - = 0,53516076;
6_;. = 2,145969210;
rin
6 \ = — 0,515245873.
6^ = 2,1721751 ; b''^ = 0,6057052 ; 6^^' = 0,2465877 ;
,(3)
^^L. 6, = 0,1107665;
,w
6^ = 0,0520855;
,(5)
6V= 0,0251378;
,(6)
[4087] 5^ = 0,0123166 ;
,P)
6\'= 0,0060633
(S)
6^;'== 0,0029287 ;
6^^'= 0,0012758.
* (25G3) From a, a [4079], we have a=-, as in [4085]. Then from [989].
[40S6a] we find, 6,, b_^, as in [4086]; from these we get b<, b, [40S7], by means of
the formulas [990, 991]. Then putting, in [966], s^i, and successively, ?:=-2. /=3,
I = 4, Sec. we obtain the remaining terms of [4037]. From these last, we get those
[40866] in [40S8], by putting, successively, 2 = 0, j* = l, Sec, and s = i, in [981]. The
same values, being substituted in [98-2], give [4089] ; also [983] gives [4090] _
Lastly, by taking the partial differential of [983], relative to a, we shall get an expression
U)
d*b s ■
[4086c] of ; in which we must put s = i ; then j'^0; /=1, &;c. ; and we
(0) a)
shall get [4091]. Again, the formulas [99-2] give ba. , bs. , [409:2]; from these two
Vl.vi.S^a.] VALUES OF b'^, AND ITS DIFFERENTIALS FOR MERCURY. 191
(0)
dbl,
da.
= 0,780206 ;
do.
1,457891 ;
dbi
do.
= 0,691487 ;
(41
dbi
do.
0,423818;
(6)
dbi
rfa
= 0,147708 ;
a)
db^
do.
0,085953 ;
(0)
dH^
da?
= 2,756285 ;
(11
dHi
da?
= 2,426165;
(3)
d"-bi
do?
= 3,381072 ;
(41
dHk
do?
= 2,826559 ;
(6)
dHi
da?
= 1,511016;
(71
dH>,
da?
= 1,014134;
(01
dHk
do?
= 11,308703;
m
dHk
da?
- 12,064245 ;
(31
da?
= 14,584366 ;
(41
dH^
do?
= 16,067040;
(61
dHk
da?
= 13,720218.
(2)
d*bK
do.*
= 69,60594 ;
(?)
d^b^
da.*
= 82,36773 ;
(51
d*bk_
da.*
= 105,33962.
r-1
db.i
do.
- 1,070071 ;
j/^
dbi
do.
= 0,252376 ;
[4088]
77"'
db^
do.
= 0,050726.
(21
dHi
do?
= 3,395022 ;
d^fl
do?
= 2,137906 ;
[4089]
Mercury
and Venue .
(21
dHi,
(51
dH^
da?
11,983424;
15,617274;
(41
d*b),
92,72610 ;
[4090]
[4091]
terms, we may obtain the others of [4092], by means of the formula [966] ; putting
s = |, and, successively, i^2, i=3, &;c. The values [4093] are found from [981],
by putting s = f, and i = 2, i = 3, he. Those in [4094] are deduced from [982], by [4086d]
using similar values of s, i ; observing to substitute, in any of these formulas, the values
of b, or its differentials, which occur, and have been found in the preceding parts of the
calculation. All the other terms of this article, §23, are found in the same manner, except
those in [4113, 4119, 4124, Sec], where a is very small ; and there is no difliculty in the [4086e]
calculation, except the ennui, arising from a long and uninteresting numerical calculation.
192
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
(!) (2)
4,214154 ; 6^ = 3,035376 ; 63 = 1,950536 ;
[4092]
Mercury
and Venus.
(3)
b^= 1,192372;
2-
0,238807.
(4)
6 , = 0,708667 ;
(5)
63 =0,413762;
[4093]
(2)
= 12,50630;
(3)
^ = 9,76666 ;
da.
(1)
dbî
7,08399
(5)
db^
da.
= 4,88781.
[4094]
(3)
dHi
= 78,09476 ;
(4)
dHi
da.^
67,14764.
[4095]
MERCURY AND THE EARTH.
hence we deduce
a = - = 0,38709812 ;
[4096]
, (0)
6^; =2,07565247;
~2
,(I)
6 , = —0,37970591.
Then we get
Mercurjr
and the
Earth.
6*^"' = 2,081980;
(1)
6, =0,411140;
2
è'f = 0,120178 ;
[4097]
(3)
6^ = 0,038900 ;
(4)
6 J = 0,013202;
ô'J = 0,004603 ;
(6)
6 = 0,001629 ;
(V)
6^ =0,000573;
(8)
b, =0,000177.
VI. vi.§:2;3.] VALUES OF T' AND ITS DIFFERENTIALS FOR MERCURY. 193
(0)
da.
(3)
dbj
do.
(6)
dbi
= 0,464378 ;
=. 0,316756 ;
= 0,026130 ;
(0)
cPbj
da.'
(3)
dH^
da.^
(2)
dHj
,(0)
= 1,672199;
= 1,852364;
= 6,49232;
6 =2,871833;
6*'' = 0,334212 ;
(I)
db),
(4)
dbj
do.
m
dbj
da.
(1)
d^bi
(4)
dHj_
do.^
(3)
dHj
da.^
.(')
1,199633;
0,141792;
0,011153.
1,220775;
1,197245;
: 5,45663;
63 =1,576062;
,w
63 =0,153779.
(2)
dbj
do.
(5)
dbis
da.
0,665739 ;
0,061433 ;
[4098]
(2)
'^l^^ = 2,235935 ;
do.^
(5)
'^^J^ = 0,670874.
[4099
Mercury
and the
Earth.
(4)
^'^^ _ 6,51373.
[410C
6^!' = 0,747619;
(3)
dbi
= 3,05535.
[4101]
[4102]
MERCURY AND MARS.
hence we deduce
a = ^, = 0,25405312 ;
,(0)
b'^ = 2,03240384 ;
, (1)
r: = — 0,25198657.
[4103]
Mercury
and Mare.
[4104]
VOL. III.
49
194
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
Then we have
[4105]
Mercury
and Mars.
[4106]
[4107]
b^'^ = 2,033500 ;
,(3)
(0)
,(1)
dbl
do.
do.
(0)
(3)
d^bj
doJ"
= 1,050458.
b =0,260462;
,(2)
6 J ==0,049765;
(5)
= 0,010546;
'.=
= 0,002331 ;
b ^ = 0,000538.
= 0,273829 ;
(1)
dbi
d a
= 1,077839;
ir^ = 0,402980 ;
do.
= 0,127139 ;
db^
do.
= 0,037781.
= 1,244725;
do?
= 0,656780 ;
(3)
^= 1,778641 ;
[4108] è^J^= 2,322536;
,(1)
rJ = 0,863876 ;
6''' = 0,272085.
[4109]
Mercury hcncc WB deduce
and
Jupiter.
[4110]
MERCURY AND JUPITER.
a = - = 0,07442555 ;
a"
f^ = 2,00277053 ;
2
&"; = — 0,07437397.
,(0) ,(1)
In computing the values of 6 , 6 , &c., by means of the formulas
[966 — 983], it is found, that the successive terms of the series become
more inaccurate, particularly if o. be rather small ; because these values
Vl.vi. §23.] VALUES OF b^^ AND ITS DIFFERENTIALS FOR MERCURY. 195
are the differences of numbers, Avhicli vary but little from each other ; so
that we are under the necessity of computing them to an extreme degree [4lll]
of exactness, to enable us to determine correctly their differences,* and
this requires the use of tables of logarithms to ten or twelve places of
decimals. To obviate this inconvenience, we may have recourse to the
value of b '\ developed in a series, by means of the formulas
[976, 984— 985],t
'ill (i±i' a2J_*-(*+l) (^+')-(^-H+l) ^4
*-=^-— 172737^::^^ — •"-'•< ^ [4112]
1.2.3 • (i+l).(i+2).(i+3) • "^
This value of 6'"' is, in the present case, very converging, on account
of the smallness of a. We shall hereafter use it, in finding the values of
b , b \ &ic.; 6'°\ &c., in ail cases where a is rather small.
i h ^
By this method we have computed, for Mercury and Jupiter, the
following values ;
(0) (1) (2)
6 = 2,002778 ; b,= 0,074581 ; 6, = 0,004164 ; [4113]
Mercury
(3) (4) and
b^ = 0,000258 ; b^ = 0,000017. '"•'''"•
* (2564) Thus, if we put s = i and i = 2, in [966], it becomes
<i) (0)
(2) (l+a').6a— ia.è , [4111a]
** = f^ —^ •
Now ht, is much smaller than h. or h. [4105], and the preceding value of
b' is divided by the small quantity J a. Hence it necessarily follows, that the terms
(1 + a^) .b, and — ^ a . è , , in the numerator of this expression, must be very nearly
equal to each other; and their difference, which is to be divided by a quantity of the r^mii
order a, must therefore be very accurately computed. The same takes place in b\, &.c.
t (2565) The quantity h is the coefficient of cos. i ê, in a-^ [976] ; and X-* is
the product of the two factors [985]. If we multiply these factors, and retain only terms
of the form 0=*='^*^, putting c'"^-' +c-'^»^"' = 2.cos.i é [12] Int., it becomes [4n2„j
as in [4112].
196
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4114]
Mercury
and
.lupiler.
[4115]
(0)
dbs
d CL
da.
m
dUi
0,074891 ;
: 0,010428;
1,018876;
Cl)
dbj_
da.
dHj
da.^
1,006269;
= 0,171781
(2)
dbi,
d a
(2)
dHj_
do.^
0,111380;
= 1,499780;
[4116]
6^ = 2,025143 ;
b'l = 0,225613 ;
,(2)
0,020984.
MERCURY AND SATURN.
hence we deduce
[4117]
[4118]
^Zr Then we find
Saturn.
,(0)
[4119]
6 j'= 2,000823:
6^^^ = 0,000042 ;
0. = - = 0,04058547 ;
6^°' = 2,00082368 ;
6^'' =—0,04057711.
—2
6^^ = 0,040610;
2
6'^ = 0,000001.
,(2)
6^^^=0,001236;
[4120]
[4121]
rf&è
da.
= 0,040662 ;
(1)
dbi
da.
1,001841 ;
(3)
dhh
da.
= 0,003085.
(0)
dH^
da.^
= 1,003904;
d<
da.^
:^ 0,091840;
(2)
dbj^
da.
= 0,060919 ;
(2)
dHi
1,469188.
VI.vi.'§23.] VALUES OF b'^ AND ITS DIFFERENTIALS FOR VENUS. 197
MERCURY AND URANUS.
hence we deduce
a= 4- = 0,02017895;
a"
6_\ = 2,00020360;
6!1', = — 0,02017792,
Then we find
6^^*= 2,000182;
6^''= 0,020183;
(0)
(1)
020196: ^^^ =1
(2)
è^'= 0,000306;
do.
do.
= 1,000913.
[4122]
[4123]
Mercury
and
Utanuf.
[4124]
[4125]
VENUS AND THE EARTH.
hence we deduce
«L = ^, = 0,72333230 ;
a
6 , = 2,27159162;
— 3
Then we obtain
6 ''! = — 0,672263]
(0)
b^ = 2,386343 ;
b'^ = 0,942413 ;
(3)
b. =0,323359;
6*" = 0,206811 ;
6 J = 0,090412 ;
▼OL. III.
.cn
60
6^ =0,527589;
2
6^ = 0,135616;
(8)
6i = 0,061101 ; 6^ = 0,041731.
[4126]
[4127]
Venus
and the
Earth.
[4128]
198
[4129]
[4130]
Venus
and the
Earth.
[4131]
[4133]
[4133]
(0)
db^
do.
(3)
djb^
do.
(6)
dbj
d a
(0)
dH^
do.^
(3)
dHj
do.^
(6)
do.^
(0)
do.^
(3)
d^
do.^
, (0)
PERTURBATIONS OF THE PLANETS ;
: 1,643709;
1,738781;"
0,867147 ;
7,719923 ;
9,112527;
: 7,842733.
: 66,55335 ;
: 62,87646 ;
[Méc. Cél.
63 = 9,992539 ;
,(3)
b, = 6,953940 ;
K3)
d_H_
do.
(I)
dbi
do.
= 2,272414;
(2)
dbi
do.
: 2,069770 ;
(4)
do.
: 1,407491 ;
(5)
db^
do.
= 1,113704;
df^
do.
: 0,668830.
do.^
= 7,531096;
(2)
d^fii
do?
= 8,558595 ;
(4)
d^i
: 9,107400 ;
(5)
dn^ _
/7«2
= 8,634030;
d^èl
do.^
(4)
dH^
do.^
= 57,35721 ;
66,32409 ;
,(i)
Ô; = 8,871894;
rt^)
b\ = 4,704321 ;
= 56,65440 ;
(4)
dbi
do.
(2)
dHi
= 58,19633;
dH
(5)
da»
i = 70,54326.
,(2)
6 y = 7,386580 ;
6 ; = 3,652052.
50,90290.
VENUS AND MARS.
[4134]
Venus
and Mars.
[4135]
hence we deduce
a = 4; = 0,47472320 ;
a
6^°J= 2,11436649;
6"j = — 0,46094390.
VI. vd.^SS.] VALUES OF i^;' AND ITS DIFFERENTIALS FOR VENUS. 199
Then we find
67=2,129668;
S
5^ = 0,521624;
3
fe'f = 0,187726;
5
6*^' = 0,074675 ;
6*^'* = 0,031127;
6'f = 0,013337; [4136]
2
(6)
6 , = 0,005829.
dh
(0)
do.
(3)
1 = 0,631752;
^ = 0,510976;
do.
(0)
^ = 2,192778;
do.-
(3)
dHh
do?
= 2,628516 ;
(0)
dH^
do?
(3)
dHi,
da?
7,65440 ;
= 10,66513.
(1)
db^
do.
1,330781 ;
do.
0,279002 ;
(1)
dH^^
do?
-- 1,815836;
do?
= 2,004429.
J .3
= 8,45655 ;
do?
(2)
dh^
do.
(5)
db^
da.
(2)
d^i
do?
0,884106 ;
0,147606.
= 2,795574 ;
i^ = 8,17676 ;
[4137]
[4138]
VonuB
and Mars.
[4139]
,(0)
6 = 3,523572 ;
2
.<3)
6, =0,722687.
6*3^ = 2,304481 ;
(2)
dH
da.
8,47521.
.(2)
&3 = 1,325959;
[4140]
[4141]
VENUS AND JUPITER.
a=- = 0,13907116;
VenuB and
Jupiter.
[4142]
200
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
hence we deduce
[4143]
b^'l = 2,00968215 ;
6''' = — 0,13873412.
Then we have
6^ = 2,009778;
6'^'= 0,140092;
6'J'= 0,014623;
[4144]
6^^=0,001695;
f4)
6^ = 0,000206;
(5)
6^ = 0,000026.
Venus and
Jupiter.
2
[4145]
\^ =0,142160;
a a
^=1,022206;
da
-^ = 0,212046 ;
da '
dh
'^ "* = 0,036783 ;
d 0.
"/* =0,006111.
da
rdi /irfîi
(0)
^=1,067532;
(1)
Ç^= 0,325869;
(2)
'^^*_ 1,575190;
[4146]
(3)
'^l^^ -0,533951.
[4147]
C)
63 =2,089736;
(1)
b^ = 0,432801 ;
(2)
63= 0,075054.
VENUS AND SATURN.
[4148]
Vennsand hcHCe WC dcduCC
Saturn.
[4149]
a = - = 0,07583790 ;
b^^ = 2,00287673 ;
b^'\ = — 0,07578334.
,w
Vr.vi.§23.] VALUES OF b'J AND ITS DIFFERENTIALS FOR VENUS.
201
riieii we obtain
(0)
6j = 2,002886 ;
(1)
= 0,076002 ;
(2)
-. 0,004323 ;
[4150]
6^' = 0,000273 ;
2
'*:
= 0,000018.
[4151]
(0)
''f * = 0,076331 ;
da.
(1)
dbi,
do.
= 1,006490;
(2)
dbi _
da.
0,114267;
[4152]
7/'"
,* -0,011085.
da
Venus anil
Saturn.
(0)
''^=1,019629;
ft a.-'
d^i
da?
= 0,172510;
(2)
dHi
do?
1,419950.
[41.53]
(0)
b\ =2,026116;
!>":-
^
= 0,229988 ;
= 0,021791.
[4154]
VENUS AND URANUS.
hence we deduce
Then we find
a = — = 0,03770634 ;
r\ = 2,00071095;
-'2
— 0,03769964.
[4155]
[41.56]
Venu3 and
Uranus.
,(0)
0^=2,000712;
,(1)
6^=0,037725;
,(2)
b\= 0,001067 ; [4157]
,0)
4
67 = 0,000034.
(0)
dbi
da.
VOL. Ill
= 0,716690 ;
(1)
dbj
da.
= 1,000829;
51
(2)
db i
do.
= 0,056634.
[41.58]
202
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
THE EARTH AND MARS.
[4159]
[4160]
a =. — = 0,65630030 ;
hence we deduce
Then
[4161]
The Earth
and Mnrs.
[4169]
[4163]
[4164]
.(0)
(3)
(6)
6, =
(0)
da.
(3)
(6)
da.
(0)
(3)
dHj
do.^
(6)
dH^
do?
(0)
dHh
do.^
(3)
dH^
2,291132;
0,224598 ;
0,046595 ;
= 1,228078;
: 1,240990;
: 0,473942 ;
4,985108 ;
6,057860 ;
: 4,388001.
29,03400 ;
33,29381 ;
, (0)
6_j = 2,22192172;
,(")
6_^ = _ 0,61874262.
= 0,804563 ,
a'^'
h
= 0,129973 ;
0)
= 0,028480 ;
(1)
dbh
do.
= 1,871211;
(4)
d b ^
do.
^ 0,920710 ;
7 7^"
db^
do.
= 0,333444.
(1)
dn^
do.^
= 4,744671 ;
(4)
d^bi
.1 .2
= 5,776483 ;
(I)
^ = 29,78930 ;
(4)
d^bi
Vf- = 36,32093 ;
(SJ
h-
= 0,405584 ;
(5)
= 0,077170;
2
: 0,0175565.
1,601236;
(5)
,7 „
= 0,666207 ;
(2)
dH^
do?
(?)
dHk
= 5,731111
5,141993;
(2)
îÇii. = 30,18848;
(5)
dH),
= 37,23908.
VI. v-i. §-23.] VALUES OF ù^'^ AND ITS DIFFERENTIALS FOR THE EARTH. 203
(0)
6 y = 6,856336 ;
6''' = 3,255964 ;
6^'' = 1,174650.
(2)
^ = 31,80897;
do.
,(')
J 3 = 5,727893 ;
fi'I' = 2,351254 ;
f^ =^ 4,404530 ;
6'^' = 1,671668;
[4165]
(3) (5)
'^ = 32,26285 ; .... 'Ill ^ 18,25867. [4166]
a a «a
THE EARTH AND JUPITER.
hence we deduce
Then
6^ = 2,018885;
6^^'= 0,004516;
,(6)
a= - = 0,19226461 ;
&'"[= 2,01852593;
6^'! = — 0,19137205.
6^ = 0,195003;
2
6'^'= 0,000779 ;
,(2)
6^=0,028195;
(5)
6^ = 0,000132;
[4167]
[4168]
The Earth
and
Jupiter.
[4169]
6, =0,000023.
(3)
0,200586 ;
= 0,070932 ;
d a
da.
1 ,043204 ;
0,016369;
(2)
da.
(5)
0,297995 ;
= 0,003448 ;
[4170]
204
PERTURBATIONS OF THE PLANETS ;
[Méc. Ct
[4171]
(0)
dHi,
1,132355;
= 0,746681.
The Earth d 0?
and
Jupiter.
(0)
[4172] 1!^ = 1,4727 14;
(1)
d^bi
= 0,466165;
Vf = 2,874986 ;
(2)
dH^
do?
1,628667;
(2)
^ = 1,418830.
[4173]
,(0)
6 ' = 2,176460;
(3)
h 3 = 0,032493.
6*'' = 0,619063;
5
,(2)
b =0,148198;
[4174]
THE EARTH AND SATURN.
hence we deduce
a = - = 0,10484520 ;
(0)
b_^ = 2,00550004 :
[4175]
(1)
6_ J = — 0,10470094.
Then
The Earth
and
Saturn
^'I'
= 2,005535 ;
[4176]
6?
'S
(0)
dbi,
d<x.
= 0,000724 ;
= 0,106155;
[4177]
(3)
dbi
do.
= 0,020779.
fe'l'^ 0,105283;
s
(4)
b, = 0,000066.
dJl
d Ol
= 1,012536;
,(2)
b =0,008282;
(2)
dbi
0,158723;
Vl.vi.§-23.] VALUES OF 6*;' AND ITS DIFFERENTIALS FOR MARS.
205
(0)
(0)
1,037816;
b = 2,050321 ;
^ 0,246193 ;
do?
: 1,526303,
(1)
(2)
b :
1
= 0,321144;
*,=
= 0,041977
[4178]
[4179]
THE EARTH AND URANUS.
hence we deduce
a = -=0,05212866;
[4180]
, (0)
6 ; = 2,00135893;
-i
.0)
6_^ = — 0,05211095.
[4181]
Then we find
. (0)
b =2,001355;
.0)
6- = 0,000089.
2
(1)
6 =0,052182;
6^=0,002040;
2
The Earth
and
Uranus.
[4182]
(0)
dbi
~ = 0,052288 ;
-(0
da.
' - 1,003060;
(2)
'^ = 0,078449.
o a
[4183]
MARS AND JUPITER.
a = — =0,29295212.
[4184]
VOL. III.
52
206
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
hence we deduce
[4185]
Then
(0)
b i= 2,04314576;
(1)
& , = — 0,28977479.
'r
= •2,045112;
b'^= 0,302922 ;
2
<-
: 0,066812;
[4186]
2
= 0,016357;
(4)
6^=0,004192;
"h
0,001109;
">
= 0,000297 ;
(7)
6^ = 0,000081.
(0)
dbi
da.
= 0,324004 ;
0)
,^ -1,105998;
do.
C2)
do.
= 0,473717;
[4187]
(3)
db^
da.
= 0,172096;
(4)
'^Z* =0,058420;
a OL
db'l
- 0 019258:
do.
Mara
and
Jupiter.
(6)
dbh
do.
= 0,006173.
(0)
dHi
do.^
= 1,338759;
3. m
^ = 0,794557 ;
(2)
do.^ '
= 1,871538 ;
[4188]
(3)
do.''
= 1,258858;
(4)
'^'\*- 0,623184.
(0)
d^bi
do.^
= 2,69358 ;
^ = 3,77722 ;
(2)
= 2,91068;
[4189]
(3)
dHi
do.^
= 5,47068.
(0)
*3 =
= 2,444762 ;
6 '=1,040206;
3
: 0,376693;
[4190]
'b%
i
=0,127942.
^
VI. vi.§23.] VALUES OF 6"' AND ITS DIFFERENTIALS FOR MARS.
207
(0)
db^
da.
= 3,48815 ;
(I)
db^
do.
= 4,80540 ;
(2)
^ = 2,99684.
a a
[4191]
MARS AND SATURN.
a = — = 0,15975187;
[4192]
hence we deduce
Then we find
(0)
(3)
6_ 1 = 2,01278081 ;
6^'^ =—0,15924060.
(0) (1) (3)
6, =2,012945; 6, =0,161305, 6, = 0,019347 ;
- 2 2
w
h-
= 0,002577 ;
= 0,000360 ;
(0)
dbi
do.
= 0,164463;
(1)
dhi .
da.
= 1,029493;
dbi
do. ~
= 0,048740 ;
do.
= 0,009065.
(0)
dHi
do?
= 1,090095;
(1)
dHi
da?
- 0,379322 ;
(3)
- n fi9nfi.S9
(5)
b^ = 0,000052.
(2)
dbi
0,244843 ;
(2)
Ç^ = 1,596248;
da.-'
[4193]
[4194]
Mars
and
Saturn.
[4195]
[4196]
da?
, !0)
6; = 2,119585;
b'l' = 0,503071 ;
,(2)
6y = 0,100136; [4197]
2
208
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél
MARS AND URANUS.
hence we deduce
[4198]
[4199]
Mars Then we find
aod
Uranus.
,(0)
[4200]
6^ = 2,003167;
, C3)
a = - = 0,07942807 ;
a"'
5'°^ = 2,00315565 ;
f\ = — 0,07936538.
—5
6'" = 0,079617;
(4)
6V = 0,000314 ; b.= 0,000022
2 2
,(2)
6^=0,004746;
[4201]
(0)
da.
(3)
d_H
d a.
0,079995 ;
0,011982.
(1)
dbk
= 1,007144<;
(2)
dH
da.
= 0,119822:
JUPITER AND SATURN.
[4202]
Jupiter
and
Saturn.
[4203]
a = - = 0,54531725;
a"
hence we deduce
(0)
6_j =2,15168241;
felj= — 0,52421272,
Then we have
6^^' = 2,1802348;
6*'' = 0,6206406 ;
•3
b'^ = 0,2576379 ;
VI. vi.§^3.j VALUES OF i'j'AND ITS DIFFERENTIALS FOR JUPITER.
209
(3)
bi =0,1179750;
(6)
5, = 0,0139345 ;
,(9)
èy= 0,0018056;
2
b^ =0,0565522;
(7)
b . = 0,0070481 ;
6*J"L 0,0008632 ;
6, =0,0278360;
(8)
6^ = 0,0035837 ;
bl'L 0,0003223.
[4204]
77'"'
db i
da.
= 0,808789 ;
(3)
db^
da.
= 0,726550 ;
(6)
dbi
da.
= 0,163506;
(9)
= 0,033083 ;
(0)
- 2.875229 :
da.^
(3)
'^^ = 3,533622 ;
dHi
do?
= 1,664586;
(9)
^ = 0,485135.
da?
da.
(4)
(7)
da.
(10)
db i
(1)
dHj
do?
(4)
dH^
da?
= 1,483154;
= 0,453285 ;
0,096019 ;
0,020265.
2,552788
(7)
dH^
1^
= 2,995647 ;
= 1,144377;
(2)
db^
= 1,105160;
(5)
db^
(8)
dbi,
da.
0,274717;
= 0,056171 ;
(2)
da?
(55
dH},
da?
(8)
d^b^
d a'
= 3,521040;
= 2,302428 ;
i = 0,760603 ;
[420.5]
Jupiter
and
Saturn.
[4206]
(0)
da.^
12,128630 ;
(3)
dH,,
= 15,454850 ;
d^bk
~~ = 14,958762 ;
VOL. III.
(I)
dHj
da?
= 12,878804;
(4)
^1^ = 17,058155;
Vf= 12,234874;
53
(9)
(Z^èj
= 12,832050;
(5)
dHk
(8)
= 16,655445; [4207]
, , = 9,566420.
210
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4208]
(0)
d^hi
(?)
d*bi^
84,40159 ;
= 89,8615;
(6)
dHi
118,6607;
(1)
d*bi
do}
(4)
d'bi,
da.*
(7)
da."
= 83,94825 ;
101,3809;
= 115,9588.
(2)
d'bk^
— 87,3027 •
da.*
(5)
dH^
da."
= 113,5238;
[4209]
Jupiter
and
Saturn.
[4210]
d'b^
da.^
(1)
= 747,480 ;
dH
(3)
d a:
I = 785,884 ;
(6)
^—^ = 912,301.
da."-
f^ = 4,358387 ;
6''' = 1,295672;
6^" = 0,273629 ;
b'^ = 0,053922.
3 '
d^bi
da.'
(4)
dHi
(3)
da.^
= 753,417;
= 819,180;
6^^^ = 3,185493;
z
f^ = 0,784084 ;
67 = 0,158799;
2
d'
H
d
a5
d^
(5)
761,843:
■■ 884,505 ;
(2)
b = 2,082131 ;
,(5)
63 =0,466047;
(8)
b , = 0,092290 ;
[4211]
(0)
^ = 14,681324;
da.
^=10,598611 ;
d a
db
i = 3,710043 ;
(1)
db$
da.
(.1)
dbj
da.
(7)
db§
d a
15,239657 ;
7,802247 ;
: 2,426079 ;
(2)
^=13,416026;
(5)
db§
da.
= 5,470398 ;
(8)
1^^=1,563695.
a a.
(0)
dHj
da.^
= 96,68536 ;
(I)
^2/,;t
VI- = 94.91 701;
d a.-'
(2)
Vf = 93,19282;
VI. vi.§'23.] VALUES OF li^" AND ITS DIFFERENTIALS FOR JUPITER. 211
(3)
dHè
d<x?
(«)
dH^
da?
(0)
d^i
do?
(3)
dH§
do?
(6)
dHi
da?
= 86,90215 ;
= 47,48185;
830,0586 ;
785,5855 ;
= 574,9115.
(4)
ll/f= 75,08115;
dHi
do?
35,74355.
do?
= 830,1580 ;
(4)
d^b3-
Vf = 740,6775 ;
do."
(5)
d^b§
do?
= 61,10115;
dH§
do?
= 810,1045;
(5)
d^b^
Vf = 666,4080 ;
do?
[4212]
Jupiter
and
Saturn.
[4213]
JUPITER AND URANUS.
hence we deduce
Then we get
6^°' = 2,038359 ;
S
6^ = 0,012879 ;
2
(6)
6, =0,000185.
do 4
a = — = 0,27112980;
a"
6'"! = 2,03692776 ;
6l'! = — 0,26861497.
6''' = 0,278966 ;
ft'"' = 0,003058 ;
da.
= 0,295410;
dix.
= 1,089551;
6 , = 0,056906 ;
6^^^ = 0,000745 ;
(2)
da.
[4214]
[4215]
Jopiter
and
Uranus.
[4216]
= 0,433630 ; [4217]
212
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Jupiter
and
UranCB.
(3)
^,^ - 0,145398 ;
a a
Ç^= 1,283434;
da''
[4218]
(3)
'^'\* =1,133359.
da''
(0)
b 3 = 2,372983 ;
[4219]
2"
6 , =0,099260.
(4)
da
rf^tT
= 0,045930 ;
= 0,714932 ;
(1)
b =-0,938794;
(5)
dbi
da.
(2)
dHi
da?
0,015410.
1,815451 ;
6 "'=0,315186;
3
[4220]
hence we deduce
SATURN AND URANUS.
a = — =0,49719638;
a"'
(0)
b 1=2,12564287;
[4221]
(1)
b i = — 0,48131675.
Saturn
and
Uranus.
Then we get
(0)
6, = 2,144440;
(')
6^ = 0,552007 ;
(2)
6 =0,208313;
2
(3)
[4222] 6^=0,086834;
(4)
6 =0,037909;
i
(5)
0^ = 0,016990;
(6)
6 , = 0,007728 ;
(V)
6 =0,003522;
i
.<8)
6 =0,001547.
2
(0)
dh 1
^ = 0,683055;
da
db.
(1)
= 1,373806;
(2)
1^=0,949128;
da
,(i)
Vl.vi. ^v,>3.] VALUES OF bj AND ITS DIFFERENTIALS FOR SATURN.
213
(3)
i^A = 0,572896 ;
do.
(6)
dbl
= 0,098799 ;
dbj
da.
(7)
dbi
do.
= 0,327198;
0,053642.
(5)
dbj
da.
0,181370;
[4223]
(0)
" 1^ = 2,377102;
= 2,881218;
d
a.-2
d°~
(3)
'b^
d
0.^
d^
(6)
H
d
0.^
rf3
(0)
d
a?
J3
(?)
bi
do?
rC)
1,067430.
= 8,798999 ;
= 11,904140;
6; = 3,750905;
(3)
b =0,872105;
d^bj
do!^
dn\
= 2,017767;
= 2,278077 ;
(1)
d 0?
9,578267 ;
d^b^
--^= 12,988670;
do."*
f^ = 2,547992 ;
6^'J = 0,482564 ;
(2)
d^bi
= 2,992245 ;
(5)
d^J.
= 1,616470;
(2)
do?
9,425450 ;
(5)
12,135721.
">
= 1,530452;
(5)
b = 0,262146.
[4224]
Saturn
and
Uranu».
[4225]
[4226]
(2)
^=9,75656;
do.
(3)
db^
-P = 7,24097 ;
da.
(4)
'iH ^ 4^95062.
do.
[4227]
YOL. III.
54
214 PERTURBATIONS OF THE PLANETS ; [Méc. Ctl
CHAPTER VU.
NUMERICAL EXPRESSIONS OF THE SECULAR VARIATIONS OF THE ELEMENTS OF THE PLANETARV ORBITS.
24. We shall now give the numerical values of the secular variations of
the elements of the planetary orbits. For this purpose we shall resume the
differential variations of the excentricities, perihelia and inclinations of the
îbVthe"' orbits ril22, 1126, 1142, 1143, 11461. To reduce these formulas to num-
computa- L ' ' ' ' J
(oln^Lc. bers, we must previously determine the numerical values of the quantities
(0,1), [^J], &c. These are obtained by computing, in the first place, the
values of (0,1), [ÔJ] ; by means of the formulas [1076, 1082],*
(1)
. _ 1 , 3 m. n a? . b_i
[4^8] CQ'^)=-4.(l-a^)^
I'onnHlas
14228'] [JM ] = o n_.2^a
(1) (D)
3m'. no.. I (l+a^).6_^ + Aa.6_è \
2.(l-a2)2
From these we have deduced the values of (1,0) ["iT|, by means of the
equations [1093, 1094].
(0) U)
* 2566. The values of mf, n, a, b_i, 6_j to be used in tliese formulas are given in
IA09R [4061 — 4222]. By the formula [4228] we must compute the values corresponding to
the exterior planets, namely ; (0,1), (0,2), (0,-3), (0,4), (0,5), (0,6); (1,2), (1,3), (1,4),
(1,5), (1,6); (2,3), (2,4), (2.5), (2,6); (.3,4), (3^), (3,6); (4,-5), (4,6); (5,6); and the
similar ones of [4228'], namely; [J^] Sic; [wi] &ic. ; [W] fee; [J±] &lc.; [JJ]
he; [Jfi^j. The remaining terms corresponding to interior planets are to be deduced
from these by the formulas [4229]. Thus, if it be required to compute (4,5), [^] cor-
responding to the action of Saturn upon Jupiter. The value of m' to be used in [4228],
[42286]
VI.vii.^^24.]
SECULAR VARIATIONS OF THE ELEMENTS.
215
(1,0)
m .\/a
7-(0,i);
m.\/a ^ 7)1. y a
Bv this means we have obtained the following results, in seconds, supposing
tlie numerical characters 0, 1,2, 3, 4, 5, 6 to refer respectively to Mercury,
Venus, the Earth, Mars, Jupiter, Saturn, and Uranus. The preceding masses
of the planets [AOG], AOGl d], hove been multiplied by 1 + f^j 1 + f^', 1 + /')
&.C. respectively, in order that these results may be immediately corrected,
for any change in the values of the masses, tohich may hereafter be found ne-
cessary.
(0,1) = (1 +,a').3",052453
(0,2) = (1 +|/').0%963818
(0,3) = (1 + /") . 0',040631
(0,4) =- (1 +(^'0.1',575473
(0,5) = (1 + f' ) • 0'>080560
(0,6) = (1 +,^'')-0',001702
mi = (1 +f^').l'',961407
Ul] = (1 +(^").0%457195
[m] = (1 +/-"')-0',012797
[ÎZ] = 0 +(^'0•0^ 146329
UKl = (1 +H-O-0~',004086
["îZJ "= (1 +(^'')-0''000042.
[42291
[42,30]
[4230']
[4231]
Mercury.
(1,0) = (1 +,a ).0^422318
(1.2) = (1 +/').7\416280
(1.3) = (1 +|j."').0',148161
(1.4) = (1 +M-'^). 4', 131 166
(1.5) = (1 +f^O-0',207370
(1.6) = (1 + ,a-) . 0%004354
(2.0) = (1 +,., ).0',097574
(2.1) = (1 +a').5',426695
(2,3) = (1 +;^-"').0',432999
rvi
=
(!+/-■■
.0,271367;
i,,.i
=
(1 +P-
/ \
6-, 174974;
[-^1
=
(1 +(-
///\
.0',085252;
iHi
==
(1+,-
v\
0',7 16427;
L^J
=
(1+'^-
V \
0 ,019641 ;
[ÎZ]
=
(1+1^
\'i\
. 0',000205.
[^]
;
(1 + 1^'
0',046285 ;
[IZ]
=
(1 +f^
\
4',5 18397;
[H]
=
(1+f^'
tl\
0',332961 ;
[4232]
VcTiua.
The Ea rill.
[4233]
is that of Saturn, ?n'- = ^l^ '"•." ^ [40611, the value of n is that of n'" = 109256'29.3
3339,40 ^^^ [4^8c]
[4077]; the value of a is 0,54.531725, [4202]; then we have è_à = 2.15168241,
b_h = -0,52421272 [4203]. Substituting these in [4228, 4228'] we get the values
of (4,5), [i£] as in [4235]. Lastly the formulas [4229] give (5,4) =
•(4,5);
[m;] = ™'^V^.[T£]; hence we obtain (5,4), \jr\ as in [4236], using the factor [4228(/]
1 +(*'" instead of 1 + /J.^ In like manner the other formulas [4231 — 4237] are to be
computed.
216
PERTURBATIONS OF THE PLANETS ;
[Méc. Ctl-
(2,4) =
(i+(^'0-
6',947861 ;
[m] =
= (l+^-'o-
1 ',662036;
TheEaiih.
(2,5) =
(l + p-n-
0,340441 ;
[iï] =
= (1 + ^^).
0%044514;
(2,6) =
(i+H-^O-
0',007095 ;
[ISJ =
= (1 + ^") .
0',000463.
(3,0) =
(i+O-
OSO 18662;
[m] =
= (1+f^ ).
0',005878 ;
-
(3,1) =
(1 + ^')-
0',491880;
[s^] -.
== (M-.').
0',283029 ;
[4234]
(3,2) =
(1+^-").
r, 964546 ;
r^] =
= (1+^").
1 ',510657;
MlTB.
(3,4) =
(i+i^-O-
14',411136;
Lm]
= (i+^-'O-
5',2 19092;
(3,5) =
(1+^')-
0%658341 ;
[^]
= (1+.-).
0,131041 ;
(3,6) =
(1+^-')-
0%013436;
[3,6]
= (ï+f").
0,001333.
(4,0) =
(l+f^ )•
0%000226
lii]
= (i+M- ).
0',000021 ;
(4,1) =
(1 + M-').
0%004291
[ÏZ]
= (i+,v).
0',000744 ;
[4235]
(4,2) =
(1+f^").
O',009862
[4,.]
= (i+fx").
0',002359 ;
Jupiter.
(4,3) =
(i+n-
0,004509 ;
[i^]
= (i+O-
0',001633;
(4,5) =
(1+pO-
7%701937
w^
= (1+,-^.
5%034195;
(4,6) =
(i + i^^O-
0',096647,
[ja]
= (1+H.^'O-
0',032446.
(5,0) =
(i + (^ )•
0',000027
; [m]
= (i+(0-
0^000001 ;
(5,1) =
(i+f^').
0',000501
; [ ^'M
= (1+.').
0',000047 ;
[4236]
(5,2) =
(1 + 1^.").
0,001123
; [^.^J
^ (1 +..")•
0^000147;
(5,3) =
- (i+O-
0',000479
; [m]
== (i+O-
0',000095 ;
(5,4) =
= (l+f^-)
17%90ô446
; Ua\
= (l+(^'0-
11%703495;
(5,6) =
= 0 +!'■') •
0,355214
■ [.6]
= (l+r'.^0-
0',2 13356.
(6,0) =
- (i+O
0',000002
; [M]
= (]+..)•
0,000000 ,
(6,1) =
= (1 + p-')
. O',000043
; [^]
= (1+.-')
0',000002 ;
14237]
(6,2) =
= (1+f^")
. 0',000096
; [^]
=^ (1 +(■.")
O',000006 ;
L '■'"-" J
1 1 rciti us.
(6,3) =
= (i+O
0\000040
; [Mj
= (i+O
O',000004 ;
(6,4) =
= (l+(^")
0%919814
U±\
= (l+(^-0
O',308803 ;
(6,5) =
= (1+^0
1 ',454 176
; un
= (1+f^')
0^873434.
[4237
25. By means of these values and the formulas [1122, 1126, 1142,
1143, 1146] the following results have been obtained; ivhich exhibit, at
the epoch of 1750, the annual variations of the elements^ during a year
of 3651 days, namely,
VI. vii.§25] SECULAR VARIATIONS OF THE ELEMENTS. 217
dl'
•2de
the annual sidéral motion of the perihelion in longitude in 1 750 ;* [42381
[4238']
= the annual variation of the equation of the centre, or that of double
the excentricity in 1750 ;t
-— = the annual variation of the inclination of the orbit to the fixed ecliptic r^omn
d t ^ [4239]
of 1750
Symbol! .
— -!^ the annual variation of the inclination of the orbit to the apparent ,.^„^,
d t ^^ [4239]
ecliptic ;
d è
-— = the annual sidéral motion of the ascending node of the orbit upon the
d t
fixed ecliptic of 1750 ;
de
— ' := the annual sidéral motion of the same node upon the apparent
d t
ecliptic. Î
[4240]
[4241]
* (2567) Neglecting terms of the order i^, we get u=^U-\-t.— — , by Taylor's [4238a]
theorem [.38-50a]. The time t is counted in Julian years [4078] and the values of n, n', n'
kx,. [4077] are taken to conform to this unit of time, so that n"i, which represents generally
the motion of the earth in the time t, will become simply n", in one year, or when
t=\. Now U being the value of m when < = 0, if we subtract it from the value for
dU [42386]
the case of ^=1, which by [42.38a] is U -\- —, we shall get the annual variation of
It equal to — . Therefore if we write successively «, 2e, ip, 9,, è, 6^, for u,
we shall obtain the annual variations of these quantities respectively, namely, -— ,
°' [4238c]
'^-T,' Ti' Ti' 77' Tr ^°^ '" [4080 — 4083] « represents the longitude of
the perihelion, e the excentricity of the orbit, 9 the inclination of the orbit, and è the longitude moqqji
of the ascending node of m, upon the ^retZ ecliptic. Moreover, 9, is, as in [1143'"], the
inclination, and ê^ the longitude of the node counted upon the apparent ecliptic. With one ^ .poo
accent above these quantities, they correspond to the body m'; and with iivo accents to the
body ni' , &«;.
t (2568) Neglecting terms of the order e^, in the equation of the centre [3748], it
becomes 2 c . sin. {nt-\- t — ra) ; the maximum value being 2 e, whose annual variation is [4239a]
^.~ [4238c].
X (2569) The formulas used for computing the values [4242 — 4248] are as follows. [4242a]
VOL. in. 65
218 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
MERCURY.
^ = 5',627032 + 3',014032 . (^'+ 0',929932 . (x"+ 0%041846 . f^'"
at
+ P,560043 . f^''+ 0',079478 . i>y+ 0S001702. f."'.
2 .j^ = 0',013690 + 0', 021948 . |x'+ 0',00651 1 . m-"— 0', 002330 . t^'"
— 0',012560 . i>>'+ 0',000116 . f^^+ 0 ,000004 . t^^
Mercury. ^ ^ ~ 0',1 19993 — 0',087951 . f^'— 0%000052 . ix"'— 0^028764 . f^^"
— 0 ,003215 . f^"— 0',00001 1 . /j-".
[4242] jf^-^ 0'' ^ ^'^^^^ + 0%068409 . H-'+ 0^000508 . f^"'+ 0^098085 . fx'^
+ 0%0 10373 . i>y+ 0%000033 . f^"'.
— = — 4",224994— P,764590. m-'— 0',963817 . p-"— 0',029951 .(x'"
dt
— P,396112 . M--— 0%068989 . (x^— 0^001535 . fx".
— ' = — 7%566802 — 0',097574 . f^ — 4',054426 . t^'— 0^963817 . (>■"
d t
— 0',143774 . fx'"— 2%1 87093 . i^^''— 0%1 17889 . m-"
— 0%002228 . fi.^'.
The values of — , —- , &ic. are given in [1126] ; 2.—, 2.—-, &c. are derived from
d t (t t (t I u t
[««, f"'-'^ '-?■ '^■■'^■'•"' 'iï- s.*"-'™" [.142, .143]. L..,y ^S f.8..
General ^nd -r-^ , — ^, &ic. are obtained from [1 146]. If we put i for the number of accents o«er
exprès- " ' " '
the'annuai <p; «) &•€. SO that (p*'', -sj^'^ Sic. represent the values of (p, «, 8ic. corresponding to the planet
of the eie- vvhicli is numbered i, according to the notation adopted in [4230] ; and suppose the sign 2
ments of,, 7.7 • 1 • i • /^
the orbits, of finite integrals to include all the values of k, contained m the series of numbers, 0, 1,2, 3,
[4242c] 4, 5, 6 [4230], excepting i =^ ]c ; then the four first of the preceding equations, may be put
under the following forms, as is evident by mere inspection,
[4242d] ^^=i;.J(i,Ar) — [Tr].^\cos.(:=<'>— «W)5; [1126]
[4242e] 2.^ = — 22.[TF].e^'i.sin.(ra«— b"'1); [1122]
_ _2%343127 — 4%315] 77 . (^ — 5',754638 . ^."+ 1 ',203777 . t^"
(It
VI. vii. §25.] SECULAR VARIATIONS OF THE ELEMENTS. 219
VENUS.
+ 6',435827 . (j."+ 0^083814. (x'+0',003269 . ,j.''.
2 .^ = — 0%260567 — 0',090479 . fx — 0%101170. fx"— 0',006378 . >'■"'
a t
— 0%061 143 . <— O',001409 . f'+ 0',000012 . m-".
^ = — 0%015950 + 0S025200 . ^ + 0%002I57 . f^'"— 0%037854 . (x- ^^^^^
— 0',005455 . fx'' -I- 0%000002 . /x"'.
^' = 0%044538 + 0',019377 . f^ — 0',004148 . fx"'+ 0%025810 . t^- [4243]
+ 0',003500 . fx'— 0^000001 . t^-.
— = _ 9,900996 + 0%342053 . fx _ 7',416280 . f."— 0%0761 12 . m-'"
— 2',66 1705 . fx-— o%085589 . K— 0',003363 . ,.''.
^ = — 18%387762 + 0', 165450 . fx — 5^426693 . m-'_ 7^416280 . v^'
— 0',286675 . fx'"— 5',133067 . jx"— 0',285519 . v"
— 0',004978 . fx".
^ = 2.[ a_] .tang. ^(«. sin. (ôO-ô^'i); ^242/]
[1142,1143]
In like manner the expressions [1146] may be reduced to the forms [4242i, fc], supposing
the orbits of all the other planets to be referred to that which is numbered I [4230] ; ?/'' [4242ft]
bebg the indination, and â|'' the longitude of the node of the orbit denoted by i referred to
that which is denoted by I; conformably to the notation [1 143''] ; the fixed plane being the
orbit of /, at the epoch 1750,
-^ = 2.{ (i,Ar)_(/,A') \ . tang. <p'*'. sin. (âio_^(B); [4242i]
-^ = -{l,i)-Mi,^) + ^A{i,^-{l,^\-~^yCO,.{è'~^-è^''^). [42424]
220 PERTURBATIONS OF THE PLANETS; [Méc. Céi.
THE EARTH.
-- = 11* ,949588 — 0',414923 . f^ + 3',813276 . (^'+ r,546163 . ^^"'
at
The Earth
4- 6^804392 . i>^"+ 0', 194066 . 1^."+ 0',006614
M-'
[4244] 2 /-^ = — 0',1 87638 — 0',008057 . (^ + 0',030435 . i^'— 0',049410 . f/-"
— 0', 159738 . i>'"— 0S000909 . f^''^- 0',000040 . f.^'.
Instead of excepting Ic = i [4242c] , we may suppose the sign 2 to include all the numbers
[4242Z] 0, 1,2, 3,4,5,6 [4230]; putting {i,i)=iO, [TTJ = 0, in all the formulas [4242«i — A:] ;
observing also that the first term of [4242Ar], namely — (^j ^) is that which arises from the
tano^. &^''^
[4242m] value A; = i, under the sign 2 ; because then ° — — = 1; cos.(é''' — Ô(''>)=:1. We may
moreover remark, that as the orbit of the planet /, in 1750, is taken for the fixed plane
[4232/t], tang. <p"' must be of the order m, and since this is multiplied, in [4242/], by quanti-
[4242n] ties of the same order, the product will be of the order m^, which is neglected ; likewise the
term depending on tang. 9''' vanishes, because it is multiplied by sin. {&'■''' — â*'') = 0. If
we now substitute in [4242f/— t] the values [40S0— 4083, 4231—4237], we shall
[4242o] obtain the expressions [4242 — 4248] For the sake of illustration, we shall give a few
examples of the numerical calculations in the following notes.
* (2570) As an example of the formula [4242(/J, we shall compute the action of Mercury
on the Earth, in which case i^ 2, A: = 0, and the corresponding terms of this formula
[4244a] are (2,0) — [Ml • -• cos. (n"— w). Substituting the values of (2,0), [aiô"], e, e", to, ra"
[4233, 4080, 4081], it becomes.
[42446]
(1 + p.) • ^^ 0-,097574 - 0',046285 . °^^lf^l- cos. (98^ 37" 1 6-73" 33'» 58') ^
= (1 + fJ^) . { 0',097574 — 0',512497|= — 0^414923 — 0^414923 . ij. ;
in which the part depending on fx is the same as in — — [4244], the other part — 0',414923
is included in the constant term 1 1',949588, which is the sum of all the coefficients of /a, ja'd,
14244c] .... dzi"
&ic. noticing their signs. This constant quantity represents the value of -r-r-, supposing (a,
/J.', &.C. to vanish, or the numerical values of the masses [4061] to be correct.
VI.vii.<^25.] SECULAR VARIATIONS OF THE ELEMENTS. 221
MARS.
^" ^ 15',677160 + 0',015944 . \>. + 0^511046 . f.'+ 2%129320 . ^'
a t
+ 12%312891 . (/-"+ 0^693878 . f^^+ 0',014082 . ^''K
2. ^"= 0%372537 + 0^002363 . (x + 0',001566 . ,x'-f 0',040492 . /'
+ 0',314982.,j."+ 0',013167 . p."— 0^000032 . f^".
1^ = — 0',293800 + 0^,000092 . ^ — 0',013146 . f^'— 0^254879 . m-'"
— 0%025790 . vy— 0',000076 . ^^K *'""
1^ = — 0%012984 — 0',000388 . (/. + 0',131893 . f^'— 0S131999 . f."
dt
— 0',0 12454 . V- — 0',000036 . p". [4245]
— = — 9%728234 + 0%052224 . \^ + 0',3 14067 . (.'— P,964546 . ^'
a t
— 7%855103 . ^i'— 0',266532 . f^"— 0%008345 . ,x^'.
^ = — 22%789674 — 0',31 8395 . fx — 8%577599 . f.'— 1 ',964546 . fx"
— 0^,432999 . fx'"— 1 P,015955 . i>'r— 0',469146 . fx"
— 0',011033.fx-.
de"
In like manner the terms of 2 . — [4242cj, depending on Mercury, become by using [4244ti]
tiie same values as above,
— (1 + fx) • [111] -26. sin. (ra"— -a)
= — (I + (x) • 0,046285 X 2 X 0,20551320 . sin. (98'' 3T" 16'- 73'' aS" 58*) [4244e]
= — ( 1 + ^) . 0',00805T = — OS008057 — 0',008057 . fx,
in which the coefficient of (x is the same as in [4244], and the quantity
— 0',00S057 forms part of the constant quantity — 0", 187638 [4244], as in the
dTS" r T T •! rfwCO
case of — — [4244c]. In like manner we may compute any other values —rrt
d_f>
dt '
VOL. III. 56
222 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
JUPITER.
'^— = 6%599770 + 0',000186 . f^^ + 0',004330 . (^'+ 0',009837 . m"
dt
Jupiter.
+ 0^002047 . f^"'+ 6%457871 . (j.''+ 0% 125498 . ix"':
0%55441 8 — 0',000008 . m- + 0',000009 . f^'+ 0',00(
— 0',000191 . /x"'+ 0',553308 . f>-''+ 0',001220 . i^",
* ^ = _0, 078140 + 0',000022.(x+0',0001 01 . |x'+0^000112.f^"'
at
— 0',078933 . M-' + O',000557 . f^".
^ = — 0^223178 — 0%009491 . m. — 0%128114 . (^'— 0',010646 . (x'"
[4346] _ 0^075444 . (^^"4- 0%0005 1 6 . f^^'.
— = 6^456281 + 0',000509 . (^ + 0%005857 . i^'— 0%009862 .(."
dt
— 0%000461 . f^"'+ 6%505571 . f^"— 0%045332 . j."'.
^^ _ 14%663377 — 0',316227 . t>- — 12S828736 . i^'— 0^009862 . t^"
dt
— œ,389153 . (^"'— 6%947861 . f^'^+ 5%877561 . t^"
— 0,049 100 . H-^'.
* (2571) As an example of the use of the formula [4242/], we shall compute the
[4946a] part of — — depending on the action of Mars. In this case i ^ 4, A: = 3, and
the corresponding terms of the formula become, by using the values [4080 — 4083.
4231 — 4237] ;
(4,3) . tang. (?'". sin. {&"— é'")
[42466] = (1 + 1^'") ■ 0',004509 X tang. I'' 51™ X sin. (97'' 54™ 22'— 47'' SS" 38')
= (1 +|i,"').0',000112 =0',000112 + 0%000112.fA"'
of which the part depending on ;*'" is the same as in -— [4246], and the other term
0' ,000112 forms part of the constant quantity — 0',078140 of this formula.
[4246c] In like manner by putting i = 4, A = 3, Z = 2 in [4242 J], and using the same data,
VI. vii. ^S25.] SECULAR VARIATIONS OF THE ELEMENTS. 223
SATURN
+ 0%000550.f^"'+ 15',790810.(x-+0%3]9768. f.'
, = 16%1 12726 + 0',000022 . t^ + 0',000496 . fx'+ 0S001080 . i^"
(It
2 . ^ = — I ',080409 — OSOOOOOO . f. + O',000000 . f.'+ 0^000001 . /-•"
a t
— 0%000016 . (^"'— P,099919 . fj^"+ 0%019524 . (x-.
1^ = 0%099740 + O',000003 . fx + 0^00001 8 . m-'+ 0%000014 . i>!" ^^^^^^^
+ 0^,096696 . H-" -f 0^003010 . fx".
^ = — 0-,155290 — 0^,010955 . fx_ o-,1939] 8 . ^— 0%012542 . f^'" [4247]
+ 0%059175 . fx-+ 0', 002950 . m-".
* 'jj =— 9%005292 + 0%000004 . ,x + 0',000042 . ^^— 0%001 123 . ^'
— 0',000323 . (x'"— 8%734249 . f^-— 0%269642 . iCK
^ = — 19^041499 — 0',1 10961 . fx — 5',883249.f^'— 0',001123 . ^."
at
— 0', 141 41 4 . p.'"— 12',292960 . t^"— 0',340441 . (x"
— 0',271351 .fx".
we get the part of —7^, or as it is called -7^- [4246], depending on Mars, equal to
\ (4,3) - (2,3) I . tang. 9'" . sin. (é-— é'")
= (1 + fx"').|0' ,004.509 — 0',432999| X tang. 1'' 51"' X sin. (9T'54"'22'— 47"' 33™ 38')
[4246rf]
= _ (1 _f- ^"') . 0',010643 = — 0',010643 - 0',010643 . ix'",
which agree very nearly with the corresponding terms of -77 [4246].
* (2572) Putting i = 5 in [4242g-], we get the expression of —, and the terms
corresponding to the action of any one of the planets, is found by using the value of k
corresponding to it ; thus for Mars k = 3, and the terms depending on this planet
become, by using the data [4080— 4083, 4231 — 4237],
224 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
URANUS.
^' = 2',454851 + 0',000003 . (^ + 0',000043 . ^'+ 0',000095 . fx"
at
+ 0',000048 . ^^"'+ r,2 10830 . i>>''+ 1 ',243833 . i^\
2 . — = — 0', 1081 84 — 0',000000 . (^ — O',000000 . f^'— 0',000000 . /x"
dt
+ 0',000000 . fj.'"— OSOl 1952 . yi"— 0',096232 . i>y.
^ = _ 0',048861 + 0',000000 . i^ + 0',000000 . f/.'+ 0%000000 . f^'"
""""'• — 0',009036.(x-— 0^039826.,x^
4^'- = — 0',027460 — 0',005492.pi + ^,010145./— 0',005907 . f^^'"
a t
[4248] + 0',05921 7 . 9^" — 0%030502 . i>.\
— = 2^,700876 +0',00001 7. f^+0S000146.fx'—0',000096.(^"
dt
+ 0',000047 . M-"'+ 0',496382 . i>:"-\- 2^204381 . i>.\
l!il' = _ 34\403396 — 0',78851 7 . \>- — 23^81 5885 . i>!— 0^000096 . f^"
dt
— 0',938767 .M-'"— 10',200902 . ,x-+ 1 ',347 866. j^"
— 0',007096.fi.".
■(5'3)+(5'3)-tïï!|^---(^'-^"')
= (I + fx."'). 5 — 0-,000479 + 0'.000479/^"^' ^,„^^'"°'.cos.(lll''30"'23'-47''3S'"38^) ]
\ ^ '^ ' I ' ' - tang. 2'^ 29™ 55' ^ ' )
[4247i]
= (1 + fJ^'") .{ — 0',000479 + 0^000156 }
= _0',000323 — 0^000323.fA"', as in ^[4247].
d é^ •
Putting 1 = 5, Z = 2, in [4242^], we obtain the expression of — j-, in the notation
[4247c] °f [4247]. The term of this expression corresponding to Mars, is found by putting k=3,
and using tlie above data, by which means it becomes,
VI. vil. § ^'5.] SECULAR VARIATIONS OF THE ELEMENTS. 226
The variations of the earth's orbit are not included in the preceding formulas ;
they may be determined by the equations *
tang. <?". sin. f = p" ; tang. /. cos. o" = q". [4349]
With resjiect to the values of p", q", we may determine them by the
formulas [1132, &:c.], and we have, by taking the ecliptic of \1 50 for the [4249]
fixed plane,i
d p"
in which t is the number of Julian years elapsed since 1750, and -rr^
[4250]
r/q" rldp'
, , &.C. are taken to correspond to that epoch. It is only necessary
to notice the first power of t in these formulas, if t be less than 300.
If t do not exceed 1000 or 1200, we may reject the third and higher powers
of t ; and we may do the same even with the most ancient observations,
[4250']
[4250"]
- (5,3)+ <(5,3)-(2,3)}.^^'.cos.(év_r)
= (!-]_ |j,"'). ) _ 0^000479 + (0',000179 — 0',432999). ^'.^^ ^■. cos. 63-' 51" 45'^ [4247rfJ
\ ^ '^ ' I IV. I / tang. S"* 29"' 55» 3
= (1 +fj-"').{— 0',000179 — 0',141035| = — 0,141514 — 0%141514.(a"',
which differs 0',0001 from that given by the author. We have thus given an example of
the numerical calculations of each of the formulas [4212(/ — k'\.
* (2573) The formulas [4249] are similar to [1032], accenting p, q, Sic. with tioo r4249„]
accents, in order to conform to the case now under consideration.
t (2574) Putting successively m =p"; U = p" ; or u = 5", TJ = q", in the
formula [3850«], we get the following expressions of p" , q",
in which the quantities p", q", and their differentials, in the second members, correspond
to the epoch of 1750. Now at that epoch we have 9" = 0 [4249'] ; substituting this in [42506]
[4249], we get p"= 0, q"= 0 ; hence the formulas [4250a] become as in [4250].
VOL. III. 57
226 PERTURBATIONS OF THE PLANETS; [Méc. Cél.
taking into view their imperfections. We obtain from the formulas [4250],
Value. the following results.*
CO r re 3-
po.ndiag
rrrlil '^ = 0%076721 + 0',008420 . m- + 0%0863 16.^'+ 0%009423 . ,a"'
odiit. a t
[4251]
— 0»,022021 .M.'"— 0^005446 . i^--+ 0s000029.|x^'.
^ = — 0%500955 — 0%008522./x — 0',309951 .,/- 0%010335.f/^"
— 0',1 58234. /x'"— 0',013821 .F-^— 0%000091 .f^^'.
theperi" 26. Wc havB seen, in [4037], that the oblateness of the sun produces, in
helion de-
the"t'n,p°" the perihelia of the planetary orbits, a small motion, which is represented by,
cily of the
sun. y-j2
[4252] 5«=.(p_X^).-:.„^.
* (2575) If we substitute tlie values p", q" [4250J, in the terms of -j-, —
d p"
[1132], depending upon p", or ç", they produce terms of the order {(2,0) +(2,l)+&ic. }• — ;
[4251a] . , , . . ., dp" dq" ,. , • , c u <• u
or 01 the order m m comparison with — , — , which occur in the first members ol these
^ dt' dt'
equations ; therefore these terms may be neglected, and then the values of — — , — •
[1132], become,
Ji" = (2,0) . q + (2,1). 2' + (2,3) . q"'+ &c. ;
[42516] . „
^ ^ _ (2,0) . p - (2,1) ./- (2,3) .p"'~ he.
[4251c] Substituting p = tang. cp. sin. é, p' =tang. 9'. sin.ô', &,c.; q ^tang. q> .cos.^, &;c. we get
[4251rf] ''■JT ^ (2.0) • tang. <p . cos. ^ + (2,1 ) . tang. 9'. cos. 6' + (2,3) . tang. 9'". cos. ()'"+ Sic. ;
[4251e] rfl" ="~ (^'^^ • tang.9-sin.â — (2,1) . tang.9'. sin. â'— (2,3) .tang. 9'". sin.ô'"— he ;
and by using the values [4082, 4083, 4233], they become as in [4251] nearly. Thus the
(l p"
term of —- , depending on Mars, is
[4251/] (2,3). tang. 9'". cos. r= (1 + |x"') .0',4.32999 X tang. l-'Sl™ X cos.47''38'" 38'
= (l+H."')-0''009423,
Vl.vii.§26.J SECULAR VARIATIONS OF THE ELEMENTS. 227
We shall consider the motion relatively to Mercury. Now q is the ratio of [43531
the centrifugal force to gravity at the solar equator [4028] ; and if mt be
the sun's angular rotary motion, the centrifugal force at the solar equator will ^oss'i
bo ni'D* Puttine; the mass of the sun enual to S, we havef -^,.l=^»"^ or ,,„,„
* 1 a^ [4254]
.S = ti"-. a"', which gives the gravity at the solar equator,
S n"2.a"3
2)3 DP '
therefore we have %
m' D^ /ot\2 /D\3
[4255]
The time of the sun's revolution about its axis, according to observations, is
nearly equal to 25'*°y%417. The duration of the earth's sidéral revolution is [4257]
365'""^S256 ; hence we obtain,
TO 365,256
n" 25,417
The apparent semidiameter of the sun, at its mean distance, is 96P,632;
which gives
[4258]
[4259]
dp"
in which the coefficient of iu-"' is the same as in the value of — - [4251]. In like manner
at
dp" dq"
we find the Other terms of ——, — — [42511.
dt dt ^ ■■
* (2576) The angular rotary velocity being to, and the equatorial radius D ; the actual
velocity of a point of the surface of the equator will be represented by to D. The square [4253o]
of this, divided by the radius D, gives the centrifugal force [54'], equal to m^D, as
in [4253].
t (2577) We have n^ = ^3 = "^^^ [3700, .3709a] ; and in like manner n"'^=^^ . ^^254a]
Now changing M into S to conform to the notation [4254], neglecting also to" in comparison
Ç /t"3
with S, we obtain -^ = n"^ [4254]; multiplying by — we get [4255]. [42546]
jt"2.a"3 .
t (2578) The centrifugal force ni'D [4253'], divided by the gravity -p— , gives q
[4253], as in [4256] ; substituting the values [4258, 4260] it becomes r4255o]
q = (^54^^)^. (sin. 961',632)=' = 0,000020926, as in [4261].
228 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4260] ^ = sin. 961%632;
a
therefore we have
[4261] y == 0,0000209268.
If the sun be homogeneous, we have p = l;q [1590', 1592'], m which case
the motion of Mercury's perihelion [4252], produced by the ellipticity of
the sun, is*
or the equivalent expression,
[4264] <5 « = ^7 . (sin. 961 ',632)-. T- Y. nt.
[4262]
If we substitute in this formula the values of n, «, a" [4077, 4079], it
[4265] becomes f>-r^ = 0 ,012250.^ ; so that it increases '-^ [4242] by the quantity
0',012250, which is nearly insensible. This must be still farther decreased
if the sun be formed of strata whose densities increase from the surface to the
centre, as there is reason to believe is the case.f Hence we may neglect this
[4266] expression for Mercury, and much more so for the other planets. The
variations of the nodes and inclinations of the orbits, depending on the
same cause, may also be rejected on account of their smallness [4045']
* (2579) The density of the sun being supposed uniform, we have 'h? =: ^q nearly
[4262a] [1590']. Moreover by [1592'] the polar semiaxis being 1, the equatorial semiaxis is
\/{\ -f- 'i^) = 1 + J^^ =^ 1 4~ i*/ nearly ; so that the ellipticity p is nearly equal to fç, as
in [4262] ; substituting this in [4252J we get [4263]. Now we have
[42621] V^-7,- 5 = ( -"• 961',632)^ g)' [4260] ;
hence [4263J becomes as in [4264] ; and by using the values of q, a, a", n
[4261, 4079, 4077], it becomes as in [4265], namely,
[4262c] 5^^ I X (0,0000209268) X (sin. 961',632f X (0,.38709812)-2 X 538101 6^ < = 0',01 250. <.
t (2580) The effect of increasing the density towards the centre is seen, in the extreme
r4266a] case, when the whole mass is collected in the centre, and p = io-tp [1732'"]; or in
the present notation f^hq [1726', 4253]. Substituting this in [4252], we get ira=Oj
so that in this case the ellipticity has no effect on the motion of the perihelion ; hence it
[42665] appears that this increase of density, towards the centre, decreases the motion of the
perihelion. We have supposed, in this example, that I) remains unaltered, the density
being considered as infinitely rare, from the suiface towards the centre.
VI. viii. V-î"?-]
THEORY OF MERCURY.
229
CHAPTER VIII.
THEORY OF MERCURY.
27. The inequalities of the planets which are independent of the
excentricities, and those which depend on the first power of the
excentricities, were computed by means of the formulas [1020, 1021, 1030],
having previously ascertained the values of ^"'*, ^^'' &c. and their
differences, by the formulas [963'^ — 1008]. The results of these
calculations are contained in this, and in the following chapters, neglecting
the perturbations of the radius vector, whose effect on the geocentric
longitude of the planet is less than one centesimal second. To determine *
[4267]
TertiiB
whieh
may be
neglected
on account
of their
ainallnc".
• (2581) Let S be the sun, E the earth, M Mercury, supposing it to move in
the plane of the ecliptic ; S T the line drawn from the sun towards the first point of Aries in
the heavens, being the hne from which the longitude v, v" are counted. Then S E =t"
F
74
(4Q(Jf''al
Hence the longitude of the sun, as it appears from
the earth, is 180''-[-f"; and if from this we
subtract the angle of elongation SEM =^ E,
we shall obtain the geocentric longitude of Mercury
V=lSO''+i'"— £. Now if SM=r be
increased by the quantity MJ\1! = or, the angle
E will increase by the quantity MEM'z^SE,'^'
while V, v" remain unaltered ; therefore the variation of the preceding value of V will
be 5V^ — f5£. If we draw .M'.Y, EF, perpendicular to EM, .S./V/ respectively,
w-e shall have in the -similar triangles J\1JYM', MFE; ME : EF :: MM : M'N; [4268rf]
EF
ME'
(5 E = — 5 V :
[42(3S6]
[4268c]
hence iV/'JV=dr.
angle
MEM'
Dividing this by M' E, or ME, we obtain very nearly the
EF
or
substituting EF = S E.sm.ESM
ME'i'
=r".sm.{v—v"), and ME^=r"^—2r"r .cos.{v—v")-}-t^=r''^.\l—2<x.cos.{v — v")■j-oJ
[6■2 Int. 4268], we get [1269].
VOL. III. 68
[4'i(>er]
230 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
the limit, which an inequality in the radius vector must attain, to produce
one second in the geocentric longitude of Mercury, we shall observe that if
[4a68] we put this longitude equal to V, and r = ?"a-, we shall have for the
variation ôV corresponding to Sr,
,. Sr sin (v — v")
[4270]
The maximum of the function
s'in.(« — v'')
1 — 2a.cos.(j;— i;")-l-a''
corresponds to^
[4271] cos.(z^-t?) =j-p- ;J
[4271'] which gives ^ [4270e] for this maximum ; therefore we shall
then have,t
* (25S2) The maximum of [4270] is found, by taking the differential, supposing v to
270 b^ t'l^ variable quantity; putting it equal to zero, and dividing by d.{v — v"). This
differential expression being multiplied by |1 — 2a. cos. (y — ii")-|-a-p becomes, without
[4270i] reduction, as in the first of the following expressions, and this is easily reduced to the
last form [4270(/] ;
[4970c] 0 = cos.(«— v") . \ 1 — 2o..cos.(« — v")-\-a?l — 2a. sin.^. (t) — v")
= (1 +o.^).cos.{v — v")—2a..{cos.^.{v—v") + sm.^.{v — v")]
[4370dJ = (1 +a2).cos.(i'— î>") — 2 a.
From this we easily obtain [4271] ; thence
. / 4a2 U_l-a2
[4270e] V {l + c>-2)V l+a2
l-2a.cos.(.-.") +a2= 1 - .^-p^, + a=== \-:^^ .
Dividing the first of these expressions by the second, we get the value of the maximum of
the function [4270], as in [4271'].
t (2583) Substituting in [4269] the value of the function [4270], at its maximum
t4271a] 6r 1
[4271'], we find &V = ; . :; :,; hence we get or [4272].
n. viii.^27.]
THEORY OF MERCURY.
231
<S,- __,•".(! —a'^).. 5 V. [4272]
If we suppose iV = ± 1"= ± 0,324, and take for r, r", the mean
distances of Mercury and the earth from the sun [4079], we shall have by ^^^^^^
\vliat precedes r" = 1 ; a = 0,38709812 [4095] ; hence we obtain*
6r=^ 0,000001335 ; [4274]
therefore ive may neglect all the inequalities of the radius vector of Mercury,
in which the coefjicient is less than, rt 0,000001. Among the inequalities of [4275]
the motion in longitude, we shall retain generally only those whose coefficients i„equaii-
e\ceed a quarter of a centesimal second [0,081]; but as the inequalities ^^^^J^»
depending on the simple angular distances of the planets can be introduced ^«JTbe
into the same table with those of greater magnitude, they are retained.
Inequalities of Mercury, independent of the excentriciiies.
iv = (1+0
+ (l+0-
0%662353 . sin. (n't — nt + e— s) \
— r,457111 . %\n.2{n't — nt + ^' — ^) j
— 0', 128075 . s\n.3(n't — n < + s' — 0 '
— 0',029264 . sin.4(n'« — nt + s—^)
— 0%008905 . sin.5(n'« — nt + i —()
0',201 688 . sin. (n"t — nt + ^"- s)
— 0', 165645 . sm.2(n"t — n i + «" — 0
— 0',016901 . sin.3(n"« - nt + s"- 0
— O',003127 . sm A(n"t - ni + s"- 0
[4276]
Inequali-
ties inde-
pendent of
the ec-
centrici-
ties.
0',569336 . sin. {n'H — nt + i"- 0
_|- (1 4- ^iv) . I _ 0-,l 1 8384 . sin.2(n'H — nt + ^'— 0
— O',003 118. sin.3(M''i — nt + s'^— 0
* (2584) Using the mean values r = a, r" = a" [4079], we get a [4095],
substituting these and 6V = ± 1", or o V = dz sin. 1" = ± 0,324 . sin. 1% we
obtEÙn [4274]
[4274o]
232 PERTURBATIONS OF THE PLANETS ; [Méc. C^l.
/ 0,0000000376* \
\ — 0,0000004094 . cos. (n't-nt + i'-s) 1
[4277] i r = — (1 + /) . + 0,0000015545 . cos.2(n't — n i + e' — i)
+ 0,0000001702 . cos.3(n't — 7i i + s' — s)
+ 0,0000000437 . cos.4(n' t — 7it + i—s)
* (2585) The parts of or, 5v [1023,1024] independent of the excentricities are,
by using T [.3702a],
[4277a] ^. = _^ . a3.(^_ j+ _ . V. I -^^,,^,,^_^ ^■cos.^ i';
[42774] 5 u = — . 2 . ) :- . a ^" -\ r-. ,. , ' -5 -r, ( • sin. i 1 ;
[4277rf]
[4277c] in which m, «, w, s correspond to the disturbed planet, and m', a n', s', to the
disturbing planet. These expressions must be accented so as to conform to the notation
[4061, 4077 — 4083], taking for i all integral numbers from i= — œ to z'^co. For
example, if we wish to calculate the action of Mars on the earth, we must, in the formulas
[4277«, &], change m, a, n, s into vi", a", n", s", &c. corresponding to the
[4277 e] disturbed planet; and m', a', n', s'. &c. into ?;/", «'", n'", b", kc. respectively,
for the disturbing planet.
As an example of the use of these formulas we shall apply them to the computation of
the perturbations of Mercury by the action of Venus. The constant part of 5r deduced
r4Q77/'l
from the first term of [4277a] is as in the first expression [4277ri]. This is successively
reduced, by the substitution of the values
•0)
fdA^o)^ 1 dbi ^g^^g^^ „ = 0,.38709812 [4079],
da ) (i''2 do-
(0)
[4277g-] -= a = 0,53516076 [4085], -^ = 0,780206 [4088], ?«'== ii^ [4061]:
a' d CL oboloU
III)
m' „ /'dA('>)\ m' a" db^
ùr =^ -—.cr. -~ — 1 = ■ a.
G ' \ da J 6 a'2 da.
[4277/t] ^ (0,
= - - . a a2. — : = — (1 +,a') . 0,0000000.370, as in [4277].
Again by putting successively i=!, i=— 1, .^'i'=^(-" [954"J, in [4277o],
and connecting the two terms, we obtain the part of &r depending on cos. T, namely.
VI. viii.§27.] THEORY OF MERCURY. 233
Iiicqual'.ties depending on the first power of the excentricities.*
0-,295201.sin.(H.'^ + .-'_^)
-4-,030852.sm.(2?i'«— nt + 2-:'— £ — ^)
- ]%686n4>.s\n.{3nt~2nt + 3e' — 2s — z^)
6-i, = (1 + ,/) . ^ + 0'/J93989 . sin. (3 n't — 2nt + 3s—2 s — ^') ) [4278]
+ (»%'293992. sin.(4 n't — 3 n < + 4 .' — 3 £ — ^)
— 0%17682;).sin.(2M/;_ nf + 2e— i' — ^)
+ 0%394 1?,6 . sin. (3 n t—2n't + 3 s —2e' — ^)
Ô r = m' n^a.) ^ '^ " ^ , "~" ( . cos. T ; [4277i]
[42774]
in which we must substitute a.^"'==a^ — a.OA, o-'. ( == a^ — a'^.
\ da J do.
[997,1000,963''], and use tlie values [4277f] coriesponding to the disturbing and disturbed
planets. Tiius in computing tiie action of Venus upon Mercury, we must use the
values o, a, m'[4-277o-], ji = 538101 6',736, ?i'-^ 210664 1',520 [4077], ^i, [4087],
(1)
j^ [4088], and we shall get (5 r = 0,0000004094 . cos. T, as in the second
line of [4277]. The terms depending on cos. 2 T, cos. 3 T, cos.4 T, &c. are found
from [4277n], by using successively, i = =p2, i==\^3. i = ^4, &c. [4277to1
In like manner, the part of '5 v [42776], depending on sin. T, is found by using i = ^l ;
hence we have
< \ da / ' n — n >
^'^-n^'.l j£^,^, . aA'-r^ -f - t^^^,;;]:^,; _ J}.~;^ ^ • sin. T. ^^^..^^
Substituting the values of the elements given in [4277^,/], it becomes 0^6623. sin. T, as
in the first line of [4276] ; the other terms depending on sin. 2 T, sin. 3 T, he. are found [4277o]
in like manner, from [4277/./], by using successively « = zt 2, « = ±3) he. The
similar terms, corresponding to the other planets, are com[)uted by means of the same
formulas [4277a, 6], altering the accents as in [4277t]. The results of these calculations
are given in [4289, 4290 ; 4305,4306 ; 4.373, 4374; 4388, 4389; 4463,4464; 4523,4524]. ^^^'^'^^
* (2586) The terms depending on the first power of the excentricities are those parts of
ir, ÔV, [1020. 1021], containing e and e. The calculation of these terms is made as [4278a]
in the preceding note ; using for e the excentricity [4C80], corresponding to the disturbed
VOL. III. 59
234
PERTURBATIONS OF THE PLANETS;
[Mée. Cél-
Inequali-
ties de-
pending
on tbetirst
power of
the excen-
tricities.
[4279]
0',09541 8 . sin.(n"i + e"— ^)
+ (1 +f^")-<| — 0',461708.sin.(2n"i_ wi + 2s"_ £_™)
+ 0 ,244148.8111.(3 n"t — 2nt + 3/'— 2 e — ™)
0',236346. sin.(n'''i + s" _ ra)
+ (1 + f^'^) . { — (r,572172. s\n.(7i'H + ^'^ — ^'0
- 3 ,278687 . sin. (2 n'H _ n < + 2 1" — e
(1+^^).
O',084]67 . sm.{n'i + s" —z^")
+ 0',395493 . sin.(2n"ï — rU +2 5"
-.)
')
3r = — (1 +p.').0,0000013482.cos.(3n'i — 2n^+3£' — 26— z;j)
— (1 + 1^"). 0,0000029625 . cos.(2 n"i— nt+2 s'"— s — ^).
Inequalities depending on the scjuares and products of the excentricities and
inclinations of the orbits.
[4280]
[4281]
These inequalities have been calculated by the formulas of [3711 — 3755].
Now twice the motion of Mercury differs but very little from five times that
of Venus ;* so that 5(n' — n) + 2n is very nearly equal to — n; we must
therefore, as in [3732], notice the inequality depending on 3nt — 5 n't.
The angle 37i't — 7it varies quite slowly, therefore it is necessary to notice
the inequality depending on it [3733]. Moreover the motion of Mercury is
very nearly equal to four times that of the earth, so that 4.(n" — n) + 2w
differs but little from — n; therefore, we must, as in [3732], notice the
inequality depending on 2nt — 4<n"t. Hence we obtain,
[4282a]
planet; and for e the value [4080] corresponding to the disturbing planet; these symbols
being accented so as to conform to these two bodies.
* (2587) Using the values [4076^] we have very nearly 2 n
.5n'=z 72° =
23"
3n' — n=:289'^ = ^, and 71 — 4 ?j" = 61° = — ; so that these three quantities are
small in comparison with 71, as is observed above. Hence 5 («' — n) -j- 2»t is very nearly
r4282tl equal to — «, and must be noticed as in [3732] ; also 3 (ji' — 74) -(- 2 n is very small,
and must be noticed as in [3733] ; lastly 4 (?i" — n) -j- 2 n is very nearly equal to — n,
and must be noticed as in [3732]. The ;enns ofiî[3745-3745"'jdependingon these angles
VI.viii.§27.] THEORY OF MERCURY. 235
^ l',690443.sin.(3n< — 5»'<+3î— 5e'— 43^18'"32'))
i r = _ ( 1 + f^ ). s o-,597664 . sin. (3 n t— n / + 3 /— £ + 4O"36™350 ( M^'Sl
V ■' second
— ( 1 + f.") . 0',263474 . sin. (2 n / — 4 n'7 + 2 s — 4 s"— 41 M 1 "" 46^
ir = (1 -f ,j.').0,0000016056.cos.(3n^ — 5n'i + 3£ — 5e' — 42^58"'04').
the
order.
[4282]
are found by pulling in the first case ?'^ 5 ; in tlie second i^3, and in the third i = 4.
The values of ^W"', iW'", M^'^^, .'V/'^', corresponding to these values of i, are successively [4282c]
obtained from [3750, 3755, 3755', 3750'"] ; and they may be reduced to terms of
U'\ - — , &:c. by means of the formulas [996 — 1001]. These values are to be
substituted separately for Jfcf in the expressions of -^, àv, [3711,3715], and we shall [4282d]
obtain very nearly the terras of — , 5 r, having the small divisors 5 n' — 2 n,
3 n' — n, 4 n' — n, which are the only ones necessary to be noticed in this place. Now [4282e]
if we use, for a moment, the abridged symbol, T.^iJn't — n t 4- ^ — e)-\-2nt4-2e
i^ [4282/]
[371 lij-], the resulting terms of — or 5r [3711, &ic.] will be of the form [4282/t].
Developing this by [24], Int. it becomes as in [4282?]; substituting .^jsin. Z?, for the
coefficient of sin. 7', also ,/2,cos. -Bj, for the coefficient of cos. T^, it changes into
[4282t], and is finally reduced to the form [4282/], by means of [24], Int. [428%]
dr = J»f/<".cos.(T— 2;n)+J/;".cos(T— a— ^')+M/2\cos.(r— 2îi') + i>7/3>.cos.(T— 2n) [4232^]
= { M}^'. cos. 2 Î3 + AJ^'K cos. (îi + -/) + Jl/'^'. cos. 25/ + M/". cos. 2 n | . cos. T,
+ { ./U;»'. sin. 2 w + .W ">. sin.(i^ + z,') + iVif \ sin. 2 ^' + M,'^\ sin. 2 n| . sin. T, ^*^^^'^
= ^1.5 cos. S, . COS. T, + sin. 5, . sin. 1] } [42824]
= A, . cos.{T-B,), as in [4282]. [4282^]
In like manner the several terms of i5 v may be reduced to the form A-3. sin.(T, — B.-,) ;
there is no other difficulty than the tediousness of the numerical calculation, arising from its [4282m]
length.
We may observe that the quantities 7^, 2 IT, which occur in [3745'"], are not
explicitly included among the data [4077 — 4083], but must be computed from the formulas [4282n]
[10.32, 103.3].
7 .sin.n =z tang. 9'. sin. â' — tang. 9. sin. é; 7. cos. n = tang. 9'. cos. â'— tang. 9. cos. Ô; [4282o]
supposing 9, é to correspond to the disturhed planet, and 9', è' to the cUsiurbing jilanet ;
these symbols being accented so as to conform to the notation [4230] ; then using the
values [4082, 4083] we get the required values of 7, n.
236 PERTURBATIONS OF THE PLANETS ; [Méc. Ct
Inequalities depending on the cubes and jirodncts of th-ee dimensions of
the excentricities and inclinations of the orbits.
The first of these inequalities, depending on the angle 2nt — 5 n't, is
[4282'] computed by means of the formula [3844] ;* the second, depending on
the angle nt — 4 /<% is found by means of [3882] ;t hence we obtain,
ÔV = —{l +f^')-8',483765.sin.(2ni — 5w'^ + 2s— 5.='+30''13'"36')
ineq^.ii. — (1 + O • 0',690612 . sin.( n i — 4 n"t + £ — 4="+ 19^02'" 13').
ties of Ihe
etder. The inequalities of Mercury's motion in latitude, may be calculated by-
means of the formula [1030] ; but as they are insensible, being less than
[4283] ^ quarter of a centesimal second, it was thought unnecessary to insert
them.
r4283ol * (2538) The first line of [4283] is obtained from the formula [3844], connecting all
the terms into one, as in \_4282h — ?].
[42836]
t (2589) The second line of [4283] is obtained from [3882], reducing all the terms
into one, as in [4282/i — Z]. We have already seen in [3883/(], that the correction, as it is
given by the author, in [4283], is rather too great ; his method of computation [3882] being
i J merely an approximation. The direct method of computation has already been explained
in the previous notes [3876a— 3833io] ; and it is unnecessary to say more upon the subject
[4283rf] ji^ jijjg place. There is a similar equation in the earth's motion [4311, 3S83i/].
VI. ix. ^^28.] THEORY OF VENUS. 237
CHAPTER IX.
THEORY OF VENUS.
28. If we put - = a, and V equal to the geocentric longitude ^4284]
of Venus, we shall find that the equation [4272],
6r = — r". (1 — a=) . (5 V, [4285]
will become, relatively to Venus,
^r'=. — r".(l— a'=).6V'. ["286]
Taking for r', »", tlie mean distances of Venus and the earth from the
sun [4079J, we shall have, as in [4126], a = 0,72333230 ; therefore by [4287]
putting 6 V = ± 1"— ± 0',324, we shall obtain,
6 r' = :f 0,0000007489. [4288]
Therefore we shall neglect those inequalities of the radius vector whose ''"'"»
coefficients are less than 0,0000007. We shall also neglect the inequalities .'"gLfeci
on account
of the motion in longitude, which are less than a quarter of a centesimal "f">,
^ ^ sinullnes*.
second, or 0',081.
Inequalities of Venus, independent of the excentricities.
'+ 5',015931 . sin. {n"t — n' t + s"— s'Y
+11',424392 . sm.2(7i"t — n' t + b"— s')
- 7%253867 . sm.S(ti"t — n' t -\- e"— s')
— p-,056720 . smA(n"t — n! t + /'— /)
iV = {\+ O . ( _ Q, 345898 . sin.5(n"^ - n' t + ."- 0 > ^'''""^
— 0% 145382 . sin.6(w" t - n' t + a"— £')
— 0',069726 . sin.7(n"« - n' t + s"— s')
— 0%036207 . sln.^n"t — n' t -\- e"- i') ^
VOL. III. 60
238
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
+ (1 + n
0%079908 . sin. (n'"t .
0% 105987 . sm.2(n"'t ■
0^010853.sin.3(n"'/
0',002332 . sm.4>(n"'t .
n't + £'" — £')
n't + s"'-s')
n't + i"' — 0
[4289]
Inequali-
ties inde-
pendent of
the ex-
centrici-
tias.
+ (l+f^'0-
2% 891 136 . sin. (n'^t — n' t + >— =')
0%877624 . sin.2(n'^i — n't + s'^— s')
0',040034 . sin.3(?i'^i — n't + s"— .')
0',002754 . sin.4(n'^i —n't + e'"— s')
0% 190473. sin. (n't-
+ (1 + 'O • <! — 0',039859 . sin.2(«^^
- 0',00 1306. sin. 3(n^r
[4290]
ir' ^ (1+f.'").
— 0,0000003145
+ 0,0000038362 . cos. (n"t - n't + s"- s')
I + 0,0000165050 . cos.2(n"^ — n't + .-"— i')
— 0,0000140155 . cos.3(n"t — n't + s"— i') \
— 0,0000024255 . cos.4(n" t — n't + s"— i')
1 — 0,0000008873 . cos.ô{n"t - n't + /'- e') i
- 0,0000004021 . cos.6(n"^ — n't + s"_ e') I
— 0,0000002033 .cos.7(n"^ — n't + ;"_ s')
— 0,0000001094 .cos.8(n'7 - n't + s"— =')
'—0,0000003106
- 0,0000048903 . cos. (n'^t — n't + s"— i')
+ (1 + \^") . / — 0,0000021911 .cos.2(n''^ — n't + ^'—s')
1 _ 0,0000001 155 . cos.3(?i'^^ — n't + s'"— e^
_ 0,0000000098 . cosA{nH — n't + b"-— ,')
* (2590) The values 5v', &r' [4289,4290], were computed from the formulas
"' [4277a, 6], accenting the symbols as in [4277c], so as to conform to the present case.
the excen-
tricitiei.
VI. ix.§28.] THEORY OF VENUS. 239
InequttlUies depending on the first potoer of the excentricities*
i r' = (1 + ,a) . 0% 800933 . sin.(2 n't —nt + 2s'—s — ^)
0',073206 . sin. {n"t + ;" — ^')
— OM 27720 . sin. (7ft + s" — ^")
-I- 0^1631 15 . sin. (2 n"t — n't + 2 s" — s' — ^')
— 0', 1 1 3443 . sin. (2 n" ? — n'f + 2 s" — a' — ^,")
/ Inequalt
— 1 ',549550 . sin. (3 n"i — 2 w'< + 3 e" — 2 s' — z^') "'',<'•>■
^ \ I / I pending
+ (1 + (..") . / + 4',766332 . sin. (3 n"t — 2 n't + 3 s" — 2 .=' — ^") ) pp'-'r'
\ ' ' V ' ' / the excen
— 0^299478 . sin. (4>n"t — 3n't + 4^ b" — 3 s' — ^')
+ 0',947648 . sin. (4 n"t — 3 n'i + 4 e" — 3 =' — t.")
— 0',69 1 744 . sin. (5 n" f — 4 n' i + 5 a" — 4 -=' — ^')
+ 2', 196527 . sin. (5 n"t — 4 n'i + 5 s" — 4 / — ^") / [4991]
+ 0% 106435 . sin. (3 n' t — 2n"t +3^ — 2 b"— ^')
— (1 + P-'") . P,092755 . sin. (3 71'" t —2n't + 3 a'"— 2/— ^"')
— P,503893 . sin. {n'H + 3'"—^'^)
0%32n08 . sin. (2 n'^t — n't + 2 b'"— s' — a')
' ^ ' '^ ^ \ ^ 0',232430 . sin. (2 n'-'i — n'^ + 2 a''— / — a-)
— 0',163470 . sin. (3 n'^t — 2 n't + 3 b"— 2 b'—^'^)
— (1 + ,a') . 0%218743 . sin. (n" t + b^ — z-^) ;
6 r' = (1 + (.) . 0,0000008831 . cos. (2 n'i — n ^ + 2 /— a — ^)
r 0,00000 1 6482 . cos. (3 /t" « — 2 n' < + 3 a" — 2 / — îj") y
+ (1 4-^") .<^_ 0,00000 11406 . cos. (5n"t — An't + 5a" — 4s' — «')> ^^^^^^
(+ 0,0000036421 . cos. (5n!'t — A n't -{-5^' — 4a' — ^");
— ( 1 _^ ijJ" ) . 0,0000019404 . cos. (3 n'" t— 2 n't -^3 a'" — 2 £' — ^"').
* (2591) The terras of &v', Sr' [4291,4292] are computed from the parts o( S v, or
[1021, 1020] depending upon the excentricities e, e'; in the same manner as the [4291o]
calculation is made for Mercury in [4278a].
240 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Inequalities depending on the squares and products of tivo dimensions
of the excentricities and inclinations of the orbits.
.5 v' = — ( 1 + ,;.) . 0',333596 . sin. (4^n' t — 2nt -{- 4,s'—2 £ —39" 30" 30')
( 1%505036 . sin. (5 ?i"t — 3n't+ôs"—3 s' + 20'' 54" 260 >
[4993] — (1 + •/) .{ }
- ^ ^ ^ ^ l^ 0',089351 . sin. (4 n"t — 2 n' i + 4 s"— 2 s' + 26" 66'" 32') )
i„c^uai,- + ( 1 + //" ) . 2^009677 . sin. (3 n"'t— n' t + 3 /"— s' + 65" 53'" 09').
tie3 of the ' V ' ' ^ ' V ' ' /
secund
order. rj,j^^ mean motions of Mercury, Venus, the earth and Mars, bear such
proportions to each other that the quantities 2n — 5n', 5 n" — 3n' and
[4393] n' — 5 n'" are very small in comparison with n';* hence it follows from
the remarks made in [3732, &c.], that the preceding inequalities [4293]
are the only ones of the order of the square of the excentricities which
can become sensible.
Inequalities depending on terms of the third order, relative to the powers
and products of the excentricities and inclinations of the orbits.
[4294] ,5 Î)' = (1 + M-) . r,184842 . sin. (2nt — 5 n't + 2s — 5e' + 30" 13"' 36'). t
Inequali-
ties of Ilic , n TT • 1 • 1
"'''•' Inequalities of the motion of Venus m latitude.
order.
The formulas of ^ 51. Book I. giv^ej
n
* (259-2) The values [4076A] give, very nearly, 2 » — 5 7i'= 72^ = - ;
[4293a] 5n"— 3 7/ = 50^= - : «' — 3 «'"= 12=' = -?^ : all of which are small. The
13 ' 54 '
first of these gives 4?i' — 2n nearly equal to — ti', and corresponds to tlie
first form mentioned in [3732]. The second quantity 5 n" — 3 )i', and the third
n' — 3 »'", being nearly equal to zero, correspond to the second form [3733]. The
[4293fc] terms of àv' [4293] corresponding to these quantities are to be computed from [3715],
and reduced as in [4282/i— Z]. The term depending on An" t — 2 n' = 300° == Jn'
nearly, is computed for the same reasons as that in [4310'].
t (2593) This is obtained from [3817], reducing the several terms to one, as
t^^^^^l in [4282A-Z].
[4295a] X [2594) If we change, in [1030], n, a, e, n', a', i', into n', a, s, n", a", i
VI.ix428.] THEORY OF VENUS. 241
0%124804.sin.(n"< + £"_0
6s'=—(l+t^").
4- 0',090932 . sin. (2 n"t — n't +2 s"-
+ 0',073443 . sin.(3 n"t — 2 n't + 3 s"— 2 e'
+ 0S081481 . sin. (4 n"t — 3 nV + 4 /'— 3 s
+ 0',312535 . sin.(5 n"t —A n't + 5 s"— 4 s
— 0',078119.sin.(2n'i— n"t + 2i'— s"— è')
Ineqoali-
tioi in tbe
latitude.
-0
-0
— '•') \ [4295]
[42956]
— (1 + ix"') . 0%148701 . sm.(3n"'t—2n't + 3 ê'" — 2 s — n'")
+ (1 +(x'"').0%161414.sin.(2n'7-n'^+2«'^ — f'-n").
respectively, we shall obtain the value of 5 s' corresponding to Venus disturbed by the
earth ; and by neglecting the term containing the arc of a circle n t without the
signs of sine and cosine, as is done in [1051] ; also excluding i = 0 [1028, &ic.] from
the sign 2, we get.
In this formula, y [1026'] represents the inclination, and n the longitude of tlie
ascending node of the orbit of the disturbing planet, above that of the disturbed planet.
These quantities for the earth's action upon Venus are, nearly y = tang. q>', and
n= \ëO''-{-è' • (p' being the inclination of the orbit of Venus to the fixed orbit of the
earth ; and è' the longitude of the ascending node of the orbit of Venus upon that of the [4295d]
earth [4082,408.3]. For Mars they become /", n'"; for Jupiter y'% U", he.
In the expression [42956] we must include all positive and negative integral values of i, [4295c]
except 1=0 [1028, &;c.]. The values of y, /, &c. II, n', &ic. are deduced
from those of cp, <p', kc. é, è', Sic. [4082, 4083] ; by means of formulas similar to
those in [4282o]. Thus if we wish to find the part of 5 s' depending on the angle [4295/"]
2n"t — n't. we must put i=2, in [42956], and the term in question becomes,
Now the factor n'^— (2n"— n')2 = 4 n".(n' — n") ; also B'''^ = ~ . b'''.^ [1006];
a •> <j
substituting these and y, n [4295c], in [4295^], it becomes,
[4295c]
-m'.n'Ka'^a" 6f.tang. m'
m.{2n"i—n't + 2^'-^—6')
2 4rt".(n'— n").a"3
(.-, [4295;.]
VOL. III. 61
242 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4295']
n'" being here the longitude of the ascending node of the orbit of Mars
upon that of Venus,* and n'" the longitude of the ascending node of the
If ill this we substitute m" ^l^' [4061], n"=: 1295977', ?»'= 2106641' [4077],
[42951"] (1) „' . , , .
63=8,871894 [4132], — = 0,72333230 [4126], <p' = 3'^ 23"" 35' [4082]; it is
J a
[4295;t] reduced to — 0',090932.(1 + fx"). sin.(2n"< — n'^ + 2 s"— s' - â'), as in [4295]. In
the same way we may compute other terms. If we suppose i= 1, there will be found
two corresponding terms in [4295i] ; namely,
[4995/] -^J^l., • ^ • tang- ?'• 1 1 — ^«" ^- ^'^^ ■ sin- («" < + s"— «'}•
to)
But by changing a' into a", in [1006], to conform to this case, we have a" ^ B''°'' ^ b ^i
[4295m] j^gjjpg thg preceding expression becomes -^ — ^ • (~) • ^''ng- <p' • ( 1 — 2-^3)- If
a' W)
use the values of m", n', n", - [4295i"] ; also 6 3^ = 9,992539 [4132]; we get
to)
3'
2
we
[4295n]
(0)
0',031231, for the part independent of b ^ ; and — 0',156035, for the part
(0)
depending on b ^ ; the sum is — 0^,124804 . sin. (71" < -{"'" — ^') > ^^ '" '''^ ^'^^
line of [4295].
* (2595) A small inequality in the mean motion of Venus, depending on terms of the
fifth order of the powers and products of the excentricities, has lately been discovered by
rAvjQni Mr. Airy, arising from the action of the earth upon that planet. This inequality affects the
mean motion, the radius vector, tiie perihelion, the excentricity, and the latitude ; its period
[429661 is nearly 239 years ; being the time required for the argument 8 7it — 13 «"^ to increase
from 0' to 360''. This appears from the values of n', n" [4077] ; from which we
[429Gf] get 8 m' — 1 3 7i" = 5427' = — — nearly; and as this quantity is very small, it follows
that tlie mean motions of Venus and the earth must be affected by inequalities, depending
upon the argument 8n'< — 13?i"^; in like manner as the mutual attraction of Jupiter and
Saturn produces the great inequalities of these planets in [1 196, 1204] ; supposing the accents
on the letters a, n, &ic. to be increased to conform to the present notation, and putting
i' = 8, i" = 13. The variations in the excentricities and in the motions of the perihelia,
similar to tiiose of Jupiter and Saturn [1298 — 1302], are in the present case nearly
insensible. The inequalities of the mean motions of Venus and the earth, ^', ^" depending
' on the argument 8n'i — 13 n"^, are of the order 13 — 8 := 5 [957^'", &c.], or of
the fifth order nlative to the powers and products of the excentricities. Now e, e" are
[4296/] both quite small, so that the largest of them e" gives e"* = . .r nearly ; but this
VI.ix.§28.] THEORY OF VENUS. 243
orbit of Jupiter upon that of Veaus.
very minute fraction is multiplied, in [1 1 97] , by .^J,'^'^ =3 X 13 X (239)''= 2200000 [4296^]
nearly, in finding the value of ^" ; and by this means the correction is very much increased.
The theory and numerical computation of this inequality are given by Mr. Air)',in an elaborate
paper on this subject, in the Philosophical Transactions of the Royal Society of London for ^ '«'"*]
1832; using the data [4061—4083]; and putting (x' = — 0,045, (ji"=0, so that [4296i]
m'= . He finds the correction 2^ of the mean motion of Venus, to be represented by [4296ft]
^r=: {2',946-r.OS0002970|.sin.{8«^ — 13n'7 4-8s'— 13£"+220''44'»34'-M0%76|. [429«]
He also obtains the following equations, depending on the same cause, and similar to those
Siven in [1298-1302] ;
5 b' = _ 5',70 . cos.(8 n't—\2n"t + Ss— 13 s") ; [4296i»]
W= — 0,000000190 , sin.(8 ?i'< — 13n"i-\- Be' — 13 s") ; [4296n]
5s=0',0151 .sin.(9n'i — 13n"< + 9£' — 13£" + 140''31"'). ^4296^,
These corrections of 5 ro', Se, S s, may be generally neglected, as insensible; as also
that in the radius vector, similar to [1197]. We shall give, in [4310c—/], the corresponding
corrections of the earth's motion. The expressions of 8,', ^" [4296Z, 43 1 Oc] , are subject t^'^^^p]
to the noted equation [1208], which in the present case becomes
7n'./a'.^'+mV«"-l"=0. ^**^^
244
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
CHAPTER X.
THEOKY OP THE EARTH'S MOTION.
[4296]
[4297]
[4298]
29. If we suppose the geocentric longitude of Venus to be represented
T
by V, and -, = <>•; V'* will be a function of a and v' — v".
Then we shall have, by [4269],
&\'= —
à ft . sin. («' — v'')
1— 2 a. COS. («'—«") 4- a2'
which gives, as in [4272], where ôV is at its maximum,
5 a.
6V'=r — ■
l-a^»
(4297ol * (2596) In strictness it is not the angle V which is to be considered as a function
of a and v — v'' exclusively, but the angle of elongation E of Venus, as seen
from the earth. This will appear by referring to fig- 74, page 229 ; supposing M to
represent the place of Venus ; S M = /, °fSM = v'. For it is evident that the angle of
elongation E=^SEM will remain the same, if the angle ESM = v' — v" and the
-^— , = — do not vary, whatever changes may be made in the absolute lengths
of the lines SM, SE. This inadvertence of the author, in using V for E does not
however affect the result of his calculation [4297. &c.] ; because the differentials only of
these quantities are used ; and we have, as in [4268c] (5 V' = — 6 E. Now in [4268, 4269]
[42976] ratio a = --
[4297c]
we have supposed r" to be invai-iable, so that the variation of — = a
or
— =: ô a ;
[4297rf]
substituting this in [4269], and accenting the letters r', v', so as to correspond to the
planet Venus, we get the expression [4297]. This Is reduced to the form [4298], by the
substitution of the maximum value of the coefficient of — Sa [4271'], in the second
member of [4297].
VI. X. ^^29.] THEORY OF THE EARTH. 245
a.5r"
Supposing r" only to vary in 6 a, we have 5 a. =^ ~ ;* therefore, [42t>y]
èr"=r"MlZ^ .6Y'. [4300]
a
If we put 6V' = ±1"= ± 0',324, and take for r' and r", the mean [43001
distances of Venus and the earth from the sun [4079], we shall get.
6r"= ±0,000001035.
[4301]
r
[4301']
If we put V" for the geocentric longitude of Mars, and — =^ a, we
shall have, by [4272],t
6f'=— r"'. (1 — a-) . 6 V". [4^02]
If we take for r", ?'", the mean distances of the earth and Mars from
the sun, we shall have,
a = 0,65630030 [4159] ;
r'" = 1 ,52369352 [4079] ;
[4303]
Terms
and if we put 6\"' ^ ± 1" = ±0'.324, we shall obtain, which
^ ' may be
i /•" = =F 0,000001363 ; [4304]
neglected
therefore, we may neglect ilie inequalities of sr", whose coefficients are ofîhciT
* (2597) If we suppose / to be invariable in the value of a [4296], we shall
get Ja = — '^= — °^ [4299]; substituting this in [4298], we obtain [4300] ;
which is reduced to the form [4301], by the substitution of âV' = ±l" [4300'],
r" ^ I [40r9] and a == 0,7233323 [4126].
t (2598) Venus, being an inferior planet to the earth, is situated in the same relative
position as the earth is to Mars ; therefore the equation [4286], which obtains relatively to [4301»]
Venus and the earth, may be applied to the earth and Mars, by substituting in [4286] the
value of a [4284], and then adding one more accent to each of the symbols r', r" , V ;
by which means we shall obtain 3r" = — 7-'" . A-^V .5 V" [4286]. In this ^^g^j^^
case 0 V" is the change of the longitude of the earth, as seen from Mars, arising from the t/^^Qi^^
increment 5 ?•" ; and is evidently equal to the increment of the geocentric longitude of
Mars, depending upon the same cause, which is represented by 5\"'; hence we get r43Qjj|
^ r"==-r"'. (\ -~\ . 0 V", as in [4302].
VOL. III. 62
246 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
less than ±0,000001.* JVe shall also neglect those inequalities of the
[4304'] earth^s ^notion in longitude, lohich are less than a quarter of a centesimal
second, or 0',081.
Inequalities of the Earth, independent of the excentricities.f
5%290878 . sin. (n't — n" t + s'— z")
— 6',015891 . sm.2(n'i — n" t + e'— î")
— 0%743445 . sin. 2{n't — n" t + i'— /')
.,,,,/— 0%225439 . sin.4rn'i — n" t + s'— n
' ^ — 0S091210. sin.5(w'^ — n" t + s'— s")
— 0%042805 . sin. Q{n' t - n" t + s'— .")
- 0',022027 . ûn.l{n't - n" t + s'— e")
— 0%0 12053 . sin. 8(n'i — n" i + i'- =")
[4305}
.„.n.aii- / 0',427214 . sin. (n"7 _ n" t + s'" — /')
ties inde- /
p™t"-' "'■ f 3^483037 . sin. 2(n"'ï — n" t + ^"' — e")
""'""" \ — 0',21 5249 . sin. 3(n"'t — n" t + s'" - ,")
+ (1 + f^'") • ( — 0',047022 . sin.4(n"7 — n" t + s'" — /')
— 0^015871 . sin.5(M"7 — n" t + b'" _ s")
— 0%006458 . sin. 6(ti"'t — n"t + £'" — £")
— 0^002923 . sin. 7(n"'t — n"t + s'" — s")
+ (1 + f^'')
7^059053 . sin. {n"t — n" t + £'"— e")
- 2'-,674257 . sin.2(n'^'i — n" t + ."—.")
— 0', 167770 . ûn.S{n'''t — n" t + ."— s")
— 0',016549 . sin. 4(?i'"/ — n" t + 5'^— s")
( 0S439410 . sin. {n't — n"t + s'— s")
+ ( 1 4- f^.'') . I — 0', 1 1 1 0 1 0 . sin. 2{n''t — n"t + £^— s")
/ — 0»,004]45 . sin. 3(n't — n"t + s'— /')
* (2599) This quantity, independent of its sign, is less than either of the values
[4301,4304], corresponding to the 7îea?-cs< inferior and superior planets ; and for the more
[4304a] distant planets this degree of accuracy is more than is absolutely requisite, in the present
state of astronomy.
t (2600) The quantities [4305, 4306] are deduced from [4277a, b] ; accenting the
[4305o] symbols so as to correspond to the present case, and using the data [4061, 8:c.].
Vl.x.^^-29.] THEORY OF THE EARTH. 247
, 0,0000015553 ^
— 0,0000060012 . COS. {n't — n"t + s'— i") \
4- 0,0000171431 . COS. 2(n't - n"t + i'— s")
Ô r" = (1 + f^') • ^ + 0,0000027072 . cos. 3(n't — n"i + s'— e")
+ 0,0000009358 . cos. A(n't — n"t + s'— e")
+ 0,0000004086 . cos. ô(n' t — n"t + b'— b")
. + 0,0000002008 . cos. 6{n't — n"t + s'— /') ^
,— 0,0000000478
+ 0,0000005487 . cos. (n"'t — n"t + /"_ s")
+ (1 4_ f.'") . ) + 0,0000080620 . cos. 2(n"7 — n"t + e'"— s") V i„e,„,u.
— 0,0000006475 . cos. S(n"'t — n"t + «"'— ^") \ 'ifEt
— 0,0000001643 . cos. 4(n"7 — n"t + £'"— £")
tricities.
— 0,0000011581 \ [4306]
+ 0,00001 59384 . cos. (n^'t — n"t + é"— e")
+ (1 + f^") . <f — 0,0000090986 . cos. ^.{n^t — n"t + s'"— f")
' _ 0,0000006550 . cos. S{n'H — n"t + 1"—^')
- 0,0000000704 . cos. 4(«'7 — n"t + s'"— s") /
-0,0000000580 ^
+ (1 -ff;^'). <[+ 0,00000 10337. cos. («7 — m"^4-£^— z")\.
— 0,0000003859 . cos. 2{n't —7i"t + £'— s"))
In the solar tables of La Caille, Mayer, La Lande, Delambre and Zach, published before
the year 1803, the chief correction of the radius vector of the earth's orbit, arising from the
action of Jupiter, is given with a wrong sign ; in consequence of taking, for n"t-\-s'', the '
sun's longitude, instead of that of the earth, in finding the argument corresponding to the
terms which were used, namely,
+ 0,0000 1 59384 . cos . {n'^i — n"t-\- £'"— e") — 0,0000090986 . cos . 2 (n-i — n't -f- e^' — e") . [4305c ]
This mistake was first made known in a letter communicated by me to La Lande, and u^did}
noticed in vol. 8, p. 449, of the Moiiatliche Correspond enz for 1803.
248
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
Inequalities depending on the first power of the excetitricities.
iric<iuali-
ties de-
pending
on the first
power of
the excen-
tricitiea.
[4307]
0^075910
— 0', 129675
- 0', 145 179
-0%168981
+ 1', 186390
— 2',342956
+ O"-, 722424
V+ 0S2 16368
— r,095603
+ 2, 137658
— 0",087400 ,
+ 0%661950
— 0', 103758 ,
+ 0',807 1 1 1
— 0-, 1349 15
+ (i+0
sin. (n' t + ^' — z,")
sin, {2 n't — n"t + 2 e' — =-" — ^")
sin. (2 n"t — n'i + 2 /' — i' — z,")
sin. (2 n"t — n't + 2 /' — s' — ^')
sin. (3 )i" t — 2n't + Ss" — 2.' — ~/')
sin. (3 n" t — 2 n' t-^3e" — 2^ — ^')
sin. (4 n" t — 3n't + 4>i" — 3 ^' — z^")
sin. (4 n"t — 3 n't + 4 s" —3 / — ^')
sin. (5 n"t — 4 n' i + 5 ;" — 4 / — ^")
sin. (2 n"'ï — n" t + 2 /" — /' — ^")
sin. (2 n"'t — n" i + 2 c-'"— :-" — ^"')
sin. (3 n"'t —2n"t + 3 i'"— 2/'— v/')
sin. (3 n"'t — 2n"t + 3 .'"— 2 ."_ ^"')
sin. (4 n"'t — 3 n" i + 4 --'"— 3 ;"_ w" )
sin. (4 H"'t — 3 n" t + 4 s'"— 3 ^"—^"')
sin. (5 /t"7 — 4 7i" t -1- 5 s'"— 4 ="— r/")
+ (!+(-"')
+ (1 + O
0',302092 . sin. (n'-'ï -1- e'"— ^")
— 2%539884 . sin. {n'H + .=" — j.")
— 1%492044 . sin. (? w'7 — «'7 + 2 .>— /'— ^")
+ 0',606399 . sin. (2 n'7 — n"t + 2 «'"— /'— ^'0
— 0',543364 . sin. (3 n"t — 2n"t-\-3 s"— 2 1"— z^"")
— 0', 148925 . sin. (2n"« — n'^'i + 2 3"— e-—^")
\— 0^093643 . sin. {2n"t — n"t +2£"— j'"— ^'')
J — 0',359921 . sin. {n't + e^ — ra') >
( — 0',151752 . sin. (2 n7 — ?i"i + 2 s'— e"— ^") ^ '
* (2601) The terms of àv", or" [4307, 4308] are computed as in the theory of
'^^"'"^ Mercury [4278«].
VI.x.§29.] THEORY OF THE EARTH. 249
r_ 0,0000030439 . cos. (3 ift — 2 n' ^ + 3 s" — 2 / — ^") y
àr"= (1 +.a').^ — 0,0000049815 . cos. (4»"/ — Sn't + 4s" — 3 e' —^")}
(+ 0,0000015895 . cos. (4ïi"/ — 3n'^ 4- 4 s" _ 3e' — ^'))
4- (1 +,x"') . 0,0000017707 . cos. {^n"'t—Sn"t + 4 a'"— 3 s" — ^"') [4308]
— 0,0000030410 . cos. (2 n'^ï— n"^ + 2£'''— £" — ra")"
4-(l 4- f^'0.<| + 0,0000012652 . cos. (27rt— n"t + 2e''— e" — ^'O!
-0,0000018101 . COS. (3n''f — 2n"^+ 3s"—2s" — ^''')'^
Inequalities depending on the squares and products of the excentricities and
inclinations of the orbits,*
Inequali-
6 v" = (1 + I'.') . r-,125575 . sill. (5 n'7 — 3n't + 5-="— 3/+ 21''02"' 1 8^ iVJi'''
order.
C + 0^993935 . sin. (4 n"'t — 2 n"t + 4 s'"— 2 s"+ 67H8"560 ) [4309]
■^ ^ "^ ^ ^ ■ ^ + 0^351 796 . sin. (5 n"'t — 3 n"t + 5 s"'- 3 a" + 68'' 25'" 09^ ) '
The mean motions of Venus, the earth and Mars bear such proportions
to each other, that the quantities 5 n" — 3 n', 4 n'" — 2 n" are small ^^gj^-j
in comparison with n" ; hence it follows, from [3733], that the two
first of these inequalities are the only ones of this order which are
deserving of notice. However we have calculated the third ; because
3h"— 5?r, being very nearly equal to ^n", it is satisfactory to show, by •[4310]
a direct calculation, that this inequality acquires by integration only a very
insensible value. t
n
[4309a]
* (2602) From [4076A] we get, very nearly, 5n" — 3 n' = 50° = - ;
4 n"' — 2 n" r= 50= = ^ ; .3 ?i"— 5 ?i"' = J 37° = ~. These angles ought therefore
0 3
to be noticed, as in [3733] ; and by making the computation, as for Mercury [4282a— jp],
we may reduce the terras, depending on each angle, to one single term, as in [42S2/t — /].
t (2603) We have already mentioned, in [4296/;], that Mr. Airy had discovered an
inequality in the earth's motion, depending on terms of the fifth order of the excentricities [4310a]
and inclinations, connected with the angle 8 n't — 13 n"i. He has given in the paper
mentioned in [4296A] the numerical values of the inequalities of the mean motion |", [43106]
of the perihelion ozi", of the excentricity ôe", and of the latitude &s", namely,
VOL. III. 63
250
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Inequali-
ties of ihe
third
order.
[4311]
hiequali-
tio3 ill the
latitude.
[4312]
Symbols.
[4313]
Inequalities depending on the powers and prodiccts of three dimensions of
the excentricities and inclinations of the orbits.
iv" = (1 +1^.) .0',069915 . sm.(ni— 4n"i + s— 4=-"+19''O2'"130.*
Periodical inequalities of the Earth's motion in latitude.
We find, by formula [1030],t
( 0%0991 80 . sin. (2 n't —n't + 2 /'— ;'— è') >
\ 0',234256 . sin. (4 n!'t — 3 n't + 4 /'— 3 ='— é') $
+ (1 + (J-") . 0',164703 . sin. (2 n"t — n't + 2 .-- — s" _ é-).
Inequalities of the Earth depending upon the Moon.
30. If we jHit
U = the longitude of the moon, as viewed from the centre of the earth ;
v" = the longitude of the earth, as viewed from the centre of the sun ;
R = the radius vector of the moon ; its origin being the earth's centre ;
r" = the radius vector of the earth ; its origin being the sun's centre ;
m = the mass of the moon ;
M = the mass of the earth ;
s = the latitude of the moon, as viewed from the earth's centre.
[4310c]
[4310rf]
[431 Of]
[4310/]
[4311a]
[4312a]
^"= (2',059 — ^.0',0002076).sin.(8n7— 13 «"<+ 8 s— 13 /'+ 40''44'"34'— MO',76);
5 ra" = 2',268 . sin. (8 n't— 13 n" / + 8 s' — 13 s" + 60'' 16"') ;
<5e"= — 0,0000001849. cos.(8n'<— 13 m" <+ S s' — 13 £" + 60''16"') ;
5s" =. 0',0105 . sin. (8 n't — 12 n"t-\- 8 ; — 12 s" — 39'' 29'").
* (2604) The direct calculation of this inequality can be made, by a process like that
which is used for Mercury, in [3881f, &c.] ; but it is probable that the author deduced it
from the similar inequality of Mercury [4283], by the method given in [3883y].
f (2605) The terms of [4312] are computed by means of the formula [4295/!»] ;
changing, in the first place, n, «', e', into n",
i", respectively. Then changing
m , n , a , s mto
earth ; or into m'", n^"
the earth.
a', s', in computing the action of Venus on the
, respectively, in computing the action of Jupiter on
VI.x.^30.]
THEORY OF THE EARTH.
251
we shall have, for the inequality of the earth's motion in longitude [4052],
produced by the action of the moon,*
ôv"^ - -.-.sin.(U—v).
The inequality of the radius vector of the earth [4051] is
ôi-"^ — jj.R.cos.(U—v");
and the inequality of the earth's motion in latitude [4053] is
„ m R
711
The
moon's
action
produces a
perturba-
tion in the
longitude ;
[4314]
in the
radius ;
[4315]
latitude.
[4316]
in the
For greater accuracy, we must substitute f —— for —,
expressions of these three inequalities.
We shall suppose conformably to the phenomena of the tides [2706,2768],
m
R^
3S_
^"3 '
[4317]
* (-2606) The moon's action upon the eartli produces, in the radius vector, the longitude
and the latitude of the earth, the ineciualilics given in [4051, 4052, 4053] ; namely,
m
. r . cos.(v — U) ;
m r
MR
.{v-U);
m
Jl
rs
[4314a]
and by comparing the notation used in [4047, 404S], with that in [4313], it appears r^^Ub]
that we must change R, r, v, U, into ?•", R, U, v", respectively, to conform
nearly to the notation of this article. By this means the preceding expressions become,
m R . ,-rT „, ™ Rs
— ^.R.cos.{U—v");
M
M r-
corresponding respectively to the formulas [4315, 4314, 4316]. In the original work the
divisor r", by mistake, is omitted in [4314], and inserted in [4315] ; we have rectified
this mistake.
f (2607) The radius r [4048] has for its origin the common centre of gravity of
the earth and moon. This is changed into R, in [4314&], to conform to the present
notation ; but as the origin of R [4313] is in the centre of the earth, the value of the radius ■■ "^
is too great, and must be decreased in the ratio of M to M -\-m; which is equivalent
M
to the multiplication of the perturbations [4314 — 4316] by ; or in other words [43164]
to change the divisor M into M-\-7n, in all three of these formulas.
252 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
S being the sun's mass. Now, bj the theory of central forces [3700],*
we have,
[4318] -^ = n/ ; - = n - ;
n,t behig the moon's mean motion; hence we obtain,
[4319]
m 3n"2
M+i
[4319] We have by observation - = 0,0748013 [5117, 4835] ; hence we get,
71,
[4320]
Mass of consequently.
m 1
the moon.
M+m 59,6 '
m 1
If we suppose the sun's horizontal parallax to be 27",2 = 8',8, and
[43221 *^^® moon's mean horizontal parallax 10661" = 3454' = 57'" 34',t we
[4323]
shall have,
R sun's lior. par.
?•
moon's hor. par. 3454,0 '
* (2608) Substituting fi^Ji-j-m [3709«] in [.3700], then changing a, n, into
[4318a] Jl^ n^, respectively, we get the first of the equations [4318], corresponding to the moon's
motion about the earth. Changing in this, iW, m, B, n, into S, M, r", n",
[43186] and neglecting M in comparison with .S', we get the second of the equations [4318] ;
corresponding to the earth's motion about the sun. MuUiplying the first of the equations
r4318cl [■^■^■^®]' hy „ , and the second by 3; then substituting the products in [4317] we
set ir?-i n2=3?j"^: dividing this by n^, we obtain [43191; substituting in this
[4318rf] ^I-\--"' a J / ' L J J 6
the value [4319'], we finally get the expression of the mass of the moon [4321]. This
was afterwards found to be too great [4631, 1190i, &c.], as we have already observed
in [.3380 J, &ic.].
Instead of supposing, as in [2706], tliat the ratio of the mean force of the moon on the
r.fj.g -, tides, is to that of the sun as .3 to 1, we may express it more generally by 3(1 — |3)to 1 ;
by which means the second members of the equations [4317, 4319, 4320], will be
[4318/] multiplied by 1 — (3 ; and the last of these expressions will become - = -r^rê '■>
[4318g-] whence we get the following expression, which will be used hereafter, — = — .
[4322a] t (2609) This parallax, taken for the mean between the greatest and least values,
VI. X. §30.] THEORY OF THE EARTH. 253
consequently,*
iv" = _ 27",2524 . sin. (C/— v") = — 8%8298 . sin. {U—v") ;
pptturba-
tionn in the
longiliide,
[43'24]
or" = — 0,000042808 . cos. (U— v"). a„d in .h..
radius.
Then taking for s the greatest inequality of the moon in latitude, which ^43251
wo shall suppose to be 18543'. sin. (f/ — f) [5308]; U — 0 being the pcnurba
tion of
moon's distance from her ascending node; we shall obtain t inlau'"'
tude-
6 s" ^ — 0%7938 . sin. (U—è), [432e']
for the inequality of the earth's motion in latitude. We must add it to
the terms of is" [4312], to obtain the complete value of 6s"; and by
taking this sitm, with a contrary sign, we have the inequalities of the sun^s utïtâde.
apparent motion in latitude. These inequalities in the latitude have an
influence on the obliquity of the ecliptic, deduced from the observations of [43'27]
the meridian altitudes of the sun near the solstices. They have also an
influence upon the time of the equinox, deduced from observations of the
sun, when near the equinoxes, as well as upon the right-ascensions and
declinations of the stars, determined by comparing directly their places in
exceeds, by .33% the constant quantity in Burg's tables [5603], and is nearly conformable to
the resuh given by La Lande in *5' 1698 of tlie third edition of his astronomy. For the
purpose of illustration, we may neglect all the inequalities of the moon's pai-allax, except [43'22?i]
those depending on the moon's mean anomaly ; then taking the coefficients to the nearest
second, we have, from Burg's tables [5603],
J) 's hor. par. = 342P' + 187" . cos. (mean anom.) -)- 10'. cos. (2 mean anom.). r4322cl
The greatest value of this expression, corresponding to the perigee, or the mean anom. = 0,
is .3421>--f 18T"+10'; and the least value, in the apogee is 342P — 187^+10'.
The ?nert« of these two values 342P+ 10% exceeds hy 10% the constant term 3421";
and it is from causes similar to this, that the difference above-mentioned depends.
[4322rfJ
* (2610) The inequalities [4324] are deduced from [4314, 4315], by using the values
[4321,4323], and multiplying the value of S v" by the expression of the radius in [4324«J
seconds 206264%8.
t (2611) Substituting the values [4321, 432-3], and s [4.325], in [4316J, we get
Û »" [4-326] ; changing M into M-fm, in all these calculations, as in [4316e].
VOL. III. 64
[432fi<»J
254
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Perturba-
tion of the
sun in
declina-
tion,
[4328]
and in
right-
aacenston.
[4329]
[4329']
Increment of O's declination = —
the heavens with that of the sun. On account of the great accuracy of
modern observations, it is necessary to notice these inequalities. It is
evident that this correction increases the apparent declination of the sun, by
the quantity,*
Ss" . COS. (obliquity of the ecliptic) _
COS. (sun's declination) '
and its ajiparent right-ascension is also increased, by the following
expression,
r c ^1 • 1 5s". sin. (obliquity of the ecliptic) . cos. (sun's riffht-ascension)
Inc. of O's nght-ascen. = ^^ — -, — , / ,. — ~ \
° COS. (sun s declination)
fVe must therefore decrease, hy these quantities, the observed declinations and
right-ascensions of the sun, to obtain those lohich ivould be observed, if the
earth did not quit the plane of the ecliptic.
* (2612) Let ECC be the ecliptic, -EQQ' the equator, P the north
equator; then if the earth's latitude, north of the ecliptic, be
5s", that of the sun will be south, and may be represented
by CU = &s" perpendicular to the ecliptic. P CL Q,
r4328al PC'L'Q^, are circles of declination, perpendicular to
the equator, and L L' is parallel to the equator. The
small differential triangle CLU, may be supposed
rectangular in L, and angle LC L'= 90' — angle E CQ^.
Then in the spherical triangle E C(^, we have, by
[1345^2], cos.i:CQ = sin.L CL'=^s\n.CE q.cos.Eq-,
COS. CE q
[43386]
sm.ECq=cos.LCL'=
COS. C Q
Now the declination
is decreased by the quantity C L ; the right-ascension is
LL' LL'
lEqaator
[4328c] increased by the quantity QQ' =
sin. PL
COS. dec.
and we have
[4328rf] LL'=^CL'. sin. LCL' = 5s" . sin. CEq. cos. E q ;
hence we get.
[4328e] Increm. dec.= — CL =^—CU. cos. L C L--
, „ COS. CEQ
OS .
COS. CQ
, as in [4328] ; and
I . , y-^ ^, L L' sin. C E Q .cos.E Q
[4328/] Increm. nght-ascen. Q Q' = -—g- ==Ss' -^ ^, as in [4329]
COS. dec.
VI. x.§31.]
THEORY OF THE EARTH.
255
[4329"
On the secular variations in the Earthh orbit, in its equator, and in the
length of the year.
31. We have given, in [4244, 4249, &c.], the secular variations of the
elements of the earth's orbit ; but the influence of these variations on the most
important phenomena of astronomy has been an inducement to compute them
with greater accuracy, noticing the square of the time t;* supposing t to
denote the number of Julian years elapsed since 1750. We have found by
the methods given in [1096 — 1126], and using the values of the masses of [4329"
the planets [4061], that the coefficient of the equation of the centre of the
earth's orbit is represented by,t
* (261.3) The values of e^, tang. « [1109, 1110], give those of e"^, tang, a";
by changing the quantities corresponding to m, into those relative to m", and the contrary.
The formulas, thus found, may be developed in series, ascending according to the powers of
t, by Taylor's theorem [.3850a] ; hence we easily deduce the values of c", •zs", in similar
forms. The calculation may also be made by the method pointed out in the following note.
t (2614) We have, by Taylor's theorem, as in [1126'"],
2e'
2 de"
= 2E+~.t +
dde"
~dfi
neglecting the higher powers of t ; the values of — ,
to the epoch 1750. The differential of
-— , -— - , being taken to correspond
de"
— [1122], taken according to the directions
dde"
[4329a]
[43296]
[4330a]
[43306]
[4330c]
in [1126"], or as in note 768, vol. I. p. 612, and divided by dt, gives -— , m terms
of e, e', 8ic. w, -n', &.c. and of their first differentials. Substituting in this expression,
the values of these first differentials, given in [1122, 1126], it changes into a function of the
finite quantities e, c', fee. tn, -s/, &,c. ; and by substituting the values of these quantities, [4330(/]
dde"
for the year 1750, given in [4030,4081], we obtain the expression of
dfi
Moreover,
de"
by similar substitutions, we get the value of the expression of "-^ [1122]. These values,
bemg substituted in [4330a], give the expression of 2 c" [4330]. The formulas
[4330—4360] are so frequently referred to in the work, that we have given the numerical
values in centesimal, as well as in sexagesimal seconds. The values given in
[4330, 4331, 4332], are altered, in [4610 — 4612], by reason of the changes in the masses
of Venus and Mars.
We have seen in vol. I. p. 612, note 468, that terms of the order m'e' are retained,
and those of the order m'e'^, which are of the Jirst order relative to the mass m', are
[4330e]
[4.330/]
256 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Coeff. eqiia. centre = 2E — t. 0",579130 — f . 0",0000207446
= 2E—t. 0% 187638 — f . 0',0000067213,
2E being this coefficient at the beginning of the year 1750, when t is
ëàrth'B nothing. We have also found the sidéral longitude of the perihelion of tlie
earth's orbit, namely,*
Long, perih. of the earth = ^"+ i • 36",881443 + t"' • 0",0002454382
= t^' + i . 1 1',949588 + t- . 0 ,0000795220.
Lastly, the values of jf, q", at any time t, have been found respectively
equal to,t
p" = t. 0",2.36793 + Ï-. 0",0000665275
= t . 0',076721 + t- . O',0000215549 ;
q" = — t. 1",546156 + t\ 0,0000208253
=^ — t. 0',500955 + i"-. O',0000067474.
[4330]
Secular
equations
of the
earth'
orbit.
[4331]
[4332]
[4330A]
d e"
[4330e-] neglected, in the expression of — [1122]. If we suppose, for a rough estimate, that
e' = -^'^, the neglected terms will be of the order of j^^y part of those retained; so that
the neglected part in the coefficient of t [4330], may be considered as of the order
j^X0%18763S =0^0004, which is much greater than the coefficient of t^ in [4330] :
and at the first vjew it might be thought strange that we sliould neglect this, and yet notice
the much smaller coefficient of fi, which is of the order of the square of the disturbing
masses. But the reason will appear very evident from the consideration, that when t is
large, the term depending on t^ becomes very great in comparison with these neglected
[4330i] terms. Thus, iÇ t= 2500, the neglected term 0,0004 1 is only one second, while the
term depending on t^, exceeds 42". Similar remarks may be made relative to the
quantities w", jj", q" [4331,4332].
* (2615) Proceeding as in the last note, we may deduce from [3S50«], by changing
[4331a] M into •zn", ^" = t^"-\- i • -j-/ — \~ i^^' "TT ' ^'^^ quantities In the second member referring
to the epoch of 1750. The difierential of —— [1126], divided by dt, gives —rur
[4331i] i'^ terms of e, e', &c. zs, zi' , and their first differentials. Substituting in this expression
the values of the differentials [1122, 1126], it changes into a function of the finite quantities
e, e', &:c. -a, -a, Sic; and by using the numerical values [4080,4081], we get the
'^ values of — - , -775-) to be substituted in [4331a], to obtain [4.331].
t (2616) The expressions of —-, -—, are in [425 1 i] ; their differentials taken
Vl.x.§31.]
THEORY OF THE EARTH.
257
We have given, in [3100 — 3110], the expressions of the precession of the
equinoxes,* and of the inclination of the equator, referred to the fixed
ecliptic, and to the apparent ecliptic. In these formulas, we have supposed
the values of p'\ </", to be given under the forms
;?"==-. c . sin. (§-f + (3) ; q" = ^ . c . cos. {g t + ^) [30686].
Moreover, we have seen, in [1133], that the finite expressions of //', ç",
appear under these forms, and we may determine, by the method explained
in [1098, &c.], the values of c, g, p. To obtain these quantities
accurately, by this method, we must know the correct values of the masses
of the planets; and there is considerable uncertainty relative to some of
them, as we have observed in [4076, &c.]. Therefore, instead of making
the tedious calculation, required by this method, it is preferable to simplify
it, so as to embrace a period of ten or twelve hundred years, before and
after the epoch of 1750 ; which is sufficient for all the purposes of astronomy.
We may easily rectify these calculations as often as the development of the
secular variations shall make known, with greater accuracy, the masses of
the planets. We shall give to the values of p" and (f the following
forms, which are comprised in those mentioned in [4334]. f
p" = 2. c.sm.(^gt-\- (3) = c. sin./3 — c.cos.f3. s'm.gt — c.sin./3. s\n.(g't-'r}'^) ;
ç"= 2. c.cos.{gt -|_/3) ^ c.cos. f3 — c.cos.(i.cos.g( — c . sin. § . COS. (g't-\- ^v) ;
ff being the semi-circumference of a circle whose radius is unity. If we
[4333]
[4334]
[4335]
[4336]
Assumed
forms of
[4337]
r", 1"-
, . , 1 1- • 1 1 1 , ■ ddp" ddq"
relatively to t, and divided by at, give -tt ' ,
dp dp
in terms oi ---, -; — , cic.
dt at
—, — , &:c. ; substitutina; the values of these last quantities [11321, we get -~- , — - ,
dt dt' ' ^ '■ L J' & (^^2 ' rf<a '
expressed in finite terms of p, p', &;c. q, q', &c. The values of p, p', &tc.
q, q', &z;c. are given in [4251c], in terms of (jj, cp', &,c. ê, &', &,c. ; and the
numerical values of these last quantities, in the year 1750, are in [4082, 4083] ; hence we
obtain the numerical values of p, p', Sic. q, q', &;c. at that epoch. Substituting
these in [4251 f/,e], and in the preceding values of
d dp" d d q"
we get tlie numerical
dp" dq" ddp" ddq"
values of — , -;— , — r^
dl dt' dt3
in the general values of p
dt^ ' dfi
-— -, at the same epoch, 1750 ; these are to be substituted
q" [4250], to obtain [4332].
* (2617) The formulas, here referred to, are [3100, 3101, 3107, 3110].
t (2618) The three terms of the second member of the value of p" or q" [4337],
VOL. III. 65
[4332a]
[4.332i]
[4.332c]
[4332rf]
[4333a]
258 PERTURBATIONS OF THE PLANETS ; [Méc. Ctl.
develop these two functions relatively to the powers of the time t, we
shall find, by comparing them with the preceding series [4332],*
Valuesof c £r . COS. |3 = — 0',076721 :
[4338]
[4339]
[4337a]
[4338a]
[43386 ;
eg', sin. f3 = — 0',500955 ;
cg\ cos. [3 = 0S0000134948 ;
cg'K sin. |3 = 0',0000431098.
Hence we easily obtain,!
g = — 36^2808 ;
g' == — 17',7502;
c. sin. f3 = 582P,308;
c.cos. f3 = 436%17.
are deduced from those of p" or q" [4334], by changing c, g, p, respectively, into
c, 0, p, in the first term ; — c . cos. p, g, 0, in the second term ; and — c.sin. p, g', J^r,
in the third term. Tliese expressions of p", q", being developed according to the powers of
[433/6] f^ and compared with those in [43.32], give, as in [4.339], values of c, p, g, g', which
satisfy the numerical expressions of p", q" , [4332], neglecting f, and the higher
powers of t : and as the values [4332] will answer for ten or twelve centuries from the
epoch, it will follow, that the forms assumed in [4337] will answer for the same period, by
using these values of c, p, g, g'.
* (2619) We have by development, using the formulas [43, 44] Int. and neglecting terms of
the order «^ sin.gt = gt; cos.gt=l — hg^i-; sm.{g't-]-iv) = cos.g'i = l — ig"^t^;
cos. {g't -\-i ■ïï) = — sin. g't = — g't ; substituting these in [4337], we get,
p" = i: .c . sin. {gt -i-^)=c . sin. p — c. g t .cos.fi — c.(l — i g'^t~) .s'm. p
^= — t . {c g . COS. p) + t^-{i cg'''^. sin. p) ;
ç"=2. c .COS. (^^ ~\-fi) = c. cos.(3— c . {I — Ig^t-). COS. p -{- c g' t . s\n. fi
=z t . {eg', sin. p) + fi. (I cg^. COS. p).
Comparing the coefficients of t, in these expressions, with the corresponding ones in [4332],
r4338cl "'6 &^^' without any reduction, the two first equations [4338]. In like manner, by
comparing the coefficients of I i^, in [4332,43336], we get the other two equations [4338].
f (2620) Dividing the square of the first equation [4338], by the third, we get
c . cos. p [4339] ; and the square of the second, divided by the fourth, gives c . sin. p [4339].
[4339a] Now, dividing the values of c^^.cos.p, cg'~. s'm.fi [4338], by those of eg. cos. g,
eg', sin. p [4338], respectively, and multiplying the products by the radius in seconds,
206265% we get g, g' [4339].
VI.x.§31.] THEORY OF THE EARTH. 259
Now we have seen, in [3100], that the precession of the equinoxes +, relative precessio..
, , , relative to
to the fixed ecliptic of 1 750, noticing only the secular variations, is, 'cU|,1"tf
■I = /^ + ^ + 2 . I [j— 1 j . tang, h + cot. /i 5 • y • sin. (ft + f3).
[4340]
First form.
To obtain ^ .c .un. {ft -\- ^), we must increase the angle gt-\-^., in
2 . c . sin. {g t + f3), by the quantity 1 1 [3073', &c.] ;* making f = g-\-l
[3113a] ; then we shall have,
2 . c . sin. {ft + f3) = c . sin. {lt + f^) — c . cos. (3 . sin. {gt ^l t)
— c . sin. p . sin. {gtJ^U + \^) ;
consequently,!
[4341]
[4342]
* (2021) If we increase the angle gt, by the quantity lt={f — g) t [3113a],
the function 2 . c . sin.(^< + P) will become 2 . c . sin. (/<-(- p), as in [4341] ; and the
first equation [4337], will change into [4342] ; observing that we have ^ = 0 [4337aJ,
in the first term, or c . sin. p ;= c . sin. [0 . t -{- js), which becomes c . sin. {It -\- p), as in
the first term of [4342].
[43410]
t (2622) The expression 2 . c . sin. (/i + P)i in the form assumed [4342], consists
of three terms. In the first of these terms, the general symbols c, f, /3, of the first [4342o]
member, become c, I, 3 ; or in other words, f is changed into /, while c, (3, are
unaltered ; and the corresponding term of [4340] becomes,
[43426]
[4342c]
I \ ') Ic
. 1 j . tang, h -f-cot. A> . — . sin.(/ t -\- ^) ; or simply, c.cot.h . sin. {It -\- ^);
which is the first term of 4^ [4343], depending on c. The second term of [4342],
— c . cos. p . sin. {gt -\- It), being compared with the general expression c . sin. {ft + p),
in the first member of [4342], shows that c, /, p, must be changed into — c . cos. p,
g -\-t, 0, respectively; and the corresponding term of [4340] becomes,
'i / ' , \ II 7 ) 'c .COS. (3 . , , , ,
— ^(^rpj— Ij-tang.A+cot.A^ .____. sm.(^i + ?0 ; [4342d]
which is easily reduced to the same form as the term of [4343], depending on the angle
gl-\-lt. Lastly, the «Aire/ term of [4342], — c .s\n.fi .s\n.{^t -\- It -{-\v), being r4342e]
compared with the general term, in the first member of [4342J, gives for c, f, (3, the
corresponding expressions, — c.sin.(3, g -\-h J*, respectively; and the resulting
terra of [4340] is,
— ^(^; — l)-tang.A+cot.A^ . ^±^ .ûu.{g' t-{-lt + 1^);
which is easily reduced to the form of the last term of [4343]. The two first terms of
[4340, 4343], represented by It -\- 1, are the same in both formulas.
[4342g]
260 PERTURBATIONS OF THE PLANETS; [Méc. Cél.
r;sft" ^ = It + ?, + €. cot. h . sin. Qt + p)
the fixed
ecliptic 01 7 r n- '\
1750. ^ ^ ^ çjjg_ o _ S çQ^_ ^ ^_ _ ^ajj„_ /j / _ gij^_ (-rrt + lt)
[4343] ^^ ^ ' •^ -"
I
Second — ^ . c . sin. f3 . > cot. h — ~_ . tang, h i . sin. (g't-j- ii-^ i'^)-
form.
l+g'
[4345]
îôfàiwe'"" Then by putting V /or the inclination of the equator to the fixed ecliptic of
ediptkof 1750, we shall have, as in fSlOll,*
1750. ' ' L J'
[4344] V = h — ^.-. COS. (ft + p).
First formj J
To obtain 2 . c . cos. (ft + /3), we must increase the angle gt -{- § in
2 . c . cos. (gt + P) by lt\ [3073, &c.] ; hence we shall have,
2 . c . cos. (ft -j- p) = c. cos. (I t-i- j5) — c . cos. f3 . cos. (gt -\- 1 1)
— c . sin. |3 . COS. (g't J^lt-^-^-n) ;
therefore, Î
second J
form. Y = h — C . COS. (I t + (3) -j- —— . C . COS. f3 . COS. (g t + 1 1)
[4346] ^^^
+ — -j.c.sin. f3.cos.(^7 4-Z< + i^).
[4347] 4^' denoting the precession of the equinoxes relative to the apparent ecliptic.
* (2623) This is the same as [3101], putting V for the part of ê, depending on
[4344a] ^ and 2 ; or in other words, neglecting the periodical terms depending on the angles
/^ + p', 2«, 2t)'.
f (2624) This is done upon the principles used in [4341, &.c.]; and in the same
[4345o] manner as [4342] was deduced from the first of the equations [4337], we may derive
[4345] from the second of [4337].
X (2625) Proceeding as in [4342« — ■/] ; and comparing the general form of the first
member of [4345], with the three terms of the second member, we find, that c, f, p,
become, respectively, c, /, p, in the Jirst term ; — c . cos. p, g -\- I, 0, in the
second term ; and — c . sin. p, g' -\- I, I "^j in the third term,
in the terms under the sign 2 [4344], we get the three terms c
the first term h, is the same in both expressions [4344, 4346].
second term; and — c.sin.p, g' -\- I, I "^j in the tAirrf term. Substituting these values
in the terms under the sign 2 [4344], we get the three terms containing c, in [4346] ;
VI.x.§31.] THEORY OF THE EARTH. 261
and V' the inclination of the equator to this ecUptic_; we shall have, as in [4347]
recesfiion
fjuiiy rola-
live ic the
I T ) . ^ , T \ ajiparciit
eclii»tic.
[4348]
[4350]
[3107,3110],* r^rz:
1' = lt +^ + 7^ .c.cos. |3. )cot. /i+ 7-— . tana;. /t i. sm. (irt-\-lt)
+ j^—, . c . sin. p . j cot. h + — ;; . tang, h I . sin. (g't-^ lt+ ^r:);
V = h — j^ . c . COS. (3 . cos.(gt+lt) — j^-^, . c.sin./3. cos.(^7 + /^+i*). [««]
The expression of 4-' gives,t
~ = Z + c ^ • COS. ^ . < cot. /t + -=—, — . tang, h > . cos. (£[14-1 1)
dt ^ I l+g i ^^
+ c §•' . sin. (3 , ) cot. h -f — — . tang, h > . cos. (g't ~\-lt -{-^■jr).
If we subtract from this value of — -, when t is nothing, its value at any [4350'j
other epoch, and reduce the difference of these two expressions to time ;
considering the whole circumference as equal to one tropical year ; we shall
get the increment of the length of the tropical year since 1750. We see, ' ^
by this formula, and by the differential of the general expression of
* (2626) Retaining only the secular inequalities in 4"', ^' [3107,3110], changing
also Ù' into V [3103, 4347'], we get, by a slight reduction in the term of -^J, under [43''''"]
the sign 2,
+' = ^^+? + 2.^cot.;i+j.tang.A^ . (^-^V c .sm.(ft + fi); [43476]
V = A + 2 . (^-^ ^ . c . COS. (/i + p). [4347c]
In the terms under the sign 2 [43476], we must substitute, successively, the values of the
triplets of terms c, f, ^, given in [4342a, c,/], and we shall obtain [4348] ; observing
I — f
that the first term vanishes, because the factor — — = 0. In like manner the substitution
r , ['4347f]
of the same triplets of values [4346a — 6], m [4.347c], gives h [4349] ;the first term vanishing,
f I
on account of the factor = 0.
/
1(2627) The differential of ■].' [4348], taken relatively to t, and divided by rf r, [4349a]
gives [4350].
VOL. in. 66
262
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4350'"]
[4351]
[4351']
[4352]
[4353]
[4353']
[4353"]
[4354]
4'' [3107],* that the action of the sun and moon changes considerably the
law of the variation of the length of the year. In the most probable
hypothesis on the masses of the planets, the whole variations, in the length
of the year, and in the obliquity of the ecliptic, are reduced to nearly a
quarter partf of what they would be without that action [31 15, 31 13;y].
^ 154",63 ^ 50', 1 ;
According to observation, we have in 1750,
but, by what has been said, we get at this epoch, |
I
~dr
I -{-eg . cos. (3 . ) cot. A +
l+.
di
tang, h
V
hence we obtain,
l-\r c g . cos. I'S . \ cot. h -{■
1 + ^
tang. /j \ = 154",63 = 50',1.
If we neglect the square of c, in this equation, we may substitute for
h, the obliquity of the ecliptic to the equator in 1750.§ This obliquity
was then, by observation, 26^,0796 = 23" 28'" 17 ,9 ; hence we deduce,**
I = 155",542 = 50',396 ;
[4350o]
[4351a]
[43516]
[4352a]
* (2623) Tills difTerential is found in [3 11 8], and by reducing it into time, as in [3118'],
we get tlie decrement of the 3'ear, using f=zg-l-l [3113a]; or the increment of the
year, by changing its sign, as in [4350"].
t (2629) This subject has ah-eady been discussed in [3113a — z] ; and we have merely
to remark in this place, that the values arbitrarily assumed in [4337 — 4339] do not produce
such essential alterations in these variations of ■]^', V, as are mentioned in [3113iy, 4351].
This ditierence is what might be expected, taking into consideration, that the results, obtained
in [4338, 4339], are restricted to values of /, which are less than 1200 [4335] ; and that
for much greater values of t, the results cannot be relied upon.
t (2630) At the epoch 1750, we have <;= 0 [4329"], and then cos. (gt-{-It)=l,
cos. {g t -\- 1 1 -{- ^ v) = cos. ;| ir ^ 0 ; substituting these in [4350], it becomes as in [4352] ;
putting this equal to 50',1 [4351'], derived from observation, we get [4353].
<§! (2631) The expression of V [4346] differs from h, by terms of the order c;
[4353a] hence it is evident that if ws neglect terms of the order C", we may substitute indifferently,
the value of V or h, for h, in [4353].
** (2632) Substituting in [4353] the values k = 23''28'"17',9 [4353"], also the values
[4354a] of eg. cos. ^, g [4338,4339], it becomes, as in the following equation, from which we
easily obtain the value of / [4354],
VI.x.>,^31.] THEORY OF THE EARTH. 263
then we have in 1 750,*
V = A — -^ . C . COS. ^ ; [4355]
which gives,
h = 26°,0796 — 3460",3 = 23'^ 28" \Tfi — 1 121', 1. [4356]
By means of these values we obtain the following expressions,! [wliich arc
altered in 4614 — 4617],
/ _ 0',076T21 . cot. 23'' 28"' 17%9 — ^^7^ • tang. 23* 28" 17%9 = 154',63. [43346]
* (2633) Putting ^=^0 in [4349], it becomes as in [4355]. Substituting in tiiis,
V = 23'' 28'" 17%9 [4353"], also tbe values of /, g, c.cos.p [43.54,4339], it becomes, [435Ga]
23'' 28"' \V,9 = A + 1 121 ',1 ; hence we get h [4356].
t (2634) Dividing the value of c.sin.|3 [4339] by that of c.cos.js [4339], we
get tang.|3=13,.34636 = tang.85''42"'54"; hence (3 = 85'' 42'" 54°' ; substituting this [4357a]
in the expression of c.sin.|3 [4339], we obtain c = 5321',.308 . cosec. (3 = 5837',6.
Using these values of p, c, and these of A, I, g, g" [4356, 4354, 4339], we get, [43576]
c. cot.A = 13646',3;
. c . COS. p . j cot. h — .j^— . tang. A ^ = — 5o52',8 ; [4357c]
— -L.c. sin. p . 5cot. h — -^ . tang. hl= — 23097%7 ;
'+g c t-f-g }
l-\-g=z 14',115 ; Z+^^=.32',645. Substituting these in the third, fourth and fifth
terms of [4343], we get the third, fifth and fourth terms of [4357], respectively. The [4357^^]
term 1 1 [4343, 4354], gives the first term of [4357]. The term ^ [4343], is to be
taken so as to render ■\,=^0 [4357] when « = 0 ; whence
^ = _ 13646-,3 . sin. 85'' 42" 54' + 23097^7 = 2''38'" 9',4. [4357e]
In like manner, we have,
' .c.cos.p=1.557V3; -L.c .sin.p = 8986',6 ; [4357/]
substituting these and h [4356], also the preceding values [4357c], in [4346], we
get [4.358].
From the same data, we have,
^ . c . cos. p . < cot. h -\- — — . tang. h>= — 4333',2 ;
i-Vs " d ' l+g
z^—,.c. sin.p .^cot./i + ,-r— ,. tang. A[ = — 9499',4
l+g i l+g >
[4.3.57e-]
[4358]
[4359]
264 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
^ = tA 55",542 + 2°,92883 + 421 1 8",3 . sin. (t . 1 55",542 + 95°,2389)
— 71289",2 . COS. i.(100",757) — 16521",! .sin. (i.43",564)
^^^^'^ = t . 50^396 + 2''38"'09%4 +13646%3. sin. {t . 50^396 + So'' 42"" 54')
Precession
-f,;bj;- — 23097%7 . cos. {t . 32%645) — 5352',8 . sin. {t . 14',1 15) ;
theeclip-
ye.^"^^'" V = 26°,0796 — 3460",3 — 1 801 7",4 . cos. {t . 1 55",542 + 95°,2389) to.""']
+ 4806",5 . COS. (t . 43",564) — 27736",3 . sin. (t . 100",757)
== 23^ 28"'17',9 — 1 121%1 — 5837--,6 . cos. (t . 50',396 + So' 42"" 54')
+ 1557'-,3 . COS. (t . 14%1 15) — 8986',6 . sin. (t . 32',645) ;
i' = f . 155",542 + 2°,92883 — 29288",3 . cos. t . (100",757)
— 1 3374",2 . sin. (t . 43",564)
=- t . 60',396 + 2''38'" 09^4 — 9489',4 . cos. (t . 32%645)
— 4333',2.sin.(f . 14', 115);
[Apparent!
orbit. J
V'= 26°,0796 — 3460",3 . ^ 1 — cos. (t . 43",564) |
— 9769",2 . sin. (i . 100",757)
= 23''28"'17',9 — 1121',l.jl ~cos.(i. ]4',115)i
— 3165',2.sin. (<.32',645).
We may determine, by means of these formulas, the precession of the equinoxes
and the obliqiiity of the ecliptic, in the interval of ten or tivelve hundred years
[4357A] sin. {g t -{- 1 1 -\- ^ -jr) = cos. (^-'^ -f- / ^) = cos. {( . 32%G45) ;
Z< = i . 50',.396. Substituting these in [4348], it becomes as in [4359], the constant
quantity 2,, being taken so as to make 4-' = 0, when t=^0 [4359] ; consequently,
t*^^^'^ ^ = 9489',4 = 2-* 38"' 9%4.
Lastly, by a similar calculation, we have,
-f^ .c.cos.p = — 112P-,1 ; ,4— , •c.sin.3 = — 3165',2;
[4357i] ' ' "
cos.(g't-\-lt-]-h'^)= — sm.{g't + It)= — s\n.{t. 32^645) ;
substituting these and [4356] in [4349], we get [4360]. The numerical values, given in
r4357n [435T— 4360], are varied by the author in [4614 — 4617], on account of the changes made
in the values of the masses of Venus and Mars. We have already given the formulas of
Poisson and Bessel, in [3380^,(7].
[4360]
VI.x.§3l.]
THEORY OF THE EARTH.
265
before, or after the epoch of 1750; observing to make t negative, for any
time previous to this epoch. We may indeed apply the formula to the
observations made in the time of Hipparchus ; taking into consideration
the imperfections of these observations.
The preceding value of -i', gives, for the increment of the tropical year,
counting from 1750, the following expression,*
Increment of the year = — O''^000083568 . {1 —cos. {t . 14^1 15) \
— 0''»^00042327 . sin. {t . 32'-,645).
Hence it follows, that in the time of Hipparchus, or one hundred and
twenty-eight years before the Christian era, the tropical year was 12'*''-,326
[= 10,65 sexages.] longer than in 1750;t the obliquity of the ecliptic was
also greater by 2832",27 = 917^66.
[4361]
[4362]
[4363]
[4363']
* (2635) Using the same data as the preceding note, we get the numerical values of
the two functions [4362c, (/], expressed in sexagesimal seconds. These are turned into time
by supposing the whole circumference, 360''= 1296000", to be described in one year, or
3g5da,s^242 ; hence we have,
c^.cos.p. ^cot.^ + — — .tang.A?= — 0',296527=:— 0''"y,000083568 ;
c g'. sin. p
jcot. h
■ -r—, . tang. A ^ =— l',501877 = — 0''^^00042327.
Substituting these and [4357c?], in [4350], we get the general expression of -— [4362/] ;
which becomes as in [4362^], when t^O. Subtracting the first of these expressions
from the second, we get the increment of the year [4350'], as in [4362], corresponding to
any number t, of years after 1750.
^=1 — 0''"5',000083568 . cos. {t . 14',1 15) + 0^''y,00042327 . sin. {t . 32',645) ;
^ = / — 0'i''^00008356S.
dt '
These numerical values are altered in [4618], in consequence of a change in the values of
the masses of Venus and Mars.
[4362o]
[43626]
[4362c]
[4362rf]
[4362e]
[4362/]
[43C>2g]
[4.3G2/I]
t (2636) In the year 128 before the Christian era, < = — 128— 17.50 = — 1878;
substituting this in the two terms of the expression [4362], we find that the first terin [43(53„n
becomes, —0^»y,00000069, and the second, + ©■'"".OOOl 2396 ; their sum is O^'^OGO 12327,
as in [4363] nearly. The variation of the obliquity of the ecliptic, in the same time, 1^43(53^1
deduced from [4360], is nearly the same as in [436.3'], being expressed by,
— 112P,1.{1 —cos.{t. 14^115)} — 3165,2.sin.(i.32^645)
= — 9^,2 + 926^9 = 917%7 nearly. I'i363c]
VOL. III. 67
equinox
[4364]
and sun's
apogee
coincide.
266 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4363"] A remarkable astronomical epoch, is (hativhen the greater axis of the earth\s
hie'as'tro- orMt ivas situated in the line of the equinoxes; because the apparent and ji.ean
Xt'the equinoxes then coincided. We find, by the preceding formulas, that this
phenomena took place about 4004 years before the Christian era, and at this
epoch most of our chronologists place the creation of the ivorld ; so that, in this
])oint of view, we may consider it as an astronomical epoch. For we have,
[4364'] at that time, t = — 5754 ; and the preceding expression of 4^' gives,*
[4365] ^'= — 79'' 04™ 04';
which is the longitude of the fixed equinox of 1750, referred to the equinox
of that time t. The preceding expression of ra", gives, for the longitude
of the perigee of the earth's orbit, or of the sun's apogee, referred to the
fixed equinox of 1750,t
^"= 80M5"'ll'.
This longitude, referred to the equinox of the year 4004 before the Christian
[4367] era, is V II '"07; J hence it follows, that the time lohen the longitude of the
suri's apogee, counted from the moveable equinox, ivas nothing, precedes, about
sixty-nine years, the epoch usually assumed for the creation of the world.
This difference will appear very small, if we take into consideration the
imperfections of the preceding expressions of 4-'? and ro", when applied to so
[4367"] distant a period, and the uncertainty which still remains relatively to the motion
of the equinoxes, and to the assumed values of the masses of the planets.
[4365']
[4366]
[4367']
[4365a]
* (2637) Putting ^=—5754, we have < ..32'-,G45 == 52''10"'39';
£ . 1 4%1 15 = 22'' 33"' SS'- ; t . 50»V396 = 80'' 32"" 59» ;
substituting these in [4359], we get the value of -.j^' [4365].
t (2638) Substituting «"= 98''37"'16^- [4081], in [4331], it becomes,
[4366a] T^" = 98''37'" 16" + i . 1 1',949588 + t^. 0^,000079522 ;
and by putting ^ = — 5754, it is reduced to 98''37'"16'— 19''5'"58'+43"' 53"=80"50'"1I',
as in [4366].
J (2639) Taking, for the fixed point, the equinox of 1750 ; the longitude of the
moveable equinox, and of the solar apogee, corresponding to the year 4004 before Christ,
[4367a] will be respectively 79'' 4™ 4'' , 80''15'"lp- [4365,4366]; the difference of these quantities
j^rfj^jm-js represents the distance of the perigee from the equinox at that time. The
[4367t] distance of these points, in the year 1750, was 98'' 37'" 16' [4081] ; so that in the period
of 5754 years, they have approached towards each other, by the quantity.
\l.x.^31.J THEORY OF THE EARTH. 267
Another remarkable astronomical epoch, is that when the greater axis of the
Another
romarka-
eurth^s orbit, was perpendicular to the line of equinoxes ; for then the apparent [4367"]
ana mean solstices were united. This second epoch is much nearer to our '"''<^°"'°
[4368]
times; it goes back nearly to the year 1250. For if we suppose t = — 500, eq:
uinox
and sun's
the preceding formulas give 90'' 1 '",* for the longitude of the sun's apogee, [4368']
counted from the moveable equinox. Hence the time when this longitude diSr'°
was 90'', corresponds very nearly to the beginning of the year 1249. The
imperfections of the elements used in this calculation, leaves an uncertainty
of at least one year in this result.
[4309]
98''37'«16'— I'' 11"'7»= 97''26"'9^; [4367c]
being at the rate of about 61* in a year ; and at this rate, the arc fll"?" will be [4367d]
described in about 69 years ; so that the equinox and solar apogee must have coincided about
the year 4004 -[- 69 = 4073 [4367'J before the Christian era, according to the data we
have used.
[4368a]
* (2640) In the year 1250, we have ^ = 1250 — 1750 = — 500 ; and for this
value of ^ we get, from [4359, 4366a], 4,'= — 6'' 57"'; z;i" = 96''5S'"; therefore
the solar apogee, in 1250, was distant from the equinox of that time, by the quantity
96<i 58"" — Q^ 57"" = 90'^ 1 "■ ; [43686]
and as the distance of these points, in 1750, was 98''37'"16'" [43676], the variation of
distance, in five hundred years, is 98''37"' 16' — 90'' ]"■= 8''36"' 16^ being about 61" in a [4368c]
year, as in [4367rf] ; consequently, the distance of these points must have been 90^, about
one year before the year 1250, or in the year 1249.
268 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
CHAPTER XI.
THEOIIY OF MARS.
32. We have, in the case of the maximum * of d V",
[4370] 6 0. = — (1 —o?).ôY"';
r"
[4370] supposing — = a. If we consider r'" as the onlj variable quantity in
a, we shall have,
ma
[4371] 5 r"' = — . ( 1 — a=) . 6 V".
[4371'] If we take for r", r'", the mean distances of the earth and Mars from
^S' the sun [4079], and suppose 5 V" = ± 1" = ± 0^324, we shall get,
[4372] 5 r'" _ ± 0,000002076 ;
may be
neglected, therefore we may neglect the inequalities of the radius vector r"', whose
coefficients are less than ± 0,000002. We shall also neglect the inequalities
of the motion in Mars in longitude, which are less than a quarter of a
centesimal second, or 0',081.t
* (2641) The earth is situated, relatively to Venus, in the same manner as Mars is,
r4'î70 1 relatively to the earth ; therefore we may obtain 5 V"', corresponding to Mars [4370],
from the calculation made for Venus in [4297, 4298], by merely changing the accents on
V, in [4298], which makes it become as in [4370], and using a [4370']. Now the
ôr"'.r"
variation of a [4370'], considering a, r'", as the variable quantities in ôa.= ^^tj— 5
substituting this in [4370], we get [4371]; and by putting r"=a", r"'=a" [4079],
using also a [4159], .5 V" [4371'] ; it becomes as in [4372].
[4373a]
* (2642) The values [4373,4374] are computed from the functions [4277a, 6],
accenting the symbols so as to conform to the present example.
VI. xi. §32.]
THEORY OF MARS.
269
Inequalities of Mars, independent of the excentricilies.
6v"'^ (l+(^')-
+ 0+O
+ (l+f^'')
0',208754 . sin. (n't — n'" t + s— e'")
- 0',024915 . s\n.2 (n't — n'" t + -'— s'")
j _ 0',005000 . sin. S (n't — n'" t + t'— ^"')
( _ 0',001368 . sin.4(n'i — n!" t + t— O
6',988832 . sin. (n" t — n'" t + £"— O
— 0',968689 . sin. 2(n"i — n'" < + s"— O
— 0', 1830 12 . sin.3(n" t — «'"f + s"— s'")
— 0',058242 . sin.4(n" t — n'" t + s"— i'"')
— 0',023099 . sin. 5 (n" t — n'" t + s"- s'")
— 0%010339 . sin. 6 (n" i — n'" t + s"_ /")
— 0',004992 . sin. 7 (n" t — n"'t + «"— ^"')
24S440843 . sin. (n'' t — «'" < + e'"— Z") ~
— 13',598063 . sin.2(n-'' t — n'" t + e'"— s'")
— r,l 80288 . sm.S(n''t — n'" t + £'"— s'")
— 0%172768. sin.4(n'''ï _ n'" t + e-— e'")
— 0',033166 . sin.5(n''' t — n"'t + s"— s'")
— 0'-,013422 . sin. 6 (n'" i — n!" t + s''— i'")
[4373]
Inequali-
ties inde-
pendent of
the excen-
tricilies.
+ (l+(xO.
P,343754 . sin. (n't-
0^,443668. sin. 2 (n^r
0%023088.sin. 3(n''«
0\001879.sin.4(n"r
n"'t + 5"— e"')
.n"7 + £"—£'")
n"'t + ^-—s"')
■ ^"7 + 5"— £'")
4r"'= (!+,.')•
0,0000016104
+ 0,0000021 947 . cos. (n' t — n"'t + e'— e'")
+ 0,0000001972 . cos. 2(n'i — n"'t + e'— e'")
+ 0,000000041 8 . cos. 2 (n't — n"'t + s'— e'")
[4374]
VOL. III.
68
270
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
Inequali-
ties inde-
pendent Oi"
the ex-
centrici-
tiea.
[4374]
+ (i+0
0,0000023860
— 0,0000187564,
+ 0,0000052387.
+ 0,0000011969.
+ 0,0000004169.
+ 0,0000001733.
+ 0,0000000796.
COS. (n"t-
COS. 2(?^"^
COS. 3(?^"^
COS. 4(n"r
COS. 5(n"f ■
COS. G(n"t ■
Inequali-
ties de-
pending
on the first
power of
the excen-
tricilies.
[4375]
+ (l+f^'0
^—0,0000066174
+ 0,0000784371
— 0,0000679436
— 0,0000069390
— 0,0000010930
— 0,0000002004 ,
. COS. (n'^t •
.cos.2(n'7
.cos.3(nH
.cos.4(n'7-
. cos. 5 (n'7 •
— 0,0000000520 . cos. 6 (n'^ -
if't + ^'—f)
■ n"'t + i"—i")
.n"'t-\-i"—s"')
.n"'t + s"—s"')
.n"'t + i"—s"')
n"'t + s"— £'")
. ,t"7 + £■"— s'")
- n"7 + £"— s'")
-?i"7-|-e"— /")
.«'"ï + ii"' — s'")
.n"7 + s»_/")
+ 0+^^^)
— 0,0000003173
+ 0,0000047062
-0,0000023275
' — 0,0000001399
-0,0000000125
.cos. (tft-
.cos.2(nU-
.COS.3(M^^
. cos.4(w''^
. n"'t + 1"—
.n"'t + s"—
£'")
£'")
n
Inequalities depending on the first power of the excentricities.
^ 1 ',082545 . sill. (2 /*'" ? —n't + 2 ='"— s' — ^"')
\ — 0S252586 . sin. (2 n'" i — w' ï + 2 s'" — .-'_ z=')
0%698649 . sin. («,'7 + e" — ^"')
— 0', 134530 . sin. (2 n'7 — 7i"'t + 2 s" — /" — t;^'")
— 10', 1 14699 . sin. (2 n"'f — n"t + 2 s'" — -=" — ^"')
- 5% 1 23062 . sin. (2 n"7 — ?z"/ + 2 s'" — .=" — ^/' )
+ (1 + p.") . ( — 6^516275 . sin. (3 n"'t — 2 jz"i + 3 /" — 2 ." — 3.'")
+ 0',846004 . sin. (3 tft — 2 n"t + 3 s"' — 2 =" — ^" )
+ 0',677748 . sin. (4 n"'t — 3 n"t + 4 s'" — 3 ^" — ^"')
— 0',0791.55 . sin. (4 n"'t — 3 n"^ + 4 s'" — 3s" — t.")
, + 0',1 19926 . sin. (5 n"'t — 4 n"t + 5 /" — 4e" — ^j'")^
VI. xi. §32.]
THEORY OF MARS.
271
+ (1 + f^'^)
+ (t+t^')-
,+ 5',490297 . sin. (irt + s'"— î^'") *
— 5',367005 . sin. (n'^l + ^" — ^'0
-23\552332 . sin. (2 «'^7 — n"'t + 2 5'"— s'"— ^"')
■ 2%593100 . sin. (2 w'7 — m'"^ + 2 s"— h'"— ^'')
+ 2%296703 . sin. (3 n" « — 2 «'" ^ + 3 s- — 2 s'"— ^"')
— 3%568875 . sin. (3 n" t — 2 n"'t + 3 e^' — 2 e'"— ^"')
+ 0%220149 . sin. (4> n'" t — 3 n'" t + 4> s'" — 3 s'" — vs'")
' — 0%352640 . sin. (4 n'" i — 3 n"'t + 4 e" — 3 s"' — a'")
- 2S868651 . sin. (2n"'t — n"i + 2 e'"— e'^— ..'")
— 0',204519 . sin. (2 n"'t — 71'" t + 2 s'"— s- — ^")
+ r,853159 . sin. {3n"'t — 2n'''t + 3s"'—2 e"— ^"')
+ 0',198136 . sin. (4> 71'" i — 3 n" t + 4> s'" — 3 s'" — z^'")
I 0-, 143758 . sin. {n't + f" — ^"')
— 0',696926 . sin. (Wf + £" — a^)
— r,798071 . sin. (2 n-t — h'" i + 2 s' — /" — ^"')
+ 0',132176 . sin. (2 nU — n'" / + 2 s' — s'" — z^^)
— 0',100246 . sin. (3 n't — 2 71'" t + 3 £" — 2 .'" — ^')
— 0', 156784 . sin. (2 n"'t — rft + 2 £'" — £' — ^3'")
\
â
[4375]
Inequali-
lies de-
pending on
the fijsx.
power of
the exceti-
tricities.
C 0,0000044700 . cos. (2 n"' t—n't + 2^" — ^ — ^"') ^
W=(\ + (^')- ^_ 0,00000097 13 . cos. (2/1'"^ — ^'^ + 2.'" — e' — ^ ) ^
'—0,0000022865 . cos. (71" t + s" — ^"')
+ 0,0000086337 . cos. (2 n"7 — ?t"/ + 2 e'"— /' — ^"')
- 0,0000031269 . cos. (2 n"'i — n"^ + 2 e"'— s" — .," )
4- (1 + f.") . y _ 0,0000200331 . cos. (3 n"'t — 2n"t + 3 e'"— 2 s"— ra'")
+ 0,0000025454 . cos. (3 n"'t — 2n"t + 3 e'"— 2 h"— ^" )
+ 0,0000030863 . cos. (4 7i"'t — 3 7i"t + 4 «"'— 3 s"— ^"')
+ 0,0000040239 . cos. (4 71'" t — 3n"t + 4 e"'— 3 s"— ^") ,
[4376]
* (264.3) The computation of the terms [4.375, 4376], is made in the same manner
as for Mercury, in [4278a] ; accenting the symbols so as to conform to the case under [4375a]
consideration.
[4376]
272 PERTURBATIONS OF THE PLANETS ; [Méc. Cél
0,0000035825 . cos. (n"'t + s'"—^'")
— 0,0000107986 . cos. (n'7 + s'"— ^"')
+ 0,0000031431 . COS. (n'H + i"— ^"•)
|_ 0,0000599470 . cos. (2 n'"/ — n"'t + 2 =-"— /"— ^"')
_j_ n 4- ^ivN ^ 7+ 0,0000069892 . cos. (2 /t'^/ — n"'i + 2 £■'— s'"— «")
^+ 0,0000114352 . cos. (3 7rt _ 2n"7 + 36"'— 25"'— ^"')
-0,0000169741 . cos. (3 n''^ — 2n"'/ + 3£"'— 2^'"- ^'^) '
'—0,0000020307. cos, (4 w'7 — 3n"'i4-4s'''— S^'"— ^'')
+ 0,0000087307 . cos. (2 n"'t — ?i"7 + 2 s'"— s'"— ^"')
y_ 0,00^0063983 . cos. (3 n"7 — 2ra'''i + 3s"'— 2£'^— ^"')
— (1 + M-') . 0,0000061906 . COS. (2 n' f — n"'t + 26" — £'" — ^"').
Inequalities depending on the squares and products of the excentricities
and inclinations of the orbits.*
iv'" = _ (1 + ^') . 6',899619 . sin. {3n"'t — n't + 3/" — e' + 65''26'"15')
( l',414532 . sin. (3 n"'t — n"t + 3^'" — ^' + 73"! l'"55') ,
Inequali- \ /
^'-„°f '"<= — (1 + f/-") . J + 4\370903 . sin. (4 n't — 2 n'7 + 4 Z"- 2 s" + 67^49"' 0') \
order. ) (
(+ 2^665900 . sin. (5 n"'t — 3 n"t + 5 f'"- 3 /' + 68''23'"00)
t4377] /_ 0',462779 . sin. {n''t + n"7 + ^'^ + ^"' — 53' 07'" 48') "
+ (1 -f ^iv^) _ ^ _ i.<,444i22 . sin. (2 n'7 + 2.- + 60^ 07"' 02')
+ r,295408 . sin. {n''t —7i"'t + s'" — s"'+ 54'' 41"' 32^ '
ïl'
* (2644) Using the values [4076A], we get vei-y nearly, 3 n'" — n' = — 12° =
18'
n
[4377a] also .3 ?i"' — »i"=238°, which is nearly equal to ii"' ; 4n"—2n" = 5l°=~;
[43776] ^''"' — 3n"= — 137'^= — — - nearly. Hence it is evident, that if we proceed in the
same manner as in the computation of the similar inequalities of Mercury [4282a, &.c. ],
we must notice the angles depending on these coefficients, in computing the terms of
[4377 — 4380]. For the second of these angles comes under the form [3732],
[4377c] { ?i" -|- (2 — i) . n" = n'", supposing i = — 1 ; and the others under the form [3733],
supposing successively, i = — 1, i = — 2, i = — 3. Lastly, as n" is small in
VI. xi. §32.]
THEORY OF MARS. 273
The last of these expressions may be connected with the following inequality,
computed in [4373], and which is independent of the excentricities,
(1 + (.'^■) . 24-,440843 . sin. (71" t — n"'t + e-' _ s'") ; [4378]
their sum, by reduction,* gives the following term of ov'",
^ ^■" = (1 + ;V') . 25%211710 . sin. (n'^t— n'" t + £'"— s"'+ 2'^ 24"'110. [43^^]
We have also,
<5 r'" = — (1 + f^') • 0,0000023461 . cos. (3 ir!"t — n't + 3/"— 3' + 64''47'" 29^ îp^u..
second
order.
[4380]
0,0000050403 . cos. (3 n"'t — ii't + 3 s'"— 3" + 72" 47'" 00')
+ (1 + (.") . j +0,0000070248. cos. (4 n"7 — 2«"i+46"'— 2-="— oB^ol'-'oO')
- 0,0000075032 . cos. (5 n"'t —3n"t+ 5 s"'—3^"— 68" 27'"280 ^
( +0,0000080002 . cos. (2 n^'t + 2 s'^ + 60'' 1 T" 52^) )
^ ^+0,0000041488.cos.(?t'7 — «"'i + s'" — 5"' + 59''8'"570^
The last of these quantities may be connected with the following inequality,
which is independent of the excentricities [4374],
(1 + fx") . 0,0000784371 . cos. {n"t — n'" ^ + s'" — £'" ) ; [4381]
their sura gives the following term of 6 r'",
&r"'= (1 + (^'') . 0,0000806432 . cos. {n"t — n"'t + £'" — £'" + 2''31'"550. [4382]
The inequalities of the motion of Mars, in latitude, are hardly sensible.
comparison with n'", their sum w"' + «"', is very nearly equal to n'", so that
this angle comes under the form [3732] in"-\-{2 — i).n"', supposing i^l; and [4377rf]
produces the term of [4377], depending on the angle ?«"'< + n"'t. If we suppose i^2,
in the same expression [4377(/], it becomes 2(i'*'; now, as this is small in comparison with
n'", it comes under the form [37.3.3], and produces the terms of [437T, 4380], depending on
the angle 2n"t. The quantity n'" — n'" differs but little from — n"' , and comes under
the first form [3732], depending on the angle n'H — n"t [4377, 4380].
* (2645) The term ( I + (x-) . 24',440843 . sin. («''' t — n'" i + s'^' — s'") [4373] may
be added to the term (1 + |x'^) . l-,295403 .sin. (m'>7 — ?r< + e-— s"'+ 51'' 41"'32^);
and the sum reduced to one single term [4379], by a calculation similar to that in [43o0a]
[4282^ — /]. In like manner the terms of [4374, 4330], depending on the angle
n^" t — r\!"t, may be reduced to one single term of the form [4382].
VOL. III. 69
274 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4383]
[43840]
[43846]
Putting n" equal to the longitude of the ascending node of Jupiter's orbit
upon that of Mars, we find,*
( 0',094394 . sin. («-"^ + e'" — n") )
[4384] dr=n+^").^ y ^ J f
^ ^ ^0%403269.sin.(2n''i — n"'i + 2£"'— s-'—n'")^
* (2646) The term of 5 s'", depending on the attraction of Jupiter, maybe derived from
the formula [4295è], by adding two accents to the quantities s', a', n', s', a", n", s", m";
also supposing y to represent the incUnation, and II the longitude of the node of Jupiter's
orbit upon that of Mars [4295c]. The terra independent of 2 produces the first term of
[4.384], and the term under the sign 2, corresponding to i = 2, gives the second term ;
using -B^''=-^3 -K [1006,4190].
VI.xu.§33.] THEORY OF JUPITER. 275
CHAPTER XII.
THEORY OF JUPITER.
33. The reciprocal action of the planets, upon each other, and upon the
sun, is most sensible in the theory of Jupiter and Saturn ; and we shall [4384']
now proceed to show that the greatest inequalities of the planetary system
depend on this cause. The equation [4371],
6r"' = y.(l-a=).6V"',
r
corresponding to Mars, becomes for Jupiter,
[4385]
6 r''' = !L . (1 — a') . 6 V". [4386]
r
If we take for r", r'", the mean distances of the earth and Jupiter from
the sun [4079], and suppose 6 V''= ± 1"= =t 0',324, we shall obtain,
6 r" = =F 0,0000409225. [4387]
Therefore we may neglect the inequalities of 6 r", which are below
q= 0,000041. We shall also omit the inequalities of Jupiter's motion in [4387]
longitude, or latitude, which are less than a quarter of a centesimal second,
or 0',081.
Inequalities of Jupiter, independent of the excentriciiies*
Itiequah'
^^ ^^ , „^ ( 0'-, 120833. sin. (n"t — n'^'t + s" —s-)^ [.^^ntr
iV = (1 +f^ ) . < > the ex-
^ ^— 0,000086. sin. 2 (n"Z — n'''i + ^" — ^'0) "''"'"'
ceotrici-
tics.
* (2647) The inequalities [43S8, 43S9], are deduced from [4277a, J], increasing by
four the accents on the symbols, to conform to the present case, and using the data
276
PERTURBATIONS OF THE PLANETS;
[Méc. CéJ.
[4388]
iaequalî-
ties inde-
pendent of
the excen-
tricities.
[4389]
+ (1 + f^O • X
+ (l+^'0
82',8117n.
- 204',406374 .
— 17%071564,
— 3',926329 ,
— P,2 10573.
— 0',428420 .
— 0% 170923,
— 0 ,076086 .
— 0%041273.
P,051737.
— 0',427296 .
— 0',044085 .
— O',005977 ,
sin. (jCt ■
sin.2(M'^
sin.3(J^^^
sin.4(n'^
sin.5(?i''^-
. sin. 6 (ii" t ■
. sin. 1 (n't-
. sin. 8{n''t-
. sin. 9 (n'' t ■
sin. [n'"t ■
sin.2(n"7
sin.3(n"7-
■ n"t -\- s'—i")
■ n''i + e" — e'^)
. n"t -\- b' — e'^)
n'^t + 6" — i")
n" t + e" — £'")
• n^t + b" — £'")
.n'''t + ^'-'—é")
. n" t + £'■' — i'")
n"t + e"'— £")
àf
— 0,0000620586
+ 0,0006768760. COS. {n't— n"t + b^— b")
— 0,0028966200 . cos.2(ift — n"t + £'— b")
— 0,0003021367 . cos. 3 {nH — n"t + b'— e'")
r^ 1 v^ /— 0,0000782514. cos.4(îi"ï — n'7 + s"— e'M -
\— 0,0000258952 . cos. 5 {n^'t — n''t + «" — «'") |
- 0,0000094779 . cos. 6 (?j" ^ — n''t + £^— s'") |
- 0,0000037560. COS. 7 (?i^7 — n'U + £^ — s") |
-0,0000014781 .cos.8(»i^^ — ?ri + E"— ^'O '
- 0,0000004799 . cos. 9 (ift — n'-t + «"— '^'O
Inequalities depending on the first poiver of the excentricities.
Several of these inequalities are of considerable magnitude, so that it
becomes necessary to notice the variations of their coefficients ; which we
[43886]
[4061, &1C.]. The term depending on sin. {n'' t — n'U -{- b" — b'"), being computed, by
means of the formula [4277n], is found to be nearly the same as in the first line of this page,
and has the same sign ; therefore the remark made in the Philosophical Transactions for
1831, page 65, that the sign of this coefficient is negative, is incorrect.
VI.xii.§33.]
THEORY OF JUPITER.
277
shall do, in those terms of the expression of 6«" which exceed 100", or
32',4. The coefficients of the inequalities depending on a'', have for a
factor the excentricity e" ;* therefore, by putting one of these coefficients
Se'"
equal to Ae'", its variation will be Ae'". -^. We shall find, in [4407],
that if we include even the quantities depending on the square of the
disturbing force [4i04,&c.], of which we have given the analytical expression
in [3910], we shall have.
[4389']
[4369"]
* (2648) The terms of 5v'\ Sr'" [4392,4393], were computed from those of
ÙV, &r [1021, 1020], depending on e, e' ; changing m, a, e, zs, i, n, into
rlfi nictiirnti,„
ni", a", e'", -a'"', e'*', Ji'", respectively. In computing the disturbing force of Saturn,
we must also change the symbols m', a', he. into m'', a", he. ; and in computing that
of Uranus, we must change them Into rri", «", &c. We shall neglect the terms containing
the arc of circle nt, without the signs oi sine and cosine, as is done in [1023, 1024]. In
this notation, the angle ra", is evidently connected with a coefficient having the factor e''';
and the angle -a^, with the factor c" ; as in [4389', 4390']. The variations of c'", e",
are given in [4407] ; and if we retain only the first power of the time t, they will be as in
[4390, 4391]. For an example of the method of computing these variations, we shall take
the largest term of ôv'" [4392], which arises from the substitution of the value of i = 2,
in the term multiplied by e, or c" [1021] ; so that this term becomes.
Substituting the values of the elements [4061,4077,4081], and that of jF® deduced from
F'*' [1019], we find that the coefficient becomes, as in [4392],
— 13S',373337 = A e'" [4389'].
ôe
This is to be multiplied by —, to obtain the expression A (5 e'". Now, i5 6'"= t . 0',329487
[4390], being divided by the radius in seconds 206265', becomes,
5 e''-=^. 0,00000 15974;
dividing this by e'"' [4080J, we get,
— = t. 0,0000.33226 ;
multiplying this by Ae" [4390/], we finally obtain,
^JÉi^ = — <.0',004598.
Connecting this with Ae'" [4390/], we get the coefficient of the term depending on the
angle 2 n'i— n'^< -f 2e^— e'" — ra'' [4.392]. In the same way the variations of three of
VOL. III. 70
[4390al
[43906]
[4390fJ
[4390(f]
[4390e]
[4390/]
[4390g]
[4390/. J
[4390t']
278
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4390] W" ^t. 0',329487.
In like manner, the coefficients of the inequalities depending on ^j", have
[4390] the factor e" ; and by putting B e"" for one of the coefficients, its variation
will be B e". ^^, and we shall find, as in [4407], that
[4391] 0 6"= —t. 0%642968.
This being premised, we obtain,
Inequali-
ties de-
pending
on tlie first
power of
the excen-
tricities.
6D-=(1 + /J.'')
[4392]
8',608489 . sin.
— 9',692385 . sin.
— (138',373337 + ï
+ (56S634099— i
— (44',460822 + i
+ (84',942569-i
+ 7%925312.sin.
— 15',629621 .sin.
+ r,047717 .sin.
— 2',78 1664. sin.
/ + 0',407251 . sin.
— 0,9 13302. sin.
+ 0', 149277. sin.
— 0',325592 . sin.
— 5',208122.sin.
— 0%569738 . sin.
+ 12',876650.sin.
— 0',352399 . sin.
+ r,287482 . sin.
— 0% 172892. sin.
+ 0-,356627 . sin.
— 0',083]89.sin.
(n^'t + E^ — si'')
. 0%003 1 398) . sin. (2 n^ i — n'" i + 2 £^ — e'^ — ^^ )
.0%0014776).sin.(3n''i— 2n'^i+3i''— 2>— ^")
. 0^0047094) . sin. (3n'' «— 2n"^+33''— 2.'"— ^' )
— 3 n'" i -f 4 £'' — 3 5'" — zi"
— 3 ?i'' ; 4- 4 £" — 3 1" — ra"
— 4 n'^f + 5 £' — 4 s" — zi'"
— 4 n'' « + 5 6^ — 4 s'" — zj"
— ôn'''t + 6i'—ôs'"'-~-u,"
— 5n*'i + 6s^ — ô^'" — T.^
— 6n'"^ + 7£'- — 6 e'"— 33'"
— 6 irt + 7 £" 6 s'" — -n"
— rft +2 £'" — £" — ra'"
— nU + 2 s'" — £" — ra"
— 2 71^1 +3 i"— 2 6'— z="
— 2 71^1 + S s'"— 2 £" — 73'
— 3 71"^ + 4 s'"— Si" — zs'"
— 3 ?ri + 4 s'"— 3 £' — ^"
4 7ft 4- 5 t-'" 4 s' — z:'"
— 4>7i''t 4- 5 £"— 4 ^^" — i^'
(4 71^1
(4^^
(ôn^t-
(ÔTl't
(6 71" t
(6 n" t
(In^t
ÇlrCt
(2 n'H
(2 7l"t
(3 n'7
(3 n"«
(4 n'H
(4 7l"t
(5 Tl'H
(5 71" t
the other large terms of [4392] are computed. The variations of the remaining ones are too
small to be noticed.
VI.xii.<^33.]
THEORY OF JUPITER.
279
(1+^")
.ir"=(l+,.-)
0% 123506 . sin. (n"< + s^' — x.")
- 0',235240 . sin. (ti'U + i"' — ^")
- 0',533079 . sin. (2 n"' t — irt + 2 1" — s" — ^")
+ 0', 102673 . sin. (2 ?r'/ — n'" t + 2 s"' — i" — a")
— 0'-, 127963 . sin. (3 n'H —2 n"i+ 3 s" — 2i"— i^''')
0,00002061 1 1 . COS. {n'H + s'"— a")
— 0,0000795246 . cos. {n" 1 + ^' — ra'")
+ 0,0000492096 . cos. (n" i + s" — «^ )
— 0,0002922 130. COS. (2 H'i— n''i + 2E''— e"— ^i'
1+ 0,000 1688085. COS. (2/1" Ï— n"t + 2t' — 'Z^"—^-'
— 0,0004584483 . cos. (3 n^i — 2n'''i+ 3 £"— 2e"— x^"
+ 0,0009047822. cos. (3 n'i — 2 n''7+ 35"— 26"— ^"
j-l- 0,000 1 259429 . cos. (4 n' ^ — 3 iVH + 4 £" — 3 s'"— t^"
/— 0,00024244 1 3 . cos . (4 n'' i — 3 n"7 + 4 s" — 3 s'"— a"
+ 0,0000268383 . cos. (5 n' t — ^n"t + 5 s"— 4s"— ^"
— 0,0000516048. COS. (5/1"^ — 4n"i -foe" — 4s''— ^^
+ 0,0000579151 . COS. (2n''t— nU-^2s"'— I' — zs"
I— 0,0001346530 . cos. (5 n'H — 2ift + 3 e'"— 2 s" — ^'^
Inequali-
ties de-
pending on
the first
power of
the excen-
tricitiei).
[4393]
Inequalities depending on the squares and products of the excentricities
and inclinations*
l',003681 . sin. (n"ï + n'^7 + e" +£"' + 45''29"220
— 5',578707 . sin. (2 nTt + 2s" + 15' 56" 24^
+ ir,724245 . sin. (3 ri't — n"t + 3^" — e'"+ 79-^ 39" 48^
—1 8',075283 . sin. (4 n' ^ — 2 n}H + 4 e"— 2 e'"— 57" 1 2" 26')
'"■•=('+-')V(,69s266896-<.0S004277).si„.(^;i-4»-|+3-^^^^^^^
+ P,647140.sin. (6 n"f — 4n"< + 6 s"_4£'"— 64''25'"480
+ 2'-,47 6404. sin. ( n't— n'"i-|- e"— £'"4-43^17'"0P)
V — 6',287997 . sin. (2 n" i — 2 n'" i + 2 £"— 2 e" + 42 '4O'"440
Inequali-
tiei of the
second
order.
[4394]
* (2649) The calculation of the six first terms in [4394] is made in exactly the same
way as for Mercury, in [4282a — 6]. The coefficient of the angle 3 n''< — 5 rCt, being [4394a ]
280 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
These two last inequalities being connected with the two following,
[4395]
( 82',8n711.sin. (n^ï — «'" ^ + s'— £'") ^ _
*" ^ ■ ^ _204',406374 . sin. 2 (n't — n'" t -\- b' — s'") ^ '
which are found in [4388], among the terms independent of the excentricities,
produce the terms,*
(_ 84',628936.sin. (?ri— n^i + s'" — s" — 1''08"'530 )
^ ^ (+209',098224.sin. (2?^'"^— 2n^'i+2£"'— 2 e''— r'09"'58^)^
Then we have [4394^/],
loequali-
sic'oôS""' / 0,0000822415 . cos. (2 w^i + 2 s" + 1 1'OO"' 550
order. I ^
+ 0,0000226252. cos. (3 n-t—n'^t + 3 s"— s''— 21''47"'18')
^4397^ 6r''=^(l + f^'').( —0,0001010533. cos.(4?i^^—2n'"«+4î''— 2=-''— 51''04"'040
-(0,0021 1 14502-^0,00000005323).cos.(^J;l-!J;;2o;:^
— 0,0000652204. cos. (2 ?i'7— 2rt'''<+2 e'— 2 .'"+ 54''08"'52')
If we connect the last of these inequalities with the following,
— (1 +f^") . 0,0028966200. cos. 2(n"i — iV't + i' — i") ;
which is found in [4389], among the terms that are independent of the
excentricities, we obtain the equivalent expression,
[4399J 6 r"= _ (1 -|- j.^) . 0,0029251 892 . cos. (2 7i'H — 2 n''t + 2 i"—2 e^'— 1''02"'080.
The preceding inequalities of iv'", are calculated by the formulas [3711,
3715, 3728, 3729] ; excepting, however, that which depends on the angle
[4400] 3^'"^ — brft\ observing that bn" — 2n", is a very small coefficient, as
appears from the ratio which obtains between the mean motions of Jupiter
large, its variations must be noticed and computed by the metiiod pointed out in
[43946] [4017 — 4021]. The other coeniclents are less than 32%4, and their variations are
neglected, as in [4389', he.]. The two last terms of [4394] correspond to [3729, 3728] ;
[4394c] using i=^±l, or i^±2; the values of A" being found, by means of the formulas
[3753 — 3755'"], and the corresponding terms are to be connected together, like those
depending on M, in [4282A — /]. In like manner, the four first terms of [4397] are
[4394dJ deduced from [3711]; the last term from [3728]; noticing always the variations of the
elements in the greatest coefficients, as is done with the terms of & v.
[4396o] * (2650) This computation is made in the usual manner, as in [4380a].
[4398]
VI.xii.§33.] THEORY OF JUPITER. 281
[4400']
[4401]
and Saturn [4076/t] ; so that the angle Sn'^t — ôn^t differs but very little
from n''f, as in [3712, &c.] ; in consequence of which, we have used the
formulas [3714, 3715], in computing this inequality, by the method given
in [4017 — 4021].
Inequalities depending on the poicers and products of three and Jive
dimensions of the excentricities and inclinations of the orbits,
and on the square of the disturbing force.
The great inequality of Jupiter, is calculated by the formulas
[3809—3868; 3910—4027]. We find, from [3836—3841],
a\ ilfw ^ — 5,2439100 . m' ;
a\M")= 9,6074688 . m\-
a\ M<-^> = — 5,8070750 . nf ;
a\M'-^^= 1,1620283. m^
a\ itfw = — 0,6385781 . m" ;
a\ M' '' -= 0,3320740 . m\ mequaii^
' tics of the
Hence we find, at the epoch 1750,* orJ"-
a\P= 0,0001093026;
a\ P' = — 0,0010230972. [4402]
We must find the values of the same quantities in 2250 and 2750. For
this purpose it is necessary to determine the values of e'", e", t^'", w*', j, n,
in series, ascending according to the powers of the time ; continuing the
series so far as to include the second power of t. We must, in the first ^ ^
place, calculate, by the formulas [3910 — 3924], the secular variations of
0 e'% ie", 5ra'', Szs", depending on the square of the disturbing force ; and
we shall obtain, for these variations,!
* (2651) The values of «'P, «^P' [4402], are deduced from [3842, 3843] ; adding
four accents to the letters m, a, e, zs, m , a', e , Stc. to conform to tlie present [4402a]
notation, and then using the numerical values [4061, 4077,4079, 4080, fcc.].
t (2652) The value of ^e'" [4403], is computed from the part of (S e [3910], depending
on the time t, without the signs of sine and cosine ; adding four accents to the letters
m, a, e, m', u', e , &:c. to conform to the case now under consideration. izi'" [4403], is
VOL. III. 71
[4403a]
282 PERTURBATIONS OF THE PLANETS ; [Méc. Cél,
[4403]
[4404]
[4405]
àé" = t. 0%052278 ;
<! ^" = t. 0^352941 ;
ie' = — i.0%102763;
61^"= t. 3',242722.
The coefficients of t, in these expressions, represent the parts of — , — ^ ,
— , — [4404a, 6, c], depending on the square of the disturbing force.*
Ut CE t
Adding them respectively to the parts of the same quantities, determined in
[4246, 4247], we obtain the entire values in 1750,
^^ = 0%329487 :
'^= 6',952808;
dt '
^ = _ 0',642968 ;
dt '
^-^ = 19%355448.
dt '
obtained from the like parts of &-a [3911]. The expressions 6 e'-', (5 a" [440-3], are deduced
from [3922, 3923], by making the same additional number of accents to the letters, and
then substituting the values of these elements [4061, 4077,4079, &ic.].
* (2653) We have, as in [4330rt], e'" ==; e'" + ^ . ~ + J t^. ^ ; e" in the second
member, being the value of e'', at the epoch ; and by putting for e'* — e"', its value ^e'",
we get,
[44046] 5e-=<.— + 1^^^^.
In like manner we have,
[4404c] 6e^ = t.^+it^'—+^o.; 6.^ =t . - + if^.^ + ^o.
The coefficients of t, \ t^, in the second members of these expressions, correspond to the
epoch. The coefficients of the first power of t, in these expressions, are composed of two
parts, namely, those computed in [4246,4247], and those depending on the square of the
[4404rf] disturbing masses, computed in [4403] ; the sums of the corresponding parts give the
coefficients, respectively, as in [4405]. Thus,
^ = 0%052278 + Jx0%554418 = 0V329487, Sic. as in [4405].
at
[4404a]
VI. xii. <^33.]
THEORY OF JUPITER.
283
We obtain, by the same method, their values in 1950, and find, at this epoch,*
^= 0',326172;
dt
IT
de"
It
= 7',053178;
= _ 0%648499 ;
~— = 19^424739.
dt
From these we get, as in [3850, &c. 3850c], the following expressions of
e'*, îj'", e", w' ; for any time whatever ;
e'* = €'■' + t. 0%329487 — f: 0^,0000082871
z," = I.- + / . 6^952808 + t-. 0 ,0002509259
e' =€" — t. 0%642968 — f: 0%0000138275
^" =^" +1. 19--,355448 + t\ 0',0001 732274 :
the values of é", to'% e", «% m the second members of these equations,
correspond to the year 1750.
[4406]
General
values of
TO", -m".
[4407]
[4407']
* (2654) The calculation of the annual variations of the elements [4406], for the year
1950 is made in the same manner as in [4405], using the expressions of e'^, e", «'", -a", [4406a]
corresponding to 1950. These elements are obtained, very nearly, by means of the annual
decrements [4405], which give, with sufficient accuracy, the required values, when t does
not exceed 200. Thus the increment of e'*', corresponding to t = 200, is
200X0',329487=:65',8 nearly [4405]; |-4406J]
being the same as the term depending on the first power of t, in the expression of e'» [4407].
The term depending on i^, in this last expression, is very small, being represented by
— 2002 X 0^000008287 1 = — 0»-,3 nearly ; [4406e]
which is about -^^sxs part of the term corresponding to the first power of t. Similar remarks
maybe made relative to the values of t" , ra", ■a'' . If these calculations were to be repeated,
in consequence of any changes in the assumed values of the masses of the planets, we could
take into consideration the parts depending on t'^, as they are given in [4407] ; and by this [4406rf]
means we might obtain, by successive operations, corrected values of the elements. This
process is the same as that so frequently used by astronomers, in re-touching and correcting
the elements of the orbits of the heavenly bodies.
Now, from [3850c], we have.
ddé"
îiTT'^
in which we must substitute
dé"
for -—', its value 0',326172 [4406] ; also for — - , its value 0',329487 [4405] ; hence
[4406f]
284 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
We may find the values of y, n, by means of the equations,*
7 . sin. n = (p" . sin. ff' — <?'"• sin. o'" ;
^^^°®^ y . cos.n = (f" . COS. r — 9'\ cos. ê'\
d y du
Then we compute the values of -—, —, by taking the differentials of
J , • • r dq,'" dtp" dù'^ dè^ . . .
[4408] these equations, and substitutmg tor — — , - — , —— , —, then* values
[4246, 4247]. We find, in this manner, in 1750,
7= IMô-'SO';
n = 125"^ 44"' 34';
^''''^ Ç = - O',000106;
dt
~=~26%09U33.
dt
The formulas [3935, 3936] give, for the secular variations of y and n,
depending on the square of the disturbing force,
ij= i.0',000184;
in = — ^.0',00763.
If we add the coefficients of t, in these equations, to those in the preceding
d y d TÏ.
values of — , — [4409], we obtain, for the complete values of these
quantities in 1750,
[4410]
rfrffi' 0',003315 „„ ^^^„ „ , . . ,. , ^ ddei^
[4406/"] we eet ,r77Tr = TTJ^^ — = — O',0000082b / . Substituting this value of -— -„, and
■' 4di'^ 40U 2dt~
d c'^
that of -7- [4405], in (:"■ [4404^], we get the first of the equations [4407]. The values
[440tig-j ^^ ^,v^ gv^ ^v^ g,,g ]om;|f| |„ ^i]g same manner, changing e" [4404«,4406e], successively,
into w", e", -a", and using the values [4405,4406].
[4409a]
[44096]
[4409c]
* (2655) Tlie equations [4408] are similar to those in [4282o], adding four accents to
ç), è, cp', ê', to conform to the present case ; and changing tang.ç)"', tang. 9", into 9", ç",
respectively, on account of their smallness. In this case 7 [3739] represents the langent
of the inclination, or very nearly the inclination itself, of the orbit of Saturn to that of
Jupiter ; and n [3746], the longitude of the ascending node of the orbit of Saturn upon
that of Jupiter. Substituting in [4408] the values of 9'% é'\ ç\ é-, [4082, 4083], we
get 7, n [4409]. Tlien taking the differentials of [4408], and substituting the preceding
values of ç)*", è", kc. ; also those of do'", de'", df, do" [4246, 4247], we get the two
last equations [4409], by making a few reductions.
VI.xii.§33.] THEORY OF JUPITER. 285
'^ = 0',000078 •
(It
V-=-26',101764.
dt
We find, by the same process, in 1950,
- 0',001487;
dy
~dt ~~
V^=— 26',402056.
dt
Hence we obtain, by the method in [3850 — 3853], for any time whatever t* inciininion
^ , and
y = y-^t. O',000078 — t\ 0',000003913 ; [4413]
n = n— ^ . 26', 101 764 — t-. 0',000750731. [4413]
longitude
The values of y, n, in the second memhers of these equations, correspond to n, of the
1 750. This being premised, we find in 2250, f ascending
niideofiho
a\P = — 0,0000801 89 ; '^;:^
[4414]
«'. F =—0,001006510; J.H,.„f
and in 2750,
a\ P = — 0,000260997 ;
«^P'=— 0,000954603-
Jupiter.
[4415]
* (2656) If we change the symbols y, XI [4412], for the year 1950, into y^, IT,
respectively, and leave those in [4411], corresponding to the year 1750, without accents, we
shall have, as in [4406e],
-^^ = jJ^ . $'^ -Jf\^ ^"^ •^'" ^''001487 - 0',000078| =— 0S00000.3913 ; [4413a]
also,
'^''" =^a^.HlL'_— |=^^^.f_26',402056 + 26',101764} = — 0',00075073.
2^,2 — tU ■ I -jj — TT ( =î*t7- i— i=;o%4U2U5(j + '^b%1017b4| = — 0%00075073. [44136]
Substituting these and the values of [4411], in the general expressions of y, n [4404«],
namely,
we get [4413,4413']; observing that the values, in the second members, correspond to the
year 1750.
t (2657) The values of a\ P, a'. F', are given in [3842, 3843], in functions of
e", e", -a'", ti\ y, n, &c. ; and their values in 1750, have already been given in [4402]. [4414a]
VOL. III. 72
286 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Hence we deduce, by the method of [3850—3856],*
a\ ^= — 0,000000387666 ;
[4416]
a\ '^ =- — 0,000000002145 ;
d t
a". ^= 0,000000000034734;
a\^= 0,000000000141280.
The part of 5v'\ given in [4023], is,t
[4417]
6v"
. „, , ^a'^'.dP 3a'^.ddP' ) \
( . rfP' , 2a"'.ddP ) , , „ . ddP'i /
Cm". ni*-2
~(5,V-2,i"f\ , 2a>''.dP' Sa'\ddP
a'\P
(5n^—2n>^).dt i5n^ — 2n"f.dt^ ^
^ cos.(57iH—2ii'H+5s^ -2;'")
(^ . dP 2a<-.ddP' ) , , 2 i,ddP\
. , ''( ' dt (ân-'—2n").dV2S
G root * ^ ^ ' ^
iiidiuiility.
This becomes, by reduction to numbers,
[4418]
6î;'^=(1263%799671— t . 0',008418 — <2.0'.000019247) . sin. (5 ii''t — 2n"t -{'5s'-— 2 s")
4- ( 1 19%526S51 — t . 0%473686 + i^. 0-,0000:8562) . cos.(5 n't — 2 n"t + 5 s^ — 2 s^').
The great inequality of Jupiter includes several other terms ; thus, it
contains, in [3844], the expression,!
To obtain a^.P, a\ P' in 2250 [4414] ^ we must put !!= 500, in [4407,4413,44131,
[44146] and substitute the corresponding values of e'", -cj'^, Sic. in [3842, 3843]. In lilce manner,
by putting t = 1000, we get their vakies in 2750 [4415].
* (2658) The values of «^ P [4402,4414,4415], being substituted, respectively,
dP d-P
[4416a] foj. p^ p^ Pi,,m [-3856], give the values of n\--, a\— [4416]. In like manner,
from a^.P' [4402, 4414,4415], we get the terms depending on the differentials of P' [4410].
t (2659) The formula [44 17], is the same as in [4023] , increasing the accents on the
[4418a] elements m, a, c, &c. m', a', e', fcc. by four, to conform to the case under consideration.
Substituting in [4417], the values [4402, 44 1 6] , it becomes as in [4418].
r4419al Î (2660) The expression [4419] includes the third and fourth lines of (5 «" [3844], the
accents being increased as in the last note.
VI. xii.§:33.]
fdP
6V"'= —
THEORY OF JUPITER. 287
n'V
7M<
To reduce it to numbers, we must calculate the values of a"". (-^ ) ;
(hW
a"-. ( 4-^ )? &;c. ; and we find,*
V da" J
\da'^\
a''^
/dAP'^'
\ '^ «" .
«"^
A?.W(3V
\ f'"'"
iC\
V ^
/f/7»/('>~
\ f^"'" ,
„v9
^«/./l/ra-
= — 26,46390 . m^ ;
= 65,75870 . m" ;
= — 50,227 14. m'';
= 1 2, 14696. m";
= _ 6,75963. m^;
= 4,13173. m^
From these we deduce the values of a^-. ( -— ^- j, a'". ( ^^ )' ^^' ' ^^^"^''
are necessary in the theory of Saturn, by means of the general equation of
homogeneous functions [1001a],t
da'" J \ d a"
[4419]
[4420]
d /J/^'A /d 1/t'A
* (2661) The accents being increased as in [4418«], the formulas [3836 — 3841] give the
a'V (2) (3)
valuesof a^.W"', a'J)f"\ &c. in terms of a:= — , b,, 6 -,, &.c. and their differentials.
Taking the partial differentials of these expressions relative to a'", and substituting the values [44000]
(2-) (3)
[420-2-4211], we get [4420]. Observing that hp h^, &c. are functions of a [964]. and
if we represent any one of them by h, its partial differential, relative to n'^, will be,
/■ dh \ fdh\ / do.\ /db\ 1
t (2662) Tlie general values of M^''\ M''\ iW^^', M'^\ M^'', M<5>
[3836£/, 3337c, 3838A, 3840A, &ic.], are composed of functions of a'", a", of the forms,
288 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Hence we find, in 1750,*
a"2. (if. V COS. (5 7iV — 2 n'^t + 5 £"— 2 e'")
— a"'~.f-—A .sin. (5ft''< — 2?i'''<+5 6^— 2;")
[4422]
2 m". n'"
5 Ji"— 2 n'" ' ^ ■ „ /d P'\
[4423]
[4424]
[4425]
= — 1 7%228862 . sin. (5 n'' i— 2n'''< + 5£'— 2 ;'0
+ 5',36001 6 . COS. (5 if t — 2 n'' < + 5 s"— 2 r) ;
and in 1950, it becomes,
— 16',836801 . sin. (5 n" / — 2 n''^ + 5 e^— 2 s")
+ 6^449839 . cos. (5 n't — 2 n'"^ + 5 £"— 2 e").
Hence we obtain the following value of this function, for any time whatever t,
6 ?)'"=: — (17',228862 — /. 0',001960) . sin. (5 n't — 2 n'^t + b^-—2 i")
+ (5',360016 +i.0',00o449).cos.(5n^7 — 2n"'i + 5s^ — 2 s"').
all of which are homogeneous, and of the order — 1, in a'^, a" [1001', 1007'] ; i
being any integral number. Hence the general value of JJi*'' is also homogeneous, and of
the degree — 1, in a", a''; and the formula [lOOlo], by changing A, a, a', m, into
M^", a'\ a", —1 becomes as in [4421].
* (266-3) The values of m^n" P, m" a" P', are found as in [4402n], by increasing
[4422a] tbe accents of the elements in [3842, 3843] by ybwr. Taking the partial differentials of
these expressions, relative to a'", we obtain the values of,
[44225] ™'«^-C^)' -^"-63'
expressed in functions of a'^, e'", &c- a^, e", &ic. and of the terms [4420]. Substituting
these in [4419, or 4422], we get [4423], corresponding to the year 1750. Repeating this
calculation, with elements computed for the epoch 1950, it becomes as in [4424] ; observing
that the functions [4420], must also be computed and taken for the year 1950. Comparing
the numerical coefficients of the terms [4423,4424], we find the increments, in 200 years,
to be respectively represented by,
— 16',8.36801 + 17^228S62=:0^392061,
[4422i] and
G^449839 — 5,-360016 = l-',089823.
Dividing these by 200, we get the annual increments, or the coefficients of t, as in the general
expression of '5 1''' [4425].
VI.xii.§33.]
THEORY OF JUPITER.
289
The great inequality of Jupiter [3844] contains also the term,*
ôv''' = — h He", sin. (on" t — 2 n"t-{- 5 1" — 2 s'^—zs^" + A);
which, in 1750, is equal to,
0^820290 . sin. (5 1^1—2 n" t + 5 b"— 2 i'")
— 1%837963 . cos.(5 n't— 2 n" t + 5s'' — 2 s'") ;
and in 1950, is,
0%701624 . sin. (bn''t — 2 n" t + 5i'—2 é")
— r, 840958 . cos. (5 n't — 2 71" t + 5 s^— 2 e").
Hence we find, that for any time whatever t, this term is represented by,
6 v''= (0',820290 — t . 0',000593) . sin. (5 n't — 2 n'" i + 5 s^— 2 £'')
— (1 -,837963 + t . 0^000015) . cos. (5 n'^t — 2 rrt + 5 e^— 2 s'').
To determine the part of the great inequality of Jupiter, depending on the
products of five dimensions of the excentricities and inclinations of the orbits,
we have computed, by the formulas [3860—3860'^], the values of iV*"', N^^\
&c. for the tAvo epochs 1750 and 1950, and have found.
In 1750.
a\iV('"= 0,00000135044
fl\iVW= 0,00000789719
a\ iV'=' = — 0,0000198552
a\iV(3)= 0,0000175127
a-\ iVH) = — 0,0000066540
a\ iV<^) = 0,0000009277
a\N^'^= 0,0000003618
a\ iV<'' ^ 0,0000003643
a\ A^<«' = — 0,0000001720
a\ iV'^' = 0,0000000730.
In 1950.
a\ iV<°' = 0,00000129983
fl^7V(')=:3 0,00000754771
a\ iV<"~) = — 0,0000196012
«\iV'="= 0,0000172415
a\ iV'" = — 0,0000066551
a\ N'-'^ = 0,0000009408
a\ N'-'^ = 0,0000003562
a\ iV'" = 0,0000003460
«v,jV(S) ^ — 0,0000001712
[4426]
14427]
[4428]
[4429]
Terms of
the fifth
order on
e, e', 7.
[4430]
a\ iVW
0,0000000732.
* (2664) The term [4426] is the same as that depending on — |/Je [3844], accenting
the symbols as in [4402a] . In this case H denotes the coefficient of,
73
VOL. III.
[4431]
290 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
By means of these values* we have computed the corresponding inequality in
[4430] Saturn, in [4487]. Multiplying it by the factor —~?~r-, we obtain the
following inequality of Jupiter,t
6 1,'" = _ (1 2',536393 — t . 0^001 755) . sin. (5 Ji'f — 2 n'" i + 5 £" — 2 s")
+ (8',120963 + t . 0-,004885) . cos.(5 ift — 2n'''t + 5 £>— 2 s").
Lastly, we have computed, by the method in [4003], that part of the great
inequality of Saturn, which depends on the square of the disturbing force,
[4431] ^j^ij jj^ q£ ^ sensible magnitude. Then we have deduced from it the
corresponding inequality of Jupiter, by multiplying it by 177^ ! which
gives, for this last inequality, the following expression,^
[4426a] COS. {5n't—-3nt-\-5s'—3s-\- A),
in the expression [3814], corresponding to Jupiter. Computing the value of — | He", for
[44266] jj^g ygj^j.g j^j5Q^ jggQ^ ^ jjj [4427,4428], we obtain its annual increment, and the general
value [4429].
* (2665) The signs of all the terms in [44-30, 4431], are different in the original work ;
[4430a] ^^g j-^^yg clianged them, iii order to correct the mistake in the signs mentioned in
[3860«].
t (2666) Changing, in [1208], ,g, <^', into iv", ôv^, which represent, respectively, the
corresponding parts of the great inequalities of Jupiter and Saturn, we get, by using
the notation of [4402«],
[44306] 5«-=_^'^.5v
[4430c] Substituting in this, the values w", m", a'', a^ r5 «" [4077, 4079,4487], we get [4431].
[4431o]
X (2667) We have already mentioned in [4006^ — 4007rt] the difficulties which occurred
in computing this part of the great inequality of Jupiter, and liave also observed, that the
numbers given by the author, in [4432], are inaccurate ; the chief coefficient having a wrong
sign, as Mr. Pontécoulant found by computing the most important terms, depending on the
[44316] arguments contained in the table [4006m], numbered from 1 to 10, and from 1' to 10'.
The parts of êv", corresponding to these terms, are given in [4431/], from the abstract,
printed by Mr. Pontécoulant, in the Connaissance des Terns, for 1833 ; using, for brevity, the
^^^^^'^ symbol T5 = 5 ift — 2 n'H + 5 s' — 2 i" [3890J] . The first line of the function [443 1/]
[4431rf] ig produced by the term 3 ci^ff. {ndt.dll ./d 7?) [5844] ; the other lines arise from the
products of the quantities in the table [4006m], marked with the numbers on the same lines
Vl.xii. ^^33.]
THEORY OF JUPITER.
( ]%6U663 — t . 0%00'l68S) . sm. (5 n't — 271'" t -\- 5 e^ — 21''')
— (18',461954+ i.0',001515) . cos.(5 if t — 2 n"t + 5 B''—2r).
291
[4432]
respectively. The sum of all these terms is given in [4431^] ; and it differs essentially from
that of La Place, in [4432] ; particularly in the term depending on cos. 1\ , which has a
difl^rent sign, though it is nearly of t)ie same numerical value ; an error in the sign having
been discovered in the original minutes of the numerical calculation of La Place.
5 u" = + 0',02489 . sin. T,, -f- 0',002G6 . cos. T5
1 + 0',08628 . sin. Tj — 0%01857 . cos. T5
1' — 2',00454 . sin. T^ + 0%4375- . cos. T^
2 + 0',07587 . sin. T5 + (y.OSlQT . cos. T^
2' + 0',39242 . sin. Tr, + 0,22555 . cos. 7^
3 + 0%28829 . sin. T5 + 0',19273 . cos. T5
3' — 0%71831 . sin. 7^ — r,5S65S . cos. T^
4 _ 0',14619 . sin. Tg — 0^09422 . cos. Tg
5 — 0%76290 . sin. T5 + 0',-7529 . cos. Tj
6, 6' + 2%16304 . sin. T5 +16',97139 . cos. T^
7, i = 2, + 6',G2968 . sin. T5 — 0%80829 . cos. Tg
7, t == 1 , — 2,49438 . sin. T5 — 0,92192 . cos. Tg
8, i = 2, + 0',22613 . sin. Tg — 0^53472 . cos. Tg
= 3',76028 . sin. Tg +14'',72286. cos. ïg .
In computing these numbers, the mass of Saturn is supposed to be, as in [4061 J], equal to
WbT^ ■> instead of t^tî^jï) used by I^a Place [4061]. To compare them with La Place's
calculation [4432], given below, in [4431it], we must increase the coefficients [4431^], in
the ratio of 3512 to 3359,4 ; by which means they will become as in [4431iJ ; the terms
depending on t, t^, being neglected ;
5 V" = 3',93109 . sin. Tg + 15',39164 . cos. Tg ;
ôv"= p-,64166 . sin. 7'g— 18',46195 . cos. Tg .
The difl'erence of the two expressions [443h', ^], which we shall denote by C'", is a
correction, to be applied to the formula [4433 or 4434] ; and we shall have,
C =3 2',2S943 . sin. Tg + 3.3',85359 . cos. Tg.
We may remark, that the number of terms of the forms 7 to 10, and 7' to 10', [4006m],
is infinite ; but it is only necessary to notice a few of them, in which S r, 5 v, â r', or ô v',
have sensible values. Moreover, the terms depending on ô i, were not computed by Mr.
Pontécoulant,when he published the above results. The effects of the correction C" [4431/],
of the terms depending on S s, and of other quantities of a similar nature, are taken into
consideration in book x. chap. viii. [9037, &ic.] ; where the final results of all these
calculations, relative to the inequalities of the motions of Jupiter and Saturn, are given.
[4431e]
Terms of
the order
of the
square of
the dia~
tuibin^
forces.
[4431/]
[443%]
[4431A]
[4431 i]
[443U]
CorrectioH
of ihe
g:reat ine-
quality.
[4431J]
[4431m]
[4431n]
[4431o]
[4431;?]
[4431g]
292 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
Now, if we connect the several parts of the great inequality of Jupiter, we
shall obtain, for its complete value,*
r (126P,569155— ^0',013495— /3.0',0000I9247).sin.(5w^<— 2n'-i'+5£^— 2si')-
[4433] (1+1^^).)+ (9G',466088— t0%474651+^2.0.00007S564).cos.(5«^<— 2«'7+5e>— 2£'^)J
C+ function C" [4431/] -}- 2 5 v'" [4431]
If we reduce these to one single term, by the method in [4024 — 4027"], we
«'"'",. shall obtain, for 5 v", the following expression,
inequality. ' ' o 1 '
\(1265',251781-<.0',037090+i-0',000036669).sin. ]/
[4434] (l+,^^)-r ' ' T J V-<.77»-,653 + r2.0',013581 A.
(+ function C'" [4431/] +2.5 î;'" [4431] )
This inequality may require some correction, on account of the coefficient i^",
depending on the value of the mass of Saturn ; and also on account of the
[4434] slight imperfection in the assumed value of the divisor (on' — 2n)^; a long
series of observations will remove this small source of error. We must apply
this great inequality to Jupiter'' s mean motion, as we have seen in [4006"].
The square of the disturbing force produces also, in 6 v'", the inequality
[3890],
[4435] (5 1)'' = ^ . ^ . sm. (double argument oi the great inequality) ;
which, in numbers, is,
[4436] &v" = — 13",238897 .sin. (double argument of the great inequality) ;
ive must also apply the inequality of a long period to the mean motion of
Jupiter.
The inequality [3921],
[4437] 6v-= 1 . ^•^"'"^""+^^^^"^^ffJr.sin.(5>r/-10»-/ + 53'--10s--i?-2),
reduced to numbers, becomes,
[4438] 6 If" = _ 4',024751 . sin. (5 n'" t —10 n-t + 5 £"—10 s^' + 51" 21'" .55').
[4433a]
* (2668) The expression [4433], is the sum of the terms contained in the functions
[4418,4425,4429,4431,4432] multiplied by (1+ l^''')- Then, by computing this expression
for the times, t = 500, and t = 1000, we may reduce the whole to one term, as in
[4434], by the method explained in [4024—4027"].
VI.xii.>^33.] THEORY OF JUPITER. 293
We have also, in [3844], the inequality,*
6 v'" = ^ . Ke'\ sin. {5 n't — 4> n'" t + 5 i" — 4> s" + z^'" -J^ B) ; [4439]
and by reducing it to numbers, it becomes,
&v" = 10',084660.sin.(4n'^i: — 5n^-^ + 4s'^ — 5 £^ + 45''21"'440 ; [4440]
if we connect this Avith the two inequalities [4392], f
P,097613 . sin. (5n'i— 4?i'^'/ + 5 s"— 4 s" — z^")
— 2',781664.sin. (5 n't — 4> n'" t -^ 5 1" — 4> s" — z^" ) ;
we obtain the single equivalent expression,
6v" = (l + t^") .1 P,506 190 . sin. (4 n'" t — 5 n" t + 4> s"— 5 6"+ OS'' 00™ 36^). [4442]
We have seen, in [3773], that the expression of d.iv'" contains a secular
inequality, depending on the equation,
[4441]
[4439a]
* (2669) The inequality [4439], is the same as the last of [3344], augmenting the
accents of e, n, n', Sic. to conform to the present example. The term K, which occurs
intliis expression is, by [3824 — 3826], equal to the constant term of the coefficient of the
part of [4394]. depending on the angle 3 n'^t — 5 ?i'' t ; or rather on the angle ôtVt — 3n"'t.
This part being nearly equal to
— 1 69^265895 . sin. (5 n" t — 3 n'^'t + 5 1"— 3 s'^— 55''40"' 49^. [44394]
If we compare this with [3826], putting i=5, we get,
Z = — 169%265895; i? = — 55M0™ 49"' ; [4439c]
and by [4081], ra"'= 10'' 21'" 4' ; hence,
•TO"- -\-B=—45''l 9'" 4.5^ ; [4439(/]
and [4439] becomes,
^.Ke'\sm.{5n't — 4 71'" t -{- 5 b" — 4 e'" —45'' 19™ 45')
= — f . Z é\ sin. (4 n'^t — 5 n" t -\- A t"— 5 s" + 45" 19" 45").
Substituting in this, the value of K [4439c], and that of c" [4080], it becomes neai'ly as
in [4440].
t (2670) These mequalities are found in the ninth and tenth lines of [4392], with a
slight and unimportant variation in the first coefficient. These terms [4441] may be [4440a]
connected with [4440], and reduced to one term, of the form [4442], by the method given
in [4282A— Z].
VOL. III. 74
[4439e]
294 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
d.dv'" m".»'^
dt 8
Hence we deduce,*
[4444] ^^"'= — 23%9441 . e''=^— 27%7951 . e'=^ + 42^,9296 . é\ e\ cos. (^-— ^").
[4444'] ^^ ''^^^Z neglect the constant part of the second member of this equation,
which is confounded ivith the mean motion of Jupiter, and then we shall have,t
* (2671) We have, as in [.3756a, è],
2 2 2 2 2 2
[4443a] h'" + l'" = e"' , A" -j- P = e" , li" h" + t' />' = e'" e" . cos. (to^— to'' ).
Substituting these in [444-3] ; also the values of .^"", A'-^\ and their differentials, in
terms of Oj, and its differentials [996 — 1001]; then the values of these quantities
[4202, Sic] ; we finally get the expression [4444] .
f (2672) We shall put E for the general expression of the second member of [4444],
corresponding to any value whatever of t, and E for its value when ^ ^ 0 ; then substituting
the values é", e", -a'", -n' [4407], we shall obtain,
[4445a] ^-#-' = ^ = E + ^-^+^°- [4444,4445a].
Multiplying this by d t, and integrating, supposing Sv'" = 0 when t = 0, we get,
[44456] 6v^''= Et + ^. — .t^ + kxi.
of which the first term Et, may be neglected, being confounded with the mean motion of
Jupiter ; then we have, by neglecting t^, t'^, &c.
s iv 1 ''^ v2 d.Sv'" dE . ^....-.
[4445c] èv'^ = i.-.f, or --^ = — .t, as m [4445].
The coefficient of t, in the second member of this last expression, represents the differential
of the second member of [4444] , divided by d t, corresponding to the time of the epoch
1750. Substituting in it the values [4405], and dividing by the radius in seconds 206265^,
we get,
d.Sv'v
[4445d] —7^ = — 0',0000013 . i, nearly.
This equation being multiplied by d t, and integrated, gives [4446] ; no constant quantity
being added, because it is supposed to vanish when t = 0.
VI.xii.<^33.] THEORY OF JUPITER. 295
il^^ _ 03-= 9441 .t.2 e'\ — — 27^7951 . < . 2 e". ^
dt (It dt
[4445]
+ 42^9296.ï.5(e-. ^ +c\ ^).cos.(ra-— ^"O— e'^e^'^^^ ^'.sin.(«^— ^»)^.
Substituting for ^ , '-^, ^"^ '-^\ their values, given in [4405], and
integrating, we obtain,
^v'- =—t~. 0',00000065. [4446]
This inequality is insensible in the interval of ten or tivehe hundred years,
and even as it respects the most ancient observations that have been handed [4446']
down to us ; therefore we may neglect it.
It now remains to consider the radius vector of Jupiter. We have found,
in [3845], that the terms depending on the powers and products of the
third degree of the excentricities, add, to the expression of this radius, the
quantity,*
i ,-- = — H a'\ é\ COS. (5 n" ^ — 2 n''' i + 5 s' — 2 è^— ^''+ A)
+ Hà\ é\ COS. (4 «'''i — 5 n" Ï + 4 a-— 5 s' — ^'"—A) ^^447]
^ Correc-
,4m\n-.a-2r p .sin. (5 w'i — 2 7ii^<+5a'— 2 6-) ) 'A°dius'""
-\ . < ■ > r • vector.
5,iv_2„.v ^_[_p'.cos. (Sn^i— 2n'''<+5 6''— 2s"')!)
Reducing this function to numbers, we obtain,
(— 0,0003042733. cos.(5n'i—2n"'^+5s'— 2s'''— 12''08'"490)
6r'^=(l + f^').< >• [4448]
( + 0,0001001 860. cos.(4/î''i—5?i'ï+4si'— 5s' + 45n6"'470 ^
If we connect this expression with the terms computed in [4393],
i,_ J 0,0000268383. cos.(5n'< — 4w"'i + 5E'—4ê''—ra''')?
àr —{^ '^^'l — 0,0000516048. cos.(5r^ — 4n'7 + 5s' — 45'"-^')^' ^'^'^^^^
* (267.3) The expression [4447] is composed of the three last terms of [3845], increasing
the accents as in [4383a]. The value of H is as in [4426a] ; those of P, P', as in
[4402] ; the other elements are given in [4061, 4077, 4079,4080] ; hence the expression [4447a]
[4447] becomes as in [4448]. Connecting this with the two terms of or", given in
[4393 or 4449], and reducing by the method [4282A— Z], we obtain [4450].
296 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
we obtain the following result,
[4450] à r"= (1 + f^O • 0,0000983161 . cos. (4 n'^t — 5 n't + 4, £'^— 5 s-— 14''23'" 19').
The semi-major axis a'", which we have used in calculating the elliptical part
[4450] of the radius vector, must be augmented by the quantity ^o'". m'" [4058].
Adding this to the expression of a'" [4079], we obtain,
[4451] a'" = 5,20279108.
Inequalities of Jupiterh motion in latitude.
34. It follows, from [3931,3931'], that the terms depending on the
square of the disturbing force, add to the values of — - , — — , the following
quantities,*
* (2674) In deducing the differentials of 5 9, 5 è, Uc. from [.3931—3932'], in order
[4452a] jQ f5|^(i jjjg increments to be applied to the values of -tt^ "tt") ^c. [4246, &tc.], we may
consider 5y, 0 II, J <p, 5 ë, to be the only variable quantities ; or, in other words, we may
neglect the variations of n, è, ç, y, on account of their smallness. For the expressions
of 5 7, i5 n [393.5,3936], which are independent of the periodical angles, are of the order
[4452!)] m'~; consequently their differential coefficients -77") , are of the ««me order, and
are therefore much greater than the terms arising from the variations of the angles n — ê,
in the differentials of the expressions [3931 — 3932'] ; because these last terms depend on the
[4452c] products ^7-jt^ ^'^'Jt' ^c. which are evidently of the order m'^ ; since -^, -— ,
[4411] are of the order m'. Hence the differentials of [3931,3931'] become, by dividing
by fl t, and increasing the accents, as in [4388o] ;
* —- : -^ ^ .]—r-^ .cos. (n — è") — y.—, — .sm.fn — ô'")} ;
[4452e]
[4452/]
dt
p. — ; — = : — -— ^ — . } '- . sm. (n — ô'>) + 7.— --• .cos.fn — Ô"-)}.
^ dt j«'^ y/a"' + '"'Va" i dt ^ ' ^ ' dt ^ '5
Now, from [4410], we have.
[4452g] 1^ = O',000184 = ^ ; ^' = _ 0^00T631 = '^ ;
substitutmg these, in [4452e,/ J, we get, — — — , — — - , which are changed mto -j-,
U Z (I- t lit
[4452;»] 1^", in [4452,4453]; and by using [44-52^], also the values of y, n [4409], m'\ m\
VI.xU.§34.] THEORY OF JUPITER. 297
^'= — 0%078213
dt
— 0',223251
dt
de"
— = 6',457092
dt '
dt
Then we find, by means of the formula [42956]
[4452]
dt ~ ;H''Va"-fm''.v/«'' l^ ' t ^ )
^ = - -"'-^"" .$i2'.sm.(n_r) + ,.i^.cos.(n-r)^- [4453]
6 J, en, beuig comiKited by the formulas [3931,3931']. Reduchig these
functions to numbers, we obtain,
ifL =_0',000073; [4454]
dt
^= 0,000811. - [4*5^1
dt
d(f"' d(f>y
The first of these expressions must be added to the values ot — , -j^
[4246], and the second to the values of -j^, '-^ [4246] ; hence we
obtain,
dtpi
[4456]
a'", a", ê" [4061,4079,4083], they become as in [4454,4455]. Adding the expression
[4454] to the first terms of -— and -y- [4246], we get their values [4456] ; also [445ai]
do" rfd'"
addmg [4455] to the first terms of — — and -~ [4246], we obtain the
correspondmg values [4456].
* (2675) Tlie terms of us'"' [4457], are deduced from those in [4295&], by adding
three accents to the symbols m", n', n", /, î", a', a", in order to conform to the case ,..._,
•' ) J J 3 ; : 3 r4457nl
now under consideration. 7, IT, are as in [4409]. The values of 5<'~'^= — . 0 „
[1006], are given in [4210,4079].
VOL. III. 75
298 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
/ 0 ,564458 .sin. (n" ^ + s" — rr)
\ + 0',663927 . sin. (2 fi't — n'"t + 2 s" — s- — n'")
[4457] 6 5'"= (1 +^^).|-j- V,\l97S2.sm.(3n''t — 2 n"t + 3s-— 2 ^^ — n-")^^;
— 0',279382 . sin. (4 n't — 3 n'H + 4 s' — 3 .'^ — n'^)
— 0',269130 . sin. (2 n'-t — ift + 2 s'' — s'— n'^)
n'', in this formula, being the longitude of the ascending node of Saturn's
[4457'] Q^}j{t iipon that of Jupiter [42956 — c]. Lastly, we have, in [3885], the
inequality,*
[4458] 6 s'" = 3',941 680 . sin. (3 n'^t — 5 Ji' ^ + 3 i" — 5 s'' + 59'-' 30» 35').
Inequali-
ties ill the
latitude.
* (2676) The quantity [4458], is deduced from [3885], reducing both terms to one, as
[4458a] in|-4282A-/].
Before concluding the notes on this chapter, we may remark, that the inequalities of the
motions of Jupiter and Saturn, computed in tliis book, are corrected by the author in
[5974,&ic.], and afterwards more thoroughly, in book x. chap. viii. [9037,&ic.] ; where he has
decreased the assumed value of the mass of Saturn [4061]. He has also computed several
[4458 ] ^^^jj inequalities, which had not been previously noticed, and has given new forms to some of
the arguments. Finally, the subject of these inequalities has been treated in a wholly different
manner, withafrequentuseof definite integrals, by Professor Hansen, Director of the Observatory
. g , at Seeberg, in a memoir, entitled, " Untersuchung ueber die gegenseitigen Storungen des
Jiipiters und Saturnsf which gained, in 1830, the prize of the Royal Academy of Sciences,
of Berlin, relative to the inequalities of these two planets. In this method, the true longitude
is computed by means of the elements corresponding to the invariable ellipsis at the time of the
[4458rf] epoch ; taking instead of t, a function of t, which corrects for the perturbations. As the
inequalities of Jupiter's motion had not been completed by Professor Hansen, when he
[4458c] published this memoir, we may have occasion to refer to it more particularly, after the
completion of his work.
Vl.xiii. §35.]
THEORY OF SATURN
299
CHAPTER XIII.
THEORY OF SATURN.
35. The equation [4386],
r
corresponding to Jupiter, becomes for Saturn,
If we take for r", and r\ the mean distances of the earth and Saturn from
the sun [4079], and suppose 6 V' = ± 1" = ± 0',324, we shall find,
6?- = ±0,000141326.
Therefore we may neglect the inequalities of àr", below =F 0,000141.
We shall also neglect the inequalities of Saturn, in longitude and latitude,
which are less than a quarter of a centesimal second, or 0%081*
Inequalities of Saturn, independent of the excentricilies*
,+ 3%156532.sin. (w''i — n"i + s'" — £
— 3r,493729 . sin. 2(«"i — 71" t + i'" — ='
— 6',56593 \ . ûn.S{n"' t — n' t + b" — i
- 1%965748 . sin. 4(/i'''i — n^t + i" — î
ii)'==(l + (.''). ^ _ 0',697047 . sin. 5(n'^^ — n^i + £" — s'
— 0,270789 . sin. 6 {n}" t — n't-\- s''' — s
— 0-, 1 1 6291 . sin. 7 (ir t — n't + £*' — s
— 0',056126 . sin. 8(n"'t — nH + i" — s"
K— 0%034097 . sin. 9 (n'^i — n't + s'" — ="
[4459]
[4460]
[4461]
Terms
which
may be
neglected.
[4462]
Inequali-
ties inde-
pendent of
the ex-
cent rioi-
ties.
[4463]
* (2677) These are computed as in [4277a — 0], increasing the accents on a, n, rt, hx,.
so as to conform to the present case.
[4463a]
300
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4463]
+ (l+f^^')
InequftTi-
ties inde-
pendent of
the excen-
tficities.
ir'=(l+f^")
[4464]
+ (l+f^'')
+ 9',248269 . sin. (n^H —n't + i"' — s^)
—14^451913 . sin. 2(ri'H — n''t + s- — s")
— l',427160 . sin. 3(»'" t — n''t + s"' — s^)
— 0 ,314960 . sin. A^Çn^'t — n" t -{- b"' — s")
— 0',090690 . sin. 5 (n^'t — n" t + s^
— 0%047444 . sin. 6 (n"' t — n't + t"
— 0',010686 . sin. 7(/j''7 — n't + i"
— 0%003942 . sin. S (if 't— n't + i"
' + 0,0039077763
+ 0,0081638400. cos. (n''i — rC / + e'^— ^\
+ 0,0013838330. cos. 2 {n"t — TCt-\- e'^— e'
1 + 0,0003200673 . cos. 3 {iVt — n't + £'"— £^
'+ 0,0000992632. COS. 4 (n''ï — n"ï + £'" — £^'
+ 0,0000355919. COS. 5 (m'"^ — m^ï + e'^—£^
I + 0,0000135999 .COS. 6 (w'^/ — «"^ + f'^ — £'
+ 0,0000( f55 1 35 . COS. 7 {n'" t — ')ft+ e'" — £\
+ 0,0000021631 .COS. 8(?r^ — Jt^^ + E'"— e"
+ 0,0000006436 . cos. 9 (n" t—7ft-^ e"_ =
'—0,0000137622
+ 0,0001491217. cos. (71" t — 7^ t + i" — i'''
- 0,00039499 1 6 . cos. 2 (n' f — »r' i + £" — e'
' — 0,0000480303 . cos. 3 {7f 1 — 7^1+^''— e' '
- 0,00001 1 8201 . cos. 4 («^ Ï — n^ 7 + £"— £"'
— 0,0000036280 . cos. 5 {if t — »" t + s^— e^
— 0,0000012501 . cos. 6 {7ft — n'H + e^ — s^
Inequalities dependiTig 07i the first poiver of the excentricities*
We shall here notice the secular variations in the coefficients of those
[4465] inequalities of Saturn, Avhich exceed 1 00", or 32',4 ; in the same manner
as we have done for Jupiter, in [4389']. Hence we have,
[4466a]
* (2678) The inequalities depending on tlie first power of the excentricities, are
computed in the same manner as for Jupiter [4390a, &c.].
VI.xiii.§35.]
THEORY OF SATURN.
301
èv"^ (l + f^'").
+ (l+f^'")
— ir,509517 . sin. (trt + i' — z^)
+ r,258041 . sin. (rrt + s'" — tz'")
— 2'-,064438 . sin. (2 n'7 — n't -\-2 i"— s- — a"
+ 2^,672881 . sin. (2 7rt — «"i + 2 e"— s" — ^i'
— 0^292291 . sin. (3 n"t — 27ft +3 s"— 2 =' — in^
— 0',223191 . sin. (3 »"< _ 2 ?i'^ + 3 s"— 2 6"— ^'
— 0%090633 . sin. (4 7rt — 3 n^i + 4 s'^— 3 1" — .3*
— (1 82%068686 — Ï. 0S0101095) . shi/_^^^^J^^l,
+ (41 7',057741 + 1 . 0^0] 38572). sin. (_^Z^_~.!!^l;}j
+ (34',341627 — ^0^0019068).sin.
Sn''t — 27i"t
_^3sV_2çiv_-,v
VOL. III.
— 17'-,654164 . sin. (3 n^i — 2 ?»"< + 3 s" — 2a" — ^^'^
+ 4',795080 . sin. (4 «"i — 3 ?^'^i + 4 s' — 3 s'" — k''
■ 2%43541 0 . sin. (4 rt'' i — 3 n'" i + 4 s" — 3 s" — ^'^
+ r,393612 . sin. (5 7ft—^n"t + 5 £"—4 s'^'—z^^
— 0-,703450 . sin. (5 ift — 4 n'H + 5 s^— 4 s'' — ra'"
+ 0S537161 . sin. (6 71" t — 5 n'" t J^ 6 s" — 5 s'" — z>'
— 0-,25651 0 . sin. (6 'ïû' t — 5 n'" i + 6 a" — 5 s'" — ^'^
+ 0V2] 6195 . sin. (7 n't — 6 n'^ t+l^' — Q s'^ — ^^
— 0', 1 07342 . sin. {l7i't — & ir t + 1 1'—G s'" — -sj"
,+ l',142398 . sin. (7tH + a" — tn^)
— P,01 1647 . sin. {ifH + s^^ _ ^")
—10^033866 . sin. (2 n" t — 7i''t + 2 s"' — s^ — z,^ )
+ 2%766173 . sin. (2 >t^7 — Ji'i + 2 s^' — s'' — t^^i-^
-16^936280 . sin. (3 71'' t — 2 7^1 + 3 s"' _ 2 5'— :3^
+ 25', 153348 . sin. (3 w^'i — 2 n^i+ 3 s" — 2 «'— «"
+ 0',559336 . sin. (4 rf't — 3 n"^ + 4 h"' — 3 s' — ^^
' — 0',758225 . sin. (4 n^'t — 3 7ft -^4, s"' — 3 £"— ^3^'
— 0-, 1 87729 . sin. (5 n'H — 4 n^' ^ + 5 s'' — 4 s' —
— 0',673817. sin. (2n^i— «^■74-2?'— «''—
+ r,521577 .sin. (3?i^^ — 2?i'''^ + 3s^ — 2e'"—
+ 0^ 151 701 .sin. (4?r^ — 3 «"'^ + 4«* — 3 £"—
76
Inequali-
ties de-
pending on
Ihe first
power of
iheexcen-
tricities-
[4466]
302 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
/— 0,0003422170 . cos. (n'i + £^ — ^n'")
I — 0,0020775935 . cos. (2 ?ft~ n'H + 2 s"— e-_ w" )
. /I, i„s )+ 0,0053861750. COS. r2?rï — n'7 + 2 s^—s"—^")
(5r ==( 1+ fji'^). / \ i /
j+0,0011594872.cos. (3n^i — 2n'7 4-3E''— 2.'"— ^'')j
[4467] j —0,0006217670 . cos. (3 n^ i — 2 n'^i + 35" — 2 £'"— ra'^)
\+ 0,00021 17893 . cos. (4 7iU — 3 n'H + 45"— 3 s"— ^^^ ) ,
+ n+ -) ^— 0,0003750767. cos. (3 r'i — 2 ?i'ï + 3£^'— 2 s^—^^)>
(+ 0,0005605490. COS. (3 n'V — 2 H-' i + 3 s^'— 2e'' — ^") ^ *
Inequalities depending on the squares and products of the excentricities and
inclinations of the orbits*
i"'n°';- / /k;,. o '.noon , n. rvno^oN • /3 n^t — n'" t + 3 s" — s''
^nS'" / -(54^847829_^.0^00362).sln.f 84^36"'45^-^.34',55
order. 1 ^ ' '
^_ i )+ 28',526709.sin.(«'''ï — n''^ + E'''-£^ + 84''16'"430
'-(669%682372-i.0%015469).sin.^-_j"5'g,j5!!5f_^^_4g.r5
,,,^^, , — 2-,935793.siii.(5n'^ — 3«"Ï+5e^— 33-— 57''O9'"O80
[4468] > ' V
( + 1 •■,923552 . sin. (3 n^'t — Sn't + S s" — 3 s^— 67" 54'"43') )
^^ ^ ^4-3^,025379. sin. (3 n"<— n"^ + 3£''— s'— 85''34'"120 )
If we connect the inequalities depending on n'^t — n^t; also those on
[4468'] Sn^'t — Sn^t, with the corresponding terms which are independent of the
excentricities [4463], we shall obtain for their sum, the following expression,
[4469]
[4470]
6r^= + (1 + ij.'^) . 28',967123 . sin. (n'U — n't + s'"— g^-f 78''03'"130
— (1 + f^'O . 1 ',9 16292 . sin. (3 n^' t — Qn-'t + S s"— 3 ï>+68''27'"07').
Then we have,t
'—0,0011710805. cos. (3 n^ï — n'''^-f3i^ — £''—90''I2'"350
^,.v^(-l^^iv-j 1-0,0005621 901. COS. ( n'H — n't+ £i''_sv_83''26'"330 (.
+(0,0151990624- 1 . 0,0000003370) .cos. (^"5J.;^^:'3;3+^^/'^^^
[4468o] * (2679) Computed as in [4394«, Sic], for Jupiter.
[4470(1] t (2680) Tliis computation is made as in [4394c/] .
VI.xiii.§35.] THEORY OF SATURN. 303
The inequality of the radius vector, depending on the angle li'^t — rft,
being connected with the similar term in [4464], which is independent of the ^ ^
excentricities, becomes,
f^r" ^ (l + i^'O . 0,0081090035.cos.(/r^ — n'^i + s-— e"— 3''57™35'). [4471]
Since 5 n" — 2 n" is very small, we have computed the inequality depending
on 2»'''^ — ^n't, by the formulas [3714, 3715]. Moreover, as Sn^' — n"
is very small, we have computed the inequality depending on the angle [-44721
on^'t — M'7, by the formulas [3711,3718]. For greater accuracy, ive
must apply this last inequality to the mean motion of Saturn, on account of the
length of its period.
Inequalities depending on the poicers and products of three and Jive
dimensions of the excentricities and inclinations of the orbits,
and on the square of the disturhing force.
The most considerable part of the great inequality of Saturn, is that which
has (5 n' — 2 n")', for a divisor, and depends on P, and P'. It is derived [4472]
from the great inequality of Jupiter, by multiplying it by — v ' wa' iv? "^ [4473]
conformity with the formulas [3844,3846].* Hence we get, for this part of
the inequality of Saturn, the following expression,
è v'= — i 2957^357566 — t . 0',01 9701 — A 0^00004505 1 . sin.(5 nU—2 n"t4-5 e^— 2 s")
' [44741
— |279%746590— / . P,1086.38 + î!2.0',00018387|.cos.(5w"<— 2m"Y+5ev_26'^).
* (2681) If we represent, for brevity, the terms between the braces in the two first
lines of [3844], by aP^, we shall find, by inspection, that the two fii-st lines of [3846],
between the braces, are equal to a' Pg ; and by noticing only those terms of 5 r, ô v, which [4472a]
have the small divisor (5 n' — 2 n)^, we shall get, by increasing the accents so as to confonTi
to the case now under consideration,
, . 6jn».n''2 15m'"'. ny^ „, r^^~oi.i
ôv'^ = —- ^-^.a'v.p ; . 5v''=-- --——.a\Pl. [44/26]
(5 7i>-— 2 n")2 - (5nv— 2n'^)2 ^
Hence it is evident that (> v" is easily deduced fi-om 5 v", by multiplying this last quantity
by the factor [4473] ; so that we shall have,
15 m'^.n"^. a" , .
^v'-=— .^„ „„-^v"' [4472c]
as in the terms of the fifth dimension of the excentricities [3868a — cj.
304 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
inec.uaii- The great inequality of Saturn is composed of several other parts: it
S. contains, in [3846] , the function,*
_j_av2 _/ — N COS. (5 rCt — 27V''t 4- 3 i" — 2 £'")
[4475] 6V'' = — ^^:;--nn^ • <
- a" 2 . f — j . sin. (5 71' < — 2 n'" < + 5 £» — 2 s'^)
[4476]
[4477]
[4478]
[4480]
[4481]
5?i' — 2 7t"
Reducing this quantity to numbers, we find in 1750,
6 v''= + 52%l3n99l. COS. (5 n't — 2 n"t + ôs"— 2£'0
— 1 P,275407 . sin. {brCt—2 1^ i + .5 s^ — 2 i") ;
and in 1950,
6 v^ = + 51Sl 92839 . sin. {5rC t — 2n"t + bs''—^ s'")
— 14^982033 . cos. (5 n^i — 2 n"t + 5 -='— 2 s").
Hence we deduce the value of this function for any time whatever t,
6 v' = + (52^138991 — t . 0%0047308) . sin. (5 7^1 — 2 n'^t + 5 £"— 2 ;'")
— (11%275407 + ^. 0',01 85331) .cos. (5 n't — 2 7i"t + ôs-'—2B").
The great inequality of Saturn contains also, in [3846], the term,
[4479] 5 w^ = — 4 //' e" . sin. ( 5n''ï — 2 n'^ i + 5 £" — 2 e'"— ^» + A').
This term, in 1750, is,
6v'' = + 7%554290 . sin. (5 ift—2 n"'t + 5 e"— 2 £*")
+ 5',321290 . cos. (5 n't — 2 n"t+ 5 s' — 2 s") ;
and, in 1950, it is,
a v" = + 7%71 1294 . sin. (5 Wt — 2n"'t + 5i' — 2 s")
+ 4^825821 .cos.(5n''t—2n"'t + ôs'' — 2e"').
* (2682) The expression [447 o] is similar to [4419], in Jupiter's theory, and is
[4475a] computed in the same manner ; namely, by finding tlie values of ( > v ')' ("TT )' ^'^^
similar to [4420] ; which maybe easily done, by means of formula [4421], and the values
[4475t] [4420]. Then from [3842, 3843], we get (t-;). (j^X^'^- It is useless, however,
to explain the details of this computation, as it is done in almost exactly the same way as
VI. xiii. § 35.]
THEORY OF SATURN.
305
Hence, for any time t, it becomes,
6 r" = + { 7'-,554290 + t . 0\000785 \ . sin. (5 n'i — 2 w" i + 5 s^- — 2 ;'^)
+ {5^321290 — t . 0%002477 \ . cos. (5 nH—2 n"t + 5 î'— 2 é").
The part of Saturn's great inequality, depending on the poivers and products
of five dimensions of the excentricities and inclinations of the orbits, is, by
[3846,4023],*
[4482]
for Jupiter ; we shall tlierefore only observe, that the expressions [4476, 4477, 4478, 4479,
4480, 4481, 44S-2,] correspond respectively to [4423, 4424, 4425, 4426, 4427, 4428, 4429].
* (2683) From the terms of R, of the third dimension, depending on P, P' [3810],
we have deduced in the two first lines of [3844], the corresponding terms of S v; which
ai-e afterwards developed in [4022,4023], according to the powers of t; and the same process
may be a))plied to the two first lines of 5 v' [3846]. We may also derive these tenns of
S 11', from the corresponding ones of 5 v, by multiplying by the factor — —- — , or
ISm'^'.n'a.a
rather by
6 m'. rfi. a
as is evident by the inspection of the formulas [3844,3846]. We may
[4475f]
[4483a]
[4483i]
[4483c]
proceed in exactly the same manner with the terms of R, of the fifth dimension, depending on
P„ PI [3863], or with those of il', depending on P„, P,/[3865]; the only change requisite
is to place the accents below the letters P, P'. Now, if we neglect the parts of [4023] ,
depending on t"^, ddP, ddP', and make the above-mentioned changes in the factor
and in the accents of the remaining terms ; also putting P, , for P„ , and Pf, for P„' [3864è], [4483(/]
we shall get, for S v'' the expression [4483], depending on quantities of the fifth order in
e", e", y. In finding the values of P, , P/, we may observe that the function R [3859]
is easily reduced to the form [386-3], by the method explained in [3842i,&c.] ; using the
values of A'"», A"*", &ic. [4430], by means of which we obtain the expressions of a^.P ,
a\P;, [4434,4485], for the two epochs of 1750, 1950. The difference of these two
expressions being found, and divided respectively by 200, give the values [448C] ; as is
evident from the formula [3723]. Substituting [4484, 4486], in [4483], it becomes as in
[4487]. The signs of all the terms [4434—4487], are different in the original work, being
changed, as in [4430a], to correct the mistake mentioned in [3860a]. JMoreover, to rectify
this mistake in the signs, it is necessary to add the expression 2 (S j;" [4487] to the second
member of the great inequality of Saturn [4492, &c.], in the same manner as the similar
value of 2ÔV''' [4431], is added to the expression of the great inequality of Jupiter
[4434, &ic.]. The numerical coefficients, in [4434, 4491], are equal to those given by the
author; but the corrections C', C, 2ôv'\ 2 «5 «% in the second members, are not
mentioned in the original work.
[4483e]
[4483/]
VOL. III.
77
306 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
fifth order. , -» -.^
m which m'^P^, m'^P/ [3863, 44836], express the coefficients of
sm. (ôn''t — 2n''t + ôc' — 2 ;"), cos. (5 n' t — 2 trt + 5 -=^— 2 s'^),
m the development of R, depending on the products of five dimensions of
the excentricities and inclinations. We find, in the year 1 750,
^4484^ a\ P, = 0,0000068376 ;
a\P;= 0,0000100087;
and in the year 1950,
^4485j a\P, = 0,0000077132;
[4486]
[4487]
[4489]
consequently.
a\P;=: 0,0000096940;
a\ —' = 0,0000000043780 ;
at
a\ i^ =3 —0,0000000015735.
dt
Hence the preceding function [4483], reduced to numbers, is,
5v''=^ + j^ 29% 144591 —t. 0%004081 } . sin. (5 n't — 2 )rt + 5 s"— 2 s'")
— \ 1 8',879594 + t . 0-,01 1 356 j . cos. (5 n't — 2 ir t + 5 ="— 2 s'').
Lastly, we have, in [4003], the sensilile part of the great inequality of
Saturn, depending on the square of the disturbing force. This, in 1750, is,*
Sv' = — 3^,816537 . sin. (5k^7— 2îi'^i + 5 s'— 2 s")
[4488] + 42^,92031 9 . cos. (5 n't — 2 n'^ f + 5 s" — 2 .='")
+ function C [4489/c] ;
and, in 1950,
6 tj" = — 1 %636772 . sin. (5 n''t — 2n"t-\-5 =^' — 2 r)
+ 43',624686 . cos. {ôn''t—2n"t + bf—2 ^")
+ function C' [4489/t'] , nearly.
* (2684) The expression of & v" [4003] , being developed as in [3842a,J], and then computed
'■ "■' as in the last note, becomes, according to the author, in 1750 and 1950, as in [4488, 4489],
Vl.xiii. §35.] THEORY OF SATURN.
Therefore, in tlie time 1750 +^ this part is expressed by,
307
respectively. From these values, tlie general form [4490] is (leiluced, by the method used
in [44S3e, &ic.] ; but these numerical values, of the function [4003], have the same defects
as the similar expression in Jupiter's motion [4432], of which we have treated in [44896]
[4005(7 — 40076, 4431 rt — A]. The corrected value of Hv^', given by Mr. Pontécoulant in
the paper referred to in [443 If], is as in the following table, which is similar to that of
Jupiter [4431/,&c.].
Ô v'= 2',17020 . sin. T^ + 0' ,23185 . cos. Tj
1 + 8^14230 . sin. T^ + P,8S43S . cos. Tg
1' + 4^891 14 . sin. T^ — P,067G9 . cos. Tg
2- _ 0,951 1 2 . sin. Tg — 0',54669 . cos. Tg
2 -f- 0',054SS . sin. Tg — 0',830G0 . cos. Tg
3 _ 0',2576S . sin. Tg — 0%80208 . cos. Tg
3' + P,74101 . sin. Tg + 3 ,84548 . cos. Tg
4 + 0',22091 . sin. Tg + 0',2.3748 . cos. Tg
5 + r,85702 . sin. Tg — r,18481 . cos. Tg
6, G' + 3',466n7 .sin. Tg -40%36260 . cos. Tg
7, i = 2, — 16^06895 . sin. Tg + ] %9591 4 . cos. Tg
7, i = 1, + 6%04586 . sin. Tg + 2',23454 . cos. 1\
8, z = 2, — 0%54808 . sin. Tg + 1%29603 . cos. Tg
= 10%7635G . sin. Tg -33',10557 . cos. Tg.
Termg of
tlie order
of the
square of
the dis-
turbing
forces.
[4489c]
This differs very much from the expression given by La Place, in [4488] ; which is connected
with the other terms of the great inequality [4491], after multiplying it by 1 -j- fx'". This
multiplication, by 1 -j-f^'") is not strictly correct; because some of the terms depend on
(1 +t^'') • (1 + /J-')) and others upon (1 + (-^''Tj ^'ut as jx''', ij.", are small, this difference
is not of much importance in this small inequality. We shall therefore adopt this method of
the author, as we have already done in the similar inequali:y of Jujiiter [4431A, &tc.] ;
where the factor 1 -{-[>■'', is used for all the terms. Proceeding, therefore, as in [443lA,&;c.],
we shall observe that the mass of Jupiter . [4061ffj, is used in computing [4489rf] ;
and the mass ^n^?:^^n [4061], is used in computing [4488] ; and if we increase the
expression [4489(/]. in the ratio of 1070,5 to 1067,09, it becomes as in [4489iJ. Subtracting
the expression [4483] from [4489t], we get very nearly the correction C" [4489A;], to be
applied to the formula [4491 or 4492]. We must also apply a correction, depending on à ^,
similar to that of 5e [443 Ip J, in the great inequality of Jupiter ;
6 v" = 10^,79796 . sin. Tg — 3.3%21 1 37 . cos. % ;
C' = 14',61450 . sin. Tg— 76',I3169 .cos. Tg.
[4489rf]
[4489«]
[4489/]
[4489g]
[4489ft]
Correction
ofihe
great ine-
quality.
[4489i]
[4489fe]
308 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
6v' = — {3',816537 — t . 0',01 08988 1. sin. (ôn't — 2 n'" ^ + 5 s'' — 2 e'")
[44903 +{42^920319 + t . 0%00352l8].cos.(5 n't— 2n" t + 5s'' — 2^'")
+ function C [4489A:].
Now, if we connect together the different parts of the great inequality of
Saturn, we shall obtain its complete value, which is to be applied to ike
planètes mean motion ;*
1+{2931%125445— <.0^0307355— ^2.0%0000450| .sin/^'g^";"^^?)
+ {223»-,252793-<.lM025051+i2.0^0001838| .cos/^g^';^^^?)
+ function C" [4489^'] + 2 <S î;^ [4487]
Great Reducing these two terms to one, by the method in [4025 — 4027'], we shall
mequality.
obtain,
C(O939»,615848-<.0^085024+i2.0',00008421).sin. .y.-gog + fa 0-012676 ^f '
[4492] ,5t>'=-(l V) < i-t.// ,b^-\-t .u,vi.4i}/i, :)\
^4- function C^ [4489^-]+ 2 5i;^ [4487] ^
The square of the disturbing force produces also, in [3891'], the inequality,!
[4493] ôv' = ^ . "' '^"""i "* '^'^ . sin. (double of the argument of the great inequality) ;
which, in numbers, is,
[4494] ^«'= (30 ,688957 — /.0',001 724).sin.(double argument of the great inequality) ;
and this must also be applied to the mean motion of Saturn.
[4489i]
[4489m]
Professor Hansen, in the work mentioned in [4458c], makes this part of the great inequality
of Saturn, in the year 1800, as in [4489m], using the masses m'", m' [4061]. The
corresponding value of La Place's formula, is found by putting t = 50, in [4490], by which
means it becomes as in [4489o]. The difference of these two expressions represents the
value of C' [4489p], corresponding to the calculations of Professor Hansen, noticing all the
terms of any importance ;
[4489n] à V- = 15',476 . sin. Tj — 47',531 . cos. Tg ;
[4489o] àv' = — 3',271 . sin. Tj + 43',096 . cos. T^ ;
C>- = 1 8',747 . sin. T^ — 90',627 . cos. T^.
* (2685) The function [4491] is the sum of the expressions [4474,4478,4482,4487,4490];
and this sum is easily reduced to the form [4 192],containingbut one term,by the method explained
in [4025—4027']. There is a small mistake in the calculation of the term 223' ,252793
[4491], which in the preceding sum is 223',900794; the difference being 0',648 = 2".
(4493a] t (2686) The term [4493] is the same as [-3891'], —H' [3891] being the great
[4489;?]
[4491a]
VI.xiii.§35.] THEORY OF SATURN. 309
The inequality [3927],*
reduced to numbers, is,
6 1"= + 8',26451 7 . sin. (4 n'" / — 9 n" / + 4 s''— 9 s^' + 51'' 49'" 37'). [4496]
We have also, in [3846], the inequality,!
ôv'= l;K' e' . sin. (Sn't — 2 n'^7 + 3 s" — 2 =" + ^' +5') ; [4497]
inequality of Saturn, or
5' =2939%61 5848 — <. 0^085024, and :ï'=4''2r' 20% nearly [4493]: [44936]
substituting this and the values of m"-', irû', a'", a" [4061,4079], and dividing by the
radius in seconds 206265% for the sake of homogenity, we get ô v" [4494]. The correction
in the value of H' [4483/], has a slight efiect on this result ; and the same may be observed '■ '
relative to the correction of H [4483/], in the term [4436] ; and in other terms depending
on H, H-
* (2687) The inequality [4495] is the same as [3927], increasing the accents as in
[4388n]. Now we have nearly as in [44936],
F=2939%615848, :3' = 4''21"'20' [44936] ; [4495o]
and by comparing the expression [3925] with the third line of [4468], we get, by neglecting
the teiTus depending on t,
K = 669%682372, B' = — bQ^ 10'" 57'. [44956]
Substituting these in [4495], it becomes,
-f 9%2107 .sin. (4?i'>7 — 9?j^-r +4 £'' — 9 £^4-51'^49'" 37»). [4495c]
In the original work the coefficient has a difterent sign, being
— 25",507770 =— 8%264517,
also the angle — B' — Â' , as given at first, is,
— 67°,3508 = — GO-* 36"' 57'. [4495rf]
These mistakes are corrected by the author in [9105], where the coefficient is made equal
to +8',264517, and the angle — B — .?= 51" 49" 37' nearly.
t (2688) This is the same as the last line of [3846], increasing the accents as in r4497„n
[4388a].
VOL. III. 78
310 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
and by reduction to numbers, it becomes in 1750,*
[4498] 6 v" = 47% 115141 . sin. (2 n'' t — Sn" t + 2 e'"— 3 s" + US'' 08'" 08^ ;
and in 1950,
[4499] 6 V' = 46',307169 . sin. (2 n"t — Sn''t-\-2 i""— 3 =' + U9' 41'" \&).
Therefore its value for any time whatever t, is,
^^5Q(j^ 5u'=(47%115141 — <.0%0040399).sin.(27r<— 3?rt-[-26i>'— 3£^+148''08"'08'+^27%94).
Connecting this expression with the following, obtained in [4466],
èv" = + (34V341627 — i.0%0019).sin. (3n^t—2nrt+3s'—2^''—z^)
^''^"^^ — 1 7%6541 64 . sin. (3 7i^t — 2 rr t + 3^" — 2 s''' — «''') ;
we shall obtain for their sum, the following inequality,!
[4502] s 1)'=:— (24^571253— < . 0',004392).sin.(2?i''Y— 3 ?i^-<+2s'^— 3 e^+14''48'"19*— M2',38).
We have found, in [3777], that Saturn's mean motion is subjected to a
secular equation, corresponding to that of Jupiter in [4446], namely,
[4503] êv" ^ — t"". 0 ,00000065.
The corresponding secular equation of Saturn is represented, as in [3777],
by,t
[4504] 6 v" = -^^ . f. 0"-,00000065 ;
and is therefore, in numbers,
[4505] 6v' = fA)%00000\51;
which may be neglected without any sensible error.
* (2689) If we retain the terras depending on t, in the values of K', B' [4495i,4468],
we shall have,
K' = 669',682372 — t . 0',015469 ; B'= — SG'' 10"" 57^ — i . 49'-,5 ;
[.4498a] ^ = ^i2\^< 20'— t . 77%629 [4492, 3926], &ic.
With these values, and those of e^, zf [4407], we may compute the function [4497], for
[44986] the j-ears 1750, 1950, as in [4498, 44S9]; hence we may deduce the general expression
[4500], by the same method as in [4017—4021].
[4502o] 1(2690) This reduction is made as in [42S2/t—r].
[4505o] t (2691) The integral of [3777 or 3785], being divided by m'\/a', ^ives,
Secular
equatiun.
VI.xiii.§35.] THEORY OF SATURN. 311
It now remains to consider the radius vector of Saturn. We have seen,
in [3847], that the terms, depending on the tliird power or product of the
excentricities, add to the expression of the radius vector of Saturn, the
quantity,*
6r' ^ — H' a\ e\ cos. (5 n't — 2 n"t + 5 £" — 2 ^'— ^' + A)
+ H' a\ e\ COS. (3 n't — 2 n"t + 3 s>' — 2 s- + ^^ + A) ^450^^
10 m". n\ a'-^ C P . sin. {5)i't — 2 n'^i + 5 s^— 2 s'") )
5 n'— 2 n" '(-\-P'. cos. (5 yt^i— 2w"7+ 5 e^— 2 s'") ^ '
Reducing this function to numbers, we obtain,
, ( +0,00351994565.cos.(5n''i— 2n'''^+5 .^— 2s-+ 13^01'"490 )
6r''^(l + (A"').< ^ >. [45071
(_0,0008553506.cos.(2/rf— 3n^i+2s"— 3s' + 35''49"'080^ ^ ^
nequQli-
ties in the
Connecting the last of these two inequalities with those we have found 1
in [4467], depending on the first power of the excentricities, namely, ^^^'j»»
V .1, i„x ^ + 0,001 1594872. cos. (3/1^^ — 2 ?r/ + 3s'— 2 s'-'—^M)
^ ( — 0,0006217670.cos.(3n"i — 2n'''^ + 3£'— 2 s"' — ^'0^
we get,t
0 r"= — (1 4- f^'") . 0,0013806201 . cos. (Zn'^t—S n't+2 s'"— 3 s"— 23'^ 19" 18'). [4509]
the accents being increased as in [4.38Sa] • Substituting Sv'" [4503], we get Sv" [4504],
which is reduced to numbers as in [4505], by using the elements m'", m"' , «'", a" [4505c]
[4061,4079], This correction is only 1*,5, in 1000 years, which is hardly deserving of
notice.
* (2692) The function [4506] is the same as the three last terms of [3847], multiplied
by a', and increasing the accents [4388a] ; the first term of [3847] being of the second
order in e, e', y, is included in [4170]. H represents the part of — -;- [3848], [4506a]
depending on the angle 4 n" t — 2 li" t ; P, P', are given in [4402, &ic.]. Hence the
expression [4506] becomes, in numbers, as in [4507].
t (2693) The function [4508] is the same as the fourth and fifth lines of [4467].
Connecting these with the similar terms [4507], and reducing the whole to one term, by the [4509a]
method in [4282A — /], it becomes as in [4509].
312 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
The semi-major axis, which is used in calculating the elliptical part of the
radius vector, must be increased as in [4058], by the quantity ^a'-m'';
and hy adding it to the value of a" [4079], we obtain,
[4510] «^ = 9,53881757.
Inequalities of Satimi^s motion in latitude.
SQ. The formula [1030] gives,*
+ ] %787358 . sin. {n" t + b" — n^)
— 0',2501 80. sin. (2 n'-'t — n't + 2 é" — s" — n^)
0',083g46 .sin. (3 n' 't — 2n'' t + 3 s"—2s'' — n")
^^^- ] + 3% 1 43523 . sin. (2 n' t — n"l + 2 j" — s" — n')
— 0%522865 . sin. (3 n't — 2 n"t + 3 .-" — 2 s" — n")
— 0',083182.sin.(4n"f — 3n"'i+4,£''— 3£'^— n^)
[4511] ^
Ç + 0%084871 . sin. (n'"' t + ^'■' — n")
^(l_^p.-) . 3 + 0',122203.sin.(2n"f— n^f + 2£^' — ^=^— n")
/ + 0',662991 . sin. (3 n^'t — 2n't+ 3 £''— 2 £^— n^')
n', being the longitude of the node of Jiipiter''s orbit on that of Satnrn,
[4512] and n"', the longitude of the orbit of Uranus on that of Saturn. Lastly,
we have, in [3886], the inequality ,t
[4513] 6s' = — 9', 163599 . sin. (2 n'^t — An't + 2s- — 4 s^+dd' 30™ 350-
It follows, from [3932, 3932'], that the terms depending on the square of the
disturbing force, add to the values ot — -, —, the quantities,!
* (2694) The terms of 5 s''' [4511], are computed from [4295i], increasing the
" accents, so that m" may be the attracted planet, and m''' or ot'' the disturbing planet.
f (2695) The inequahty [4513] is the same as [3886], reduced to one term, as in
[4513a] [4282/t— Z].
t (2696) The values [4514,4515], are deduced from [3932,3932'], in the same
[4514a] manner as [4452, 4453], are derived from [3931,3931']. We may also derive [4514]
from [4452], and [4515] from [4453], by the following method. The expressions
VI.xiii.§36.] THEORY OF SATURN. 313
■rf,v ,«ivva^ Sir,^os.(n-n-'-^.sm.(u-é^)h [4514]
de" m'\ y/»'" iSj .^^ ^^ ^^^ ^_ ySn
dt m'\ s/a'"-^ m\ /«" ( t
'.sin.(n— (r)+ ^— .cos.(n— â')^
^7, 6n, being determined as in [3935,3936]. Reducing the functions
[4514, 4515] to numbers, we get,
do?''
[4515]
= _|_ 0',000154; [4516]
==._0\001873. [4517]
[4518]
dt
de"
77
Tire expression [4516] is to be added to the valuesof -p, -^ [4247];
d è^ d è^
and the expression [4517] is to be added to the values of -jj- , — ^ [4247].
Hence we obtain,
^ =. + 0',099894 ;
dt
'^' = — 0% 155136;
dt
— =_9,007165;
dt
1^ = — 19%043372.
dt
[3931,3931'], become the same as [3932,3932'], respectively, by changing, in the second
members, è into à', and multiplying by — . This is equivalent, in the present [4514A]
notation, to the change of ê", into ê", and then multiplying by the factor — .
Therefore, if we perform this operation on the fonnulas [4452,4453], they become ^4514^1
respectively, as in [4514,4515]; in which we must compute (5 7, 5 IT, as in [4452A] ;
and then, as in [4452/t, &c.], we obtain the other quantities [4516, 4517,4518].
We have already remarked, that the inequalities of the motion of this planet are again
noticed by the author, in book x. chap. vili. [9037, &c.], and the subject is also resumed in
the notes on this part of the work.
VOL. III. 79
314 PERTURBATIONS OF THE PLANETS ; [Méc. Cél
CHAPTER XIV.
THEORY OP URANUS.
37. The equation [4460],
[4519] 6r^=^.(l—a?).ô\\
corresponding to Saturn, becomes for Uranus,
[4520] ôr^'^'^.n— a=) . <S V".
r
If we take the mean distances of the earth and Uranus from the sun, for r",
and r", and sup]30se â V"'=: ± 1"= ±0,324, we shall find,
[4521] Ô r" = ± 0,00057648.
l^™^ Therefore we may neglect the inequalities of 5 r", below ± 0,00057 ;
"sLwd. and we shall also omit the inequalities of the motion of Uranus, in
[4522] longitude or latitude, below a quarter of a centesimal second, or 0%081.
Inequalities of Uramis, independent of the excentricities.*
+52%306055 . sin. {n"'t — n'H + e" — £^')
— 0', 190366 . sin. 2(n'U — n'H + s" — ^'")
6«''=(1 + H'") . { — 0',026023 . sin. 3(n"^ — n''H + s- — s^*)
^'^^^^^ ' — 0%003593 . sin. 4 («'" t — n'H + s" — i-')
— 0',000768 . sm.b{n"t — n^H + «'"— s"')
* (2697) Computed as in [42T7«, &ic.], changing the accents on a, ii, n', &,c. to
"•' conform to the case now under consideration.
Vl.xiv. §37.]
+ (1+^-")
THEORY OF URANUS. 315
+2P,371379 . sin. (n't — rûH -\- i' — i")
— 4--,220972 . sin. 2{n''t — n'H + b" — e^')
— 0 ,8621 15 . sin. 3(n''t — n" t + s^ — s")
— 0%2444U9 . sin. 4 (ît' t — n" t + e" — s") \ . [4523]
^ _ 0',08U21 1 . sin. 5 (n" t — n"" i + e" — s^')
_ 0-,028931 . sin. 6 (h" t — 7t" < + s" — £"0
— 0',01 0929 . sin. l{nH — ri" i + s' — s")
— 0%004148 . sin. ^{ift — n'H + ^' — e")
Inequali-
ties inde-
pendent of
thd exGÊQ~
0,0063473160 \ fi"«i«-
+ 0,0048914790 . cos. {ii}'t — rf't + s'''— ^'")
5rvi=(H-^-) . / + 0,00002361 84. cos. 2 (7i"i — n^'^ + ^'^—^'0
+ 0,0000030669 . cos. 3 {n'H — n'H + s-_ £")
+ 0,0000005044. cos. 4(w'''i — nH + s'"— ^'0
+ 0,0023641285
+ 0,0035433901 . cos. (n" t — n'H + B" — s"')
_(.-(l + ^v) _ I ^ 0,0004061682 . cos. 2(n't — n"i + s" — s")
+ 0,0000889425 . cos. 3 (n" t — n^'t + e- — s^')
+ 0,0000255870 . cos. 4> (n" t — n"t + ^''—s'")
Inequalities depending on the first power of the excentricities*
— 1%233612 . sin. (n'" t + s" — ^^')
+ r,25954B . sin. (2 n'H — n'H + 2 8'" — /'— w'")
«jy- = (1 + (^'O . j _ 3.^g3g663 _ sin. (2 r^^ — ?r< + 2 s^' — s'" — t."')
— 0^221997 . sin. (2 w^'f — n'H + 2 s" — a" — to*")
[4524]
[4525]
* (2698) These inequalties were computed in the same manner as those for Jupiter [4525a]
in [4375a].
Inequali-
ties de-
pending on
tho first
tncities.
316 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
— P,402359 . sin. (n^'t + i" — w")
+ 0',214857 . sin. (n" t + s^ — ^^ )
— 0^219788 . sin. (2 n" t — if H + 2 s'— s" — îj^')
&1 \ + 0',878763 . sin. (2 rVt — if H + 2 e>— s^'' — ^^)
— (44S051575 — t . 0',000247) . sin. (_5!v,'^^^,
[4525] + (1 + f^O • ( _j_ (1 49',807764 — i . 0^008306) . sin. (_^S7~J[^^
+ 2%486191 . sin. (3 «"i — 2 ?i^'^ + 3 s"— 2 s" — ^")
— r,642451 . sin. (3 ji^'/ — 2 ?j'7 + 3 s''— 2 s" — ^^)
+ 0',422729 . sin. (4 n"ï — 3 n-' ^ + 4 s" — 3 s' — «^')
— 0',281 800 . sin. (4 n'H — 3 «"i + 4 e'' — 3 s' — ^^ )
+ 0^ 1 26493 . sin. (5 n"'^ _ 4 »^7 + 5 s^' — 4 e^ — ::i")
( _ 0,0016092001 . COS. (2n'H — ift + 26^' — s^— ^^') )
[4526] Ar^'=('l+M''). < >.
^ ^ J + 0,0061835858. COS. (2n^'i — ?r^ + 2 s^'—E"—.^-) C
Inequalities depending on the squares and products of the excentricitîes and
inclinations of the orbits.*
^(132^508872-t0^0145205).sin.(i-37 -^^^^^^^ f^Tg'^^) \
[4537] 6r^"=(l+0.j _!_ i^-,7i3455.sin.(4?i^'i — 2?r^+4/'— 25'— 38''34"'54'')1-
J^oAt ( + 8^380157 . sin. (n" t — ii''H + a' - /' + 88"29"' 40^ j
second
The first of these inequalities must be applied to the mean motion of the
planet, on account of the length of its period. The last of these inequalities,
being connected with the corresponding one in [4523], which is independent
of the excentricities, gives the following,!
[4528] & r^' = (1 + f^') • 23^156281 . sin.(n^^ — ii''' t + i^'— e^' +21^1" 05^.
[4527o] * (2699) Computed as in [4377a, &c.], for Jupiter.
t (2700) The term + (1 + f^"). 2^,37 1379. sin. (n'< — n^V + s'— £>') [4523],
[4528a] being connected with the last term of [4527], by the method used in [4282A — /], becomes
as in [4528].
Vl.xiv..^3S.] THEORY OF URANUS. 317
Tiieu we have,*
ô ,•'■' = _- (1 + (x^) . 0,0007553840 . cos. (3 n'H — n't + 3 s"— 5^+75" 00" 42^. [4529]
Inequalities depending on the poiver.t and products of three dimensions
of the excentricities and inclinations of the orbits.f twîd'""'
order.
S r'- ^ — (1 + M-^) . 0',964688 . sin. (5 n" t — 2n't + ô s"— 2 s' + OS'' 23" 3P). [4.53oi
Inequalities of the motion of Uranus in latitude.
38. From the formula [1030], we obtain,t
6 5^' = (1 + ,.-) . 0%638393 . sin. (ra'^ t + s"— n'^)
Inequali-
ties in the .
latitude.
( 0',9 15741. sin. (w^^ + s^—n") ) [4531]
^ ( + 2',921052.sin.(2w^-i — n^ï+2£"— s''— m)^
n" being here the longitude of the ascending node of Jupiter's orbit upon
that of Uranus, and n' the longitude of the ascending node of Saturn's orbit ^ ^
upon that of Uranus.
* (-2701) Computed as in [4394«, Sic] for Jupiter. r4529al
t (2702) This computation is made as in [4417, &c.] for Jupiter ; changing the accents
to conform to the present notation. [4530a]
t (2703) The terms [4531] are computed from the formula [4295è], altering the accents
to conform to the present case. [4531a]
VOL. III. 80
318 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
CHAPTER XV.
ON SOME EQUATIONS OF CONDITION BETWEEN THE INEttUALITIES OF THE PLANETS, WHICH MAV BE
USED IN VERIFYING THEIK NUMERICAL VALUES.
39. The inequalities of a long period, produced by the reciprocal action
of two planets m, and m', are nearly in the ratio of m'\/a' to — m\/a
[1208] ; so that to obtain the perturbations of this kind, corresponding, in the
motion of m', to those in the motion of m, loe need only to multiply the last
[4532] lyy ^^_ T\ù% result is most to be relied upon, in those cases, in which
the ratio of the mean motions of the two planets is such, as to render the
period of these inequalities great, in comparison with the times of their
revolutions. We shall now, by means of this theorem, verify several of the
preceding inequalities.
The action of the earth on Venus produces, in [4291], the two following
inequalities, whose period is about four years,
àv'^— 1%5 49550 . sin. (3 n"t — 2 w'^ + 3 s"— 2 s'— ^')
+ 4',766332. sin. (3 n"t — 2 n't + 3=" — 2 / — ^").
By multiplying them by — ^^^ryr,? we have, for the corresponding inequality
of the earth,
hv"^ P, 1 33838 , sin. (3 n"t — 'Un' t -\-S i' — 2 .' — ^)
— .3'-,487666 . sin. (3 n"^ — 2 ?î'i + 3 ;" — 2 / — ^■').
We have found, by a direct calculation, in [4307], that these inequalities are,
hv"= r,186390 . sin. (3n'7 — 2 n'i+ 3s" — 2s'_^')
^3%667112.sin. (3h"^ — 2n'i + 3.-" — 2=-'— ^");
[4533]
Venus
and
the Earth
[4534]
[4535]
VI. XV. §39.] VERIFICATION OF SEVERAL INEQUALITIES. 319
which difTers but little from the preceding expression [4534].
The action of the earth upon Venus, produces also, in [4293], the following
inequalitv, whose period is about eight years,
6v' = — \ -,505036 . sin. (5 n" i — 3 n' i + 5 s"— 3 e' + SO'' 54"' 26'). [453G]
Multiplying it by, ,, „, we obtain, for the corresponding inequality of
the earth,
ôv"= l',101277.sin. (5 n"t — 3 n't + ôs" — 3 s' -\-20''54>'"2&) ; [4537]
and, by a direct calculation, we have, in [4309],
6 v"= r, 125575 . sin. (5 n"t — 3n' ^+5 /'— 3 s' + 2 1-; 02"' 18'). [4538]
Mars suffers, by the action of Venus, as we have seen in [4377], the following
inequality of a long period,
è v"'= — 6',899619 . sin. (3 n"'t — n't + 3 £'"—£'+ 65" 26'"15'). [4539]
„T , • , • -, m"V«"' 1 •
Multiplying It by y—- , we obtain, fnd'
™ \/« Mars.
5 v" = 2%078266 . sin. (3 n'" t—7i't + 3 /" — / + 65' 26"* 1 5') ; [4540]
and the direct calculation [4293] gives,
6 v' ^ 2',009677 . sin. (37i"'t — n't + 3 /" — s' + 65' 53"' 09') ; [4541]
which differs but little from the preceding.
Mars suffers, from the action of the earth [4375], the two following TheEarti,
inequalities, whose period is about sixteen years, mIL
S v"'= — 10',1 14699 . sin. (2 n"'t — 7i"t+ 2 s'" — a"— t,'")
+ 5', 1 23062 . sin . (2 n'" t — n" t + 2 a"' — e" — ^") . ^''^''^^
m' \/a"'
Multiplying them by — "^;^;^ ' ^"^'^ obtain, for the corresponding inequalities
of the earth,
6 v" = 2',2293 . sin. (2 n'" t — n" t + 2s"' — i' — •=='")
— 1 -', 1 292 . sin. (2 jj'" t — n!' t + 2 a'" — a" — t^") ;
and the direct calculation gives, in [4307],
[45431
320 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4544]
6 v" = 2% 1 37658 . sin. (2 n'" t — n"t + 2 s'" — s" — ^"')
— P,095603 . sin. (2 n"'t — n"t + 2 b'" — s"— v!') ;
which differ but little from the preceding.
Mars also suffers, on the part of the earth, in [4377], the following
inequality of a long period,
f4545j i v"'= — 4%370903 sin. (4 n!" t—2n"t + 4> s'"— 2 «" + 67'' 49"' OOO-
m"'\/a"'
Multiplying it by iryui we obtain, for the corresponding inequality of
the earth,
f 4546] <5 v" = 0-,9634 . sin. (4 n'" t — 2 n" t + 4 -="' — 2 e" + 67'' 49" 00») ;
which differs but little from the expression, given in [4309],
[4547] 6 v" = 0-,993935. sin. (4 n'" t — 2n"t-{-4> s'" _ 2 s" + 67" 48™ 56').
^"and" The two great inequalities of Jupiter and Saturn, are also to each other,
^""'"' nearly in the ratio of — 77f\/a'' to m'''\/a"', as is evident by comparing
[4548] |-4434^ 4492].
Saturn Lastly, Urauus suffers, from the action of Saturn, the following inequality
and
Uranus, of a long period [4527],
[4549] è V'' = — 1 32',508872 . sin. (3 n"' t — n't+Ss^'— i' — SS" 19" 05").
Multiplving it by — , we obtain, in the motion of Saturn, the
inequality,
[4550] 6 v" = 32-,368 . sin. (3 7^1 — n't + 3 e^'— e" — 88'' 19'" 05^) ;
which differs but little from the inequality, given in [4468],*
[4551] év" = 30%888288 . sin. (3 n'H — n" t + 3 e-— s' — 87" 25™ 07') ;
40. We shall now consider, in the development of R, the term of the
form [3745'],
* (2704) The term here referred to is the last one of the expression [4468] ; which
[4550a] differs, however, a little ; the coefficient being 31',025379, instead of 30%888288 ; and the
constant angf»- 85''34"' 12% instead of 87'' 25"' 07'.
VI. XV. §40.] VERIFICATION OF SEVERAL INEQUALITIES. 321
R = «I'.ilf ". e e'. COS. [ / . (71' ï — n ï + e'— <=) + 2 n < + 2 s — ^ — ^') I ; [4552]
supposing i. (11 — 11!) + 2 n to be very small in comparison ivith n or %'.
We find, in [1286, &c.], that this term produces, in the excentricity e, of
the orbit of the planet m, considered as a variable ellipsis, the following
inequality, which we shall represent by,*
ie=—., /'''t" — .M''\e'.co^.\i.(n't — n i + a'— 0+ 2 n i + 2 s — « — ^'|; [45531
i.{n — n)-\-2n '
and in the position of the perihelion ra, an inequality [1294, &c.], which we
shall represent by,
a^=_ /"'•"" .M<".-. sin. \ i. (n't _ ,i i + 6'— 0 + 2 n i + 2 e — ^ — ^' 1 . 14554]
i.(n — ?i)-|-2 7t e ^
The expression of v contains the term 2 e . sin. Çnt -j- e — w) ; and the
variation of the elliptical elements, produces, in this quantity, the following
expression,!
[4.555]
6v =^ 26e . sin. (n t + ^ — w) — 2e 6-si . cos. (nt+e — «) ; [4.5.5t)]
* (2705) If we take the partial differential of R [1281], relative to e, and multiply
it by -7;—, — . , it will produce the corresponding term of e, represented by (5 e [4553a]
[4553i]
[1286]. Now, if we perform the same operation on the assumed value of R [4552], and
put fx = 1 [-3709] ; changing also i', i, into i, i — 2, respectively, we shall get (5 1
[4553]. Again, if we multiply the same partial differential of R [1281], relative to e, by
— .andt, putting |j. = 1 , it becomes like the expression of cdzi [1294]; and by
the same process we deduce, from R [4552], the expression,
e d-a = — m'.andt . JH''\ e'. cos. \ i.{n' t — nt-\- s — s) -\-2nt -\- 2 s — ra — zs'\. [45.53c]
Dividing this by e, and integrating, we get the part of ra, represented by 0 w [4554] ;
observing that we may consider the terms 31, e, e', of the second member, as
constant quantities, in taking this integral ; always neglecting quantities of a higher order
than those which are retained, and such as depend on different angles.
t (2706) Since v [3834] contains the tenu 2 e . sin. {yit -\- s — ro), it is evident that
the variation of v, corresponding to the increments Se, <)■&, in e, zs, respectively, is as in
[4556]; and by using the symbol JV:=nt-{-s — zs [3702»], it becomes,
8v=:2Se. sin. W — 2 e ô a . cos. W. [4.557a]
Now, if we put, for brevity,
VOL. III. 81
322 PERTURBATIONS OF THE PLANETS ; [M^c. Cél.
which gives in v the inequality,
2 jn ft vt
s •
It follows, from § 65 of the second book, that in the case of i.(n' — n) + 2n
[4557] being very small, the expression of R, relative to the action of m upon m',
contains also a term, of the following form and value, very nearly,*
[4558] R :^m. M'". e e'. cos. {i.(n't — ?t ^ + s'— e) + 2 n ; + 2 s — w — ^^'1 ;
since, by noticing only the two terms of this kind, in R, and R, we have,
as in [1202], very nearly.
[45576] T,^i.{n't — nt-\-^—i)+^nt-\-2-= — zs — zi'; M,=^ '"'" — .M'\e':
i.(n — n)-|-2n
the expressions [4553, 4554] become,
[4557c] (5e = — ^j.cos.T,; e <5 w = — Ji, .sin.T, ;
substituting these in [4557a] , we get,
[4557rf) èv = 2 Jlii. { — cos.Tj. sin. JV -{- sin-Tj . cos. W\ = 2 JW,. sin.(Ti— JF)
= 2 iVii- sin. { i.{n't — nt-\- s' — s) + « t -\- s — ra' |
[455re] ^2M,.sm.\{i~l).{n't — nt + s'—E)Jf-n't+i'-u'\, as in [4557].
* (2707) Using the symbol Tj [45576], we get, from [4552],
[4558a] B. = ni. JW">. e e' . cos. Ti .
Its differential, relative to d [37056 — c], is,
[45586] à.R = rri. M^^\ e e'. (i — 2) . n dt . sin. Ti ;
substituting this in the differential of [4559], which gives m'. d'iî'=: — 7n.AR, and
dividing by m', we obtain,
[4558c] d'iî'= — m . M'-^K e e'.(i —2) . n dt . sin. Tj .
Now, i.{n' — n)-\-2n, being very small [4557'], we have, very nearly,
[4558rf] {i—2).ndt=^in'dt;
hence,
[4558e] à'R= — m. iVi">. e e'. in'dt. sin. Tj .
Integrating this, relative to the characteristic d', which does not affect n t r3982al, we
[4.5.58/] , r^rron
•■ ■' ^ obtam, as m [4558],
[4558g-] ^'= "* • -^^"'- ^ ^'- COS. Tj .
VI.xv.§40.J VERIFICATION OF SEVERAL INEQUALITIES. 323
m.fàR+ m'.fd' i?'= 0 ; [4559]
therefore we have, in v', the inequality,*
S v'= .,!"''"'!'' -ili'". e . sm.l(i—l).(n't — 7it -{- i' — s) + nt 4-s — 7,\. [4560]
t.{n' — «) + 2n ( V / \
These two inequalities of v and v' [4557,4560], are in the ratio of m'.e'.\/a' [4560]
to ?ft.e.y/« ; so that the second may be deduced from the first, by multiplying
the coefficient of the first by "1;}^^^ [4660«-]. ^4560"]
•^ ?»'.\/« e' *- '^
The quantity 5n" — 3 n' being small, in comparison with n' or n",
we have, in v' [4557], by supposing i ^ 5, an inequality depending on the
argument 5 n" ^ — 4 n' t + 5 s"— 4 /— ^" ; and in v" [4560] , an inequality [4560'"]
depending on the argument 4n"i — 3n't + ^s" — 3s' — -a'. The first of
these inequalities is, by [4291],
6 v' = 2^,196527 .sin. (5 n"t — ^n't + b f— 4 ^'— ^")- [45Ci]
Multiplying its coefficient by / „ . -, we have, for the earth, the ve"j
"* V*^ ^ thcEarlh.
inequality,
iv"=0\6580.sm.(^n"t — 3n't-\-4^i" — 3e—zs'). [4562]
By a direct calculation, we have found, in [4307], this inequality to be,
Ô v" = 0',722424 . sin. (4 n" t — 3n't + 4^s" — 3e' — ^'); [4563]
which differs but little from the preceding.
* (2708) We may obtain ô v' from R', by a similar process to that used in the two
preceding notes ; or, more simply, by derivation, in tlie following manner. If we [45600]
change, in [4552], i, m, a, n, e, v, &ic. into — i-\-2, m', a', n', e', v' he. respectively,
without altering Jlf"', R changes into R [4558a,^], and the factor r.(»i' — ti)-\-2n, becomes,
(— ' + 2).(m — 7«') + 2n'; [45606]
which, by reduction, is easily reduced to its original form ; so that the angle T, [45576]
remains unaltered. The factor M^ [45576], changes into
^^-d^^^n-^'"''-^-^ [«60c]
W changes into W [.3726a]; and the second expression of 5 d [4557 </], becomes as in
the first of the following expressions of lî v', which, by successive operations, is reduced to
the form [4560e], as in [4560] ;
324 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
In like manner, 4 n"' — 2 n" is rather small, in comparison with n"
or n'" [4076/i] ; and if we suppose i = 4, we obtain in v" [4557], an
inequality depending on the argument
4 w"'i — 3 n"t + 4 s'" — 3 s" — ^"' ;
[4564] and in v"' [4560], an inequality depending on the argument
Sn"'t — Inl't + 3 £'"— 2 e"— ra".
The first of these inequalities is, by [4307],
^45g5^ & v" = OS8071 1 1 . sin. (4 n"'t — 3 n" t + 4 e'" — 3 -="— ^"').
m"\/a" e"
Multiplying its coefficient by ,„ ,— . — [4560"], we get, for Mars, the
The Ea- til 1 J to J m \/ o! t
and *^
*'^" inequality,
[4566] ^ «'" = 0',661446 . sin. (3 n'" ^ — 2 ?î"/ + 3 £'" — 2 e" — ^") ;
and by direct calculation we have, in [4375],
[4567] <s v'" = 0', 846004 . sin. (3 n'" i — 2 n" « + 3 .'" — 2 e" — i.") ;
the difference is within the limits of the error which may be supposed to
[45681 exist, taking into consideration, that the ratio 4 ?i"' — 2 m" to n'", instead
of being very small, is nearly equal to |.
41. It also follows, from § 71, of the second book, that if i.{n' — n)+2 n
[4569] ^^ ^''^^'2/ s»«a// in comparison with n', the inequality of m, in latitude, depending
on (i — \).{n't — nt-{-s — s) -{- n' t -\- s' , is to the inequality of m', in
[45<39'] latitude, depending on (i — \).{n't — nt^s — i)-{-nt + i, in the ratio
of m! \/a' to — m\/a.*
[4560(i] 'î d' = 2 Ma . sin. (Tj — W) = 2 .¥o . sin. \ i.{n't — n < + e'— s) + 2 « i + 2 £ — n't — s'- ra ?
r4.5<jOel "^^ 2^2 • sin.{(i — \).{n' t — nt-\-i' — i) -\- nt-\- s — la^.
Dividing the value of ^v' [4560] by that of &v [4557], we get, successively, by using
^"^^^^■f^ an = a-i, a'n'= «'-* [3709'],
i5«' m.a'n' e m.a'~i e m.ai e ■ r^rz-^wn
In applying this formula to numbers, we must vary the accents in the elements, so as to
conform to the notation used in this book, as is done in [4560"', &;c.].
[4569a] * (2709) The inequality of s, here referred to, is given in [1342] ; that of s', depending
VI. XV. §41.] VERIFICATION OF SEVERAL INEQUALITIES. 325
If we suppose i = 5, we shall have, in the motion of Venus in latitude venu»
[4569o-, 4295], the inequality [4295], thlEani..
ôs' = — 0',312535 . sin. (5 n"t — 4 n't + ôi"—4>e'—è'). [4570]
Multiplying the coefficient of this inequality by ■;yy^ [4569'], we get,
in the motion of the earth in latitude, the inequality [4569/],
& s" = 0',22869 1 . sin. (4 n" i — 3 n' Ï + 4 s" — 3 s' — 0 ; f"*^^^ ^
and, by direct calculation, we have found, in [4312], the inequality,
6 s" = 0S234256 . sin. (4 n"t — 3 ti' Ï+4 s"— 3 s'— 6') ; [4572]
which differs but little from the preceding.
on the same angle, is similar, the accents being changed so as to adapt them to the value of
s'. Instead of this formula, we may use [4295J], observing that the second line of this
expression is used in computing the inequalities which are taken into consideration in ^ '
[4569 — 4576]. The expression of 5 s, deduced from this part of [4295i], may be
simplified; because the divisor n^ — \n — i.{n — «')P' '^^'^Y ^^ reduced to the form [45G9c]
(' . ()i — «') .\i .[n' — n) -\- 2 n] . Hence this part of i5 s becomes,
_B(.-i)
6s=im'.n^.a^a'. -^ tttT-: --r-r.y .sin. h'.(»i' i — ?i i + s'— s) + n <+ s — nj ; [45G9d]
7 being the inclination, and II the longitude of the ascending node of m', upon the orbit
of m. This expression may be simplified, from the circumstance, that, in the terms here [4569e]
taken into consideration, the divisor i.{n — n') is very nearly equal to 2n [4569].
Substituting this, and ?irt^=l, in [45^9 (Z] ; making also a slight reduction in the ["^^'^^l
arrangement of tlie terms depending on i, we get,
ôs=+im'.^a'.{aa')i (■_l)_(j!^^"^^>^,^ -7 •sin-K''-l)-K^- ^^ i + s'-s)-\- n't + b'-UI [4569g:]
Changing the elements ?«, a, n, s, U, &c. into m, a', n', s', n + 180'^. Sic. M,-pq.-i
respectively, and altering the sign of i — 1, which does not affect 2i"~'' [956,956'],
we get,
5 s'=— my a .{act') . rr—rr:-, -—, • 7 .sin.U?:— l).(?l'^— « ? + /— e) + ?i C + £ — H^ [4569i]
Hence we evidently perceive, that S s is to as' as m'\/a' to — m\/a, as in [4569']. [4569A;]
Now, the values n', n" [407 6A] make 5 n" — 3 ?i' quite small, in comparison with ?i'.
This corresponds with the value assumed for i.{n' — 7))-{-2n [4569], supposing i = 5 ; [45G9i]
hence we get [4.570 — 4572]. In like manner, 3 ?i'' — n" [4076A], is very small, in
VOL. III. 82
326 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4573] The quantity 3 n" — if is small in comparison with n"' ; therefore,
[4573] by making i =: 3 [456%, z], we obtain in ôs", an inequality depending on
3 n''t — 2 7f t -{- S s''' — 2 i" ;
and in 6 5", an inequality depending on
2n''H — n''t + 2!'' — s\
The first of these inequalities is, by [4511],
[4574] 6 s" = 0',662991 . sin. (3 n^'t — 2 n^t + 3 s"'— 2 1^ — n''').
n"' being the longitude of the ascending node of the orbit of Uranus upon
[4574'] that of Saturn. Multiplying the coefficient of this inequality by -^y— ,
and we obtain in ôs"', the inequality,
Uranus. ' ^ J '
[4575] 65"= — 2'',714213. sin. (2ra'"^ — w^^ + 2£^'— s"— n^');
and by [4531], this inequality becomes, by putting n^ = n"' + 1 80''
[4531', 4574'],
[4576] is'' = — 2',921052 . sin. (2 n^'t — nU + 2 1"'— e" — n^') ;
which differs but little from the preceding.
42. It follows, from § 69, of the second book, that if we suppose
[4576'] i' n' — in to be very small relatively to n and n', and represent by,*
[4577] R =3r m'.P. sin.(i'n't — int + i'e'—is)-}-m'.P'.cos.(i'n't—int + i's'—is),
the part of the development of R, depending on the angle
i'n't — i nt-\- i's' — i s ;
it will produce, in 6v, the inequality,
..-„„ , comparison with ?r or n"' ; and this comes under the form [4569], by putting f = 3;
hence we get [4574—4576] ; observing in [4576], that n^ = n" + 180<f.
* (2710) Using the value Tg = i'n' t — i nt + i'^ — i s [4019a], and (j, = 1 [3709] ,
we find that the tenns of i2, Se, eSzs, which correspond to each other in [1287,1288,1297],
become,
[45776] -R = "*'• P- sin- ^9 + '">''• P- c°s- ^9 ;
14577c] , e = "^4^ . 5 _ (^) . Sin. T,- (f) . cos. T, \ ;
m—m ( \de J \de J )
2 m. an , , . .
^V=rr-r-r7.-< , h [4578]
Vl.xv.H2J VERIFICATION OF SEVERAL INEQUALITIES. S27
— ( — V COS. {i'n' t — int-\-i' ^ — is — nt — s + ra)
and in 6 v', the inequality,*
, , ( — ( -—A . COS. (i' n' t — int4-i's' — is — n't — s' + ra')
, 2m. an' \ \de'J ^ ^ ' \
6v'^ T-; ' ^ , >. [4579]
^^" = ^$S. • 1 O •^°^- ^«- (S) • ^'"- ^^ 1 • f4577d]
Substituting these in 8v [4556], using for brevity, W=znt-\-s — w [3702a], and
reducing, by [22, 24] Int. we get, as in [4578],
„ , (— ("— Vrsin.Tg.sin.fF+cos.rg-cos.fF),
tn-in }_|./^V(sii,.2'9.C0S.fr— C0S.r9.sin.fF)<
2m'. an ( /dP\ ,_, „^, , /dP'\ . ,^ „n >
* (2711) Proceeding in the same manner as in [4558a — c], and using Tg [4577a],
we have,
àTg= — indt, à'T^==i'n'dt; [4578a]
hence the differential of It [45776], relative to the characteristic d, becomes,
dK = — m'.in.{P.cos.T9 — P'.sin.rg^ [45786]
Substituting this in m'.à'R'= — m.dR [4558i — c], we get,
d' «' = m . i n . { P. cos. Tg — P'. sin. Tg } . [4578c]
Integrating this, relatively to d', and observing that the divisor i'n' is, by hypothesis,
very nearly equal to in [4576'], we get, for the corresponding terms of R', depending
on the angle Tg, the following expression ;
R' = m.{P. sin.Tg + F. cos. Tg]. [4578rf]
From this value of R' we may compute 5 v', in the same manner as we have found S v
[4578], from R [45776]. It will, however, be rather more simple to use the principle of [4578e]
derivation, by observing, that if we take the differential coefficient of R [4577è], relative
to e, multiply it by 2andt, then take its hitegral relative to t, and change Tg into
Tg — W, it will become equal to S v [4577e]. In like manner, if we take the differential Mgyg^-,
coefficient of JÎ' [4578rf], relative to e', multiply it by 2a!n'dt, take its integral relative
328 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
It follows also, from ^71, book ii. that the same terms of R [4577]
produce, in 6 s, the inequality,*
[4580] 5 s = .—. :
' — ^ .cos .(i'n't— i u t4- i' s' — i I — n t — e-|- n) )
— (--\ . sin. (i' 7i't — int-^i's' — is — nt — s + n) \
y being the tangent of the respective inclinations of the orbits of m and m',
[4580'] and n the longitude of the ascending node of the orbit of m' upon that
of m [42956— c].
If we increase the argument of the inequality of 6 v [4578], by
[4580"] nt -{- s — ra, and multiply its coefficient by e ; also, if we increase the
argument of the inequality of 6v' [4579], by n't + s' — ■us', and multiply
[45811 its coefficient by — ^-— . e' ;t lastly, if we increase the argument of the
■■ ■' m \/a "^
inequality of 6 s [4580], by nt-\-i — n, and multiply its coefficient by
— 2-/, the sum of these three inequalities will be.
[4582]
Qm'.an
'+^•(f)+«■•(")+^Q^™'■<'■■»■'-'•»'+^"■-")l
to t, and afterwards change Tg into Tg — TV [.3726a], it will produce the following
expression of & v', which is equivalent to [4579] ;
l"»^l *"'=l5^ h (© ■ »■ (''.-"'■) + Q • ''■•■ (T,-rr)f
* (2712) If we put, for brevity, T.2 — i'n't — int-{'A—gè;, also 7 = tang, ip/
[4580a] ["1337'^ 3739] . the assumed value of R [1337"] becomes, R = m'.k .jK cqs.%.
[45806] Substituting this in the expression — fi — ] . andi, we find that it becomes equal to the
expression of s or 5 s [1342] ; provided the angle T, be decreased, after the integration,
by the quantity v — Ô/, or by the angular distance of the body m from the ascending node
of the orbit of λ' upon that of ?« [1.337']. In the present notation v — 0/ is represented
by the quantity nt-\-s — n, neglecting terms of the order e [429.5i — c]. The same
process being performed upon the assumed value of R [4577], produces the expression of
6s [4580].
[4581a] t (2713) This factor is equal to ^^^'-.«'[4560/].
VI. XV. §42.] VERIFICATION OF SEVERAL INEQUALITIES. 329
Now, P and P' are homogeneous functions of e, e', 7, of the dimension
i' — i, and i' is supposed greater than i ; therefore the preceding function
is equal to,*
'2m!.an.{i'—i) ^ ^_p ^^^(■^^,f_ j-,j^_)_^v^.,_ • ^^ _^p,_ sin. {i'n't—int+i's'—i 0 1 . [4583]
Now we have, in àv, the inequality, [1304],
'^'^— p""^ — ^ • S -f*- ^°^- (*'"'^ — î'ni-f iV — i £) — P'. sin. (iV/ — int+i's' — is) \ ; [4584]
hence it follows, that if we represent by
6v = K. sin. (i' n't — i nt-\-i' i'—iB — nt — i-\- O), [4585]
the inequality of 6 v, depending on the angle i'n't — in t + i'^' — i s — n t — £ ;
and by
5 v' = K'. sin. {i' n' t — int + i s' — z ; — n' t — s'+ O),
the inequality of &v', depending on the angle i'n't — int-\-i'i' — z's — 71' t — s'; [4586]
lastly, if we represent by
6s = K". sin. (i! n't — int-^i' s — z s — nt — e + 0"), [4587]
the inequality of à s, depending on the angle i'n't — i nt + i's' — is — nt — s,
we shall have,t
Ke . sin. (i' n't — int -{- i' i — is — « -f- O)
+ '^'. K' e'. sin. (i' n' t — int + i' s' — is— ^'+ O')
—2 K" 7 . sin. (i' n't — int + i's' — is — n-{- O") ^''^^^^
= _ U^Illl.H. ^''"'~"'^ . sin. (if n't — i nt + i' s' — is + Q);
* (2714) From [957'"] it appears, that any part of B, depending on angles of the
form i'n't — int, must be composed of terms in e, e', 7, of the orders i' — i, ^ ^^
i' — Ï + 2, &IC. ; and by neglecting all, except the first, on account of their smallness, they
must be of the order i' — i; and therefore homogeneous in these quantities. Now, if we r-rpoii
put, in [1001a], a = e, «'= e', a"=y, m = i' — i, and then, successively, .^''':= P,
d^')= P', we get,
«•(f)+''-(S)+-(f)=P-^)-
Substituting these in [4582], we obtam [4583].
t (2715) The first member of [4588] is equal to the sum of the inequalities ôv, èv',
VOL, III. 83
330 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4588']
n
IS
6v = H. sin. (i' n' t — i n t -\- i' s' — i « + Q) being the inequality of 6 v
depending on the angle i' n' t — int + i' s — i^ .
The quantity 5 n' — 2 n [4076/i] is very small in comparison with
^'^^^^"^ and we have, in àv [4282], the inequality,
[4589] àv = l %690443 . sin. (5 n' < — 3 n i + 5 s' — 3 s + 43" 1 8 "■ 32').
^,^^^^^, The inequality 6 s [4283'], depending on 5n't — 3nt + ôs' — 3e,
Venus. insensible; and we have, in 6v' [4293] , the inequality,
[4590] Sv' = — 0',333596 . sin. ( 4 n' < — 2 n/ + 4 s' — 2 ^ — 39' 30'" 30').
Lastly, we have, in ô v [4283], the inequality,
[4591] 6t)= 8',483765 . sin. (.5 n't — 2nt + 5.-' — 2.- — 30" 1 3- 36').
In this case ?''= 5, i = 2 [4584,4591]; and we have, by what precedes
[4585 — 4591], the following equation of condition;
r,690i43. e . sin. (5 n' t — 2nt + 5 s' — 2 i —^ + 4^3' 18'" 32')
[4592] — 0',333596.e'. -"'^.sin.(5n'i — 2ni + 5£' — 2s_^'_39''30'"300
= _ 8',483765 . ■^"'"'^"^. sin. (5 n7 — 2 n^ + 5 / — 2 . — 30" 13"' 360-
71
The first member of this equation is,*
[4593] 0%359753 . sin. (5 n' i — 2 w i -f- 5 e' — 2 s — 28" 27"" 33') ;
the second member is,
[4594] 0',3605 . sin. (ôn't — 2nt + ô s'— 2 s — 30" 13'" 36') ;
and their difference is insensible.
Ss, [4585,4586,45871; multiplied respectively by e, — - — . e', and — 2y; the
[4588a] '»•«
arguments being also increased by nt-\-s — w, n't-^e' — ts', nt-\-s — IT, respectively,
according to the directions in [4530"— 4531]. Now, it is shown, in [4530"~4583J, that this
sum is equal to the expression [458.3], which is the same as that of &v [4584], multiphed
by — — . ( ) ; and if we suppose this expression of Sv to be reduced to the
[45886] '^^ \ " /
form [4588'], this product will be represented by the second member of [4588].
* (2716) This is easily olrtained, by reducing the two terms of the first member
[4593a] of [4592] into one, by the method [4282/t — /], after substituting the values m, m',
a, a', he. [4001,4079,4080].
VI. XV. §43.] VERIFICATION OF SEVERAL INEQUALITIES. 331
We may verify, by the preceding theorems, many of the corresponding
inequalities of Jupiter and Saturn ; but as all the inequalities of these two [4594']
planets have been verified several times, with much care, by different
computers, this last verification is unnecessary.
43. The inequality of m, produced by the action of m', and depending
on the argument n' t + -' — ^'j is expressed as in book ii. ^ 50, 55, by,*
à V = ^-^ — - . (0,1) . e'. sin. (n' t + i' — «')• [4595]
The inequality of ?«', produced by the action of m, and depending on the
argument nt-\-£ — ct, is,
6 v' = " . (1 ,0) . e . sin. (Jl / + e — ra). [4596]
n.{n- — n-') ^ ' ^
[4596']
The coefficients of these two inequalities are, therefore, in the ratio of
— (0,1) . 7i\ e' to (\,0).n'\ e; now we have, in [1093],
therefore, if we put Q for the coefficient of the inequality 6 v [4595],
we shall find, that the coefficient of the inequality ôv' [4596], will be
represented by,
f^.î.Q [4595/]. [4598]
m. a ■' e
* (2717) The term of 5 ti depending on n't-{-s' — zs', is deduced from that in [1021],
depending on G'', by putting i := 1 ; whence we obtain,
Sv='^ . G<". e'. sin. {71' t + s'— ^'}. [4595„]
Now, from [1018, 1019, 1073], we have, in the case of i=l,
^"=-«H-^)-*«H7^)=™r„-(«'')= [45956]
G-^-,£^..^-==-„7:(^)-(0.1)- [4595.]
Substituting this value of G<'\ in 5v [4.595rt], it becomes as in [4595]. The value of Sv'
[4596] may be directly computed in a similar manner ; or it may be obtained more simply by
derivation from [4595] ; changing to, a, n, e, he. into m', a', n', e, &lc. ; and [4595rf]
the contrary ; observing, that by these changes, (0,1) becomes (1,0), according to the
332 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
The inequalities of this kind have been verified, either by means of this
equation of condition, or by that of the preceding expression of Q. Thus,
the action of Jupiter produces, in the earth, the sensible inequality [4307],
[4599] 6 v"= — 2',539884 . sin. (n" t + s"— ^").
This inequality, by what precedes, is represented by [4595],
[4600] 6 v"= ^^,^^.,_^..,^ . (2,4) . e\ sin. (n- 1 + a-- «-) ;
TheEarth ^^d wc havc (2,4) =6^947861 [4233]. If we substitute this, in [4600],
juiuer. also the values of n", tf\ é" [4077, 4080] ; then multiply the result by
the expression of the radius in seconds, we shall obtain,
[4601] âî)" = — 2'-,5401.sin.(?i'-'/; + e"'— ra'^).
The action of Uranus upon Saturn, produces, in the motion of Saturn, the
inequality [4466] ,
[4602] 5 î;v ^ _ p^oil647 . sin. {itH + i"— :='')•
Saturn
and
Uranus
„v „vo pV
Multiplying its coefficient by ^v73"' Ti [4'598], we obtain, in Uranus,
the inequality,
[4603] 5z;'i=0%214852. sin. (m'^ + s' — ^') ;
and the direct calculation has given, in [4525],
[4604] à v"' = 0',2 14857 . sin. (nU + i"— ^').
notation in [1085, &;c.]. Comparing the values of o v, & v' [4595, 4596] , we get the first
[4595e] expression of [4595/] ; and by substituting the value of (1,0) [4597]; also n^=zcri,
_9
n'^=a' ^ [-3709'], we get successively the last expression [4595/], which is equivalent
to [4598];
. , (1,0) n'^e . m\/a n'^e m . a^ e
rdw^n à v= • . .01) = ; — , • —r-, .ov = — -— — r- . -, .ov.
L^^^^.'J (0,1) Ji3e' viVa' n3e' m'.a'^ e'
[4600o] * (2718) The expression [4600] is similar to [4595], changing m, m', &ic. into m",
m'", Stc.
Vl.x%'i. 5,41.] ON THE MASSES OF THE PLANETS AND MOON. 333
CHAPTER XVI.
OiN THE MASSES OF THE PLANETS AND MOON.
[4604]
44. One of the most important objects in the theory of the planets is the
determination of their masses ; and we have pointed out, in [4062 — 4076'],
the imperfections of our present estimation of these values. The most sure
method of obtaining a more accurate result, is that which depends on the
development of the secular inequalities of the motions of the planets ; but
until future ages shall make known these inequalities with greater precision,
we may use the periodical inequalities, deduced from a great number of
observations. For this purpose, Delambre has discussed the numerous
observations of the sun, by Bradley and Maskelyne ; from which he has
obtained the maximum of the inequalities produced by the actions of Venus, [4604"]
Mars and the moon. The whole collection of these observations of Bradley
and Maskelyne, makes the maximum of the action of Venus greater than
that which corresponds to the mass we have assumed for Venus [4061], in
the ratio of 1,0743 to 1 ; hence the mass of Venus is ■gyVe^-aa of that of [4605]
the sun. The observations of Bradley and Maskelyne, when we take them Mass or
Venue.
separately into consideration, give nearly the same results ; therefore, it is
probable, that this estimate of the mass of Venus is not liable to an error of
a fifteenth part of its value. [460.5']
Hence it follows, incontestably, that the secular diminution of the
obliquity of the ecliptic approaches very near to 154"=49',9. To reduce
it, as some astronomers have done, to 105"= 34% we must decrease the [4606]
mass of Venus one half;* and this is evidently incompatible with the [4606]
[4604"
* (2719) This appears, by substituting q"= — M', t = \00 [4606], in [4074c];
whence we get, very nearly, — .34'= — 50' — 31V' j consequently, i>-'= — h, nearly.
VOL. III. 84
[4606o]
334 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
observations of the periodical inequalities, produced by Venus, in the motion
[4606"] of the earth. The best modern observations of the obliquity of the ecliptic
are too near to each other, to determine this element with accuracy. The
observations of the Arabs appear to have been taken with much care. They
[4607] made no alteration in the system of Ptolemy ; but directed their attention
particularly to the perfection of the instruments, and to the accuracy of their
observations. These observations give a secular diminution of the obliquity
r.„«-„ of the ecliptic, which differs but very little from 154"^49%9. This
[460/ ] .... .
diminution is also confirmed by the observations of Cocheouking, made in
China, by means of a high gnomon ; and it appears to me, that these
observations may be relied upon for their accuracy.
Delambre has also determined, by a great number of observations, the
[4607"] maximum of the action of Mars upon the motion of the earth. He has
Mass of found this action to be less than that which corresponds to the mass we
have assumed for Mars [4061], in the ratio of 0,725 to 1 ; making the
[4608] mass of Mars ^TTis-a^? of that of the sun. This value is probably not
quite so accurate as that of the mass of Venus, because its effect is less ;
but, as the data [4076], from which we have determined the mass of Mars,
[4608] in [4075, &c.], are very hypothetical, it is important to ascertain the error
which might result from this cause, in the theory of the sun's apparent
motion. Now, the observations of Bradley and Maskelyne, combined
together, or taken separately, concur in indicating a diminution in the mass
of Mars ; therefore, we shall decrease the preceding inequalities, produced
[4609] by Mars, in the earth's motion, in the ratio of 0,725 to unity.
These changes, in the masses of Venus and Mars, produce sensible
alterations in the secular variations of the elements of the earth's orbit.
Longitude We find the longitude of the earth's perihelion to be represented by the
'p^ihe'Ln. following expression ;*
[4610] Long, perihelion © = ^"+ 1 . 1 P,807719 + t\ 0',00008 16482 ;
the coefficient of the equation of the centre of the earth's orbit is
represented by.
* (2720) The expression [4610] is computed as in [4331], changing the masses of
[4610a] Venus and Mars, as in [4605—4608]. The formulas [4611,4612] are computed in hke
manner as [4330, 4332], respectively.
VI. xvi. •§, 44.] ON THE MASSES OF THE PLANETS AND MOON. 335
Coeff. equat. centre © = 2^— ï.0',171793 — ^-.0',0000068194. [461 1]
Lastly, the values of p" and (f [4332], become,
p'= t. 0'-,080543 + f. 0',000023 1 1 34 ; ^^^^2
q"= — ^0%521142 + i% 0',0000071196.
Hence it follows, from [4074c, 461 3«], that the secular diminution of the
obliquity of the ecliptic, in this century, is equal to 52',1142.* Using these
data, we find, by the formulas of ^ 31, f
^^t.\ 5b", 5921 + 3M 1 0 1 9 + 42556",2 . sin. {t . 1 55",5927 + 95°,0733)
— 73530",8. cos. (i.99", 1 227) — 1 7572",4. sin. {t . 43",0446)
[4C14]
= t . 50%412 + 2H7"- 57^+ 13788^2 . sin. {t . 50%412 + 85''33"'570
— 2382.3-,98.cos.(^.32%l]58)— 5693%5. sin. (i.l3^9465) ;
tPixcilT
. —^v, ,.... ^v..^ ,.. -. , ..y...^^ ,^.^. , .^ ,^.^^j """'-J
[4615]
+ 5082",7. cos (^ . 43",0446)— 28463",6 . sin. (^ 99", 1227)
Corrected
= 23'' 28*" 17%9— 1191',2 — 5892',8.cos. (i.50%412 + 85'* 33" 57^ o7'.r
precessioii
+ 1646%8 . cos. (i. 13%9465)— 9222»-,2 . sin.(/.32'-,1158); JJJS
P-
tic for the
year (,
ailer the
|'= i .155",5927 + 3°,11019 — 3°,11019 .cos.(ï.99",1227) etocV
1750.
— 14282",3 . sin. (t . 43",0446)
[4616]
= ï.ô0%4120 + 2'^47'"57^ — 2'*47'"57^cos. (L32%1158)
— 4627^5 . sin. (M3^9465) ;
pApparent"!
L orbit. J
V'= 26°,0796 — 3676",6 .^1— cos. (< . 43",0446) ^
— 10330",4. sin. {t. 99", 1227)
[4617]
= 23''28'"17%9— 119P,2.p— cos.(Ll3%9465)|
— 334^,05 . sin. (^.32%1 158).
* (2721) The chief term of the value of q" [4612] is — i.0%521142, and by
putting < ^100, it becomes q"^ — 52',1142. This represents, by [4074a — c], the secular [4613a]
variation of the obliquity of the ecliptic, corresponding to the second formula [4612]; in the
original work it is printed 160",85=52',1154, and it is thus quoted in [3380n].
t (2722) The formulas [4614 — 4618], are computed in precisely the same manner as
336 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
The increment of the tropical year, counted from 1750, is, then,
Increment
of the represented by,
[4618] Increment of the year = — 0'"'^000086354 . { 1— cos. (t . 13',9465) I
— 0''=^000442198 . sm.(t . 32^1158).
Hence it follows, that, at the time of HIpparchus, the tropical year icas
[4618'] 10^9528 sexagesimal seconds longer than in 1750. The obliquity of the
ecliptic was then greater by 955%2168. Lastly, the greater axis of the sun'' s
[4618"] orbit coincided with the line of equinoxes, in the year 4089 before our era ;
it tvas perpendicular to that line in 1248.
The mass of the moon has been determined by the observations of
the tides in the port of Brest ; and, although these observations are
[4619] far from being so complete as we could wish, yet they give, with
considerable precision, the ratio of the action of the moon, to that of the
sun, upon the tides of that port. But, it has been observed, in [2435 — 2437'],
that local circumstances may have a very sensible influence on this ratio, and
also on the resulting value of the moon's mass. Several methods have been
pointed out, in the second book, to ascertain this influence ; but they require
very exact observations of the tides. The observations which have been made
at Brest, leave, in their results, such a degree of uncertainty, as makes us fear
that there may be an error of at least an eighth part, in the value of the
moon's mass. Indeed, the observations of the equinoctial and solsticial tides,
^^'^^^^ seem to indicate, that the action of the moon upon these tides is augmented
one tenth part, by the local circumstances of the port. This will decrease,
[4621] '^y one tenth, the assumed value of ihe moon's mass ; and, in fact, it appears,
by several astronomical phenomena, that the assumed value [4321] is rather
too great.
The first of these phenomena is the lunar equation, in the tables of the
[4692] sun's motion. We have found, in [4324], 8',8298 for the coefficient of this
inequality, supposing the sun's parallax to be S^S [4322]. It will be
[4357— 4360, 4362], ahering the masses of Venus and Mars, as in [4605,4608]. We
have previously spoken of this change of the masses of tliese two planets, in [3380rt,&;c.],
[4614a] ^^^ j^^^g ^jg^ given the formulas of Poisson and Bessel [3380p,y], for the determination of
the precession and the obliquity of the ecliptic.
Vl.xvi. §44.] ON THE MASSES OF THE PLANETS AND MOON. 337
8',5767,* if the sun's parallax be 8^56, which is the value deduced [4622']
from the lunar theory, as will be seen in the following book. Delambre has
determined the coefficient of this lunar equation, by the comparison of a very
great number of observations of the moon, and has found it equal to 7',5. [4623]
If we adopt this value, and also the second of the above estimates of the
sun's parallax, which several astronomers have deduced from the last transit "lass.
of Venus over the sun's disc, we find the mass of the moon to be ^-i ^ of [4624]
the earth's mass [4622&].
The second astronomical phenomenon is the nutation of the earth's axis.
We have found, in [3378a], the coefficient of the inequality of the nutation
to be equal to 10',0556;t supposing the mass of the moon, divided by the [4625]
cube of its mean distance from the earth, to be equal to triple the mass
of the sun, divided by the cube of the mean distance of the earth from the
sun [2706]. This makes the mass of the moon equal to ^i^^ of the earth's [4626]
mass [4321]. Maskelyne has found, by the comparison of all Bradley's
observations on the nutation, that the coefficient of this inequality is equal
* (2723) The coefficient of this inequality, neglecting its sign, is -rv-—, multiplied by
the radius in seconds 206265' [4.3141; and by substitutinsr — = f3-r-,and — =. ^.^'
■- -' ' •' ^ M 58,6 ' r" 3454"
1 m'snai- [4622a]
[4321,432.3], it becomes ^^ X ^^A- ' ^ 206265'. Putting this parallax equal to 8',8,
the coefficient becomes nearly equal to 8%8298 [4324] ; and by using the value of the
parallax 8", 56 [5589], the coefficient becomes 8',58 nearly, as in [4622']. To reduce
this to 7', 5, the value obtained by Delambre, we must decrease the moon's mass in the
7 5 1 1
ratio of the numbers 7', 5 to 8', 58, so that it will be equal to — ^x— ^— =7;=-, [46226]
8,58 58,6 6/
instead of -p^rK'f given by the author in [4624].
t (2724) The coefficient 31",036 = 10%0556 is computed, in [3376e], from the
formula — — . — = 10',0556 ; in which X=3 [3376,-3079] represents the assumed [4625a]
ratio of the lunar to the solar force on the tide. This value of X is used, in [4319], in
computing the value of m [4321,4626]. Now, substituting X = 3, in [4625a], we
obtain,
~ = \x 10',0556 = 13%4074 ;
VOL. HI. 85
338 PERTURBATIONS OF THE PLANETS; [Méc. Cél.
[4627] to 9',55 ; and this result makes the moon's mass equal to y'y of the earth's
mass.
Lastly, the third astronomical phenomenon is the moon's parallax. We
shall see, in [5605], that the constant term contained in the expression of this
parallax, when developed in a function of the moon's true longitude, is
[4628] 3427%93; supposing the moon's mass to be ^^-g- of the earth's mass. Burg
has computed this constant term, by means of a very great number of
[4629] observations of the moon. He finds it equal to 3432',04 [5605] ; and, by
the formulas given in the next book, this result will be found to correspond
[4629'] with a mass of the moon, which is equal to ^l-g- of that of the earth.*
[46256]
substituting tliis value in the first member of the equation [4625rt], we get — -— . 13',4074,
for the mitation, corresponding to any assumed value of X. If we put this equal to the vahie
Q'jSS, obtained by Maskelyne [4627], we get,
X 9,5500 , 9,5500 ^ ,.„ . , r , o j i.
[4 625cl r~rT^= .o ,r.~,. ; hence x= — -— -r = 2,4i6, instead oi X=:3, used above;
^ !-{->■ 13,4074 cf,o574
and as the mass of the moon is proportional to X [3079] , it mil be reduced, from r^-^
[^•^^^1]'^° .5-^X3-5=71' as in [4627].
* (2725) The constant term of the parallax is — .(l-f-ee) [5311] ; and by substituting
D / M \i
[4629a] the value of — [5324], it becomes of the form A . f j ; A being a function of the
known quantities a, e, he, which are independent of M, m. Now, by using the value of
— =^j— [4628], we obtain the constant term [5330'], corresponding to the latitude
whose sine is \/^ ; also the constant term 3427',93 [5605] of the horizontal parallax ;
hence we have,
[46296] ^. ^^^V=3427%93, and ./2=3447V32;
so that the constant term of the horizontal parallax is,
[46290] 344r,32.(^-^)*.
Putting this equal to the constant term of Burg's tables 3442',44— 10S40^3432'',04
[5605], we get,
[4629^] '^= gig) = 1,01341 = 1 +^ nearly, as in [4629'].
Vl.xvi.§44.] ON THE MASSES OF THE PLANETS AND MOON. 339
Hence it appears, from all three of these phenomena, that we must decrease
a little tlic mass of the moon, deduced from the observations of the tides
at Brest ; therefore, the action of the moon on the tides in that port, is [4630]
sensibly increased by local circumstances. For the numerous observations,
both of the heights and intervals of the tides, do not permit us to suppose
this action to be less than triple the action of the sun.
The most probable value of the moon's mass, which appears to result
from these various phenomena, is -g-i^y of the earth's mass.* By using this [4631]
value, we find 7',572,t for the coefficient of the lunar equation of the solar [4632]
tables, and 3430%88,t for the constant term of the expression of the [4033]
moon's parallax. We also find 9',648 . cos. (longitude of the moon's node), [4634]
for the inequality of the nutation, and — 18%03.sin. (long, moon's node), ^ [4C35]
*■ (2726) Subsequent observations of tbe tides at Brest, induced tbe author to reduce
this value of X [3079], from X = 3 to X= 2,35333 [11905]; making the mass of the [463ia]
moon equal to jj.Vjrir of that of the earth [11906]; as we have ah-eady remarked in
[33806', &ic.]. We may observe, that the value of X= 3 [4318,4319] corresponds with [46316]
71» 1 ..ml
5ri=rr-; [4321], and that X is proportional to m ; hence we get, m the case of — =— —
M 58,6 L J' IF ' o ' M 68,5 [4631c]
[4631], the value x=3. ^ = 2,566, as in [4637].
t (2727) This equation of the earth's motion is proportional to — [4314] ; and if
m 1 [4632a]
we suppose — = — - [4321], it becomes 8^,58 nearly, as in [4622'] ; but if we use
Jrl. OQjO
>K 1 58 6 [46326]
Ti.=^TT [4631], this equation becomes 8^,58 X ;;3V = 7'%34; which differs a little
Jn ob,o oo,o
from [4632].
X (2728) Substituting M=68,5.m [4631c], in the constant term of the moon's
parallax [4629f], it becomes 3447%32 . r^y= 3430^8, as in [4633]. Moreover, by [4633a]
substituting X= 2,566 [4631c], in the coefficient of the nutation [4625 J], it becomes,
'^ .13S4074=|^^.13',4074 = 9%648, as in [4634]. [46336]
1+X ' 3,566"
§ (2729) The coefficients of the inequalities in the nutation and precession are
represented, in [3376e,/, 3378,3380], by _f^'' „„ — ^/^^ ,.cot.2A ; which are to [4635a]
[4638]
340 PERTURBATIONS OF THE PLANETS ; [Méc. Ce].
for the inequality of the precession of the equinoxes. The ratio of the
[4636] moon's action on the tides to that of the sun is then 2,566 [4631c] ; and
as the observations of the tides in the port of Brest make this ratio equal
to 3 [46316], it appears evident that it is increased, by local circumstances,
[4637] in the ratio of 3 to 2,566. Future observations, made with great exactness,
will enable us to determine, with precision, these points, in which there
remains, at present, some slight degree of uncertainty.
Jupiter's mass appears to be well determined ; Saturn's has still some
degree of uncertainty [4635c], and it is a desirable object to correct it.
This may be done by observing the greatest elongations of the two outer
[4638'] satellites, in opposite points of their orbits, in order to have regard to the
ellipticity of the orbits. We may also use, for this purpose, the great
inequality of Jupiter [4417], when the mean motions of Jupiter and Saturn
shall be accurately determined ; for these mean motions have a very sensible
influence upon the divisor (5 n" — 2 n"y, which affects this inequality. It
appears probable, that the mean annual motion we have assigned to Jupiter,
must be increased, one or two centesimal seconds ; and that of Saturn,
decreased, by nearly the same quantity. The periodical inequalities of Jupiter
and Uranus, produced by the action of Saturn, afford also a tolerably
accurate method of determining the mass of Uranus.
The value we have assigned to the mass of Uranus, depends on the
[4641] greatest elongation of its satellites, which were observed by Herschel.
These elongations should be verified with great care.
With respect to Mercury's mass, we may use, in ascertaining its value, the
inequalities it produces in the motion of Venus. Fortunately, the influence
[4642] of Mercury on the planetary system is very small ; so that the error,
depending on any inaccuracy in this estimate of its mass, must be nearly
insensible.
[4639]
each other as 1 to — 2.cot.2A. Hence, if we suppose the inequality of the nutation to
[46356] ^^ 9^,648, as in [4634], that of the precession will be — 2x9%648.cot.2 A; and by
using 2A = 52°,1592 = 46''56"'35S8, it becomes — 18',03, as in [4635].
Before concluding this note we may observe, that the late estimates of these masses,
[4635c] ^^ different astronomers, have already been given in [4061 (/—m].
VI. xvii. <§> 45.] ASTRONOMICAL TABLES. INVARIABLE PLANE. 341
[4643]
CHAPTER XVII.
ON THE FORMATION OF ASTRONOMICAL TABLES, AND ON THK INVARIABLE PLANE OP THE
PLANETARY SYSTEM.
45. We shall now proceed to explain the method which must be used in
constructing astronomical tables. We have given the inequalities, in
longitude and in latitude, to a quarter of a centesimal second ; but the most
perfect observations do not attain to that degree of accuracy ; so that we may
simplify the calculations, by neglecting the inequalities which are less than
a centesimal second. We must form, by means of a great number of
observations, selected and combined in the most advantageous manner, the
same number of equations of condition, between the corrections of the
elliptical elements of each planet. These elements being already known, to
a considerable degree of accuracy, their corrections must be so small that we
may neglect their squares and higher powers ; and by this means the
equations of condition become linear.* We must add together all the
equations in which the coefficients of the same unknown quantity are
considerable ; so that from these sums we can form the same number of
fundamental equations as there are unknown quantities ; and then, by [4644]
elimination, we may obtain each of the unknown quantities. We can also
find, by the same method, the corrections which may be necessary in the
assumed masses of the planets. If the numerical values of the planetary
inequalities be accurately calculated, which may be ascertained by a careful
verification of the preceding results ; we may, with each new observation,
* (2730) We have given the form of an equation of this kind, in [849(f] ; and have
shown, in [84 9a — r], how to combine any number of them together, by the method of the [4644a]
least squares ; which process is now generally used, in preference to that in [4644].
VOL. III. 86
342 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
form another equation of condition. Then if we determine, every ten years,
the corrections resulting from the combination of these equations with all the
preceding ones, we may, from time to time, correct the elements of the
orbits ; and by this means obtain more accurate tables of the motions ;
[4645] supposing that the comets do not produce any alteration in the elements ;
and there is every reason to believe that their action on the planetary system
is insensible.
46. We have determined, in [1162'], the invariable plane, in which the
sum of the products of the mass of each planet, by the area its radius vector
describes about the sun, when projected upon this plane, is a maximum. If
we put 7 for the inclination of this plane to the fixed ecliptic of 1750, and
n for the longitude of its ascending node upon that plane, we shall have, as
in [1162'],
2 . OT . \/a.{\ — ef) . sin. (p . siu. ê
[4646]
[4647]
[4648]
[4649]
tang. 7 . sm. H:
tang. 7 .cos. 11:
2 . m.^a.(i— ee).COS. 9 '
2 . JW . ^«.(1 — ee) . sin. (p . COS. ê
2.OT.\/«.(1 — ee).COS.(p
The integral sign of finite differences 2 includes all the similar terms relative
to each planet. If we use the values of m, a, e, cp, and ê, given for each
of these bodies, in [4061 — 4083], we shall find, by these formulas,
7= l''35'"3P;
n=]02''57"'29\
Then, by substituting for e, tp, 6, their values, relative to the epoch 1950
[4081—4083, 4242, &c.], we shall obtain,
7= l''35'»3P;
n= 102'^ 57™ 15';
which differ but very little from the preceding values [4648]. This serves
as a confirmation of the variations we have previously computed in the
inclinations and in the nodes of the planetary orbits.
VI.xvm.§47.] ACTION OF THE FIXED STARS. 343
CHAPTER XVIII.
ON THE ACTION OF THE FIXED STARS UPON THE PLANETARY SYSTEM.
47. To complete the theory of the perturbations of the planetary system,
there yet remains to he noticed those, which this system suffers, from the [4G49']
action of the comets and fixed stars. Now, if we take into consideration,
that we do not accurately know the elements of the orbits of most of the
comets ; and, that there may be some, which are always invisible to us, by
reason of their great perihelion distance, though they may act on the remote
planets ; it must be evident, that it is impossible to determine their action.
Fortunately, there are many reasons for believing, that the masses of the
comets are very small ; consequently, their action must be nearly insensible.
We shall, therefore, restrict ourselves, in this article, to the consideration of
the action of the fixed stars.
For this purpose, we shall resume the formulas [930, 931, 932],
C / fl 7?\ ^ "\ exprès-
a.cos.vfndt.r.ûn.v. j 2/di2 + r.r^j \ ) ^7^1,
— a. sin.v .fndt.r. COS. V. < 2fdR-}-r.(--—\
[4650]
General
6r:^ 1_£_2_^. ^x) [4651]
t^ • \/l — ee
2r.d.ôr+dr.Sr 3a jr> , 2a „ ,^ /^7?\
5 — 7 .ffndt.dR-] .fndt.r. ( — )
6V^ '^^^ ^—7— '— ^^; (Y) [4652]
V/l— ee ' ^ ^
^ ,, . /dR\ . . . /dR\
a. cos. V.J ndt.r.Hva.v.y — 1 — a. sm.v. J ndt. r .cos.r.( — j
S s = /./1 ^-^- (^ [4653]
f^.y/l — ee
344 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
[4654]
We shall put m' for the mass of the star ; x', y', z', its three rectangular
co-ordinates, referred to the sun's centre of gravity ; r', its distance from
that centre ; a:, y, z, the three co-ordinates of the planet m ; and r, its
distance from the sun. We shall have, as in [3736],
[4655] R = >»'-(^^'+yy'+~^') _ m!
Developing the second member of this equation, according to the descending
powers of r', we shall have,*
[4656] ^=—7 + -jTT—i^'- ^^ ,-5 ' &C.
[4656']
We shall take, for the fixed plane, that of the primitive orbit of the planet ;
and we shall have, by neglecting the square of z,f
[4657] X = r . cos. V ; y = r . sin. v ; z = r s.
[46581 Putting I for the latitude of the star m', and U for its longitude, we
obtain, Î
[4659] a;' =r'. cos. Z. C0S.C7: ?/'=?•'. cos. /. sin. C/; 2' = r*. sin. /.
* (2731) Putting, for brevity, xx' -\-yy'-\-zz'=zrr'.f; and, as in [914'],
[4655a] a;2-|-/-f22 = 7-2, x'^+y'^ -}-z'^=r'%
we find, that the last term of [4655] becomes, by successive reductions, as in [4655c] ;
[46555] _„/.|(.,'_:,)2+(y'_y)^+(z'-_^)«f*=-_m^{r'2-2r'r/-^r«r*=-p.{l-2('^^')J"*
[4655c] __-'_-7?:;^^^^U?.™'.f^i^*'^^f _ &c.
r' r \ r'^ J 2 r' \ r'2 /
Substituting this in [4655], we find that the first term of [4655] is destroyed by the second
term of [4655c], and the whole expression of R becomes, by a slight reduction, as
in [4656].
t (2732) The values of x, y [4657], correspond with those found in [926'— 927].
[4657a] The value oî z^rs [4657] is the same as that in [931"], changing as into s, to conform
to the present notation.
\ (2733) The radius vector of the body m' is /, and its latitude above the fixed
[4659a] plane /. Hence it is evident, from the principles of the orthographic projection, that the
projection of r', upon the fixed plane, is /.cos./; and the perpendicular z', let fall from m'.
Vl.x™i447.] ACTION OF THE FIXED STARS. 346
Hence we deduce, by neglecting the descending powers of r', below Z"^,* [4059]
/?= — ^' + '1^' • )<2— 3.cos.=Z— 3.cos.=/.cos.(2i;— 2C/)— 65.sin.2/.cos.(v-t7)|. [4(iG0]
Now, ?"', /, and U, vary nearly by insensible degrees ; hence, if we put R^ [4661]
for the part of R, divided by r'^, and neglect the square of the excentricity
of the orbit of m: also, the term dependine; on 5, which is of the order of
lb' [4661']
the disturbing forces, that m suffers by the action of the planets ; we shall
have,t
fAR==R- ^-^' . (2 - 3 . cos.^Z) ; [4662]
r.(g) = 2i?, - [4662-]
upon the fixed plane, is equal to /.sin./, as in [4659]. Now, this projected radius r'.cos.l,
makes the angle U with the axis of x [4658, Sic], and 90'' — U with the axis of «/.
Hence we easily obtain expressions of x', y', similar to those of x, y [4657], and which
may be deduced from them, by changing r into r'.cos.l, and v into U, as in [4659].
[46596]
[4659c]
* (2734) Substituting the values of J", y, &c. [4657,4659], in the first member of
[4660a], reducing, developing and neglecting terms of the order s^, we get, by using
[24, 6, 31] Int. the following expressions,
\xx'-\-yy'-\-zz'l^^^r"r'^.\cos.I.(cos.v.cos.U-\-s\n.v.sm.V)-{-s.sin.I\^ [4660a]
=r^/^. ^cos.Z.cos.(w — [7)-)-s.sin. /}^
^7-^r'^.{cos.^Lcos.^(« — [7)4- 2s. sin. Z.cos.Z.cos.(« — U)l
=r2r'2.^cos.2Z.[|+icos.(2i; — 2Z7)]+«.sin.2/. cos. («—[/)}. [4660i]
Now, the first and second terms of [4656], are the same as the first and second terms of
[4660] respectively ; so that if we neglect terms of the order mentioned in [4659'J, we
shall find, that the remaining part of [4656] becomes,
— -^ -{xx'+y y'+z 2' p. [4660c]
Substituting in this the expression [4660e], it produces the three last tenns of il [4660].
t (2735) If we use the symbol R,, we shall have, from [4660,4661],
iî^=^-Ç.j2 — 3COS.2/— 3cos.2Z.cos.(2y — 2f7) — 65.sin.2/.cos.(j;— Z7)i ; [4662a]
^= — ~'+^' [46626]
VOL. III. 87
346
PERTURBATIONS OF THE PLANETS;
[Méc. Cél.
[4662"]
Then, if we put n = 1, which is nearly equivalent to the supposition, that
the sun's mass is equal to unity [3709], we shall obtain from the formula
[4651],*
[4669e]
[4662rf]
[4662rf']
[4662e]
[4662/]
[4662e-]
[4662^1]
[4662i]
[4662fe]
[4662 i]
[4662m]
[4662n]
[4662o]
[4662o']
[4662p]
[4662fl]
The characteristic d affects the elements of the orbit of the body m, namely, r, », s, inc. ;
but does not affect those of the body ?»', as r , I, U,hc.; hence the differential of [46626]
becomes, àR=àR^. Integrating this, and adding, as in [1012'], the constant quantity
m'g, to complete the integral, we get /dfi=/d/î,+ m'^. Now, as r', I, U, are nearly
constant, we may neglect their variations, and then the quantity di?, will be the complete
differential of /?, ; so that we may write R, for fdR/, hence the expression [4662f/]
becomes fdR^R^-j-m'g. If we neglect terms of the order e^, in the expression of
r [1256], it becomes as in [4664]; and if we substitute this in the expression of r^.dv
[1256], we easily obtain the expression of ndt [4664]. By inadvertence, the author has
given a wrong sign to the term depending on e, in the value of r [4664] , wliich in the
original work is r = a.{l-j-e.cos.(D — ro)|. This affects the numerical coefficients of the
formulas [4666,4666',&tc.], but does not alter the general results [4669',4673,&.c.]. Putting,
for brevity, h equal to the coefficient of r"^, in the expression of R^ [4662a], we have.
h= ^.{2— 3. cos.^l— 3. cos.-l.cos.{2v — 2U) — 6s. sm.2l. COS. {v—U)l
R.= h.i
whence
dR\
:2A r = .
r
we obtain the
Substituting this in the partial differential of R [4662è], relatively to
following expression,
\d^) \d^) ~ T '
multiplying this by r, we get [4662']. If we determine the constant quantity g, as in
[1016",&.c.],by making the coefficient of t vanish from the expression of ôv, we shall find,
by putting fj.=l, and neglecting e^, that the terms of 5v [4652], necessary to be noticed
in finding the constant quantity, are,
a.f{3fàR+2r.(~y.ndt.
Substituting the values [4662e, 4662'], it becomes, a ./{I R,-{-3m'g).ndt ; and if we
retain only the constant part of R,, the preceding expression will vanish, and we shall have
the constant part ot Sv equal to nothing, by putting 7 Ri-\-3m'g = 0; or m'g=^ — i-H^r
Now, the constant part of R^ is evidently obtained, by putting r^=a, and retaining only
the two first terms of [4662a]. Hence we get,
, 7 m'. cfi ,- n o 7\
^*^=-"Ï27T-(^~^-'=°'-')'
and fàR [4662c] becomes as in [4662]. In the original work the numerical coefficient
is — \, instead of — ^^.
* (2736) From [4662e, 4662'], we get.
VI.xnii.§47.] ACTION OF THE FIXED STARS. 347
6 r = 4 a . COS. v .fn dt.rR,. sin. v — 4.a. sin.i; ./n dt.rR,. cos. î;. [4663]
Substituting the following expressions [1256, 4662/, &c.],
r = a.\\-e.cos.(v-^)\; n dt = dv .{I— 2e .cos. (v — ^)\; [4664]
and neglecting under the sign /, the periodical terms, affected with the angle [4665]
V, we shall have,*
ndt.r.R,cos.v=-^^\{(l-hcos.H).e.cos.^-hcos.H.e.cos.(^-2U)]; [4666']
2/d /Î + r . (^) = 4 i?,+ 2 m'g. [4663a]
Substituting this in [4651], also ii=l, and neglecting c^, we get,
— =4. COS. ■y./'ri(/«.r iî,. sin. i; — 4.sin.i). AitZ^.r-R^.cos.v
a "^ ' ^ ' [4663a ]
-(-2m'^.cos.t)./nf/<.r.sin.î) — 2mg.sm.v.f7idt.r.cos.v.
This differs from [4663], in the terms multiplied by g. The two expressions would agree,
if we were to take the arbitrary constant quantity g [4662d] equal to nothing ; but this J
would be inconsistent with [4662?t, 4668].
* (2737) From [4662/], we obtain ndt.rR,^h.ndt.r^. Now we have, by
neglecting e^ r^ = a^.\l — .3 e.cos.fw— «)| [4664]; multiplying this by ndi [4664], [4666a]
we get,
ndf.r^=:a^.dv.\l — 5e.cos.(« — ■a)\; hence, ndt .rR = h.a^.dv.\\ — 5e.cos.(«j — ro)}. [46666]
Multiplying this successively, by s'm.v, and cos.d, we get, by reduction,
ndt .r R^. sm.v^= h .a^.dv .\s\n.v — f e.sin.-a — f e.sin. (2« — ra)|; [4666c]
ndt . r R^ . cos.t) =h.a^.dv .\ cos.t) — | e . cos. is — f e . cos. (2 v — o) | . [4666rf]
The second of these expressions may be derived from the first, by augmenting each of the
angles v, zs, U, by 90'; as appears, by making this change in the second members ; no [4666e]
alteration being made in /, /, &c.; so that h [4662^-] may remain the same. If we suppose
the plane of x y, to be the primitive orbit of m, the latitude « will be of the order of
the disturbing forces of the planets, which is neglected in [4661'] ; and then A [4662A:] is
composed of the two terms,
^.(2 — 3.COS.2/), _|^3.3.cos.2Z.cos.(2t;— St;-). [4666^]
Tliese are to be substituted in [4666c], and those terms retained, which do not contain the
348
PERTURBATIONS OF THE PLANETS ;
[Méc. Cél.
[4666"] which gives, by considering ïs, Z, r', U, as very nearly constant,^
5r 3m'.a^.v
[4667] - =
.{(1 — I . COS.-/). e . sin.(z) — w) — a. cos.'Z. e . siu.(t)4-w — 2C7)|.
[4666i]
[4666fe]
[4667a]
[46676]
[4667c]
[4667d]
[4667e]
[4667/]
[4667gr]
[4667A]
angle v, or its multiples [4665] ; consequently, the first of these terms of h must be
combined with the second of [4666c] ; and the second of these terms of h, with the third
of [4666c] ; hence we shall have,
m'. a?, dv
[4666A] ndt. rR, . sin. v = •
4,. '3
.f— (2 — 3.cos-2Z).Ae.sin.«— V-e.cos.2/.sin.(«— 2t7)|;
which is easily reduced to the form [4666]. In like manner we may compute [4666'] ; or,
we may obtain it much more easily, by derivation from [4666], by increasing the angles
V, «, V, by 90'', as in [4666e]. These results are free from the error in the value of r
[4662^] ; and if we compare them with those given by the author, in the original work, we
find, that we must multiply his expressions by — 5, to obtain those in [4666,4666'] ; or,
in other words, we must change e into — 5e, in his formulas.
* (2738) Putting, for brevity,
. 5m'. a^ ,, „ „,,
I5m.a? „
B^ —^—- .COS. I . e ;
16r'3
we find, that the integrals of [4666, 4666'] become, very nearly,
f7idt.7-R^.sïn.v= — ^î'. sin. 13 — 5u.sin.(^3 — 2U) ;
fndt.r R^.cos.v= — Av .cos.zi-\-Bv.cos.(-Gs — 2U).
Multiplying the first of these expressions by 4.cos.i', the second by — 4.sin.K, and taking
the sum of the products ; putting
— sin. ra.cos.v-j-cos.a.sin.i) = sin. [v — to) ;
— sin.(« — 2U).cos.v — cos. (to — 2t/).sin.i) = — sin.(w + TO — '2U) ;
we get, for the terms in the first line of [4663a'], the following expression,
4 . cos. v.fndt .rR^. sin. i' — 4 . sin. « .fn di .rR^. cos. v
^4:.A.v.sm.(v — to) — 4.B .v.s\n.{v-\--a — 2U).
Again, if we multiply together the expressions of r and ndt [4664], neglecting e^, we
obtain,
ndt ,r =^ adv .\l — 3e .cos. [v — ^)}.
Multiplying this, successively, by sin.r, cos. d; reducing and retaining only the terms,
which are independent of the angle i', we get.
ndt.r.s'm.v = — adv. ^e.sm. -a ;
fii dt .r.s'm.v = — a w . | e . sin. to ;
ndt .r.cos.v
-adv .^e. COS. zi.
fndt.r. COS. d = — av.^c. cos. i
Multiplying these integrals, respectively, by 2?n'^.cos.D, — 2m'g.s'm.v ; taking the
sum of the products, and reducing, by means of [4667 J]; then substituting the value of
VI. xviii. § 47.] ACTION OF THE FIXED STARS. 349
Now we have,*
Sr
a
— ôe . COS. (v — ^) — e (5 3 . sin. (v — ^). [4668]
Secular
variations
Comparing together the two expressions [4667, 4668], we obtain,t iriïè
excentrici-
ly ami
1 'Î m' /7^ï' perihelion.
5 e = '-""■" . cos.^Z . e . sin. (2 ^ — 2U) ; [4669]
S^= _ iî^!i'.p_3 . cOS.^Z — I . COS.^Z. COS. (2^ — 2U)]. [4669']
Thus the action of the star m' produces secular variations in the excentricity and
in the longitude of the perihelion of the orbit of the planet m; but these variations
are incomparably smaller than those arising from the action of the other [4669"]
planets. For, if we suppose m to be the earth, r' cannot, by observation.
m'g [466-2y], we finally get, for the second line of [4663a'],
2 m'g . cos . V .fn dt .r.s'm.v — 2 m'g . sin . î) .fn dt.r. cos. v
= 2m'g.^.ave . j — sin.w.cos. v + cos.ra.sin.i) | [4667i]
= m'g.^ave.sm.{v — «) = 9~^3~'^^ — f .cos.'^/j.e.sin. (w — «). [4667fc]
Adding together the expressions [4667e,^-]; re-substituting the values of A, B [4667cr],
we get the complete value of — [4663rt'], as in [4667]. In the original work, the author [466/^]
.3 m'. a^v
makes the factor, which is without the braces, equal to V^ — ) instead of „ ,, »
and the numerical coefficient of the second term within the braces is erroneously printed
— f instead of — J. These mistakes are the consequences of using erroneous values of
g and ;■ [4662o', p].
[4667m]
* (2739) In finding the variation of r [4664], we must neglect that of v, arising from
the constant quantity g' [4662/i], and the expression becomes as in [4668] ; which is MQgg;,-]
similar to [3876]. The signs of the terms in the second member of [4668], in the original
work, are incorrect, by reason of the mistake mentioned in [4662^-].
[4669o]
t (2740) From [21] Int. we have,
sin.{f + 3— 2f7] =sin.[(y — îs) + (2w — 2t7)}
= sin. (v — ra) .COS. (2zs — 2U)-\-cos. {v — a) .sin. (2zi — 2U).
Substituting this in the last term of [4667], and then comparing separately, the coefficients
of sin. (i- — 3j) and cos.(i' — w), in the two expressions [4667, 4668] ; we get, by a slight [46696]
reduction, the values of 6 e, ôtz [4669,4669']. These expressions agree with those given
VOL. III. 88
350 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
t 3
[4670] be supposed less than lOOOOOa, and then the term '^^-^, does not exceed,*
[4671] m'^.0%000000001 ;
t denoting the number of Julian years. This is incomparably less than the
[4671'] secular variation of the excentricity of the earth's orbit, resulting from the
The ao- action of the planets, which, by [4244], is equal to.
Stars has
[4672] — ^.0',093819,
no sensi-
ble effect
ontheex- uiiloss we suDDOsej that m' has a value which is wholly improbable. Hence
centrici- J L
ties and
perihelia
of the
we may conclude, tJiat the action of the stars has no sensible influence on the
p'ian'e"ts. secular variations of the excentricities and perihelia of the planetary orbits.
[4673] In like manner, it is evident, from the development of the formula [4653],
that their action has not any sensible influence on the position of these
orbits.f
by Mr. Plana, in the Memoirs of the Astronomical Society of London, vol.ii. p. 354 ;
which he deduced from the formulas [1253a]. Hence we see, that the method here
f4669cl proposed by La Place, to find 5e, Sw, when it is correctly followed, leads to an accurate
result ; and is not liable to the objection made by Mr. Plana, in the same page of that
volume, namely; that it is nowise fit for the intended purpose, without taking into view other
circumstances, which render the calculation more complicated. We may remark, that in
[4669(/] the original work, the factor -y- [4669], is printed J ; and, in [4669'], the factors —,
[4669e] — f.cos.% are changed into , — J. cos.®/, respectively
*■ (2741) The value of ?-=100000a [4670], corresponds to an annual parallax of
[4671a] about 2'; and we have nearly «=1295977'.^ [4077]; substituting these in — j^-
[4670], it becomes as in [4671] ; or simply, by supposing 7ft'= the sun's mass = 1,
^0',000000001.
The secular variation of c" [4330a], is nearly represented by,
[46716] -^.<= — i.(0%187638).^=— 0',093819.!: [4244,4672];
which is much greater than the expression [4671].
t (2742) If we substitute rs = z [4657], in R^, R [4662i, «], and retain only
*■ "•' the terms of R, containing z, we find,
[46736] R= j-j^ .sm.2l.cos.{v — U), and f— j = — — ^ .sm.2/.cos.(t) — U) .
VI.xvrii.§47.] ACTION OF THE FIXED STARS. 351
PFe shall noio examine into the influence of the attraction of the stars on the
mean motion of the planets. For this purpose, we shall observe, that the
formula [4652] gives, in d.6v, the term* dAv = ^andt .R/, from [4674]
which we deduce the following expression,!
d.iv='^ .ndt. {2 — 5. co^.H]. [4675]
We shall put
r'=r;. (1-aO; /=Z^. (1-/30; [467G]
r' and / being the values of r' and /, in 1750, or when / = 0; we shall [407G']
have, in &v, the variation, f
6 V = ^-^ . ri — # . cos.%) . a . ,1 f— ^-^. sin. 2l.^.nt\ [4677]
Substituting this in S s [4653], we find that the terms are multiphed by the very small
factor of the order [4670,4671], which renders them insensible [4671'].
* (2743) This expression arises from the last term of 5v [4652], which, by neglecting
quantities of the order e^, and putting jj-^I [3709], becomes,
2afndt.r. (^\ = 2afndt.2R, [4662'] . [4674a]
Its differential gives, in d.5v, the terra Aandt.R^, as in [4674]. This would be
increased to landt.R^, by noticing the term depending on fàR [46-52], as we have [46746]
seen in [4662o']. This increases the terms [4675, 4677] in the ratio of 7 to 4.
m' r-
t (2744) The two first and chief terms of R, [4662a], are -^ • (2— Z.cos.^l) .
Substituting the value of r [4664], we obtain the part rrj • (2 — 3. cos.-/), which [4675o]
does not contain v ; hence, the term of d.Sv [4674], becomes as in [4675].
Î (2745) The value of / [4676] gives cos.? =cos.(/,— p^/,) = cos./,-f-p^sin./,, [467Ga]
by using [61] Int. Squaring this, neglecting i^, and putting 2 . sin. /,. cos. /,= sin. 2/, [31] Int.,
we get cos.^I = cos.^/^ -\-^t- sin. 2 Z, ; whence,
2 — 3.cos.2/ = 2.(l— f.cos.2/,)— 3(3/.sin.2/,. [46765]
If we now substitute the value of r' [4676], in the first member of the following expression,
and then develop it according to the powers of a, neglecting a*, we get,
-^.ndt = —j^ .7idt.{l-lr3o-t). [4676c]
352 PERTURBATIONS OF THE PLANETS ; [Méc. Cél-
We cannot ascertain, by observation, the value of aï, but may determine that
of f3t. Now, if we suppose, relatively to the earth, f3=l"^ U%324, and
[4678] ,,' ^^ 100000 a ; the quantity -^ . ^nf becomes, very nearly,*
^/
[4679'] which is insensible, from the time of the most early observations on record.
The expression of d.&v, contains also, by what precedes, the terms, f
[4680] dM^=—i.m\a\ndLfAy-^~.cos.{v—U)\—&tri;M\ndt.Sp^l.co^^^^^
Multiplying together the expressions [4676i, c], we get the value of d .&v [4675], nearly,
[4676d] tZ.5u=z — JL ,ndt.{\ — %.cos.H\A —- .(1 — J.cos.^/J.an^t/i 7T-.sm.2Lsnic/;.
We may neglect the first term of this formula, because we have taken the constant quantity
[4676c] ^ so as to make the coefficient of t vaftish from the expression of 5« [4662«]. Integrating
the other two terms of [4676c/], we get the value of h%) [4677].
* (2746) The assumed values of (3, r/, are taken within reasonable limits ; since the
value of p corresponds to an annual variation in tlie latitude of the star, of about a third of a
[4679a] sexagesimal second ; and the value of r/ to an annual parallax of nearly two sexagesimal
seconds. To reduce the expression [4678] to numbers, we have, in the case of i=\,
nt = circumference of the circle = C,2S.31 ; hence, generally,
[46796] „ < = 6,2831 .t; also, p t = 0',324 . t.
The product of these two expressions is,
[4679c] fi7i(^ = 2',0357 . t^
Substituting this, and rf=l0^.a, in the first member of [4679], it becomes as in the
[4679d] second member of that equation. This is wholly insensible in observations made 3000
years ago ; since, by putting t = — 3000, and 7«'=:1, it becomes less than O',00000002.
t (2747) If we now notice only the terms of R, R, [4662rt, b'\, depending on s, we
obtain,
[4680a] Rz= — f ."^ .«.sin.2/.cos.(i' — U) ; whence, r.f—j^ — 3.^^.s.sin.2/.cos.(t) — [').
If we substitute the value of r [4664], and neglect terms of the order es, we get,
[46806] R= — f.m'.a^. } ~^ — .cos.(« — Z7)| ; r. f — 1 = — 3. -^ .s.sm.2/.cos.(«— t').
Now, if we put (J.= l, and neglect e^ ; noticing only the terms of [4652], where R
VI. xviii. §47.] ACTION OF THE FIXED STARS. 363
Now we have,*
s = t .— . sin. I' — t--r-- cos.i) ; t^^^^]
(It (It
which gives, by neglecting the quantities multiplied by the sine or cosine of
the angle v,f
s.sin.2/ , y.^ ^ sin. 2/ (. dq . jj dp ^^^ tj\ .
[4682]
[4683]
consequently, t
,. , s. sin. 21 . rr\ , sin.2Z ^dq . ^r '^ V tt}
i d . -^3- • COS. (t' -C/) = ^ . -^^. I ^ . sin.t/-^^ . cos.f/ ^ .
Hence we obtain, in d.6v, the term,^
^.,^,=_l^î^^„^^^.sin.2^5^.sin.C7-^.cos.f7^ [4684]
4 r^ i dt dt 3
[4G80c]
explicitly occurs, we get, for its differential,
dJjv = 3a.7idt.fdR~\-2a.7idt.r. (-77 )•
Substituting, in the first term of this expression, the value of R [46S0J], we get the first
term of [4680] ; and we obtain the last terni of [4680], by the substitution of the second L •' 1
expression [4680&] in the last term of [4680c].
* (2748) This expression is similar to that in [3802, Sic.]. We may remark, that the
author, in this article, has interchanged the usual signification of the symbols p, q [3802]. [4681n]
We have rectified tliis, by changing jj into q, and q into p, in all the formulas [4681 — 4685]
of the originLil work.
sin.2i
t (2749) If we multiply the expression [4681] by — ^.cos.(i' — U), and reduce
the products by [19, 20] Int., we shall obtain the equation [4682], by retaining only the [4682a]
terms which are independent of 1;; or in other words, by retaining only the terms |sin.f/,
^cos.f-^, of the expressions sin. v. cos. (» — U), and cos. v . cos. (u — U), respectively.
X (2750) If we neglect the variations of r', /, U, in the second member of [4682], the
sign d may be considered as the complete differential, and then the signs /d, mutually [46e3aJ
counteract each other, and they may be prefixed to the first member of [4682], without
altering its second member; hence we get [4683] from [4682].
§(2751) Multiplying [4683] by — ^ .m'.d^n dt, nnd [4682] by —Q.m'.({\ndt,
we find, that the sum of the products, or the second member of [4680], is as in [4684]. l'1684«|
Integrating this, we get, [4685].
VOL. in. 89
354 PERTURBATIONS OF THE PLANETS ; [Méc. Cél.
consequentlj, we have, in 6 v, the secular inequality,
[4685] 5x, = . _^,nt\ sm. 2Z. \ -/-.sm.U — — - cos.t/ S .
8 r* I (It dt 5
We have given the values of -— , -j-, [4332], relatively to the earth.
If we substitute them in the preceding term of &v [4685], we shall find
that it is insensible,* even in the most ancient observations.
* (2752) From [4332] it appears, that — -, — -, are each less than F, and sin. 2/,
sin.ZJ, cos.f^, do not exceed unity : tlierefore, sin.2/. < •—- .sin.t/ — —- .cos.U )■ , maybe
[4685a] ■' ' i dt dt <, ■'
considered as less than V ; and then, the expression [4635], neglecting its sign, becomes
21 m'.a?
less than — . —7 — .nt^.V ; which is found to be insensible, in [4679'].
8 r'3
Other terms of the like nature with those which have been particularly examined, in this
[46856] chapter, may be deduced fiom the formulas [4651 — 4653] ; but it is evident, from what we
have seen, that they must be excessively small ; so that it is hardly worth the labor of a
r../-o^ 1 more thoroudi examination. The author himself, seems to have considered the subject as
[4685c] ° ... ...
not deserving much attention, and has been quite negligent in the numerical details of this
article ; so that it has been found necessary to correct the text in several places, as we have
[4685(/] already remarked. In writing the notes on this volume, soon after its first publication by the
author, I pointed out the mistakes in this chapter. It has since been done by Mr. Plana, in
vol. ii. p. 351 of the Memoirs of the Astronomical Society of London, for 1826; and
[4685e] subsequently by La Place, in the Connaissance des Terns, for the year 1829, page 250. The
method used by Mr. Plana is more direct and simple than that of the author. It consists in
[4685/] substituting the value of R [4660], in the formulas [5787—5791], and making the necessary
reductions ; but, as the process is simple, it is unnecessary to enter minutely upon it.
Mr. Plana remarks, in page 355 of the work above-mentioned, that the action of the
fixed stars affects the mathematical accuracy of the equation [1114],
[4685g-] ''^- '» • \/» + e'^. m'.\/a'-\- kc. = constant ;
as we have already remarked in [11 146]. This is evident ; for, if we increase the quantity
e, in the first member, by the expression &e [4669], the second member will be increased
by the quantity,
[4685?,,] ■2f.5e = ^^^ . cos.a/.t-.s.n.(2:3 — 2t/), nearly;
which destroys the constancy of the second member. The same defect exists in the
equation [1134 or 1155].
VI.xviii.§47.] ACTION OF THE FIXED ST^VRS. 355
It is easy also, to satisfy ourselves, that the preceding results hold good,
relatively to those planets which are the most distant from the sun. Hence
it a|)pears, that the action of the stars upon the planetary system, is so much [4680]
decreased, by reason of their great distance, that it is wholly insensible.
It now remains to compare with observations, the formulas of the
planetary perturbations, given in this book, and particularly those of the
two great inequalities of Jupiter and Saturn. This comparison requires too
much detail for the limits of the present work; we shall, therefore, merely
remark, that before the discovery of these great inequalities, the errors of the ^
best tables sometimes amounted to thirty-five or forty minutes ; and now they
do not exceed a minute. Halley had concluded, by the comparison of modern
observations, the one witli the other ; and also, by comparing the modern with
the ancient observations, that Saturn's motion is retarded, and Jupiter's
accelerated, from age to age. On the other hand, Lambert ascertained, from
the comparison of modern observations alone, that Saturn's motion was then [4088]
accelerated, and Jupiter's motion retarded. These phenomena, apparently
opposed to each other, indicated, in ihc motions of the two planets, great
inequalities of a long period, of which it was important to know the laws and the
cause. By submitting to analysis their mutual perturbations, I discovered
the two principal inequalities [4i34, 4492] ; and perceived, that the
phenomena, observed by Halley and Lambert, naturally arise from them ;
and, that they represent, with remarkable accuracy, both ancient and modern
observations. The magnitude of these inequalities, and the great length of
the period of revolution, to complete v.'hich requires more than nine hundred
years, depend, as we have seen, on the nearly commensurable ratio which
obtains between the mean motions of Jupiter and Saturn. This ratio gives
rise to several other important inequalities, which I have determined, and these
inequalities have given to the tables the precision they now have. The same
analysis, being applied to all the other planets, has enabled me to discover, in
their motions, some very sensible inequalities, which have been confirmed by
observation. I have reason to believe, that the preceding formulas, computed
with particular care, will give a still greater degree of precision to the tables
of the motions of the planetary bodies.
IVrii-iS cl'
diflcit'iii
ufilerà.
[4693]
SEVENTH BOOK,
THEORY OF TriE MOON.
The theory of the moon has difficulties peculiar to itself, arising from
the magnitude of its numerous inequalities, and from the slow convergency
of the series by which they are determined. If tlie body were nearer to
the earth, the inequalities of its motion would be less, and their approximations
more converging, But, in its present distance, these approximations depend
on a very complicated analysis; and it is only by a very particular attention,
and a nice discrimination, that we can determine the influence of the
successive integrations, upon the various terms of the expression of the
disturbing force. The selection of co-ordinates is not unimportant for the
[4692] success of the approximations. The sun's disturbing force depends on the
sines and cosines of the moon's elongation from the sun, and on its multiples.
Their reduction to sines and cosines of angles, depending on the mean
motions of the sun and moon, is troublesome, and has but little convergency,
on account of the moon's great inequalities. It is, therefore, advantageous
to avoid this reduction, and to determine the moon's mean longitude in a
function of the true longitude, which may be usciul on several occasions.
We may, then, if it be required, determine accurately, by inverting the series,
the true longitude, in a function of the mean longitude. It is in this way
we shall consider the lunar theory.
To arrange conveniently these approximations, we shall divide the
inequalities, and the terms which compose them, into several orders. We shall
consider as quantities of the first order, the ratio of the sun's mean motion to
that of the moon, the excentricity of the orbit of the moon or earth, and the
inclination of the moon's orbit to the ecliptic. Thus, in the expression of the
mean longitude, in a function of the true longitude [5574 — 5578], the
principal term of the moon's equation of the centre is of the first order
[5574]. The second order includes the second term of that equation : the
vil. Introd.] INTRODUCTION. 357
reduction to the ecliptic ; and the three great inequalities, known under
the names of variation, erection, and annual equation [5575]. Tlie ^ *' ■'
inequalities of the third order are fifteen in number [5576]. The present
tables contain all these inequalities, together with the most important ones of
the fourth order; and it is on this account, that they correspond with the
observations made on the moon, with a degree of accuracy that it will be
difficult to surpass ; and to this great correctness we are indebted for the
important improvements in geography and nautical astronomy.
The object of this book is to shoio, in the first place, that the law of
universal gravity is the only source of all the inequalities of the lunar
motions; and then, to use this law as a method of discovery, to perfect the
theory of these inequalities, and to deduce from them several important
elements of the system of the ivorld ; such as the secular equations of the
moon, the parallaxes of the moon and sun, and the oblateness of the earth.
A judicious choice of the co-ordinates, and well conducted approximations,
with calculations made carefully, and verified several times, ought to
give the same results as those derived from observation ; if the law of
gravity, inversely as the square of the distance, be the law of nature.
We have, therefore, endeavored to satisfy these conditions ; which require [4(i95]
the consideration of some very intricate points ; the neglect of which is
the cause of the discrepances, that have been observed in the previously
known theories of the moon. It is in these points, that the main difficulty
of the problem consists. We may easily conceive of a great many
diffijrent and new methods of expressing the problem by equations ; but
it is the discussion of all those terms, which are of themselves very small,
and acquire a sensible value, by the successive integrations, which constitutes
the important and difficult part of the process, when we endeavor to
make the theory agree with observation ; which should be the chief
object of the analysis. We have determined all the inequalities of the first,
second and third orders, and the most important ones of the fourth order,
continuing the approximation to quantities of the fourth order inclusively ;
and retaining those of the fifth order, which arise in the calculation. For
the purpose of comparing this analysis with observation, w^e may observe,
that the coefficients of Mason's lunar tables are the result of the comparison
of the theory of gravity with eleven hundred and thirty-seven observations [\g^m;-]
of Bradley, made between the years 1750 and 1760; that the eminent
VOL. III. 90
[4697]
[4698]
358 THEORY OF THE MOON ; [Méc. Cél.
astronomer Burg has rectified these tables, by means of more than three
thousand of Maskelyne's observations, from 1765 to 1793; and, that the
corrections he has made are small ; with the addition of nine equations,
indicated by the theory. The tables of both these astronomers are arranged
in the same form as those of Mayer, of which they are successive
improvements : and we ought, in justice to this celebrated astronomer, to
observe, that he was not only the first, who constructed lunar tables,
sufficiently correct to be used in the solution of the problem of finding the
longitude at sea, but also, that Mason and Burg have deduced, from his
theory, the methods of improving their tables. The arguments are made to
depend on each other, in order to decrease the number of them. We have
reduced them, Avitli particular care, to the form which is adopted in the
present theory ; that is, to sines and cosines of angles, increasing in proportion
to the moon's true longitude. By comparing these results with the
coefficients of the present theory, we have the satisfaction of perceiving,
that the greatest difference, which, in Mayer's theory, one of the most
accurate heretofore published, amounts to nearly one hundred centesimal
seconds [=32',4], is here reduced to thirty [9',8], relative to the tables
of Mason, and to less than twenty-six centesimal seconds [^8',3], relative
to the still more accurate tables of Burg. We could diminish this difference,
by noticing quantities of the fifth order, which have some influence, as may
be known by inspecting the terms of this kind already calculated. This is
proved by the computation of the two inequalities [5286'", &c.], in which we
have carried on the approximation to quantities of the fifth order. The
present theory agrees yet better with the tables, relative to the motion in
latitude. The approximations of this motion are more simple and converging
than those of the motions in longitude ; and the greatest difference between
the coefficients of my analysis and those of the tables, is only six centesimal
seconds [= l',9], so that we may consider this part of the tables as being
founded upon the theory itself. As to the third co-ordinate of the moon, or
[4700] its parallax, we have preferred, without hesitation, to form the tables by the
theory alone, which, on account of the smallness of the inequalities of the lunar
parallax, must give them more accurately than they can be olitained by
observation. The differences between the results of the present theory and
those of the tables, express, therefore, the differences between this theory and
that of Mayer, which has been adopted by Mason and Burg. These
differences are so small that they are hardly deserving of notice ; but, as the
[4699]
vil. Introil.] INTRODUCTION. 359
present theory agrees better with observation than Mayer's, in the motion in
longitude, there is also reason to believe, that it possesses the same advantage ' '
relative to the inequalities in the parallax.
The motions of the perigee and nodes of the lunar orbit, afford also a
method of verifying the law of gravity. In the first approximation to the
value of the motion of the perigee, by the theory of gravity, it was found,
by mathematicians, only one half of what it was known to be, by observation ;
and Clairaut inferred, from this circumstance, that we must modify the law of r4702'|
gravity, by adding to it a second term. But he afterwards made the important
remark, that by continuing the approximations to terms of a higher order, the
theory would be found to agree nearly with observation. The motion,
deduced from the present analysis, differs from the actual motion only a four
hundredth part [5231 J ; the difference is not a three hundred and fiftieth part ^
in the motion of the nodes [5233'].
Hence it incontestabhj follows, that the laiv of universal gravitation is the
sole cause of the lunar inequalities. Now, if we consider the great number
and extent of these inequalities, and the proximity of the moon to the earth,
we must be satisfied, that it is, of all the heavenly bodies, the best
adapted to confirm this great law of nature, as well as to show the power of [47041
analysis, that wonderful instrument, without the aid of which it would be
impossible for the human mind to penetrate into so complicated a theory, and
that can be used, as a means of discovery, as sure as by direct observation.
Among the periodical inequalities of the moon's motion in longitude, that
which depends on the simple angular distance of the moon from the sun is [1705]
important, on account of the great light it throws on the sun's parallax. It
has been determined by the theory ; noticing quantities of the fifth order,
and also the perturbation of the earth by the moon, which are indispensable [4'OC]
in this laborious research. Burg found this inequality to be 122,38, by the
comparison of a very great numl)er of observations. If we put this equal to
the result by the theory, we obtain 8',56, for the sun's mean parallax ; being [4707]
the same as several astronomers have found, from the last transit of Venus
over the sun [5586].
An inequality, which is not less important, is that which depends on the
longitude of the moon's node. Mayer discovered it by observation, and
Mason fixed it at 7',7 ; but, as it did not appear to depend on the theory [4708]
360 THEORY OF THE MOON ; [Méc. Cél.
of gravity, it was neglected by most astronomers. A more thorough
examination of this theory led me to the discovery, that its cause is the
oblateness of the earth. Burg found it, by a great number of Maskelyne's
[4709] observations, to be 6,8 ; which corresponds to an oblateness of -a-ëi.-ôT
[5593].
We may also determine this oblateness, by means of an inequality in the
moon's motion in latitude ; which I discovered also by the theory ; and
[4710] which depends on the sine of the moon's true longitude. It is the result of
a nutation in the lunar orbit, produced by the action of the terrestrial
spheroid, and corresponds to that produced by the moon in our equator ; so
that the one of these nutations is the reaction of the other : and, if all
the particles of the earth and moon were firmly connected together, by
inflexible right lines, void of mass, the whole system Avould be in equilibrium
about the centre of gravity of the earth, in virtue of the forces producing
these two nutations : the force, acting on the moon, compensating for
its smallness, by the length of the lever to which it is attached. We
may represent this inequality in latitude, by supposing the lunar orbit, instead
of moving uniformly on the ecliptic, with a constant inclination, to move,
with the same conditions, upon a plane but little inclined to the ecliptic, and
which always passes through the equinoxes, between the ecliptic and
equator : a phenomenon which occurs in the theory of Jupiter's satellites,
in a still more striking manner. Thus, this inequality decreases the
[4712] inclination of the moon's orbit to the ecliptic, when the ascending node
of that orbit coincides with the vernal equinox. This inclination is
increased, when the ascending node coincides with the autumnal equinox,
which was the case in 1755; in consequence of which, the inclination, as
it was found by Mason, from 1750 to 1760, is too great. This point has
been determined by Burg, by observations made during a much longer
interval, noticing the preceding inequality ; and he has found the inclination
[^^^"^1 to be less, by 3 ,7. At my request, this astronomer has undertaken to
determine the coefficient of this inequality, by a very great number of
observations ; and he has found it to be equal to — 8". The oblateness of
[4714] the earth, deduced from it, is ^^t.t [5602], being very nearly the same
as that which is computed from the preceding inequality of longitude,
Thus, the moon, by the observation of her motions, renders sensible to
modern astronomy the ellipticity of the earth, whose roundness was made
VII. Introd.] INTRODUCTION. 361
kiioAvu to the early astronomers by her eclipses. The experiments on the
pendulum seem to indicate a less oblateness,* as we have seen in the third
book. Tills difference may depend on the terms by which the earth varies
from an elliptical figure ; m hich may have some small effect in the expression
of the length of the pendulum, but is wholly insensible, at the distance of
the moon.
The two preceding inequalities deserve every attention of observers ;
because they have the advantage over geodetical measures, in giving the
oblateness of the earth, in a manner which is less dependant on the
irregularities of its figure. If the earth were homogeneous, these inequalities
would be much greater than they are found to be by observation. They l"*'^^!
concur, therefore, with the phenomena of the precession of the equinoxes,
and the variation of gravity at the surface of the earth, to exclude its
homogeneity. It results also, that the moori's gravity towards the earth, is
composed of the attractions of all the particles of the earth; ivhich furnishes
another proof of the attraction of all the particles of matter.
Theory combined wàth experiments on the pendulum, the geodetical
measures, and the phenomena of the tides, make the constant term of the
expression of the moon's parallax less than by Mason's tables. It differs but [4716]
very little from that which Burg computed from a great number of observations
of the moon, of eclipses of the sun, and of occultations of stars by the moon.
It is only necessary to decrease a little the mass of the moon, which was
determined by the phenomena of the tides, to make this constant term
coincide with the result of that skilful astronomer. This diminution is also [4717]
indicated by the observations of the lunar equation of the solar tables, and
by the nutation of the earth's axis. This seems to prove, that in the port of
Brest, the ratio of the moon's action on the tides to that of the sun, is
sensibly increased by local circumstances. Future observations of all these
phenomena will remove this slight degree of uncertainty.
One of the most interesting results of the theory of gravity, is the
knowledge of the secular inequalities of the moon. Ancient eclipses
* (2753) Later and more accurate observations give a different result, as may be seen, [4715a]
by referring to [201 7ji, 2056î, &:c.].
VOL. III. 91
[4718]
362 THEORY OF THE MOON ; [Méc. Cél.
[4719]
indicated, in the moon's mean motion, an acceleration ; the cause of which
was sought for a long time in vain. Finally, I discovered, by the theory,
that it depends on the secular variations of the excentricity of the earth's
orbit. The same cause decreases the mean motions of the perigee and nodes
of the moon, while her mean motion is increased ; so that the secular
equations of the mean motions of the moon, the perigee and the nodes,
[4720] are always in the ratio of the numbers 1, 3 and 0,74 [5235]. Future ages
ivill develop these great inequalities, which are periodical, like the variations
of the excentricity of the eartli's orbit, upon which they depend. These
will finally produce variations which amount, at the least estimate, to
a fortieth part of the circumference [d''], in the moon's secular motion;
[4721] and to a twelfth of the circumference [30''], in that of the perigee.
Observations have already confirmed these secular inequalities in a
remarkable manner. The discovery of them induced me to believe, that
we must diminish, by fifteen or sixteen centesimal minutes, the present
secular motion of the moon's perigee, which astronomers had determined,
[4722] by comparing modern observations with ancient ones. All the observations,
which have been made during the last century, have put beyond doubt, this
result of analysis. We see, in this, an example of the manner in which the
phenomena, as they are developed, throw light upon their true causes. When
[4723] the acceleration of the moon's mean motion only was known, it could be
attributed to the resistance of the ether, or to the successive transmission of
gravity ; but analysis shows us, that both these causes produce no sensible
alteration, either in the mean motion of the nodes, or in that of the lunar
perigee : this is a sufficient reason for rejecting them, even if we were
ignorant of the true cause. The agreement of the theory with observations,
proves, that if the moon's mean motion is affected by any causes, besides the
action of gravity, their influence is very small, and is not yet perceptible.
[4724]
This agreement establishes, with certainty, the constancy of the duration
of a day ; which is an essential element in all astronomical theories. If
this duration were now one hundredth part of a centesimal second [or 0',864]
[4725] more than in the time of Hipparchus, the duration of the present century
would be greater than in his time, by 365i centesimal seconds [or 315',576].
[4725] In this interval, the moon would describe an arch of 173',2, and the present
mean secular motion of the moon, would appear to be augmented by the
VII. Introd.] INTRODUCTION. 363
same quantity. This would add 4,4* to the secular equation, which is [472G]
Ibuiid, by the theory, to be 10',1 81621 [5543], in the first century after the
year 1750. This augmentation is incompatible with the best observations,
which do not permit us to suppose, that the secular equation can exceed, by
V,62, the result of the analysis [5543]. We may, therefore, conclude, that
the duration of the day has not varied a hundredth part of a centesimal [4727]
second, since the time of Hipparchus ; which confirms what has been found
a priori, in book v. ^ 12 [3347,&.c.],by the discussion of all the causes which
could alter it.
To omit nothing which can have an influence on the moon's motion,
we have considered the direct action of the planets upon this satellite, and
have found, that it is of very little importance. But the sun, by transmitting
to the moon the action of the planets on the elements of the earth's orbit,
renders their influence on the lunar motions very remarkable, and makes it much [4728]
greater than on the elements themselves ; so that the secular variation of the
excentricity of the earth's orbit is much more sensible, in the moon's motion,
than in the earth's. It is in this manner, that the moon's action on the earth,
which produces, in the earth's motion, the inequality known by the name of
the hmar equation, is, if it may be so expressed, reflected back to the [4729]
moon, by means of the sun, but decreased in nearly the ratio of five to
nine [5226]. This new consideration adds some terms to the action of the
planets on the moon, which are of more importance than those depending
on their direct action. We have investigated the principal lunar inequalities,
resulting from the direct and indirect actions of the planets upon the moon ; [4/30]
* (2754) If we neglect tlie term of the secular equation [5543], depending on P, and
put (7=10', 181621, we may represent the moon's mean motion, in i centuries after 1750,
by ni -{-ai-. If we substitute in this successively, i^ — J, i = -j-|, and take the "'
difference of the two results, it will be found equal to n, whicii must, therefore, represent
the motion between 1700 and 1800. In like manner, by putting successively i= — 20,
i^ — 19, and taking the difference of the two results, we get n — 39 a, for the motion in the
century included between the years 250 and 150 before the Christian era. The difference
of these two results 39 a, represents the augmentation of the secular motion between these
two epochs; and, if this quantity were increased 173%2, as in [4725'j, we must increase the ' '^■'
value of a by ^VX 173',2 = 4%4, as in [4726].
[4720i]
3^ THEORY OF THE MOON ; [Méc. Cél.
and, if we take into view the accuracy to which the lunar tables have been
carried, it must be considered useful to introduce these inequalities.
The moon's parallax, the excentricity and the inclination of the lunar
orbit to the apparent ecliptic, and, in general, the coefficients of all the lunar
inequalities, are likewise subjected to secular variations ; but, up to the
[4731] present period, they are hardly sensible. This is the reason why we find
now, the same inclination, that Ptolemy obtained from his observations ;
although the obliquity of the ecliptic to the equator has sensibly decreased
since the time of that astronomer; so that the secular variation of the obliquity
affects only the moon's declination. However, the coefficient of the annual
equation, having for a factor, the excentricity of the earth's orbit, its
variation is sufficiently great to be noticed, in computing ancient eclipses.
[4732]
The numerous comparisons, which Burg and Bouvard have made, of Mason's
tables, with the observations of the moon ; at the end of the seventeenth
century, by LaHire and Flamsteed ; in the middle of the eighteenth century,
by Bradley ; and the uninterrupted series of observations of Maskelyne,
from the time of Bradley to the year 1800, give a result which was wholly
[4733] unexpected. The observations of LaHire and Flamsteed, being compared witli
those of Bradley, indicate a secular motion, exceeding by fifteen or twenty
centesimal seconds, that which is inserted in the third edition of La Lande's
astronomy ; which, in a hundred Julian years, exceeds a whole number of
[4734] revolutions, by 307'^53"'12^ Bradley's observations, being compared with
the last ones of Maskelyne, give, on the contrary, a smaller secular motion,
by at least one hundred and fifty centesimal seconds. Lastly, the observations
[4735] made within fifteen or twenty years, prove, that the diminution of the moon's
motion is now decreasing. Hence, it becomes necessary to vary incessantly
the epochs of the tables ; and it is an object of importance to correct this
imperfection. This evidently indicates the existence of one or more unknown
^ ^ inequalities of a long period, which the theory alone can point out. By a
careful examination, I have not been able to discover any such inequality,
depending on the action of the planets. If there were one in the rotation
of the earth, it could be perceived in the moon's mean motion, and might
introduce the observed anomalies : but an attentive examination of all the
causes which can alter the rotation of the earth, has more fully convinced
[4737] ^^^ ^j^^^ j^g variations are insensible. Returning back, therefore, to the
VII. Intiod.] INTRODUCTION. S6b
exaniin;itioii of the sun's action on the moon ; I have discovered, that this
action produces an inequality, whose argument is double the longitude of the
node of tiie lunar orbit, jdus the longitude of its perigee, minus three times
the longitude of the sun's perigee. This inequality, whose period is 184 [4738]
vears, depends on the products of these four quantities, namely ; the square
of the inclination of the moon's orbit to the ecliptic ; tlie excentricity of that
orbit ; the cube of the excentricity of the sun's orbit, and the ratio of the
sun's parallax to that of the moon. Hence it would seem, that it ought to [4739]
be insensible ; but the small divisors it acquires by integration, may render it
sensible, especially, if the most important terms, of wliich it is composed,
are affected with the same sign. It is very difficult to obtain its coefficient
by the theory, on account of the great number of terms, and the extreme [4740]
difficulty of appreciating them ; the difficulty being much greater in this than
in the other inequalities of the moon. This coefficient has, therefore, been
ascertained by means of the observations made during the last century ; and
I have found it to be nearly equal to 15',39. Its introduction in the tables [4741]
must change the epoch and mean motion ; and I have also found, that we must
decrease, by 31'',964, the mean secular motion, in the third edition of [4742]
LaLande's astronomy, and have determined the following formula, which
must be applied to the mean longitude given by these tables, the epoch [4743]
of which, in 1750, is 188" 17'" 14',6 ; Equnt,,,,,
nf 184
Correction of moon's mean long. = — 12',78 — 31 ',964 . i + 15',39 . ûn.E ; T4744]
i being the number of centuries elapsed since 1750, and E the double of the
longitude of the node of the lunar orbit, plus the longitude of its perigee, [47451
minus three times the longitude of the sun's perigee. This formula represents,
with remarkable precision, the corrections of the epochs of those tables,
which have been determined, by a very great number of observations, for the
six epochs of 1691, 1756, 1766, 1779, 1789 and 1801. By a most scrupulous
examination of the theory, I have not been able to discover any other lunar
inequality Avith a long period ; hence, it appears to me certain, that the [4746]
anomalies observed in the mean motion of the moon, depend on the
preceding inequality ; and I do not hesitate, therefore, to propose it to
astronomers, as the only means of correcting these anomalies.*
* (2755) It has not been found necessary to introduce this equation in the new tables
of Damoiseau, pviblished in 1824; since the elements lie has used, give very nearly the L4<46a]
VOL. III. 92
366 THEORY OF THE MOON ; [Méc. Cél.
We see, by this exposition, how many interesting and delicate elements
hai^e been deduced, by analysis, from observations of the moon, and how
[4747] important it is to multiply and improve them. Since, by the greatness of
their number, and by their correctness, we may more and more confirm the
various results of analysis.
The error of the tables formed from the theory, which is given in this
book, does not exceed a hundred centesimal seconds, except in very rare cases;
[4748] therefore, these tables will give, with sufficient accuracy, the longitude at
sea. It is very easy to reduce them to the form of Mayer's tables ; but, as
in the problem of the longitude, it is proposed to find the time corresponding
[4749] to an observed longitude of the moon, there is some advantage in reducing
into tables, the expression of the time in a function of the apparent
longitude. Considering the extreme complication of the successive
approximations, and the correctness of modern observations, the greatest part
of the moon's inequalities have heretofore been better determined by
observations than by analysis. Thus, by deriving from the tlieory those
coefficients which it gives with accuracy, and also the forms of all the
[4750] arguments ; then rectifying, by the comparison of a great number of
observations, the coefficients which it gives by approximations, with
some degree of uncertainty : we must finally obtain very accurate tables.
This is the method which has been used with success by Mayer and
Mason, and lately by Burg, who, by pursuing it, and profiting by the
late improvements in the lunar theory, has constructed tables, whose
greatest errors fall short of forty centesimal seconds. However, it would
be useful, for the perfection of astronomical theories, if all the tables
^^^^^^ could be derived solely from the principle of universal gravity; without
borrowing from observation any, except the indispensable data. 1 am
induced to believe, that the following analysis leaves but little wanting
to procure this advantage to the lunar tables ; and that, by carrying on
farther the approximations, we may soon obtain the required degree of
correctness, at least, as it respects the periodical inequalities ; for, however
great the accuracy of the calculations may be, the motions of the nodes and
same mean longitudes, at the epochs 1756, 1770, 1801 and 1812, as Burckhardt has
deduced from the observations made in that interval.
VII. Intiod.] INTRODUCTION. 367
perigee will always be best determined by observation.* [4752]
* ('2756) Since the publication of tliis volume, two very important works on the lunar
theory have been published ; the one by Baron Damoiseau, in the first volume of the
Mémoires présentés par divers sai'ans à F Académie Royale des Sciences ; the other by
Messrs. Plana and Carlini. We shall have occasion to speak of these works in the notes L^'^^"]
on this book, and shall now merely remark, that the object of them is to carry on the
approximation to such a degree of accuracy, as to be able to deduce all the inequalities from
the theorv alone.
368 THEORY OF THE MOON ; [Méc. Cél.
CHAPTER I.
INTEGRATION OF THE DIFFERENTIAL EOUATIONS OF THE MOON'S MOTION.
1. Resuming the differential equations [525], we shall put them under
the following forms,*
[4753] dt= "^^
General
"•'••\/' + ^/(^?)-S'
dv
1 /dq\ s fdq
h^u'\du) h^w^ '\ds
In these equations, t denotes the time, and we have, as in [499', 397] ;
M-\- m m', (x .■?/+ y ij'-\- z z') m!
(L)
[4756] Q =
i/(x'-xy-{-(y'-yr+(z'-zr
* (2757) The equation [4753] is the same as the first of [525], and if we multiply
the other two equations [525] by
they willbecome as in [4754, 4755].
VII. i. §1.] GENERAL DIFFERExNTIAL EQUATIONS. 369
M is the mass of the earth ; [4757]
m the mass of the moon ;* [4757']
m' the mass of the sun ; [4757"]
T, w, ~, the rectangular co-ordinates of the moon, referred to the centre of
[47581
gravity of the earth, and to the ecliptic of a given epoch, taken as
^ , , Symbols.
a fixed plane ;
x, y', -', the rectangular co-ordinates of the sun, referred to the same centre [4758']
and plane ;
r the radius vector of the moon ; [4759]
r* the radius vector of the sun ; [4759]
s the tangent of the moon's latitude above the fixed plane ; [4759"]
- the projection of the moon's radius vector r, upon the fixed plane ; [47G0]
V the angle formed by this projection of r and the axis of x ; [4760']
h^ a constant quantity [518 — 519], depending chiefly on the moon's [47G0"]
distance from the earth [4825, &c.].
In the value of Q [4756], the earth and moon are supposed to be spherical.
To obtain the true value, corresponding to the actual forms of these bodies,
we shall observe, that, by the properties of the centre of gravity, we must ^ ' J
transfer to the moon's centre of gravity the following forces ; first, all the
forces by which each of its particles is urged by the action of the particles of
the earth, and divide the sum by the whole of the moon's mass ; second, the
force by which the centre of gravity of the earth is urged, by the moon's
action, taking it in a contrary direction. This being jjremised, it is evident,
that (131 being a particle of the earth, and dm a particle of the moon, whose
distance from the particle dM is /, we shall have the forces by which the
moon's centre of gravity is urged, in its relative motion about the earth, by
means of the ])artial differentials of the double integral, f
(M+rn) ^ dM.dm
Mm -^-^ / '
* (2758) This value of to is used in the two first sections of this book ; but its
signification is changed in [4793], so that, in the rest of the book, 7nt represents the sun's
mean motion.
[4762]
[4762']
t (2759) If we substitute, in [455], the value of dJ\l [452], also
VOL. III. 93
370 THEORY OF THE MOON ; [Méc. Cél.
taken relatively to the co-ordinates of the moon's centre. Therefore, we
[4764] must substitute this function for , in the expression of Q [4756].
If the moon were spherical, we might suppose the whole mass to be collected
in the centre of gravity [470'"] ; and then, by putting V equal to the sum of
[4765] the quotients, formed by dividing each particle of the earth by its distance
from the moon'' s centre, we shall have [4767«],
[4766] ^ ff ^^m.V.
[4763a] f = \/\{=o'-^f+{y'-yf + {^-zf] [455«], it becomes, F=:/y;
and then, the corresponding force of the body M on the particle dm, in the direction — x,
— ) [455']. This accelerative force, acting on the single
particle dm, is to be decreased in the ratio of dm to m, to obtain the corresponding effect
[47636] Qp {]jg whole body m, of which it forms a part ; by which means it becomes — f — — .
Integrating this, so as to include all the particles dm, of which the body m is composed,
it becomes,-
pdm ^ dM 1 ^-dM.dm
[47636'] J — J ^^ O'"' »^./ ^^ '
which represents the value of V, to be used in finding the accelerative force of the body m,
from the attraction of the body M. If we change m, M into M, m respectively, we
shall get — Cr '- — • , for the value of V, to be used in finding the accelerative force
of the body M, from the attraction of the body m. Adding these two parts together, we
[4763c] obtain the complete value of F= T- + ^ j .yy '——, corresponding to the whole
accelerative force of m towards M, supposing M to be at rest. This is easily reduced to
the form [4763] ; and its partial differentials, relative to the co-ordinates x, y, z, give the
r -«/.<% ., accelerative forces parallel to those co-ordinates respectively. Now, when the bodies M, m
[4763a]
are spherical, these accelerative forces -— 7-, -—, -—^, are represented by the ^
[4763rf'] partial differentials of Q, taken relatively to x, y, z [499], retaining in Q [4756] only
the term Q^^= ", which is independent of the disturbing mass m' . Therefore,
r
[4763e] to notice the non-spherical forms of the bodies M, m, we have only to substitute the
expression [4763], m the place of , in the function Q [4756].
VII. i. § 1] EFFECTS OF THE OBLATENESS OF THE EARTH AND MOON. 371
* V would be equal to — if the earth were spherical ; hence, if we put
ôV= V ; [4767]
m,&V will be the part of the integral ff ^— , depending on the non- [4768]
sphericity of the earth. In like manner, if the earth be supposed spherical,
and we put V equal to the sum of the quotients, formed by dividing each
particle of the moon by its distance from the centre of gravity of the earth,
we shall have,
rr ^J^-^^ ^ M.V; [4770]
[4769]
and if we put
m
6 F = V , [4770']
r
M. sV will be the part of the integral ff — - — , depending on the non- [4771]
sphericity of the moon ; hence we shall have, very nearly,!
^TT — • / / ? = ■ h (M4-m) . { ^Ti -\ ■ > . [4772]
Mm ^-^ f r ^ ' ■' \ M in S
* (2760) If the mass m were collected in its centre of gravity, the integral ff — - —
dM . dM [4767a]
would become mf -^ ; and, by putting f —-^V [4765], it changes into m.V, as
in [4766]. The expression [4770] is found in a similar manner.
t (2T61) If we suppose m to be spherical, we shall have
/•^dM.dm „dM . ^ ,^„„ -,
JJ ^ — = "U-T' as in [4 /67a];
and if ^f also be spherical, [4772o]
.dM M , ^^dM.dm m M
/ y = 7 ; hence, ff—j^— =
Adding to this the parts m.SV, M.SV [4768,4771], depending on the non-sphericity,
we obtain the complete value of
ff — - — = — \-m.ôV-{-M.5V'. [47721]
■»«■ 1 ■ 1 • 1 ■ , M-\-m , . , , , M-\-m ^^dM. dm .___, , . ,
Alultiplymg this by — — , we obtain the value of -jr. — .JJ — [4/72]; which
372 THEORY OF THE MOON ; [Méc. Céî.
Therefore, in the preceding expression of Q [4756], we must augment the
M-\-m
term — ■ — , by the quantity,
[4773] <^M+m).\~ 4-~l= increment of Q [4756],
J^n=/«- in order to notice the effect of the non-sphericity of the earth and moon.
fron^t'be
foEï ^' ^^ shall, in the first place, suppose both bodies to be spherical, and
"frîhMd shall develop the expression of Q in a series. Now, we have,*
moon.
[4774]
[4775]
1 1
II we develop the second member of this expression, according to the
descending powers of ?', it becomes,
1 (xx'+yy'+z^^-lr^) {xx'+yy'+zz'-lr^f
+..^^y^.z^^^+^,.
Taking for the unit of mass the sum M-{-m of the masses of the earth and
■' moon, we shall have,t
j\t-\-7tl
is to be substituted for — ; — in the function Q [47636,4756]; and by this means the
general value of Q [4756] will be increased by the function [4773].
* (2762) The development [4774,4775], is the same as in [4655?», c], rejecting the
factor — m', which is common to all the terms. We may remark, that if we use the values
[4774a] 0Ï R, M-\-m [4655,4775"], the expression of Q [4756] becomes Q = ^ — ^,
which will be of use hereafter.
I (2763) If we put I for the latitude of the moon, we shall have, as in [4759"],
^''''"^ [31',34"'] Int.,
[4776t] tang.Z=.; sin./=^^^; cos.Z=^^^^.
If we proceed, as in [4659, Sic], changing r' into r, and U into v, we get,
[4776c] a; = r.cos.Z.cos.t); y = r.cos. Z.sin. ij; s = 7'.sin.Z= ?-5.cos.?.
[4776rfl Now, the projection of?-, upon the plane ot xy, is represented by r.cos.Z = - [4659a,4760];
VII. i. §2.]
DEVELOPMENT OF Q.
373
1 = M+ m = iJ- ;
r =
.r ^
2/ =
u
COS. D
sin.f
U ^
y/I+Tg
M
u
[4775"]
[477C]
Lunar co-
ordinates.
[4777]
[4778]
[4779]
We shall mark toith one accent, for the sun, the quantities u, s and v, u^-g^r
relative to the earth.* Then we have,t
1 + f.
{« »'. COS. (i)'— v)-\-uu'.ss' — hu'-. (l-)-ss)|2
(H-s'9)3.m4
Q__ » I '»'•"' / , ^ |mM^C03.(«'— «)+««'.Ss'— àM'2.(l-|-ss)|3
2.(1+s'2).m3
Value of
[4780]
substituting in this the value of cos. I [4776e], we get [4776] ; moreover, by substitutino-
the value of r.cos. Z [4776dl] in the expressions of x, y, z [4776c], they become as in
[4777—4779].
* (2764) By this means the solar co-ordinates become,
r' the radius vector of the sun ;
s' the tangent of the sun's latitude above the fixed plane ;
— the projection of the sun's radius vector upon the fixed plane ;
v' the angle formed by the projection of ?•' and the axis of x, or a;' ;
r
\/i+«y
m'
COS. v'
x'
«' '
sin. v'
y
=^
/ 5
t (2765) Substituting the value of R [4656], in [4774a], we get,
VOL. III. 94
[4777a]
[47776]
[4777c]
[4777i]
[4777e]
Solar co-
ordinates.
[4777/]
[4777^:]
[4777/i]
374 THEORY OF THE MOON ; [Méc. Cél.
[4781] "^'^^ sun's distance from the earth is nearly four hundred times as great as
that of the moon ; so that îi' is very small, in comparison with u ; and we
[4782] may, therefore, neglect terms of the order u'^, in the lunar theory. We may
also simplify the calculations, by taking the ecliptic for the plane of projection.
It is true, that this last plane is not fixed ; but, in its secular motion, it carries
the moon^s orbit with it ; so that the mean inclination of the moon'' s orbit,
upon the variable ecliptic, remains constant, and the phenomena, depending
on their respective inclinations, are always the same.
[47S3]
3. To prove this, we shall observe, that, from 5j 59, book ii., s' is equal to
[47841 . i ' ' 1
a series of terms of the form A; . sin. {v' -\-it -\- s) ; we shall represent it by*
^ 1 , m' m'.r- , „ , [xx'-\-yu'4-zz' — è r2)2 (xx'-\-yif-\-zz' — i r"2)3
L J ^ )■ J-' 2/-3 ' - c'o ' - r'^ '
Now, if we substitute the values [4776 — ^4779,4777e — A], in the first members of [47S0i,c],
they become, by shght reductions and using [24] Int., the same as in the second members of
those expressions,
[4780i] *^'+yy+~-^'= — -, • {cos.i!.cos.«'+sin.'y.sin.r'+«s'| = — 7.{cos.(r' — v)-\-ss'\;
, ,o cos.{v'—v]-^ss' A.(l+s2) n ii'. cos.( n'— v)-\-%i u'.s s'— I uK( \^ss)
[4/80C] xx^yyArZz'-lr-= — ■ —— = :;^^^^ .
By means of these values the expression of Q [4780a] becomes as in [4780] . For the
first and second terms of [4780a] correspond, respectively, to the first and second of [4780] ;
[4780rf] jjjg jj-ii,.^ Qf [4780a] gives the last of [4730] ; finally, the terms of [4780«], connected with
the factors | ot', ^m', by the substitution of [4780c], become respectively equal to the
terms connected with the factors f , J, in [4780].
*, (2766) Using the same notation as in [4230], we shall have, for the earth's latitude
s", above the fixed ecliptic, the expression,
[478g„] ,"=r/.sin.."-iy'.C0S.^" [1335'].
Substituting in this the values of jj", q" [4334], and observing, that
[4785a'] sin.t)".cos.(,§-< + (3)— cos.j)".sin.(^< + |3) = sm.{v"—gt — p),
we get the earth's latitude,
[47856] s" = ^.c.sm.{v"-gt-^).
Changing v" into the sun's longitude v' [4777f/], we get the sun's latitude,
[4765c] s' = S.c.^m.{v' — gt — fi).
This is of the same form as [4785], the constant quantities c, g, p, being changed into
[4785c'] k, — i, —s, respectively. Hence, the coefficient i is of the same order as the quantities
VII. i. §3.] INCLINATION OF THE LUNAR ORBIT TO THE ECLIPTIC. 375
s' = 2 . ^ . sin.. (v'-\- it -}- s) ; [4785]
i being a very small coefficient [4785f?], whose product, by m'î«'^ we shall
neglect. The value of s, neglecting quantities of the order s^, may be [4785']
represented by*
s = s, + 2 . A; . sin. (v + it + 0 ; [4786]
s^ being the tangent of the moon's latitude, above the apparent ecliptic. This
being premised, we have,t
[4780']
g, g", he, which are very small [4339,3113^]. The values [4339] are nearly g= — 36% [4785d]
g'= — 18*; these quantities may serve to give an idea of the magnitude of g, g', Sic.,
though they are not computed strictly by the method given in [1098, &ic.].
* (2767) If the moon were to move in the apparent ecliptic, her latitude above the fixed
plane, or its tangent, corresponding to the longitude v, would be ^.k.s'm.[v-\-i(-\-s) [4785]. M-gf i
Adding to this the quantity s, [4786'], we get, very nearly, the tangent of the moon's latitude
s, above the fixed plane, as in [4786].
t (2768) The quantity Q occurs in the first member of [4787], under a linear form
only ; therefore, we may take each term of Q [4780] separately, and compute its
effect. In making the substitution of any term of Q, we may consider the quantity
M.(l-|-««)^, and its powers, as constant. For, if we put (^=A.\u.(l-\-ss)~^'', for any
terra of Q, neglecting, for a moment, the variable parts contained in Jl, and taking the
differential of log. Q, we shall get,
(i Q J du , s ds
~Q ~ 'Tt~ ' l+ss '
hence.
T?)-o^
du J u ^ '
ds
l + ss ^
[4787a]
[47876]
[4787c]
[4787(f|
Substituting these in the first member of [4787], we find, that the terms mutually
destroy each other. Hence, it is evident, that we may neglect the first term of Q [4780],
which corresponds to b^\, A=l; the second term, which corresponds to b = 0,
and the last term, which corresponds to b= — 2, A = -
(!+*'«')»' """-"-"- ' -I- ^- -' -— 2.[\-\-s's')û'
Then using, for brevity, the following abridged symbol B, we get from [4780],
\uit'. COS. (w — v')-\-uu'.s s'—hu'^.[\-\-ss)\
B =
dq
(l + s's')i
3 m'. u'
(i+s'TjJ'
{l+s's').u^ '
\^B^ + ^B^+hc.l;
\B-\-^B^-^kc.].dB + hc.
[4787e]
[4787/]
[4787g:]
[4787/i]
376 THEORY OF THE MOON ; [Méc. Cél.
[4787]
[4788]
„ , ,, CcOS.ft! v') ") Cs. COS. (v 1'')
U^ À , 5«' ,' ■ o i i ''* • / /\ M
f -4--— .cos.-(v — t)') + &c. 1 f -.sin.(i) — v ) — s
Substituting, in the second member of this equation, the values of s', s,
[4785,4786], we get,*
Substituting the partial difterentials of Q, in the first member of [4787], it becomes,
3m'. «' ,„ , „^, C</s (dB\ fdB\ . , , /rf-B\ >
[4787V] — — --. B + iB^ . \ - . (—)—"« -(-r ) — (! + ««)• (-7-) \ •
^ •■ (l+«s')} ' ' - ^ \dv \dv) \du) ^ ^ ' \ds)')
The part of this expression depending on lŒ, in the last factor, is of the same form as the
first member of [4787], changing Q into 5; therefore, it has the property mentioned in
[4787i] [4787 i] ; that is to say, we may consider the powers of m.(1 -{-««)"" as constant. Now,
the last term of J9 [4787/] corresponds to the power — 2 of that quantity ; therefore, we
may neglect its partial difterentials, and, in finding AB, may use the remaining terms as in
the following expression ;
[4787/fc] B = 7^x77, • 1 ""' "'• COS. {v — v')-\- M-' m'. s s' I .
The partial differentials of this expression give,
ds /'dB\ «' C ds . , ,.7
[4787m] -"^-O^dzS'T.»- l^-cos.(i— i-O+^'-^^'l
du J (\-\-s's').u'
^.'l.
[4787,v] -(^+'')-(^) =
(i-j-^yj-M
Adding these three expressions together, we find, that the terms depending on s-s' destroy
each other, and we get,
ds /dli\ fdB\ ,, , . /dB\ u' ( , ds ,. ,-)
Now, if we retain, explicitly, the terms of B [4787/], w hich do not contain s, s', we obtain,
[4787;,] B + fS^ = '^'. ^^cos. {v — v')-~+^£. cos.2(^ _ ^') + &ic. ^ .
Substituting the expressions [4787o,p] in [4787A'], and neglecting terms of the third order in
s, s', it becomes as in the second member of [4787].
* (2769) If we substitute the values of s', s, [4785,4786], in the last factor of [4787],
VII. i. ^ 3] EFFECT OF THE SECULAR MOTION OF THE ECLIPTIC. 377
*"' Jcos.fî' — v') — 7: \-'— .cos.^ft! — î)') + &ic. ;.<«,. COS. (î) — V) — '.sin.(i; — v')>. [4789]
ifi I ^ ^ 2u ' 2u ^ ' ' ) ( ' ^ ' dv ^ '))
Hence the equation [4755] becomes,*
elds , , i.ml.u'^Si-\-hc.
^^lv-^'+—^ Z^T^— ; [4790]
or,
dds
'¥+^sO^ '
[4791']
0=1-2 + 5 + ^-IT^ + &C. [4790']
If we neglect the excentricities and inclinations of the orbits, we shall have
M = -, u'=— [4826,4833]; a' and a being the mean distances of the [4791]
sun and moon from the earth. We shall see, in the following article [4826],
that h-== a, very nearly ; therefore, we shall have [4791 (/],
we shall find, that the terms depending on k mutually destroy each other. For these terms
produce, without reduction, the following expression, neglecting quantities of the order
mentioned in [4785'] ;
2.t.|sln.(t)-|-i^-|-s).cos.(j; — v') — cos.(« -|-*'^ + ^) -s'"^- (" — ^') — sin. ('u'-|-i<-|-e)|.
The two first terms, between the braces, are reduced by [22] Int. to
sm.{{v-{-it-{-s) — (t) — v')l = sin. [v'-\-it-\-s) ;
which is destroyed by the third term. The remaining terms of [4785, 4786] are «'=0,
s z^ s, ; substituting these in the last factor of [4787], we obtain the expression [4789].
[4789o]
[47896]
[4789e]
* (2770) Multiplying together the two factors of [4789], we find, that the product of
the term cos. (« — v') by x,.cos. (d — v'), produces Js, , disconnected from the periodical [4791a]
angle v — 1; ; so that we may put the expression under the form -2 — '. JJl — '; as we [47916]
shall soon see, that it is not necessary for the present object to mention particularly the parts
included in the general term + &c. This represents the value of the function in the first
member of [4787], and if we divide it by h^u-, it produces the three last terms of [4755] ;
which will, therefore, be represented by J-^Ll^^!/ + ^- . Substituting this in [4755], [4791c]
and dividing by 1 + 75''/(t)--^' we get [4790]. Reducing the denominator of
the last term of this expression into a series ; neglecting m'^, and observing, that
idi) l'^^^^^ 'S of the order m'u'^, it becomes as in [4790']. Finally, substituting in [4791rf]
this the values of u, u', h^ [4791, 4791'], we get [4792].
VOL. III. 95
378
THEORY OF THE MOON ;
[Méc. Cél.
[4799]
[4793]
Change
ia m.
[4794]
[4795]
[4796]
[4797]
[4798]
We shall put mt for the Burl's mean motion ; so that m will no longer denote
the moon's mass; we shall have, by ^ 16 of the second book,
»r =
Then, if we sui^pose the time t to be represented by the moon's mean
motion, which can always be done, we shall have -^ := 1 ; therefore,
0 = ^. + 5 + l.m^s,+ &c.
Substituting, in this equation, the value of s [4786], and observing, that we
may, in this case, change it into iv, we shall have,t
0 = — ; + (1+ f . m-) .s^+2.k.{l — {i +\yi. sin.(« + i î) + 0 + &c.;
which gives, for the part of s, relative to the secular motion of the ecliptic, Î
* (2771) If we change, in the equation [605' or 3700] , a into a', and n into m, to
[4794a] conform to tlie notation [4791, 4793], we get m^^= tx.a'~^ ; ij. being tlie sum of tlie masses
of the sun and earth. If we neglect the mass of the earth, in comparison with that of the
sun, we have |A = )ft' [4757"], and the preceding expression becomes as in [4794]. In
the moon's motion about the earth, the equation [605'] becomes n-=^{M-\-m). a~^
[4757,4757']; and, as the moon's mean motion nt, is here represented by t [4794], we
have M=l ; substituting tliis, and M-\-m = \ [4775"], in the preceding value of n^,
we obtain 1 =«"3 ^g in [4795]. Dividing the value of nv'' [4794] by this last expression,
[4794c]
[4794d]
we get iir
substituting this in [4792], it becomes as in [4796].
[4798a]
[47986]
[4798c]
t (2772) The terms neglected, by writing iv for it, are of the order of the
excentricities and inclinations, multiplied by the very small quantity i, and connected with
terms containing sin. cv, s'm.gv, and their multiples, as is evident from [4828, 4794c].
All the neglected terms are considered as being included in the general expression +&ic.
Now we have,
(Ills
s—s,-\--2.ksm.{v-\-iv-\-s) [4786,4797]; hence —
substituting these in [4796], we get [4798].
:'^-2.t.(»+l)=.sin.(«+n.+£);
J (2*73) This equation is of the same form as [865], which is solved in [871] ;
changing y, fl^ t, m into «, , l + |m^ v, 1 + i, respectively ; and putting for a Q, or
[4799a] aK, the terms under the sign 2 [4798]. These changes being made in [871], it becomes
as in [4799], by a slight reduction, and changing
a; the signs in the numerator and denominator.
vil. i. § 3.] DEVELOPMENT OF Q AND ITS DIFFERENTIALS.
379
S.{'ii-\-P).k.s\n.{v-{-iv + i)
This last quantity is insensible ; for i v, at the most, does not exceed fifty
centesimal seconds [ = 16',2] in a year;* and ^nrv expresses very nearly, as
we shall hereafter see [4800f/J, the retrograde motion of the nodes, which
exceeds 19' [3373] ; therefore fm^ is at least four thousand times as great
as i ; so that we may neglect the term,
^.k.\\—{i+\f].ûn.{v + iv-^î),
in the differential equation [4798] ; and then this equation becomes
independent of every thing connected with the secular motion of the ecliptic.
The mean inclination of the moon's orbit to the apparent ecliptic, is one of
the arbitrary quantities of the integral of this equation ; hence we perceive, that
on account of the rapidity of the motion of the moon'' s nodes, this inclination
is constant; and the latitude s^ of the moon, above the apparent ecliptic, is the
same as if the ecliptic loere immoveable. We may, therefore, suppose s' ^= 0,
in the following investigations ; which will simplify the calculations.
[4799]
[4800]
[4801]
[4802]
Inclination
(if the lu-
nar orbit
tu the
apparent
ecliptic.
[4803]
[4804]
Therefore, we have, by neglecting quantities of the order m' u'^ s\ m' ?<'^,t [4805]
* (2774) This agrees nearly with the remarks made in [4785(/], relative to the value
of i. Moreover, tlie retrograde motion of the nodes is expressed by (g — 1) .v [4817], and
the values of m, g [5117], give g — l^^m^ nearly ; therefore, the retrograde motion of
t!ie nodes is nearly equal to ^m^.v, as in [4800]. The same result may be obtained
analytically; for, if we neglect terms of the order p"^, e'^, the motion of the nodes [5059]
becomes \p".v. Now, by comparing the coefficients of sin.(^D — <)), in [5053, 5049], and
retaining only the first term of each of them, we get,
V
$ m
[5094] ;
[4800a]
[48005]
[4800c]
[4800rf]
[4800c]
whence, the motion of the nodes becomes iii".v = ^nfi.v. This exceeds 19'' in a year
[3373] ; which is more than 4000 times the value of iv, assumed in [4785rf] ; hence the
term of s, [4799] must be insensible, and we may, therefore, neglect the corresponding terms
of [4798], which are given in [4802]. Then all the remaining terms of [4798], which
are included in the expression -|-&c. [47986], maybe considered as independent of the
secular terms arising from i.
t (2775) Substituting s'^0 [4304] in the value of Q [47S0], it becomes, without
any reduction, as in [4806a]. Developing the powers, and neglecting terms of the orders [4800/]
mentioned in [4805], it becomes as in [48066]. This is reduced to the form [4806c] by
380 THEORY OF THE MOON ; [Méc. Cél.
[4806]
+ !!LL!i_.^3.(l— 4s^).cos.(îJ— 'd') + 5.cos.(3î;— 3t0|.
[4807] Hence we get, by neglecting quantities of the order m'u'^s^,*
\duj ^ u \clsj (1+s')^ 2m3 ' ' ^ ^*
[4808]
_ ^'^'* . I (3 _ 4s"-) . COS. (v—v') + 5 . COS. (3 ^— 3i;') | ;
using [6, 7] Int. ; and if we connect the terms depending on the same powers of ?t' it
becomes as in [4806J ;
Cl+^^-[uu'.cos.{v—v') — iu'^.{l+ss)f
[4806a] q= - \-m'u'.{ " n_L ws
[48065] =^ + mV. j ,3 _^,,
3,,'2 3ji'3
0+^2-[i+àcos.2(î,-^')]-^-(l+")-cos.(.-î)'),
«
[4806c] =-—^ + rm,
+ ^.Ucos. (,,-,;') +icos.3(i;-i;')]--—7,—
* (2776) The partial differentials of Q [4806], taken relatively to v, s, u, become,
without any reduction, as in [4809,4810,4810a], respectively. Multiplying [4810] by
-, we get [48106] ; adding together the expressions [4810a, Zi], and making some slight
u
reductions, we get [4808] ;
1 m' m'3
[4810a] V(i«y ~ \ 3 m' «'1
^ ^ ■* ^-.[(3— 12s2).cos.(t) — t)') + 5cos.(3^ — 3u')]'
s /dQ\ ss m'.u'^s^ 3 m'. «"Isa . ,,
[48105] û-UJ=-fî+^)î ^ ^^.cos.(.-.)
[du) i
VII. i. §4.J APPROXIMATE VALUES OF s, u, t. 381
I -^] =z -— .sin.C2t> — 2i))
\(lv J 2»- ^ •^
[4809]
— '!l^^-.l3.(l—As-).sm.(v—v')-^l5.sm.(3v—3v')\
(1Q\ us m'.u'^s 3m'. u"^ s , ,, r^oim
4. 7*0 integrate the equations [4753 — 4755], we shall observe, that, by
excluding the sun's disturbing force, the moon will describe an ellipsis, in
which the earth occupies one of the foci. We shall then have, as in [532,533],
[4810']
S = /.sin.('y — 0; [4811]
u^j^,j^y\n + ssy+e.co..iv-.)\. ;^4812]
5, 11, in an
iiiv iriahle
eliipsia.
In these equations, y is the tangent of the inclination of the lunar orbit ;
d the longitude of its ascending node [533"] ; e and w are tivo arbitrary
quantities, depending chief y , on the excentricity of the orbit, and on the ^
position of the perihelion [534']. y and e are very small quantities. If we
neglect the fourth power of 7, we shall have,*
[4813]
[4815]
U =
/,3.(l + y3)-n + l>' + e.cos.(p— ^) — iy^cos.(2t;-20|. [4816]
In this value of u the ellipse is supposed to be immoveable ; but we shall
soon see, that in consequence of the sun'' s action, the nodes and perigee of this
ellipsis are in motion. Then putting,
(1 — c).v = the direct motion of the perigee ;
{g — \).v = the retrograde motion of the nodes ; ['iS,\é]
* (2777) Developing (l + s.s)i, according to the powers of s, substituting [4811],
neglecting s^, and reducing, by means of [1,3] Int., we get, successively,
(l+,s)i = l+J,a_,,4
= l + |4i — 2Cos.(2y— 20)^ — Ç.^f — |.cos.(2« — 20)+|cos.(4j;— 4â)f
= (1 + ^7"— sV/) — (ir'— tf/)-cos.(2« — 20) — ^ij^4.cos.(4t,— 4ô).
Substituting this in [4812], and neglecting y'^, it becomes as in [4816]. We have retained
the terms of the order 7^. in [4812a], because they are required hereafter. '■ ^
VOL. III. 96
[4812a]
382
THEORY OF THE MOON :
[Méc. Cél.
we shall have, from [4811,4816],*
[4818] s = '/.sin.(gv—è);
1
[4819] u^—j---^.{l + l7'+e.cos.(cv — ^) — {';~.cos.(2gv—2^)\.
Assumed ^ \ I J
forma of
movcabiJ" If we substitute this value of «, in the expression of dt [47531, observing,
ellipsis. L J o
[4820] x}asX, if we neglect the solar attraction, \f] vanishes ; we shall have,
( l + l.fe'^+j.^)— 2e.(]+|e^+f7")-cos.(c«— ^)
[4821] dt = h^. dv . ) +|.e'.cos.(2cv— 2?^)— e^cos.(3ci)— 3n)+i7^cos.(2^i'— 2i') \.
—^.e7^.{cos.{2gv+cv — 2t) — i^)+cos.(2^î;— ct' — 2d+j:) \
/
[4891a]
[48216]
[4821c]
[482W]
[4821e]
[4821/] c'd —
[4821gr]
[4891A.]
[4821i]
[48914]
[4821?]
[4821m]
* (277S) The object of this article is to obtain approximate vakies of m, w', s, v',
expressed in terms of v ; for the purpose of substituting them in Q, and in its differentials ;
as is observed in [4838']. Now, s, ii [4818,4819], are the approximate values of s, v,
corresponding to the equations [4755,4754], noticing two of the most important perturbations,
namely ; the mean motions of the perigee and nodes. Substituting these in [4753], we
get tlie approximate values of dt, t [4821, 4822], which are afterwards corrected in
[5081,5095]. In finding the approximate value of dt [4821], from [4753], tlie term
glected, and then [4753] becomes dt = -—;^; in which we must substitute the
value of u [4819]. In making these substitutions, we shall put for a moment, for brevity,
f — 1 y2 A^^. cos. (2^ « — 2d) ; and, during the process of the calculation, we shall omit the
symbols ê, w, ra', ivhich are connected respectively with the angles gv — è, cv — ra,
^'^ c' mv — zi'; taking care to re-substitute them at the end of the o2}eration. This
abridged form of writing the angles, will be used frequentlij, in the notes which follow ; it
saves considerable labor, renders the formulas more simple, and cannot be attended with any
inconvenience. Hence, the preceding expres.?ion of tZ^ [4821rf] becomes as in [4821A];
developing the factors, and neglecting Z^, fe^, eS 7^ he, we get successively [4821 i,fc,Z].
Substituting the value of / [4821e], and reducing, by means of [6,7,20] Int., we get
[4821w]: connecting together the terms depending on the same angles, we obtain [4821];
whose integral is as in [4822] :
dt=P.{l+7'')-.dv .\l + {f+e .cos.cv)]-^
^p(^lJ^2y^}.dv.\l—'2{f+e.cos.cv)-j-3(f+e.cos.cvf—'i(/+e.cos.cv)^
^p(^lj^2y^}.dv.\l — 2e.cos.C!) + 3e2.cos.^c«— 4e^.cos.3ct) — 2/+6/e.cos.c?)}
=P.dv.{{l-^2y^) — 2e{l + 2f).cos.cv-{-2e~.cos.^cv — Ae^.cos.hv— 2 f-\-6fe. cos. cv\
(l_|-Oya)_2fi(l-f27^) .cos.ciJ + ae^. (1 + C0S.2CI')— e^(3cos.c!;+cos.3ci))^
=li^-^'i'-^_i^^,^j^^^2_f,f^^^2gv+iey^.cos.cv—^ef.[cos.{2gv4-cv)+cos.{2gv-cv)]<^'
VII. i. § 4] APPROXIMATE VALUES OF s, u, t. 383
This gives, by integration,
/=constant+/i^f.(I+|e^+|r)— — .(l+|e=+|r).sin.(CT— ^)
.sin.(2c!; — 2^) -— .sin.(3cj; — 3sî)+— ^.sin.(2^tj — 2^) [4822]
4c 3c ^ ^S
— r7T-f-l-sin-(%»'+<^«— 2^— '')— T7^--N-sin.(2ffî}— «;— 20+ra) ;
4. (-2^+0) ^ ^ '^•(2^ — 0
the coefficients of this equation are modified a little by the sun's action, as
we shall hereafter see [5081, 5095].
In the elliptical hypothesis, the coefficient of v, in this expression, is, by Mgoo'i
3,
[541' — 543], equal to a^ ; which gives,*
/t3.(l + |e^+|^2)=a^; [4823]
« being the semi-major axis of the ellipsis ; hence we have, [4824]
h = a^,(l — ie"'-hy); [4825]
consequently,
u = --^.{l + e-+lf-\-e.(l + ec).cos.(cv—^) — ly-.cos.(2gv—2ê)}. [4826]
_3_
Then, by putting n= a ^ [482Sa], we get,t [4827]
* (2779) Substituting (x=rl [4775"], in [541'J, we get n= a ^; hence [543] gives
«-f-a^e=«-y + &c. ; in which the coefficient of îj is a-". To make this conform to the ' "''
result of the elliptical theory [4822], we must put the coefficients of v equal to each other;
hence we get [4823]. Dividing this equation by the coefficient of h^, and taking the [48236]
cube root, we obtain h [4825], neglecting terms of the fourth order in e, j. This value
of A gives A=.(l + y^) = a.(l-eS); whence, ^^-_ ==:l.(i+e^) ; substituting ^4823^]
this in [4819], we get [4826].
t (2780) Multiplying [4823] by 1— 1-/, and neglecting /, we get
substituting this in the third term of the second member of [4822] ; also [4823], in the
3
second term, and putting the constant quantity equal to — oF^c-, we shall obtain for these
384 THEORY OF THE MOON ; [Méc. Cél.
[4828] nt + s=^ v — ~.(l—^f).sm.(cv—z,)+'^-^.sin.{2cv—2^)
Ac
2
mate .^ .Sill, (o CV O 'CJ ) H . oui. \ ^ii u -i-o i
value of 3C '^ 4^ ^ ^ ^
nt-\-s.
Q 2 O 2
~ 77iï^T~^ • sin- (25- î' + c tJ — 2 d — ^) — — -""-^^^ . sin. (2 £• 15 — c I' — 2 1' + ta ) :•
4.(25-4-0) vol ^ 4.(2^— c) ^ ° ' ^'
£ being an arbitrarj constant quantity. In substituting nt-\-e, we may
r482Ql
suppose c and g to he equal to unity [5117], and neglect quantities of the
order e^, or ey^, in the coefficients of the sines. Thus we shall have, by
retaining the term depending on sin.(2^t' — cv — 2i-\-vi), which will be
useful hereafter [4828f/] ;
[4830] ^*^ + ^ = « — 2e.sin.(ctJ — a)-f-ie^sin.(2c?;— 2^) -\- \y^.sm.{2gv — 2è)
— fey^ sin.(2^w — cv — 2<) + ^).
[4831] If Yve mark Avith one accent for the sun, the symbols relative to the moon
Approxi- •^
riucsof [4779'], and observe, that /= 0 [4804], we shall have,*
t, u'.
[4832] n't + s' = v'—2e'.sm. (c'v'—^Z) + 2 e'^.siu. (2c'v'— 2^') ;
u'= -,. n + e'^+e'.n + e'^).cos.(c'v'—z,')\.
[4833] a' ' ' ' \ ' -> V /5
[4834] The origin of the time t being arbitrary, we may suppose ; and s nothing,
3 3 2e a
[48286] three terms, the expression — a- e-\-a- v .«^.(1 — j}^).sin.(fi.'— to). Substituting this in
[4822], then multiplying the first member by n, and the second by its equivalent expression
a ^ [4823a], it becomes, by slight reductions, as in [4828] ; observing, that, in the second
_3
[4828c] ^n*^! 'h'^*^ ''"^^ °f [4822], we may put h^a - ^ 1 [4823], since these terms are of
the second or third orders in e, j. Now, putting c and g equal to unity, in the coefficients
of [4828], and retaining terms of the second order in e, /, also the term depending on the
r4828(/l angle 2^t) — cw, we get [4830]. The reason for retaining this term, is on account of the
smallness of the divisors introduced by it, in consequence of 2°- — c being very nearly equal
to unity. For the values of c, g, m [5117], give very nearly,
[4828e] c=l— #w^ ^=l+f'K^ 2^— c=l+3m-.
* (-2781) The values [4832,4833], relative to the sun, are deduced from those of the
[4832»] moon [4830,4826], by merely accenting the symbols, as in [4779']; observing also, that
s'=0 [4804], corresponds to y'=^ [4818].
vu. i. §4 ] INVESTIGATION OF v', u', IN TERMS OF v. 386
If
and then putting - = m, the comparison of the values of nt and n't will [18351
give,*
n
*
v' — 2e'. sin.(c'i''— ^') + f e'^ sin.2(c'îj'— ^')
= in V — 2m e . sin. (c v — ^) + t »« e-. sin. ('2 c» — 2 ra)
^ ^ ■* ^ ^ [4836]
+ { m.y. sin. (2gv — 2 o) — ;^ mey~. sin.(2o-r — cv — 2 â + w).
Hence we deduce, by observing, that c' varies but very little from unity,t [4836']
* (278-2) If we take, for the origin of i, the moment when the bodies are in their mean
conjunction, or ni-\-s equal to n't-\-i', we sliall have s^e'=0. Substituting these in [4834a]
[4830,483:2], we get the values of 7it, n't. Multiplying the former by m, and substituting
mn = n' [4835], we get an expression of n'l, wiiich is to be put equal to that in [4832] ; [48345]
hence we get [4836].
t (2783) We may obtain v' from [4836], by means of the theorem of La Grange
[629c], which, by changing ■\'X into x, then x into v and t into t, becomes,
v'—F{v') = t; ^4837„^
,_,,,,, d.Fiif , d2.F(t)3 „
i,'=t+F(t) + è.^ + i.-Ai+&c. [48376]
Comparing the equations [4836,48.37a], we find, that t represents the second member of
the equation [4836], and, tliat
F{v') == Se'.sin. {c' v'—-:) — ^é^.sm.{2c'v'—2i^'). ^483^,
Changing v' into t, we get F{t) [4837e], its powers [4837/], and the differentials [4837^],
omitting, for brevity, the symbol — tt/, which is connected with c't ; the reductions being
made by means of [1,2, 17] Int. Substituting these in the second member of [48376], we L^SJ/^/]
get v' [4S37A] ;
F(t) = 2fc'.sin.(c't — ^') — ^e'-. sin.(2c't — 2ûj') + &ic.; [4837,]
F{xf=2<:'K (l-cos.2c't) — |e'3. cos.c't+&:c. ; F{lf=6e'\ sin.c't + &c. ; [4837^]
è--^=2É'-2.sin.2c't + fe'='.sin.c't + &c.; è- -5^ =— e'^. sin-c^t+Sic. ; [4937^]
„' = t + (2e'-^ «'=*). sin.(c't-t.')+|e'2.sin.(2c't- 2^')- [48.37;^]
Now, t represents the second member of [4836], and the substitution of this value in the
first term of [4837A] produces the four first terms, or the two first lines of the second [4837i]
member of [48-37]. The last term of [4S37AJ produces the last term of [4837], by putting
for t the first term mu of the second member of [4836] ; it being unnecessary to take any
other term of t, because m is of the same order as e, or e'. To obtain the value of the [403711
second term of v' [483TÂ], we must have the expression of sin. (c't — -a'). Now, as this
VOL. III. 97
386 THEORY OF THE MOON ; [Méc. Cél.
[4837]
v'^m V — 2me . sin. (c v — in) + i m e". sin. Ç2.C v — 2 ra)
+ i m 7 -. sin. (2gv — 2è) — ^ m e f. sin. (2gv — c v — 2 è -{- -.->)
Approxi-
TilTesof + 2e'. (1— ie'^) .sin. (c'mv— ^') — 2mee'. sin. (c?; + c'mv — ra — xa')
r , M .
— 2 m e e'. sin. (cv — c'mv — ^ + ^') + f ^'^' ^i"* (^ c' w î' — 2 -/)
, 1 C l + e'.(l— ie'2).cos.(c'?»i'— w') + «'^-cos.(2c'mî; — 2ra') )*
«' ' ( -t~''''fi6'.cos.(c« — c'mv — zs-^-z}') — me e'. COS. {cv-{- c'mv — s — u') ) '
5. We must substitute these values of u, u', s and v', in the expression
[4838'] of Q [4806], and of its partial differentials [4808—4810], which will, by
this means, be developed in sines and cosines of angles proportional to v ; but
it is necessary, for this development, to establish some principles relative to
term is of the order e', it will be sufficient to take the two first terms of [4836], namely ;
[4837i] tz=mv — 2me.sm.{cv — a); whence, f't — ■ûi'=:(^c'mv — ra') — 2c'me.sm.{cv — «).
Developing the sine of this expression, by means of [60, 18] Int., neglecting e-, we get,
successively,
[4837m] sin, (c' t — ra') == sin. (c'm v — to') — 2c' m e . sin. {c v — zs) . cos. {c'mv — ra')
[4837n] :=sin.(<''mf — ~/) — c'ine .sm.{cv-\-c'mv — ra — a') — f'me .sln.(f v — c'mv — ■a-\-z!').
Multiplying this by its coefficient 2 e' — i e'^, or 2 e'. (1 — ^ e'^), neglecting terms of the
fourth order, and putting c'= 1, we get the sixth, seventh and eighth terms of [4837].
* (2784) To obtain u, we must substitute the value of v' [4837] in [4833] ; and,
as we retain terms of the third order in e, c', /, m, in [4838], it is necessary to retain
those of the second order in v' [4837]. Hence, if we put for a moment, for brevity,
[4838a] ~ = 2 c'. sin. {c' mv — ra') -|- J e'^. sin.(2c'?rt v — 2 to') — 2m e . sin. {cv — to) ;
and observe, that c' is very nearly equal to unity, we shall have, from [4837],
[48386] v'=7nv+z, and c' v'—zi'—{c' mv — to') + s.
Its cosine, reduced by formulas [23,4.3,44] Int., becomes, by neglecting z^,
[4838c] cos. (c' v' — w') = cos. z . cos. (c' mv — to') — sin. z . sin. {c' m v — to')
[4S38(i] = (1 ~ 2 ~^) • cos. {cm v — to') — z. sin. {dmv — to') ;
hence,
e. (l + e'2).cos. (c'jj — to')
[4838e] =e'. (1 + ê'-) .cos. (c' mv — to') — | e' z^. cos. (c'mv — to') — e' z . sin. {c'mv — to').
Now, substituting the value of z [4838a], in the first members of [4838^, A], neglecting
VII. i. §5.] REMARKS ON THE DIFFERENT ORDERS OF THE TERMS. 387
the magnitudes of the quantities which enter into these functions, and on the [4839]
influence of the successive integrations upon the different terms.
The value of m [o\\l\ is very nearhj equal to the fraction ^\ \ loe shall [4840]
consider it as a very small quantity of the first order. The excentricities of o,do,. or
the orbits of the sun and moon, and the inclination of the lunar orbit to the
ecliptic, are nearly of the same degree of smallness [5117, 5194]. Thus, %oe g^^
shall regard the squares and products of these quantities, as very small
quantities of the second order ; their cubes and products of three dimensions,
as very small qua7ititics of the third order; and so on for others. The sun's
m' «'•*
disturbing force is of the order* -A5-, and we have seen, in ^ 3, that this [4842]
quantity is of the order m", or of the second order. The fraction -, being
very nearly equal to ^i^, may be considered as of the second order. We [4843]
shall carry on the approximation to quantities of the third order inclusively ;
terms of the fourth order, also those depending on the angle Sdrnv — 3 ro', we get,
successively, by using [31, 17,2] Int., the following expressions; omitting, for brevity, the [4838/]
symbols -n, n', as in [4821/] ; y
— y'z^. COS. (c' mv — •n') z= — e' '. (2 sin. d mv . cos. c m v) . s'm. d m v
= — e'^. sin. 2 c' m V . sm. c' m V :=^ — i e' ^. cos. d mv:
[4838g-]
— e'z.sin. {d mv — z/) = — e'^. (1 — cos. 2 c' m «) — |e'^. cos. cm «
-\-me e'.cos. [cv — d inv) — mee'. cos. (cv -\- dmv).
Substituting [4838^, h'\ in [4833e], we get, by connecting the terms,
e'. (1+ c'2) . cos. (c » — w') = — c'^+ e'. (1— i e'^) . cos. d mv -\- e'"-. cos. 2 d m v
-\-mcd. cos. {cv — dmv) — mc e'.cos. (cv + dmv).
Finally, by the substitution of this, in [4833], we get [4838].
[4838/i]
[4838i]
* (2785) The accelerative forces [4763rf'J, are represented by the partial differentials of
Q, relative to the co-ordinates. Thefe partial difltrentials occur in the general equations
[4753 — 4755], and are computed in [4807— 4810J. Now, if we compare the part of ^^ '* "^
[4808 or 4810], which does not contain the disturbing mass m', with the chief term of the
same equation, depending on this disturbing mass, we shall find, that it is of the order
^4-' °^ '^ [4791]; which, by means of [4794, 4795], is of the order m^. l'*S'*26]
388 THEORY OF THE MOON ; [Méc. Ctl.
and in the calculation of these inequalities, toe shall take notice of quantities
[4844] of the fourth order;* but we must take particular care not to omit any
quantities of that order in the integrals.
The equation [4754] becomes, by development, of the following form,t
„ (Ida , -.-TT
[4845] 0 = — ^+iV-. M + n ;
[4845'] N^ differs from unity but by a quantity of the order «r [4845c], and n is a
series of cosines, of the form /t.cos. (/ y + ;) [4961]. The part of «,
[4o4o]
relative to this cosine, is represented, as in [870', 871], by
[4847] „__A_.cos. (it' + O-
Now, it is evident, that if r differs from unity by a quantity of the order m,
[4848] j^j^g jgj.,^ k.cos.Çiv -{- ô) acquires, by integration, a divisor of that order;
which increases the term considerably ; so that it will become of the order
^ ^ r — 1, if it be of the order r, in the differential equation. We shall see
[4850] hereafter, that the greatness of the inequality named the evection, arises from
this cause. t
* (2786) The angles connected with coeflicients, as far as the third order inclusively,
[4844a] are retained ; and, in computing the coefficients of these terras, the approximation is carried
on, so as to include terms of the fourth order.
t (2787) The chief inequality of M [4819], is that depending on cos. (ci; — ra), which
we shall represent by e.cos. (cd — ra) ; putting the other terms equal to Su, so that
iAaA^„-\ . dilu „ , d-. (]u
iwioai „^e_cos_(-c!) — w) + <Sit. Its difierential gives — = — c^ e. cos. (cv — îi) + -^ .
Multiplying the first equation by c^ and adding the product to the second equation, we get,
(Wu , „ d'^.Au ,
U c~ u= \- c~. au.
[48455] dfi ^ dr2 ~
Putting the second member of this last equation equal to — n, we get,
''''" 12 IT.
\- c" u = — IT
[4845c] dl^ '
and this is of the same form as [4845] ; N^ being changed into c-, which differs fiom unity
by a quantity of the order 3 m- [4828e].
X (2788) The evection depends on the angle 2v — 2mv—cv-\--m, and its cosine is
multiplied by ./3/"e, in the expression of <5k [4904]. Now, in finding ^/i>, from the
[4850a] equation [4999], we must divide by the factor 1 — (2 — 2/« — c)^ which is of the order
m ; and by this division its value is very much increased.
VII. i. § 5.] REMARKS ON THE DIFFERENT ORDERS OF THE TERMS. 389
The terms where i is very small, and which depend only on the sun's [4850']
motion, do not increase, by integration, in the value of u ;* but, it is
evident, from the equation [4753], that these terms acquire, by integration, [4850"]
the divisor /, in the expression of t ;t we must, therefore, pay great
attention to these terms. It is on them, that the magnitude of the annual [1851]
equation depends.
The terms of the form k .dv.sm(iv-{-!), in the expression of (~^)-~Tj [4852]
[4753, 4754] acquire, by the integration of that differential expression, a divisor
of the order i, in the value of u. Hence, it would seem, that in the expression
of the time t, these terms ought to acquire a divisor of the order r, which
would render them very great when i is very small ; but, it is essential to [4853]
dbserve, that this is not the case, and that, ifive only notice the first poioer of
the disturbing force, these terms will not have the divisor r, in the expression
of the time. To prove this, we shall observe, that by [1195, Sic], the
expression of v, in a function of the time, cannot acquire a divisor of the
order r, except by means of the function — 3af)idtfdQ;t in which the [4854]
[4853']
* (2739) When i is very small, the divisor i^ — JV^ [4847] becomes nearly equal to
— JY'^, which is of the order — 1 [4845'] ; consequently, the term [4847] is not
increased by this division.
[48506]
t (-2790) If the development of the denominator of ilt [4753] contain a term of the
form A,-.cos.(/ 1'-|~-')> arising from u^, it would introduce in dt a term of the form
Jc [4851a]
k.dv.cos.{iv-{-;) ; whose integral would introduce in t a term of the form 7 .sin.(w'-[-;),
having the small divisor /, as in [4851].
I
I (2791) The differential of Q [4774fr], relative to the characteristic d, gives,
d-R=— ^— dQ; hence fdR=^-.—fdq. [4854a]
Substituting this, and ij.=1 [4775"] in ^ [1195], we get,
^ = 3 « .fn dt.~—3a .fn dt .J\\ q. [45546]
Now, the first term of this expression has only one sign of integration, and can, therefore,
introduce only the first power of the divisor i [1 196', &c.] ; and, if we neglect tliis term, we
shall have,
I = _3 a ./n dt ./d q, as in [4854]. ^^^^*'^
VOL. III. 98
390 THEORY OF THE MOON ; [Méc. Cél.
differential dQ refers only to the co-ordinates of the moon. If Q contain a
[48o5] term of the form k .cos. (it -{-;), i being very small; this term cannot acquire
a divisor of the order r, except dQ does not acquire a multiplicator of the
order i. The part of this angle it, relative to the moon, must depend solely on
the mean motions of the moon, and on those of her perigee and nodes, when
we neglect the square of the disturbing force. If i be very small, this part
of / does not depend on the moon's mean motion ; it must, therefore, depend
only on the motions of the perigee and nodes. In this case, dQ acquires a
factor of the same order as the motions of the perigee and nodes, that is, of
[4850] the second order [4817,4828e] ; which causes the term in question to lose its
divisor of the order r. Therefore, the angles increasing slowly have, in the
expression of the true longitude in a function of the time, a divisor of the
order i only ; and it is evident, that this likewise holds good, in the expression
[48571 ^^ ^''^ time in a function of the true longitude. But, if Ave notice the
square of the disturbing force, the part of the angle it, relative to the moon's
co-ordinates, may contain the sun's mean motion ; and then, the differential
[4657'] dQ acquires only a factor of the fust order, or of the order m. From
these principles ice can judge of the order, to which the several terms of
the differential equations are reduced, in the finite expressions of the
co-ordinates.
6. Upon these considerations ive shall develop the different terms of the
equation [4754]. In the elliptical hypothesis, the constant part of u is
represented by,*
[4858] - • { 1 + C" -f 5- 7^ + 13 5 := constant part of u ;
[4858'! [3 being a function of the fourth dimension in e, y, we also have,
[4859] ^^' = « . { 1 — e-— 7' + |3'};
[4859] ^' being likewise a function of the fourth dimension in e and 7. The sun's
[48C0] action alters this constant part of u [4858, 4964] ; but a being arbitrary,
* (2792) Neglecting terms of the fourth order, we have, in [4826], the constant part of
[4858a] u equal to - -{l + e^ -\-ll^; and, from [4825], h~ = a .\\ — e- — y^. Adding to
these the functions of the fourth order, depending on p, (S', they become respectively, as
in [4858,4859].
Vn.i.^^6.] TERMS OF q IN THE DIFFERENTIAL EQUATION IN u.
391
we may suppose, that - .p + f'~ + T7" + (3| [4858] always represents the
constant part of u. In this case, we shall no longer have
h- = 0.(1— e=— >H|3') [4859] ;
and we shall then put,
a being an arhitrarij quantity which becomes equal to a, if we exclude the
sun^s action. We shall then put,
m
This being premised, the term
m'.u'^
Q,o 3 5 or the expression
becomes, by development, as follows ;*
1
Â2'
dq
du
s
JFu
'§) [4808],
'3
m. Il
2li\u^
m
2a.
l + e^^+i^^ + le'^
— 3e.(l+ie^+fe'^).cos.(cj; — t.)
+ 3e'. (l + e= + ir2+|fi'2).cos.(c'OT» — ^')
— f . (3 + 2 m) . e e'. cos. (cv-{-c'mv — ■is — ■^')
— f. (3 — 2m).ee'. cos.(cj; — c'mv — -^ + w')
+ 3e-.cos.(2cv—2^)
+ ^7--cos.(2gv — 2è)
-f- |-e'^cos.(2c'mt?— 2'o')
— I- e y-. cos. (2^« — cv — 2^ + ra)
[4861]
[4862]
a,.
[4863]
[4864]
m.
[4865]
[4865']
[4866]
* (2793) If we separate the terms of the expression of - [4826], into different
classes; using the abridged symbols Xi, x^, x^ [4866i], whose indices represent respectively
the orders of the terms, we shall have u [4866c], from which we obtain — r4866c?l,
neglecting terms of tlie fourth order in e, 7 ;
Xi= e. COS. (cv — zi); x.2 = e^-\-iy^— {7^.cos.{2gv—2ê) ; X3 = e^.cos. (cv—-a)', [48666]
[4866e]
[4866rf]
u = a-\\l-\-Xi-\-x.2-{-X3\ ;
■.a^l—3.{x^-\-X2+X3)-\-6.{Xl''-{-2x^X2)—l0x^^.
392 THEORY OF THE MOON ; [Méc. Cél.
To develop the term
^"'"''^ 'SS-^°^-(2'^-2^0, of the expression of- ^,.('J)-^^.('||)[^^^^^
Now, substituting the values of x^, x^, x^ [48666], in tlie first members of [4866/— i], and
reducing the products, by means of [6,20, 7] Int., we obtain the second members of these
[4866e] expressions respectively ; always neglecting terms of the fourth order, and those depending
on the angles 2gv -\-cv, 3cv, which are not retained in [4866] ; and using the abridged
notation [4821/] ;
[4866/] 1 — 3.(.Ti+Xo+.i:3) = l—3e^—f7'^—3e.{l-lre^).cos.cV'}-iy^.cos.2gv ;
[iS66g] + 6<-= +3e^ + 3e^cos.2ci;;
[4866fc] + 12a;ia;3= — 3e.(— 46^ — 7-).cos.ct) — f e7^cos.(2^i'-a') ;
[4866i] — 10^1^= — 3c.(|ca).cos.ci-.
The sum of these four expressions being multiplied by a^, gives the value of m"^ [4866(/,fc].
Moreover, from [4863], we get J/t"^ [4866/] ; the product of the two expressions
[4866A:,?] gives [4866??t], neglecting terms of the fourth order ;
[4866fc] u-^=^a\ { l—iy''—3e.{l—ie''—7^)-cos.cv-\-3c"-.cos.2cv-{-h^.cos.2gv—^ef-cos.{2gv—cv)l;
[4866i] ih-^=har\\l + e^+7
21 .
r4866ml A- ^.^' .{l+e^+j7^-3e.(l+èe')-cos.Cf+3e^cos.2«,+372.cos.2^«-|e72.cos.(2^r-ft>) }
«3
[4866n] ~2â'^'"^^^'
X beino- put, for brevity, to denote all the terms between the braces in [4866m], except the
first, or unity.
We may proceed, in the same manner, to find u'-^. For, by using the symbols y^, y^.
j/3 [4866^], the expression of 7t' [4838] becomes as in [4866;-]; omitting, as above, the
[4866o]
[4866p]
angles •a, ■a', in the rest of the calculation. From this value of u' we get u'-^ [4866*].
The terms, composing the factor of this expression, are found in [4866< — ?«] ; whose sum,
multiplied by a'-^, gives m'= [4866s], as in [4866x] ; neglecting the terms depending on
the angle Sc'mv — 3ra' ;
y, = c'.cos.c'?ftt) ; y^=e'^.cos.2c^mv;
[4866g] y^ = Jc'3. cos.c'TOti+OTee'.cos.(cD — dmv) — mee'. cos. {cv-\-c'tnv) ;
[4866r] «' = «'-' .\l + yi-\- 2/2+ y 2 ] ;
[4866*] m'3 = «'-3.^ i+3.(yi+y2+y3)+3.(yi=^+2y,y,) + </i=h
VII. i. §6] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 393
we shall first give the development of
3 m' . u' ''. COS. (2 ?; — 2 v') . [4866"]
This term, being developed, becomes,
[4866/]
1 + 3. (yi+ y-i + i/s) = 1 + 3 m c e'. cos. (cv — c'm v)—3 m ee'. cos. (cu+cW)
+ (3e'— |e'3).cos.c'mu -\-3e'^.cos.2c' mv;
3i/^^= +2e'2 +|e'2.cos.2c'mr; [4866u]
6)/i(/o= +-2/.e".cos. c'tod; [4866d]
J/i^= + 1 e' 3. cos. c'm 2'; [4866u>]
, , Cl + 3e'2 + 3e'.(I + |e'2).cos.c'mt>+8e'3.cos.2c'mr )
M'3 = a'-3.) y- ~ V ' \ o ' ^ I ' ^^ L4866a:]
( -["3»ict -cos-^CD — c mv) — 3mee. cos. {cv-\-cmv)\
= «'-3. Sl + r|; [4866)/]
l+Y being used, for brevity, to denote all tlie terms between the braces, in [4866x]. [48662]
Multiplying together the expressions [4866», y], and their product by m' ; then substituting
_a
m
[4865], we get.
2 A2. ((3
= ^ .{1 + X+ Y+XYl. [4866a]
Now, XY is of the second order; and, in finding its value, retaining the same angles and
terms as in [4866], we may use the following expressions, which comprise the chief terms of
X, Y [4866«,y];
X^e--j-^7^ — 3e.cos.cv; F=|e'^+ 3e'.cos.c'mv. [4866p]
Now, taking the terms of Y. and multiplying them separately by X, we get,
|ê'2. X= — lee'^.cos.ci' ; [4866y]
.3e'.cos.c'mD.X=.3e'. (c-4-ï7^) . cos. c'm r — % ee'. cos. {cv-\-c'mv) — |ee'.cos.(CT — c'mv). [48666]
The sum of the expressions [4866y, (5] is equal to the value of X Y, which is to be
substituted in [4866a] ; moreover, the sum of the terms hctiveen the braces in [4866m, a?],
decreased hij unity, is equal to the value of l-{- X-\- Y. Hence we find, that the terms of
[4866a, or 4866], between the braces, are equal to the sum of the terms between the braces [4866e]
in [4866m, 1], added to the second members 0/ [4866y, 5], and decreased by unity.
Connecting the similar terms, we find the result of this calculation to be the same as in
[4866].
VOL. III. 99
394
THEORY OF THE MOONj
[Méc. Cél,
/(l_|e'2— 4mV) .cos.(2i'— 2my)
+ 1 e'. cos . (2 V — 2 mv — c'mv-\-^')
-^e'.cos.(2 V — 2mv-j-c'mv — -'')
+ 2me.cos.(2î; — 2mv-\-cv — w)
— 2 me. COS. (2 « — 2m v — cv-\--^)
\ + y e'l COS. (2 V — 2 m v — 2 c'm d+2 i^')
I — y mee'.cos.(2î; — 2?tt« — cv — c'mw+^+^')\
[4867] 3»i'.M'^cos.(2u -2t;')=^-( + V ^^«'- cos.(2w— 2mt)+c«— c'm«— ^+53') ^*
\+ ^mee'. cos.(2'«; — 2m w — cw-f c'm«+ra — -n')/
- 1 mee'. cos.(2« — 2 m tJ+c i)-f c'm?) — œ — 33') )
' + |m.(3+8m).e^cos.(2cz;— 2v+2»ii;— 2^)
— Jm.(3— 8m).e-.cos.(2c«+2«— 2m?;— 2^)
+ ^my". cos.(2gv—2v-{-2mv—2à)
— J:m7^cos.(2^?;-f2y — 2mv — 2è)
\ — |-me>^cos.(2«— 2mîJ— 2^-?;+ci;4-2t'— c=)/
3 m'
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[4867a]
[48676]
[4867c]
[4867rf]
[4867e]
[4867/]
[4867g-]
[të67h]
[4867«]
* (2794) Using, for brevity, the value of v^ [4867e], putting also «^ equal to all the
remainin'^ terras of the second member of [4837], except the first mv, we shall have v', as
in [4367/] ; always omitting, for brevity, the symbols a, ra', as in [4821/]. Substituting this
value of v' in the first member of [4867^], and developing by means of [24,43, 44] Int., it
becomes as in [4867A] ; observing, that v^ is of the first order, v.^ of the second order,
and, that some terms of the third order are neglected. Substituting in [4867/t] the value
2v^= Sm^.e^. sin.^cu + 167« ee'.sin. c«. sin. c'?n« — 8e'^.sin.2c'?»i) [4867e],
and reducing it, by means of [l,17]Int. ; also, 2v^+2v2 = '2v'—2mv [4867/], it
becomes as in [4867 i] ;
tijr= — 2me.sin.ci) + 2e'.sin.c'm!;;
cos.(2i) — 2«')=cos.{(2t) — 2mî;) — (2ui+2î)o)|
= cos. (2w,+ 2i'2).cos.(2« — 2m«)4-sin.(2ri4-2î)2).cos.(2a— 2mj))
= (^l—2vi^).cos.{2v — 2mv) + {2v,-i-'2v.2).sm.{2v—2mv)
_f (l_4»iV-4e'2)+4mV.cos.2CT+4c'2.cos.2cW) ^^^ __2,ftt,)
I +8mee'.cos.(c« — c'mv) — Smve.cos.{cv-\-c'mv) y
+ {2v' — 2mv\.sin.{2v — 2?fti').
VII. i. >^S 6] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN w. 395
We must multiply this function by
2/1=
; and we have this factor, by [4868]
We must substitute, in the last line of this expression, the value of 2v' — 2m v, whicli is
easily deduced from the second member of [4837], by neglecting the first term mv, and [48674]
doubhng the remaining eight terms. We must then reduce the products of the sines and
cosines of this function, by means of [17, 20] Int., as in the following table ; in which, the
terms of column 1, corresponding to the different angles, are taken in the same order as
in [4867;'], namely ; the first five terms in the same order as in the first and second lines of
[4867i] ; and the remaining eight lines as in 2 i'' — 2mv [4837, 4867A:]. We may observe,
that a term is neglected in line 9, depending on the angle 2v — 2mv-\- 2gv — cv, which is [4867i]
not expressly retained in [4867] ; also a term, of the order e'^, inline 10,&ic.;
(Col.l.)
1
2
3
4
5
6
7
8
9
10
11
12
13
(Col. 9.)
( 1 —4)n^e^ — 4c'2).cos.(2y— 2m«)
4-|-mV.cos.(2CT — 2v-\-2mv)-\-^.m.^e^.cos.(2cv-\-2v — 2mv)
-\-2e'^.cos.{2v—2mv—2c'mv)-'r2e'^.cos.{2v—2mv-i-2c'mv)
-}-4mee'.cos.{2v — 2mv — cv-{-cfmv)-{-4mee'.cos.{2v — 2mv-\-cv — c'mv)
— 4mee'.cos.{2v—2niv — cv — cfmv) — 4mee'.cos.(2u — 2mi;-f-c«+c'»i«)
4-2?«e.cos.(2ii — 27nv-\-cv) — 2me.cos.(2y — 2mv — cv)
-|-?'«c^.cos.(2ci; — 2v-\-2mv) — fme^.cos.(2c2;+2i' — 2mv)
-{-imy^.cos.{2gv — 2v-{-2mv) — i7Hy~.cos.{2gv-^2v — 2mv)
— .^mej'^.cos.(2f — 2mv — 2^y-j-cu)-j- Sic.
-|-2e'.cos.(2y — 2mv — c'mv) — 2e'.cos.(2r — 2mv-j-c'mv)-}-hc.
—2mee'.cos.{2v~2mv—cv—c'mv)-^2mee'.cos.{2v — 2mv-\-cv^c'mv)
— 2ffiee'.cos.(2« — 2mv—cv-{-c'mv)-{-2mee'.cos.{2v—2mv-{-cv — c'mv)
-{-îe"^.cos.{2v—2mv—2c'mv)—^e'^.cos.{2v—2mv-^2c'mv).
[Terms of
C03.(2i7-20'). J
[4867m]
Toobtain the expression [4867], we must multiply this value of cos. (2d — 2d') [4867m],
by Sm'.u^, or 3m'. a'-^. (l + Y) [4866(/] ; by this means all the terms will have the common
. 3m' ,., , . , • r . -, [4867n]
lactor —, like that without the braces m [4867] ; and the terms of this expression within
the braces will be obtained, by multiplying the function [4S67m] by 1 + F; or, in other
words, by multiplying the functions [4867m] by Y [4866x, y], and reducing the products [4867o]
as in [4867r], then adding together the two functions [4867?«, r]. In the first column of
[4867?-], we have given the terms of Y [4866-r,y] ; and, in the second column, the terms
of [4867m], by which they are multiplied: the third column contains their products,
respectively. The numbers in column 2, refer to the numbers in the margin of the lines t^^^'^-P]
of [4867m], putting one accent to denote the first term of any line, tivo accents for the
396
THEORY OF THE MOON ;
[Méc. Cél.
putting e' equal to nothing, in the preceding development of
2h\i
[4866],
[4860] aiTid \^y multiplying this last quantity by —, We shall thus have, very
nearly, by neglecting quantities which remain of the order m' after the
[4869']
[4867c]
integration,*
[4867r]
second term of the same line, &i'C. Thus, 6' denotes the term 2ffîe.cos.(2i' — 2mv-\-cv)i
and 6", the term — 2me.cos.{2v — 2mv—cv). This method of distinguishing the terms
ivill he frequently used.
(Col. 3.)
Products of these terms.
-j-fe'^.cos.(2?; — 2 niv)
-(-|é'.cos.(2ii — 2mv — c'mu )-|-fe'.cos.(2t) — 2mv-\-c'mv)
-\-'3mee' .cos.{2v-2mv-\-cv-c'mv)-\-^mee' .cos.{2v-2mv-\-cv-\-dmv)
— ^mee'.cos.(2,v-2mv-cv-c'mv)—Zmee'.cos.{2v-2mv-cv-\-c'mv)
4— 3e'^.cos.(2y — 2mv — 2c'mt>)-|-3e'^.cos.(2i' — 2mv)
— 3e'^.co3.(2u — 2mv-\-2dnn^ — 3e'^.cos.(2y — 2mv)
4-|e'2.cos.(2i;— 2my— 2c'my)+fe'2.cos.(2u— 2?nj)4-2c'?ni;)
-\-^mee .cos.{2v-2mv-cv-{-c'mv)-\-^inec' .cos.(2,v-2mv-\-cv-c'mv)
— i^mee' .COS. {2v-2mv~cv-c'mv) — ^mee .cos,.[2,v-'2'mv-\-cv-{-c mv^ .
Connecting together the terms of [4867m, r], depending on the same angles, we find, that the
coefficient of cos.(2y — 2?nu + 2c'»i«^) vanishes, and the rest become equal to the function
between the braces in [4867], conformable to [4867o].
* (2795) The method given by the author, in [4869], is evidently correct. For, if we
m'.u' 3
(Col. 1.)
(Col. 3.)
Terms of Y [48C6r].
Terms
of
[4867,n]
+ |e'2
r
-[- 3 e'. cos. c'm y
1'
6'
6"
10'
10"
|c'^.cos.2(;'7rtw
1'
-|-3?î2ee'.cos. (ct) — dmv)
1'
— 3mee'.cos.(cy-|-c'mt')
1'
I
[4869a]
[48696]
, we get u = -,, whence, ——7
—, ; multiplying this by
2/t9.„3
We shall not, however, be under the necessity of using this process,
[4869c]
put e' = 0, in
1
eives J77T — ;.
1 a3
because we have already given the value of 2^;5~^ = ^ • (1 + -^) [4866m, 71] ; and, if
we multiply this by the function [4867], we shall obtain [4S70]. In the first place, the
factors without the braces -^, ;^, being multiplied together, produce,
3 m'.a3 .3 _2 ,
— . -— - = — . «I [4865] ;
VII. i.{.G] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u.
397
/
il+e^+iy^—èe'^).cos.{2v—2mv)
1
_i(34-lm).t.(l-}-^e-— Je'2).cos.(2y— 2mi'— CD+n) \ 2
— i(3 — 4m).c.cos [2v — 2mv-\-cv — w) ;{
-|-Je'.cos.(2r — 2m v — c'm v-j-:^') 4
— ie'. cos.(2u — 2mv-]-c'mv — ra) / 5
— --^J-(14-2?«).ee'.cos.(2i' — 2mv — cv — c'mn-f a-f-^')
-'\'~{ 1 — 2 m) .e c'. COS. (2 v — 2 m «-j-c v — c'mv — ra-f-^)
6
7
^-.COS.(2«-2w')=±!^./ + i (3+2m).ce'.cos.(2t)— 2mj;— CD+c'mD-J-za— Î3') \ . 8 [4870]
2li\u
2a
+ T (3 — 2/?«).ec'.cos.(2^; — 2 m d-(-c y -f t''« ^ — « — si')
l+-V.e'2.cos.(2r— 2?nt)— 2c'mz)+2îî')
+i(6+15m+8ffl2).e2.cos.(2ct)— 2t)+27«i'-2i3)
+K6— 15m+8waj.g2_£.os.(2cr4-2t)— 2mj;— 2-51)
+Ï (3+2 ?«) .y-. cos.(2^ j;— 2 t)-f 2 m i-— 2 è)
+} {3—2m).y~.cos.{2g v-\-2v—2mv—2ê)
9
10
II
12
jl3
14
The term
^^^,.cos.(.-.), of the expression -,^.(f
— f (2+m).e7a.cos.(2z!— 2m«— 2^c4-ct;+2()— ^) / 15
hh
lis
[4808],
[4871]
which is the same as the common factor of [4870] . Moreover, the terms between the
braces in [4870], are represented by the product of the terms between the braces in [4867],
by l-{-X [4866n] ; or, in other words, this product is equal to the terms between the ^ ■'
braces in [4867], added to the function [4S69c]. TJiis last function being the result of the
product of these terms of [4867] by the the quantity X; and it is obtained in the following
table, which is similar to [4867r]. The first column contains the terms of X; the second, [48fi9t/']
the terms of [4867], and the third, the corresponding products, reduced in the usual manner,
and using the accented number 1', to denote the first term of the first line of [4867], as
in [4867 5] ;
VOL. III.
100
398 THEORY OF THE MOON ;
[4871'] gives t!îe following ;*
[Mtc. Cv\.
[4869e]
[4869/]
[4869e-]
(Cl. 1.)
Terms of X [4866jn,n]
— 3e.cos.cv
-f-3e^. cos.âcw
-{-^■y^.cos.2gv
— |cy^.cos.(2^y — cv)
(Col. 2.)
Terms
of
[4867].
1'
1'
1'
2
3
4
5
13
1'
1'
1'
4
1'
(Cul. 3.)
Products of these terms.
[4870a]
[48706]
[4870c]
[4870rf]
[4870e]
-f-e^.cos.(2 V — 2 mv)
-j-.l7^-cos.(2); — 2mv)
-Jc.cos.(2t>— 2 m y+CD)—fe. (1—1 e'2).cos.(2t>— 2m r—cr)
-?^^ee'.cos.{2v-2mv-cv-c'mL-) — --j'-ee'.co3.(2tf — 2mv-}-cv—c'mv)
4-.?Ée'.cos.(2« — 2mt;-cz)-fc'mi')+3ee'.cos.(2î) — 2mv-\-cv-\-c'mv)
-3 m e^. COS. (2 V — 2/»!;) — 37ne^.cos.{2cv-\-2v — 2 m»)
-\-3me^.cos.(2v — 2m,v)-\-3me^.cos.(2cv — 2v-\-2mv)
-f m e y^.cos.(2 v — 2 m v — 2^- v-\-c v)
■|(?.cos.(2i' — 2 my — cv)
-|-5 e-.cos. (2 c I' — 2 1'-|-'~ ni v) -\-^ e^.cos.(2f y-j-2 v — 2 m v)
-\-^y-.cos.{2gv — 2v-{-2mv)-\-^y^.cos.(2gv-{-2v — 2m v)
-|-|-me7^.cos.(2!; — 2mv — 2gv-\-cv)
-|e7^.cos.(2D — 2mv — 2gv-\-cv).
Now, adding tlie function [4839e] to the terms between tlie braces in [4867], we get very
nearly, the expression between the braces [4870]. Tiiere are some shght differences, of
the same order as that of tlie terms which we have usually neglected. Thus, the term
— 'im^e^, in the coefficient of line 1 [4867], is neglected in [4870]. The term — 2me,
in hne 5 [4867], is connected with the ftctor (l+lt^ — i^'^) in line 2 [4870], which arises
from the chief terms of this coefficient in [48G9e] ; but this merely introduces terms of the
sixth order. Finally, we may observe, that a similar factor might be introduced in the
coefficient of line 3 [4870].
* (2796) Proceeding in the same manner as in note 2793, and retaining terms of the
second order only, we get, from [4866c] u~*=câ.\l — 'i.{xi-\-x.2)-\-l0xi^\; substituting
in this the value of 10ccj^:= 10 e^.cos.^<:« = 5 e^+5e^cos.as; ; also the value of x^-j-x.^
[4866i], we get,
u~' = a'. \ l-\-e^ — j^ — 4e . cos. cv-\-5 e^. cos. 2 c j; -[-}'"• cos. 2gvl.
Multiplying this by
9 m'
■ \l + e"~^y~) [4863], we obtain,
9)
„,„ , = -—^ — .ll-\-2e^ — 4e.cos.ct)-)-5e^.cos.2ct)4->'^.cos.2fi-i'i
8^2. ,j4 8(,^ ( I I I / to s
Again, from [48667,?-], we have successively, 6w^^ = 3 e'--j--3e'^.cos. 2c'm y ;
= a'-'.|l+3e'2-|-4e'.cos.c'm»-[-7c'2.cos.2c'm»}.
VIl.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 399
8«, ■ a
__.cos.(«— r)=<^^ji^ji_e'_cos.(i'— mz)+c'm«— ^') /• 2 [4872]
, 2~lh a , . t . i\
-1 -.e. cos.(i' — m V — c m v-I-t^ )
8 a, a
[4870/]
If we denote tlie factors between the braces in [4870c, e] by 1-|-X,, l + ^u respectively,
their product will be l-f-^i+î^'i+^i^V» ^Y noticing only the chief terms of X,, F,,
we have,
Xi Yi= ( — 4e.cos.ci').(4e'.cos.c'HM') = — Sec'. cos. (c« — c'mv) — 8e e' .cos. [cv-\-c'mv). [4870g]
Adding these terms of Xj Fj, to those of l+^u l + ^i [4870c, e], and decreasing the
sum by unity, we get the expression of 1-j-Xj-f- Fj + Xj Fj, to be used in the product of
the functions [4870c, c], which becomes,
C 1 -j-2 e--)-3 e'^4-4 c'. cos. c'm« — 4e.cos.cv '\
9m'. u"^ 9m'. a' ),-g ^ ,0 „ 1 n "> 0/ V
g^-^ =g^^^ • ) +^^-cos.2cv-j-f.cos.2gv-frle~.co5.2c'mv'> . [i870h]
(_ — 8 c e'. cos. (c V — c'in v) — 8 e e'. cos.(c v-{-c'm v) )
m . w
Substituting the value of —7— [4865], in the first factor of this expression, it becomes,
9m'. a* a — ^O'
8^4- l'A; -o [4870,-]
which is of the fourth order [4842,4843] ; therefore, in finding the value of cos.(«) — v'), we
need only to retain, in general, the terms of the first order ; except in those depending on
the angle v — mv ; in which greater accuracy is required [4874]. Hence we may neglect [4870/i;]
«0 [4867/"], and we shall have the value of cos.(«; — v') [4870»i], by proceeding as in
[4867^,A]. Substituting in this the value of I'l^^ 2e'. sin.c'wt) [4S67e], it becomes as in
[4870n]. It being unnecessary to notice other terms of a higher order, or such as depend on [4870i]
angles which differ from those in [4872] ;
cos.(r — r')=(l — irj^).cos.(i' — ?«t))-|-i'i-sin.(v — mv) r4870 1
= (I — fc'^).cos.(t) — mv) — c'.cos.(« — mv-{-c'mv)-\-c' .cos.[v — mv — dm v). r4870n]
The four terms of which this expression is composed, being multiplied by the terms between
the braces in the function [4870/i], produce respectively the terms in the four lines
[4870o — r]. Their sum is given in [4870^]; to which we must annex the common factor
9m'. vJ^
[4870ï] , and we shall obtain the corresponding terms of ' ,^.cos.{v — v), as in [4872]. We
shall hereafter, in [4870/ — w], see, that the neglected terms have much less effect, in the
value of u, than those we have explicitly retained ;
400 THEORY OF THE MOON ; [Méc. Cél.
[4872'] - being, hy the preceding article [4843], of the order m' ; the two first of
[4873] these terms become of the order »t^ by the integrations. The inequality,
depending on the angle v — mv, is remarkably loell adapted to the determination
of the Slut's parallax, by means of the ratio -. It is, therefore, important
[4874]
[4870o] 1
[4870;>] 2
[48709] 3
[487 0<]
(Coi.i.) {Col. a.)
(l-j-2c^-f-3e'^).cos.(«) — mv)-\-2e'.co'3,.[v — mi'-f-c'mt')-l-2e'. cos.(r — mv — c'mv)
— e'^. cos.(i) — mv)
— e' . COS. (v — m v-\'C'm v)
[4870r] 4 j J^e' .cos.{v—mv—c'mv).
[4870«] (l-|-2e"-4-2e'-).co3.(i! — mv)-\-e'.cos.{v — mv-\-c'mv)-\-^e' .cos.{v — mv — c'mv).
If we compare the terms [4872] with the assumed form [4846], we find the values of i,
corresponding to them respectively, are i^l — in, i:=l — m-|-c'm, i=l — m — cm; and,
as c' hardly differs from unity, they are very nearly represented by i^\ — m, i=l,
i = \—2iii. The corresponding divisors, in the value of u [4847], are of the orders
[4870m] (1 — m)~ — JV^ \—N^, (1— 2m)2 — N~ ; and, as JV- differs from unity by quantities
of the order m^ [4815'], these divisors will be respectively of the orders m, m^, m. In
consequence of these divisors, the part of the first term [4872] which is independent of c, e',
is reduced from tlie fourth to the third order ; the second term is reduced from the fifth to the
third order ; and the third term is reduced from tlie fifth to the fourth order. Several terms
of the function [4870t", or 4872], are not increased so sensibly in the value of m, and they are
[4870tt] therefore neglected. Thus, the term — 4e.cos.cv [4870/(1, being multiplied by the first
term of [4870»], produces, in the function [1872], the following expression,
[4870jr] |.^ . -.( — 4c.cos.cy).cos.(!; — mv) = — ~.^ . -.2c.{cos.(ci' — v-\-mv)-\-cos.{cv-\-v — mv)\.
r4670vl corresponding, in [4846], to i = c — l-f'"> i^c-\-\ — m, and as c=l — f»«~ nearly
[4828e], these terms will not render the divisor i^ — JV^ small [4847].
We may observe, that the term treated of in [4871], occurs in [4S08], under the form
3wi' w'M
.(3— 4«-).cos.(v — v'), and in [4754], with a dilierent sign, and under the !brm
[4870^']
8ui
3m'.i('-l ,_ , o, , ,, y»ï'.!('' / , . n\ / /\ 1 • 1 1 1 • o
.(3 — 4s-).cos.(t' — v), or, .(1 — *;.s-j.cos.(y — v ) ; whicli, by neglectmg s",
becomes as in [4871]. Now, by [4618], wc have,
[4870Z] _4,a__£-/.cos.2(5-f-d) = _f ,.2_£y5.cos.(25-.-2<!) ;
which contains the constant quantity — f^-; so that we might multiply the function [4871]
[4870z'] by 1— §r, which would change the foctor (l+2(;'-+2e'-2) [4872] into l-l-2e-+2c'2— f/^.
VII. i. §6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 401
to determine this inequality with particular care ; and, for this purpose, we [4875]
shall carry on the approximation so as to include terms of the order m*.
We shall now develop the term \j)-j;r~Yj~^ of the equation [4754].
In the first place, this term contains the following,* — ^-^^.--.s'm.(2v—2v'). tl87C]
.sin. (2d — 2v'), by increasing 2v by a right angle, f [4876']
We shall have
* (2797) This is produced by the first term of [4809].
t (2798) We may change 2y mto any other angle, as 2 v in [4867^ — r, 4867,4870],
without altering the angles mv, gv, cv, c'mv, as is evident by the mere inspection of the
process of calculation in [4767^, &.c.]. This change being made in [4870], and then
putting 2 V = 2 1"-]- 90'', its first member becomes,
-^u^-'^-^-'-^''^^ ^^ In [4876'].
In the second member of [4870], we must, by the same process, change any term of the
form cos.(2!;-l-p) into — sin.(2«4-|3) ; and any one of the form cos.(3 — 2r) Into
-|-sin.(3— 2v). Hence we get, by changing the signs of all the terms of [4870], and
neglecting the symbols è, ct, -ô/, as In [4821/],
?^';sin.(2«-2.')=|^
2h\u^ ^ ^ 2a,
/(I +e2+i 5-2—1 e'2).sln. (2 v— 2 m«)
— |(3+4»?).e.(l+|e2— Je'2).sin.(2î>-2mD-cî))
-5(3 — 4m).e .sin. (2 \i — 2 m vA-c v)
-|-Je'. sin.(2t) — 2mv — (^mv)
I — J e'. sin. (2 v — 2 m v-\-(!m. v)
\ — -i-{\ -|-2 ni) .e e'. sin . (2 « — 2 mv — c v — dmv)
I — %'-( 1 — 2 ni).ee'. sin .(2 v — 2 m v-\-c v — c'mv)
-\- 1 (.3-l-2OT).eÊ'. sln.(2y — 2mv — cv-{-dmv)
\ -\- i (3 — 2 m) .e c'. sin . (2 v — 2 m v-\-c v -f c'm v)
I +-¥• e' ^. si n. (2 r — 2 7n v — 2 c'm v)
-l(6+15?n+8m2).e2.sin.(2ct)— 2i}+27n«)
+i{6—l5m4-8m^).e^.s:ii.{2cv+2v—2mv)
-i (3+2 m).y~. s\n.{2g v—2 v-\-2 m v)
+^{S—2m).y-.sm.{2gv+2v—2mv)
\ — ^ (6-|-3 in).ey'^. sin. (2 v — 2 m v — 2^r-f-cy)
[4875a]
[4876o]
[4876t]
[4876c]
[4876rf]
1
2
3
4
5
6
7
8 [4876c]
9
10
11
12
13
14
15
VOL. III.
101
402 THEORY OF THE MOON ; [Méc. Cél-
3m' u'^
ill the preceding development of rr^— 5.cos.(2« — 2v') [4870]. We must
then mutiply this development by,*
•— ce.(l+ie^ — J-7-).sin.(ct; — 3j) v i
+ icelsin.(2ctJ— 2^) I 2
[4878] rfM I , ^ . ,„ q . ( „
«</«; 1 I
+ ia-7^sin.(2^i;— 20) | 4
, — ±ef.sm.(2gv—cv—2ê+z!) I 5
* (2799) The differential of [4826], relative to v, gives, by neglecting «, 6, as in
[4821/],
[4878a] ~ = a-K{—ce.(l+e^).sm.cv+^gf.sm.2gvl;
dv
and if we neglect terms of the third order in all the coefficients, except those which are
connected with the angle 2gv—cv, we obtain from u [4866c], the following value
[48786] of - [4878c, tZ], by observing, that a-i^E^.cos.^cf =ie2+ie^.cos.2cy [4866è].
w
We may remark, that the author has retained, in the coefficient of cos. cj;, a term of the
third order e^, but has neglected others of the same order, as will be seen in [48846] ;
[4878c] i^a.{ l-(x^+x,+x,)+{x,i-x,+x,y-{.T,-j-r,+x,f+hc.l
u
[4878d] =a.\{l—ie-—lf)—e.(l-\-e").cos.cv+he^.cos.2cv+if.cos.2gv\.
Multiplying together the two expressions [4878a,rf], we find, that the factor without the braces
becomes a-^.a = 1 ; so, that we have only to notice the product of the factors between the
[4878e] jjj.j^(.g3 -pj-jjg jg jjQj^g jjj (i^g following table ; in which is given, in column 1, each of the four
terms of the function [4S78(f] ; and the corresponding products, by the function [4878a],
are given in column 2, on the same lines respectively ;
(Col. 2.)
— ce.(l-\-ie'^ — ^■)'^).sm.cv-\-^g'y^. sin.2^v
4-i c e^. sin.2 c v — | g c -f. sin. (2^ v — c v) — &;c.
■\-\c e^. sin.cu — \ ce^. sin.3c v
-\-\ct -f. sin. (2^ t — ci))-f-&ic.
Connecting together the similar terms, and putting c = l, g^=\, in those of the order
e 7-, it becomes as in [4878].
(Col. 1.)
[4878/]
l-|c"-i7'
[4878^]
— c.(l4-«^)-cos.cw
[4878/.]
-|-Je2.cos.2c«;
[4878i]
4-^7®.cos.2^w
VII.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u.
Then we shall have,*
/cc.(l-f^.[2— 19m].e=— |e'2).cos.(2u— 2m»— cv+ra)
/ — ce.cos.(2« — 2mv-^cv — «)
I + J.cce'.cos.(2t) — 2mv — cv — c'Mt)-j-ra-|-w')
\ — |-.cee'.cos.(2i; — 2mv-\-cv — cfmv — ra-j-w')
I — ^.cee'.cos.(2D — 2mv — cv-\-c'mv-\-zi — •ra')
3m'A3 du . ,^ ^ ,, 3in +h-cee.cos.{2v—2mv-\-cv-{-c'mv—-a—a')
— nT^T" -T •sin.(2u — 2t)')=: — ./
2''-"' ''" 4"' \— 2c.(I+m).e2.cos.(2c?;— 2«+2mu— 2«)
403
4-2c.(l— m).e^.cos.(2ci)4-2t) — 2mv — 2a)
-{- 4 m c. e^. COS. (2 v — 2 m v)
— tgf.co5.{2gv—2v-\-2mv—2è)
-}-igy^.cos.{2gv-\-2v—2mv-\-2è)
_|_ J.(2 — 5m) .e y^. cos.(2u — 2mv — 2gv-^cv-\-2ô — ro)/
1
2
3
4
5
6
7
8
9
10
11
12
[4879]
* (2S00) If any term of [4876t;], be represented by
. A . sin. V,
2a, '
and any term of [4878], by .^'.sin.F', the product of these two terms, changing its sign,
will represent the corresponding part of — ' ^ ■— .sin.(2t; — 2v') [4879], which, by
reduction, becomes,
^^.{AA'.cos.{V-\-V')—AA'.cos.{Vy^V')l.
The factor of this expression, without the braces, is the same as in [4879] ; consequently,
the terms within the braces, must arise from the terms
A A', cos. ( V+ V) —A A'. cos.( F«= V) .
These terms are computed in the following table, neglecting quantities of the third order
in e, e', y, except they depend on the angles
2 V — 2 m vzhc v-\- zs, 2 v — 2 m v — 2g u-j-c v-\-2 ê — «.
The numbers in the first column refer, respectively, to the five terms or lines of [4878] ;
and those in the second column, to the terms or lines of [4876e] ; in the third column are
the corresponding terms of the function [4879/"] ; and the sum of all of them represents the
terms between the braces in [4879] :
[4879o]
[48796]
[4879c]
[4879rf]
[4879e]
404
THEORY OF THE MOON ;
[Méc. Cél.
[4880]
The terms,*
'«'4
m.u
8}Au=
-..I3.sïn.(v—v')+lô.s'm.(3v—3v')]
du
dv
[4879/]
(Col. a.)
Function [4870(/].
ce.cos.(2y — 2mv-\-cv)-l[-ce.(l-\-^e^ — Je'2).cos.(2« — 2mv — cv) 1
+J(3+4w).fe2.cos.(2y-2my)— i(3+4ïK).ce2.cos.(2CT-2«+2«y) 2
i(;3— 4w).cc2.cos.(2y-2mi')+^(3-4m).cc2.cos.(2cy4-2i;-2/Hw) 3
— icce' .COS. (2v-2mv-\-cv — c'mv)-^?rcee' .cos. C2v -'Hmv- cv — c'mv) 4
-f-T7Ccc'.cos.(2i'— 2);u'-|-fi'-[-f'mi') — ^'^cc' .cos. {^v —2mv —cv-^c'inv) 5
6
7
8
9
10
11
12
13
14
15
(Col. 1.)
(Col. 3.
A' [4878].
[4876c
-ce{\+le^-m.
sin.cz'
1
o
3
4
5
11
12
13
-j-Jce^-sin.2c»
1
o
3
— ^ce^.s'm.Scv
. .
-{-igf.sin.2gv
1
3
— ^e'y^.s)n.{2gv-
—cv)
1
[4879^]
[4879fc]
[4879q
[48794]
— |(G-(-15?k).cc3.cos.(2« — 2mv — cu)-j-&c.
-[-i(6 — 15m).ce^.cos.(2y — 2mv-\-cv)-\-&c.
-i(3_[_2,„).ej,2.cos.(2y— 2my— 2^v+cd)+&,c.
-f-ice^.cos (2eD-j-2u — 2mv) — ice-.cos.(2cu — 2v-{-2?nv)
— J-(3-j-4»i).fe3.cos.(2y — 2mv-^cv)-\-&L,c.
-j-J-(3 — im).ce^.cos.{2v — 2mv — c«)-)-&,c.
. . neglected.
-\-igy~-cos.{2gv-\-2v — 2mv)—lg7-.cos.{2gv — 2v-\-2mv)
-|-i(3 — im).c}'^.cos.[2v — 2niv — 2gv-\-cv)-\-&,c.
-|-|e7".cos.(2y — 27iiv — 2gv-\'Cv)-\-&bc.
Connecting the terms of this expression, we obtain the factors between the braces in [4S79],
neglecting terms of the tliird order, connected with the angle 2v — ■2 m v-^-cv, or with other
angles differing considerably from v. To estimate roughly one of these neglected terms, we
shall observe, that y^ « ^ e' [51 H, 5120] ; therefore, the greatest product of the third
order, which can be made of these three quantities, and can occur in the above function, is
ey-; and, if this be multiplied by the factor i^ [4879], or its equivalent expression
|m^, it becomes |m^. ey®. Substituting the values [.5117, .5120], and multiplying by the
radius in seconds 206265", we get ^ m^. e7^ = 0",38 ; whidi represents the order of the
greatest neglected term in [4879]. This may be somewhat increased by integration in this
value of u [4847], by means of the divisor i^ — JV^ ; for which reason the author has
retained the last term of the function [4879], which depends on the factor ey~. We may
observe, that the factor l-|-|e- — f e'~, which occurs in the second term of the first line of
[4879/], might also be connected with the first term in that line.
* (2801) Substituting, in Cy)'W^v [4'754], the term of [4809], depending on ?A
[4880a] it becomes as in [4880] ; neglecting the very small teim depending on s-. We have, in
VII. l.s^6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 405
in the expression of ( '-7^ i.,~-i-) produce no inequality of the third order [4881]
'■ \dv J Irirdv
in the integrals.
Lastly, we shall develop T~^-f'^'-\ [-4754]. This function contains [4881']
dv u
'3 m' y,u'^.dv
the following term,* — j2~-J^ — i — .sin. (2îj — 2v'). The development of [1883)
ft It
3m'. U'^ ._ -, ,, ^)rl^/^^ • 1 r 3m'. u'^ ■ ,n ^ ,^
___ . COS. (2 v-1 V') [4870] , gives that of p- - . sin. (2 v—2 v'), [4883]
by increasing the angle 2*; by a right angle [4883rt], and multiplying it byf [4883']
[4872], the expression of " ^.cos.(t) — v') ; in wliich we may change v into t)-|-90'',
as in [48765, c], witliout altering m v, c'mv ; and we shall obtain the expression of
~8Â5:^-""-("~^)- [4880J]
This being multiplied by one third part of the expression [4878], gives the value of
-S5-^-^'"-(^-''')-S [4880]. ^4g,0c]
Now, the chief term of [4872] has the factor ^.nfi.-, [5094] ; and that of [4878] is ce,
or c, nearly, neglecting its sign. Hence, the greatest coefficient of this product, is,
i-.w2.^,.e = 0,0000004 [5117,5120]; [4680rf]
which, in seconds, is less than 0',09. This is insensible, and it is not increased by
integration in [4847]. The same may be inferred, relative to the term of [4880], depending [4880e]
on the angle 3v — 3v'. Hence, we may conclude, that the expression [4880] maybe
neglected, as in [4881].
* (2802) The first term of (^^) [4809], being substituted in [4881'], produces
the expression [4882] ; and we have already seen, that the expression [4870] gives that in [4883a]
[4876e]; by changing 2» into 2i)-f90'', according to the method proposed in [4876']
or [4883'].
t (2803) Retaining terms of the third order in [48786], and multiplying by 2, we get,
2
— = 2a.\l—{xi+x,,+.r.j)-^xi'i-^2xi x.^—x^^. [4884a]
Substituting the values [48666], we obtain,
VOL. III. 102
406
THEORY OF THE MOON;
[Méc. Cél.
— e.(l — le^ — lf).cos.(cv — ^)
[4884] ï^2a.^ +ie^cos.(2ct; — 2^)
+ i?2.cos.(2g-?;— 2ô)
—lef.cos.(2gv—cv—2è-Jr-:s)
Hence we shall have,*
[48846]
[4885a]
[48856]
[4885c]
1 — {xi-{-x.2-\-X3) = 1 — c^ — i>^ — c.[l-\-e^).cos.cv -\-ly^.cos.2gv
2xxi= — c.( — 2c^ — i7~)-cos.cv -rriey^.cos-i^gv — cv)
— a:j3__ — j,_^ Sga ).cos.cv.
The sum of these, gives the terras between the braces in [4884a, 4884].
* (2804) Multiplying together the second members of [4876e, 4884], we obtain the
expression of
3»i'.m'3
--^ a
-.sin.f2D — 2v'): and the factor witliout the braces becomes 3 m ■—, as in
Ifi.u* ^ ' a.
[4885] . The products of the terms between the braces, are found in the following table ;
in which the first column contains the terms of [4884] ; the second column, the terms of
[4876e] ; and the third column, their respective products, reduced by [ 1 8, 19] Int. ; using
the abridged notation [4821/] ;
(Col. 1.)
[4884].
1
— f.cos.(;«
-j-e(ie^+j72)cos.c«
-j-ie^-cos.2cy
-}-57®.cos.2^y.
— Je72.cos.(2jO-y-cD)
(Col. a.)
[487Ge]
All the
terms.
1
2
1
o
3
4
5
11
13
1
1
3
1
3
1
(Col. 3.)
_. ,. . 3m'.«'3 . ,„ „ ,.
Correspondinsr terms of -r= .sin.fiJw— at) ).
:■ the whole function [4876e] between the braces
(-
4^2— i72).sin.(2D— 2»i«)
1
2
-j->(3-|-4;rt).e.(|e24-iy2).sin.(2i'— 2»u'— ft') 3
-|_|c.(l_}-e2-|-|y2-|e'2).{— sin.(2y— 2mw+cw)-sin.(2K-2»i«-cy)] 4
-|-i(3-]-4,„).e2.|sin.(2y— 2?nu)— sin.(2cD— 22)+2)rM))} 5
_f_.(:3_4„j).f2^sin.(2y— 2mw)4-sin.(2cw+2»— 2»iu)} 6
■\-\ce'.\ — sin.(2u — 2mw+ct) — Cmw) — sin.(2u — 2nîD — cv — drnv) \ 7
_j_ic(;'.^_|-sin.(2u— 2«u'-j-c2J+c'OTt))+sin.(2« — 1mv—cv-\-c'mv)\ 8
9
10
11
12
13
14
15
16
__i_(G+1.5/n+8M2).e3,gin.(2y— 2my— cv)
— _i^(3-|-2/n).cy2.sin.(2w — 2mu — 1gv-\-cv)
+(^c3+iey2).sin.(2y— 2/«K— cr )
— ica.sin.(2cw— 2tf+2»!!;)+J:e2.sin.(2cv+2i'— 2m')
_|(3_4Hi).e3.sin.(2w— 2;ny— cu)
+^>2sin.(2g'î,-j-2t'— 2mw)— ^y2.sin.(2^y_2t,_j-2»iy)
_^i^(3_4,n).e72.sin.(2H— 2mw— 2^«+cd)
— i-cy2.sin.(2i'— 2»!y— 2§-y+cv).
VlI.i.^G.J DEVELOP^rENT OF THE DIFFERENTIAL EQUATION IN u. 407
-^ n
(l_}-2e2— |e'2)
o o
\m
.COS. (2d — 2m2;)
2.(l+>»)
2— 2/«— c
2.(1— m)
2— 2m+c
7e'
{1+fe-— i/'— |e'-|.e,cos.(2t)— 2mî?— cî;+ra) \ 2
e.cos.(2« — 2mv-\-cv — -ji)
+ oTS o— r.C0S.(2v — 2ot» — c'mV-^-:r!)
~-{.i — dm) ^ ^
. COS. (2 V — ^mv^c'mv — si')
cos.(2v — 2 mv — cv — c'inv^^-\-ui')
COS. (2 V —2 m t^+c v — c'm v — ra+ro)
2.(2— m)
7.(2+3 m).ce'
2.(2— 3ot— c)
~.(2— 3m).ee'
' 2^(2— 3 m+c)
(24-m).ee'
■?;:;.« / , ['■-i+m).ee'
-é.m.~./J^ .A^^^-J^^,cos.(2v—2mv—cv+c'mv+z^—z.')
(2—m).ee'
+ "oTo r-T-.cos.(2t) — 2m«+cu+c'mv — •a — îî')
(10+19/«+8m2) „
■ 4:(2^2-+^-^-^°^-C2'^^-2t'+2m.-2.)
, (10-19m+Sm.2) „
+ -4:(2^^-^^-'^««-(2-^+2.-2m.-2.)
— 4:^^24:^ •'''•cos-(2^«-2«+2mt;— 20
I (2— m)
"^ 4.(2o-+2-2m)-^^-^"^-^^^"+^^~^^^— ^^)
17e'3
+ 2. (2 —4m) • ^Qs- (2 D— 2 m ?;— 2 c'm v+2z>')
I (5+m) „ /
4.(2-2m— 2g+c) • ^ '^ • cos-(2«— 2mt;— 2g-z;+c^+2a-^)/ 15
The first line of this table includes the terms of the function [4876e], and by adding them to
... ,, , 3m' «3 [4885rfl
theremammg terms of [4885c], we get the terms of —-^.sm.(^v—2v') ; which ought to be
408
THEORY OF THE MOON ;
[Méc. Cél
The terms of this formula, depending on the angles 2cv — 2v-T-2mv — 2ra and
[4886] 2gv — 2v-{-2mv — 2^, have divisors of the order m ; and they again acquire
these divisors, by integration, in the expression of tlie moon's mean
longitude ; wliich reduces them to the second order ; and this, it would
seem, ought to make the inequalities relative to these angles become great.
But we must observe, that, by [4853, &ic.], the terms having for a divisor
the square of the coefficient of v, in these angles, nearly destroy each other,
in the expression of the mean longitude ; so, that the inequalities in
question, become of the third order, conformably to the result of observations,
as will be seen hereafter [5576]. We may, therefore, for this reason, dispense
[4887] with the calculation of the terms multiplied by e*
[4886']
[4886"]
„9 9
e 7 ,
because the
[4885e]
[4885/]
[4885g]
[4885A]
[4885;]
[4885fc]
[4885i]
equal to the differential of [4885] divided by — dv ; or, in other words, it ought to be equal
to the terms between the braces in [4885], changing cos. into sin., and neglecting the
divisors 2 — 2m, 2 — 2m — c, he, which are introduced in [4885], by the integration. The
comparison of the sums of the terms of [4876e, 4885c], with those of [4885], may be made,
in most cases, by inspection, or by very slight reductions ; and they will be found to agree,
neglecting some terms of the third order, depending on angles which are not expressly
included in [4885] ; or, on angles, whose coefficients are not much Increased by integration ;
as 2v — 2mv-{-cv, 2v — 2)nv-\-c'mv, he. The reductions, relative to the terms depending
on the angle 2v — 27nv — cv, are rather more complicated than the others, on account of the
great number of its terms. We have, therefore, placed these terms in the following table
[4885/], in the order in which they occur in the functions [4876e,4835[] ; and have found
their sum in [4885?n]. Comparing this sum with the corresponding coefficient
— 2.(1+ÎH,),(1+Î<
-h7'-
\e^).e,
in the second line of [4885], we find that they nearly agree ; their difference being equal to
the very small quantity 2me.^^e~, which maybe considered as of the fifth order ; and,
as this is to be multiplied by the factor without the braces, which is of the order nfi, or of
the second order, it becomes of the seventh order, which is usually neglected in this
coefficient :
— 2e.(5+,»5e2 _.|e'2)— 2me.(l+Je2 — |e'2)
[4876e],line2
[4885c], line 3
4
9
11
13
-2e.{—i,e"—i,y^
— 2e.a+ite2+A7^.
—2e.{+,%e^
— 2e.(-,\e^— A7=^
— 2e.
)—2me.{ —I «2—172
c'-2)
)— 2me.( +I^e2
)
)— 2me.( — i1;e2
)
)•
[48S5m]
Sum is =-2e.(l + ie^-i?^-|e'2)-2me.n+jJe^-|7--ie'-).
VII.i.§6.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 409
quantities of the fourth order, which result, after integration, nearly destroy
each other.
The intes;ral tt^-TI ]-'^ [4754], contains also the following
° h- -^ \ dv J ir
[4887']
term,*
-T-/-^;^-Si»-(^'— ^')- [4888]
4 II
This quantity, by development, produces the following expression,!
* (2805) The second term of (—] [4809], namely, — ^ . sin. (tJ — v'), being
, . ,. , , 2,fo , . 2 /dq\ dv . 3m' u't.dv . , ,.
multiplied by — ,, produces, m "^^i" • (^^^j ,7> t^e term, — — .-^.sm.(«— 1> ) ; [4887a]
whose integral is as in [4888].
t (2806) We may change v into v-\-^(f, in [4872], in the parts which are not
connected with m^v, or c'mu, upon the same principles as in [4876a, &c.]. By this means, Mggg^n
the expression [4872], with the addition of tlie two terms [4870.c], becomes as in [48896].
Multiplying [4884] by ^, we get [4889c] ; always using the abridged notation [4821/], ..gg
which ivill frequently be done, in the commentary oil this hooJc, without any particular
notice, that the angles -a, •s/, ê, are omitted;
_2 r{l-\-<2e^-\-2e"^).sm.{v—mv)
9)ii'. «'■* . 9 nt rt ) . \ ^ • / 1 \
— ^-p, — r.sm.ti; — V )^ — ; — . — . < +2e.sm.(cu — v-\-mv) — 2e.sm.[cv-\-v — mv) , , ^. „„,
87i2.«4 ^ '' 8a, a' \ ' '^ ' ^ ^ ^ C [4889i]
' (_ -j-e'.3in.(y — mv-\-c'mv)-\-3e'.sm.{v — mv — c'mv) ,
3-== 3 « • Î (1-* e2_i7.2)_e.cos.c .+ &c. S . ^^ggg^^
The product of these two expressions, retaining terms of the same form and order as in
[4889], becomes as in [4889A]. For the product of the two factors without the braces, is
evidently equal to — •-• -, as in [4S89A]. We shall now multiply the terms between [4889rf]
the braces in [48895], by those in [4889c]. The first line of [48895], being multiplied by
the factor (1 — 56^ — iy^) [4889c], produces the expression,
{\ + le^—lf-+2e'^).sm.{v-mv) ; ^^^^^^^
and the term — c.coscd [4869c], being multiplied by each of the terms depending on e,
in the second line of [48895], produces a term of the form e-.sin.(y— mi') ; adding these
two terms to those in [4889c], we get,
{l + ie^-iy^+2e'^).sm.{v-mv), as in [4889A]. ^^gg^^^
VOL. III. 103
410 THEORY OF THE MOON; [Méc. Cél.
2 I I— m
.cos.(w — mv) \ 1
. , .u„..,^ „ , . ' -\-p'.r.nfi.Cv — «??
[4889] — — .r — —.sin.ft; — w)= — .-.-,.( +e.cos.(« — mt-+c'mw — -') ^- 2
[4890]
[4891]
3 ft' \
-4 .cos.fi' — mv — c'mv-X--J) j 3
the other terms of the integral [4887'] may, in this part, be neglected. This
being premised, if we observe, that the expression of u [4826] gives,*
l+e"+l/ \ 1
^'1,, _i) +(1— c-).e.cos.cî)— ^) . 2
a ) g I
the term ( -^ + m ) • y-a'/f -y^) • '"i ? of the equation [4754], will
produce, by its development,!
Lastly, the first term, or unity [4889c], being multiplied by the terms in the third line of
[4889i], produces those depending on e', in [4889A] ;
r{l+le'^—ly^-}-2e'~).sm.{v—mv)]
3m' u'i - a a y , . . , , '
[4889A] ~\}fi-^'^^'''^^"~'^'^~^'''^^"^''^"\ + c-sin.(t!— ?w«)-j-cw«— n)
\ -\-Ze! .%\\\.{y — mv — c'mi;-(- si')
Multiplying this by ih, integrating, and putting in the divisors c'=l, it becomes as in
[4889i] [4889]. We may remark, that the term — §7^, which we have connected with the factor
(l-|-o<^^2_|_2g/2j^ i„ [4870^', 4872], ought also to be connected with that in [4889/i, 4889] ;
'■^^^^^^ so that, instead of l+Je^—i 5.24.2 e' 2, we may write l+J t^— J-i^2_|_2e/a_
* (2807) The second differential of u [4826], taken relatively to d, and divided by
dv^, gives,
[4890a] ''^'^^=^.l-r^e.(l+.s).cos.(c.-.) + ^V-cos.(2,o-.-2ô)r
Adding this to the expression [4826], and neglecting terms of the fifth order (1 — <?).^
[48906] [43O8e],weget[4890].
t (2808) The terms of the integral yJr/;^-^' are contained in [4885,4889].
[4892a] These two functions must be multiplied by the expression of ——-\-u [4890]; and the
Vll.i.^se.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 411
ddii \ 2 ^f/Q dv
dv
Ifi '-^ dv ' «3
(l4-3ea+j-v3— ge'a)
2— 2m
.cos.{2v—2mv)
^H(l— "0 2-2m-c V T^* 2 >'\ ^ ' ^
2(1 — m) /r» rt .
'^ -.e.cos.(2îJ — 2'mv-{-cv — ra)
— :^~7^ — K • COS. (2 V — 2mv-\-c'mv — -a')
2.(2 — m)
\ — „ ,!l'^ —.eé.cos.(2v — 2 mv — cv — c'mtj+ra+ra')
\ 2.(2— 3 m — c) ^ ' ' /
' o '/7^~^ — r~: • ^ ^'- COS. (2 v —2 m «4-<^ « — c'm v — ro+w)
2.(2 — Jot-j-c) ^ '
, (2+ot) ,
+ -^5 -.ee.cos.(2» — 2m« — cv-\-c'mv-{--a — ra')
/*. { Ài Tïl C)
. (2-m)
2 /+ ITTTi — t--^^-cos.(2î; — 2'mv-{-cv-\-c'mv — tx — n')
"' \- 4:(2^2+2^-^'-^°«-(2c.-2.+2m.-2.)
+ X(2^2=:2;^-^-'^''^-(^^^'+^^— 2»^^— 2^")
+ ^-6:(ï=^3 -i:(^qk)5-^""-^«^-C2â-.-2«+2m«-20
c (4^2 — 1) /2 m) )
+ Jï6:n=7,ô"^ 4.(2^^+2-2,«)] •'''•cos.(2^«+2i'-2mv-20
Ij (5-j-m) 3.(1— m) ) ^
"U.(2-2«-2^+7)+4(2:^^5-''"-^°^-(2^~2mi;-2gt;+ci;+2«-^)
+ — rfi T— .-.cos.(zj — mt))
4.(1 — m) a ^ ■'
+ J- . — .e'. cos.(t' — mv+c'7nv — ^')
\^
3
4.(1— 2/«)
-. e'.cos.(w — mi' — c'mt;+ra')
[4892]
18
sum of the products will be equal to the function [4892]. In finding the products of the
[4892a']
412
THEORY OF THE MOON ;
[Méc. Cél.
[4893]
[4894]
7. The term
>.(l+,sjl'
of the expression
_1 f'^3\__L f'R
h~ \ du J h^
ds
[4808],
[ 48926]
[4892c] _ __ [4889], which is of the ybwr^A order ; by this means, these terms become so small,
functions [4889,4890], we may neglect the second and third lines of [4890] ; for (1 — c^).e
is of the third order, 7® is of the second order ; and these are to be multiplied by the factor
2 a
[4892rf]
[4892e]
[4892/]
[4892g:]
that they may be neglected, and the function [4890] is reduced to its first term
-.(l+e^-j-iy^). MultijDlying this by the terms in [4889], lines 1,2,3, we obtain respectively
the terms in [4892], lines 16, 17, 18. In the term depending on cos.(j; — mv), in line 16. we
may, for greater accuracy, decrease the factor l-|-|e"+2c'^, by |>^, as in [4889i].
We shall now compute the product of the functions [488.5, 4890]. In the first place, the
product of the factors, without the braces, is
2
3 wf . ^ X - = — ; as in [4892].
a, a a,
The multiplication of the factors, between the braces, is made, term by term, as in the
following table ; in which, the first column contains the terms of [4890] , the second column
the terms of [4885], and the third column the corresponding products of the terms between
the braces, in these lines of the two functions respectively: observing, that 4^^ — 1^3, nearly:
[4892^1]
(Col. 1.)
Terms of [4890].
1
(1 — c').e.cos.cv
(4g^-l) 2
.y^.C0S.2^D
(Col. 2.)
Terms of [4885].
whole of [4885]
1
2
1
1
3
(Col. 3.)
Products of these terms.
whole function [4885] between the braces
.cos.(2î) — 2mv)
2— 2nt
2.(l-}-jrt)
2-2m-c
•{e^-\-i')'^)-e.cos.(2v — 2m v — cv)
(1— cS)
■-TTz --.e.coa.C^v — 2 mi' — C1O+&C.
4.(1 — m) ^ ' '
1
2
3
4
^y ^^ .; 2. {cos.(2^i)-2y+2)«D)+cos.(2^f+2r-2mî)) \ 5
— — .c>^.cos.(2d — 2wv — 'i. g v-\-cv)-\- hx.. 6
4.(2— 2wi+f)
Connecting the terms from lines 2 to 6 of this table, with those in line 1, or the lines between
the braces of [4835] ; we get the corresponding terms between the braces, of the function
[4892].
VII. i. §7.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 413
becomes, by neglecting quantities of the fourth order,*
a. ( 4 ) h~
|3" being a function of the fourth dimension in e, y ; and 6s the part of s
arising from the disturbing force. We shall see, in [5596], that ^5 is of the
following form ;t
[4895]
[4896]
[48936]
* (2809) Developing the expression [4893], according to the powers of s, it becomes
— /i~-.(l — J«^+-r«'* — Sic). If we substitute in this the value of 5 [4818], augmented by [4893a]
the term ôs, and neglect terms of the order ôs^, which are noticed in [4958, &;c.], we
shall find, that the part of the function [4893], depending on Ss, is equal to the differential
of the expression [4893«], relative to S, which is — Jr^.{-3sSs~\-J^-s^Ss-&Lc.). Neglecting
terms of the order s^5s, it becomes 3h~^.sSs, as in the last terra of [4895]. Now, the
value of s [4818] gives, by means of [1, 3] Int.,
l—^s^={l—iy^)+if.cos.2gv; -'^s^ = M7^— i|7^cos.25-ô+&c.; [4893c]
1— §s^+W— &ic.= {l—t/-)+Î7^-{l—i7^)-cos.2gv-\-teTms of the 4th order. [4S93rf]
And, from h^ [4863] , we get,
— ^-2. =: .^i_|_e2-]-y2^_j_ terms of the 4th order}. [4893el
Multiplying together the two expressions [4893£/,e], we get the part of the function [4893a],
which is independent of Ss, as in [4895].
f (2810) The form here assumed for 5s is easily obtained from a comparison of the
equations [4754, 475-5], by which u, s, are determined, with the preceding development of [^^'''"J
the terms of M. Forthe equation [4754] contains the function — r3[~r ) — rô^-( T )' whose [48976]
terms have been developed in [4866, 4870, 4872, &tc.] ; and the equation [4755], by which
s IS determined, cont3.\ns the same function, multiplied by -. Now, the chief term of r4897ci
the factor - is equal to a-,.sm.{gv — ê), as is evident from [4818, 4791] ; and, if we
multiply the terms we have just mentioned [48G6, 4870, 4872, fee] by a7.sin.(o-j) — ê), we [4897rf]
shall obtain the most important terms of [4755], depending on the function [4897c]. Thus,
the first term of [4866] produces a term depending on sm.(gv—ê), which may be
considered as being included in the form [4818]. The second term of [4866] produces the '■ ^'
angles gvzhcv [4897], lines 3, 4. The third term of [4866] produces the angles
gv±(fmv [4897], lines 8, 9. The first term of [4370] produces the angles 2v — 2mvzizgv
[4897], lines 1,2. The second term of [4870] produces the angles 2v — 2mvdizgv — cv
[4897], lines 6, 7. The third line of [4870] produces the fifth line of [4897] ; and so on, [4897^]
VOL. III. 104
414 THEORY OF THE MOON ; [Méc. Cél
6s=^ B^^^K J'. sin. (2 V— 2m V— g v+ô) 1
-i-B^^'\y.s'm.(2v — 2mv+gv—ê)
+ B^^-^ . e -/.sin. {gv-\-cv — ê — ct)
-j- B^'-^\ ey .sin.Çgv — cv — ^+^)
-{-B^^''\ey.sin.(2v — 2mv — gv~\-cv-^Ê — to)
~i-B^'-^\er.s'm.(2v — 2mv+gv — cv — ^+ra)
-\-B^^'^\ej.sin.(2v — 2mv—gv — ct)+â+ra)
-{-B^'-''\e'-)'.s'm.(gv-\-c'mv — ê — z=')
+B^^^Ke'r.sm.(gv — c'mv — ^+ra') 9
-\-Bf\ eV-sin.(2« — 2mv—gv-i-c'mvi-ê — ra) 10
+B[^''Ke'y.s'm.(2v — 2mv—gv—c'mv-\-É-{-^')
+Bl''\e"-y.sm.(2cv—gv—2zi+è)
Atisumed
form of
2
3
4
5
3+^) 6
7
8
') 11
12
16
+Bl''\e"-y.sm.(2cv—gv—2zi+è)
+ B\'^Ke^y.sïn.(2v—2mv—2cv+gv+2r,—D) 13
+B<'-^Ke''y.s\n.(2cv+gv—2v+2mv—2^—è) 14
+5^''''.-.7.sin.(^ti — v+mv — ê) 15
+ 5''^^.-,.7.sin.('£-«+î? — mv — ^).
a ^^
for other terms. Hence we see, that the forms of the angles in [4897], are given a priori
[4897ft] by the theory; and they agree with the results of observation [5596]. The differential
equation in s [4755], is similar to that of u [4754], and may be reduced to the form
[4897m], which is similar to [4845]. For the chief term of s is given in [4818], and if we
[4897i] suppose the other terms of « to be represented by 5s, we shall have «=7.sin.(^i' — â)+i3s.
dds d-.Ss
Its difFerential gives "7^= — g^.y.sin.(gv — ^)-\-~pr- Multiplying the fu'st of these
[4897A] expressions by g--. and adding it to the second, we get -—-\-g^.s=-—-j~g^.5s; and if
[4897t] we put the second member of this expression equal to — n', we shall get,
[4897m] ^^+^9., + n'=0.
This is of the same form as [4845] , g taking the place of JV, and differing from unity by
quantities of the order m^ [4828c, 4845']. Moreover, n' may be considered as a series of
terms, whose general form is k'.sm.{iv — 6), like that in [4846] ; and the part of s, relative
to this sine, is represented as in [4847, Sic] by
VII. i-sW] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m.
415
The number placed beloio any one of the letters B, indicates the order of that
letter. Thus, Bf is of the second order ; ^t"' is of the first order ; and t'*^^^!
B'-g'^ is finite. We may observe, that this takes phice according as the number [4898']
by wliich v is multiplied, in the corresponding sine, differs from unity, by a
finite number, by a quantity of the order m, or by a quantity of the order nf, [4899]
respectively ; because the integration [4897o] causes the terms to acquire a [4900]
divisor of the same order. This being premised, we shall have,*
-JV2
. sin. {iv — 6) ;
so that these terms may be much increased by this integration, when i is nearly equal to
unity. From the similarity of the equations [4754, 4755] it is evident, that the terms of n'
_2
[4897m], depending on the disturbing force of the sun, must have the same factor m , as
the functions [4866, 4870, 4872, &c.] ; and in is of the order «i^ [5094], or of the
second order. This factor is divided by i^ — JV~, in finding the value of s [4897o], or that
of &s [4897] ; and, as i^ — JV^ may be considered as of the same order as i~^g'^^i^-\-^m^
[4828e] ; the order of the symbol B will ie represented by
Hence, it
i2— 1— 3m2
appears, that if { differs considerably from unity, tlie corresponding symbol B will be of the
second order, as in [4897], lines 2, 3, 4, 5, Sic. ; using the values of c, g [4828e]. In the
first term of [4897], the coefEcient of u is i=2 — 2m — g^=l — 2m nearly; hence,
i^ — 1 — I'm^ is. of the order m, and the corresponding value of B [4897r] is of the order
m, represented by Bf; and the same occurs in lines 8 — 11 [4897]. In line 12 we have,
i=^2c—g = 1 — if'-rri^ [4828e] ; hence, the divisor of the expression [4897 r] becomes of
the order m^, and the corresponding value of B is reduced to the order m", or a finite order,
as it is called by the author in [4898'], and is represented by i?},'". If we compare the
indices of B [4897], with their values, computed in [5122—5214], we shall find they
generally agree ; but the term B'i^' [5179] is nearly oï the first, instead of the second
order ; i?i'-' is of the second order, fee.
* (2811) Substituting in the firstmember of [4901], the values of A-s, s [4893e,4897i],
and neglecting terms of the order &^ we get [4901a]. If we also neglect terms of the fifth
order, it becomes as in [4901e] ;
3s.6s 3 , . ^
— = -•7<^*-sin.(^r— â)x |l+e^+/2+terms of the fourth order}
= -.Yh.sm.{gv—è).
We must substitute in this last expression, the value of vs [4897], and we shall get [4901].
If any term of & be represented by C.sin.F, the two corresponding terms of [49016]
[4897o]
[4897p]
[4897?]
[4897r]
[4897s]
[4897<]
[4897u]
[4901a]
[4901i]
[4901e]
416 THEORY OF THE MOON ; [Méc. Cél.
^==-~l Bf -B^^ ] .y^cos.(2 v-2 m v) 1
+ ^.B^p.7~.cos.(2v—2mv—2gv-\-2è) 2
— --.Bf\e y^cos.r2 &■ v—c v—2 é+^) 4
2a, ^
+ -.B^,'Key^cos.(2v—2mv—2gv+cv-^2ê—^) 5
[4901] -i- ~AB^'^—B'i'>l.ef-.cos.(2v—2mv—cv+^) 6
+ ^.lB[-^+B['^.e'7''.cos.(c'mv—^) 7
— — .£f'.e'7^cos.(2 ?;— 2 m v+c'm v—z,') 8
^.S'"').e'7^.cos.(2 1>— 2 m v—c' m v+^') 9
2a,
— —.B^^'\e^y^.cos.(2cv—2 ^) 10
+ —SB'-}*^+B^!'^.-.7^cos.(v—m v). 11
2a, "^ "a.
will be
[4901(i] ^.y.C.cos.{(^D— â)«>F| —^.y.C.cos.^^t)— â+F^ ;
but it is not, in general, found to be necessary to notice more than one of these terms. The
[4901c] only cases in which the author has noticed both terms, are those depending on Bf\ Bf*
[4897], lines 1 — 4. The neglected terms are generally smaller than those which are
retained, or they are such as depend on angles that have not been usually noticed, because
their coefficients do not increase by the integrations. For, the function [4901] forms part of
r490in ^^^ expression of n [4902, or 4845] ; and its coefficients may be increased by the divisor
t2 — JV^ [4847, 8iC.], when i differs but little from unity ; as is the case in lines 3 — 6,11
[4901]. To estimate roughly the order of the terms, which are not increased by the
integrations, and are neglected as in [4901], we may observe, that they produce terms of a
[4901g] similar order in u [4847], and in the lunar parallax [5309, &.c.]. Now, if we put - equal
VII. i.'^^ T.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 417
If we connect together the different terms which we have developed, we
shall find, that the equation [4754] becomes of the following form,*
cJc/u
0 = "IP + " + " ; [4902]
n being a rational and integral function of constant quantities, and of sines
and cosines of angles proportional to v ; but, as loe propose to notice all the [-1903]
to the constant term of the lunar parallax 3424', 16 [5331], and use the values of c, e', 7
[5194,5117], also
f^u [5221], we shall get, very nearly,
_3
2 a,
:40'
TTZ-n-'
:2%3; :^.e'Y'
0%7:
— - e^ y~ := 0' 1 • — - . - >2
2 a.
2 a, a'
OM.
The first of these expressions, being multiplied by the very small quantity .Bf,') [5177],
becomes insensible; and it is retained in [4901] line 1, merely because there is no
inconvenience in doing it, since it is found necessary to notice the angle 2d — 2mv, in
consequence of the magnitude of the other term i?J'\ In like manner, the term
-^ .e>=2.^|'=— 0',01
2o,
[5178, 490 U],
[490U]
[4901i]
is nearly insensible ; but it is retained in [4801] line 3, because the coefficient c, in the
angle cv — -us, diflers but very little from unity [4828e], and it is increased by integration ;
which is not the case with the coefficient depending on the other angle 2gv-\-cv — 2d — ra,
with which -B^-' is connected. One of the largest of the values o{ B, is that denoted by
3
J?f> = 0,07824 [5183]; multiplying it by the coefficient — .e'j2 = 0',7, with which [4901A]
it is connected in [4901] line 7, it becomes 0',05 ; this is retained in the angle c'mv — ■n'
[4901] line 7, because the divisor i- — N^ [4847] is nearly equal to unity ; but it is
neglected in the angle 2gv-\-c'mv — 2d — ra' ; because it is considerably decreased by the
divisor i- — JV^, which is nearly equal to 3. We may also observe, that it is of more importance
10 retain the terms depending on the angle c'mv — -a, than those on 2gv-\-c'mv — 2d — -sj' ;
because the terms introduced by the former, in the value of dt [4753], are increased by
integration, in finding the value of t, in consequence of the smallnessof the coefficient c'm
of the angle v. Similar remarks may be made relative to the other terms, which are
neglected or retained.
[490K]
[4901»;
* (2812) Connecting together the terms [4866,4870,4872,4892,4895,4901, &c.],
depending on Q, and putting the sum equal to n; then adding it to the terms of [4754], [4909o]
which are independent of Q, it becomes as in [4902].
VOL. III.
105
418 THEORY OF THE MOON ; [Méc. Cél.
inequalities of the third order, and the quantities of the fourth order connected
[4903'] with them, ive must add to the preceding terms all those which depend on the
square of the disturbing force, and become of these orders by integrations.
We shall now examine these new terms.
r4<»03"i ^' ■^°^' *^"^ purpose xve shall suppose ou to be the part of u arising from
the disturbing force ; and, that we have,*
aàu = AJ-^K COS. (2» — 2 mv) 1
+ J/^'. e . COS. (2 w— 2 m v—c v+^n) 2
+A^'-^\e.cos.{2v — 2mv-\-cv — ra) 3
+^/'. e'. COS. (2 V — 2 m v-{-c'mv — ^') 4
Assumed ^ ,,-, , ,^ ^ * . /x r
form of + J,< ^ e. COS. (2 V — 2 m v — cmv-\-^) o
ÔU. ~ \
-^A^'-^le'. cos.(c'mv — W) 6
-|-^/'^'. e e'.cos. (2 v — 2 m v—c v+c'm ?;+« — -') 7
+^/''.ee'.cos.(2 «; — 2 m v — cv — c'mtJ+ro+za') 8
+^i''*^e e'.cos. (c v-^-c'mv — « — t^') 9
+yi/^'.ee'.cos.(cD — c'mv — -zs+ijj') 10
[4904] +4'°). e^cos.(2ct;— 2^) 11
+4'^'. C-. cos.(2cv—2v-i-2mv—2^) 12
+4'->. '/. cos.(2 o-v— 2 Ù) 13
+4'='.7^ cos.(2^«— 2«+2ot«— 20) 14
+4»'. e'^cos.(2 c'mi)— 2 ^) 15
+4'^'.e7^cos.(2o-i)— CÎ)— 20+ra) 16
+4'«.e7^cos.(2j;— 2mw— 2^î)+cv+2â— T.) 17
+^["'. -.cos.(î; — mv) 18
+4^'- -.e'.cos.(« — tnv-\-c'mv — -n') 19
a
+4°). -,. e'.cos. (î; — m^; — c'miJ+ts') 20
a
[4904a] * (2813) The terms of a<5tt [4904] are evidently of the same form as those of tlie function
Vll.i. §S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 419
The number 0, 1, or 2, placed below any one of the letters A, denotes,
that it is of the order zero, or of the order m, or of the order m^, respectively. [4905]
We shall here take into consideration the inequalities of the third order, and
those of the fourth order, which can produce terms of the fourth order in
the coefficients of the inerjualities of the third order. We shall continue the
approximation to a greater degree of accuracy, relative to the inequality [4906]
depending on cos.(j' — mv). This being premised, we find, that the term
*"' "'^ , . ,3ot' u^ (5m
[4865'] gives, by its variation, the expression ^~T~ ? f^""'^ [4907]
[4905']
which we deduce the following function ;*
n [4902a]. Tlie order of the coefficient A may be found by the formula
ia— l-f3ma ' [49046]
whicli is similar to that in [4S97/-], using for JV~ the value of c-=l — Zm", instead of g^,
which is used in [4897 5-, r] ; i being the coefficient of v, in the angle corresponding to the [4904cl
coefficient.^. Thus, for «/if"' [4904], we have i = 2— 2/«; hence ^<"> is of the order m^,
or 2. For A'^^, we have i = 2 — 2 m — c=l — m, nearly; hence ^^'^ is of the order m,
or 1; and so on, for the other coefficients of [4904]. If we compare these indices of A,
with the values obtained by numerical calculation in [5122 — 5213], we shall find, that in [4904rf]
general they are correctly marked.
* (2314) The expression [4907], whose value is to be determined, may be put under
the form
3 2 m'. m'3
~Ya^7r'i}fi.u^^"' "' [4908a]
in which the second and third factors have been already computed in [4884, 4866] ; we shall
3
first find the product of these two factors, and then multiply it by and a Su. Now,
if we multiply the factors without the braces, in [4884j 4866], by , the product
becomes
2 _a
m ^3^ 3 m
as in the second member of [4908/] . The products of the terras between the braces, in
[4884, 4866], are found in the following table ; in which the first column gives the terms of
[4884] ; the second column, the terms of [4866] ; and the third column, the products of
these terms respectively ; using the abridged notation [482 1/J, and neglecting the same terms
and angles as we have usually done ;
[49086]
[4908c]
420
THEORY OF THE MOON ;
[Méc. Cél.
[4908]
[4908']
[4909]
Sm'.u'^ôu .3j7r.(l+fe'2)
2 F.
2a.
a,ôu [4904] ^ 1
—2A<^'\e.cos.(2v—2mv—cv+^) |2
-2A\'\e".cos.(2v—2mv—2cv-{-2^) 13
+1 ^J" .ee'.cos. (2v—2mv—cv+c'mv+zi—ô,') J 4
+|J('>.ee'.cos.(2w— 2my— cy— c'mi>-f-a-f^') v ;,
+|J|'^'. - . e'.cos.(t' — mv-j-c'mv — -53')
«
f7
+|-J,"^\ -.e'. COS. ft» — mi; — c'mt'+ra') to
' ' a ^ ' ^°
+14"'- -,.e'-.cos.(i'— m«) / 9
?{' varies by means of the variation of v' , Avhich depends on the time ^, and
on its inequalities in functions of v [4822, or 4828] ; but these inequalities
are multiplied by m, in the exjjression of v' [4837], and also, by e', in the
expression of v! [4838] ; we may, therefore, at first, neglect (5m', without
[4908d]
(Col. 1.)
Terms of [4884].
1
-e{ l-ic"2-i72)cos.«'
(Col. 2.)
Terms of [4866].
dioleof[4866]
1
— '3c.cos.cv
-\-2e'.cos.c'mv
— 3c.cos.cv
-\-3c'. COS. c'mv
-%-ce' .COS. {cv-\-c'mv)\
fee'.cos.(cy-c'»iy)
+3e2.cos.2fi'
-^^y^.cos.2gv
I — 3e.cos.fi'
1 — 3e.cos.cv
(Col. 3.)
Products of these terms,
whole of the function [4866]
— ic2 a.72
-|.(_[_je3+fc;2).cos.cy
-}-( — ^c~e' — ^e'}'~).cos. c'niv
— ( l+f 62_j.y2_|_^e/2) .c.cos.cr
-|-fc2 -\-^e^.cos.2cv
— §ee'.cos.(c!) — c'7itv) — ^tc'. cos. (cv-^c'iiiv)
-j-fe-e'.cos.c'my-|-&c.
-\-^e~e'.cos.c'mv-\-&c.
^C'^.COS.CV-^&LC.
-|«»/2.cos.(2g'i'-cr)-)-&.c .
-j-5e^.cos.2c« 1 — 3e.cos.cv — ^e^.cos.cv -\-ie^.cos.2cv~\-&Lc.
-\-\y^.cos.^gv. 1 — 3e.cos.c« ly-.cos.2gv —^ey~.cos.{'2gv-cv)-\-&i.c.
Connecthig together the terms which are e.Kphcitly given in this tahle, with those between
r4908('1 *'^*^ braces in [4866], wliich are included in the first line of this table; the sum becomes
equal to the expression between the braces in [4908/"] ; and the factor of a ou [4908a]
becomes as in the second member of [4908/"] :
VII. i. ^8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 421
any sensible error. We shall hereafter [4947, &c.] notice the term of this
variation, which depends upon the action of the moon upon the earth.
[4909']
3 m
27,
~\- ( — 4 e — 3 e' — 6ee' ®+e y^) -cos.ct
+3e'. (l+2e2+fe'^).cos.c'mj;
— 3.(2-|-m) .ce', cos. (cv-\-dm v)
-3.(2 — m).ee'.cos.(cD — cfinv)
-\-5e^.cos.2cv
-\-y^.cos.2gv
+fe'^.cos.2c'7«t)
K — |e7^-cos.(2^« — cv)
Muhiplying this by a Su, we obtain the value of the function [4908c(, or 4907]. To reduce
this to the form [4908], we may divide the terms, between the braces, by l-j-#e'^, and
connect this with the factor without the braces; and, by neglecting terras of the fourth order
in e, e', y, between the braces, we get,
l+2e2
-j-( — 4e — Bt^-\-e 2^).cos.cv
+3e'. {l-\-2e^—^e'^).cos.c'mv
-3. {2-\-m).ee'. COS. (cv-{-c'mv)
— 3.(2 — m).ee'.cos(c« — c'mv) \ . aàu.
-f-5e^.cos.2cD
-|-7^.cos.2^i;
-l-Se'^.cos.2c'mD
3m'.u^.U
31h.{l+ie'^)
2a. '
„, . 3m'.(l+|e'a) . ,
1 he factor —- — - is the same as
2a
le 7^.CGS. (2^ t) — cv)
[4908]. The term 1, between the braces in
[4908/]
[4908g:]
[4908^], being multiplied by the external factor aSu, produces the term aSu in the first
line of [490S]. Now, if we neglect this term 1, between the braces in [4908^], and [4908A]
multiply the remaining terras by aSu [4S04], it will produce the terms of [4908],
between the braces, which contain A explicitly. In performing this multiplication, it will
only be necessary to retain the two following terms of [4908jg-] ; namely,
— 4e.cos.cr-|-3e'.cos.c'mu. [4908t]
For, the other terms, between the braces, are of the second order ; and these are multiplied
VOL. III.
106
422
THEORY OF THE MOON ;
[Méc. Cél.
[4909"]
[4910]
3 m' m"
The term „,„' , . cos. (2 y —2^') [4870], has, for its variation,
9m'.M'3
3m'.u'
/„'3
, ^ . 6u. COS. (2v — 2v') + -— ^ .iv\sm.(2v—2v').
If we substitute the preceding value of &u, we shall find, that the first of
these terms produces the function,*
[4908fc]
[4910a]
[49106]
[49i0c]
[4910rf]
[4910e]
[4910/]
[4910^]
[4910A]
[4910{]
by m, of the second order, and by a 5m, of the «etorao, order ; proaucing terms of the
sixth order; some of which may be reduced to the Jifih by integration [4847]. The terms,
depending on the angle » — mv, of higher orders, are retained as in [4874, &lc.]. The two
terms [4908J] evidently produce those in [4908], which depend explicitly on the symbol
A, neglecting the terms which have been usually rejected.
* (2815) If we take the differential of [4885], relative to dv, and multiply it by
ia.dv
, we shall obtain the expression of
9ot'.u'3
' ihKu->.a
.sin.(2« — 2v'). The effect of this
2
operation will be to change the factor 3 m.— [4885] into — - — , as in [4910 Ar] ;
moreover, it will take away the divisors 2 — 2m, 2 — 2m — c, he, which were introduced
by the integration, and will change, in tlie second member, cos. into sin. When the
function is reduced to this form, we may change 2v into 2 y + 90'', as in [4876a — d] ;
and we shall obtain the expression of
9m'. u' 3
4hKu'>.a
.cos.{2v—2v') [4910k'].
If an angle, in the second member of [4885], be of the form cos.(2î)-}-(3), it becomes, in
[4910f/], sin.(2i;+(3); and, in [4910c], it changes into sin.(2«4-|3-|-90''), or cos.(2j)-f-^) ;
which is the same as its original form in [4885]. But, if it be of the form cos.(3 — 2v),
the successive changes are
sin.(p — 2v), sin.((3— 2« — 90''), and — cos.(p — 2v) ;
this last being the same form as the original, but with a different sign. Hence we easily
derive the expression [4910^] from [4885], by using the factor
neglecting the denominators 2 — 2 m, Sic. [4910c], and changing the signs of the terms
depending on angles of the form cos. ((3 — 2d) ;
VIL i. «5- s.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u.
423
^'"'■%.5u.cos.(2v—2v')
'2h^.u*
9m
4a.
4°>.(l-|e'=^)
+ { Ji"-44''^ +4='-Mf -e'^+J^r'-f ''} •«•(1-f «'')-cos. (c» -«)
+ { S^o'+^^'J+Jt^' I . e'. COS. (c' m v—^)
i_[_^4') — iJ<'>].ee'.cos.(cw+c'/«i' — ^ — ^)
+ -^f • e e'. COS. (2 v — Imv — c t) — c'mî;+«-}-a')
+ ^<^' . e e'. COS. (2 « — 2 m w — c v-\-dm v-\--a — to')
\+{^J""+i(2+?tt). J;"-2( 1 +«0.^"' I .e/^cos.(2o-zj-a'-2i)+TO)^'
1+4'^'. 67^. COS. (2 z? — 2mt5 — 2^w+ct)+2â — a)
+ { Jf^— i^'^'.e'^j. -,. cos.(«— OT z))
+ {4"'— è4"'l.-.e'.cos.(i;— m2;+c'm?)— to')
+ {4'3)+^'J(/^)^.-,.e'.cos.(«— mtJ— c'mtJ+TO')
1
2
3
4
5
6
7 [4911]
8
9
10
11
12
4 A^ vr.a ^ '
9m_
4a,
/(l+2e2— |c'2).cos.(2d— 2mD) V 1
-2(l+m).(l+Je2-iy2-fe'2).e.cos.(2«-2mj;-c!;) \ 2
— 2(1 — m).e.cos.(2y — 2mv-\-cv) I 3
+Je'.cos.(2i; — 2m« — c'otw) I 4
-Je'.cos.(2j; — 2m«-j-c'?n«) f 5
^ — J(2+3m) .ee'.cos.(2v—2mv—cv — c'mv) I 6
-f(2— 3m).ee'.cos.(2u — 2mv-\-cv — c'înv) 1 7
+|(2-j-m) .ee'.cos.(2« — 2«t)— cjj-fc'mv) \- 8 [4910/t]
1+1(2 — m) .ee'.cos.(2«) — 2mv-{-cv-{-c'mv) / 9
/+ï(10+19/«+8OT2).e2.cos.(2ct;— 2j;+2mi') 10
'+i{l0~19m^8m%e^cos.(2cv-\-2v—2mv)\ll
+i(24-m).j^.cos.{2gv—2v-\-2mv) Il2
+i(2—m). 72.003.(2^ D+2 v—2mv) Il3
+^.e'3.cos.(2î)— 2mi;— 2c'7«») /l4
\ — î(5-j-m).ey®.cos.(2î; — 2mt) — 2gv-\-cv) / 15
424 THEORY OF THE MOON ; [Méc. Cél.
[4911'] aèu contains a term, depending on cos. (3t) — Smv), which we have
[4910i]
Multiplying the first member of this expression by 2.a6u, and the second by its equivalent
expression [4904], we shall obtain, by making the usual reductions, the value of the first
term of [4910], as in the second member of [4911]. For, the factor, without the braces,
a
97/1
[4910m] — ' — , is the same in both these functions ; we shall, therefore, neglect the consideration
4 a,
of it in the remainder of this note ; and, in speaking of the functions [4910A:, 4911], shall
[4910n] refer exclusively to the terms between the braces ; and, shall separately investigate the results
arising from each line of the function 2-a&u [4904], by the ivhole of the function [4910A:].
First. We shall take into consideration the product of the term 2 ^g'*. cos. (2 a; — 2mv),
by the whole of the function [4910fc] ; and shall reduce the products by formula [20] Int.,
retaining the same angles as in [4911]. The first line of [4910A:] produces the term
(l-[-2e^ — Je'^)..42""; the part depending on cos.(4t) — 4 ??(«;) being neglected. This
2
corresponds to the first line of [4911], neglecting the part depending on 7/t .c^.^,'*, of
[4910o] the sixth order, as is done generally in the rest of this calculation ; the term, depending
on fe'®, is retained, on account of its importance in the secular equations of the moon's
motion [4932, 5059, 5087, &.c.]. Again, if we neglect e^ y^ in the factor [4910/1]
line 2, and introduce the factor (1 — Je'~) in [4910À:] line .3, according to the directions in
[4869^, fee], we shall find, that these terms, when multiplied by 2A.j°\cos.{2v — 2mv),
produce respectively the terms
— 2.(l+m).(l— Je'S).^/'. e.cos.ct;, — 2.(1— m).(l -Ae'^) .^^w.e.cos.ct; ;
whose sum is
— 4.(1— fe'2).^2°'icos.ct), as in [4911] line 2.
In like manner, the terms in [4910^] lines 4, 5 being multiplied by 2jÎ^°\cos.{2v — 2mv),
produce respectively the terms
^A.;^°\e .COS. cm V, — l^^'^'.c'. cos. c'mo ;
whose sum is
3AP. e'. cos. c'm v, as in [491 1] line 3.
the remaining terms of the function [4910fc] may be neglected, on account of their smallness,
and the forms of the angles.
Second. We shall now compute the terms produced by multiplying
2.^,<'\ e .cos.(2v— 2 m v—cv) [4904],
by the terms of [4910/*:]. The first line of [4910A-] produces .^i<'\ e . (1— Je'^) . cos. c v,
as in [4911] line 2. The second and third lines of [491 OAr] depend on c^, which is neglected.
[4910/)] The fourth line of [4910^::] gives iee'.Ai^'\cos.(cv—c'tnv), as in [4911] line 4 ; the fifth
line, — ^ e b. Jli-'\ COS. {cv-]-c' m v), as in [4911] line 5; and the twelfth line
} {2-\-m) .ey^.cos.{2gv — cv), as in [4911] line 8.
VII. i. ■§ 8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 425
neglected,* on account of its smallness in [4904] ; but, as it may have an
influence in the term depending on cos.(îj — mv), we shall take notice t*^^^'^
The other terms, depending on ►?/'>, are neglected, on account of their smallness, &ic.
Third. The product of 2J.r-''.e.cos.{)îv—2mv-\-cv) [4904], by the first term of
[4910t], produces the term A./~le. {l—ie"'}.co5.cv, as in [4911] line 2. This is the [^910?]
only term depending on ^.f\ which requires attention ; the other terms being small, or
of forms which are unnoticed.
Fourth. The product of 2 Jl/^Ke'. cos. {2 v— 2 7JIV+ cm v) [4904], by the first term of
[4910A], produces the term Ji.^^^'.e'. cos. c'mv [4911] fine 3; the other terms maybe
neglected. In like manner, 2^a<".e'.cos.(2«—2/?iu—f'mî;) [4904], produces w3^«.e' cos.c'mw [4910r]
[4911] line 3; and 2.^2'^'. e'. cos. c'otd [4904], gives nothing deserving of notice.
Fifth. The terni 2.,'î/''' ce'.co3.(2(; — 2mv — cv-\-c'mv) [4904], being multiplied by the first
term of [4910Â:], produces ^/'".ee'.cos.(ct' — c'mv) [4911] fine 4; and the same term,
being multiplied by the fifth term of [4910A-], produces — \ee'^.A['^\cos.cv; which is
nearly the same as in [4911] line 2. In like manner, the term [4910s]
2.4/''. e e . cos.(2î) — 2mv — cv — c'niv),
being multiplied by the first and fourth terms of [4910A-], produces the terms
Ap.cc'.cos.{ci'-\-c'mv), and -{-lJlp\ee'^.cos.cv; as in [4911] lines 5, 2.
Sixth. The terms depending on Ai^\ .4/°' [4904], being combined with the first
term of [4910A-], produce the terms [4911] lines 6, 7. Those depending on ./^a'"^ ./3/'",
^.2"^', produce small terms, which are not noticed. The term
2A\^'>K',^.cos.{2gv—2v+2mv),
being combined with the term — 2.(l-j-7?i).c.cos (2« — 2mv — cv) [4910A-]line 2, produces
the term depending on ^4,"^' [4911] line 8. The term depending on A.^^*^ [4904], produces
nothing of importance.
Seventh. The terms 2.â^^^^\ey-.cos.{2gv—cv), 2Ai^"^\ef.cos.(2v—2mv—2gv-j-cv)
[4904] , being combined with cos.(2y — 2mv) [4910/i:], produce respectively the terms in [-lOlOu]
[4911] fines 9, 8, depending on .^J'^', .4/"".
Eighih. The term|,-2.,3/'^'.cos.(D— mu), being combined with the terms in [4910/^-]
lines 1, 5, 4, produces the terms depending on .4/'"', in [4911] lines 10, 11, 12, [4910r]
respectively.
JVinth. The first term of [49101], being combined with the terms of 2.aûu [4904],
depending on .^o*'^'' -^o™. produces the corresponding terms of [4911] fines 12, 11. [4910u>]
[4910<]
* (2816) This term occurs in [4808], and must, therefore, be found in the differential
equation in u [4754] , and in its integral 5u, or a ou.
VOL. III. 107
[4911a]
426
THEORY OF THE MOON ;
[Méc. Cél.
of it. For this purpose, we shall put it under the following form ;
[4912]
[4912']
[4913]
[4914]
[4914']
[4915]
[4916]
Term of aiu=^'^^
Substituting this in the expression
it produces the term,*
2
9 w (
Aa, ^ i
- .cos.(3t;— 3«').
^^,.6u.co^.{2v—2v') [4910],
To develop the variation
A2.m3 •
-. cos.f» — mv).
a ^
iv'.ûn.{2v — 2v') [4910], we shall
observe, that iv' contains, in [4837], the same inequalities as the expression
of the moon's mean longitude, in terms of the true longitude ; but they are
multiplied by the small quantity m. It is sufficient, in this case, to notice the
terms in which the coefficient oïv differs but little from unity;t and it is evident
that as the term e.cos.(ct) — ^n), of the expression of «i< [4826], gives, in v', the
termj — 2me.s'm..{cv — ra) ; any term, whatever, of af>u, such as A;.cos.(ù"-|-£),
r4913ol * (^Sl"^) Substituting the values of m, u' , [4791], and h^ = a, [4863], also
v'=^mv [4837] nearly, in the expression [4912'], it becomes
[49136]
9)
2a,.<i'3
—.aSii.cos.(2v — 2d') = — "-— .a5u.cos.{2v—2mv) [4865].
If we now substitute the term of aSu [4912], we obtain that in [4913], and also one
depending on the angle 5 v — 5 m v, which may be neglected.
t (2818) We shall see, in [4918], that the terms of this form, in which the coefficients
[4914a] of V are nearly equal to unity, produce only small quantities of the fifth or sixth order.
These terms are noticed, because they are much increased, by integration, in finding the
[49146] value of u [4841] ; but this does not happen with the terms in which the coefficient of »
differs considerably from unity ; and we may also observe, that, in this last case, the terms
[4914c] may also be decreased by the integration in [4822]. Hence, we see the propriety of
noticing only the terms mentioned by the author in [4915].
X (2819) If we inspect the calculation in [4812 — 4837], we shall find, that the term
[4915a] c.cos.(c«;— ra), which occurs in u [4812,4816, 4819, 4826], is introduced into dt [4821], and
by integration, produces in t [4822], or rather, in nt-\-s [4830], a term — 2c. sin. (en — w).
[49156] rpjjjg jg j-,;iu]^piied by m in the second member of the equation [4836] ; and it finally
produces in v' [4837], the term — 2?»e.sin.(cj; — 13), as in [4916]. This may be derived
[4915c] f,.Q,j^ {],g preceding term of u, by changing cos. into sin. and multiplying the result by
VII.i.§8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m. 427
ill which i differs but little from unity, gives very nearly, in dif, the term
2mk.sm.(iv-{-s). Thus we find, that the preceding term [4914] gives, by
its development, the function,*
[4917]
— 2 m. Tlie same method of derivation may be used with any other term of u, in which
ihe coefficient of v diflers but little from c, or from unity [48286] ; as is the case with the [4915d]
term ^•.cos.(^•»-)-s) of u [4916], which produces, in Sv', the term — 2mk.sm.(iv-{-s)
14917].
* (2820) Instead of the angle iv-\-s [4916, Sic], we shall, for brevity, use iv, omitting
s, as we have to, -n', 6, in [4821/], and re-substituting it at the end of the calculation.
Then, if we represent any term of ai5w [4904], in which i differs but little from unity, by
a5u = Ji:cos.iv [4916], the corresponding term of 5v' will be very nearly represented by
^i)'= — '2m k.s'm.iv [4917]. Moreover, if we represent any term between the braces of the
second member of [4876e], by As'm.V ; or, in other words, any term of the function
A2.u3
— .sin. (2» — 2v') by
.^.sin.F;
and then multiply it by the preceding expression of 5v', we get, by using [17] Int.,
—^ .5i''.sin.(2c — 2î)')= .^AmJc.cos.(iv'-^T^) — Am'k.cos.{i v-{-V )\ .
The factor, without the braces, is the same as in [4918] ; consequently, the terms, between
the braces, in [4918], must arise from the other factor of [4918/] ; namely,
A mk.cos.{iv>xy^ — A mk.cos.{iv-}-V) ;
in which we must substitute the terms of a&u [4904], for k.cos.iv; and, the terms between
the braces in [4876e], for .^.sin. K; neglecting the terms which are insensible from their
smallness, or those, where the coefficients of v, in the angles, vary much from unity [4915].
We shall, in the first place, compare the terms of the function [4918^], with the terms
between the braces in [4918], taking successively, for k, the coefficients of the terms [4904],
which are retained by the author. First. The term .^/".e.cos.(2« — 2mv — cv) [4904],
corresponds to /i;=./î/'^e, iv = 2v — 2mv — cv; combining this with the first line of
[4876e], neglecting e^+i)-^, we find that this first term of [4918^] produces the first line
of [4918]. If we combine the same term of [4904] with the first term in line 13 [4876e],
we find, that the second term of [4918^] produces the second hne of [4918]. It is
unnecessary to notice the products of the other terms of [4876e], by the term [4918A:] ;
because the coefficients are small, or the angles are different from those which are usually
retained. Second. The term Af)^^^\e'y^.cos.[2gv — cv), being combined with the first
tenu of [4876e], produces, by means of the first term of [4918o'], the third line of [4918].
[4918a]
[49186]
[4918c]
[4918rf]
[4918e]
[4918/]
[4918gr]
[4918^]
[491 8i]
[4918A:]
[4918i]
[4918m]
[4918n]
428 THEORY OF THE MOON 5 [Méc. Cél.
m.J/".e.(l— |e'-).cos.(ci'— î3) . 1
+^.m.A[^\ef.cos.(2gv — cv — 20+13) | 2
3wtt^ , . .^ ^ . 3w / +'/».^''"'.e?^ .COS. , , CI. ° 1 I 3
[4918] ■^j^.'V.sin.(2t;— 2i;')-=— ^^-^ ' " ' V +cv+2l—^ J \
-\-m.A\"\-.cos.(v — mv)
a '
-j-m.Al^^K-.e'.cos.(v — mv — c'm^7+3')
The other terms of this development are insensible.
The terms
a m' ni' i
.\2.cos.{v—v')+5.co&.(3v—Qv')\,
[4919] Qh^.u^
of the expression
|.^gjg Third. The term AP''\ -, . cqs.{v — mv), [4904], combined with the first temi of
[4876e], produces, in like manner, the fourth line of [4918]. Fourth. The term
r4918ol •^o"*'- ",-fi'-cos.(i; — mv-\-c'mv — is') [4904], combined witii tlie same first term of [4876eJ,
produces the fifth line of [4918].
[4918c]
[4918r]
It appears, from [4840, &c.], that the terras in the five lines of the function [4918], are of
the orders 5, 7, 6, 6, 6, respectively. The integration [4847], introduces divisors of the
order m^ [4828e], in the first and second lines of [4918], and of the order m, in the other
three lines. By this means, the first line of [4918] produces, in the value of u, a term of
the third order, and the other lines produce terms of the fifth order ; which are within the
limits proposed in [4905', &c.]. With respect to the order of the terms which have been
neglected, we may observe, that, in calculating in [4918Z] the quantity produced by one of
[4918s] the ^rea<es< terms of [4904] ; namely, .^"\e.cos.(2y — 2mu — cv), when combined with
the greatest term of [4876e], contained in its first line, we have noticed only the first term
of the function [4918^], and neglected its second. This second term has the same coefficient
of the fifth order, as in the first line of [49! S], but the quantity cos.c» is changed into
cos.(4w — Amv — cv) ; making 2^4 — 4m — c^=.3, nearly [4846] ; and the divisor P — JV^
[4918u] [4847] becomes so large, that the corresponding term is much decreased, so that it may be
neglected. Similar results will be obtained relative to the other neglected terms.
[4918<]
VII.i.^3] DEVELOPiMENT OF THE DIFFERENTIAL EQUATION IN u. 429
have, for variation,*
2
— '"^ "'/""". -, . {3.cos.(«— miO+5.cos.(3 v—Smv)]. [4921]
Substituting AfKcos.(2v — 2mv), for aôu, we obtain the term,t [4921]
2
a, - «,
The variation of the term [4876],
3 m'. U'^ du . ,n r, IS
jj^^.-.sm.(2v-2v), [4923]
* (2821) The variation of [4919], relative to u, which is the most important part of
this expression, as we shall see in [4922;], is
—'-rrr^-^uA3.cos.(v—v')4-5.cos?(v—v')]. [4921al
If we neglect terms of the order e, we may substitute the values of u, ii' [4791], h^ = a,
2
[4863], and 7/1 [4S65], in the factor, without the braces, and it will become,
2
3m'.u'*.Su „ m'.«3 aiiu a 37n .aôu a . ,..„„ -,
o^o 5 =—i-— 7^ • — ■-,= 7, •-' as m [4921]. [49216]
2/i2.M5 ■^ a 3 a, a' '■Za^ a ■■ -■ '■ -'
]Moreover, by putting v'^mv [4837], in the terra between the braces [4921 «], h becomes [4921c]
as in [4921].
t (2822) Taking, for aSu, its first term [4904]; namely, aSu^A':?\cos.{2v—2mv),
we get, by noticing only the angle v — mv, which requires particular attention, as is observed [4922a]
in [4874, &ic.], we obtain,
n5«.3.cos.(u — mv) = ^Aj^Kcos {v—mv) ; aki.5.cos.{3v—3mv)z=^A:i°\cos.{v—mv); [49226]
whose sum is AA.2^''\cos.{v — mi-). Substituting this in [4921], it becomes as in [4922]. [4922c]
The remaining terms of aùu are of the second, third, &c. orders; and, when multiplied by
2 a
the factor »» • ^, they become of the sixth, seventh, &ic. orders, which are usually
[4922rfl
neglected. If we notice the variation of v', in [4919], it will produce terms of an order
equal to those in [4921], multiplied by the factor — , which factor is of the order m
" [4922e 1
[4916,4917]; so that, the terms produced by oV, will be less than those retained in
[4921,4922], and may, therefore, be neglected.
VOL. III. 108
430
THEORY OF THE MOON ;
[Méc. Cél.
[4924]
may be reduced to the following terms ;*
6:ii.iP (In ÔU . ^ ^ ,^ 3m'. u'^ dSii
dv u
+
Sm'.u'Uv' du
--.cos.(2î;-2î;');
A^.i
dv
these terms, bj development, produce the following expression ;t
[4922/]
[4923a]
[4923i]
[4923c]
[4923rf]
[4923e]
[4923/]
[4923^]
[4923/i]
[4923i]
* (2823) The term [4923], is the same as that whose approximate value is computed
in [4876,4879]. Its variation, considering u, du, v', as variable, and neglecting &u', as in
[4909], becomes as in [4924].
t (2324) Multiplying the equation [4S84] by — 2 Su, we get, by using the abridged
notation [4821/],
4 rÎM 4 nSu f c . , , „ J
or i^ftdM.j — 4-4-4 e.cos.ci'+&ic. (.
Multiplying this by the function [4879], we get the expression of tlie first term of [4924].
Now, the function [4879] is of the third order, and ahi [4S04] is of ihe second order;
therefore, if we retain only the two terms — A-\-Ae.cQs.cv of tlie factor [4923«], the final
product will be correct, in the sixth order. We may even neglect the term 4 c.cos.îj ;
because, when it is multiplied by tlie two greatest terms of [4879] lines 1, 2, it produces
terms depending on e-. cos. (2d — 'i.mv), which mutually destroy each other; also,
terms of the order c-, connected with the angles 2;; — 2/««j;2c«, which do not increase by
integration, and are neglected in [491 1 ,&c.]. Hence, the first term of [4924], is represented
as in [4923a, i], by the following function ;
6m. «3 du iu
-TTT-r ■ -r- ■ — .sm.(2i' — 2v) =■
-4.« (5m X function [4879].
It is only necessary to notice the terms A.^\ Ji[^^, ^J'^\ in the value of a i5m [4904];
because, the function [4879] is of the third order, and the other terms A'w^'e, A.^^^c, &.C.
are of the third, or higher orders; so that their products are of the sixth, or higher orders,
which are neglected. The reason for retaining the term .^/'^' is, because it is connected
with the angle 'igv — cv, and is much increased by integration [4828r/]. Now, the part of
— 4. «(5m [-1904], depending on A"", is — 4^o"'\cos.(2u — 2mv). If we multiply this by
the first line of [4879], between the braces, neglecting c^, we shall get the term
— 2 cc^2W'.(l— fe'2).cos.(c«)— ra) ;
and the second line of [4879], retaining the factor [4879^'], produces the same term, with a
different sign ; so that these terms mutually destroy each other. The other terms produced
by .^o'"', are too small to be noticed, or depend on angles which may be neglected. The
product of the term — 4.^i'''e.cos.(2w — 2mv — cv), in — 4.aSu [4904], by the tenth line
of [4879], between the braces, produces gA^'^ ej^. cos. {2 gv — cv). Finally, the product of
VII. i. §8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 431
2.(1— m).2r.(l— |e'2)
l+^{^-3»i—c).J'p.e'^—l{2-m-c).^f'.e"' S
+ J6.(1— mJ.^f+(2— M)-2f+(2— 3m)..^|'"}.e'.cos.(c'm«-^')
_)_ I (2_3m_c) .^;')_à (2— 2m— c) ../^i" I .ee'.cos. (cD+c'my -a -ra')
\-\-l{2—m—c).J.f-\-l{2-~2m—c).J'-p}.ee'.cos.{cv—c'mv-a+-a')
'-j-(c — m)..4i^'.ec'.cos.(2w — 2m!; — cv-{-c'mv-\--a — to')
3ml -\-{c-}-m).^f\ee'.cos.{2v — 2mv — ct— c'mr+tô+ra')
^"'"\ , Ci(4-+4+m-2c)^f"— 2(1— 2m).^i'3'j
' " ^ _[_ (2— 2m— 2^-f c)-^l'«' <
/+^i5).C"^2_cos.(2i>— 2m«— 3^w+c«+23— n)
. ey^. cos.(2o-i; — c» — 2è-\-a)'
-{-\{\-2m).^™—h{'^—m).A[^''^.-,.e'.cos.{v—mv+c'mv—-a')
, _|_^ JTO1+ J {l—m).Â[^'^ .-,.e'.cos.{v—mv—c'mv-^T^')
4^/'3y2.cos.(2^r— 2r4-2mr), in —A.aSu [4904], by the first term of [4S79],
between the braces, produces — 2^,'^'^''e-y^.cos.{2gv—cv). Substituting these two terms
in the second member of [4923e], we get,
6m'. u'3 du du
3
h^. u* dv u
.sm.{2v — 2v) = ~ .{{gA^'''^—2AP^^^).ey^.cos.{2gv—cv)].
[4923ft]
The third term of [4924],
A a,
3m'.M'3.'Iu' du /r, n r\ i i
. — . COS. {2.V — 2v), produces only a very [4923^]
/Au'î ' dv
small quantity, depending on the same angle as in the preceding' expression [4923^]. Now,
without taking the trouble to compute the whole development of this third term, we may
form a satisfactory idea of its value, by taking the product of the two functions [4878,4918];
which gives the expression of
3m'. u'3. iv' du . ,_ _ ,,
. -- .sm.(2D — 2v) ;
A3. M-»
dv
[4923m]
and, as this differs from [49237] only by the change of cos. into sin. in its last factor, it is
evident, that the two functions will produce terms of the same forms and orders ; so that,
what may be neglected in the one, may also be neglected in the other. Now, the greatest
term of [4878], independent of its sign, is ce.sin.cw ; and, if we multiply it by the terms
432 THEORY OF THE MOON ; [Méc. Cél.
[4926] The expression of (j^)-j^j [4754], contains also the following
of [4918], we obtain only quantities of the sixth order, depending on angles which may be
neglected. The remaining terms of [4878] are of the second or higher orders, producing
terms of the seventh or higher orders ; therefore, they may all be neglected, excepting one,
depending on the angle 2gv-cv, which is retained for the reasons stated in [4828(/]. A term
of this form is produced in the function [4923m], by multiplying the term in line 4 [4878],
which is nearly equal to ^■)'^.s'n\.2gv, by the term depending on ^'^h, in the expression of
^'"'" "\sin. (2y— 2u') .5v' [4918] line 1.
[4923ȕ]
Hence, it is evident, by a similar process, that the terms of the function [4923/], depending
[4923o] on the angle 2gv — cv, may be found, by multiplying ^-y^.s'm.Qgv, by the terms depending
on A'-^''e , in the function ., - ,-,
[4923^.] 3m^ _ ^^^_ (2«_2t,') . &v'.
Now, the term depending on ^/"e, in the expression of aSu [4904], is
a 5u=: ^/*'.e .cos.(2y — 2mv — cv) ;
the corresponding term of &v' [4916,4917], is
[4923;?'] &v' = —2 ^/I'.m e.sm.{2v—2mv—cv).
Multiplying this by the chief term of
Âa " •<^os.(2i) — 2v') [4870], which is, ^-^.cos.(2« — 2/?ii'),
we get, in the function [4923j(], the term
_a
. A.^^'.me.sm.cv.
Finally, multiplying this by the factor iy^.sm.2gv [4923o], we get, for the third tenu of
[4924], the following expression ;
[4923g] 3,Môv' du _ ^ ^ .lm.A,^^\ef.cos.(2gv-cv)].
We shall now develop the second term of [4924], which is the most important. It may
be put under the following form ;
3m'.«'3 dhn C 3w'.«'3 . ,„ ' „ > d.[ahu)
The factor between the braces, in the second member of this expression, connected with the
negative sign, is evidently equal to the differential of the first member of [4885], divided by
2.aàv ; and if we perform this process on the second member of [4885], we shall find, that
2
o —
[4923s] the division by 2a, makes the factor, without the braces, become — — . Moreover, by taking
VlI.i.^,8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 433
variation ;
the differential of the terms between the braces, the divisors 2 — 2m, 2 — 2m — c, &ic.,
which were introduced by the integration, are effaced, and cos. is changed into — sin. ;
so that, if we represent any term, between the braces in ['1885J, after effiicing the divisors,
by Ic'.cos.v' the corresjionding term of the first factor of the second member of [4923r],
will be represented by a series of terms, of the form
_2
3 »i
.— IJlL.k'.sm.v' [4923«,m].
2a
Now, putting aSu equal to a series of terms of the form k.cos.(iv-\-!) [4916], or, for
brevity, A-.cos.?"« [49I8i], the corresponding term of
d.{a6u)
dv
will be
■ik.
Multiplying this by the first Aictor, which is given [492;3y], we get the following expression
of the function [4923r], or, of the second term of [4924] ;
2
^m'.u'^ d 6u . , 3 in
— . sm. (2 v—2 v)— —. \ikk'.cos.(iv w v')—{kk'. cos.{iv-\-v') ] .
_s '
3 m
The factor without the braces
is the same in all three terms of the functions
[4923.1-, y.x] ; and is equal to that in [4925]; we shall, therefore, neglect wholly the
consideration of this factor; and, in speaking of these functions, shall limit ourselves
exclusively to the terms within the braces. These terms, of the function [4923i], are
represented by,
ik. jt'. cos.(/y m v') — t'.cos.(it)-f-v') | ;
m wliich k.cos.iv represents the terms of [4904], cui'l k'.cos.V the terms between the
braces in [4885], rejecti7ig the divisors 2 — 2m, 2 — 2m — c, he. which ivere introduced by
the integration.
We shall now take, for ^.cos.iy, the terms of the function [4904] ; so as to combine
successively each of the symbols Xj'"', .^/", &.C. with all the terms of [4885]. We shall
neglect the terms which appear to be insensible, and shall compare those which are retained
with the function [4925] ; taking the terms, depending on .^a"", ^^o'") -^é^'K ^c. in the
order in which they occur in [4904] ; and, noticing also the terms [4923Ar, y], depending
on the angle 2^w — cv.
First. The first line of [4904] gives k = J.f\ i=2—2m; substituting this in
[4923r], it becomes, {2—2m).A.i''K\k'.cos.{[2—2m\omv')—k'.cos.{2v—2mv-\-v')\.
The first line of [4885], neglecting e^, gives A:'= 1 — ^e'^, v'=2u — 2mv; substituting
these in the first term of [4924c], we get the first line of [492.5] ; the other term of
[4924c] depends on the angle (Av — 4;nu), which is neglected. In like manner, the
second line of [4885], gives k'^ — 2(l-)-m).(l — |e'2).e; v'= 2;; — 2mv — cv ; hence,
the first terra of [4924c] becomes,
—{2—2m).Ai''\2[\+m).{\—y).e.cos.cv=—A{\+m).\{\—m)Jl.p.{\—^e'^).e.cos.cv\;
and, by the same process, we get, from the third line of [48S5], by using the factor 1 — Je'^
VOL. III. 109
[4926']
[4923t]
[4923u]
[4923w]
[4923u>]
[4923x]
[4923j,l
[4923:]
[4924o]
[49246]
[4924c]
[4924<i]
[4924e]
[4924/]
434
THEORY OF THE MOON
[Méc. Ct
[4927]
— - — -.\3.sia.(v — m r) +15. sin. (3 1) — 3m.v)l.-
8a,a'
civ
[4924^]
[4924i]
[4924*]
[4924?]
[4924m]
[4924nl
[4924o]
[4924j>]
[4924?]
[4924r]
[4924«]
[4924<]
[4924<']
[4S79)t], the term — 4(1 — m).\{l — m).A.}°\{l — Je'^).e.cos.cy} . The sum of tliese two
terms is — 8|(1 — m).Jl2^°\{l — ^e'^).e.cos.cv], as in the second hne of [49'25]. It is
unnecessary, in this case, to notice the second term of [49:24c], because tlie coefficient of v
is so large, that the term becomes insensible. Proceeding in the same manner with the
fourth line of [4885], which gives A:'=J«', v'=2v — 2mv — c'mv ; also, with tlie fifth
line of [4885], which gives k'^ — ie', v'=^2v — 2mv-\-c'mv, we find, that the terms
corresponding to the first of the functions [49'24c], are, respectively,
-\-{2— 2in). A2^^\^e'. COS. c'mv, — {2— 2m). A^^^lie'. cos. c'mv ;
whose sum is 6.(1 — in) . ^n"'. e'.cos.c'm v, as in [4925] line 4.
The remaining terms of the function [4S85], being of the seco?u/ or higher orders in e,
e', 7, multiplied by Wt of the second order, and ^o"" of the second order, produce only
terms of the sixth and higher orders, which may be neglected.
Second. The second line of [4904] gives Ar=.^/".e, i^2 — 2m — c, hence
[4923^] becomes,
(2—2m—c).A^'^\e.\lc'.cos.{[2—2m—c] VMv')—'k'.cos.{2v—2mv—cv-^v') \ .
Substituting, in the first term of this function, the values [4924f/], corresponding to the first
line of [4885], we get the term (2 — 2n — c)Jl^''>.r.{l — |Ê'^).cos.cit, as in the second line
of [4925]. The second and third lines of [4S85], produce terms having the factor
A[''.m.e^, of the fifth order; but they do not increase by integration, and are therefore
neglected. The fourth and fifth lines of [4835] correspond to the values [4924A], and by
substituting them in the first term of [4924/], we get the two terms,
ie'.{2—2ni—c).J['''.i .cos. {cv— c'mv), —ie'.{2—2m—c).â['Ke.cos.{cv+cmv),
as in [4925]lines 6, 5. All the remaining terms of [4885], excepting that in line 12, ma}'
be neglected as in [4924A:]. This line corresponds to ^"':= — i(2-j-m).y^, v'^=2gv-2v-J[-2mv,
and produces, by means of the second term [4924/], the expression,
+i{2-{-m).{2—2m—c).A^'\ey^.cos.{2gv—cv).
Connecting this with tlie terms, between the braces in [4923^, q], depending on A';'\
they become \g-\-m-{-l{2-\-m).(2 — 2m — c)l.A['Ke-)'^.cos.{2gv — cv) : and, as c is nearly
equal to 1, we may, by neglecting m^, put jm.(2 — 2m — c)=:.^m; consequently, the first
-c)+]^ = i{4g + 4-}-m-2c)
factor of the expression becomes, ^-i-'«+f(2 — 2;«-
which is the same as the coefficient of A'l\ in [4925] line 9.
Third. The term Jlf\e.cos.{2v—2mv-]-cu) [4904], combined with [4885] line 1, gives
the term depending on .^2'-' [4925] line 2. In like manner, we may combine the terms
of [4904], depending on ^.P\ A"') ^^'tli ''^e same terms of [4885], to obtain the terms
depending on Ai'^\ A^''^ [4925] line 4 ; observing, that, as c' is nearly equal to 1, we have
very nearly 2 — 2m-\-c'm.=:2 — m, 2 — 2m — c'm=2 — 3m. The term depending on ^j^^'
produces nothing of importance.
VII. i. § 8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 435
*hence, wc obtain the quantity, [4927']
Fourth. The term depending on ^f [4904] gives k^Af^ce', i=2 — 2m — c-\-cm ,
or nearly t=2 — m — c. Substituting tiiis in [49232] it becomes, ^ "'
(-2 — 7n—c).ji'°\ec'.\f<f.cos.([2—2/)i — c-\-c'm] v r» v') — A;'.cos.(2« — 2mv—cv+c'mv-{-v')l. [4924i)]
The first line of [48S5] produces, in the first term of [4924y], the quantity depending on
.4''' [49-25] hne 6 ; and the fifth hne of [4885], produces the terms depending on Jl['^\ in
Une 3 [4925]. In like manner, the term depending on A'-p [4904], combined with [4885] "''
lines 1,4, produce tliose in [4925] lines 5,2, depending on A'-p. Also, the terms depending
on Af\ A'p [4901], being combined with the first term of [4885], produce the [4925a]
corresponding terms in [4925], lines 8,7.
Fifth. The terms of [4904] depending on ^2<"», ^;"\ Ai''^\ produce nothing of
importance. The terra in line 14 [4904], gives k = A[^^>.y^; i = 2g—2+2m^2m [4925t]
nearly; and the first term of line 2 [4885], gives ]i'= — 2e, v'^2v — 2mv — cv.
Substituting these in the second term of [4923:r], it produces 4m.C7'^.^,"3'. cos.(2^t' — cv). [4925c]
Connecting this witii tlie second term of [4923/c], we obtain -2(l-2m).A[^^''.ey^.cos.{2gv-cv), Mg^Sj^i
as in [4925] line 9. The term depending on -^a^'^'.e'^ [4904] produces nothing of importance.
Sixth. The term in [4904] line 16, gives k=A^-^^Key^, i=2g—c=l nearly; and the [4925e]
first term of [4835] line 1, makes k'=l, v'—2v—2mv ; hence, the first term of [49232] [4925/]
produces ^„'''''. e7^.cos.(2v — 2mv — 2gv-\-cv), as in [4925] line 11. The same values of
]c', v', being combined with the term in [4904] line 17, produce
(^2—2m—2g+c).AP'^\ey\cos.{2gv—cv), as in [492.5] line 10. [4925g]
Seventh. From [4904] line 18, we have k = A["\-, i=l — m. Combining these
[4925A]
with k', v' [4925/], we get the term {l—m).J["\-.cos.{v—mv) [4925] line 12. If
we combine the same values of k, i, with the term in line 4 [4885], we get the term rjqor-i
depending on A^''' [4925] line 14 ; and if we combine them with that in line 5 [4885], we
obtain the term depending on A^p\ in [4925] line 13.
Eighth. From [4904] line 19, we have k= A^^^\-,.e', i = \ — m-^c'm=l nearly.
Combining this with k', v' [4925/J, we get the term depending on ./îo"*' [4925] line 14. [4925t]
If we combine these values of k, i, with the term in [4885] line 5, we get the term
depending on A'^^''> [4925] line 12.
JVinth. From [49041 line 20, we have fc=./3P\-,.e', i= 1— ?«— c'ot=1— 27« nearly.
"• ^ " a' ' •' [4925/]
Combiningthis with the values yt', v'[4925/'],we get the terms depending on ./3,"'' [4925] line 13.
Tejith. The term of a (5m [4912], gives k=\.-,i^3 — 3m. Combining this with r^^^Sm]
the values [4925/], we obtain the term depending on Xn, in [4925] hne 12.
Thus, we have obtained all the terms of the function [4925], as they are given by the
author ; and, it is evident, from the details of the calculation in this note, that, in general, [4925n]
the neglected terms are such as have been usually rejected.
* (2825) Having found, in the preceding note, the variation of the first term of
[4928]
* 2
9/«
436 THEORY OF THE MOON ; [Méc. Ce).
2
9 »" 1 N Jim " ^ \
-;; — . ( 1 m) .^'"' .-.COS.fv — OT V).
4o, ^ "^ a ^ ^
( 7 ) ■ Z^^iv ' '^°"'*'"^'^ '" [4876], we shall now proceed to the calculation of the next
■ temi, which is given in [4860] ; and, if we put, for brevity,
t'*^^^"] A=— ^^.|3.sin.(t>— t,')-f 15.sin,(3y— 3i;')};
this part becomes ^. — . Its variation, considering u, du, v', as variable, and neglecting i5m',
[49276] as in [4909, fee], is ('L^] ^ ^^^/l' , f^) s,' ^a ■ ^ ^
\ du J dv '~ \ dv' J ' ' dv ' dv '
The factor ^^. in the value of A [4927rt], is of the order iïi.- . -, [4921 è],
[4927c] which is of the /oMrtA order ; therefore, (-r-j, (-p) are of the same order. Moreover,
5m [4904] is of the «cconrf order ; — [4878] is of the^îr^i order; Sv' is of the third order
[4916,4917]; consequently, (-— ).(5m. — is of the ici'OiiA order; and (—].Sv'. —
V ait / du \ dv/ dv
of the eighth order ; so that, by rejecting these terms, the function [4927i] is reduced to
A. --7- of the sixth order. Then, by neglecting terms of the seventh order, we may use
in A [4927a], the values [4921a — c], and the preceding expression becomes as in [4927].
* (2826) The differential of [4904], divided by dv, gives,
-^^ =— (2— 2m).^i''\sin.(2i;— 2mt))
— (2— 2m— c) ../?/". e . sin.(2u— 2mM— cy) — &c. ;
which is to be substituted in [4927]. In the first place, the terms depending'on .^o"" [4928a],
produce, in [4927], the following expression ;
[49286] 5"^'-(2 — 2m)..^o°\{3.sin.(t) — M«)-]-15.sin.(3i; — 3mt>) |.sin.(2t) — 2mr)).
As this is of the sixth order, we need only notice the resulting terms which depend on the
angle (y — mo). Now,
3.sin.(t) — mr).sin.(2u — 2mu) =^.cos.(i' — mv) — kc. ;
15.sin.(3y — 3mw).sin.(2u — 2my)^-L5-.cos.(4) — mv) — Sic.;
whose sum is 9.cos.(« — m«) — Sic;
hence, it is evident, that the term [4928i] is equal to
[4928c] ^^.(2— 2m).^2<''i.9.cos.(D— mu) ;
[4927d]
[4928a]
VII. i. •§. 3 ] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 437
The function [4891],
contains, in the first place, the term,
- (S + ") • / ^^ • ^•"- (2 «-2 «') [4882] ; [4930]
its variation is,*
which is easily reduced to the form [4928] . We may proceed in the same manner with the
terms of «(5it [4901], deiiendins on /7'j".c, Jl:?\c, &ic. ; but, as these terms produce only
cjuantiiie? of the sixth, seventh, &ic. orders, they may be neglected.
* (2827) We shall put, for brevity,
V=~+u, JV='-'i^.sm.(2v—2v'); [4929a]
then, we shall have the development of F', in the second member of [4890] ; and the
expression [4930] will become — V.fW.dv. Now, as V, W, contain the variable
quantities u, u' , v , the variation of the function — V.fW.dv, will be denoted by
— ^^■■fl O ■'^" + (?)•''"' \ .dv~5V.fJ'V.dv— V.f(^-£).âu'.dv. [49296]
The three different integrals, of which this expression is composed, correspond respectively
to the three integrals in [4931], as we shall find by the following investigation ; in which we
shall U5e the abridged notation [4821/].
If we substitute the values of (y-j, (yr)) deduced from that of JV [4929a], in the
first of thc^ integrals [4929'], it becomes,
-^•'^K^}'"+(^j-'"T''"=-i:^-/-,7r- J -•^'"•(2«-2''')+|5«'-cos.(2t;-2«')^ ; [4929c]
in which the terms under the sign /, are the same as in the first term of [4931]. If we
substitute the values of c, g [482Se], in V [4890], and neglect terms of the order m%
m^'r, e^ iA we obtain, ['•^29^]
V=-^.ll-^^f.C0S.'2gv]. [4929e]
Substituting this in the factor, without the sign / [4929c], it becomes as in the first term of
[4931]. As the terms of nôu [4904], are of the second or higher orders, it follows, from
[4908°-]. that the terms depending on Su, under the sign / [4939c], are of the fourth or
higher orders ; and when these are multiplied by the terms of V, which we have neglected [4929/]
m [4929f/], they will produce only terms of the siith or seventh orders. Those of the sixth
VOL. II[. 110
438 THEORY OF THE MOON ; [Méc. Cél.
^^'"' ' 1+f / ^-008.(2^^-2 Oj./—,- . 5 -.si.i.(2«— 2i;')-|-èV.cos.(2o— 2i0?i
U f It J
h^a
r,.r,oii /(hUu , \ .Zm'.u'^.fh . ,r, o '\
[4931] — ( _f oM \ . f — — .sin.(2i'— 2t7'_)
rr- . / ■ .dv.s\n.(2v — 2v). *
order are produced by c^. ^ ^- ['IGSOc/], and do not depend on the angles v — m i\ and
'\lgv — CD, whose coeflicients are required to a great degree of accuracy; hence, we see the
propriety of neglecting the above-mentioned terms of V [49ii9(/].
In making this estimate, we have omitted the consideration of hv' [4929c], because it is
[4929g-] of the order tn.aou [491G, 4917], and must, therefore, produce terras of still less importance
than those of «<3m, which we have neglected.
Again, the value of J^ [4929«] gives 5V='-—--\-&u; substituting this in —(SP'.//r.rfw
[4929i], it becomes as in [4931] line 2.
Lastly, taking the partial di.Terentlal of ÏV [4929fl], relative to m', and substituting it in
the third integral [49296] , it becomes ,
Now, from [4833], we have nearly, a' a' = e'.coscv'^^ whose variation is,
a'&u' = — c' e'. (]v'. sin. c' v' ;
and, as 6v' is of the order m.ahi [4929j], this quantity will be of the order me'. a Su. or of
[4929fc] the fourth order [4904]. If we retain only the chief term of [4929e], we get V= -
and, by using the value [4921i,&.c.], we find, that --— is of the order
[4929Z] ^^^ • a a' —m .a a' [4865] ;
[4929t']
V. f (^~\ . Su!, de = —V.f ^-^ . Su'.dv sin. (2«— 2u').
consequently, the function [4923/] is of the sixth order ; and, by neglecting terms of the
seventh order, we may subnltute the value of V [4929;.], in [4929J] ; by which means it
becomes as in third line of [4931].
* (2828) In computing the value of the function [4931], we shall retain termsof the fifth
[4931o] order in e, e', y, (V); also, in the coefficient of cos.cv, we shall retain the factor l—^e'^.
[49316] In the terms depending on the angles '2gv—cv, v—mv, v—mv±c'mv, we shall retain terms
VII. i. §8] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 439
The development of these terms, observing, that c is nearly equal [4932]
of the sixth order ; observing, that the divisors, arising from the integration, 2§* — 2-|-2m,
2c — 2-(-2m, which occur in the terms depending on -^3'"', -^j,"'*' [4934], are of the
order m ; so that, independent of these divisors, these terms must be taken to include
quantities of the sixth order.
We shall first compute the term
12 ni'
./iq;l".^.sin.(2.-2.') [4931].
To obtain this, we shall take the differential of the equation [4885], and then multiply it by
2
-, neglecting such terms as we have usually done, and using the abridged notation
[4821/] ; hence we get,
[4931c]
[4931rf]
[4931e]
[4931/]
6 m'
2— .sm.(2« — 2v) =^ .dv.
^(1— I e'2).sin.(2t)— 2m«)
— 2(l+w).(l — Je'2).e.sin.(2«— 2mt)— cr)
— 2(1— m).(l— Je'2).e.sin.(2D— 2mt;+ci')
-|-Je'.sin.(2« — 2m j) — c'mv)
I — i d . sin. (2 v — 2 m v-\-c'm v)
\ — J (2-|-37n) .ee'.sin.(2D — Imv — cv — (imv)\
-^ (2 — 3m) .ee'.sin.(2t) — 2mv-\-cv — c'mv)
~\-l (2+m) .ee'.sin.(2v — 2mv — cv-{-c'mv)
\~\'i (2 — m) .ee'.sin.(2« — 2mv-\-cv-\-c'mv)
I_|(104-19m).e2.sin.(2cj)— 2«+2mr)
+i(10— 19m).e2.sin.(2ct!+2t)— 2?n?;)
— ^ (2-fm) . 7=.sin.(2^tJ— 2 y+ 2m v)
-f-î(2 — m) .^-.sin.(2^«-|-2D — 2mv)
4-^^-.e'^.sin. (2v — 2mv — 2c'mv)
\ — 1 (5-{-m) . e 7^. sin. (2 v — 2 m v — 2gv-\-c v)
1
2
3
4
5
6
7
8
9
10
11
|l2
1 13
14
15
This is to be multiplied by the expression of - [4884] , to obtain the value of the function
in the first member of [4931fc]. By this means, the product of the factors, without the
braces, becomes.
a
12 7n
dv, as in [4931 A:] ;
and the products of the terms, between the braces, are found as in the following table ; in
which, the first column contains the terms of [4884] ; the second, those of [4931^] ; and the
third, those of [4931A:], respectively ;
[4931g]
[493U]
440 THEORY OF THE MOON ; [Méc. Cél.
[4933] to 1 — fm^ and, that g is very nearly equal to \-\-^m^ [4828e], is,
[4931t]
[493U]
(C..1.3.)
Products, or terras of [49314].
whole function [4931^] between the braces
. . . .neglected
— J-e.(l— |e'a). \ sin.(2y— 2H;u-(-a))+sin.(2i)— 2mi;— «>) \
-(l+«i).t2.sin.(2cD— 2î)4-2nu))-l-&c.
-|-(1 — în).c2.sin (2cî)-j-2i) — 2inv) — &c.
— j-ec'.Jsin.(2u — 2«iD-|-fi;— cm«)-|-sin.(2i)— 2mti — cv — c'mv)\
-\-iee'Asm.[2v—'imv-\-cv-\-c'mvY\-sm.[2v — 2mv—cv-\-c'mv)\
. . . .neglected
-)-J-c2.|sin.(2cu-(-2v— 2mD)— sin.(2cu— 2«+2mu)|
-|-gL>2.{sin.(2g'i)+2u— 2mw)— sin.(2g-u— 2i;-t-2mD)}
— jfi 2.(i_j_,„jsin_(2j, — 2mD-|-2g-i- — cv) — &,c.
— je}2.(l — Hi).sin.(2« — 2niv — 2gv-\-cv) — &c.
Substituting, in the third column of this table, the value of its first hne, which is equal to the
terms between the braces in [4931^] ; and then connecting together the terms of tiie same
forms, it becomes equal to the terms between the braces in the second member of [4931 A:] ;
and the external factor is as in [4931A] ; hence we get. by retaining terms of the usual forms
and orders,
(Col. 1.)
(Col. a.)
Terms of [4884].
Terms of [4931f J.
1
whole of [4931g-]
-y^-if
same
— e.cos.cw
(l_|e'a).sin.(2r— 2mi>)
-2(l+m)f.sin(2v-2Hit)-a')
-2(1-7)1 )e.sin{2iJ-2nn; + cii)
4-Je'.sin {2v—2mv—c'mv
— 4e'.sin.(2D— 2;)iD-[-c'mD;
-c{-^e2-i72)cos.«)
whole of [4931g-]
-\-^e'^.cos.2cv
-t-sin.(2y-2mD)
+ly,Kcos.2gv
-(-sin.(2t)— 2m»)
-2( 14-)n)e.sin(2i)-2mv-CD)
-2(l-m)e.sin(2D-2nn)-|-cv)
12)ra' u'Hv 1 . ,^ „ ,, 12m ,
7r-5 •-^-•-•sin.(2» — 2v )= . dv.
(1— Je'2).sin.(2«— Smt))
— (t+2ff?) (1 — fe'^).e.sin.(2« — 2mv — cv)
—{i—2m).{l—he~).e.sm.{2v—2mv-{-cv)
-|- Je'. sin {2v — 2mv — c'wd)
— 2 e'.sin.(2» — 2mv-^c'mv)
— f (|-|-3/n).ee'.sin.(2« — 2mv — cv — c'mv)
— i (I — 3 m)- e e'. sin. (2 v — 2 m v-j-c v — c'm v)
\ _j-i(A-[-m).eÉ'.sin.(2 Î) — 2 jn v — cv-\-c'mv) f
-)-^(J — jn).ee.im.{2 v — 2mv-\-c v-\-c'm v)
—l{\5-\-2^m).e^.im.{2cv—2v-\-2mv)
J^l{\b—23in).e^.sm.{2cv-\-2v—2mv)
— {{h-\-m).y^.sm\2gv—2v-\-2mv)
-\-l{h — m) 7^.sin.(2^t)-|-2y — 2 mv)
-}-y- e'^. sin. (2 v — 2 m v — 2 c'm v)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
[49311]
This is to be multiplied by a&u [4904], and then integrated, to obtain the value of the term
[4931e]. Now, if we suppose any term of aiu to be represented, as in [49l8i], by
a<5u=A:.cos.iu ; and any term of the second member of [4931fcJ, by .dv.Icsm.i'v ;
VII.i.§S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN m-I 441
3 M
4«,.(1 — m)
{4.(1— m)^— 1 l.A°\(l—ie'^)
Z_!!L) 4.(1— m) ' < V IS l2-2,n~c ' 2-2H^-[-c3 ~ >.c(l-Je'»).
6m
n.
+ ^
{4J/'+J/'-4'^'-10^/'>e=+|(J/^-J/^'.).6^
_ — . ( 4 . 1— ,« —1 ..Jf'. ] — — - — S
^^ .if2=;,'_l\^(3)4.('ôi:3;^'_l\^(4)
4a,.(l — m)
3 w
a.
>.e'.cos.(c'm«; — to')
"*■ ' .e e'.cos.(2 « — 2mv — cv — c'mw+is-(-TO')
ee'.cos.(2t) — 2m a; — c«+c'?«u+w — -n')
a,.(2— 3 m— c)
o
2
cos(cu-w)
3
4
6
6
[4934]
a,.(2 — m — c)
Bevelop-
7 nient of
the varia-
tion(4931).
8
9
the product of these two terms will be represented by
.dvJck'. \ sin.(i't) — i v) -\-sm.{i'v-\-iv')
Its integral gives the corresponding term of [4931c] ; namely,
[4931m]
l'2m' u's.rfc ill . Chi C kk' ., . kk' /•/ ■ • \>
"Ai^--^"lir----^'"-(2^— 2?')=— • J— ^Tr^.-cos.(ii;— It))— — .cos.(it) + ii;) J; [4931n]
all of which have the common factor , and the terms between the braces ; namely,
kk' k k'
— T7-:-cos.(J'y — iv) — ^r--cos.{i'v-\-iv), are computed in the following table; in which, [4931o]
the first column represents the terms of a 5m ; the second, the terms of [4931^]; and the third,
the terms of the function [4931o] : the operation being performed for each term separately,
putting c and g equal to unity, in several of the small coefficients. When i' = i, the first
term of [4931m] vanishes, and the function [493 1 o] is reduced to its second term
kk'
.cos.Sii,'. This case occurs in the first line of [493I7j1, which is reduced to a term, [4931o']
2i
depending on the angle 4v — 4my, that may be neglected.
VOL. III. Ill
442
THEORY OF THE MOON ;
[Méc. Cél.
[4934]
Continued*
2
6 vi
a^.[c — 7Ji)'
{^f +J J,'"] . ec'.cos.(c?;— cVtt I'— ro+T^i')
10
Develop-
ment of
tlie first
term of
the func-
Uon[4931].
[4931;?]
(Col. 1.) Terms o( aôu [4904].
Ji^ ' .e.co3.(2v—^mv — cv)
A ^ Ke'.co9.(2y — ^mv-\-dmv)
A ^ '.e'.cos.{2i; — ?mi' — cmx)
^j .ee'.cos.CSw— 2m«— cv-t-cwii')
A ^"^.ee'.cog.CSu— 2fflw— cw— c'nitJ)
A ^ .ec'.co3.(cr4-c'wip)
A ^ .ee'.co3.(ci* — c'twv)
^ f '^'.c".co3.(2cîî— 2«-i-2mr)
^(13) ,j2 ^.^g_^3^„_Oy_j_nn[v)
a (14) g— gpg Oc'my
2
__ ^'^^e>**.cu3.(2^v— cr)
^ f '^\c>~.cos.C:!u— 9/nu— 2uru-f-cv)
^l'^\",.C03.(i'-Jni.)
1 a
^ ' ^^ , - .e'.co3.(u — ïnî)-j-c'mu)
0 a'
(19) a
X ,-.co3.(3u — 3mtJ)
(Col. 2.) Termg of [493JA].
A— l-.c'-Vsin.Coy— 2011))
second term
third term
-t-^.c'.sin.(2u — 2mw — c'mv)
^X.e .sin.(2« — 2mv-|-c'ffiv)
/'l_|..c'^Vsin.(2o— 2mv)
-|-X.e'.3in.(2u — 9m I' — c'mr)
^X.e'.3in.(*3ii — 2mi;-|-c';nu)
4-^.ec'.sin.(2a — 9mw — cw-l-c'm«)
— |. /|.-|- m) .■),^.sin .(2«^>— 2u-f-2mv)
(\ — A.c }.sin.('2« — 2jnu)
sin. (2)1 — 2hi«)
8in.(2i; — '^mv)
sin.(2u — 2mt5)
sin.(2H — 2mzj)
^i. e . sin.(2(' — 2H(y — ce)
^_l..e'.sin.{2u — 2;rty-|-c'nn))
Rin.(2u — 2fflv)
-|-l..e^.sin.(2t'— 2mu— c my)
3in.(2u — 2m(,')
sin.(2y— 2Hiy)
8in.(2y — 'imv')
sin.(2u — 2ïnu)
sin.(2y — 2/hu)
^A.e.9in.(3u — 2)nw — cy)
[terms of 41)3IA]
8in.(2y — 2my)
Bin.(9u — 2my)
3in.(0]' — 2»ni)
-[-- Z..e'.sin.(2y — 2m y — c'my)
^X.e'.sin.(-3(5 — 2mu-|^'ïnu)
sin.(2« — 2mi')
^-l-.e.sin.(2u — 2wu-j-c'mp)
sin.(2ïi — Omy)
sin.(2y — 2my)
(Col. 3.) Factors of ^^^ [493l7i].
. . . .neglected
+|.^<°'.^,.c,„.c'„,»
4"4-'*^0 •- .C05.C'7ffU
-Jj(').e.(:_|.e2|eo3.c« '
7 fl(l) «C , , .
— -2-^j '.^-3;^.COS.(ci:— c'ïniO
+4-'*^, •— r- .cns.(cr'4-c'7ni')
^ ] (■-t""ï
~~-r'rl ', .COS.C/BV
* 1 )«
_5 ^(M e^f'
^j--*^ '. .C0S.C7WU
(+|-+l.'»)..4^j'^.É>°.cos.(2û^y— cy)
+^^~^.c.Ci— A.c^^.cos.cy
+.4^' ' S- .C09. c'/nu
— -ï ' .- .cos.f j/iy
. . .neglected
— A jî' .^^ — ^ .cos.c'?«i?
I 1 _^(6).«c'^.cos.cw
^2" 1
^^^f^) ,_^^ .co3.(cîî-|-e'mv)
1 r-j-/i/
+4-^'
û).'-
—l-.A
1
1 2-3'»-c
(9) gg'
•*1 "a-m-c
«(10).
..co3.(2y — 2mi'— CI! — c'mu)
.co9.(9« — '^mv — cy-f-c'mu)
. .cos.(2cv— 2y+9mi')
"2 9C-2+2/I
. . . .neglected
Jf-A^ '-\ ^ ' .cos. (2;fv— 2u4-27nr)
-f-A.^j'3)/.j2c^s.(2^î)— cu)
. . . .neglected
All thpse N
terrnshiive J
I the fcim- f
mon factor ^
-Jt
(13)
"0 "s-a»!
.79
..COS. (aï— 2mf— S^'-|-c!))
— .>?( ^ .€ y^.cos. (2^u — cu)
J 17) 5 1
— .y? ■ '/^^ ^_^ — .cos.fv — Tnu)
« 1 — m
2 1
■^o^'-S'Trsr,,;'^"''-^''""''-'^'""')
ia(18)a c'a
.^ .cos.(t' — my — c'mv)
a'
.e'.'i .cos.(i' — mu+c'my)
~«('0!.'i,..<-m.(,._,
('• — ïttr-f-c'mu)
+^5-S'TÎ;:-=°'-("-'"'')-
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
96
27
28
29
30
31
33
33
34
35
36
37
V1I.1.§8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 443
^"' _^jP)_ijo)l,ee'.cos.(cv+c'7nv—-^—^') 11
a^.{c-{-m)
2
_2 [4934]
H !'""l"'l X -r- cos.(2o-t^— 2«+2OTt>— 20 13 cu„.inuc,i.
a,.(25-— 2+2/n) ' ^ * ' ^
+ ^.^5 2J','=''-J[«"+^.4"i.e^^.cos.(2^r— ci'-2é+^) 14
«, 8
2
fi - ^(15) Continua.
— ,, ; . ,.ef-.cos.(2v—2mv—2gv+cv+2ê—v^) 15 .rdfvei-
' ^ '^ ' ' • Ihe func-
2 tion[4931J.
— . ^/r ,.?a+3tfO-4"^— 24"'-e'-— l.[l— (1— m)n.x,].-, cos.(î;— m«) 16
+ ^.M'/') — 2 4'^)}.-.e'.cos.(y— miJ+c'mî;— t.') 17
a, ' ' ' ^ a' ^
We may remark, that the sum of the terms in lines 2, 3, is reduced to
4m. (I ^e'-).^fe.COS.CV; (38) continued.
the sum of tiiose, in lines 4, 5, to 4^'2?.-.cos.c'/«« ; and the sum of those, in lines 9, 10,
" m
e9e' . ''"'
to —10.^/"'. — . cos. c'm r. Moreover, the term neglected in line 25, of the form
m
e2
— -(4/"\-— .cos.(2c!; — 2n), will be used hereafter in a different calculation ; also, the term [AdiXq]
2
i.^/".e2.cos.2cy, arising from the combination of [4904] linel, with the first term in [493ir]
line 3 [493 iq.
The function [4931^] is also multiplied by ^y^.cos.{2gv — 2é), in [4931]; but the
only term of [4931p], which requires any notice, is — ^y\e.cos.ct), in line 6 ; because the r493j,i
product of these two terms produces a quantity, depending on the angle 'igv — cv, of the ugsj^i
following form ;
° ' SeconrJ
o term of
-^-.|f .cos.(2^,;-2J)./ !^ . — .sin. (2.-2.')=- — • \\.'Ai'\ef.cos.{2gv-cv)\- -"H»n
''^a -^ «'' « a, <-« ) [4931«]
444 THEORY OF THE MOON ; [Méc. Cél.
[4935] We must observée, that Cf'.sin.(2j; — 2vm) is the inequality depending on
[4931i] The next term of [4931] is -^^ . f ^^-^.iôv'.cos.(2v—2v'] : which is of the
Sv'
order — j-, or ot [4922f/, c], in comparison with the terms produced by a du in
[4931^]; and, as tliis last function may be considered as of ihefoaith order, that in [4931t)]
[4931m'J may be supposed of the fifth or a higher order, in all the angles which require any notice ;
so that it will only be necessary to retain the terms depending on the angles, whose
coefficients increase considerably by integration ; as cv, 2gv — cv, v — mv. These are
produced by the terms of aoii [4904], depending on ^4/'', ^J'''; which give, by the
process in [4916, 4917], the following terms of M ; namely,
W^^A Sv' = — 2;n.^/"e.sin.(2t)— 2my— ci')— 2m.^i'''.-,.sin.(î>— mi>).
Now, if we multiply — ^.Sv'.dv by the first member of [49107c], and prefix the sign /, it
produces the term [4931 !)]. Performing the same operation on the second member of
[4910A:], we find, that it becomes,
2
'^ ^' — .y"^(5y'. f/tiX terms between the braces in [4910^] I .
The first term of ôv [4931:c], being combined with the first line of [4910A:], neglecting e^,
[4931:1 produces the term [4932a] line 1 ; the same term, combined with 1^^. cos. (2^1- — 2i'-f-2mD)
[49 10A-] line 12, gives [4932a] line 2. The second term of [4931 x], being combined with
the first of [4910A;], produces [4932«] line 3 ; hence we have,
Third
t^'otoc - f — m.AJ".e.(\ — Pje''^).cos.cv ~\ ^
the fuiic- 2» '\-/ i A
[4932a] ./ .|5u'.cos.(2y— 2i) )= . < -Ti.^-^^ .cy .cos. (zgv—cv)
''"''' "' "' I ™ /J>17)" / N \
I ; .Ji .^".-.cos.iv — mv) \ 3
I 1 — m a ^ 'I
2
These terms are the most important ones of those depending on Sv', and they are only of
[4932i] ti^e fifth or sixth order; therefore, it will not be necessary to notice the terms arising from the
multiplication of these by the factor ^■y^.cos.'igv [4931].
_,, . ^, /dd6u , , \ -3 m'. u'3. (/w
[49326'] The next terms of [4931] are —f— -+oi«j./ — —- — .sin.(2«— 2d') ; which
will evidently be obtained, by multiplying the function [43S5], by the factor ( -,—-{- <5m j.
[4932f] Now, any term oï mhi [4904,4912], being represented by aSu = k. cos. {((-{-s), the
[4932c'] corresponding term of this factor will be ■ {r — l).cos.(i<-|- f) ; ^ind the product of
the terms of this kind, by the corresponding ones in [4885], are computed in the following
table ; putting c^l, ^=^1, in some of the small terms; but, in the term depending on
VII. i. §8.J DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 445
siu.(2t' — 2mv), in the expression of the moon's mean longitude in terms of [4930]
the angle 2gv — cv [ 193-2/line 7], we must use f=l — |m-, g^zlJ^îm^ [4932,4933],
which give, very nearly.
(0_^,„,),-2 -(2+«)-)'
l+jni\ (l—im).
4.(2o-— 2-t-2/H) 4\-2m-{-^m-) 4m. Vl+lmy 4m
by which means the coefficient of the tenn, in col. 2, line 7 [4932/], becomes —(l-im).~.
•I in
Moreover, the fictor — (1- — l)./i: [4932c'] becomes, in this case, by neglecting dv',
— {(2-2/«-c)--ir.2/"e=— Kl-2'«+l'«')'-U--4/'^c = (4m-7«/2).^/n,^4^(l_Z;„')^(,)g_
Multiplying tills by the factor _^^*^ [4932e], we get —{l—2m).A'-^\ef for the
factor of cos.(2^y — cv), in line 7 [4932/].
(ci. 1.) (Col. a.) (Ci.i. 3.)
Terms of adit [4904]. Terms of [4885]. Corresponding terms of the function [49326'].
.,-2
•Sill
[4939a'l
[4932d']
[4933e]
[4932e']
.Î2"".cos.(2i;— 2mr)
^i"e.cos. (2v — 2mv — cv'
w3.J'e.cos.(2u— 2/nt)+ci;)
^Ip'e'.cos (2«-2m«+c'?rtî) ;
A'^^e .co5[2v-'2mv-c' mv)
A[^^Y.cos{2gv-'2v+2mv)
•-I, '.-,.cos.(î) — mv)
-^ ' — .cos.f2i'-2mj;)
2— 2m ^ '
-2(1 -f-w)
-9(l-,«)
2-2m+c
7c'
,e.cos(2u — 2mv—cv
e.cos(2w— 2mi'-f-ci''
>.a.-.cos.(3y — 3mv)
2(2-3m)
e'
~2(2-m)
(i-yg;
2-2m
- .c os(2t! — ^mv-c'm v ]
Thrse terms have tlie factor
J4(l-m)3-]|
2(1— jn)
,^^»>.(1— |e'2)
cos.(2u-2nM)-[-c'ni!j)
.cos. (2d — 2mv)
--- !— 1- .cos(2g-i;-2u+2mv
4(%--2-f2m)
2— 2fft
1
2— 2ffi
1
cos.(2d — 2mv)
cos.(2ti — 2mv)
— 2e-cos.(2«— 2mv—cv)
.cos.(2y — 2mv)
2—2/,
'2_o cos (2y — 2mv)
+ |4(l-;»)2-l } .gb^^^^fo, e.eos.c. 2
+ {4(l-m)2-l }.^<i=^.^|.)e.cos.c« 3
<c—^m-\-c
.l4{l-my'-l}.^^^.AI'V.cos.c'mv 4
+ {4( 1_m)2— I |.-i— .^^o,e',cos.c'm« 5
( (2-2m-c)a-l ) ^. , ,
— (1— 2m).^i"c;.2_cos.(2^y_ct,)
— 4(1— Ae'2).^.^(2^e.cos.cy nearly
..^,'3',
e .COS. cmv
.cos.c mv
VOL. III.
112
^ (2-m)g-l ^
i 2(l-m) ^
C (2— 3m)2— 1 )
( 2(l-m) ^ ^-^
—2A\'^'>ey^.cos.{2gv—cv)
, C 1 — 9.(1— m )9) a
+ { -2=2;;r~ \ ■'^-^■^^A^^^^)- 13
7
8
9
10
11
Develop -
inent of
the fourth
term of
tlio func-
tion [49J1 I
[4932/J
446 THEORY OF THE MOON ; [Méc. Ctl.
[^•^3fi'] its true longitude [5095].
The last term of the function [4931] is,
[4932gr]
9m' „H'2.(5tt' -, . ,^
r4933A1 '^° develop it, we have, by retaining only the first power of e', a'M'=l-)-e'.cos.c't;'
[4833], whose variation is a'&u'= — c'e'.Sv'.s'm.c'v'^ — e'&v'. sin. c'mv, nearly; and, by
substituting the value of &v' [4931 r], we find, that Su' is of ihe fourth order; consequently,
[4939i'] the expression [493'2o-] is composed of terms of the sixth and higher orders ; and, as the
integration, in [4932§-], does not have the effect to increase essentially these terms of the
sixth order, the whole expression may be neglected.
AVe have thus computed all the terms of the function [4931]. Nothing now remains, but
to connect togetlier the terms which depend on the same angles, as they are found in the
[4932A:] functions [493 1|j, w, 49323,/"]. The sum of these four functions ought to be equal to the
development of the expression given in [4934], neglecting, for a moment, the consideration
of the terms depending on C [4935, &lc.], which will be noticed in [4937«,8>:c.]. In
finding the sums of these coefficients, it will be necessary to make some slight alterations, to
reduce them to the forms adopted by the author in [4934]. This will be done in the
remainder of this note.
[4.9321]
[4939m]
[4933)t]
First. The term in [4932/'line I], which is independent of any angle, corresponds to
[4934 line I], without any reduction.
o — "
Second. The second term of [4934] has the factor — c.(l — ^e'^) .cos. (cv — to)
common to all its terms ; and the terms by which this factor is multiplied, in the functions
which we have mentioned in [4932Ar], are collected in the following table, in the order in
which they occur, without any reduction, except, that the two terms [4931p lines 2,3], are
reduced to one in line 38.
[493 Ip] lines 38, 6, 12,18,21
[4932rt] line 1
[4932/] lines 2, 3
[4932/] lines 6, 8
+8;n..^o"'+2.^;"— 2^^->— ^i«.e'2+7./î<;'.e'2 1
+2m.^;'i 2
_H,(,-,.).-,s.^,-i£-^+,j=:^|,A». 3
The coefficient of A[^\ in this table, is
^ , ^ , (2-2m— c'
4.(1— Î"
and, by neglecting the term m^, in the numerator, which produces only terms of the sixth
(2-2m-c)2-l 7-8m5-H2-2m-c)a
[4ft52o] 2+2m+— --^-- = ^^j-^^^j
VII. i. ^^8.] DEVELOP.AIENT OF THE DIFFERENTIAL EQUATION IN u. 447
C,<«'.e'.sin.(2 v — 2 m v j-c'm v—^') and C^"''.c'.sin.(2 v — 2 m v—c'm v-{-a) [4937]
order in [4934], which are usually rejected, it becomes equal to the coefticient of A'-p, in
[4934 line 2]. We may also omit the term Qm.A':^ [4932?i line ]], which is of the same
order; and then, the remaining terms, connected with A^^\ in line 3, are the same as in
[4934 line 2J. The terms depending on .^/~' [4932ij lines 1,4], mutually destroy each
other. The remaining terms, depending on ./3/''', A'-p, are as in [4934 line 3].
Third. The third term of [4934] has the factor e'. cos. {c'liiv — w') common to all the
terms. The coefficients of this factor, in the functions mentioned in [4932Z:], are given in
[4932«], in the order in which they occur ; observing that the two terms in [4931/3 lines 4, 5],
as well as those in lines 9, 10, are reduced to one in [4931^ line 39]. Moreover, the terms
of [4931yj], depending on the angle c'mv — ro', have the divisor m, wiiich is introduced
by the integration ; and they have also the common factor — ; so that they are all
multiplied by
6 m
a,.m
fi.fma— em") fim 3m3 ,^^^,^ 6m
= = — 1 5094 ; or — nearly
3m3
neglecting the term , which produces only terms of the sixth order in [4934].
Hence the factor of c'.cos.(c'm2; — ra') becomes, without any other reduction, as in the
following table ;
Cm , „,
[4931/7] hues 39, 13,14, 17,20
[4932/] lines 4, 5
[4932/] lines 9, 10
.\AA'f—\OA'^\e~-\-A.}''^-
3SÎ
'4
-. j A.(\-mf—\\jU->\ \ — î_ \
By altering a little tlie arrangement of the terms in the first line of this table, it becomes as
in [4931 line 4] ; the second and third lines of the table, correspond respectively to [4934]
lines 5, 6. The terms relative to C, in [4934 line 7], are discussed in the next note.
Fourth. The eighth and ninth lines of [4934], correspond to [493 Ip lines 22, 23],
respectively. The tenth line of [4934], depends on [4931/? lines 7, 16]. The eleventh
Une of [4934]. depends on [4931/; lines 8,19]. The twelfth and thirteenth lines of [4934],
correspond, respectively, to [493Ip lines 24, 26].
_a
Fifth. The factors of — .672.003.(2^1)— cy), in the functions mentioned in [4932it],
are contained in the following table;
[49.32/)]
[4932^]
[4932r]
[4932»]
[4932<]
[4932u]
448 THEORY OF THE MOON ; [Méc. Ce!.
[4937'] *are the inequalities depending on tlie angles 2v — 2inv-'rc'mv — 35' and
[493^!'']
[4932i>]
[493 1 p] lines 11, 27, 30
[493 lit]
[4932a] line 2
[4932/] lines 7, 1 1
+im.A['^ 3
Sum = +|m../?,''4-2^^'3'— ^["5>.
This sum agrees with the coefficient in [4934 line 14], except in the term depending on
«^'j'', which is Jm.^'/' instead of ^m.Jl[^K The difference is of the seventh order only,
and is hut of little importance, producing only terms of the fifth order, after integration, in
[4847]. This discrepancy appears to have arisen from putting ^=1, c=l, in the
[493ar'l calculation [4932e, e'], instead of the values [4932,4933]. For, by using ^ = 1, the
/ I-l-A m)
factor r4932rf',e] becomes . -, and the fiictor r4932e'l is
4ni -'
— \{2—2m—cf—l\ = _{(l_2m)2— 1* =4in—4m^=4m.{l—m).
The product of these two factors is nearly equal to — (1 — i»i), instead of • — (1 — 2//i)
[4932w] [49;3-2y Ijne 7]. Hence, the coefficient of m is decreased to one quarter part of its
former value, and the term |m..^j'' [4932y], will be decreased in the same ratio, so as to
become Jjn-.^'j'' ; by which means, the sum of all these terms ^m.A\'^ [4932i)], is
reduced to |m.^,'\ as in [4934 line 14].
Sixth. The term in [4934 line 1.5], corresponds to that in [4931p line 29]. The
factors of — . — .cos.(y — mv), in the functions mentioned in r4932/i:], are
2«,.(l— m) a ^ '
contained in the following table. The sum of these factors corresponds to that in [4934]
line 16, neglecting terms of the order m^.A["\
+ 4.4f/'>-2.4<'«.e'^— 4>2
+4m...^i>^i
-I { l—{l-mf\..f,^'> —X,. \ i—l{l-my^
[49.32.T]
[4931p] lines 31, 35, 37
[4932«]
[4932/] lines 12,13
Sum = {4+3m).J[''^—2J]',"'\e'^—i\_. \ 1— (l-m)*2| .
Seventh. The terms in [4934 line 17], correspond to those in [4931/? lines 33, 36] ; and
[4933i/] the terms of [4934 line 18J, correspond to [4931^ lines 32, 34]. Hence it appears, that
all the terms we have computed, agree with those in [4934].
* (2829) If we compare the value of nt-^s [4828] heretofore used, with the form
[4937al finally adopted in [5095], we shall find, that the terms depending on ft", O,'', hc.C^P,
p.qg-,, have been neglected ; and, if we put C for the sum of these terms, we must add C to the
value of nl-\-e [4828], which will introduce in the second member of [4836] the term
VIL i. ^^ 3.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u.
449
2v 2mv — c'mv-'r^', ia the same expression. We may also observe, that [4937"]
the term,
Cm ; and the same quantity in the second member of [4837] ; and we shall represent this
increment of v', by the expression 5v'=^ Cm. Substituting this in a'Su' [4932^], we
get a'ôu'^ — Cme'.sm.c'mv. Now, if we select the chief terms of [4910, 4931], depending
on Sc', Su', they will become
J«'.sm.(2!) — 2y)+— r- . / • — —.tM.cos.{9,v — 2«') — — ./ — — .rfv.sin.(2u-2w').
fc3. u3
[4937c]
[4937rf]
[4937e]
We have neglected the last term of [4924], depending on &v', because it is multiplied by
— , which is of the order e [48T8] ; so that this will be of the same order as the product
dv
of the first term of [4937e] by e, which, as we shall soon see, may be neglected [4937À:].
Now, substituting the values of &v', Su' [4937c, rf], in [4937e], it becomes, by merely
altering the arrangement of each of the terms, so as to bring them under the forms we have
already computed,
{ 'âS-^^-^^^-^'^ ] •2C^+4^:./'^".cos.(2.-2.').f C.
, 9?»' «M- . ^ /\ "'C ^ . ,
+ --— . / —.dv.sm.(2v — 2v).—r.C.sm.cmv.
The value of C, to be substituted in this expression, is easily deduced from [5095,4937tl,
and is represented by
C=Cf.sm.(2v—2mv)-{-C[''\e.sm.{2v—2mv—cv)-\-Cf\e.sin.(2v—'2mv-\-cv)
-\-C'^\e'.5m.(2v~2mv-j-c'mv)-\-C^"'\e'.s'm.{2v—2mv—c'mv)
-\-Ci"Ke'.sm.c'mv-{-hc. . . .-{-C['^\~.cos.{v—mv)+hc.
If we multiply together the two functions [4876e, 4937A], and the product by 2m, we
shall get the first term of tiie function [4937^J. These terms of this product are of the
Jlfth and higher orders ; so that it will only be necessary to retain those which depend on the
angles cv, v — mv. These tenns are found by multiplying the first term of [4876e],
2
O —
namely, '^.sm.{2v—2mv), by the terms of 2mC [4937/;] depending on C,"', C['^^;
4a,
from which we get,
in which we have neglected some terms of the sixth order, depending on C^'^', and on the
angle cv.
[4937/]
[4937g]
[4937/i]
[4937»]
[4937*]
VOL. III.
113
450
[4938]
THEORY OF THE MOON ;
[Méc. Cél.
6m
.{^A'^'>-\-Af--Afl.e'.cos.(c'mv—^'),
[4937«]
The next term of [49375-] is found by multiplying together the functions [4910Â:,4937/i],
and the product by — ^m.dv ; and then integrating the result ; as in the following table ;
Terms of [4937/t].
-\-Cf\sm.{2v—2mv)
[4937m]
+ C-p .e.s\n.{2v — 2m« — cv)
+ Cf''e'.s'm.{2v—2mv+c'mv)
-f- C™^e'.sin(2u — 2mv—c'mv)
-\-a^'>\-,.sm.{v—niv)
Terms of [4910Â:].
4-cos.(2« — 2niv)
— 2e .cos . {2v — 2mv — cv )
— 2e.cos.(2w — 2mv-{-cv)
+ Je'.cos.(2î) — 2mv — c'i7iv)
— I e'.cos.(2i) — 2mv-\-cmv)
-j-cos.(2y — 2mv)
-|-cos.(2t; — 2mv)
+cos.(2i) — 2mv)
+cos.(2« — 2mv)
Terms of [4934]. _3
These terms have the factor — •
a,
. . . .neglected 1
-\-2Cf\7nc.cos.cv 2
— 2Cj'-''.OTe.cos.ct) 3
— i Cf^.e'. COS. dmv 4
— 2 C f'.e'. COS. c'mv 5
4- O^^ .me.cos.cv 6
— CfKe'.cos.c'mv 7
-(-C^^°'.e'.cos.c'?n« g
-\-C[^^\m.-,.cos.{v — mv). 9
The last term of [4937^] being very small, we may substitute in it the values
[4937n]
14 = - ; u'=z -, ;
■ mv\
h^=a, [4921«— c];
by which means it becomes,
[4937o]
[4937p]
[4937g]
'""" .me', f dv.s\n.(2v — 2mv').sm.c'mvy, C ;
and, by using [4S65], it may be reduced to the form,
— . me'. fCclv. {cos.{2v — 2mv — c'toî)) — cos.(2t) — 2mv-\-c'mv) \ .
Now, substituting the value of C [4937A], it produces terms of the sixth order, before
integration ; and some of them may be reduced to the fifth, after integration, if they be
connected with the angle c'mv; we shall, therefore, retain this angle only. These terms
are found, by substituting, in [4937o], the part of C [4937A] represented by
C^''\sin.(2î) — 2mv). Combining this with each of the terms of [4937o], it produces a term,
rhdv.sm.c'mv = .cos.c'mv ; so that both terms, taken together, produce the following
expression ;
— . f-l dv.s\n.(2v-2v').^ . C. sm.c'm v = — A— ^Cf\e'. cos c'mv].
VII. i. §8.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 451
[4938']
appears to be of the order ?«', which would produce a quantity of the
order w^^ in the expression of the moon's mean longitude ; but this term
is, in fact, only of the order m^. For, we shall see, by means of the values
of 4°', Jf, 4^'* [5157,5160,5161], that the function 4Jf +4'— 4<'' [^«39]
is of the order m^ ; which produces, in the expression of the mean [4939']
longitude, a term of the order m'^ only. We shall, however, retain it
here, because we have imposed on ourselves the condition of including terms
of that order, in the calculation of the terms of the third order.
For this reason, it is indispensable, in the development of
j^— .J —^ .sm.(2v — 2v') [4930], [4940]
to carry on the approximation to terms of the order 6ii^ ; hence we
obtain the terni,t
Connecting together the quantities contained in [49.37A:,??î,y] , we get the terms of the function
a
[4937 e] depending on C. The coefficients of — .Cf\é.cos.c'mv, in [4937m lines 4,5],
and in [4937(7], being connected together, become,
— J— I— 1 = — ^S as in [4934 line 7] ; [4937r]
and the terms in the same line, corresponding to Ci''^ C^"', agree with those in [4937?»]
lines 7, 8. The term depending on Cf' [4937ot lines 2, 3] , mutually destroy eacli other.
The quantities we have mentioned include all the terms retained by the author ; who has
not noticed those in [4937^], and in lines 6, 9 of [4937?«], whose sum is
— .m. } C[''\c.cos.cv-\-0^^'>,-.cos.{y^mv) > . [4937s]
These neglected terms are of the fifth or sixth order, increasing also by the integration in
[4847] ; and are of the same orders as the terms which are usually retained with these [4937t]
angles ; but, as we did not wish to alter the numerical calculations of the author, we have
not introduced them into [4934].
* (2830) These values are nearly represented by ^/''i= 0,0071, .;32")__o^o030,
^»iz=: 0,0285; whence, AAf-\-Jl'-^^—A'-^=—Qfi^Z, nearly. This is less than m~ [4938a]
[5117], but can hardly be called of the order m?, as in [4939'] ; however, as it is
multiplied by e', which is much smaller than e, 7, in, we may consider the whole term [49385]
[4938] as of the order m^.
t (2831) The factor eu- is of the fourth order [4904], and, as all the terms we have
452
THEORY OF THE MOON ;
[Méc. Cél.
[4941]
30m'. u „u'3.fe2 , . ,^ „ ,.
This term produces the following ;*
computed [4910, 4924, &IC.] have the factor ?n', or m, except where the sign of integration
[49327-)-] has introduced the divisor m; it follows, that these terms depending on Su^, are generally
of the sixth order ; but some of tliem may he reduced to the Jifth order, by the integration
we have just mentioned. Tlierefore, we need only notice those terms wliere the variations
are connected with the signs of integration ; so that we may neglect the second powers or
products oftlie variations in the terms [4909", 4921, 4924,4927 ,4931, &c.], and, in fact, only
retain the chief term of [4930 or 4931], which depends on Su^. For, we need not notice
the terms depending 5u.Sv', Su.oii. Sv'^, Su'^, &;c. ; because &u is of the second
order [4904], 5v' is of tlie third order [4929^], Su' is oï the fourth order [4929i — it] ;
hence, the terms depending on hi.dv, Su.Su', &ic. must generally be much less than those
depending on Su^ ; therefore, we shall only notice this last quantity. We have already
[4941e] found, by Taylor's theorem [610. &c.], in [49296], the increment of the function — V.ffVdv ,
arising fi'om the increments Su, Sv', Su', in the values of m, v, u', respectively ; and,
by the same theorem, the term depending on om^, will evidently be represented by
—J V.f (^-^'\ . Su\dv [610, 49296] .
Substituting the value of W [4929«], it becomes,
[4941a]
[4941i]
[4941c]
[4941rf]
[4941/]
[4942a]
[49426]
[4942c]
-30m'. F ^u'^.ixfi , . .^
7l2
mO
and, by using the value of F = - = w, nearly [4929Zr,4937w], it becomes as in [4941] ;
neglecting in V terms of the order em^, e^, y^.
* (2832) As the function [4941] is of the sixth order, before integration [49416] ; we
may, by neglecting terms of the seventh order, substitute in it the values [4937n] ; by this
means, it becomes.
30)n'.a3 . .„ .
■;^.J dv.^aSuy. %in.{2v — 'Hmv) ■■
30?
.fdv.{aSuf.sm.{'iv—'imv) [4865].
If we retain only the term of {aSuf, of the fourth order, we may neglect all the expression
[4904], except the two fiist lines, and we shall have,
aSu=Jlf\cos.{2v—2mv)-\-Ap.e.co5.{2v—2mv—cv).
Squaring this, and reducing, by means of [20] Int. we get,
(«'5M)-=(^f)'-B+è-cos.(4îJ— 4fflD)|-f^^o).^<').c.{cos.c«+cos.(4r— 4mi'— a.)|
+ (^i'O^-e^- B+è-cos.(4i;— 4m!;— 2a.) \ .
This must be multiphed by sin.(2u— 2mu), and the product substituted in [4942»], after
VII. i. §S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 463
2
15 m {A^py^.e^.cQS.{2cv—2v-\-2mv—2^) _ ,^^^2]
"207 * 2c— 2+2m '
although it is only of the fifth order, yet, as it acquires by integration, in the
expression of the mean longitude, the divisor* 2v — 2+2m, it is necessary [4942]
to notice it.
The function
gives the following ;t
~ Vd^ +N' F '-^ — Âû^ {3.sin.(«— u')+15.sin.(3w— 3<;')^ [4944]
Its variation produces the terms,!
making the reductions by [18] Int. Tiie only term of this product, in which the coefEcient
of V is small, is that produced by multiplying the last term of [4942c],
i(^','')2.e2.cos.(4i' — 4m!; — 2c«), by sin.(2!; — 2mv) [4942a], [4042rf]
which produces the term ^(.^','')^.e^. sin.(2cy — 2v-\-2mv); and, by substituting this in
[4942a], it becomes equal to the following expression ;
__.(A"')-V ./-sm.(2a.-2«+2..) = _ . 2c-2+2>» ' ' ^'''^^^
as in [4942]. The terms we have neglected are of the sixth or higher orders ; the term
[4942] is reduced to the fifth order, by means of the small divisor 2c — 2-\-2m, which is
neai-ly equal to 2m [4828e].
* (2833) The term of u, resulting from the substitution of [4942] in [4961], is to be
added to u [4812 or 4819] ; and this produces in dt [4753] a term depending on the
same angle. The integration gives, in t, and in nt-\-s [4828], a term of the same form
with the new divisor 2c — 2-\-2m ; and, by this means, it is reduced to the fourth order.
[4943a]
t (2834) The terms [4809], depending on the angles v — v', 3v — ^v, are retained
in [4944] ; because they produce, in [4946], some terms depending on the angle v — mv, [49440]
which require a greater degree of accuracy than the others [4906, &ic.].
X (2835) Since 5m', 5«', are much smaller than Hu [494 If/], we may neglect
them in finding the variation of the function [4944], and consider u as the only variable ^ '
quantity; by this means, the variation of [4944] becomes,
VOL. III. 114
454 THEORY OF THE MOON ; [Méc. Cél
I /ddSu , \ J, vi'.u'*.dv
•(-^^+ôttj .J —^ — .|3.sin.(îJ— î))+15.sm.(ou— ot)}j
[4945] ' ^
+ — .- . fa6u.dv.\3.sin.(v — i)')+15.sm.(3« — 3v')\;
4 a, a'
hence results the terra,*
[4945a]
/ddâu. , . \ 1 ^ m'.u'^.dv ,„ . , ,s , , _ . /„ <^ ,> i
- {1;^ +^V • P •/— 4-,.T— •l3.sm.(.-.')+15.sm.(3t-3.')|
Substituting, in the first line of this expression, the vakie h^=a^ [4937n], it
becomes like the first line of [4945]. Again, by substituting, in the second line of
[49456] [4945a], the values of M, u', P [4937?f], and for '~+u, the chief term - [4890],
it becomes
r4945cl ■ — '—;:; ■ — • fa5u,dvA3.sm.(v — y')4-15.sin.(3!; — '3v') \ .
'■ 4a,. a-' a i ^ ' ' 1
This is easily reduced to the form in the second line of [4945], by the substitution of
m' [4865].
* (2836) The terms [4945], being of the sixth order, independent of the integrations,
it is only necessary to notice the terms depending on the angle v — mv; and, we may,
therefore, substitute the values [4937h], in [4945], and they will become, by using [4865],
.( -^7-j- ^" ) • 7 •"> •/{ 3.sin.(y — ?OT)+15.sin.(3i> — 3mv) j ,dv
[4946a] ' a
-\ . — .faSu. { 3.sin. (u — m!;)-j-15.sin. (3!) — 3m!;) I .dv.
In this we may substitute, for « Su, its two chief terms [49426] ; and a little consideration
will show, that we may even neglect the part depending on ./2/'^ because it does not
[49466] produce, in [4946], any term connected with the angle v — mv ; so that we shall finally
have aôu ^Af\cos.{'2v — 2>nv). Substituting this in [4946a], it becomes,
— — -.jjw. \ 1—^.(1— mf\xos.{2v—2mv).f{3.sm.(v—mv)-{-]5.sm.{3v—Smv)\.dv
4o, a
[4946c] J
^ — ,-.^'M.fcos.{2v—2mv).l3.sm.{v—mv)-{-l5.sm.{3v—3mv)l.dv.
Now we have.
VII. i. §S.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN u. 455
"' ,{\S-\-^.(\—m)-\.Af\-,.co^.{v—mv). [4946]
2a,.(l— ?«) ' ^ ' ^ ^ a!
We must here make an important observation relative to the terms
depending on cos.(^' — m v), which we propose to determine with accuracy.
The expressions of the radius of the sun's orbit, and its longitude, contain
terms depending on the angle v — mv [4324], resulting from the moon's [4947]
action upon the earth. These terms produce others, in the expression of u, J"^,'^'^^^^
and in the moon's mean longitude ; and it is essential that we should notice
these terms. For this purpose, we shall observe, that, in consequence of the
moon's action, the sun's radius vector contains the term
6r'= -.cos.(t;— «') [4315, 43166] ;* [4948]
3 5
riS.sm.(v—mv) 4-l5.s'm.( ov—3mv)\.dv = — .cos.(v—nw)—- A:os.{3v—37nv). [4946d]
Multiplying this by cos. (2u — 2}nv), and retaining only the terms depending on
cos.(y — mv), we find, that the product becomes,
( ; ).icos.(î) — mv) =— -.cos.(îJ — mv) ;
\ 1—m 1 — m J " 1 — ni
hence the first line of [4946c] becomes,
m
'- — ^, . i —24-8.(1— m)^ ! .Jli'>\-.cos.(v—mv). [4946e]
Again
2o,.(l-m)
cos.(2f — 2mw).3.sin.(î; — mu) =— â.sin.(i) — mv)-\-iic.
cos.(2u — 2mw).15.sin.(3î) — 3mv) =-y*-.sin.(«) — mv)-\-&iJC.
whose sum is 6.sin.(j; — mv) -\- &ic. Substituting this under the integral sign of the second
line of [4946c], that line becomes,
— - _2
5«i a .,p, 6.cos.(î>— ;ni») m , c ^ ,01 « / x
— 7— . -../2o™' = = — --— .làAé'^'. - .cos.fi) — mv).
4 fl, a ~ \—in 2a. (1—m) ~ a
Adding this to the part [4946e], it becomes as in [4946].
* (283T) The inequality of the earth's radius vector, arising from the action of the
moon, is
[4946/]
m
^''--M^^'-^'-^'^'-i^-''") [4315,4316J]. [4948„3
To conform to the present notation, we must change U into v [4313,4760'], R into r,
466 THEORY OF THE MOON ; [Méc. Cél,
r4948'i '^ being the ratio of the moon'' s mass to the sum of the masses of the moon
and earth. This gives, in u', the term,*
14949] &U'=—'^.C0S.(V-V').
u
The longitude of the sun v' contains also the term [4814],t
[4950] i.v'=-^.ûn.(v—v').
u
This being premised, the term ' 3 [4865'] contains the following :t
[4951] ____.cos.(^^— Î)).
3m' u'^
[4951'] The term --^ .cos.(2t?— 2i'') [4866'], contains the two following §
— pn*5
[4952] _ ^^^ .cos.(?;— z)').cos.(2i;— 2v') +^^~.sin.(«— '«').sin.(2ij— 2^;') ;
[49486] [4313,4759], r" into ?•' [4313,4759']; moreover, the longitude v" of the earth, seen
[4948c] from the sun [4313], is equal to 180'' + '^' of the present notation [4777rf] ; lastly
[494erf] ^=_^ [4757,4757', 4948']. Substituting in [4948a] , we get 5r'^,j.r.cos.{v—v'};
and if we neglect the square of the inclination of tlie moon's orbit to the ecliptic, we may
[4948e] put r = - [4776], and then the preceding value of S/ becomes as in [4948].
* (2838) From r4777el we have, very nearly, r'=: - ,; whence 5r'= -.
[4949a] ^ ^ ^ ■• ' } : „" „'2
Substituting the value of &r' [4948], we get Sti [4949].
t (2839) This term is given in [4314,43l6i], under the form
M-\-m r' ^ '
and, by making the changes in the symbols, as in [4948J,&,c.], it becomes,
[4950a] (5v'=-(-fji.— .sin.(t) — ■«'), or nearly 'V=fx. — .sin.(j; — i)'), as in [4950].
X (2340) The variation of the term " [4865'], taken relatively to ?t', is
[4951a] ^ ,^ ^^'■'^
" .i5m' ; and, by substituting hC [4949], it becomes as in [4951].
§ (2841) Taking the variation of the term [4951'], relatively to «', v\ and then
*■ substituting the values of 5m', h' [4949, 4950], we get [4952].
Vll.i.^8.] DEVELOPxMENT OF THE DIFFERENTIAL EQUATION IN u. 467
which produces the term,*
'3m^.^ [4953]
Connecting it with that in [4951], we obtain,
9?«'. fA.?«'* ,. [4954]
whence results the following terms ;t
2 2
9râ.(x , . 9m. fA a ,. ,
^Scos.fi' — mv) . -.e.cos.(« — mv+cmv — w ) 1
4a, «^ ^ ^ 4rt ff' ^ ^
.-.e. cos.(« — mv — cmvA-vi). z
4a, a' ^ ^
[4955]
3 m' m'^ dv
The term — .y ^— .sin.(2« — 2w') [4882] gives, in like manner, [4956]
the following ;t
* (2842) If we retain only the angle cos.(« — v'), and reduce the products by [17,20]
Int., we may substitute, in [4952], the values
cos.(« — i;').cos.(2u— 2w')^J.cos.(y — t)')-j-fcc. ; ' [49531
sin.(«; — i;').sin. (2u — 2y') = |.cos.(r — v') — &,c. ;
and, since — f-s+f-i = — tj the expression [4952] becomes as in [4953].
t (2843) Multiplying [4872] by — 2,a, and neglecting e^ c'2, we get [4955]. [4955a]
f (2844) The variation of the term [4956], is as in [49566]; substituting the values
of 6u', ôv' [4949, 4950], it becomes as in [4956c] ; reducing the products of the sines [4956o]
and cosines, by [18, 19] Int., retaining only the angle v — v', it becomes as in [4956(Z].
— ./ { .sm.(2i' — 2« )H ; — oi;.cos.(2u — 2y ) } [4jao6j
= -j^'f] — ^— .sm.(2«— 2i)).cos.(j;— d')H ^ .cos.(2t)— 2z)').sin.(ii — i)') [ [4956e]
3m'. {J- ^( 3u"'.dv . , . !<'lrfu > 3m'. ^ ^m'^. rfy . ,. r4.q';fi//1
This last expression is evidently equal to the first member of [4889] multiplied by — 2/x ;
and, if we multiply Its second member by the same factor — 2fJ., we shall get the '■ ■■
development [4957] ; neglecting the small terms e^, 7^, e''^.
VOL. III. 115
458 THEORY OF THE MOON ; [Méc. Cél.
3
^ - . -, . COS. (v—mv) -r- . - . -.e'.cos.{v—mv+c'mv—^) 1
[4957]
2.{l—m) a, a'
2
9m.iJ' a a , , , i\ c,
2.(1— 2m) a, a' ^ ^
There remains yet to be considered the part of the development of
[4958] . [4893], depending on the square of the disturbing force.
W.{\-\-ss)'^
g
[4959] rpj^jg development contains the function* — . {hsf, which produces the
following terms ;t
* (2845) We have, by Taylor's theorem
[495G/] -pC^^^*) = ^'^'' • -Ï +*("-> • rf.2
,4.5,)=,,+,3..^'+H«'^-)^-'^ + &^c. [617]
where the terms of the second order are represented by i(fîs)^. — r-^. Now, putting the
function [4958] equal to 9(4), and developing it, we get,
[4956^] <p(,)__A-2.(i+,,)-i-=_/r2.(l_3s2+^V-&c.). .
Its second differential gives,
[4956/.] ^=-'^''"-1?''~'*'+^<^-=^^ = ^ "^^^'y [4937»];
neglecting s-, &ic. Substituting this in the terms depending on {hs)^ [4956/], it
3
becomes — •('5*)"j as in [4959]. The terms of the order ^^.{i)s)^, which we have
[4956i] here neglected, are of the order 7^ [4811], in comparison with those which are retained
and developed in [4960] ; they must, therefore, be of the sixth or seventh order, and are
not usually noticed.
t (2846) If we separate the terms of h [4897] into classes, of the second, third and
fourth orders, by putting
[4960a] 6;=Bi°>.7.sin. (Sv— 2m«— ^«) ;
S3=jB^'>.y.sin.(2î) — 'imû-\-gv)-\-B'^^.t'j.ûn.{gv-\-c!mv)-\-'Bf^.e'j.ûx\.{gv — c'mv)
-\-lif' . t' y .s\'a..{^v — 2mii — gv-\-cm.v)
[49606] ■\-Bf^.e'y.^m.{2v—S>mv—gv—c'mv) -^Bf\(?y.sm.. (2ct-^d) ;
S:^=B^\ty.%\x\.{2v—'i,mv-\-g'o — czi)-[-the remaining terms of & [4897] ;
the index of S denoting the order of the terms ; we shall have &=-S'3-[-<Si3-[-'S4 • Its
[4960c] square is (&)2=.S'2. ^o-|-2S2.(Sa-j-2So. 5'4+»S3 . 1S3; neglecting terms of the seventh
Vll.i. §9.] DIFFERENTIAL EQUATION IN M. 459
£-.(5{»)/.j« 1
-\-~.\Bf^^B["'^\.BfKeY.cos.{c'mv—^) 2 [4960]
/ill
+ ~ . Bl'K 5f . e7^.cos.(2gv—cv—2è+z^). 3
9. We shall now collect together and reduce the different terms which
we have calculated ; and, bj these means, we shall obtain the following
development of the equation [4754] ;*
order. Substituting the values of .S, , S3 , S^ , and tlien reducing, by means of
[17 — 20] Int., retaining only the usual angles and terms, we get, by observing, that the
terms depending on Bî,^' maybe neglected, on account of its smallness [5177],
S,.S,^UBrfr-;
2S.2.S3 = \B^;\B'f+Bf\B[">^\.eY.cos.(fmv;
2So.Si = B^^KBi'\cf.cos.{2gv-cv) ;
S3. 83^ terms which may be neglected.
The sum of these terms gives the value of (&)^ [4960c], which being multiphed by
3 3
2^ gi^es 2^-{^^f' as in [4960].
[4960rf]
tioti lU
u.
* (2847) We have thus finished this elaborate development of the terms composing the
equation [4754] ; and we must now connect together the different terms ; namely, those
which are contained in the twenty Junctions [4866,4870,4872,4879,4892,4595,4901, [4960«]
4908,4911, 4913,4918, 4922, 4925,4928,4934, 4942, 4946,4955, 4957,4960], and add
' -" Function»
ddll wliich
to the sum the tu'ojirst terms of [4154]; namely, — -|-m, as in the two first terms of J"!?'""'
tial equa-
[4961]. In performing this part of the operation, we shall take the terms depending on
each angle separately, in the order in which they occur in [4961].
First. The constant terms of [4961 line 1], are found in [4895, 4866 line 1], without
9
any reduction. The tenns having the common factor — - — ..^^''^(1 — |e'2) are found by
adding together the terms in the first lines of [4911,4925,4934] ; namely, 3, — 2-j-2m,
4.1 1— m)- — 1 . 1
~^~Y^ • Their sum is l-j-2m-j-4.(l — m) =4 — 3?» — m'^, neglecting m^ r4961oT
and the higher powers of m ; this agrees with [4961 line 2]. Lastly, the term depending
on B;' [4960 line 1], is as in [4961 line 2].
460 THEORY OF THE MOON ; [Méc. Ce).
a
[4961] 0= ^ + «-i.n+^^+i/^+|3''} + £;{l+e^+i>^+|e-| 1
+2e+e^+3e'2-2.(5f)+5f>).^^+(l+2m-c).^^2>(l-|0\ ^
DLfferen- I ^'^^
liai equa-
tion in
u.
3m
4a
_. -4.|i+2-+(4.ri-i).(^+^,)^4«(>-iO 4
-.{ , f(l+6m+c).(l-m)+7+(-2-2m-c)a^ ^ \.e.cos(c.-«)
2«,
^.cos.(2w — 2mîj) 9
3 m l4-(l+36^+iy^-fe'^)
1 ?B
m
Second. We shall now collect together all the terms which are connected with cos.c?;.
For brevity, we shall divide all the terms of the twenty functions [4960e] containing
_2
this quantity, by the common factor — - — . e . cos.cv, retaining only the quotients which
ought to correspond to the terms, between the braces, in [4961 lines 3 — 7]. T/ie same
[49616] process will be used with the other angles in the rest of this note. Then we have, in
[4866 line 2], the terms 2-{-e^+3e'~, and, in [4901 line 3], the terms —2{Bf^-\-B^J'^).^;
m
these agree with [4961 line 3]. The rest of die quantities depend on the different terms of
A, which we shall examine according to the order of the indices. The coefficients of
—4^j,''>.(l— Je'-2), in [49111ine 2, 49>.5hne2], are, respectively, +3, and — 2+2m,
whose sum l-|-2m is the same as in the two first terms of line 4 [4961] ; the last terms
of the same line being found, without any reduction, in [4934 line 2]. The coefficients of
^i<".e.(l-fe'2), in [4911 hne2,49181ine 1, 4925 line 2], are respectively 3, 4m, -(2-2m-c),
whose sum is {l-\-6m-\-c') ; multiplying and dividing this by 1 — m, it produces the
three first terms in [4961 line 5], connected with the factor (1 — m) ; the remaining terms
VII. i.-^' 9] DIFFERENTIAL EQUATION IN M. 461
ic. {1+1(2— 19w)-e-—|e'='} \ 11
— i(3+4m).(l+ie=— |e'=) ) 12
,3m.) . „ cif, I \ \.e.cos.(2t) — 2mv — cv-\-a)
_ — J 3+c— 4m+ ^^^~"') +24-' J . e. cos.(2«;— 2mt)+ct)— t.)
15
[4961]
Q^V^ «J „9 / ^ Differen-
.— ./^ l4-95;9)/_4-2J<3' \ .e'.cos.(2v—2mv+c'mv—zi') 16 '-i?i"a-
cootinued.
3m ^ \ 7 (4 3m) _2]^ao);^_2A'*) i .e'.cos.(2v—2inv—c'mv+^') 17
' 4«, ) 2— 3m 1 a 2 (- <.
7M
l+c^+i7"-+l«"+(^l"+^?04-Kl + 2m)-^l'" \\ 18
2
_2
3™ _2(l-O„0.(3-O,.).[3-m) ,, /
I 2a, \ (2-3m).(2-m) - ~ ^ '' - / 1
"T~\ Il /7>.'9) I T>l-[m\ ÏX01 '5'
.e'.cos. (c'mzj — ro')
+(S;=>+Bi"»)..B';i.^ ^f— llCf— 2Cf+2a"» I 20
_2
m
+ ^.{4J^(0)+4;3,_J^(4,_|Q_J^(l)e2^_|(^^^(T)_J^(6)).g2^( ^ 21
of that line are found in [4934 line 2], without anjr reduction. The coefficients of
^f e.(l— Je'2), in [4911 line 2, 4925 line 2], are 3— (2— 2m+c) =1+2/»— c, as in
[496Uine3]. Tlie coefficients of —è^,W)g'2;n 1-4911 jj^go^ 4905 iine3, 49341ine3], [496I6']
neglecting the factor 1 — Je'^, are 3, — (2 — m — c), 8; whose sum is 9-\-m-\-c
[496Iline6]. The coefficients of ^.^'Jle'^, in [4911 line 2, 4925 line 3, 4934 line 3],
give 3— (2— 3m— c)+8 = 9+3m+c [4961 line 7]. Lastly, the terms in [4908 line C]
give, without reduction, 3.{Af''-{-A^p).e'^, as in [4961 line 7].
2
Third. The terms in [4961 lines 8 — 10] have the common factor — .cos.(2« — 2mv);
and, if we divide the corresponding terms of the functions [4960(] by this factor, we shall
obtain, in [4870 line 1], the ternis l+e^-^^y^—^e'^, and, in [4879 line 9], the term i^^^^'^]
-\-2mt^ nearly; the sum of these gives [4901 line 8]. The terms [4892]inel] are the
same as [4961 line 9] ; those in [4901 line 1] are the same as those depending on Bf\ B'-^''
VOL. III. 116
462
THEORY OF THE MOON ;
[Méc. Cél.
[4961]
Differen-
tial equa-
tion in u
continued.
3 m
3+2m-c
^l_.jlv._A<{^ ,
^ 3-t-m— c
I 2 '2— m
-_-|.A-
.ee'.cos.(2t! — 2mv — cv-\-c'mv-\-zi — to')
22
23
[4961 line 10]. Lastly, the first term of aSu [4908 line 1, 4904] gives the term depending
on ^f [496 nine 10].
Fourth. The terms in [4961 lines 11-14] have the common factor — .e.cos(2i--2??it'-c«).
Dividing the conesponding terms of the functions [4960e] by this, we obtain, in [4870 lineS],
the terms in [4961 line I2J ; in [4879 line 1], the same terms as in [4961 line 11] ; in
[4961rf] [4892 line 2], the same terms as [4961 line 13] ; in [4901 line 6J, the terms depending
on BJ'\ 2?!,"' [4961 line 14] ; lastly, we find, in [4908 lines 1,2], the terms depending
on A^^°\ ^/i' [4961 line 14].
Fifth. The terms in [4961 line 15] have the factor — ■'— .e.cos.(2u — 'imv-j-cv).
Dividing the corresponding terms of the functions [4960(>] by this, we obtain, in [48701ine3],
the terms 3-4;m ; and, in [4879 line 2], the term -\-c ; the suin of these is equal to the three
• ; and [4908 line 1]
[4961e] g^g^ jg^,^^_^ ^ç |.^ggj j.^^g ^gj_ ^^^.^^^ |- jgg^ line 3] gives
8.(1— m )
2— a»i-|-c
gives 2Af'' ; which are the remaining terms of [4961 line 15].
Sixth. The terms in [49611inel6] have the factor
4a,
. e'. cos. {2v — 2 m v-\-c' m v) .
[4961/]
Dividing the corresponding terms of the functions [4960e] by this, we obtain, in [4870]
2
hne 5, the term 1; and, in [4892 line 5], the term -\- ; the sum of these is
4 — ~ 7tl
, as in the first term of line 16 [4961] ; the term depending on B[^^ is deduced
/£ — 111
from [4901 line 8], and, that depending on ^f , from [4908 line 1].
Seventh. The terms in [4961 line 17] have the common factor — .c'.cos.(2t)-2m?)-c'mi;).
Dividing the corresponding terms of the functions [4960c] by this, we obtain, in [4870]
line 4, the term +7; and, in [4892 line 4], the term ^--—•, the sum of these is
2 — Sm
[4961g-] 7 (4-.3nO
2 , as in [4961 line 17] ; then we have, in [4901 line 9], the term
and, in [4908 line 1], the term — 2^^^> ; all of which agree with [4961 line 17]
Eiirhth. The terms in [4961 lines 18 — 20] have the common factor — . e'. cos.c'mt).
2a,
VII. i. §9.] DIFFERENTIAL EQUATION IN u. 463
-Wm-c) 7(2t3!îi)+4^n) ) 24 [4961]
^"^ ^ _ >.ce'.cos.(2t!— 2mi)— cy— c'my+ra+ro') .'^If:',';,"^.
I I /tr-\ I ^"^ — "1 — '' I 4 J /l/a^ V /^y tion in u
Dividing the corresponding terms of the functions [4960(:] by this, we obtain, in [4866]
line 3, the terms \+e'^+ly^+%e^; in [4901 line 7], the terms +(J5W-j-5»>) . L;
m
these include the terms of [4961 line 18], except those depending on •^j*"'. The terms
depending on ^/">, in [49IHine 3, 49-25 line 4], are — Mo*»'. j3 + (— 2+2m) j , or,
— a.(l+2«).^,"'^ as in [49GI line 18]. The other terms depending on A^'-'\ in [4961]
line 19, are the same as in [49341ine5] ; observing, that 4.(l-m)^ — 1 =(l-2m).(3-2m),
7 1 4.(3— jn)
[4961A]
2— .-3;» 2— m (2— 3;n).(2— m)
. The factors of Af\ in [491 1 line 3,4925 line 4],
are, respectively, — |, 1 — im; that in [4934 line6] is — - — = — l+l^^i
neglecting terms of the order in- ; the sum of these three terms gives, — 2A^^\ as in
[496lhnel9]. The factors of ^i^', in the same three functions [491 1, 4925, 4934],
and reduced in the same manner, are — a, 1 — ^m, — â-)-|-m; whose sum is — 2-f-3m,
as in [4961 line 19]. The term depending on Af [4908 line 1] is as in [4961 line 20].
The remaining terms of [4961 line 20] correspond, without any reduction, to those in
[4960 line 2, 4934 line 7]. Lastly, the terms in [4934 line 4], are the same as in
[4961 line 21].
Ninth. The terms of [4961 lines 22, 23] have the common factor
2
O — ;"
— — . e e'. cos.(2y — 2mv — cv-\-c'mv).
Dividing the corresponding terms of the functions [4960e] by this, we obtain, in [4870 line 8],
the terms |(3+2ot) ; in [4879 line 5], the term — ic; in [4892 line 8], the term
-^— ^ — ; and, in [4908 lines 4,1], the terms _ 3^/i)_^ffi) . these terms, connected
together in the same order, form the part in [4961 line 22]. In computing the terms which [4961i]
are multiplied by —A'^\ we have, in [491 1 line 7], theterm f; in [4925 line 7], the
tenn ihn—c); and, in [4934 line 9], the term -— ; the sum of these three parts
is as in [1961 line 23].
Tenth. The terms of [4961 lines 24, 25] have the common factor
_9 -
— ~.ee . COS. (2v — 2mv — cv — c'mv).
464 THEORY OF THE MOON ; [Méc. Cél.
2
3m
"Ô7,'
3+2m C 1+2OT+C
4 c-)-?«
26
[4961] ''^«, / +j(8)+ iiL:^^+_^ .j(;) V 27
2 ' ( 4 c-m ) ' f
5-.^ (i-^m+c 4 ^ ,,,, >.ee'.cos.(ct>-c'mt;-«+-') _
l^ ^ 2 c — m ) J
Differen-
l!o!.°ru / 3-2m , ^,c, , „ ( ]+2m+c , 2
QOQtinuecI. o
3wt
Dividing the corresponding terms of the functions [4960e] b)' this, we obtain, in [4870]
line 6, the terms f(3-f-6m) ; in [4879 hne 3] the term — J<- ; in [4892 hne 6], the
term -! — *— '- — ; in [4908 line 5], the term f^i'* ; the sum of these terms is as in
' 2— 3ot — c
[4961^] [496nine24]. Tliere is also, in [490S line 1], the term A'p, as in the first term of
[4961 line 25]. The coefficients of A[^^ are as follows; in [4911 line 6], -(-f ; in
4
[4925 line 8], — K'"+0 ! in [4934 line 8], ^_g^^__^ ; the sum of these is the same
as the coefficient of A^\ in [4961 line 25].
Eleventh. The terms of [4961 lines 26, 27] have the common factor
_9
3 HI , , , I \
.ee .cos.{cv4-cmv).
2a, \ I /
Dividing the corresponding terms of the functions [4960c] b}' this, we obtain in [4866]
line 4, the terms J(3-f2m), as in the first part of line 26 [4961]. The coefficients of
— ^i'> are as follows; in [491 nine 5], +3; in [4925 line 5], J-(— 2+2m+c); and
in r49341ine 111 + ; the sum of these three parts is the same as the coefficient of
L c-\-m
[496U] ^ (1) [4961 Hne 26]. The coefficients of A'p, in the same three functions
4
[49111ine5,49251ine5, 4934rmell], are J, 4(-2+3/»+c), + ^r;;;;; ; whose sum is
equal to the coefficient of A'p, in [49611ine 27]. Lastly, the term depending on ^f
[4908 line 1], is the same as in [4961 line 27].
Twelfth. The terms of [4961 lines 28, 29] have the common factor
2
.ee .cos. (cv — c mv).
2a, ^ '
Dividing the corresponding terms of the functions [4960e] by this, we obtain, in [4866]
lines, the terms J(3— 2m), as in the first part of line 28 [4961]. The coefficient of ~A\'-\
in [4911 line 4], is +f ; in [4925 line 6], is i(— 2+2m+c) ; in [4934 line 10], is
VII. i.-^ 9.] DIFFERENTIAL EQUATION IN u.
+ ^'. ) i_5^i').2::_^('0) ( , e\cos.(2cv—2^) 30
' '" ^ [4961]
.(.2+llm+8m2) (10 + 19m+8»l^) n Differon-
q ;;; 1 5* ~C a-j-am A tion in u
+ 7^0 ?8^ <--^iOM<'>^^J Le"-.cos.(2c.-2.+2m^-2«) «— •
V ' 2c — 2+2ot " ' J
; tlie sum of these three parts is the same as the coefficient of Jl['\ in [4961]
Une 28. In like manner, the coefficients of Af^, in the same hnes of these three
4 . [4961m
functions, are #, -\-i( — 24-m+c), -| • ; whose sum is the same as the coefficient
of .4i« [496Uine29]. Lastly, tlie term of [4908 hne 1], depending on Af\ is the
same as in [4961 line 28].
Thirteenth. The coefficients of -^^.62.003.2^, in [4866 line 6, 4901 line 10, 4908 line 1],
v2
are, respectively, 1, — -B[,"\ — 5, — A2°^ ; whose sum is as in [4961 line 30].
m
Fourteenth The terms of [4961 lines 31, 32] have the common factor
_2
— — . e'-*.cos.(2cy — 2v+2mv).
4a, '
Dividing the corresponding terms of the functions [4960e] by this quantity, we obtain, in
[4870 line 11], the terms i(6-\-l5m-\-3m^) ; and, in [4879 line 7], the terms — 2c.(l+m),
or, J( — 4 — 4m) nearly; the sum of these two expressions is i[2-\-llm-{-8ni^), as in
the first term of [4961 line 31]. The term in [4892 line 10] is the same as the second
temi of [4961 line 31]. The term in [4908 line 3], neglecting e'^, is 4^/", as in the
first term of [4961 hne 32] ; and the term of [4908 line I], depending on ./S';'", is the
same as in the last term of [4961 line 32]. The term [4934 line 12], is the same as that
depending on ./2o'"' in [4961 line 32]. Lastly, [4942] is the same as the term depending
on {A.'Y [496Uine32].
3
Fifteenth. The coefficients of ——.f.cos.2gv, in [48661ine7,4895,49081ine 1],
2 2
are, respectively, — Jm, l-(-e^— ly^^ -\-2in.Al^-^; whose sum is as in [4 961 line 33].
Sixteenth. The terms of [496 1 lines 34, 35] have the common factor
2
o ~
— .7^.cos.(2^îJ — 2v-\-2mv).
VOL. Ill 117
[4961 n]
[49610]
[iQGip]
466 THEORY OF THE MOON ; [Méc. Cél.
tial equa-
tion in u
continued.
[4961?]
[4961?]
[4961«]
— -J- . ^ \+e'^—\y^—\.M+2m.A'-p \ .y"". cos. (2gv—2ê)
■ 3-\-2m—2g {4g^—l) (2+?»)
33
34
3-M 4 '4(l-m) 2g—2+2m
+ 'yi 9«(") s/?<ia> }.7"~-cos.(2gv—2v+2mv—2è)
[4961] 4a, ) \^^i 0y/(i3) I ^-^a I ^° S'î
Difieren- m ° '
2
. a+g'+"+'-Y-""'>.4"-(10+5,»).<>) 37
Dividing the corresponding terms of the functions [4960e] by this quantity, we obtain, in
[4870 Hne 13], the terms J(3+2m) ; ia [4879 hne 10], the term — |^; the sum of
these two expressions is |(3-|-2m — 2^), as in the first part of [4961 line 34] ; the
remaining terms of this hne are given in [4892 line 12]. The term depending on B'-p
[4901 Une 2], that depending on A^^^^ [4908 line 1], and that depending on A^^^'^
[4934 line 13], correspond, respectively, to those in [4961 line 35].
n
o — "
Seventeenth. The coefficients of — ^.e'^.cos. 2c'mv, in [4866 line 8, 4908 line 1], are
f, —A.}^^\ as in [4961 line 36].
Eighteenth. The terms of [4961 lines 37, 38] have the common factor
— -— — .ey^'.cos. (2^« — cv).
Dividing the corresponding terms of the functions [4960e] by this quantity, we obtain, in
jg3)
[48661ine 9], the term J ; in [4901 line 4], the term --|- ; these agree with the two
m
first terms of [4961 line 37]. The coefficient of lA^\ in [4911 line 8], is 3+> ; in
[49181ine2], is +3m; in [4925 line 9], is — 2^— 2— ^m+c ; and, in [4934 line 14],
is — 14m; the sum of these terms is l-f-c — 2^ — 10??;, as in [49611inéî7]. The
coefficient of —^i'^\ in [4911 line 8], is 3+3m; in [4925 line 9], is — l+2m; in
[4934 hne 14] , is +8; the sum of these is 10+5??i. as in [4961 line 37]. The
coefficient of .^™, in [49111ine 8], is +f; in [4925 line 10], is nearly — f+m ;
and, in [4934 line 14], is +4 ; the sum of these is 5+m, as in [4961 line 38]. The
VII. i. ^9.] DIFFERENTIAL EQUATION IN M. 467
C -2 ^ 1— 2m ; ^"
M,-2.).(,+2e«+2e«)+?fc5tKttf±?^n
m" I (36+2l7n — 15m^) .,,-, , 3(l+m) .„o> ,„ I « , \ ao
rrt ; V — x^ ^ — y _ J"^+ ^ , <• J<'^^ e'2 ) . -, . COS. («— m?;) 42
' a ] 4(1 — m) ' 2(1 — m) " / a' ^
_(57— 33m) ,^ ^(,5) y^ \ ^g
4(l_„l) ^ -l-2-V^2 ^-"2 ^_2
Differen-
tial equa-
tion in u
continued.
term +.^{,'^' occurs in [4908 line 1]. Lastly, the terms in [4960 line 3], are the same
as in [4961 line 3SJ.
Nineteenth. The terms of [4961 lines 39, 40] have the common factor
— — . C7^.cos.(2« — 2mt) — 2gv-\-cv).
4a,
Dividing the corresponding terms of the functions [4960e] by this quantity, we obtain, in
[4870 line 15], the terms |+f m ; in [4879 line 12], the terms — i+i"*» the sum of
these is \-\-2m, as in the two first terms of [4961 line 39]. The terms in [4892 line 15],
by putting c=:l, ^=1, become - — ^ \- „ — , as in [4961 line 39]. The function
2R(4) [496K]
[4908 line 1] gives 2./3f ' ; and [4901 line 5] gives — ^ , as in [4961 lines 39,40].
In
The coefficient of Ai^^\ in [491 1 line 9], is +3; in [4918 line 3], is +4m ; in
[4925 line 11], is — 1; the sum of these is 2-J-4m = 2(1+2ot) ; and, by neglecting
2
m^, it may be put under the form - — ^ ; adding this to the term [4934 line 15], which
8 30
is nearly equal to - — 5—, the sum becomes , as in [4961 line 40] .
Twentieth. The terms of [4961 lines 41 — 43] have the common factor
— . — .cos.(t) — mv). [4961*']
a, a'
Dividing the corresponding terms of the functions [4960e] by this quantity, we obtain, in
[4872 line 1], the term | (l+2e2+2e'2); and, in [4892 line 16], the term -^-J|— T^ — L
these are the same as the terms of [4961 line 41], independent of [>■. The term depending
468 THEORY OF THE MOON ; [Méc. Cél.
4-^:— • < > • -,.e'.cos.Cv — mv4-c'mv — la')
[4961] 2a, ^ _ (5+^).^(.9, S " -iô
Differen-
S2. sd U(15-8m).(l-2.)-K76-33m).Jr' ) « , ^ ,46
+ 7; — ; — r— ^ • < > • --e .COS. (v-mv-c mv+zi').
2a,.(l-2'«) ^_ 5^(18) _(i_2^).jci9) ^ « 47
on fi, ill [49551ine 1], is — Ija; and, if we neglect terms of the seventh order, we may
connect it with the same factor as the other part of this term, putting it equal to
— f f-(l+~^^+~^'^)' ^s '" '^'is fit'st part of [4961hne41]; and, we may incidentally
[4961m] remark, that this factor might be changed into l+2e^+2e'^ — fy^ [4870r']. In like
manner, the term depending on (ji, in [4957 line 1], is — «—j ; and may be connected
with the corresponding factor l-j-fe^-j-^e'^, and then it becomes as in the last term of
[49611ine41]. The coefficient of — i^i'"', in [490Sline l],is +6 ; in [491 J line 10],
• is +9; in [4918 line 4], is 12m; and, in [4925 line 12], is — 3-j-3m ; the sum of
these is 12+15 m =—^^ — ; ; adding this to the term in [4934 line 161
1 — m ' 'J
^ '" — - = —^ , it becomes ^ , as in [4961 line 421. The coefficient
\—m 1— m 1— m '" -"
14961«'] of ^^f'.e'2, in [4908 line 9], is — | ; in [4911 line 10], is +f ; in [4925 line 12],
is — f ; the sum of these is — 3 = — j ; adding this to the term in [4934 line 16],
6 , , 3+3m 3(l+m) • , , r r^or-ii
namely , the sum becomes = -— , as m the last term oi [4961
^ 1 — m 1 — m 1 — m ^ •"
line 42. The coefficient of — i-î?', in [4922], is +24; in [4928], is — 9+9w ;
the sum of these is 15+9m = -r , neglecting ?re-; adding this to the term [4946]
— II — "—, nearly: the sum is — ^ , as in the first part of [4961 line 43]. The
1— m ^ \-m
terms in [4901 line 11], are the same as those depending on jB^"', BP' [4961 line 43].
Lastly, the coefficient of ix„, in [4913], is — 9; in [4925 line 12], is +9 — 9w; and,
in [4934 line 16], is 27^1— (1—mf | = 54m— 27m2 ; the sum of all these is 45m— 21m^;
the terms ±9 mutually destroy each other ; so that the whole term becomes of the order
[4961d] m---.K.'m, or of the seventh order, as in [4962].
Twenty-first. The terms of [4961 lines 44, 45] have the common factor
2
3m a . I / \
—- .— .e.cos.iv — mv-]-cinv).
2a, a
Dividing the corresponding terms of [4960(] by this quantity, we obtain, in [4872 line 2], the
Vll.i.§10.] INTEGRATION OF THE EQUATION IN u. 469
Wc have not noticed the terms multiplied by \ , because they mutually
destroy each other, except in quantities of the order m'' [4961»].
10. To integrate this differential equation, we shall observe, that, by
noticing only the parts which are not periodical, it gives,*
s
2
We have denoted this by M=-.(l + e-+i-/+[3) [4861]. Now, if we [4005]
[4!)f)2]
[4963]
[49C4]
[4961j)':
term -\-^; and, in [4892 line 17], the term f ; the sum of these two terms is f , as in the
first term of line 44 [4961]. The term depending on fx., in [4955 line 1], is — |fji ; and,
in [4957 line 1], is — ij- ; the sum of these two expressions is — Jja, as in the second
term of [4961 line 44J. The term depending on A\l^^ [4908 line 1], is as in [4961 line 44].
The coefficient of l^['~\ in [490Sline7], is —6; in [4911 line 11], is +3; in
[4925 line 1.3], is — 1+'« ; and, in [49-34 line 17], is +8; the sum of all these is
4+m, as in [4961 line 44]. The coefficient of —.^i"", in [4911 line 11], is -f-f. in
[49-25 line 13], is — 5+»»; and, in [4934 line 17], is -f-4; the sum of all these is
(5 -(-;«), as in [4961 line 45].
Twenty-second. The terms of [496 1 lines 46, 47] have the common factor
_2
r — -- — -— - . — . e .cos.(t) — inv — cmv).
2a,.[l — im) a ^ '
Dividing the corresponding terms of the functions [4960e] by this, we obtain, in [4872 line 3],
the term f — ^?n; and, in [4892 line 18], the term f; whose sum is :f( 15 — 18m), as in
the first term of [4961 line 46]. The term in [4955 line 2], is — %{\ — 2ni).v.; in
[4957 line i], is — 3(i; whose sum is — (^ — 9/«) (^ = — |(15 — 18m).2.a, as in the
first part of line 46 [4961]. The coefficient of — i^5^''\ in [4908 line 8], is 6— 12ot;
in [4911 line 12], is 21— 42m , in [4925 line 14], is — 7+21/«, neglecting m^ ; in [49Glti']
[4934 line 13], is +56; the sum of these terms is 76 — 33m, as in [4961 line 46].
The coefficient of —A^f^, in [4911 line 12], is |— 3m ; in [4918 line 5], is +2m,
nearly; in [4925 line 14], is — J+m ; in [4934 line 18], is 4; the sum of these
tenus is -(-5, as in [4961 line 47]. Lastly, the coefficient of Af^ [4908 line 1], is
the same as in [4961 line 47].
* (2848) The equation [4961] being linear in u, we may compute the terms
VOL. III. 118
470 THEORY OF THE MOON ; [Méc. Cél.
[4966] neglect the sun's action, we shall have - = - [4864] ; so that we may
[4967] suppose* (3 = |3" ; therefore, we shall have.
Equation
between
[4968]
Equation 2
between - - __ q — o
The action of the planets produces a variation in the excentricity of the
earth's orbit e', without altering its serai-major axis a', as we have seen
in [1051', 1122, &c.]. Therefore, the value of - suffers corresponding
2
Q — / 2
[4969] variations on account of the termf — - — '- — [4968], which it contains.
4a.
depending on each angle separately; and, if we put A for the constant terms of that
equation, we may compute the corresponding part of u by means of the equation
diJ u . .
[4963a] 0 = -p- -\-%i+A;
which is evidently satisfied by putting u= — A. Hence it follows, that the constant part
of M, is the same as the constant part of [4961], changing the signs ; this agrees with
[49636] [4964]. We may remark, that it is not necessary, in making this integration, to add an
arbitrary constant quantity to — A ; because it is implicitly included in the arbitrary
quantity a, or «, [4860,4864].
[4964a] * (2849) If we neglect the sun's disturbing force, we have o,= « [4864] ; and the
[49646] expression [4964] becomes, in this case, «= - .^ l+e^-f"î7^H"(3"l • Comparing this with
the assumed value of the constant part of u, in the same hypothesis ; namely,
M= -.{l+c^+iy^-f-p^ [4861], we get p = 3" [4967]; which is to be substituted
r4964c] '" t'^^ second member of [4964]. We must also substitute, in the first member, the value
of M = - . (l+e^+iy^+p) [4860, 4861] ; hence we get,
[4964d]
i.(l+e^+i7^+p)=i.(l+e2+iy^+3)-£-.(l+e3+XyS+3e'3)
2
+~ . (4-3m-m-2).^i»>.(l-|e'^)- £■ . (5i°')V^-
Dividing this by l-{-e^-\-ly^-\-p, and neglecting terms of the sixth order, we get the
value of - [4968].
t (2850) The variation of the term
Vn.i.§10.] INTEGRATION OF THE EQUATION IN M. 471
Moreover, as the constant term of the moon's parallax is proportional to -,
" [4970]
it is evident, that it must suffer a secular variation ; but, upon examination, it
is found always to be insensible.*
The part of it, depending on cos.(cv — 3), is represented, in [4826], by [4971]
-.(1+e-). COS. (ci' — zi). If we substitute it, in the equation [4961], and then
compare the sines and cosines of cv — w, neglecting quantities of the
d^'
order — -, which can be permitted, considering the slowness of the [4972]
secular variations of the earth's orbit, we shall obtain the two following
equations ;t
_2 _9
_3m^ [4969], is -|.^.e'Je'=— #."!e'.cV [5094]; [4969a]
therefore the whole value of -, is to this variation, as 1 to —§m^.e'.Se'. Substituting
a
the values of m, e' [5117]; also 25E, or 25e'=—t.0\l87638 [4330]; or, in
parts of the radius, 6e'= — i.0,00000045, nearly; we get
_2
7/Î
—i.-.e'.Se'=t. 0,00000000006, nearly ; [49696]
which, in 1000 years, will not produce a single unit in the seventh decimal place of the
moon's distance from the earth, taken as the unit of distance. If we multiply this
expression by the constant term of the moon's horizontal parallax 3424%16 [5331], we [4969c]
shall obtain the secular effect on the parallax, equal to t0*,0000002; which will not [4969^1
amount to a second in a million of years. We may remark, that the similar term of [4968],
depending on «4^"', is much less than that we have estimated, as is evident from the
smallness of the value of A^ [5157].
* (2S51) We shall see, by the estimate made in [4987A — I], that this quantity is |..g„^ ,
insensible.
t (2852) If we put for a moment, for brevity, E=: , and use the values of [4973a]
p, q [4975], we shall find, that the term, depending on cos.(ct — w), in [4961], is
jE.(— p— 9.e'^).cos.(c« — «), and the corresponding part of the equation [4961], is [49736]
0 = ~-\-u-\-E.{—p—q.e^).cos.cv—w). [i973c]
472 THEORY OF THE MOON; [Méc. Cél.
e.(l+e^) dd-a ^ / d^\ 'C' a S
[4974] 0 = 1 — (^c— — J —p—q-e ;
the quantity — p — q.e"^, being supposed equal to the coefficient of cos.(cv-ts),
[4975] (l+e^).e
in the differential equation [4961], divided by — ; where we must
observe, that the values of A^°^ , Jf-", Bf\ and Bf^ contaha already
[4976] the factor 1 — |e'^* The equation [4973] gives, by integration.
If we consider e, zs, as variable, and c constant [4986], we may satisfy this equation
by assuming for u, an expression of the same form as in the purely elliptical hypothesis,
which is ^i = E.cos.{cv — ra) [4826,4973«] ; substituting this in [4973c], we get,
ÇddE , , , ^dB A.\cas.{cv—tz)\ rf2.5cos.(cv— ej)^ ^
0= ■'i-— .cos.fCT— w)+2.— . , —\-t.. — \
Idv^ ^ ' dv dv ' t/l'2 >
[4973 e]
-\-E.cos.{cv — ■!s)-{-E.{ — p — 5'.e'®).cos.(c(.' — is).
Now. by neglecting quantities of the order mentioned in [4972], we may reject ddE, and
we shall also have,
^^^"^^■^^ rf2.^cos.(c«— ^)^ dd^ . . . ( d^Yî
hence, the equation [4973e] becomes,
l.W3.lO=<E.^-<c-'£).f^.,in.(..-») + 5E-E.(c-3V.(-l-î..-)^c«.(..-=,.
To satisfy this equation, for all values of cv — w, we must put the coefficients of the
[4973/t] sine and cosine of c«— sJ, separately, equal to zero. The first of these conditions gives,
without any reduction, the equation [4973] ; the second, divided by E, gives [4974].
* (2853) The chief terms of ^'f, A['\ deduced from [4998,4999], evidently
contain the factor 1 — |e'^, and the expression of B[''\ obtained from [5062], contains
[497 a] jg^.j^^g ^^.jjj^ jj^g g^i^g factor ; by this means it is introduced into the equations [5064, 5065],
from which 2?L-\ -B„^' are derived. Hence, it appears, that the quantities A':,"\ A[^\
.fiP) jç(3)^ which occur in the coefficient of cos.(cr— ra) [4961], contain the factor
1 — |.e'2, as in [4976]. We see, in this article, [4982,&ic.], the importance of retaining
VII. i. § 10.] INTEGRATION OF THE EQUATION IN u. 473
I A;.e''.(l+e2)2
dv
[4977]
k being an arbitrary constant quantity*. Neglecting the square of g.e'^
we obtain, from [4974] ,t
d'us , , i<7.e'^
_=C-v/l-p + p.^= [4978]
Therefore, if we consider p and q as constant, which we can do here,
without any sensible error,J we shall have, by putting g''= ,~^=, [4979]
the term depending on e'^, of which we have aheady spoken in [4910o] ; since the secular
inequalities of the moon's motion depend on this quantity [4984, &ic.].
* (-2854) We shall put for a moment, c — y z=W, and then, by taking its differential, [4977o]
we get T^= r- Substituting these values, and that of E [4973«], in [4973], [49776]
we obtain
^dW „,dE dW ^dE
0 = -^-7-^^^-lÂ^' '' -F = ^-Ë- t4977c]
Its integral is
log.-^ = log.E2+log.fc, or - =A:.E2, asin[4977] ; [4977c']
k being the arbitrary constant quantity. This satisfies the first of the equations of condition
[49731; and, if we deduce from it the value of JV=c — , and substitute it in the
■- -■ dlf
second of these equations [4974], it becomes,
0=1 "^ p-n.e'^ [4977rf]
This might be satisfied, if all the elements e, e', 7, &;c. were invariable, by taking the
arbitrary constant quantity Ar, so as to correspond to these elements ; but e', or E [4977e]
[4330], being subject to a secular inequality, it will produce secular terms in the value of e.
deduced from [4917(1].
t (2855) From [4974], we have
' - S = ^(^-^-9-^") = \/(l-i')-^) + ^- t4978a]
If we neglect the square and higher powers of q.e'^, it becomes, by reduction, as in [4978].
X (2856) The quantities p, q [4975], are functions of e, y; whose secular variations are [4979a]
VOL. III. 119
474 THEORY OF THE MOON ; [Méc. Cél.
[4980] ^ = cv—v \/T^ +lq'-f<i' '• dv + £ ;
£ being an arbitrary quantity*. From this equation Ave get,
[4981] cos.(cz) — ui) = cos.|t'y/l— P — iq'.fe'^.dv — £|.
Hence it follows, in conformity iviih observation, that the lunar perigee has a
motion, lohich is represented by
[4982] (\ — ^1 — p).v-\-\q'.fé^.dv=^vî\oûo\\ of the moon's perigee.
This 7notion is not uniform on account of the variableness of e' ; and, if we
suppose, in counting from a given epoch, that e' is represented by
[4983] e'=£'+/t;+/D^ [4330,&c.];
Motion E' being the excentricity of the earth's orbit, at the same epoch, the motion
of the perigee ivill bej
of the
moon'â
perigee.
[4984] {l—^îZ:^-{-^q'.E'^).v+lq'.E'fv^ + ^.7'.(2E7+/2).«;3 =: motion of the moon's perigee.
insensible [4937,5061]; we may, therefore, consider p and q as constant quanthies, in
makbgthe integrations.
* (2857) Muhiplying [4978] by civ, integrating, and substituting q' [4979], we
get [4980] ; whence,
[4982a] cv—zi = v\/{l—p) —\(fft' ^.ffo— £ ;
whose cosine is as in [4981]. Now, we have supposed, in [4971, &ic.], that cv — a
represents the moon's anomaly, and v the moon's motion ; their difference is
[49826] v — v^{\ —p) + 1 q'.fe' ^. dv+s ;
so that, while v varies from 0 to v, the corresponding motion of the perigee is
represented by
[4982c] v—v\/{l—p)+^q'.fc'^dv;
the integral fe'^.dv being supposed to commence with t)=0. This is easily reduced
to the form [4982].
1(2858) By using the value of e' [4983], we obtain,
[4984a] fe'^dv=^fdv.\E'^-\-2E'f.v-\-{2E'l+P).v^-\-hc.l=E'^v+E'f.v^-{-i{2E'l-lrr).v^-{-iic.;
substituting this in [4982], we obtain the expression of the motion of the perigee [4984].
The part of this expression, depending on the first power of v, represents the mean motion
of the perigee, which we have put equal to (1 — c).v [4817] ; hence we get
[49846] {l—c).v = {[—\/T=^ + hq'-E"').v.
VII.i.>5>10.] INTEGRATION OF THE EQUATION IN M. 475
This expression may be used for two thousand years before or after the epoch
[4984/, /]. The part of it, included in the following formula, expresses the '■ ^
secular equation of the motion of the perigee, which is decreasing from age
to age [5232] ;
^q'.E'.fv''-+}^.(2E'l+f^).v^ = secular equation of the perigee [4984(Z]. [4985]
The value of the constant quantity c may be represented by vaiueof
c = v/ÎH]^ — I q'. E ^ [4984c] ; [4986]
the angle « is then equal to the constant quantity s, increased by the secular [4986']
equation of the motion of the perigee [4985].*
Secular
variation
The cxcentricity e of the lunar orbit is subjected to a secular variation, similar "nseLib'e.
to that of the parallax, and like it is insensible\ [4970] ; these variations [4987]
Dividing by v, and reducing, we obtain
c=v/(l— i')-è2'.-E'' [4986]. [4984c]
The remaining terms of [AQ8A'\, defending on v^, v^, give the secidar motio7i [4985] ;
in which temis of the order v* are neglected. To make a rougli estimate of the value of [4984rf]
these neglected ternis, without the labor of a direct calculation, we shall observe, that the
secular motion of the perigee is about three times as great as that of the moon's mean [4984e]
motion [5-235] ; and this last quantity is very nearly represented by 10'.j^-|-0*,018.i^
[5543] ; Î, being the number of centuries elapsed from the epoch of 1750. If we
suppose i = 20, corresponding to 2000 years [4984'], these two terms, of the orders
t'^ v^, respectively, will become 4000% 144*; which are nearly in the ratio of 28 [4984g]
[4984/]
to 1 ; and, if the term of the order v'^ decrease in the same ratio, it will become , or
28
5% nearly. Now, a term of this order, in the secular motion of the moon, or one of three [4984A]
times that value in the motion of the perigee [4984e], is wholly undeserving of notice in
such distant observations; and, we may, therefore, restrict ourselves to the terms of the
orders v^, v^, included in the formula [4984]. This is conformable to the remarks of the [4984i]
author in [4984'].
* (2859) Substituting the values of c s^nd fe'^dv [4986,4934a], in [4980], we
get as in [4986'],
^=-= + {J9'--E'/»-+^?'.(2£'/+/2).î,3] = ç^ secular equation [4985J. [4986o]
t (2860) Using the value of q' [4979], we get, successively, from [4978, 4983, 4986],
by neglecting terms of the order I and f^,
476 THEORY OF THE MOON ; [Méc. Cél.
being proportional to — , which become sensible only in the integral
[4987'] "
[4988] If we represent any termwhatever of the equation [4961] by -.cos(m[3),
and denote the corresponding part of u by
'-'S = V{l-P)-W- e" = v'{l-p)-hq'-{E'^-+2Efv)
[4987a]
= ^{l-p)-hq'.E'^-q'.E'fv=^C-q'.E'fv.
Substituting this in [4977], and neglecting e*, e®, in its second member, we get
[49876] e^=^. J^^, or e = ^^^ . (l + -^), nearly ;
consequently, the secular variation of e is represented by
[4987c] ^, = __.1_^, or 5e = e.^..;
observing, that, if we neglect this secular variation, we have, very nearly,
[4987c'] e=— ^ [49S7J], and c = l [4828e].
If we compare this with the chief term of the secular motion of the perigee, which we shall
[4987rf] represent by fe=|ç'.jEyi;** [4985], we shall get 5e = -.5îï. Now, from [4984e,/], we
have, by neglecting the signs,
„„ ., , SeO"' . 1396000'.i , , , SOm.f.t
[4987e] ôîJ:=30\i^ and v= .i = , nearly; hence & = ^ ;
■■ •• m m ' •" 1296000 '
and, by substituting the values of m, e [5117,5120], it becomes
['^987/] 5e=i. 0,0000001, nearly.
[4987g-] "^'^'^ •^ wholly insensible, since, in 20 centuries, which corresponds to i=20, it only
amounts to 0,000002.
If we retain terms of the order v^, in the calculation of e [49S7J], its value will be
[4987ft] increased by a term of the form al^.v- ■. /^ being of the same order as /-, or I; and
[4987i] this value of e gives d-.- = l, . Hence it appears, that the quantities neglected in
[4972] are of the order f^, or /. Now, we have seen, in [4987/], that the expression
[4987ft] of the part of (5e, depending on the first power of f, is insensible; and, by proceeding
as in [4984 e — i], it must be evident, that these terms of the second order /^, or /, will
[4987i] be still less, and may, therefore, be neglected, as wholly insensible, even in the most ancient
observations.
VII.i.^10.] INTEGRATION OF THE EQUATION IN u. 477
« = P.cos.(iv+^)+Q.sm.(iv+^), [4989]
we shall have the two following equations to determine P and Q ;*
Q =
[4990]
\ ' (h / dv (Iv^ [4991]
The variations of (3 and P being extremely sloio, and i very great,
relatively to —, the value of Q is insensible [4990?], and we have, [4992]
dv
from [4990],
P- —ry — :ii;^ r ; [4993]
in which we must observe, that, as « + — is the coefficient of dv, in
dv
* (2861) If we substitute the assumed value of u [4C89], in
ddu H
^='Â7a+"+^-'=°'-^'''+'') [4961,4988], [4990a]
supposing V, P, Q, p, to be variable, it will become as in [4990J] ; observing, that
3 is composed of terms of w, -a', kc, similar to [4986a] ; and P, Q, of terras
e, e', y, &c., whose secular variations are similar to that in [4987/] ;
To satisfy this equation for all values of the angle ««+(3, we must put the coefEcients of
sin.(u'+p), cos.{u-\-^), separately, equal to nothing ; hence we have,
«=i-('>in-'?-'^-(.-+£)--*+^?-
If we neglect the term ddq [4990fZ], which is very small, as we shall soon see, and
VOL. Ill 120
478 THEORY OF THE MOON ; [Méc. Cél.
the differential of the angle rà+f3, ive may suppose (3 to be constant in
that angle, provided we take, for i, the coefficient of v corresponding to the
epoch for which the calculation is made. Thus, we shall determine the
[4995] coefficients ^f , A['\ Sic, in the expression of a ou.
Relatively to the terms, where the coefficient of v differs from unity, by
a quantity of the second order, and which depend on the angles
[4995'[ '2-g V — c V — 2(3 + * and v — mv-{- c'mv — a',
the consideration of the terms, depending on the cube of the disturbing
force,* becomes necessary ; but, by carrying on the approximation as we have
[4996] done, to quantities of the fourth order inclusively, the terms depending on
the cube of the disturbing force, which might become sensible, will be found
to be included in the preceding results.
This being premised, if we substitute, in the equation [4961], instead
of u, the following function ;t
divide the remaining terms of that equation by the coefficient of Q, we get its value [4991].
Now, the secular variations of 3, P, being of the order i5ra, Se, fee. [4987e,/, &ic.],
r4990el t^i^y must be very small ; and their products and differentials, which occur in the expression
of Q [4991] must, therefore, be insensible. Neglecting the quantity Q, and the second
differential of P, in [4990c], it becomes as in [4990] ; which is easily reduced to the
form [4993].
* (2862) Terms of this kind have been noticed in the differential equation in ?*. Thus,
for example, the term multiplied by m . {A[^'')^, in the coefficient of
[4995a]
cos. {2cv — 2v-\-2mv — 2u) [4961 line 32],
is of the order of the culte of the disturbing force; because m, A'^\ are each of the
same order as the first power of this force.
"f (2863) The function connected with 5u [4997], is the same as the value of m
[4826], augmented by the term p, of the fourth order [4858, &,c.], and taking the
[4997a] coefficient of cos. (2^!; — 2ê), so as to include terms of the fourth order. These neglected
terms are easily computed. For, in the first place, the term s* [4812a] introduces
the factor 1 — ^y^, in the coefficient of cos.(2« — 2^) [4816], by which means it is
changed from —if [4816] to —lf.{l—ly^) [4S12a]. The same change being
made in the coefficient of cos.(2^« — 26) [4819], it becomes, by using [4S23c],
[4997t]
VII.i.§10.] INTEGRATION OF THE EQUATION IN M. 479
1 ( 1+e''+i7^+^-\-e.{l+ee).cos.(cv—^))
u = - .{ }+6u; [4997]
" ( -i-r.(\+e'-if)-cos.(2gv-2é) S
the comparison of the different cosines will give the following equations ;*
^ TA7''-(i-i7'')-'^os.{2gv—26) = -l.{\+e%iy^.(l-iy^).cos.{2gv-2è)
= — - -iy^- ( 1 -\-e^—lf).cos. {2gv—2ô) ,
[4997c]
as in [4997].
* (2S64) If we take the value of i, corresponding to the epoch, as in [4994], and
neglect the variations of d^, we may put the equation [4990] under the form
0=11— i^.P-\-" ; [4998a]
or, as it may be written,
0=\l-i^\.Pa+-.H. [49986]
Now, muhiplying [4989] by a, and neglecting Q, as in [4992], we get, for nu, the
expression am = Pa.cos.(«i;-f-p), corresponding to the term — .cos.(Jv-|-|3) [4988], in [4998c]
the equation [4961]. Hence, it appears, that the coefficient Pa, corresponding to any
angle ù'+p, is found by muhiplying the expression [4997] by a, and substituting the
value aSu [4904]. These values of Pa, together with the corresponding ones of
H [4998rf]
— [4961], being substituted, successively, in [499S6], give the equations [4998 — 5017].
For the constant part of au [4997] ; namely, l+e~+T7^+(3 satisfies the equation [4998e]
[4961], as has been proved in [4964 — 4968]. The term of «2/. [4997], represented by
e.(l+e2).cos.(cî; — to), satisfies the equation [4961], as in [497.3f/, Sic.]. The term of au
[4997], depending on 2gv—2ê, is \—iil-i-e^— iy^)-{-A^'^^.y^cos.{2gv— 2è) ; hence,
P«=[_i(l+e2-^y2)+^a'i,| .
the corresponding value of H, [4988, 4961 line 33], is
[4998/]
[4998ff]
H=^—i\{l+e^—iy^)—i7h+2vi^i'^~^}.y^; and i = 2g; [4998A]
substituting these in [49936], and dividing by 7.2, we get,
= (l-4^=').^«|^-(l+e^-i.^) • { l-g'+h^ ] + f A^. Jè--2»U'^' } ■ [4998A]
480 THEORY OF THE MOON ; [Méc. Cél.
■l+(l+2m).e^+i?^-|e'^'
[4998] 0 ={l-4(l-m)^}.4^'+|m. - . / -' ^,_7; —
— 4»'— (5<;'— 5f'>). ^
2
Equations \ '^
for the de-
irri. / ic . { 1 +i(2— 1 9»i).e^— |e' ^ \
\ -|(3+4m).(l+ie^-|0+^
[4999] 0=ll-(2-2m-c)^i.4"+3//»'.^{ 2(l+m) ,
2— 2m— c ■'' "^* ^ ^
-i. ( j(;)— 2 jf )+i. (5f -5f ). ^
[5000] 0 = { 1— (2— 2^+0=^] .Af—irn.- . \ 3+c— 4m+/^^~"^^ +2Af' I ;
[5001] 0 = n-(2-«0'}.4''-|m'. ^ ) i^+25f>;4+2-4f' V ;
[5002] 0={l-(2-3m)^|.4^)+|m'.- Jl^M_25i"').4-24^' > ;
' 7/i _)
,,.,3, o=(i-„..).4"+iM.î .} -'^:?ii:|3|f' ■4"-^.»-(^-^»')-^=
m
+6m.^44c>+4^='-4'^'-10.J/'^e^+|(J/'>— J/''").6^|;^
[4998i]
On account of the smallness of the terms 1 — g^, -' [4828e, 4968], we may change
the factor 1+e" — {y" into 1, or - [4968], and then the equation becomes as in
[5010]. The rest of the terms of au [4997] depend wholly on a&u ; therefore, the remaining
terms of Pa [4998c, h], will be represented by the coefficients of aSu [4904] ; and,
by takingthem in the order in which they occur, we shall obtain, with but very little reduction,
the equations [4998—5017].
f5003ol * (2865) This line might, for greater accuracy, be multiplied by the factor -, like the
Vn.i. §10.] INTEGRATION OF THE EQUATION IN u. 481
0 = n-(2-»l-c)^l.Jf>+|/«.-.<; ^3^„_ 4 ^^ ^; [5004]
^7(S+6m-c) , 7(2+3m) 3
2X4 "'"2— 3nj— c~f~^ '
(+-'^' 1-^ 2 +2-3m-cS*^'
" I 2 ) 4 c+OT S ■ ' ,
2 ' ( 4 c-m ) '
0 = |l_(c-m)^^4'^)_|m.-.<^ y; [5007]
0 = (l_4c^).^f>+|m . - . ^ 1—51"'.^-— 4'»' S ;
7A
'(2+llOT+8m2) (l0+19m+8m-)
[5008]
2 2c 24-2»»
C^*^' ^ 2c-2+2m ^^> ;
0 = jl-4-^Mf'+^ . |â'-l-|-(-^) + f'«-|^'-4''' I [4998A;,/]; [soiO]
S-\-2m—2g (4g^— 1) (2+m) s
2 „ \ 4 ~*"4(l— m) 2g—2+2m I
0 = i 1— (2§'— 2+2my-i .Jf '+^m . - . < g^o, q^ (,2) } ; [son]
[5012]
other terms of these equations, as is evident by comparing it with the corresponding terms
of [4961 line 21].
VOL. III. 121
482 THEORY OF THE MOON ; [Méc. Cél.
[5013] o=ll-(2g-cyi,Ai-^-i^\0 '" ^<„,^,,>
[5014] 0 = I l_(2-2m-2^+c)^i J.f'^'-fm. - . j^ g^,, ^q_^,,,
^ _2 ' 1— 2m
m
/|.(l-2,a).(l+2e^+2e-)+ifc?Î^Hl+t!!±2!:i)'
r ^ ^ ^ ^^ 4(1— m)
2^ ) (36+21m— 15?ft2) ,,,,,, 3(1+7») ^,,,, ,
[5015] 0= 1 i-(i-m)^Mr"+^.M ^4(ï=;« )— -• ^r^+ 2(ï=;^)- ^o'^ • ^"
[5016] 0 = 5.(1— 2f.)_^^,3,^(4+m) _ ^(,,,_ (5+^). j(i9) .
3,«' « ( Klâ-8m).(l-2F-)-i(76-33m).^S-
11. fVe shall noiv take into consideration the equation [4755]. The
function
which occurs in this equation, produces the terms,*
* (2866) Multiplying the equation [4808] by —^5 also [4810] by _-p— ^;
** and taking the sum of the products, we find that the first member of the sum is equal to
the function [5018] ; consequently, the second member of this sum will express the
a
[50196] development of this function. The first terms of these products, uith the divisor (l+ss)^,
mutually destroy each other. The remaining terms of this sum, being wriuen down in the
order in which they occur, without any reduction, become
VII.i.«^ 11.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN .. 483
These terms are successively calculated in the following manner. The
quantity ^ \, ^ becomes, by development,^
[5019]
[5020]
1
— 2e.s'm. (gv-\-cv — è — ro) I 2
Sm'.u'^s J' a ] — 2e.sin.(^W — C'U — é + a) f 3
==|/».-.7./ ). . [5021]
■2h^.u* ^ a, ^ -|-|-e'.sin.(§-îJ+c'mzj — ^ — ^') [ 4
+ fe'.sin. (^v — c'mv — è + Ts') \ 5
— fe^ sin. (2ciJ—^j; — 2^ + 0/ 6
'.ii^ . I l+3.cos.(2«— 2«') I + ^|g^ .{ (3-4*2).cos.(»;— i;')+5.cos.(3i.— 3«') | [5019c]
If we neglect the terms of the order s^, and connect together the other terms, it becomes
as in [5019].
* (-2867) LTsing always the abridged notation [4821/], we have ^s = ^y.s'm.gv,
nearly [4818]. Multiplying this by the function [4884], and reducing the products by rgQgQ^n
[18, 19] Int., we get tlie following expression, which corresponds, line for hne with the four
first lines of [4884], neglecting terms of the fourth order ;
(1— Je^— i7^)-sin.^u \ 1
3s _ \—ie-{i—ie^—h7^)-\sin.{gv+cv)+sin.{gv—cv)U 2
•~ "•''■ l+Ae2.|sin.(2ci;+^i;)— sin.(2cy— ^y)^ (' 3 ^^"^°*^
+ï7^-l — sin.^D-|-sin.3^!;)| / * 4
The coefficients ofsin.gy, between the braces, by connecting the terms, become l-^e^-fy^. 5
Multiplying together the two expressions [4866, 50206], we get [5021]. The detail of
the calculation is in the following table [o020d — ■/] ; in which the first column contains the
terms between the braces in [.5020e], the second, the terms between the braces in [4866], [5020c]
the six remaining columns contain the coefficients between the braces in [5021], corresponding
to each of the smes, marked at the top of the columns [5020f/]. The sums of the coefficients
[5020/], agree with the coefficients between the braces in [5021].
484
THEORY OF THE MOON ;
[Méc. Cél.
3m'.M'3.s
15022]
[5023]
-l vyy H Ç
The development of ^^^-^.cos.(2î;— 2tj'), is obtained bj multiplying
/ o,'3
, , - 3m'. u'
the value of — — —
and we shall have,*
cos.(2tJ— 2îj'), which we have given in [4870], by
Sm'aP.s
2h^Aâ
.cos.(2w — 2v')=pii
^a
/— I l+2e2-|.(24-m) .y2_ 6 e'2 1 .sin (2i»-2mu-^D+a)
+sin.(2î) — 2mv-]-gv — é)
— 2.(l-j-OT).e.sin.(2y — 2mv-{-gv — c«— ^-j-w)
-|-2.(l+m).e.sin.(2î) — 2mv—gv — cv-\-ô-{-a)
+2.(1 — m).e.sm.{2v — 2mv—gv-\-cv-^ê — to)
— 2. (1 — m) . e .sin .{2v — 2mv-^gv + cv — 6 — zs)
— 5 . e'. sin.(2t) — 2mv~gv—c'7nv-\-ê-\-zi')
+ J . e'.sin.(2i> — 2mv-\-gv — c'mv — 6-\-zs')
+|.e'.sin.(2« — 2mv — gv-\-c'7nv-\-ê — tn')
— ^.f'.sin.(2y — 2mv-\-gv-\-c'mv — ê — ra')
, sin.(2D-2mv-2ci'+ffD4-23T-,
-fsin(2ei)+g^-2i)+2nit)-2«-:
l+J(10+19m+8m2)e
2
3
4
5
6
7
8
9
10
11
12
(Col. 1.)
[5020d] ''«'"" »'' [50206].
[5020e] -ie.sm.igv+cv)
-^e.sin.(gi; — cv)
-le^.sm.{2cv-gv)
[5020/]
fCol.S.)
Terms of [5021],
having the
common factor
|.m .;,.}
•
Terms of [4866].
sin.ffo
sin{gv-\-cv)
siii(^u — cv)
sin.(gv-\~c'mv)
sin.igv — c'ïfiv)
sin.(2ci' — gv)
l+e2+|72+fe'2
l+le2_^j.2+3,/2
— 3e.cos.cy
—ie
-ie
-\-3e'.C0B.c'mv
+fe'
+¥
+3e2.cos.2c!;
-ie-^
-\-iy^.cos.2gv
■ -17^
1
.
-¥
— 3e.cos.CD
+fe2
1
-ie
— 3e.cos.e«
+f«^
-ie^
1
-i«^
Sum
l+2Ê2-ly2f3c/2
—2e
—2e
+¥
+¥'
-Je^
3«
(2868) Multiplying one third part of the expression of — [5020i], by that of
VII.i.§ll.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN s. 485
The term
33 m'.u"^. s
.cos.(t' — v') [5019], produces the following ;
[5024]
?"''^- .cos.(2u— 2w') [4870] , we shall evidently obtain the value of ^j^ .cos.{2v-2v') ; [5023o]
which we shall find lo agree willi the expression [5023], as will appear by the following
calculalion. If any term of [5020i] be represented by Say.A.sm.V, and any term of
[4870] by — .^'.cos.F, one third part of the product of these two terms, or the
corresponding pari of -|^^.cos.(2iJ— 2f'), will be represented by
im'-.y.AA. sin. V. cos. V'= | m'. - .y. \ AA'. sin.( F+ V')+AA'.sm.(V-F') | ;
where the factor ¥ /» . -.y is the same as that without the braces [5023] ; consequently,
the terms between ihe l)races [5023], must be represented by the function
AA.Ûn.{y-'rV')+AA!.ûxi.{V—V') ; or AA'.sm.(V+V')—AA'.sm.{V'—V) ;
^.sin.F rcp-cscnting ihe terms hf.iwecn theùmces in [5020i], and A'.cos.V ihe terms
hetiocen the braces m [4870]. By means of this formula, we may compute the terms
between ihe braces [5023] in the following manner.
[50236]
[5023c]
[5023rf]
[5023c]
[5023/]
-1— Je2_[.,y2 .|..,,2
_3«3
1 !
2
3
4
Sum=— 1— 2e2-]-iy2^|„,y2^Ae'S2 5
First. The coefficients of sin. (2c — 2mv — gv) are contained in ihe four lines of ihe
annexed taljle. The first is obtained by
combining (1 — Ic'^ — i-/).sin.,^'-j; [5020ilino5]
with (]+c2-fiy2_5.c'2).cos.(2y— 2///y) [4870]
line 1, and using the second term of [5023c].
The second is produced by sin.^« [5()20'> linel]
and i(3-}-2,w).-/2.co3.(25-w— 2i)+2»««) [4370]
line 1.3. The third line is produced by
— ^e.s\n.(gv-]-cv) [5020/- line2] and — fe.c.os.(2«; — 2mv~'rcv) [4370 line 3]. Lastly,
the fourth line is produced by — 2'^.sin.(^f-CT) [5020iline2] and —§e.cos.(2v-2mv-cv)
[4870 line 2]. The sum of these four terras is given in line 5, and is the same as in
[5023 line 1].
Second. The term s'm.gv [5020/; line 1], combined with cos.(2j; — 2mv) [4870 line 1],
and using the first of the forms [5023f], gives [5023 line 2].
Tiiird. The terms of [5023 lines 3 — 6] aie computed in the following table; in which
the first column contains the terms of A.s'm.V [50206]; the second, the terms of -4'.cos.F'
[4870] ; the remaining columns contain the corresponding terms of [5023e], connected with
the sines of the angles marked at the top of these columns [50237i."] respectively. The
VOL. III. 122
I5023g]
[5023fe]
[5023;]
486
THEORY OF THE MOON ;
[Méc. CéL
[5025]
* 3
33 m
16
a, a
- .7. 1 sin.(g-« — v-\-mv — '^)-\-sm..{gv-\-v — mv — è) j .
[5023n]
[5024o]
sums of these terms, in the bottom line of the table, agi-ee with the coefficients in
[5023 lines 3—6].
[5023i]
(Col. 1.) A.ain.y
[50206].
(Col. 9.) A'.cos.r'
Corresponding terms or[5023e orSOaS].
[4870].
ain (2y-2;;iy-j-^-cu)
sin(au-2;nu-£-o-cy)
sin(2y-amu-^i*-|-ci')
sin(ao— 3mc+|T)-[-o,)
sin.g-u
-à(3+4m)e.cos(2i>-2mt)-cw)
(— 1— 2m).e
(f+2»t).e
[5023A-']
sin.gT
-he.sm[gv-\-cv)
-i(3-4m)e.cos(2«-2nw-t-cv)
cos.(2t; — %nv)
[h )-e
(J— 2m).e
(-i+2m).e
(-4 ).e
-ïe.sm.{gv-cv]
cos.(2ti — 2mv)
(-^ ).e
(i )-e
[5023^]
Sums
(— 2-2ni).e
(2+2H0-e
(2-2m).6 1 {_2+2;n).c
Fourth. The term sin.^y [50206] combined with +Je'.cos.(2»— 2?mj; — c'mv)
[502:3m]
[4370 line 4]
gives, by [5023e], the
terms in [5023 lines 7, 8]
. In like manner, the same
[50241]
term sm.gv, being combined with — |e'.cos.(2y — 2mv-\-dmv) [4870 line 5], gives
[5023 lines 9, 10].
Fifth. The terms [5023 lines 11, 12] are computed in the following table, which is
arranged in the same manner as that in [5023A-'] ;
(Col. 1.) A.sm.V
[50206].
sm-gv
— .Je.sin.(gn)-j-TO
— ie.sin.(g-w — cv)
— |e2.sin(2cf — gv )
-\-^e^.s\T\[2cv-\-gv )
(Col. 2.) A'.'ios.V
[4870].
K6-}-15m+8m2).ea.cos.(2cîj— 2î)+2mj;)
+ie2.(3+4m
+M(1
— à(3-(-47»).e.cos.(2ti — 2mt) — cv)
— à(3-[-4m).e.cos.(2y — 2mv—cv]
cos.(2« — 2mt')
cos.(2d — 2mi')
Sums
This sum agrees with the two last terms of [5023 lines 11, 12]. The other terms of the
development of the function [5023], of the fifth and higher orders, are neglected.
Corresponding terms of [5023e or 5023].
siu.(2« — 2m« — 'icv-{-gv)
-J-e2.(6+15m+8m2;
ie2.(10+19Hi+8m2
sin. (2cu-f-£"o— Su-l-SniiT)
+ie2.(6-|-15m+8m2)
-H-c2.(3+4h. )
+je2.(l
+Je2.(10+19m + 8m2)
* (2869) Substituting successively the values [4937?i, 4865, 4818], and reducing,
we get,
_2
33m'.7<'-*.s , , a3m'.a5.s , , 33m.o3.s
— -r- .cos.fu — mv) = —
8a,.a'-i ^ ' 8a,.a'
8A2.m5
.cos.(t) — V ) = -— .cos.(u — mv) = — - — -—.cos.iv—mv)
= -«-~-7--sin.(ë-i'— ^)-cos.(r— miO
33)». a'"'.}'
8a,.a'
33m . o2.7
= / .{sin.(g'» — v-\-mv — â)+sin.(§-i'-}-« — mt — d) \.
This last expression is the same as in [5025]. It is of the fifth order; moreover, the
VII.i.§ll.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN s. 487
The term dejiending on cos. (3« — 3v') is insensible.* We have noticed [5026]
the two preceding terms solely on account of their having a little influence
on the argument of the moon's longitude, depending on t' — mv.
The function ^ ' 7 • ( ? )' contained in the equation [4755], gives [5027]
the following term;t
— ^j^^.g7.cos.(gv—i).sm.(2v—2v). [5028]
We shall have the value of this term by increasing, in the development of
3m'. u'^. s
— 7^ — ^.cos.(2i) — 2v') [5023], the angles gv and 2v, by a right angle, [5029]
and then multiplying it by g, which gives, f
neglected terms of the siith order, do not depend on the angle v — ?»y [4875] ; therefore, r5024ci
it is unnecessary to notice them.
* (2870) By means of the values of [49-37;i, 486.5, 4818], which are used in the last
note, we find, that the term of [5019], depending on 3v — 3v', becomes
iSiii a a .
T 'a'af''^' ^'"•(&''~^) ■cos.{3v—3mv). [5026a]
This term is of the fifth order, and depends on the angles 3v — 3mvdzgv=0, which have
not been noticed in these calculations ; and a little consideration will show, that if we develop
it so as to include terms of the sixth order, it will not produce any quantity connected with [50266]
the angle v — mv [4875]. With other angles, the terms of the sixth order are usually
neglected.
t (2871) The differential of [4818], using the abridged notation [4821/], gives
da gy /ilQ\
■j^=g/-cos.gv; substituting this m [.5027], It becomes ■^^^■^^^•g'' •i'^) i and, by [50280]
using [4809], we get the three terms in the second member of the following equation ;
I ds /■:iq\ gy.cos.irv ( 3m'.u'3 . , m'.it'l '\ i , ^ • /o o'M^
AT^-T.-U)=-ftr„l^- i -^;^ .s^n.{2v-2v)—^.[3sm.{v-v)+loM3v-3v)]^. ^,^,,^
The first of these terms is noticed in [5028, &c.], the others in [50.31,50326],
t (2372) Substituting the value of s [4818], in the first member of [5023], and
omitting è for brevity [4821/], it becomes -^-7^.sin.^y.cos.(2o — 2u'). Now, a slight r^Q^Qa]
attention will show, that the process made use of in [5023a — n], in computing [5023], will
be the same, if we change 2t) into 2î)-f90'', and gv into g'D-1-90'', without altering [50295]
488
THEORY OF THE MOON ',
[Méc. Cél
—: a
f ^ 14-2e2-j(2+m).y2_|e'2J .sin(2u— 2mu— g«+â)\ 1
+sin.(2t) — 2inv-\-gv — Ô) \ 2
— 2.(l+m).e.sin.(2u — 2mv-\-gv — cv — ^-|-to) I 3
-2.(l-j-m).e.sin.(2« — 2mu — gv — CT-J-^-j-ro) / 4
-2.(1 — m).e.sm.{2v — '2mv — gv-\-cv-[-ê — tn) I 5
-2.(1 — m).e.sin.(2» — 2mv-\-gv-\-cv — è — ra) \ 6
i-|-J.e'.sin.(2y — 2m«; — gv — d mv-\-è-\-T^) I 7
|-l-|.e'.sin.(2« — 'imv-\-gv — cmv — ^-j-ra') I s
-2-.É'.sin.(2i." — 2mv — gv-\-c'inv-\-è — -a!) l 9
— g. e'. sin.(2« — 2mv-\-gv-\-c'mv — è — ra') I^q
+K10+19/n+8«i2)e2.
sm.(2v-'imv-2cv-\-gv-\-2TS-E) 1 1 1 j
( — sjni
i(2ci;+gT;-2«+2m!)-2^-«j5 / ^2
[5031]
[5032]
The terms of the function
I dsdg |-4755 or 50286], which depend
on m'\
h-.v? dv dv
produce the following ;*
2
-—.-.- . 7. { sin.(gv — v-{-mv — <)) — sm . (gv-\-v — mv — d j .
[5029c]
the angles mv, cv, c'mv ; by a melhod of dérivation similar to that in [4876« — d\.
These changes being made in sin.^v, cos. (2t) — 2u'), they become cos.^jj,
— sin.(2tr — 2v'), respectively; and the function [5029«] becomes
3m'.u'3.y
2/12.^4
-.cos.^!; .sin. (2 y — 2v').
Multiplying this by g, it becomes similar to [5028]. Hence vs^e see, that the method
of derivation [5029] is correct.
[5032a]
3);i'.«'4
* (2873) The second term of [5028i] is — ^7^.^7.cos.^u.sin.(v— d') ; and, by
substituting the values [4937/j, 4865], it becomes
o a, a
which is easily reduced to the form [5032], by using [19] Int. This term is of the fifth
r5032tl *^'''^^'"' ^°^ those of the dxth may be neglected, as in the similar term [5024c]. Moreover,
the term of [50286], depending on the angle Zv—Zv', may be neglected, for the same
reasons as in [5026«, 6].
Vll.i.^^U.] DEVELOPxMENT OF THE DIFFERENTIAL EQUATION IN s. 489
[5033]
llie product ( -j^ -\-s\.-— .J ( -~ \ . — contained lu the equation [4755],
is reduced to*
I . (l-o^).r.sin.(^.-.0,/ C^) . ^: . [5034]
1 — g- being of tiie order m" [4828e], we shall retain, in this product,
only the term depending on sm.(2v—2viv—gv-\-è) ; and it follows, from
the preceding development of Ta --^ ( 77 ) • ~^» ^'^'''^ ^^i^ term is equal to [5035]
2
— ?iililllÇl .^.y. sm.(2v—2mv-gv+ê). [5036]
4.(1— m) a/ ^ o T /
Thus, the equation [4755] is reduced to the following form ;*
Difffrcii-
,, liaUqua-
n I I r' lion 111 s.
"t^^ + *+ ' [5037]
1' being the sum of the terms we have just considered. But, for greater [5037']
* (2874) Using the abridgments [4821/], we have s = y.sm.gv [4818]; whence
ue obtain dds , , as •
substituting this in [5033], it becomes as in [5034]. Now, (1 — g^)-y is of the order
m^y [4828e], orof the <Am/ order; and f— 1 [4809] is of the second order; hence,
the function [5034] is of the fifth order; therefore, we need only notice its chief term.
Now, the chief part of Tz' f [~7~] • ~ [4831', 4882] has been computed in [4885], and [50346]
its chief term is r, — " " ^ /« r> \
MuUiplying this by the factor (1 — g^).y.sin.gv, we get the corresponding part of [5034],
2
3jrt.(l— g-2) a . ,^ ^ X
2(i_,„) ■-•7-sin.gv.cos.{2v—2mv). [5034^]
Reducing this by [19] Int., we get the term [5036], and another similar term, depending on
the angle 2v — 2mv-\-gv ; but this is neglected, because it is of the ffth order, and is not r5034(/i
increased by the integration of the equation [4755], as in [4S97o,&c.].
* (2875) Substituting, in [4755], the development of the terms given in
[5021, 5023, 5025, 5030, 5032, 5036], it evidently becomes of the form [5037]. [5037a]
VOL. III. 123
490
THEORY OF THE MOON;
[Méc. Cél.
accuracy, we imist add the terms depending on the square of the disturbing
force, ivhich might have a sensible influence.
3m' m" s
[5036] 12. The term -^„ — - [5020] gives, by its variation, the following
[5039]
2^2. M^
ones ;
3 m' . u' ^. Ss 6 m'. u'^.s &u
from which we obtain the function,*
h^-.u'
* (2376) In finding the variation of the function [5038], s, u, u, are the variable
[5040a] quantities; but we may neglect Su', on account of its smallness, as in [4909, 4932i,&ic.] ;
and the variation becomes as in [5039]. We shall now separately compute the two terms of
3m'.?/'
which this function is composed. The first of these terms ^.-g — r-''*) is evidently equal
10 the first member of [4908/], multiplied by
a.Ss ; and, as the factor without the
[5040i] braces, in the second member of [4908/], is — ^— , the required function will evidently
_2
be equal to the product of -~.-.Ss by the terms between the braces in the second
member of [4908/]. We shall now compute this product in the following table ; in which
the first column represents the terms of & ; the second, the terms between the braces in
'- '^ the function [4908/] ; and, in the third column, the corresponding terms of the function
[5040] ; rejecting such terms as have been usually neglected.
[5040rf]
(Col. 1.)
Terms of &s [4897].
Whole value of 6s
B ^^''^y.sm.{2v—2mv-gv)
By^y-s\n.{2v-%mf-\-gv)
B/*e'ysin {gv — c'mv)
(Col. 9.)
Terms of
[4908/].
1
— Ae.cos.cv
-\-3c' -COS c'tnv
-\-5e^.cos.2cv
all its terms
-j-3e'.cosc'mt)
-|-3e'.cosc'mi'
(Col. 3.)
Corresponding terms of [5040].
o
I All these terms must be multiplied by -^,7/i ,- 1.
OS [4S97] I
-i-2B^^''^cy.\-s\n.{2v-2mv-gv-ci^)-s\n[2v-2mv-gv-\-ev) \2
-\-^B^'■''1e'y\s\n(2v-2mv-gvic'mv)+slu{2v-2mv-gv-c'mv)\3
—^B^^'^'>e^'y.s\n{2cv-{-gv—2v-\-2mv)-{-6Lc. 4
. . . .neglected 5
-\-iB^'-'''>e'~y.s\n.gv 6
-{-iB^'■»\'•^y.sm.gv. 7
This table contains all the terms of the function [5040] depending on the coefficients B.
Thus, [5040rfrine 1] is the same as [5040 line 1]; the terms in [5040rflines6,7] are the
[5040e] g^ij^g j^g j^ [5040 line 2]; the terms in [5040rf line 2] are in [5040 lines 6,4] ; the terms in
[5040(/line3] are in [5040 lines 7, 8] ; lastly, the term in [5040(Z line 4] is the same as
in [5040 line 9].
VII. i.^ 12.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN s. 491
o
— .-.0* [4897] 1
+ 37rt'. - . j J',"'— I J("'.e2 j .j.sin.(2v—27nv—gv-\-'^) 3
~ n
— 'îiih . - . Bf.e,.€va..(2v — Imv — gvj^cv-\-(> — a) 4
2 _
— 3 m . - . J('\e7.sin.(2» — 2mv+gv — cv — (3+^) 5
_9 « [5040]
Develop-
meiil of
the vuria-
tioo[5039J.
a- i sin.(2i) — 2mv — gv+c'mv-i-è — ra') ) 7
^ "' ( +sin.(2w — 2mw— ^i'— c'my+o+jj') ) 8
+ ?^ . - .{5A['^—2A[''^—'jBf^l.e',.sm.{2cv+gv—2v+2mv—2^—s) 9
2
_j_^ . - .{5A['^—2Al"^\.e^}.sin.(2v—2mv—2cv+gv+2^—i). 10
'Z a,
- Cm'.u'3. «.(5i( . . , ,
The second term oi [oOdyj —— , is evidently equal to the product of the
4 3* ' , [5040/]
first member of [490S] by -X — ; therefore, the development of this term will be
3s
obtained by multiplying the second member of [4908] by — [5020i], and the product
by ^. This process is performed in the following table ; in which the first column contains
the terms of — between the braces in [50206] ; the second column contains the terms of L5040g-]
[4908] between the braces ; the third column, the corresponding terms of the function
[5040/ or 5040], retaining terms of the usual forms and orders ; observing moreover, that
the product of the above factor ^, by the terms without the braces [4908,50206], is
~3^ 2^ .3a7=— 67.W.- nearly. [5040^]
492
THEORY OF THE MOON ;
[Méc. Cél.
3m'. u'^. s
[5041]
The term „ [ . cos- (2 v — 2î/) [5022] gives, by its variation, tlie
following terms ;^
Sm'.u'^.Ss ^ _ „ Gm'.u'^.s.iu
COS. (2v—2v') -j-^—^ COS. {2v—2v')
2 ,A
2/i^M
L5042]
+
3m'.u'\s.W
If.u'
sin.(2«; — 2v') ;
(Col. 1.) (Col. a.)
Ter ms of [50206] . Terms of [490S] .
Bin.gv
[.5040ft]
— lie.sm.{gv-\-cv)
— à:.sin.(gT) — cv
siB.gv
»4,(i'.c.cos.(2k— 2m» — cv)
.4,'")e2.cos.(2cii— 2o-|-2mii)
.4,t'»le2.cos.(2cw— 2i;-|-2mv)
2.4,(»>e2.cos.(2u-2mD— 2cu)
.4,*'i.e.cos.(2i! — 2mv — CD)
.4 ^''.e.cos.(2i; — 2m« — cv)
— 2^/i>.c3.cos.(2«— 2m«)
(Col. 3.)
Corresponding terms of [5040].
.-2 a
]•
I AH these terms must be multiplied bj
—h^^J-^i-y-sin-i^o—^mv—gv) I
^A,^^^ey\sm.{Qv—2mv-\-gv—cv)—sm.[2v—^vtv—gv-cv)^2
iJl^^ii\eSy.sin.{Qcv-\-gv—Uv-{-2mv) 3
^^^(u ).e^.s\n.[2v—2mv—2cv-\-gv) 4
•.4 0)e2,,.{sin.(2ci;+gu-2u-f2mD)-|-sin{2y-2m?;-2cv+g-i')j5
— ^.^/i\e2>.sin.(2cD+g-i!-2«4-2mt)) 6
—lJ]Me^y. I sm{2v—2mv-2cv-\-gv)—sin(2v—2mv—gv)l7
Jl^('\e^.sin.{2v — 2mv—gv). 8
The last term of the function [4908],inckKlecl in this table, is — 2./3/".e^.cos(2» — 2mv),
'-' •' whicli is not expressly given in [4908], though it is produced by the term — 4e.cos.c«.rti5M
[4908^ line 2], neglecting, for brevity, the consideration of the factor — .(1+Je'^),
without the braces. For, by substituting the term aôu=A['\ e.cos. (^2v — 2mv — cv)
[4904 line 2], and reducing by [20] Int., we get,
— 4e.cos. cv.aôu = — 2-4',''.e^.cos.(2f — 2»ir — 2cv) — 2A['Kc-.cos.{2v — 2m v).
The first term of the second member, is given in [4908 line 3], but the second is not given ;
[5040ft] we have, however, introduced it, because it is necessary to make the development [5040]
agree with [5039]. This table contains the remaining terms of [5040] depending on A.
Thus, the term depending on ,3\°^ [5040 line 3] is the same as in [5040Aline J]. The
coefficient of .4'j".sin.(2y — 2nv — gv), in [5040 line 3], is equal to the sum of the two
terms in [5040A lines 7,8]. The coefficients of J'^'") [5040 lines 5,6] are as in [5040Aline2].
[5040/] The coefficient of ^','" [5040 line 9] is the same as [5040/t line 3]. The coefficient of
^('" [5040 line 10] is the same as [5040^ line4]. The coefficient of v^J" [50401ine9]
is the same as the sum of the two corresponding terms in [5040Alines5, 6]. Lastly, the
coefficient of -^/'* [5040 line 10] is the same as the sum of the two corresponding terms
in [5040Alines5,7].
[5041o] * (2877) In finding this variation, we neglect the terms depending on on', as in [5040«].
VII.i.§12.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN «. 493
hence results the function,*
2
4 a,
3li a ^ l(l+m).Br-~Af^l.sm.(gv-cv-é+^)^ 2
-2 -a-''
dm a ,
. - . e-
4 fl. '
/^ Develop-
tioD(50'l2].
■ + \Bf^+iB^^^.sm.(gv—c'mv—ê-{-z/) ^ 4 [5043]
5
6
• + B^'l sin. (2v — 2mv — gv — c'ntv+6 -|-n') -^ ^
^ + \Bf^+lB'-p\.s\n.{gv—c'mv—H--J) ^
) +5',;'. sin. (2v—2mv—gv+c'mv-\-è—^') Ç
V _1_ R(-) çîn f9.ii 9.WJJ) D-») /•'m1lJ-â-4-^'^ >'
* (2878) If we multiply the first member of [491 OA;] by — -^-Ss, it produces
the first term of the expression [5042] '^ .cos. (2t) — 2)/). Performing the same [5043ol
process on the second member of [4910A;], we find, that the preceding term will be represented
by the product of —.—.Ss by the terms between the braces in [491 OA:] ; or, in other [50436]
words, it will be found, by multiplying the expression of i5s [4897] by the terms between
2
the braces in [AdlOk], and theii an?iexing the common factor — .- to all the terms. r5043c]
Taking now, successively, the difl^erent terms of & [4897], multiplying, reducing and
retaining terms of the usual forms and orders, we shall find, that this fii'st term of [5042]
produces all the temis of [5043], which contain the symbol B; as will appear by the
following calculation.
First. The product of J5f .-/.sin.(2y— 2»ii;— ^u) [4897 line 1], by the first line of
[4910^], gives
2
'— .-.(1+26^ — fe'2).^.sin.(4u — imv^gv) — J.sin.^-v}. [5043rf]
The first of these terms, which depends on the sum of the two angles is neglected, as
usual, because it produces nothing of importance ; and the same happens with the sums of
all the other angles, which deserve notice, in this first term of [5042] ; provided we
change the signs of the angles in [4910A:lines 10, 12] ; which does not alter their cosines ;
so that the term between the braces, in [4910A; hne 10], may be put under the form
J(10-|-19?«-j-6ffl^).e^.cos.(2« — 2?wu — 2cv),Uc. Taking, therefore, the second term of [5043c]
VOL. Ill 124
494
THEORY OF THE MOON
[Mtc. Cél.
[5043]
continued.
a
a.
e'y.
5 ja)_2J(»5-l-Sf )
—1(1 0-1-1 9m+8j«''^).51'
(0)
. sill. (2c/; — gv — 2i3-f^)
8
9
[5043f/], depending on the difference of the angles, and neglecting the quantities depending
on c^, e'^, it becomes
— '— . -.B^°\')'.sm.gv, as in [5043 hne 1].
[5043e'] We may remark, thai the circumstance of only using the difference of (he angles in this
function, enables us to apply the priuciple of derivation with much facility, in finding the
development of the function treated of in [;j04G«, &ic.]. The same term of [4897 line 1]
being combined with [4910/f line 2], produces the tenu ilepending on Bf^ [5043 line 2] ;
|5043e"l ueglecting (?, 'f, e'~ ; moreover, the term in [491 0^- line 3], gives, in like manner, the
term depending on Bf^ [5043 line 3]; the term in [4910/c line 4] pioduces that in
[5043 line 4] : lastly, the term in [4910/.rliue 5] gives that in [5043 line 5]. The remaining
terms of [4910/i-] produce quantities of ihe sixth and higher ortk^rs, which are neglected, with
[5043/] the exception of that in [4910/j line 10 or 5043e], which produces the term in [5043 line 9],
[5043/']
[5043^:]
[5043A]
depending on the angle 2cv — gv.
Second. The terms in [4897 line 2 — 7] produce only terms of (he fifth and higher orders
which may be neglected. The term cos. (2!; — 'Hmv) [4910/; line 1], being coml)ined with
[4897 line 8], produces the term depending on Bp [-5043 line 7] ; with [4S97 line 9], it
produces the term depending on iî|^* [5013 line 6] ; with [4897 line 10], it juoduces the
term depending on B'-p [50431ine4] ; with [4897 line 11], it produces the term depending
on B["''> [5043 line 5]; with [4897 line 12], it produces the term depending on J9^">
[5043 line 10] ; lastly, with [4897 fine 13], it produces the term depending on J5[,'2i
[5043 line 8]. The terms [4910A:line4] and [4897line 11] produce the torn depending
on .Bi"» [5043 line 1]. The terms [491 Ofc line 5] and [4897 line 10] produce the term
depending on B'-p [5043 line 1]. This includes all the terms connected with the symbol JB
in [5043].
GîJî • u "tStOit
The second term of [5042] is j^^-^ .cos.(2i' — 2;;'); this is evidently equal
A3. «5
3s
to the continued product of the first member of [4911] by the function — [5020i],and
by the factor |. Now, this factor f, being multiplied by the factor —
4a,
without the
[5043t] braces [4911], and by the factor Say [50206], produces — 3m. -.y. Hence, it is
evident, that the second term of [5042] will be obtained, by multiplying together the
functions between the braces in [4911,50206]; then reducing, and annexing the common
factor — 3 Hi . - . 7. This calculation is made in the following table, which requires no
particular explanation.
VII. 1.-5.12.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN s. 495
4 -«/^^ •
e^}. sill. (2» — 2mv—
-2cv+gV-\-2z:—ô).
10
[5043]
concluded
(CoM.)
Terms of [50206],
between the braces.
(Col. 2.)
Tevms of [-1911].
between the braces.
(Col. 3.)
Corresponding terms of the function [5043].
r All Ihcso terms must be rauUiplied by — S îft^.^ .y J •
sin.^-y
.^^(I-Ae'^)
Ai,°K{i—ie'^).sm.gv
1
— |c.siu.(5-«+<-«)
A[^ c.cos.cv
A'pe.cos.cv
^A[^'>e.lsm.{gv—cv)+sm.{gv-\-cv)}
. . . .neglected
2
3
[5043ifc]
— ie.s\n.{gv—cv)
^3J''e.cos.tt)
+A^(."e2.sin.(2cu— g-«)
4
siu-gn
—2A^^^ e^.cos.2cv
+ ^«'e2.sm.(2CT— ^d)
5
s'm.gv
^^"'e^.cos.an'
—iAf'>e^sia.C2cv—gv).
6
The two lower terms of column 2, lines 5,6, correspond in [4911] to
9m
— j^. {— 2.3<'i-|-.^f"'} . e^. COS. 2cr ;
4a,
which are not expressly mentioned in [4911] ; but are easily computed, as in [4910Ar, &,c.].
For, the function [4911] is found, in [4910/], hy mulliplying the function [4910yt] by the
expression of 2uôu, deduced fiom [4904]. Now, the teim
O
Qtïî
— . 2e.cos.(2t>— 2m?;+ci') [4910X: line 3],
4a,
being multiplied by the term 2.^/".e.co3.(22) — 2mi' — cv) oi 2a5u [4904], produces the
term
Ojïï
— ~.\-2J<i'\c\cos.2cv],
4a, * '
which is used in [5043Â:hne 5]. In like manner, the terra
9m
40,
COS. (2v—2mv) [4910^- line 1],
being multiplied by the term 2.^/"'fc2.cos.(2cD — 2v-\-2mv) of 2aSu [4904], produces,
in [4911], the term f,-^
-~.\A[^^'>e^.cos.2cv], as in [5043^- line 6].
If we now compare the terms of [5043^] with those in [5043], depending on A, we shall
find that they agree. For, the term in [5043Arlinel], depending on ^g'"', is the same as in
[5043 line 1]; those in [.5043A:line 2], depending onA['\ are the same as in [5043 lines 2,3];
the sum of those in [5043 lines 4. 5] is — 3m. - .7. ^J»^/". e^.cos. (2c« — ^*')|) as in
[5043 line 8]; lastly, the term in [5043^' line 6], depending on .^/"\ is the same as in
[5043 line 8],
[5043i]
[5043m]
[5043re]
[5043o]
[5043/)]
496
THEORY OF THE MOON;
[Méc. Cél.
>ni Sm'-'u'^ lis . ,^ ^ ,.^ . , . . .
[5044] i he term -— ^— . — .sin.(2î? — 2v)* gives, by its variation,.
[5045]
2/Au'^
dv
3m'. u'^. d.6s . ,^ o A 1 Qm'.u'^ , ds . ^^ ^ ,.
. sm. (2 ÎJ— 2 î;') + — yg— 5 .5u.-f-.sin.(2v — 2v'}
2h\u* dv
dv
, 3m. îi^ ds . , ,_ _ ,.
fr.u^ dv ^ ■'
Hence results the following function ;t
[5043?]
[50466]
[5046c]
[5046rf]
[5046e]
[5046/]
The third or last term of [5042]
3m'.«'3.s.5!)'
^ .sin. (2» — 2î)'), is evidently equal to the
continued product of the functions in the first members of [4918, 50206] by the factor ^ ;
and, as this product gives terms of the sixth and higher orders, it may be neglected ;
consequently, the value of the terms in [5042] is accurately given, vvitliin the prescribed
limits, by the function [5043].
* (2879) This term is the same as [5027], substituting [4809 line 1] ; its chief part is
[5045a] computed in [5028, Sic.]. Taking the variation of [5044], and neglecting 5m', as in
[5040a, fee], we get [5045].
[5046a] t (2880)
The first terra of [5045] — ohi.A • ~T~ -sin -(21) — 2w'), may be computed
2Aa.u4 dv
in the same manner as the first term of [5042], in [504.3a — g] ; and, by this means, we shall
obtain all the terms depending on the symbol B, in [5046]. But, we may obtain the
same result in a more simple manner, by the principle of derivation used in [4876a — d] ;
deducing the terms of [5046], depending on any symbol £''"', from those in [504.3],
depending on the same symbol, in the following manner. If we denote any term of &
[4897], by h = B'-'"''.sm.iv, we shall have, by taking its differential,
d.&s
dv
= £('"). i . cos. iv = jB>'1 . i . sin.(2't)+90'') ;
d.is
[5046g]
so that -— may be derived from Ss, by increasing the angle iv by 90'', and
multiplying the coefficient by i. Moreover, if we increase 2v by 90'', in the same
manner as in [4876a — cT], the term cos.(2w — 2v'), which occurs m [5043a], will change
into — sin.(2« — 2v'), as in [5046a]. This increase of the angles iv and 2v by
90'', does not alter the differences of these angles in the terms [5043fZ — g], which
depend solely on these differences [5043e']. Hence it follows, that the terms of the
function [5046a], may be deduced from the corresponding ones of [5043rt], by merely
multiplying by the coefficient i, corresponding to each term respectively ; or, in other
words, we must multiply each of the terms of [5043], depending on B''"\ by
the coefficient i, which corresponds to this term of 5s = £'"*'. sin. îî> [4897]. Thus, in
VII.i.>5>12.] DEVELOPMENT OF THE DIFFERENTIAL EQUATION IN s. 497
3 m a
T ' a.
(2-2m-g).B[''^+i(2-3m-g).B^^'>\e'- ) 1
} .(l—ie").Y.s\n.{gv—ô)
, 3iH a
3 m «
3m a ,
. — .C'y
{{\-\-m).{1—2m—g).B^^^—A^{^\.sm.{gv—cv-è+^)
+ { {\—vi).(2—%n—g),Bf + J';> ] . ûn.{gv-\-cv—^—^)
(2— 3m— ^).5<"*>
. sin.(§-t) — c'mv — ^4-^')
+ < > . sm.{gv+c'mv—(: — -a)
-\.(2~2m—g) B^P S
-\-(g — m) . Bf\ sin. (2v — 2mv — gv-\-c'mv+^ — w')
^+{g+in) . £f' .sin. (2v — 2)nv — gv — c'mv+6-{-^')
2
3
** Develop-
ment of
the varia-
r lion|50451.
6
7
8
9
10
11
sin. (2cv-gv-2-ss-^ê)
-l(lO+\9m+8m'').(2-27n-g).B'^^^ S 12
I [5046]
_3m_a mu)^^'iy^^i^^^ç2v—2mv—2cv-rgv+2-.—ê).
4 a.
13
the first term of [4897 line 1], we have B'-'^^ = B'^^y, i = 2 — 2m — g ; therefore, the
terms depending on B'^ [5043] must be muhipHed by 2 — 2m — g, to produce the
corresponding terms of [5046] ; by this means, the term in [5043 Hne 1] produces that in l^OiGh]
[5046 line 1] ; and those in [5043 lines 2, 3, 4, 5] give those in [5046 lines 3, 4, 6, 8],
respectively. Again, in the third term of [5043 line 1], corresponding to B[^'", we have
i = 2 — 2m — g — c'ni = 2 — 3/« — g, nearly ; and this produces the term depending on
^(10) [5046 line 1]. In like manner, the terms depending on S','', 5™, Bf\ B[">\
B^o"\ B['-\ produce the corresponding ones in [5046] ; observing always, that c, g and
</, are very nearly equal to unity.
Substituting the value of
ds
dv
[5028a], in the second term of [5045], it becomes,
..5M.sin.(2y — 2v' ).gy. cos. gv;
which, by putting ^ =1, is evidently equal to the differential of the first member of the
VOL. III. 125
[5046i]
[5046/fc]
498
THEORY OF THE MOON ;
[Méc. Ce).
[5047] Lastly, the function (^+s) • | •/ (^) • % [5033] gives, by its
variation, the terms*
[5046i]
|5046r]
[5046m]
[5046n]
[5046o]
[5046p]
[5046g]
ay
function [4931/1 or 493 Ip], multiplied by — .cos.g-w; and, as the resuhing function is
composed of terms o( the JlftJi and higher orders, we need only notice the chief terms of the
differential of the function [4931p]. These, after reduction, are contained in [4931p line 6],
and in [4931c,?'], whose differential, divided by 2dv, produces the following terms,
nearly ;
2 9
^ . j ^;"esin.CT+.4i'"e2. sm.2cv \ — ~. A'k^ sm.2cv ;
2a^ 2a,
which must be multiplied by the factor ay. cos. gv. The first term of [5046»] evidently
produces the two terms, depending on A['-'' [5046 lines 3, 4] ; the second term produces
that depending on -^^/'^ [5046 line 11] ; the third term produces that depending on -4','\
in [5046 line 11].
As the last term of [5045] is very small, we may substitute in it the values [4937?^],
and [4865, 5028a] ; by which means it becomes
_2
3m .a
-.(h'. COS. (2v — 2mv). [gy. cos.gv } ;
and, as Sv' is of the third order [4931a;], the whole expression must be of the sixth or
higher orders. Now, as it does not contain any quantities of the sixth order, depending on
the angle v — mv [4875], it may be neglected ; therefore, the function [5045] will be
represented by the quantities depending on its two first terms, which are given in [5046].
[5048o] * (2381) The chief term ro-fC^)-^ [4809] is represented by —fW.
dv
Ms
[4929«], and if we put, in this case, V^^ — ^ -j-*i ^he function [5047] will become of the
dv'
[50486] form — V^ .fW.dv ; ivhose variation is given in [49296], changing V into V,. Now,
we see, in [5049], that Vi= "rr + * '^ o^ ^he order m.y, or of the third order ; also
W [4929«], or its differential coefficient, as well as ahi, are of the second or a higher
order; hence it appears, that all the terms of the variation [50486, 4929i], excepting
[5048c] — 5V^.f W.dv, may be neglected, as of the seventh or a higher order. Now, the
function — fW.dv [4929a] is evidently equal to the first member of [4885], and
[5048rf] 5y^=^^_i^Ss; hence, the function —SV^.fW.dv [5048c], or the chief part of
the development of the function [5047], will be represented by
VII. i. ^ 13.]
DIFFERENTIAL EQUATION IN s.
499
_2
3m>| a
y.-.|(2-2«-s-)M}.B<'».(l-je'^).<
1— m
.■y.sm.(gv — è)
, (10+19m+8m2) „ . ,^ ^ , x
I Develop-
^ ment of
[5047].
The terms depending on the cube of the disturbing force are insensible.*
13. Connecting together all the terms of this development, we find, that
the equation [4755] becomes.
[5048]
del 5 s
OS j X by the second member of the function [4885] ;
in which we must substitute the value of 5s [4897]. Now, the first term of this value
gives, in -^ + ^^> the term — {(2 — 2m— ^)2— 1|. jÇ'»'. y . sin. (2d — 2?«w— ^d) ;
multiplying this by the terms in [4885 lines I, 10], it produces the terms in [5048 lines 1,2],
respectively; neglecting terms of the order e^, e'^, in the factor (l-J-2e^ — f «'^)
[4885 hnel]. The factor (2 — 2m— g)^ — 1, being of the order m, renders the term
in [5048 line I] of the fifth order; which is retained, though small, because the term
connected with the angle gv gives the motion of the nodes in [5050, &c.] ; and the term
in [5048 line 2], depending on 2cv — gv, is retained for reasons similar to those in [4828rf].
The term depending on £f^ [5048/], being multiplied by the remaining terms of [4885],
produces terms of the sixth and higher orders, connected with angles which have been usually
neglected. The next term of Ss [4897 line 2] has the coefficient Bi^Ky, which is
marked of the third order; but, if we examine the value of B[^i [5177], we shall find
it to be so very small, that it may be neglected. The terms in [4897 lines 3 — 7] are of
the fourth order, producing in [5048e] terms of the sixth or higher orders, which
may be neglected. The terms [4897 lines 8 — 11] are of the form -BJ"*'. e'y.sin.io ;
in which i differs from unity, by a quantity of the order m ; so that 1 — i^ is of the
order m. This gives, in ~hr-\-^^i a term of the form Bfi.e'y.{l — i^).sin.iv,
which is of Ûie fourth order; producing only terms of the sixth order, in [5048e]. In like
manner, we find, that the remaining terms of [4897] may be neglected, and the whole
function [5047] is reduced to the two small terms [5048].
* (2882) If we compare the value of n [4902, 4961], with that of r [5037, 5049],
we shall easily perceive, that the terms of r are of the order n.y ; and, as the terms of
n, depending on the cube of the disturbing force, are of the fifth or a higher order
[4995a, 4941 , 4942, &ic.], the corresponding ones of r must be of the sixth or of a higher
order, which may be neglected.
[5048e]
[5048/]
[5048ff]
[5048A]
[5048t]
[5048/t]
[5048^]
600 THEORY OF THE MOON ; [Méc. Cél.
av a ] [
' i^^,l3Sm—g\.B'-^''\e'^+i.(3—m—g).BfKe'A 3
[5049] (L")
— fîw.-. ) /i_ ''^ V .7.sin.(2t) — 2mt) — gv-\-^)
lion in s.'_ fl^
+ fm". -.\Bf^—2+(l—m).(3—2m—g).Bl'^l.e).sm.(gv-i-cv—c—z,) 8
*"/
+ |m'.- .{5f' — 2— 2J',"+(l+m).(3— 2m— ^).£f' }.ev.sin.(^î;— cf— «+^) 9
+ |m'. -. S (1+^).(1— m)— 25<")+5(^) \ .€y.sm.(2v—2mv—gv+cv+ê-^) 10
"'/
+ |OT . -{(g—l).(l+m)^Bf^—2A'^^'>\.ey.sm.(2v—2mv+gv—cv—ê+m) 1 1
* (2883) The equation [5049] is the same as [5037], taking for- r all the terms we
[5049a] havecomputedinthe!;e»functions[5021,50-23,5025,5030,5032,5036,5040,5043,5046,5048].
In finding the sum of these terms, we shall proceed as in note 2847 [4960e, &c.], taking the
quantities depending on each angle separately, in the order in which they occur in [5049] ;
after dividing them by the factor which is common to all the terms as in [4961i].
First. The terms depending on sin.^u [5049 lines 1 — 4] have the common factor
fm .— .y.sin.^i) ; and if we divide all the terms of the functions [5049a], depending on this
angle, by this factor, we shall obtain in [5021 linel] the terms l-j-2e^ — 17^+1^'^ as in
[5049 line 1]. The terms in [5040 line 2] are the same as in [50491ine4]. The coefScient
of £<«'.(!— ie'2), in [5043 linel], is —i; in [5046 line 1], is —^{2—2m—g);
their sum is _i,^s-2.n-g) = -i.(t^pyi-m).
[50496] Lastly, the term in [5048 line 1] is
VlI.i.>^U3.] DIFFERENTIAL EQUATION IN «. 501
4.|j|.!L. ? (1 _l_o-).(l +m)+5f +2J;»-2Bî''> } .ey.sin.(2v-2mv~gv-cv+ô+z,) 12
-j-llïï^.{S+2Bf^—i(3-2m-g).BfW^3-m-g).B['^].e'y.sm.(gv-c'mv--è+z,') 14 ^.^^^^^^_ ,
/ t lal eijua-
tion in s^
o ~ concluded.
+fm .-. ^ J (1 +^) +2Sf >+35f^— ( 1 +g —m) .BW } .e'y.sm.{2v—2mv—gv-\-dmv-{-ô-zi') 1 5
4-f Hl°.-. {2jB<,'«- J(l+^)+3-Bf°i— (1 +^+?»)Bi"} .eV.sm.(2z>— 2mu— 5-i;-c'm«+â+w') i6
^ 2Bi,'"— 5— 10.3«+4^ii»— (3— 2m— 2c+g').jB<''i ^ 17
+?M.-. / ^o o Wo 9«, „-\a T î 1 / .e^y.sin(2cj;— e-î) — 2a+ra)
3„ (2B'r-^+i{l-g).{lO+l9m+8nv')^ 19
-f-f m .-. ■? > .eV.sin.(2y — Smc — 2cv-\-gv-\-2TS — é)
2^ ri.(10+19m+8m2)_|_2J5;'3) i 21
-|-|w .-. < ? • e^7' sin. (2cî)-}-,§''' — 2t)-|-2mr — 2a — fl)
"' ( +10^<'i— 4^.""— 5B<") ) 22
^ a ( ) 1
+|-m.-. ] 3+2i?^"> S . -.7.sin.(^«— î;+wî2; — ê) 23
+T»î-- • \ |+2Sr f • ^,-7.sin.(^t)+î;— m?;— ^). 24
adding this to the sum of the two preceding terms, it becomes
-^ {T^rT-) ■ (^ + '") ' [50496']
which is the same as the coefBcient of B'p [5049 hne 2]. The term depending on ./^^'"
[5043nnel], is the same as in[5049hne2]. Tiie coefficient of B[^\e'^l—je^), in
[5043 line 1], is i; in [5046 Une 2], is a(2 — m — g); wliose sum is ^{3 — m — g), as in
[5049 hne 3]. The coefficient of B['°\e'^.(l—ie% in [5043 Hne 1], is —J; and, in
[5046 hne 1], is —l(2—3m—g); their sum is — J(3— 3?«— ^), as in [5049 line 3].
The term depending on A'-°'' [5043 line 1], is the same as in [5049 line 2].
Second. The terms in [5049 lines 5, 61 have the common factor „,„ ,
ji a [50i9c]
— ^WL . - . 7 . sin. (2v — 2mv — gv) ;
VOL. lit 126
502 THEORY OF THE MOON ; [Méc. Cél.
[5049] 14, In finding the integral of the equation [5049], we must proceed in
[5049c']
[5049d]
and, if we divide the corresponding terms of the functions [5049a] by this factor, we shall
obtain, in [5023 line 1], the terms l+2e«— i(2+m) .7^— |e'2 ; and, in [5030 line 1], the
same terms, niuhiplied by g; their sum is the same as in [5049 line 5]. The expression
[5036] is the same as the first term in [5049 line 6]. The terms depending on A'^\ A'^'>
[5040 line 3], are the same as in [5049 line 6]. The terms in [5040 Hire 1,4897 line 1]
give the term depending on &l^'> [50491ine6].
Third. Of the three terms in [5049 line 7], the first is found in [5023 line 2]; the
second, in [5030 line QJ; and the third, in [5040 line 1,4897 line 2].
Fourth. The terms in [5049 line 8] have the common factor
J »rt . — . e 7 . sm. {g v-f-cv) ;
and, if we divide the corresponding terms of the functions [5049a] by this factor, we shall
get, in [5021 line 2], the term — 2 ; m [5040 line 1], the term JB!,-' ; as in the second
and first terms of [5049 lines]. The coefficient of (l—m).Bf, in [5043 line 3], is 1;
and in [5046 line 4], is 2 — 2m — g ; whose sum is 3 — 2m— g, as in [5049 line 8]. Lastly,
tlie terms depending on .^/'' [5043 line 3, 5046 hne 4] mutually destroy each other.
Fifth. The terms in [5049 hne 9] have the common factor
f m . — . ey .sm.[gv — cv) ;
and, if we divide the corresponding terms of the functions [5049a] by it, we shall obtain, in
[5021 line 3], the term — 2; in [5040 line 1], the term B':p ; as in the two first terms of
[5049e] ^g^j^g jj^g g-j_ rpjjg coefficient of (l+m).Bf , in [5043 line 2], is 1 ; and, in [5046 lin63],
is (2 — 2m — g) ; whose sum is (3 — 2»! — g), as in [5049 line 9]. Lastly, the terms
depending on A[' > [5043 line 2, 5046 line 3] , being added, give -2^< ■' [5049 line 9] .
Sixth. The common factor of the terms in [5049 line 10] is
f wi . —.e'y.s}n.(2v — 2mv — gv-\-cv).
The term connected with it, in [5023 line 5], is 1 — m ; in [50301ine5], is ^(1 — m) ;
whose sum is (1+^)'(1 — "Oj 3s in the first part of [5049 line 10] ; [5040 line 4] gives
— 2Bf'; and [5040 line 1] gives B'-^'' ; as in [5049 line 10]. In the same manner we
[5049/] obtain the terms connected with %ni . - .ey.s'm. (2v — 2mv-\-gv — cv) ; namely, in
[50231ine3], — (l+»0; in [5030 line 3], g{l+m); whose sum is {g — l).(l-|-m);
in [5040 line 5], the term — 2^/" ; and, in [5040 line 1], the term Bf'' ; all these agree
with [5049 line 11].
VII. i. § 14.]
INTEGRATION OF THE EQUATION IN s.
603
a similar manner to that in [4971, &c.]. We shall, therefore, suppose [5049'
Seventh. The common factor of the terms in [5049 line 12] is
§711 . — . cy. sm.(2u — 2mi' — gv — cv).
"■1
The term connected with it, in [5023 hne 4], is (l-{-m); in [5030hne4], is g{l-^m) ; [5049^]
whose sum is {l-\-g).{l+m); in [5040 hne6], is — 2(5^°'— ^i'>) ; and, in [5040 hne 1],
is Bf\ These agree with [5049 hne 12].
Eighth. The terms connected with the common factor
f m
- . ey. sin. (2v — 2mv-\-gv-\-cv),
are as follows. In [5023 line 6], — (1 — m); in [5030 hne 6], ^(1 — m) ; whose sum [5049;»]
(^ — l)-{l — m) is of the second order [4828c] ; or, of the sixth order in [5049] ; and, as
this is not increased by the integration [4897o, Sic], it is neglected.
JVinth. The common factor of the terms in [5049 line 13] is
.e'y.sin. {gv~\-c'mv).
The term connected with h, in [5021 line 4], is 3; in [5040 line 1], is 25f> ;
_- a
*'" a.
as m
1 .
2 5
[5049t-]
the two first terms of [5049]ine 13]. The coefficient of B'f\ in [5043 line 5], is
in [.5046 line 8], is ^(2 — 2m— g); whose sum is ^(3 — 2m— g), as in [5049 line 13].
The coefficient of ^i'"' , in [5043 line 5], is —1 ; in [5046 line 7] is — (2— 3ot— ^) ;
whose sum is — (3 — 3;n — g), as in [5049 line 13].
Tenth. The common factor of the terms in [5049 line 14] is
^iit .- . e'y.sin.^gv — c'mv).
The term connected with it, in [5021 line 5], is 3; in [5045 line 1], is 2B[^\ as in the
two first terms of [5049 line 13]. The coefficient of Bf\ in [5043 line 4], is — J; in
[-5046 line 6], is —i{2—2m—g); whose sum is — J(3— 2ot— ^), as in [.50491ine 14], [5049A]
The coefficient of Bf\ in [5043 line 4], is —1; in [5046 line 5], is —(2—m—g);
whose sum is — (3 — m — g), as in [5049 line 14].
Eleventh. The common factor of the terms in [5049 line 15] is
f m . — .e'y.sm.{2v—2mv — gv-\-c'mv).
The term connected with it, in [5023 line 9], is i; in [50301ine9], is ig; in [5040 line 1],
is 25^»; in [5040 line 7], is 36,°'; in [5043 line 6], is — iî^^^ ; and, in [5046 hne9],
is — {g — m).B'f''. These terms, taken in the same order, are as in [5049 line 15].
504 THEORY OF THE MOON ; [Méc. Cél.
[5049"'] y and ê to be variable ; in consequence of the variation of the excentricity
Twelfth. The common factor of the terms in [5049 fine 16] is
I m .—.e'7.sm.{2v~2mv—gv — c'mv).
The term connected with it, in [5023 line 7], is — J; in [5030 Une 7], is — J^; whose
[5049m]
[5049re]
[5049o]
[5049p]
sum is — i{^-\-g)> as in the second term of [5049 line 16]. The term, in [5040 line 1],
is 2B<""; in [5040 line 8], is 3Bf; as in the first and third terms of [5049 line 16].
Lastly, the coefScient of B^p, in [5043 line 7], is — 1; and, in [5046 line 10], is
— (S'\'^) 5 whose sum is — (l+â"+"')' ^' '" ''^^ '''^'^ '^•''^ °f [5049 line 16].
Thirteenth. The term connected with the common factor
a
^ m . — . c' y . sin. (2y — 2iiiv-{-gv — c'mv),
in [5023 line 8], J; and, in [5030 line 8], is —ig; whose sum i{l—g) is of the
order ni^ [4828e], producing terms of the sixth order, which may be neglected. In like
_- a
manner, the terms connected with the factor f m . - . e'y. sin. (2 v — 2 m v-\-g v-\-c'm v), in
[5023 line 10], is — J; and, in [5030 line 10], is -}-ig ; whose sum — h{^—g)^
is of the second order, producing terms of the sixth order, in [5049], which may be
neglected.
Fourteenth. The common factor of the terms in [5049 lines 17, 18] is
_2 a
f w . — . e^y . sin. (2cv — gv).
The term connected with it, in [5021 line 6], is —5 ; in [5040 line 1], is 25^"' ; as in
the two first terms of [5049 line 17]. The coefficient of ^/'^ in [5043 line 8], is —5 ;
in [5046 line 11], is — 5; whose sum is — 10, as in [5049 line 17]. The coefficient
of A[''\ in [5043 hne 8], is +2, in [5046 line 11], is +2; whose sum is +4,
as [5049 line 17]. The coefficient of JS'/=', in [5043 line 8], is —1; in [5046 line 11],
is —(^2 — 2m—2c-{-g); whose sum is — (3— 2m— 2c+^-), as in [5049 line 17]. The
coefficient of {l0-{-l9m +8m^).B'f\ in [50431ine9], is J; in [5046 line 12], is
l[2—2m—g); whose sum is j(3 — 2m — g), as in the first term of [5049 line 18] ; the
remaining term is as in [5048 line 2], neglecting the factor (1 — fe'^).
Fifteenth. The common factor of the terms in [5049 lines 19,20] is
- a
x m . — . e"-/-sin.(2i! — 2mv — 2cv-{-i!;v).
a
The term connected with it, in [5040 line 1], is 22?,'=', as in the first term of [5049 line 19].
The coefficient of i{l0-{-l9m+8m^), in [5023 line 11], is 1; and, in [5030 line 11],
is — g- whose sum is (1— 4r), as in the second term of [5049 line 19]. The terms
depending on ^i'\ .^i"^ [5040 line 10], are as in [5049 line 20]. The coefficient of
VII. i. § 14.] INTEGRATION OF THE EQUATION IN s. 505
of the earth's orbit. Then, by substituting, for s, the expression [4897z]
s = y. sin.(gv—ù)+5s, [5050]
and comparing at first, the sines and cosines of gv — ^, we shall obtain the
two following equations ;*
[5049g]
^J'' [5043 line 10], is — 1; and that in [5046 line 13], is also — 1; whose sum — 2,
is asm [5049 line 20].
Sixteenth. The common factor of the terms in [5049 lines 21, 22] is
f TO . — . e-'y.sm.(2cv-\-gv — '■2v-\-2mv).
This is multiplied by the factor i(10-}-l9m-j-8m^), in [5023 line 12] ; and also, in
[5030 line 12]; their sum is as in the first term of [5049 line 21]. In [5040 line 1], we
have 2Bj'2', as in the second term of [5049 line 21]. The terms, in [5040 line 9], give
those in [5049 line 22].
Seventeenth. The common factor of the terms in [5049 line 23] is
fm.— .— .y. sin.fFW — v-\-mv).
^ a^ a ^^ '
The terms connected with this, in [5025], is ^; and, in [5032], is :^ ; whose sum is 3, [5049j]
as in [5049 line 23]. Lastly, the term [5040 line 1], depending on Bl"', is as in
[5049 line 23].
Eighteenth. The common factor of the terms in [5049 line 24] is
im ■— . —.y.5\n.[s'v-\-v — mv).
The term connected with this, in [5025], is g! ; and, in [5032], is — I; whose sum
is f, as in [5049 line 24]. The term [5040 line 1], depending on B^^^\ is as in the [5049«]
last term of [5049 line 24]. Hence it appears, that all the terms of the equation [5049]
agree with the preceding developments.
* (2884) The quantities B[°\ ^f, in the factor of 7.sin.(^«—a) [5049 line 1-4],
are multiplied by 1— fe'^, and some of the other terms of that factor are multiplied by
c'2 ; so that we may put the whole factor under the form p"-\- q". e'" [5053]. Moreover, as
the equation [5049] is linear, we may notice the terms depending on sin.(^?; — è) separately;
and, by restricting the value of « to this term, we shall have, from [5049, 5053],
^ = S + ' + ^V"+'i"- <=")-y- sin.fei'-^). [50516]
Substituting in this, the value s = '/.sin.(^f — è), and its second differential,
dd3 ddy . .\ , n^y f ^^\ / ,x <W^ f ,s ( 'i^\^ ■ f
d^- = J^-^'^-(S^-^)+^%\S-J,)-^os.{gv-ê)-Y.^.cos.(gv-6)-^. [g--j .sm.{gv-6); ,[5051c]
VOL. III. 127
[505:a]
506 THEORY OF THE MOON ; [Méc. Cél.
ddù r. '^y f ^^
[5051] '^^'^■■Â^-^-I.Xê-J.
[5052] - ^^'^ ^^ ^^^^
0= ■d^^-''\[^-dv)-^\+'<P+^''^-'-^
[5053] p"-Ji-q".e'^ denoting the coefficient of -y.s'in.Cgv — 6) in the differential
/'' 9"- equation [5049] ; in which we must observe, that B['^^ and Af^ contain
1^°^^] inplicitly the factor (1— |e'^) [4976a, 6]. The first of these equations
gives, by integration,*
[5055] d^ — -"-7 1
// being an arbitrary constant quantity. The equation [5052] gives, by
neglecting — -j, and the square of q".e'%-f
[5056]
do io".e'=
considering v, y, é, as variable quantities, we get,
1505«, 0=-!^St-..^:.(^;)^.cos.(g.-,) + ^'^,-.4(^-2f-l]-Hy'«".>^,infe.-.,.
This equation is satisfied by putting the coefficients of cos.(^w — è), sin.(^y — è) separately
equal to nothing ; by which means we obtain the equations [5051, 5052], respectively.
'■ The whole calculation being similar to that for the motion of the perigee, in note 2852
[4973rt— A].
* (2885) The equations [5051,5052] are similar to [4973,4974], and are solved in
dv
dé
the same manner as in [4977a, Sic.]. Putting, in this case, g — — W^, we get, for
[5055a] _ jji) jjfT
its diflerential, — JT = "T^- Substituting these in [5051], we obtain,
. dW, _„. dv dJV ^ dy
[50556] ^ = -^-rf^-2^^'-^5 or, --^ = 2.-;
whose integral is ^ = H-V^ ; H being the arbitrary constant quantity. This is the
same as [5055], and is similar to [4977 or 4977c].
t (2886) In like manner as we have neglected ddE, or dde, in [4973e — h], we
may neglect ddy, in [5052] ; and then, dividing by y, we get.
[5058]
VII. i. «5- 14.] INTEGRATION OF THE EQUATION IN s. 507
therefore, if we consider p", q", as constant, which may here be done [5056]
without any sensible error [4979a, &c.], we shall have,
ê =gV—^ï+^".V— y^.fe'\dv+>. ; [5057]
>■ being an arbitrary quantity. This gives, [5057']
s\n.(gv—è) = sin. I ^/rf^".v+ ^^^^ Je'"~.dv -^ | .
Hence it folloivs, in conformity ivith observation, that the nodes of the moon'' s
orbit have a retrograde motion upon the apparent ecliptic, which is represented
by,*
i„"
Retrograde motion of the nodes = {^i-j-^" — l}.v+ /^j^, -fe'^.dv.
Motion of
This motion is not uniform by reason of the variableness of e'; and the «he nodes
secular equation of the longitude of the node is to the secular equation of the
It
perigee as ,-^— ^, is to — / ■ .f [5060]
or, by reduction,
S-'i =v/(l+i'"+?".e'-^) =v/(l+i>")+|rg-)+^- [5056a]
Neglecting the square of q".e'^ [5055'], and reducing, we obtain [-5056]. Multiplying
this by dv, and integrating, we get [5057] ; or, as it may be written,
[5059]
g„-é=[^{l+p").v+-^^.fe'^dv-xy.
[50566]
and, by taking the sine of both members, it becomes as in [5058].
* (2887) In [4818or505lJ] the quantity gv—6 represents the moon's distance
from the node, which is equal to
{^i^+P"U-+^^l^yfe'^-^v->.] [5056i]. ^,^^,^^
Subtracting from this the moon's longitude v, we get the expression of the retrograde
motion of the nodes [5059] ; observing, that by taking the integral fe'^.dv from d=0..
where the motion of the node is commenced, we may neglect the quantity >..
t (2888) The term of the expression of the motion of the moon's perigee, upon which
its secular motion depends, is represented in [4982] by lq'.fe^.dv=^—T^..fe'^.dv [4979]. [5060a]
508 THEORY OF THE MOON ; [Méc. CéJ.
The tangent y of the inclination of the moon's longitude to the apparent
ecliptic [4813], is also variable, since it is represented by,*'
[5061] y= ^H.(g-^£iy' [5055].
The seca-
IVonofT But it is evident, that its variation is insensible ; and this is the reason why
bie'"'^"" the most ancient observations do not indicate any change in the inclination,
although the position of the ecliptic has varied sensibly, during that interval.
We shall then have the following equations ;t
The similar term in the motion of the node is , .fe'^.dv [5059]; the negative sign
being prefixed, because the motion is retrograde. This last expression is to the former in
the ratio mentioned in [5060].
* (2S89) We may observe, that the equations [5051 — 5059] are similar to those in
[4973 — 4982], and maybe derived from them. Thus, by changing c, to, — p, — y^
^ , into ^, ê, p", g", 7, respectively, we find, that the equations [4973]
a *"
and [4974] change into [5051, 5052], neglecting ddy ; [4977] becomes like [5055];
[5061i] j-^g^gj j.j.g |-5Q5g-]. [4980] like [5057]; [4981] like [5058], changing cos. into sin.;
lastly, [4982] like [5059]. Hence it is evident, that we may apply the same method,
to prove, that the secular inequality of y is insensible, that we have used for c, or
'^■1, in [4987, Sic] ; observing, that both these inequalities depend on terms of a
similar form and order.
I
[5061a]
[5061c] ^
[50626]
t (2890) If we suppose any term of Ss [4897] to be represented by i?.<;^u.(ii) + s),
[5062a] it will produce, in Ti+«) the term |l — i^\.B .&iA^. (iv -\-s). Substituting this in
[5049], and putting the coeflicient of each sine equal to nothing, we shall obtain the
equations [5062 — 5077] ; taking them in the same order as they occur in [5049] ;
and reducing them, by dividing the equations by the factors depending on e, e', y,
without the braces. No other reduction is necessary in any of the terms, except in that in
[5049 line 18] ; in which we must substitute
[2—2m—gy—l = (3 — 2m — g).{l — 2m—g) ;
by which means we have,
1'^ + ~%{2c-2+2m) ~ ^^ ^'•i^^ 2.(2c-2+2m) S
— W ~"i '=^-2.{2c-2+2m)
VIL i. ^ 14.] INTEGRATION OF THE EQUATION IN s. 509
o=!
0=
0=^
0=1
0=
0=
0=
0=1
0=1
0=1
0=1
0=1
o={
0=1
0=1
[5062]
2 a C(l+§-)-{l+2eW-(2+w»).'/— |e"h
—(2—2m—gy\.B['^—^m.~.) ,. .. C.
«, i + ^-^=^^— 4J<»'+104".e^— 25f' C'
V, 1 — w " ' ^ J
_(2_2m+^)-^i.5<"+ |m!^. Si.(l-o-)+5y' i; [5063]
—(g+cy-\.Bf+im~. -.\Bf^—2+(\—m).(3—2m—g).B[''^; [5064]
—(g—cy I .^f + 4 wi . - . l^f — 2— 24"+(l+m).(3— 2»i— ^). 5f '} ; [5065]
_(2— 2»i— o- +c)-] .B;^'+ f Jrt'. - . I (l+«-).(l— m)— 25f '+5^ i 5 [5066]
a,
—(2—2m+g—cy\.Bf+iîïï. - -K^— l).(l+m)+5f— 24"}; [5067]
_(2 _2m-o--c)^ I .Bf +!«'■- -Kl +^). (1 +»î)+5f +2Af-2BT \ ; [5068]
^/
— (o-+m)2 } .B^P+lL-. 1 3+25f '+J-(3-2m-^).5f )-(3-3m-^).5î'°' } ; [5069]
_ {g—mf] . fif +f î^.-. 1 3+25f' — |(3-2m-^).5f -(3-»i-^). 5f' } ; [5070]
_(2_m— ^)^^5f'+p.^{Kl+S•)+2Si'"+3■Bl°Ml+^-'»)-Si''}; [5071]
_(2— 3;«— ofi .iBi'O'+fm'AlSBi'»'- J(l+o-)+35p-(I+^+m).B7'l ; [5072]
'2B<'»— 5— 10^«4-4^1i"— (3— 2m— 2c+g).B','=^
—(2c—gY\.B'~''^-{-lm.-.l nn-Lio™ I a "x >; [5073]
(^ V. 6/ v6-r y 2.(2c-24-2»«) ' ^
_(2— 2m— 2c+^)^|.5i'^'+âm.-.^ J; [5074]
"' ( +I04'>— 4.4V"— 25o"" >
—(2c+g—2-{-2mf\.B['^^Jrlm.-.{ f; [5075]
"' ( +10^"— 4.4/"'— 52?r> )
_(g+m_l)2^.5^"')+3 ,7'." . ^ 3+2^(»' \ ; [5076]
_(^+l_^)2|.5(.5)+.^.^. s 1+2 ^<'^' I . [5077]
0=
VOL. III. 128
510 THEORY OF THE MOON; [Méc. Cél.
16. It now remains to determine the value of t, in terms of v. For
this purpose, we shall resume the equation [4753],
dv
[5078] dt= „. ,^ =:: .
We must substitute in it the value of u [4997] : namely,
] ( l + c2+i7^+f3+e.(l+e=).cos.(a'— t.)^
[5079] M = - . < > +*« .
" ( -i7'.(l+e^-i7').cos.(2gv-2ù) S
âv
We shall have, in the first place, by developing the factor — , a term
independent of the cosines, which, by the nature of the elliptical motion,
2 7
[5080] must be equal to* — Jr— [5081o,j:;].
* (2891) If we put, for brevity,
[5081a] ^ < 1 , 2 /■ /'''Q\ '^''■' ) 4
dv
[5081a'] the expression [5078] will become, dt=~.Q^. The development of Q', in a
[50816]
series, gives, O'— i_l f f'!S:\ ^ 1 _L {r'^S *\2 fcp .
which is of the same form as the factor of [5081], depending on Q. The terms of u
[5079], independent of Su, have been heretofore denoted by u [4826, 4861, 8ic.; 4997] ;
and, by retaining this vahie, the second member of [5079] will be u-\-&u. Substituting
this complete value of u in cit [5081n'] it becomes,
dv.q dv.q / ^èu\-2 Cl 2fa , 3fc9 , > , ^,
[508id] ''^=T^H^=Tii^- •(,!+, 7; =\ï^-ï7s + ^^;^-^''■l■^'■^
[5081e] =l,i-2(«^»)-^i^ + 3(«^«)^-^a.- ^- 1 •''-^'
observing, that we must substitute in [508 le], for u, all the terms of the second member
of [5079], excepting &u. Now, by neglecting terms of the fourth order, we have,
[5081/] î=i^.(l+Je3+è7,^) [4866/]; whence, I = ±-^.{1+^,^+1 f).
Multiplying this by m"^ [4866^], we get,
[5081g] _L_— "J^.|(i_f_Je2_^y2\_3g_Q_i^2).cos.Cî)+3e'-^.cos.2cî)+f7^cos.2^i;-|cj'2_cos(o^^,_^j,)|.
VII.i.§15.] DIFFERENTIAL EQUATION IN t. 611
Then we shall have,
1 — 2e.(l — {y^).cos.{cv—-a)
-)- 3g2_ ( 1 _|.^e2— J/2) .cos.(2cu— 2ra)
3 r /dQ\ rf-w-12
iioû in I.
[5081]
(U=^-( \ -1er |cos(2^i)-cy-2t)4-w)+cos(%y+c«-2()-5i)} )\ — hc.
( l+àe2-^72— 3e.cos.(cu— a)+3c2.cos.(2c»— 2ra) p C,_L /•/'f^\ ^
-2ai« J > . < A2 -y ^ d„ j • „2
( -H>^-cos-(2gi»— 24)-ie>2xos.(2gy-c»-2e+TS)) ( + &c.
+3.(a 5u)-2. 1 1— 4e.cos.(cu— ra) j . { 1— &ic. \
Substituting this and Q' [5081J], in the terra of [5081e] depending on the first power of
aôu, we get the corresponding terms of [5081 lines5,6] ; neglecting the very small term of
the fifth order, depending on ey^. cos.cv. Again, we have in [48706],
M~^= a'*.\l — 4e.cos.cu-|-^c. j. fSOSlfcl
Multiplying this by — =: ^ nearly [5081/], we get,
1
= -7.|l — 4e.co3.c«+&z,c.|. [5081«]
hui. a2 a
Substituting this and the value of (^ [50816], in the term [5081e] depending on («iÎm)^,
we get the corresponding terms of [5081 line 7]. We may observe, that aSu [4904] is
of the second order ; so that, in these terms of [5081 lines 5 — 7], we have explicitly
retained terms as far as the fourth order inclusively. The only remaining term of di
[5081e] is the first, " ^ ; and the quantity Q' is represented by the terms depending
on Q, in [5081 lines 1 — 4]. The factor connected with Q', is of the same form as
dv
''kvfi
[50814]
[508K]
the value of dt, in the first of the equations [531]; namely, dt =~; from which [5081m]
the elliptical value [534c, 535] is deduced. This has the constant factor a^. If we
compare this factor, or — , with the calculation in [5346, &c.], we shall easily perceive,
that the numerator a^, is introduced by the term u^ [508b«], which is not altered in
the disturbed orbit [4861] ; but the denominator \/n, which is deduced from h [5346,&ic.], [SOSln]
is changed into \/«,i in the disturbed orbit [4863] ; and, by this means, it becomes .
/i, [5081o]
If we take the differential of [4828], and divide it by n==ar^ [4827], it becomes, by using
the abridged notation [4821/],
•512 THEORY OF THE MOON ; [Méc. Cél.
[5081'] That part of the second member of this equation, which is not periodical, is
„a ( 1 — 2e.(l — l-)'^).cos.cv-\-^^.cos.2cv — e^.cos.3cv-\-iy^.cos.2gv 'i
[508ip] dt=--.dv.< >•
V"- ( —iev^.cos.{2gv—cv) — iey^.cos.{2gv-\-cv) }
Now, changing the term — into — [5081o], we ought to get the factor which is
independent of Q, in [5081 Unes 1 — 4] ; and, upon examination, we shall find they
[5081g] agree ; except in some terms of the fourth order, connected with cos.Scw, cos.S^î), which
were neglected in computing the function [5081p or 4828]. To prove this, we shall repeat
the calculation [4821/i — m] ; retaining only the terms which produce quantities of the fourth
' ''J order in e, 7, and are connected with the angles 2cv, 2gv. By this means, [4821i]
becomes as in [5081?<] ; observing, that the last term arises from -|-5(/+e-cos.ct')'', which
[5081s] is omitted in [4821i). Now, from f:^ly^—i7^.cos.2gv [4821c], we obtain, by noticing
only the angle 2gv,
f=—^y^.cos.2gv; f2 = — l7\cos.2gv .
[508U] The first of these expressions ought to be changed into /= — (r^y® — j^v^)-cos.2^r, in
order to notice the term of the fourth order, which was neglected in [4812a, 6]. Finally,
the term —4 [Sfe^.cos.'^cv) gives, by noticing only the terms depending on cos.2^i), cos.Scd,
_6/e2_6/e2. cos. 2c« = fe2i.^cos.2^'-«— 3eV.cos.2cy [5081s].
Hence [5081m] becomes as in [5081«] ; and, by substituting cos.^t^ |cos.2cy-]-&ic. ;
cos.''<;« = icos.2cv-\-k.c. [6, 8] Int., we obtain [5081 w] ;
[5081u] dt^P.{l+2y^).dv.l—2f-Jr3{c^-.cos.^cv-\-P)—4.Qife^cos.^cv) +5{e\cos.''cv) \
„, , iSfiv^ il=-/).COS.22-«+3(e'3.COS.^CT iy*.C0S.2!rv)
[5081»] =hm-^2y^).dv.] ^'"^ l\{ ^ ^3^3 ^l, f [
( -|-Je^y .cos.2o-« — 2^-'y^.cos.2cv-f-oe^.cos.cv)
[5081«,] =P.{l-\-2y^).dv.\{ie^-^t^y^+^e%cos.2cv+(iy^+ie^y^—iy').cos.2gvl.
The terms between the braces are of the second and higher orders ; therefore, in finding
the terms of this function, of the fourtli order, we must obtain the factor h^.{l-\-2y^)
correctly, in terms of the second order. This value is easily found from [4823] ; which gives,
[5081X] f''- (1 +2"/^) = «^- a-l^'+hy')-
If the factor I — ^e^-{-^y^ be connected with the two terms of the second order in
rSOSlw], it will produce some terms of the fourth order ; and, by retaining terms of
this order only, we obtain the expression [5081y], which is easily reduced to the form
[5081z] ;
VII. i. >5, 15.] DIFFERENTIAL EQUATION IN t. 613
represented by,* [508l"J
r $*e2(— |e2+iy2)— |eV+|e*^cos.2ct>)
dt=a^.dv.{ } t5081y]
( +||y=.(— fe2-fiy=)+feV-4y'Kcos.25-i; )
= a^.ch. J (J-e"— feV).cos.2ci)+(feV-èv*)-cos.2o-y^. [5081i]
The terms between the braces in this expression are tlie same as the terms of the fourth
order in [50811ines 2, 3]. Hence it is evident, that the development [5081] is correctly
made.
* (2892) The function ri-fCi)- ~, wliose powers and multiples occur in rgQgg.
[5081], has already been developed in [4881', 4885,4889, Sic], and in the variations of
these quantities [4930, Sic.]. If we put the function [4885] equal to M^; and the [50824]
function [4889] equal to J\l^, the indices denoting the order of the functions; we shall rrnpo.q
neglecting terms of the sixth order ; hence, Q' [5081&] becomes,
We must add to this value of Q' the terms arising from the variations of the function
— ^M., ; the variations of the other terms being so small, that they may be neglected.
The chief term of the value of — iM„ is that which is noticed in [4929, 4930] ; namely,
/ Q,3„4 •sm.(2.-2.');
2 h^. u*
whose variation, relative to the characteristic 5, is evidently represented by,
Su' being neglected [5040«]. The function in the first memberof [4931m], is developed in
[4931_p], and we shall put this last expression equal to JV^, and that in [4932a] equal to
JV^; then we shall evidently have, for the two terms of the variation [50i?ls], the following
expression
[5082rf]
[5082e]
[5082/]
[5082^]
The second variation of the same function — i-M^, is easily deduced from that of
Jltu [4942], by dividing it by — 2?< = — 2(r^, nearly [4826]; using also [5082A]
VOL. in. 129
514 THEORY OF THE MOON ; [Méc. Cél.
'--> ^- \ >+âî|ë^ + !^;+i- Mr)-+(4'U)'] I [6092»].
[5082fe'] !!L^-= „j2 [5094] ; whence we get,
j-gQgg.., ISm^ {A[^^fe^.cos.{2cv—2v+2mv)
The other terms of — ~.f (~ ] • -3, which are noticed in [4944, 4945], produce the
following terms, which may be deduced from [4945 line 2], by dividing by — 2a~\ as
in [5082A] ;
[5082A]
— -^^ . -',.faSii.dv.\3.sïn.(v—v')-^l5.sm.{3v—3v')\.
The terms resulting from this expression may be obtained in the same manner as
[4946/] is deduced from [4945 line 2] ; or, more simply, by dividing [4946/] by — 2a~'
[5082A]. By this means it becomes, by using the value of m^ [50S2h'],
15 m^ a .,., cos.(t) — mv)
[5082^] -—— . - . Ai'K \
4a' I — m
Now, adding together the functions [5082c, g-, i, 1], we obtain
^ ~ h^'-^ \dv J ' xfi I" 2/1-1 • V^
^^^■S)-^-
1 5082m]
15m2 (^p)2es.cos. {2cB—2v~\-2mv)
4~ ■ 2c— 2+2/»
15?ft- a ..Q cos.('y — mv)
A a' - I— m
Substituting m^ [5082A'], in the value of iVl [4885, 5082e], and neglecting terms of
[5082m'] {|-|g second order, between the braces, which produce only terms of the sixth order in M^,
it becomes of the form,
[5082n] J|//^-=3^2 5_L_.cos.(2t)— 2mi-) +2P,.cos.(2y— 2m«+F)?;
as is evident, by mere inspection ; the symbol P^ being the coefficient of the first
order of any term between the braces in [4885], and 2v — 2mv-\-V the corresponding
VII. i. «5. 15.] DIFFERENTIAL EQUATION IN t. 516
The coefficient of dv, in this function, is not rigorously constant. [5082']
angle. The square of this gives, by neglecting P^, and the angles Av — Amv ,
Av — imv-\-V,
21 m^ CI , „ ^^} [50820]
Now, it is evident, by inspection, that the terms between the braces in this last
expression, are easily derived from those between the braces in [4885], by rejecting
2î) — 2m« from all the angles, and taking half of the first term in [4885 line I] ;
hence we get,
1 / 2 (1+m) , 2 n — m) \
I ^ 1 ^ _j V i_).e.cos.cw
„ _ 27 m" j 4(1— to) V2— 2m — c 2— 2m + c
^^2' =^ 16(1 — m)- ) / 7 1 \ ( ■ [50^*^?]
/ + ( I . i e'.cos. c'mv
I. ^\2 — 3m 2 — mJ
Substituting this in [5082m], and for M^, M^, JV^, JV, , writing the functions to
which they correspond [50826, 6',/], we obtain.
'?='-r.-/(f)-S+«-[/(S)v:]-«'«-
=1— i.function [4885] — i.function [4889] — ia.function [4931^]— |a.function [4932a] 2
/ 1 /2(l+m) 2 (!_,«) \
i ' ' . ^ i_ I e cos cv
21m* \4(1_m) V,2— 2to — c 2-2m + c/'
+Î6Tl"=^) • ) f 1 M > ' ' C 3
( +V2=3^-2^n)-i'-''''-'"'''
Ibjrfi {A[''>)^e^.cos. i2cv — 2v-\-2mv)
[.5082?]
4
4 2 c— 2+2m
15 m2 a ... cos.(f — mv)
4 o ^ 1 — m ^
The expression is now reduced to so simple a form, that we can; by the mere addition of the
terras, obtain the complete value of Q', as in the following table ; rejecting such terms
and angles as have beenusual'y omitted ; and putting
_" a
TO . - = m-, as in [5082/t'] ;
616 THEORY OF THE MOON ; [Méc. Cél.
[5082"] yjQ I^.^yg sgg„^ jj^ [4968], that the expression of i contains the term
Expres-
sion of
Terms of [5082fy].
[5082^ lines 2, 3]
[4885 line 1]
[4885 line 2]
[4885 line 3]
[4885 line 4]
[4885 line 5]
[4885 line 6]
[4885 line 7]
Corresponding terms of Q=l_L./^Î^YlV-/r^-tT-&c.
[4885 line 8]
[4885 line 10]
[5082«] [4931pline24]
[5082çline4]
[4885 line 12]
[4931pline26]
[4885 line 13]
[4885 line 14]
[4885 line 15]
[493 Ip line 29]
[4889 line 1]
[4931^ line 31]
[4932« line 3]
[50829 line 5]
[4831ijlines39, 13]
[4831^ lines 14,17,20]
[5082<7line3]
[4931iJlines7,16]
[4931_pline6]
27 m4
"*" 64 ■ [\-mf
(l+2e2-|e'2) _ ^ ,
— -^-^--^.cos.{2v—2mv)
[Tliis line has no factor.]
2(l+m)
'2-2711— c
(l+fe^— i7^— Je'-).e.cos.(2y— 2OTÎJ— cr)
2(1 -m)
.cos.(2t) — 2mv — c'mv)
2(2— 3m)
+
2(2- m
7 (2+3m).ec'
2 (2_3,„_c)
7 (2-3;n).ec'
2 (2— 3ot+c)
(2-|-m).ce'
cos.(2v — 2mv-\-cmv)
cos.(2u — 2mv — cv — c'mv)
.COS. (2y — 2mv-\-cv — c'mv)
COS. (2t) — 2mv — cv-^c'mv)
All the terms
except the
first line have
the conuiion
factor ^m'.
2{2—m—c)
ri(10+19m+8m2)^
+ )-2Ar > .^~^.cos.i2cv-2v+2mv)
-4- \ f'o o I o •cos.(2gi; — 2v-\-2inv)
(2— m).v2
4(2o--f-2-2m)'
17e'2
.cos.(2gv-\-2v — 2mv)
.COS. (2« — 2mv — 2c'mv)
+
+ ■
Cy^
.cos.(2v — 2mv — 2gv-]-cv)
+
2(2— 4m)
U(5+'«)
\ +2^o'i''' S 2-2OT-2â-+c
-]-2^i'^' ( 1 cos.(î> — mv)
+ 2?«.^fi f^'a'
....neglected '
- SAP +20.5i"e2 -2.^^31 -
J^2Ai'^^bAfe^—bS1h'' { £_'
1 — m
. COS. e mi'
9 m3_ C _7 ^
+161— m' (2 -3 m 2^
+ \ lA^^^-^2Af } . ~XQS.{cv—c'mv)
-\-2A[^'>e.cos.cv.
1
2
3
4
5
6
7
8
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
VII. i. ^15.] DIFFERENTIAL EQUATION IN t. Ô17
Q— '2 -
— — — ; which gives, in o", the term* ^Wiaf.e'-: thus, the quantity [5083]
— — contains the term |«/. <?«.?« .e'- [5083(?] ; now we have nearly, [5084]
«/=-, m = m- [5092', 5093] ; [5085]
Q 2
therefore, the expression of the time t contains the term — — .fe'~.dv; [50861
consequently, the value of the moon's true longitude, in terms of the mean
We have omitted, in tlie preceding table, several terms on account of their smallness.
Tims, we iiave neglected, in line 7, tlie terms depending on Af^ [493 Inline 22] ; in line
9, the terms depending on ./5/9' [1931^7 line 23] ; inline 20, the terms depending on
•^o^\ \ [4931/J lines 35, 37] ; in line 27, several terms of [4931p, 5082y], of the fifth
and sixth orders. Besides these, there are others depending on the angles
2v — 2mv-\-cv-{-c'mv, 2cv-^2v — 2mv, v — mv±c'mv, 2gv — cv, cv-^-dmv, 2cv.
These are neglected, because the terms arc of the fifth or sixth order, or are connected with
angles which do not increase the coefficients by integration in finding t, from (It [5081].
In the terms depending on cos.(y — mv), we have retained the terms dependin'^ on ^ '"'
[5082sline 21], and neglected a term of the same order, depending on A^"'> [50S2sline 22]. [5082»]
This is done, because ^-j''^ is required to a great degree of accuracy in [4874, 4904 hue 18].
The function Q' [5082s] is to be subnituterl in [5031], and then we may obtain the [5082t>]
constant terms [5082], as we shall see in note 2898 [5093m].]
a^=a;'.(l + im~.e'^+hc.y, hence, ~=a,^.dv.fl + §m'. c'^+ &icX
VOL. III. 130
[5082<]
* (2893) If we put, for a moment, — to represent the terms of the second member
of [4968], exclusive of the first and third, we shall have,
-=-.(l-~.e'^+m\ [SOSSa]
a a, \ 4 '/
The quantity m, contains another term, depending on e's, of the order in.JÏ? c'^,
2
which may be neglected, in comparison with the retained term ^ e'^ [5083a]. Involving rrngoi,-]
[5083a] to the power — 2, we get,
[5083c]
518 THEORY OF THE MOON ; [Mtc. Cél.
[5087] longitude, contains the term — ^mr.fe'^.dv, or — %m". fe'-.ndt. Hence
Kalin or _ ° " . - .
liVmo-"" it follows, that the three secular equations of the mean longitude of the moon,
[,nS^ iis perigee and its nodes, are to each other as the three quantities*
llie longi-
tude, pel Î- 2 „
[5089] sm, ^-^, -4=.
It is true,t that the terms, depending on the square of the disturbing force,
This expression contains the term ^af.dv.m.e'^, as in [5084], and by using the values
'■ ■' [5085], it is reduced to the form —.e'^.dv, which evidently represents the chief term,
depending on e'~, in the value of dt [5081]; and, by integration, we get, in t, the
[5083e] term —-.fe'^.do [50S6]. Changing its sign, and multiplying by 7i, we evidently obtain
g„ja
the corresponding expression in the moon's apparent longitude v [5095], — -^.fe'^.du ;
[5083/ ] 'i
which becomes — -—.fe'^-ndt [5087], by substituting the mean value of c/d = ?i(/< [4828].
* (2894) The secular equation of the moon's longitude is
— îm^.fe'Kdv [5087];
[5089a] that of the perigee is
i.--^ — .ft'^.dv [4982,4979]
and, that of the nodes is
[5089il — 1. — " . /"e's . dv [5060rt— 5061a].
Dividing these three expressions by the common factor — | ./e' -. dv , we find, that these
three secular motions are to each other as the quantities
I (2895) We shall, in this note, make some developments of the functions which occur
in [5081], preparatory to the calculation of the values of C'f, C^J\ &c. [5096 — 5116].
f 5090a.]
We shall commence with the computation of the terms of the Jirst pari of dt, or that
which is independent of a5u, and arises from the product of the two factors included in
[5081 lines 1 — 4]. These are found in the following table, which does not require any
particular explanation ;
VII. i. §15.]
DIFFERENTIAL EQUATION IN t.
519
produce a little alteration in the secular equation of the mean longitude ; [5089]
Terms of the first factor in] Factor Q'
[5081], between the braces. [5081 or 5082«]
whole of [508:2«]
-2e.(l — ^y-).cos.cy
e2.(|-j--|-e2_ jy2) .cos.Scr
.|e^.cos.2CT
i-/(l+te2-i72)cos.%z>
— e'.cos.Scu
— |c-/2.cos.(2g'B — cy)
— £e?^.cos.(2o'f+ci')
1
[50S2« line 2]
[5082* Ibe 3]
[5082s line 4]
[5082* line 5]
[5082s line 6]
[5082s line 10]
Corresponding terras of [5081].
whole function [5082s] multiplied by
\/a,
— 2c ( 1 — \y-).cos.cv
.J!t ,n_l_Oea_,^2_s,/2^ ^+cos(2v-2mv-cv)]
.3^(1+^) C-cos(2.-2m.)
•2_2m_c'^^+^^ ^^ ^' '^ l-cos(2i>-2»it,-2c«)
3m2(l— m)
2-3 m-^c
c^.cos.(2t) — 2mi')
3m2
"4(l-m)
cos(2t)-2mi')
[5082s line 2]
1
1
1
-\-î'>n^- K — 7r--ee'. cos.(2w — 2mv — cv — c'otd")
' J. — Am ^ '
-Tr — .ce'.cos.(2î) — 2m« — cv-\-c'mv)
2— m '
3m2.(10+19m+8m3).e3
8.(2c— 2+2m)
.cos.(2« — Imv — cv)
9niS
e2.cos.(2ci; — 2t)4-2»nv) \ \/a, '
First part
of the ex-
Q pression
16(1— m)
+( 1 +ie^— Jv-) • J7-- COS. 2^u
■f|S) ■ (l+Je2-Jy2_5e/3) .cos.(2^«-2« +2mr)
— e^.cos.Scj;
— fe7-.cos.(2o-j; — cv)
— ^ej^. cos.(2^i'+cd).
10
11
12
13
14
15
16
I'l
[50906]
In the next place, we shall compute tlie second part of the value of dt, depending on
aSu, which is contained in [5081 lines 5, 6]. Now, a Su is of the second order;
therefore, in calculating the product of the two factors by which aSu is multiplied, we [5090c]
shall not want any terms beyond the fourth order, and, in general, it will suffice to compute
them to the second or third order. We shall find, in the following table, the product of the
520 THEORY OF THE MOON; [Méc. Cél.
[5089"] but, it is evident, that the terms which have a very sensible
[5090e]
two factors of — 2a5u [5081 lines 5, 6] ; or, in other words, the product of the expression
Q' [5082s], by the following function, contained in [5081 lines 5, 6] ; namely,
[5090a;] J +jfc2_^2_3e.cos.ci'+3fi2. cos.2cv-\-i7^.cos.2gv—§e7^.cos.{2gv—cv).
Terras of
[5090rf].
il
—3e.cos.CT
Factor Q' [5082*].
1
1
3m2 (l+2e2-*e'2) .^ ^
3m2 2(l+m)e _ „
T-2:2;;r-7"°<2"-2'«^-^")
3e'2.cos.2CT
^y^.Q.os.2gv
3ot2 2(1— m )e
2~'2-2m+c'
cos(2y-2)ft!)+ct!)
1
3'»2 1 /^ ^ N
3m2
2 2(1— ;n)
,cosf2u— 2?«i))
Corresponding terms of Q', multiplied by
the factor {5Q<-Md\.
whole function [5032s]
— 3c.cos.CT
L-^l- f l_LOe2_ie'9V \ +cos(2«-2m«-cv) >
^8(l-m)-^' ' ^' .e ;e. ^ 4-oos(2r-2my+™) 5
9»t2.(l-|-m) ^ C _cos.(2u— 2mD) )
2(2— 2m— c)' *^' ■ ( — cos.(2u— 2mv-2ct)) \
~2(2-2m+c) ■ '^ • I — cos.(2y— 2mt)+2ciO \
+3e2.cos.2cî)
9m2. g c — cos.(2v— 2ffiii-t-2ct)) •)
"^8(1—7») ■ ^ ■ ( — cos.(2«— 2niy— 2ci') \
-[-|y2.cos.2g-D
9m2
32(1— m)
C —cos. (2^1) — 2'i)-|-2nir)) ")
■ I — cos.(3g-iJ-[-2y— 2mi)) ^
1
2
3
4
5
6
7
10
11
12
13
[5090/]
a'^.dv
This function [-50906] is to be multiplied by — 2a(5M.— — , to obtain the second part
of dt, contained in [5081 lines 5,6]. This process is performed in the following table
[5090^]. In the first column are given the terms of — 2 a 5m [4904] ; in the second, the
terras of [5090e], which includes, in its first line, the function [5082s] ; these terms are
taken in the same order in which they first occur in [5082s], and then in
[5090e lines 2— 13], omitting those terms and angles which are usually rejected ;
VII.i.>§,15.]
DIFFEx'.ENTIAL EQUATION IN t.
effect on the equation of the perigee, have but a very small and
521
[5089'"]
Terms of -iaiu [4904].
whole of — 2(/')M
— 'Mi\cQS.{2v—^mv)
Terms of [5090e]
— 2./3i'^e.cos.(2i' — 'imv—cv)
—'Mi^e.cos.{2v—2mv+cv)
— 2A.?^e'cos[2v — 2mv-\-c'mv)
— 2A'-^''e'cos{2v — 2m« — cfmv)
-2 Af^e'. COS. c'mv
— 2^i''Ve'cos(2i'-2/)ii;-Ci)+c'?»t))
-2w3'i''ee'cos(2«-2OTi;-ci'-c'm«)
— 2^f 'ec'.cos. {cv-\-cmv)
— 2Af'ce' .cos.[cv — c'mv)
— 2»4i'"e-2cos(2cj)— 2«+2«i')
—2Afhy^.cos.{2gv—cv)
-2A[^^''efcos{2v-2mv-2gv+cv)
— 2^i'"'.-,.cos.(i— mt')
3ni2
1
.cos.(2y — 2nir)
4(l-m)
21'«'-e' ,^ ^
"4"(2=3"»7f°'(^'"^'"''-'^'"^)
■.COS (2u-2mî)-f-c "îî')
4(2-7n)
12 -I 3
-3e.cos.c«
-|-3e^.cos.2co
os.2^'-j;
cos(2«-2/n?;)
-\'l-y'^.cos.2gv
3m2
4(1— m)
1(,.2 12
— 3e.cos.CT
-|-3!;^.cos.2fy
— 3e.cos.c«
3»!2
4(1 — ))!
— 3e.cos.c»
3ni2
cos.(2f-2mi')
4(1— m)
— 3e.cos.cf
— 3e.cos.c»
— 3c. cos. cy
— 3e.cos.cy
— Se.cos.cu
— 3c.cos.CT
— 3e.cos.c«
— 3c.cos.cy
— 3e.cos.CT
3nfi
cos.(2y-2mr)
VOL. in
4(1— m)
131
cos.(2y-2mi')
IfiS 'v^
Terms of [5081 lines 5, 6].
— 2aSn [4904]
AllUioso
Tonus
liavo tlio
, common
3m2 „, , \ factor
.5f?> i a^.dv
4(1— ,«
2— 3i«
^/a,
>.ecos.cmv
2— m
— 2^.^»^ { èe2-iv2| .cos (2u-2mi';
, , C +cos(2t)-2mv-ciî) )
+3Sfc. ] ^ ^i
C -\-cos{2v-2mv-\-cv) )
C +cos(2CT-2«4-2mD) )
(_ +cos(2CT-|-2«-2;nj)) )
— t^|V.cos.(2o-y-2D+2Mi')
3m2
— .^S'^e.cos.cw
4(l-m:
-2^</i.(^cMv^)-e.cos(2«-2m'-CT)
+003(21) — 2mv) j)
+cos(2ct-2«-|-2wî)) )
— 3.4/''e3.cos.(2i;— 2m'4-Cî))
+3.32®e3.cos.(2i;— 2/ni')
+3^SV.-
-Jj'.P^e'. COS. c'mv
' 4(1— wi
-{-3A^^''eecos{2v-2mv-cv-\-c'mv)
. 3m3
, ■Aj'^^e' .cos.dmv
' 4 (1— )«
+3^Sj'"ce'.cos(2i'-2mf-cv-c'mt!)
+3.4-ce'.|^'°"^^^+^''""^l
1^+cos.(ct — c'mv))
— 2^f.(|e2— iy2).e'.cos.c'/nw
4-3^fe2e'.cos(2y— 2my+c'mi')
-[-3.5<'ieV.cos(2«— 2my - c'mv)
-{-3Aft"e'.cos.c'mv
-\-3Afc-e'.co5.dmv
+3^i"'e='. cos.(2y— 2/««— ct)
+3A'^-'^th^.cos.2gv
-\-3A[''''':'^y%cos{2gv-2v-\-2mv)
1V'\—,.cos.[v — niv)
4(1— m
-2A'PK(lc^.i./-).-,cos.{v-mv)
5
6
7
8
9
10
11
12
13
14
15
16
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Second
part ofxUn
c.xurcssioti
of di.
[5C90g]
522
THEORY OF THE MOON ;
[Mcc. ca
[5089"] insensible effect on that of the mean motion [5090c, &c.]^
The third part of dt, which depends on the second power of a 5m, is contained in
[5081 line 7], and, by neglecting terms of the sixth order, it may be put under the form.
[5090^]
a2. dv
e .cos.cv
We shall, in the first place, compute the first of these terms, by means of [4904], as in the
following table ;
Terms of aiu [4904].
A'o°Kcos.{2v—2mv)
TliirJ part
oi the ex-
l)rcs.jion
<.f dt.
A[^''e.cos.{2 V — 2 m v — cv)
[5090i]
A'-^''e'. COS. (2 V — 2 mv-{-c'mv)
A-^h'. COS. (2 V — 2 mv — dmv)
Afhe'cos{2v-2mv-cv-{-c'mv)
^î'^ee'.cos. (2v-2mv—cv-c'mv)
A[^'^\~.cos.(v — 7)iv)
Terms of 3.aôit [4904].
3^^'".cos.(2t)— 2^1))
3.^i''e.cos.(2y— 2my— ct')
3-4^^e'.cos.(2!,' — 2mv-\-c'mv)
34i?'e'.cos.(2u— 2/ni'— c'mti)
3A^'''K".cos.(v-mv)
a
3A.2^'>''.cos.{2v—2mv)
3^'i'V.cos.(2u — 2inv—cv)
3A'-^''ee'cos{2v-2mv-cv-{-c'mv}
3A"''ee'.cos.(2v-2mv-cv-cmv)
3Js^''\cos.{2v—2mv)
3jl.2^'>\cos.{2v—2mv)
3.^f/V.cos.(2t) — 27nv — cv)
3^<'^e.cos.(2u — 2mv — cv)
3^/»'.cos.(2y— 2mr)
Corresponding terms of the
function [5090/t or 5081 line 7]
^(A'"A^
iA':KJl['\e. COS.CV
iAfKAf^c'. cos. c'mv
§A'-^\A^h'.cos.c'mv
^A(:\.^[''K-,.cos.{v—mv)
a'
sAl-KA['^e.cos.cv
All these N
lerm^ liiive
the factor
a-.dv
iA['\A['''é'e'co5.c\iv^°-'
§A['KA'^''c^c'. cos.c'mv
^A^"\A?^e'.cos.c'mv
§Af\A^*^e'.cos.c'mv
lA[^\A['^eh'. cos.c'mv
^A['\A^pe^c'. cos.c'mv
HA["\A;''K-',.cos.{v—mv).
«9. dv
I
2
3
4
5
6
7
8
9
10
II
12
13
14
[5090A;]
This table contains the development of the _^Ki! term of [5090^], ""' .3.{aoti)^. The
second term of [5090/i] is deduced from the preceding, by multiplying it by — 4e . cos. cv ;
but, we may neglect this part, because it produces only terms of the fifth and higher orders,
and of the forms which have been usually neglected.
In computing the part of Q', [50S2e — g], we have neglected the term depending on
VlI.i.'.MS.] DIFFERENTIAL EQUATION IN t.
The part of -, which is not periodical, is equal to - [4828,&c.5095]; [5
623
090]
hi or C [5082c, &:c.,4937J] ; also, that part of (3(/, which depends on the same quantity C
[4937f, A]. We shall now compute the effect of these terms, noticing only those arising from
the variation of the quantity y -^j— -^. sin. (au — 2ii') [5082rf], which is the most
important part. From this we get, by taking the variation relative to &u' , 6v',
— ^ -f"-^'- '5«'.cos.(2i;-2.')-{-^4i]. f'^idv.sm. (2v- 2v'). [5090„^]
These two terms of Q' are equal to the product of the two integrals in [4937e] by — \a. posom]
Now, the terms of [493Te] are developed in [4937m, ç]; and their sum, reduced as in
[4937r, Sec], becomes, by retaining only the most important terms, which increase by
integration, o--
_ï^ . ^ n ç^6)_|_ çp)_ Qm ^ _ g'_ COS. i!mv. [5090n]
a (J
Multiplying this by the factor — \a [5090m'], and substituting, for ni • - , its value »r
[5082/i'J, we get the following expression of these terms of Q' [5090m] ; namely,
lrr?.{'iCf^ C,3i— a'»>}. e'.cos.c'm«. [5090o]
This is to be multiplied by the conmion factor — \ — • [5081], and the product added to
the other terms of the second member of this value of dt. Hence, the complete value of
dt is found, by connecting together the terms of [50905,^, »,o]. This may be reduced
to the following form ;
dt = function [5082*] X— ;— + function [50906 omitting line 1] l
— function [4904] X 2.-^ + function [5090»- omitting hue 1] 2 [SOOOp]
+ function [5090/] +^" . |»r .{nq»)+ q»'— C^'CJ.e'.cos.c'mr. 3
We shall use this expression in the rest of this article, always taking the functions in the same
order in which they occur in [SOSOjj].
* (2896) In finding the chiefpart of the secular equation of the mean motion in [50S3,&c.]
we have only noticed the first term of the non-periodical part of dt [5082], and
*^"' . . [5090g]
have neglected the remaining terms of the fourth order, which are evidently much less than
the retained part. But this is not the case with the terms, on which the secular motion of
the perigee depends [4982,4979], since the term of g' [4974, 8.LC.], of the fourth order, [5090?-]
depending on the square of the disturbing force, is as great as any other of the retained
quantities. This is evident, by the inspection of the coefficient of e.cos. cu [4961],
upon which — p — ce'- depends [4975]. For, the term depending on A^^ [o090i]
524 THEORY OF THE MOON ; [Méc. Cél.
[5090'] and, if we neglect quantities of the order m\ this coefficient will be* — -.
[5091] We then have - = -.(1 — hn") [4968]; which gives - = l + im^ and
art, 0/
[5092] — = - = «^(l+im^)t. Moreover, we have, by [695'], n' ~ a'-.\/m';
therefore, t
[4961 lines, 4999, 5158], which arises from the square of the disturbing force, is as large
as the other terms of q.
* (289T) If we collect together the terms of dt [5090^], which are independent of
cfi.ih
the cosines, we shall find, that they all have the common factor • — ; — • ; and the
quantities connected with this factor are,
l-j ?l!î^ [50825line 1] ; f'"' JW [5090-hne 2] ;
[5092a] ^64.(1— ?«)2 •- ^' 4(1— m)
%{ •^'Cf^ii'^ff- c^ [5090; lines 1,7];
whose sum is as in [5082]. At the epoch, the constant term of —, assumed in [4828]
or [5095], is ^ ; putting this equal to the fictor of dv, in [5082], and neglecting terms of
the fourth order, we get - = ^ [5090,5090].
t (2898). Neo-lecting terms of the fourth order, as in the last note, we have, from
[5093a] [4968], - = -.(1— iwr), as in [5091]. This gives '^=\-\-^i,fi: whose square
root, multiplied by a-, is
^50031] -^] ==«*.(l+im^l = \ [5090'], as in [5092].
Now, by neglecting, in [605'], the mass of the earth, in comparison with that of the sun,
we get ?^ = f^-.v/M; and, by changing n, a, M, into n, a', m', respectively,
[5093c] ^^ ^^^^(.^j,^^^ j^jjjg present notation, we get n' = a'"^.v/w', as in [5092].
X (2899) Multiplying together the values of - ^ a^.(l+im2), n'=^a'-.^m'
[5093d] . r.ioon "' n-'./m' , . n\ c
[5092], we obtain the expression of m [4835], or - =m=— ^ .(l+im-). Squanng
VII.i.<^15.] INTEGRATION OF THE EQUATION m t. 525
^3 „,l
- = m^ = -
n
Hence we deduce,*
— = m^ = —j^.(l+lw') =m .(l+^rn^). ' [5093]
m = »i-.(l — km") : in.- = mr. [5094]
We shall now suppose the value of ?iï + £ to be of the following form ;t
this, and neglecting terms of the order m^, we get the first part of [5093] ; and, by using [5093 i
the value of m [4865], we get the last expression [5093].
* (2900) From [5093], we have m^ ^i .{l-{-im^) ; dividing this by (l+iî/i-), [5094a]
and neglecting terms of the order m*, we get,
m=m''.{l—W) [5094]. [509ib]
Moreover, by substituting the value of l-\-im^ [5091], in m^ [5094fl!], we obtain the
second equation [5094].
t (2901) If we examine the functions which form the expression of dt [5090p], we
shall. find, that it is composed of terms depending on the cosines of the angles included in the
function [.5095], with a few otiiers, whicli will be noticed hereafter [5239,5244, fee.]. I^^^^^"!
This expression, being multiplied by n, and integrated, gives the terms depending on
the same angles in [5095]. Moreover, the expression of -— , has the constant
dv
term - [5090]; therefore, ncit contains the term dv; and, its integral nt-{-s,
the term v; as in [5095 line 1]. Again, the expression of t has the secular term,
3)ft2
-—.fc'^.dv [5086] ;
2" [5095c]
and, by multiplying it by n, we find, that the quantity 7it + s contains the term,
§m^.fe'^.dv. [5095^,-,
At tlie epoch, when e':= E', the secular terra is supposed to vanish ; and this is effected
by putting it under the form,
§mKfie'^-E"^).dv, ^,095rf]
and making the integral commence with the epoch.
VOL. III. 132
526 THEORY OF THE MOON ; [Méc. Cél.
wi+5=î;-f I vf.f{e'~-E'').dv+ q»>.e.sin. {cv—^) 1
+q".e^sin.(2ct;— 2:=) 2
+Cf'.e^sin.(3ct'— 3^) 3
+ C^='.-/.sin.(2^ij— 2ti) 4
+C^^).e7-.siii.(2§-t5— CD— 2^+^) 5
+ Cf.sin.(2y— 2mv) 7
+Cf>.e,sin(2«— 2/«i;— ct'+î=) 8
e'ïà.rrf- +Cf>.e.sin.(2«— 2m^;+cr— ^) 9
-]-C'-p.e'.ûn.{2v—2mv+c'mv—^') 10
[5095] +C^"".e'.sin.(2z7— 2/«y— c'm«4-73') H
+ C<'".e'.sin.(c'mi'— ^') 12
+C;'-'.ee'.sin.(2i)— 2mi;— c»+c'm«+K— ra') 13
+C'/=).ee'.sin.(2r— 2mu— ci'— c'mi)+î3+ra') 14
+C<"'>.ee'.sin.(cw+c'mt7— s — ^) 15
4-C('='.ee'.sin.(cî; — c'm«— ra+^a') 1 6
+ Ci"^'.e^sin.(2cw— 2?;+2/«2J— 2n) 17
+ C('').7lsin.(2^i;— 2?;+2mw— 20 18
+Ci"^'.e'lsin.(2c'«i«;— 2=:') 19
+C''°'.-.sin.('«— »iii) 20
a ^
+Cf°'. -.e'.%'m.(v — mv-\-c'mv — ra'). 21
a
Then we shall have,*
* (290'2) Using the sign 2, of finite integrals, and putting any periodical term of
.. [5095] under the form C.sm.{iv-\-'^), the expression of nt-\-i becomes of the form
I i)uyu£/ J
[5096i]. Its differential, multiphed by -=— [5092f], becomes as in [5096f].
[50966] «<+-= = v+^m\f{e'^-E'"-) rf«+2C.sin.(«.+p) ;
[5096cJ i/a, * ' - ^ ^ ' V 1 W)
VII. i. ^ 15.] INTEGRATION OF THE EQUATION IN t. 52?
15ra- '>/(''
C<°'= ^^ ^ ; [5096]
3._L 1 p^ -T -, 9 O /1(10)
0
3c
2é
-J— 2^^'^)
4(1— m ) "*~ 2— 2m— c
cr>= .
[5097]
Cf>=: i- : [5098]
^'(4) 4 -^0 . [5100]
"^0 — 2g—c '
C'-'^'m: i— • [5101]
2g+c '
-3m^(l +2^^-10'") r 1+m 1-m 1) ^
' 4(1— m) '^'"'^•|2— 2ot— c"^2 — 2m + cj '
Values of
c.
V -2J^>.(l+ie--iy-)+3Jr'-^''+3Jf .e^ ^)_ 2
^" - 2— 2/rt
1
8(a>:— 2+2ffl) \ ~ i J 2 1 y ^^^^^^
2_2»i— c
Comparing this with the expression of (It [5090p], we evidently see, that the coefficient
C, of any term C'.sin.(it)-j-p) of the second member of [5095], may be deduced from
the term depending on the cosine of the same angle in the second member of [5090p], by L^"^°"]
rejecting the common factor — — , and dividing by the coefficient i, corresponding
to the proposed angle n'+p. By this means, we obtain the values of CJ,"', C*,", &c.
[5096 — 5116], as will appear, by collecting together the terms of the six functions [5096«]
[5090/^], relative to each of the angles separately, taking the terms in the same order as
they occur in [5090p].
First. Comparing the general form C.sm.(iv-\-p,) [5096i], with that depending on
Cf^ [5095 line 1], we get C=C^^\, i=c. The terms of C, taken in the order in
528 THEORY OF THE MOON; [Méc. Cél.
[5104] C(a)^4.(l-»»)^ 2-2,»+c ' ^ ' ^ ;
^ ^ ^ 2— 2/«+c
[5105]
^(9)^ 4-(2->«)
^'"' 2Jf>+3^f.e^
2 — m
21ms
[5106] C(""= "^'^^ ''^"';
_2JW+3J(^).e^
Values of
2 — 2— 3m
C 3m^^^°' _27m^ ) C _7 1_ > |
^ ^ 4 "^32.(1— m) 5 ■ i 2— 3m 2— m > /
+ 3.(A['^+Jf^).e'+3A['>.e^.(Af^+A^J') \ 4
+ f m^(ll.C'^'+2Cf— 2^"») / . 5
[5107] C'/i)^
which they occur in [5032sline 27, 5090Mine2,5090o- line 11, 5090ihnes2, 6], give,
without any reduction,
[5096/] c.C = 3m^.A[^^e- 2c.(l-lf) +^-.JI,'k+iAi\^[^^e+^A['>\A['^e.
Connecting together the first and third terms, also the two last terms of the second member ;
substituting also C = C'°'> e, and dividing by ce, we get Q"' [5096], neglecting
terms of the order m^.A^"e.
Seco7id. In the term [5095 line 2], we have C= C'J''e'^, i^2c , and then we get,
by connecting the terms depending on tlie angle 2cv, in [50906 line 11, 49041ine 11],
[5097a] -ic.l.^ e :^ [^ -f-^e —^j ).e — ^/i^ e.
Hence we obtain C^" [5097]. In like manner, Cj,~> [5098] is obtained from
[50906 line 15].
Third. In the term [5095 line 4], we have C^=Cl'''y^, i= 2^; and the terms
in [50906 line 13, 4904 line 13, 5090o- line 29], being connected together, give,
[5099a] 2g.C^Y={^+l<^''-i7'')-h^-^A^''7'+3Ai''' e^7^ \
whence we get [5099]. In like manner, from [50906 line 16, 4904 line 16], we obtain,
{2g—c) .q»ey^=—^e 7^— 2^<„'5) ^ ^s .
VII. i. §15.] INTEGRATION OF THE EQUATION IN t.
Cf-^--
4.('2— HI— f)
4.{-2—m)
— 2J|«'+34^>
'21wr
Ci'="=
21/»2.(2+3/h) .
4.(2— 3 m—c) 4.(2— 3/7i)
-2Jr'+34^)
2 — 3;« — c
-2Jp'+3Jf .
— 2Jf)+3Jf .
C?^>=
c — m
3m^(I0+19m+8m2) 3m2.(l+?n)
9m2
S.(2c— 24-2m)
— 3Jf>+3J("— 2Ji
C(16) = -
2_2m— c .16.(1— »i)
2c— 2+2m
2c— 2+2m
1
2
529
[5108]
[510!)j
Values of
c.
[5110]
[5111]
[5112]
whence we get [5100]. Also, from [50906 line 17],
{2g+c).C^J'Uf=—ief-, as in [5101].
Fourth. In the term [5095 line 7], we have C^=C'f\ i^ 2 — 2»?,; and, by
connecting togetlier the terms depending on the angle 2v — 2mv, we shall obtain, for the
expression of (2 — -2111). Cf\ the same expression as in the numerator of the value of
C'^'' [5102]. For, the first term of this numerator, with the factor — 3;»-, is the same
as in [5082$ line 2] ; the second term, with the factor — 3m^e^, is as in [5090Jline 5],
neglecting terms of the order m^ e'* ; the third term, with the same factor, is as in
[50906 line 7]. The terms depending on ^["\ are as in [4904 line 1,5090g- line 5] ; that
connected with A[^\ is as in [5090^ line 13] ; lastly, that depending on Jlf\ is as
in [5090^ line 16].
Fifth. In the term [5095 line 8], we have C^C'-^^'e, i = 2 — 2m — c ; hence we
get, for (2-2m-c).C-''e, the same expression as is given by [5103]. For, of the two terms
of the first line of the numerator of [5103], the Jirst is found in [50906 line 3] ; the second,
in [.5082s line 3]. The first term of the second line is found in [50906 line 10] ; the terms
depending on A['\ are in [4904 line 2, 5090^ line 12] ; that on ^™, in [5090^ line 6] ;
lastly, that on A["\ in [5090^ line 28].
Sixth. In the term [50951ine9], we have C=C'^h, i=2 — 2m4-c; hence we
get, for (2 — 2m-\-c). Cf'e, the same expression as is given by [5104]. For, the first term
VOL. III. 133
[5102a.]
[5103a]
[5104a]
53P THEORY OF THE MOON ; [Méc. Cél.
^.-,113^ It would seem as if this value of C['''^ ought to be of the order zero ; for,
of [5104] Is obtained from [5090i line 4], neglecting quantities of tiie order vi^e^ ; the
second term from [5082s line 4] ; the third term from [4904 line 3] ; the fourth from
[5090^1ine7J ; the fifth from [5090^ line 15].
Seventh. In [5095 line 10] we have C = Cf e', i=z2—2m+c'm = 2—m, nearly ;
,„- -, hence we get (2 — ni^.Cf'e', corresponding to [5105]. The terms being found in
[5032s line 6, 4904 line 4, 5090O- line 24], respectively. In like manner, [5095 line 11]
gives C^a^^'V, ^■=2— 2m— c'ot=:2— 3/«, nearly; and the terms of (2—3/»). C^^'e'
are found in [5082s line 5, 4904 hne 5, 5090^ line 25].
Eighth. In [5095 line 12] we have C= Ci'"e', i=^c'm^^m, nearly; hence we
[5107rt] ggj ?«.C^"'e', corresponding to [5107]. For, by comparing the terms of the five lines of
the numerator of [5107], with those in the preceding functions, we shall find that they agree,
as will appear by the following examination. The terms in [5082s lines 23, 24] give those
in [5107 line 1]. Those in [5082s line 25, 5090^ lines 3, 4] give [5107 line 2]. The terms
in [5090^- lines 17, 19] are -^-—^.{Af-^A-^^), as in the first term of [5107 line 3].
The two terms in [5090nines 3,10] make ZÂfK Af'> ; and those in [5090ilines4, 11],
3^„'".^!,"" the sum of these two expressions is ^A.^'^\{A^^'^-\-AiP), as in the second
term of [5107 line .3]. In [4904 line 6] we have —2Af\ and, in [5090i,'-line 23],
— 2^^5).(i(;a_iy3) ; whose sum is — 2.4^5'. (l+i-e^—iv^), as in [5107 line 3]. The
terms depending on A^\ A[o'' [5090jg- lines 26, 27], give those in [5107 line 4]. The
sum of the two terms [5090t lines 8,12] gives 3A['\A['^'e^; those in [5090» lines 9, 13] give
3^<''.-47'e^; the sum of these two expressions is 3A[^hK{A[''''-\-A['''), as in [5107 line 4].
Lastly, tlie terms depending on Cf, Cf\ C^'»' [50902jline3], give the terms in
[5107 hne 5].
JYinih. In the term [5095 line 13] we have C=:C['-''ec'; i=2— 2/»— f-j-('ï;i=2— ?n— c,
nearly ; hence we get (2 — tn — c).C[''^ec', corresponding to [510S]. For, the four
[5108a] terms of tlie numerator of [5108], correspond respectively to [5082s hne 9, 50906 line 9]
and [49041ine7, 5090^- line 18]. In like manner, [,5095 line 14] gives C^Cfee',
( = '2 — 2:71 — c — cin^=2 — 3m — c, nearly; corresponding to [5109] ; the four terms in
the numerator being obtained from [5082s line 7, 5090J line 8, 4904 line 8, 5090^ hne 20].
Tenth. In the term [5095 line 15] we have C = Cl^^^'ee', i= c+c'm = c+m, nearly;
hence we get {c -\- m) . C[^*^ e e' , corresponding to [5110]; the two terras of the
' numerator of C','^* being deduced from [4904 line 9, 50905-hne 21]. In like manner, we
get [50951ine 16 or 5111] from [4904 line 10, 5090^1ine 22] .
Eleventh. In the terra [5095 line 17] we have C= C['^^e^, i = 2c- 2+2m ; hence
vil. i. s^MÔ] INTEGRATION OF THE EQUATION IN t. 531
its numerator contains several terms of the order m,* and its divisor is of [5112]
we get {2 c — 2 -|- 2»i) . Cf"'' t-, corresponding to [5112]. For, tlie terms in
[50SJ* lines 10, 11, 12] give the first term and two last terms of the numerator of [5112].
In [50906 line 6], we get the term of [ôll2Iinel], having the factor (1+'") ; and in [5119a]
[50906 line 12] the last term of the same line ; in [.^OSO^-linesS, 14], the terms depending
on Ai\ A[''>; in ['1904 line 12], the term depending on A\^^\
Twelfth. Inthe term [5095 line 18] we have (■=^C['~'j^, i=2^— 2+2m; hence
we get (2g — 2-J-2»i). Ci'''7^, corresponding to [5113]. For, the terms in [5032«] lines
13, 14, give the first and last terms of [5113]. In [50906 line 14], we get the second
term of [51 13], neglecting terms of the fourth order [5112'"]. In [4904 line 14] we have
— 2.'J,'-> ; and, in [.5090^ line 10], the term — 3^^"', as in [5113].
[5n.3o]
Thirteenth. In the term [5095 hne 1 9] we have C =^ Ci^^h'^, i = 2c'm^2m,
nearly; hence we get 2 m .C','**'e'-, corresponding to [5114]. For, the term in
[49041ine 1.5], gives— 2^^'V2; whence we get Cf> [5114]. [BlUa]
Fourteenth. In tlie terra [5095 line 20] we have C=C[^^K-, i=:l — m ; hence
we get (1 — m).C['^K-, corresponding to [51 15]. For, the first term of [5082« line 19] [susa]
gives the first term of the numerator of [5115]. The terms in [SOSSs lines 20,21] give
'3m-. ./3""'
-r-^ — ^.(4+4m); adding this to the term deduced from [5090o- line 31], namely,
— '- — ^, the sum becomes ; !— — .(5-)-4«!). This difiers a little from the author,
4(1— m) 4(1— m)
who makes the factor equal to 5-j-3?H, instead of 5-{-4m. The term [4904 line 18]
gives —2A[''''; and [5090°- line 32] gives —2.A\'''\{le^—ly^); the sum of these is
— 2A'^~'>.{l-\-^e^ — iv^), as inthe third term of [5115]. Lastly, the sum of theterms in
[5090nine3 5, 14] gives 3A<-p.A[''\ as in [5115].
[51156]
a
a
a
" a . __ , . ._....- [5H6a]
a'
Fifteenth. In [5095 line 21] we have C =Cf '>'>.-,. e', i = 1— m + c'm= 1, nearly;
hence we get Cf^K-.e', corresponding to [5116]; this term being deduced from-
[4904 line 19], — 2^««.-.e'. Hence, the values of C;;", C['\ &c. [.5096— 5116] agree
with those given by the author, except in the small term of the fourth order, mentioned in
[51156].
* (2903) The two terms 3j1[^\ 2A["\ of the numerator of the value of €/»"
632 THEORY OF THE MOON ; • [Méc. Cél.
the same order. But, we have seen, in [4855], that if we retain only the
first power of the disturbing force, the value of C<"^' cannot have, for a
divisor, the square of 2c — 2-f 2m ; it must, therefore happen, that all these
terms, taken together, destroy each other, except in quantities of the order
m; which is a fact confirmed aposteriori by calculation. Hence it follows,
that, in the values of J<'' and J<"' [4999, 5009], in the expression
[5112"'] of Cj"^' [5112], we OM^A^ to re/ed i/je ie/ras depending on the squares of e,
e' and y. Each of these terms introduces in C{'°' quantities of the order
e^ ,• while their sum produces only a quantity of the order me^, which we
may neglect.* There is, therefore, a disadvantage in retaining only a part of
[5ll2'v] jj,ggg terms, and it is best to reject all of them. This is one of those
to- singular cases of approximation, in lohich ive deviate more from the truth, by
noticing a greater number of terms.
We then have.
[51166] [5112], are of the order m [4999,5009], and the denominator 2c— 2-f-2,77, of the
same expression [5112], is also of the order m, being very nearly equal to 2m — Sm"
[4828e].
Remark;
ble case
approxi
ntion.
[5116c]
* (2904) Several terms of the order e^, e'^, 7-^, have been neglected in the
investigation of the analytical expression of C/*"' [5112] ; as, for example, the factor
l-|-|e^ — \y^ — 1^'^ [50906 lineG] is omitted in [5112a]; hence, it becomes necessary,
I'llGJl "P°" ^''"^ principles adopted in [5112'"], to reject terms of the order e~, c'^ y~, in
computing the values of A'-^\ Ap, Jl[;''\ ^<'", &c., which are to be used in [5112].
Therefore, if the expression of A'^ be deduced from [5009], and put under the form
[51l6e]
^tul_3^;. ^. I /,^+ he^Jr1c/^+ Kf } ;
tj being independent of e, e', y, we must use
a,
in finding the va'ue of A'-^^ [5212,5112]; observing, that the terms k^, Jc^, kc.
have the divisor 2c — 2+2w in [5009 lines 1,2] ; and this introduces, in C|"' [5112],
[5116a-] 2^'"'
the divisor (2c — 2-f 2w)2, by means of the term — -— !— — , &c. Now, as a
xc — x-f-^m
divisor of the order (2c— 2-(-2m)- cannot occur in the first power of the disturbing
[5116A] forces [4855], it is necessary, that the terms of which l\ is composed should mutually
balance each other, so as to reduce it to the order m. The same is to be observed relative
[5116i] to k^, k,, 1%, Similar remarks may be made upon the value of Q"'' [5113], and
upon those of Ai'\ A,"", A\'^\ B'-'^, &c., which occur in [51 12, 5113, &ic.].
VII.i.§i6.] INTEGRATION OF THE EQUATION IN ï. 538
^(„, _ 8(90— 2+2m) 16. (!-;«) "^ ' ^ ^ 2g--2+2m [s^g^
We must apply to this value of C[''' a remark analogous to that made on [5ll3'j
C(i5) [5112'— 5112'^]. Lastly, we have, v.iue.of
e(i6)^_f?_ .^ [5114]
' m
„,_ )8.(l-m)^ 4.(l->») ■ ■ ^ - 4/;i- . . ^ ^^^^^^^
' 1 — m '
CÇ30) ^ _ 2 J('8). [5116]
16. We shall now determine the numerical values of these different
coefficients. For this purpose, we shall remark, that we have by
observation ;*
ohserVQ-
tjoii.
[5117]
m= 0,0748013; log. m = 8,8739091. Da,afr,.n.
c = 0,99154801 ; log. c = 9,9963137.
g= 1,00402175; log. ^ = 0,0017431.
e'== 0,016814, at the epoch of 1750; log.e'= 8,2256710.
y = 0,0900807 = tang. 5'' 8" 50',4 ; log. 7 = 8,954631 8.
According to observation, the term C''^\e.sm.(cv — to) is nearly equal to rsjigi
— 22677^5. sin. (ctJ — to) [5574]. We have given the analytical value of
* (2905) The values [5117] agree very nearly with Burg's tables ; observing, that
the moon's motion is represented by v ; the motion from the perigee is cv, and, from [^'l''']
the node, gv [4817] ; the sun's motion, neglecting the periodical terms, is mv
[4835,4836]. The excentricity of the solar orbit is represented by e'; it is the
same as e" [4030], taken to six places of decimals ; the neglect of 5, in the eighth decimal
place of e', produces a small difference in the logarithm of e" or e, given in
[4080,5117]. Lastly, y represents the tangent of the inclination of the lunar orbit to "^^
the apparent echptic [4813, 4818, &ic.]. The value of m [5117] gives m^ = 0,0055952, [5117rf]
which is frequently used in this volume.
VOL. III. 134
634 THEORY OF THE MOON ; [Méc. Cél.
[5119] ^(0) in [5096] ; and, if we substitute in it the values of A[°\ A'-^', given
Assumed^ by a first approximation, we obtain,*
[5120] e = 0,05487293.
This value is sufficiently accurate for the determination of the coefficients
Af\ JJ", Jf, &c. We have supposed, in conformity with the phenomena
^^^ ^ of the tides, that the moon's mass is y|y of that of the earth. f This
being premised, the equations between these coefficients [4998 — 5017,
5062—5077] become,î
[5122] J(») =0,00723508— 0,00501814.{£i'"—5y>|;
[5123] J_» = 0,204044— 0,0660894.4°'— 0,0480577. { Bf—Bf> \ ;
[5124] Jf __ 0^00372953;
[5125] J(3) =_o,00315160— 0,00449610.fi<^>;
[5126] 4') = 0,0289026— 0,00564793.5<'°> ;
[5127] J_6) =— 0,193315+0,104996.J;"+0,372796.4«);
[5128] ^(-) = 0,538027+0,0334044.Ji"+0,135144.^p) .
[5129] ^(8) = —0,0908432+0,139071.^1"— 0,280299. Jp' ;
[5130] ^(0) = 0,0791 193+1,055799. J<')+0,270902. Jf ;§
[5120a]
* (-2906) The assumed value of e [5120] differs but very little from that finally
adopted in [5194].
[.5121a] "t" (2907) This value agrees nearly with the result obtained in [4.321] ; the author
afterwards decreased it to t-j.^j^ [4631 « — J].
X (2908) The equations [5122—5140] are obtained from [4998—5002,5004-5017],
by taking them in the same order, and dividing by the coefficients of »4o"", A'-^\ -^é^\
[5122a] §jc. respectively. The equation [5003] is afterwards used in finding ^^5) [5205] ; and,
in like manner the equations [5141—5156] are derived from [5062-5077], using the values
[512261 of m, c, g, e', y, e [5117,5120]; also — ^ ==??i^ [5082^']. Upon examination
it will be found, that the numerical results obtained by the author are, in general, very
correct ; the differences being rarely more than one or two units in the last decimal place.
The few cases, in which a greater difference was discovered, will be mentioned in the
following notes.
■^ (2909) It will be found, by examination; that the coefficient of .^'^'K in this equation,
VII. i.^ 16.] COMPUTATION OF ^(."'', B;.""), €<."•>. 635
jc^'») = 0,00285368— 0,0041501 8. /?(">; [5i3i]
J<") = 0,366100— 0,01 72.338. J'/'— 0,259744.4'")— 0,324680. (^i'>)'; [5i32]
Jf' = 0,00265066 ;* [5i33]
4^') = 0,0523335—1 ,555935.S<»> -0,22021 G-A"^^'^ ; [5i34]
4»)=— 0,0129890; [5135]
4'5) =. — 0,1007403+0,0385084.4"+2,09016.^î'="
[5136]
— 1,022473. Jf)—36, 11 032.^5^^)— 5S°'.£^=' I ;
4'^' = 0,114623+0,166591.4"^'— 5,07811.5^^); [5137]
Fuiid.i-
4") = — 0,121028t+0,937593.4°'— 0,000031563.4'«> eTua",L
[5138]
— 0, 1 39767 . { Bi''^+Bi^'^ ] ; "> le.er J
' i 2 ' 2 J ' mine .4,5.
4^' = 1,208124+1,018700.4'"'— 5,074801.4'^' ; [5i39j
4>3) = — 0,121 295+0,675879. Ji'^'+0,183834.4'«' ; [5140]
^f"' = 0,0287031— 0,0574772.4"'+0,000432665.4" ; [shj^
5y' = —0,00000236395 ; [5142]
Bf =-0,00564433+0,0048210.5»'; [5143]
Bf =0,0166486+0,0166486.4"- 0,0165194.5;"'; [5144]
5'^' = 0,00656716—0,00708386.5;°' ; [5145]
5f' = 0,0000147361—0,00681821.4' ; ^5146^
5f =—0,0183098— 0,01700]3.{4"—5("'|; ^gj^^^
5i" = 0,0809777+0,0249192.5^'— 0,0478194.5'"" ; ^gj^g^
5(8, _ —0,0868568+0,1 87099.5f'+0,0556224.5'/' ; ^5^4^^
5(9) = —0,0263090— 0,0787687.5',°'+0,0506541 .5f' ; ^g^g^]
5["" = 0,0712575— 0,03047765.5(,'"+0,021 1 192.5</> ; ^gjgj^
ought to be increased about one tenth part ; but, as this difference does not materially affect
1 • • 1 r • [5130al
the results, no notice is taken oi it.
* (2910) Upon repeating the calculation of this value of A^^"^, it is found to be
greater by about -^-^ part, or five units in the sixth decimal place. This difference is l5133a]
unimportant.
t (2911) The numerical values of the coefficients [5138] agree with the equation
[5015]. A very small change in the constant part — 0,121028 would be made, by
introducing the term depending on — §7^ [496 Ih] ; but the effect is insensible. [5138a]
536
THEORY OF THE MOON ;
[Méc. Cél.
5;"> = 0,421270+0,842540.4"— 0,337016. J<'^'+0,586564.£1'"
[5152]
+0,157666.5f);
[5153] ^('2) = 0,000194141— 0,168403.4"+0,0673614. S Ji")+i5<"'j;
[5154] 5('" = 0,0847889+0,147896.{4')—i5;'"}— 0,0591586.4"^ ;
[5155] 5^") = — 0,01 2561 9;
[5156] 5f' = 0,00386625.
From these equations, we have obtained the following values ;*
[5157] 4«' = 0,00709262 ;
[5158] 4') := 0,202619;
[5159] 4=' == — 0,00372953 ;
[51 60] 43) = _ 0,00300427 ;
[5156o]
[5156i]
[5156c]
[5156rf]
[5156e]
[5156/]
[5156g-]
[5156;i]
* (2912) Substituting the value of B^^^ [5142] in [5122], we obtain a linear equation
in ^'"i -Si"'- Combining this with the four Knear equations [5123,5141,5146,5147],
containing the five unknown quantities ^'^\ A'^\ B[°\ B^\ B^^'', we obtain Jive linear
equations ; from which we may deduce these five unknown quantities, by the usual rules, as
in [5157,5158,5176,5181,5182]. Substituting these values in [5143,5144,5145], we
get Bf\ B^^\ B^^ [5178—5180]. Using the value of B[''> [5176], we obtain
from [5148,5151] two linear equations, for the determination of ^7'= ^î'°' [5183,5186];
and, from [5149,5150], two linear equations, to find B'f\ Bf> [5184, 5185]. Hence
we easily obtain, from [5125, 5126], the values of Af\ 4^' [5160, 5161]. Substituting
w^l" [5158] in [5128, 5129], we get two linear equations, to find A'-p, 4« [5163,5164];
and, in like manner, [5127,5130] give Af\ Af'> [5162,5165]. We may remark,
that these values of Af\ Af\ are both affected by the small correction [51o0o] ; but
the effect of this correction is insensible. Substituting the values of A'i\ B\''^
[5158,5176] in [5131, 5132, 5152, 5153], we get four linear equations, for the
determination of .^<'<», .4<"i, SJ,"', B['-'' [5166,5167,5187,5188]. Substituting
4'2)_ jB'^) [5168,5176] in [5134], we get A['^^ [5169]. Substituting, in [51.36,5137],
the values of A[^\ A^^^\ &c., which we have already investigated, we obtain two
Ztnear equations, for the determination of ./3o''^^ ./3*"' [5171,5172]. In like manner,
the three equations [5138 — 5140], are linear in A['''\ A^^^\ A\^\ and give their
values [5173, 5174, 5175] ; which would be altered a little by the introduction of the
correction [5138a]. This correction is, however, quite unimportant. Finally, with the
values we have already computed, we easily obtain, from [5154], that of i?','^' [5189].
This completes the investigation of the series of terms contained in the equations
[5157—5191].
Vll.i.§16.] COMPUTATION OF ./?„«", B'ir\ CTl 537
[5161]
[5162]
[5163]
[5164]
[5165]
[516G]
[5167]
[5168]
[5169]
[5170]
[5171]
[5172]
Values of
[5173]
JÎ, B.
[5174]
[5175]
[5176]
[5177]
[5178]
[5179]
[5180]
[5181]
[5182]
[5183]
[5184]
[5185]
[5186]
[5187]
[5188]
4'' =
0,0284957 ;
A'^^ =-
-0,0698493 ;
A['> =
0,516751 ;
Af^ =-
-0,207510;
yif =
0,274122;
Jf =
0,00081065;
jni) =
0,349068 ;
4"' =
0,00265066 ;
J(I3) _
0,0075875 ,
4"' =-
-0,0129890;
4-) =-
-0,742373 ;
^(16) =_
-0,041378 ;
^<>'' =-
-0,113197;
4«' =
1,08469;
^(19) =
0,001601;
5<«) =
0,0283831 ; ,'
5^>' =-
-0,00000236395;
jBf =-
-0,00550748 ;
^f' =
0,0195530;
B','' =
0,00636608 ;
£(5) =.
-0,00136676;
/?f =-
-0,0212720;
B'-p =
0,0782400 ;
B['^ =
—0,0833684 ;
Bf^ =
0,0327678 ;
Bf) =
0,0720448 ;
£(..) =
0,491954;
^(12) =
0,0061023;
VOL. Ill
135
538
THEORY OF THE MOON ;
[Méc. Cél.
[5189]
[5190]
[5191]
[5192]
[5193]
Corrected
value of
e.
[5194]
[5195]
[5196]
[5197]
[5198]
[5199]
[5200]
5;") = 0,0920621 ;
iJ(H) __0,0I256]9 ;
Bf^ = 0,00386625.
By means of these values, we have corrected the expression of e [5120],
making use of the equation,*
q''e= — 22677%5.
The expression of C^"' [5096] gives,
C(0) _ —2,003974;
hence we obtain,
e = 0,05486281 ; log.e = 8,7392781 ;
which differs but very little from the value before used [5120]. Then
we find,t
q» = 0,752886:
Cf =—0,336175:
C„3) = 0,243118:
q^) = 0,722823
e(5) =—0,250034:
Cf) =—0,00919876;
* (2913) Comiiaiing the expression C^''e.s\n.(cv — ra) [5095 line 1], with its value,
[5192a] deduced from observation, — 22677%5. sin. (cv — zs) [5574], and adopted in Burg's
tables [5574a], we get the expression of C^^'e [5192]. Now, substituting in [5096],
the values of m, c, y, .4«>, ^i" [5117, 5] 57, 5158], we get the value of C^"»
[51926] [■5J93J . jjnd (^en, from [5192], we obtain the corrected value of e [5194].
t (2914) Substituting the values [5117,5157—5175,5194], in [5097— 5106], we
get [5195— 5204]. Having thus obtained Cf, Cf, C^"» [5200, 5203, 5204], we
[5195a] j^^^y gQj^^^jy^g .^f [5205], by means of the formula [5003]. The values C','", C','=',
Cp\ are derived from [5107,5103, 5111], which contain A'f, Jlf'; but the effect
of the correction [5156(/] is insensible. The expressions [5208, 5209], are deduced from
[5109,5110].
VII. i.sM6.] COMPUTATION OF A\r\ B„"'\ Cf- 539
C<-p =_ 0,4 14046; [520i]
Cf = 0,0129865; [5202]
Cf = 0,00392546; [5203]
q'») =—0,0387853; [5204]
JÇ5) =—0,00571628; [5205]
C.")= 0,196755; [5206]
C['~^ = 0,127650; [5207]
C('^)=_ 1,081734; [5208]
C<"')= 0,373115; [5209]
Cf'= — 0,616738. [5210]
We must, by the preceding article [51 12'", 51 13'], in calculating the values of
ç.(i6)^ C,"', use the values of A['\ J<"\ A[''''\ determined by
nefflBcting the squares of the excentricitj and inclination of the lunar orbit.
We have thus found the following values of J<'>, J(">, J<") and Bf\
which must be used in this calculation ;*
4" = 0,201816; [5211]
4")^ 0,349187; [5212]
jn3)= 0,0077734; [5213]
B(o) = 0,0282636 ; [5214]
hence we deduce,
q'°' = 0,272377 ; . [5215]
C('^> = 0,033825. [5216]
* (2915) The principles upon wiiicli these quantities are neglected have been
explained in [511-2', &c.; 51 16c— ij. The quantities ^^% .4™ [5157,5166], being [5211a]
very small, tlieir corrections are unimpartant ; and the author seems not to have noticed [52116]
these corrections in [5211, S>:c.]. The calculation of the terms [5211 — 5216] is made in
the following order. Af is given by [4993] ; then A\'\ by [4999] ; ^V»', A[}^'', by ^^gjj^^
[5008, .5010] ; .^(">, by [5009] ; JSj% by [5032] ; and .^i'^)^ by [5011]. The values
thus found, differ but little from those in [5211 — 5214]; and, by substituting them in '^ -
[5112, 51 13], we get [5215, 5216], neglecting always e^, e'^ f.
540 THEORY OF THE MOON ; [Méc. Cél.
Then we have,*
[5217] Cr = 0,173647;
[5218] Cr = 0,236616;
[5219] Cf' =-2,16938.
This being premised, the expression of nt + e [5095], becomes, by
reducing its coefficients to seconds,!
nt+s = v+^,m^f(e'^—E").dv 1
—22677^5 .sin.(c«— 33) 2
+ 467%42.sin.(2c«— 2^) 3
— 1 1',45 . sin. (3cD— Ss:) 4
+ 406',92 . sin.(2^v— 20) 5
+ 66^37.sin.(2§-i;— ct-— 2^+^) 6
— 22%96.sin.(2^-2;+cf— 2c'— ^) 7
— 1897%38.sin.(2w— 2m?;) 8
— 4685^^45. sin. (2u—2w?f—ci-+^) 9
+ 146^96. sin. (2f—2m«;+ctJ—^) 10
+ I3',6l . sm.(2v—2mv+c'mv—z/) H
— 134',51 . sm.(2v—27}iv—c'mv+^') 12
+ 682^37. sin. (c'mv—T;^') 13
+ 24',29.sin.(2«;— 2mf— cz;+c'7ni'+-— ^') 14
— 205%82. sin.(2«— 2mii — cv—c'mv-\-^-{--!^') 16
+ 70%99 . sm.{cv+c'mv—-—'/) 16
— 117^35. sin. (cî; — c'mv — tô-f^') 17
+ 169'10.sin.(2c2;— 2v+2mtJ— 2î:) 18
+ 56',62.sin.(2^f— 2«+2mr— 20) 19
+ 10',13. sin.(2c'mv— 2^') 20
+ 122',014.(l+z).sin.(f— mil) 21
— ld%809.(\-\-i).sm.(v—7nv+c'mv—^'). 22
Formula
for Ihe
determiii
ation of
t.
[5220]
* (2916) The values of C['^\ C['^\ CJ,^»^ deduced from [5114— 5116], agree
^^^ very nearly with those given by the author in [5217 — 5219].
I (2917) Substituting, in [5095], the values of e', 7 [5117], e [5194], and those
[5220a] of C»', Cf\ he. [5195—5219] ; also ^, [5221], we get [5220].
VII. i. ^5< 16.] EQUATION BETWEEN t AND i-. 541
The two last terms were determined by supposing - = -tttz- • This [^2'ii]
fraction depends on the parallaxes of the sun and moon; it differs hut very
•J * ■' I • • I 7 Sun's
little from -^\-^ ; hut, for greater generality, ive have connected it with the p"!!""-
indeterminate coefficient 1 +/ ; and, by comparing the term depending on
sin. (r — mv), with the result of observation, we shall hereafter determine the
sola r para llax [5589].
[.5222]
[5223]
[5224]
[5223o]
It is evident, by what has been said, that the perturbations of the earth's
orbit, by the moon, introduce in A[^'^, the quantity 0,25044.tA ;* and,
therefore, in C{'°\ the quantity — 0,54139.1^; whence arises, in the
expression of the moon's apparent longitude, the inequality,!
« (-2918) Using the value of m^ [5082^'], we find, that the coefEcient of Jl\^'\
in [5015], is 1 (1 rn)3 M2.(36+21m-15mg) .
^ ' 4(1— »7l) '
and, the term depending on (ji, is
-2m^.f.. [ |(l+2e2+2e'^)+;^^^^.(l+fe3+2e'2) | . [5223fc]
Dividing this last expression by the preceding, and changing its sign, we get the term of
^'/"', depending upon (x. Substituting the values of ?«, e', e [5117,5194], it becomes rsooQ n
0,25044. (A, as in [5223] ; /x being the ratio of the moon's mass, to the sum of the masses
of the moon and earth [4948'].
t (2919) Tlie symbol fi. is introduced into the expression of C','^' [5115], by
means of the value of A[^~\ Now, the coefficient of .4^''^ in [5115], is
4(1 — my 1 — m 1 — m
and,if we use the values of ?«, y, e, .5^°' [51 17, 5194, 5157], it becomes —2,1326.
Multiplying this by 0,25044.fj. [5223], we get — 0,534.(/., instead of — 0,54139.|(a [5224].
This part of C,'""-" produces, in the expression of 7it-\-s [5095 line 20, or 5220 line 21],
the terra r> co/i " ■ / \ [522561
— 0,534 . fj^ . - . sm. [V — mv) ; \_o<.^oui
and, by changing its sign, we get the corresponding term of the moon's longitude v [5225].
The inequality of the earth's motion, depending on the direct action of the moon [5225c]
[4314, 43166], using the same symbols as in this article, is
fA.-, . sin. (v — mv) [5225'], nearly ; [5225rf]
as is evident by comparing the notation [4313] with that in [4757, Sic.]. The ratio of the
two inequahties [5225, 5225'] is as in [5226].
vol.. III. 136
642 THEORY OF THE MOON ; [Méc. Cél.
[5225]
a
0,54139 ./^ . -, . sin. (v—mv).
a
icUoTof The direct action of the moon upon the earth produces, in the motion of the
earth, the inequality,
[5225'] f^ • -, • sin. (v — mv) ;
this action is, therefore, reflected to the moon, hy means of the sun, but decreased
[5226] in the ratio of 0,54139 to unity.
The preceding expression of nt-\-i, contains the coefficients c and g,
[5237] which depend on the sun's action. We have given their analytical values in
[4986, 5228a:], and, by reducing them to numbers, we have,*
[5228] c = 0,991567;
[5229] g = 1,0040105.
* (2920) Dividing the coefficient of cos.(n)— ra) [4961 lines 3— 7], by i — -,
we "et ^+</s'^ [4975], as in the following expression, using the value of m^ [5082^'];
24-c2^3e'2_2(B«'4-Bra).ll^_(l_^2„j_e).^(2).(]_5e'a^
m
[5228a] p-\-qc'
, ... ; -4 [ i+2»+(4.r^'^-i).(^:-+^y ] . ^r>. (1-f.-) I
'"^''^ '-y^.Kl+6m+c).(l— «)+7+(2— 2m— e)2},^(').(|_|e'2)l
[52286] We have seen, in [4976o, h\, that the quantities Âi\ Ap, Bf\ B'„:\ £«> contain
implicitly the factor 1 — Je'- ; whicli must be particularly noticed when finding the
[5228c] values of f, q, from [5228a]. Thus, if we neglect terms of the sixth order in the
equation [4998], we shall find, that the term [4998 line 1] may be put under the form
[5228d] f »'• ^{l + (H-2«0.c^+i7^|.(l-fe'^).
The factor 1— |e'- is equal to 0,99929322 [5117]; and, if we put, for brevity,
[5228e] - = 0,99929322, we shall have 1 = lc.{\ — 4e"~). Hence it is evident, that, if we
ft
have found, by a previous computation, the numerical value of the first line of [4998],
[5228/] which we shall represent by A^, we can put it under the form A^Tc . (1 — |e'^) ; and,
VII. i.^ 16.]
EQUATION BETWEEN / AND v.
543
The motion (\—c).v of the lunar perigee [4817] is, therefore, by the
preceding theory, equal to 0,008433. w [5228]. This motion is, by
[5230]
[5228A]
[5228i]
[5228ft]
[5228/]
15228m]
by this means, it is reduced, by a very simple method, to the form —p — qc'^, adopted
in [4975]. In Hke manner, the second line of [4998], which may be represented by A^,
can be put under the form A,k.{l—U'-). The term Bf, which occurs in the third
line of [4998], can be put under the form Bi'''A'.(l— |e'~) ; as is evident, from the [5228g-]
inspection of the formula [50G-2], neglecting the small terms, similar to those omitted in
[5228c]. Lastly, the term B':!\ which occurs in the third line of [4998], is nearly
equal to — 0,000002 [5177] ; and, as this is so very small, we may put it equal to
B^''>k.{l — 2f'^)- Hence it appears, that, if the analytical value of Af^ be deduced
from [4998], the terms depending on e'-, will appear very nearly under the form of the
factor (1— |e'^); so that we may deduce, from the numerical value of ^i,"' [5157],
the term depending on e'^, by changing A^^ into A'-°^k.{l — ^e'"). Proceeding in
the same manner with [4999], we find, that the terms depending on e'^ may be obtained,
by changing ^i"' into ./?,"it.(l — |e'^), and using the numerical value of ^[''' [5158].
In the equation [5000], from which «/Z,'-' is deduced, the terms depending on e'- are
omitted, on account of their smallness. But, if we inspect the functions which are
enumerated in [4961(Z, c], and used in the formation of the equations [4999,5000], we
shall see, by noticing the terms depending on e'~, that the chief terms of •/1[^'', •^f\
are formed in the same manner, with the factor 1 — Je'^, as in [4879A', 4879/line 1] and
[4876e lines 2,3, Sic.]. Hence, it is evident, that we may proceed with Af' as we have
with .4','> [522S/r], and put Jli' = Afk.(l—le'-). The terms of e' 2, which occur
in the values of B'f\ B?' [5064, 5065], produce not much efiect in the computation
of i^e'-, or iqE'^, in the value of c [4986] ; so that we may, without any
sensible error, change B'f into Bf'k.{l — ie'^), and jB«^ into Bf A.(l— fe'^),
as the author has done. Hence, it appears, that if we neglect terms of the order e'*, we
shall obtain very nearly the terms depending on e'^, in the second member of [5228a],
by substituting
A^\(l—^e'^)=^fk.{l-5c'^) ; ^;'\(1— |c'2) =:^',»7c.(l— 5c'2) ;
^f .(1 -Je'2) =^«>t.(l— 5c'2) ; B'i' = 27,->A-.( I— fe'2) ; 5f = Bf'>k.{\—^e'^) ■
and then putting the terms independent of c'^ equal to p, and the rest equal to qe"^.
Having thus obtained the analytical expressions of p, «7, we must substitute in them the
values of A° , A^\ he. [5157 — 5179J, and we shall obtain very nearly,
p=: 0,01678 1; 9 = 0,04973.
Substituting these values, and E' = c':= 0,01 6814 [5117], in the expression of c [4986],
it becomes very nearly as in [5228]. From this we obtain the expression of the motion of
[5228n]
[5228o]
[5228;>]
[52289]
[5228r]
544 THEORY OF THE MOON ; [Méc. Cél.
fJa'peri-*^ obscrvation, cqud to 0,008452.î; [5117 line 2]; which differs from the
gee.
[5231] preceding hut by its four hundred and forty-fifth part.
[5231']
The motion of the perigee is subjected to a secular equation, whose
analytical expression is given in [4982, &c.]. Reducing it to numbers, it
becomes,*
[5228r'] the perigee (1 — c).v [4817,5228], as in [5230] ; which agrees very nearly with that
deduced from observation 0,00S45199.d [51 17 line 2].
[5228«] The coefficient of y.sin. (g-u — Ù), in [5019] is put equal to y+î"fc'^ [5053];
hence we get, by using [5082A'],
\-\-2e^—ly^-\-%e'"~ ^
-I I '^^"-^^;-'g+"\z?f>+4^^°)^ .(l-le'^)
[5238f] p"^_y"e'2 = |m'
a
[5928t/]
Substituting the values of Bf%.(l — Je'-), &:c. [5228o-,o]; and then putting the terms
which are independent of e'^ equal to p", and the rest equal to q''e'^; we shall
get the analytical expressions of p", q". Reducing these values to numbers, by means
of [5157—5186] we get, very nearly,
[5228t)] p" = 0,0080337; ç"^ 0,0123967.
These values and that of E' [5228)], are to be used in finding the retrograde motion of the
nodes [5059], which becomes, by retaining only the terms depending on the first power of v ,
[5228H |^/(l+y')_l+ _|_,^.£'a| .,.
Putting this equal to the expression {g — l).v, which is assumed in [4817] we get,
[5228.] ^, = ^(l+p'')+J^,^ ;
and, by substituting the values of p", q", E' [5228f,r], we obtain g [52i29].
* (2921) The secular motion of the perigee depends upon the term \q' .fe'^.ilv
[5232a] [4982] ; which may be put under the form iq'-fu''i^'^ — E'^).dv [5095c — </]; supposing
the integral to commence at the epoch where e'^E'. Using the value of q' [4979],
[52325] and multiplying by g^, it becomes, L ^ J Vf ?ft^./(,"(fi'^— E'^)^ Substituting,
in the factor between the braces, the values of p, q, m [5228y, 5117], we obtain very
nearly the same expression as in [5232]. The secular motion of the moon's longitude is
[5232c] _3„j2y^^(f'2_£'3)_j„ [5089a, 5232a], corresponding to [5232'].
Vll.i. .^16.] EQUATION BETWEEN t AND v. 546
0^ = 3,00D524.m~.f(e'-—E'-).dv. [5232]
It has a conlrari/ sig7i to the secular equation of the mean motion [523?c], [5232']
and is nearly three times as great.
The retrograde motion of the node of the moonh orbit, (g — l).v [4817], is,
by the preceding theory, 0,00401 05. v [5229]. This motion is, by observation, [^233]
equal to 0,004021 75.?; [5117 line 3], Avhich does not differ from the preceding, ^'^'^,';^»'^ ^
by its tliree hundred and fiftieth part. hm^'ù'Â.'
pertgre
iitli] ilniif.
This motion of the node is subjected to a secular equation, whose analytical
expression is given in [5059]. Reducing it to numbers it becomes,*
6ù= 0,735A524.m-.f(e'^—E"').dv. [5234]
Jt has a contrary sign to that of the moon'' s mean longitude [5232c]. Hence it
folloics,that the motions of the nodes and perigee are retarded, whilst the moon''s
mean motion is accelerated ; and the secular equations of these three motions
are always in the ratio of the numbers 3,00052, 0,73542 and 1 [5235]
[5232,5234,5232c]. Therefore, in the preceding expression of nt-\-s, we
must substitute, for the angles cv, gv, the following quantities ;t
«
(2922) The secular motion of the node depends upon the term,
^-./e'^.rf. [50565]; t^233a]
[5233i]
which may be changed, as in the preceding note, to
Substituting, in the first factor, the values of p", q" [5228;;], it becomes very nearly
as in [52.34].
t (2923) The motions of the perigee and node (1 — c).v, (g — \).v [4817], ai'e l5'>3ea^
changed, by means of the secular equations, into
(1— c).i.-i-.3,(]00J2.f/ft2./^'(e'2_JS'2).f/y [5232],
{g-l).v+0,-35A524nv'f,'{e'^—E'^).dv [5234], ^^^^^''^
respectively. This requires, that we should change cv Into
CT_3,0C052.|<7i2./^''(e'2_£'2).c/f, as in [5236] ;
and, go into [523flf]
^»+0,735452.|m2./o'(e'2— i;'2).(/r, as in [5237] .
VOL. III. 137
546 THEORY OF THE MOON ; [Méc. Cél.
[5236] cv— 3,00052 4.mKf(e''~—E"').dv ;
[5237] gv+0,13ôiô24.m\f(e'-—E'^).dv .
Hence, the secular equation of the mean anomuhj is,*
[5238] — 4,00052.|.m-./(e'~— £'-).^v ;
or, nearly four times that of the mean motion.
17. We shall now pi-oceed to determine some oj the most sensible inequalities
of the fourth order. One of these inequalities depends upon the angle
2d 2mv — 2gvArCV^2(: — ^, and we have determined, in [4904 line 17, 5014],
the part of ahu, which depends on the cosine of this angle. Then we find, by
^15, that the expression of nt-\-i, contains the inequality,!
( 3nt'-^.(2-|-w) 2^*'^'+3^*'^^ >
{ s.(2o— 2+2w) '_ [ L.e>^sin.(2i;— 2mi;— 2u-r+ci)+2d— ^).
2 — 2m — 2g-\-c
This inequality, reduced to numbers, is
8%67.sin.(2i; — 2mv — 2gv-\-cv-{-2è — n).
We shall now consider the inequality, relative to the angle {2cv ^2v-2mv-2zi).
If we connect all the terms, depending on the cosine of this angle, in
Invest iga
lion of
some
lor m s
of tUo
fourth
order.
[5239]
[5240]
[5238a]
* (2924) Subtracting the secular equation of the perigee [5232], from tliat of the mean
motion [5232c], we get the secular equation of the mean anomaly, as in [523S].
+ (2925) The part of dt, which would correspond to the term of nt-\-s [5239], may be
1 a~
deduced from it by taking the differential, and multiplying by - =-^ [5092c], by which
means it becomes
,.o^„ T \ _ J!f^±]!!l 2^f'«4- 3 ^<;'^ I . "4^" . ey\ co^.[2v—2mv-2gv-\-cv).
[o239a] ;^ 8.(2g-— 2+2m) ' ' ' ^ V^a, ^ ù "T ;
r"239tl Now the three terms of this function are contained in the expression oi dt [509ap], as we
shall see, by the following examination. The^)-s< term, between the braces, — - '" _^XV-
[5239a], occurs in the table [5O90è] ; by multiplying the term — 2e. cos.cj; in its first
column, by that of [50S2s line 13] in its second column. The second term — 2A\^^\ arises
[5239c] ^^.^1^^ [5090^ line 1, 4904 line 17]. The third term S^''^' is deduced from the table
[5090O-]. It corresponds to — 2A\^^^y-.co5.{2gv—2v-\-2mv) in its first column, or in
[4904 line 14]; and to — Se.cos.c» in the second column. Substituting in [52-39] the
t^^^''^ values of m, g, c, y [5117], e [5194], and ^.">, .3/« [5169,5172], we get [5240].
Vll.i.§n.] TERMS OF THE FOURTH ORDER. 647
the development of the equation [4754 J, which we have made in ^ 6,
this equation becomes, by noticing only these terms,*
_ a
dfJii ,3m (10 — 19 m 4-8 m^) . ('2 — m-i-c) „ ^^ , r. r» o \ r-^.,-,
0 = l-u +■ — . -!^ P^ —i ^^- .e-.cos.(2cv-ir2v — 2mj;— 2^): [5241
dv-^^ ^2fl, 4.(c+l-m) ^ ^ ^'
therefore, by putting ^'^'''.e-.cos.(2ci'+2iJ— 2hw' — 2a), for the corresponding [5242]
term of «'« [4904], we shall have,!
|m-. (10— 19m+8m2) . (2— m+c)
/J'C) — -^ '' 1 ^^ 1_. r'>24Hl
^2 4.(c+l— m).{4.(c+l— ?«)— M
Then, if we put C"^'".e-.sin.(2cz;+2?; — 2inv — 23), for the corresponding [5244]
term of the expression of nt-{-:, we shall find, by §15,t
* (2926) The terms depending on the angle 2cv-\-2v — 2mv,'m the equation [4961],
are included in the functions which are enumerated in [4960e], and if we divide these terms
3»t2 [5241a]
by the common factor -^ .e2.cos.(2ci'-|-2L' — 2mv); we shall obtain in [4S701inel2] the term
I (G-1 5m+8m^); and in [4879 line 8] the term ^(4-4ffi) nearly. The sum of these two expressions
Is 1(10— 19m+Sm2)'; addingthisto ^(10— lOm+Sm^).— ^ [4892 line 11], we
obtain i(10-19m-|-8m^).~-T-- . Connecting this with the two first terms of [4754] —^+u, [524lcl
according to the directions in [4960c, Sic.], we get [5241].
t (2927) Integrating the equation [5241], by the method in [4998a— c], we find, that if
-.cos.(?y+|3) represent any term of [5241], the corresponding term of «m or a5u rgoio i
[4998c, a] will become,
In the present case, we have,
i _ o (c+l-ni) ■ ?= ^ ^IfL (10-19m+8,ng).(2-m+c)
~^ ^' a, '^' a, ' 4(c+l— j«) [5242c]
Substituting these in [52426], and putting the result equal to the assumed expression
[5242], we get, by using nr [5082A'], the value of .4'^°' [5243].
X (2928) If nt-\-i contain a term of the form [5244], its differential will give, in
nii, the expression
ndt = {2c-{-2—2m).C'<-^h'^.cos.(2cv-}-2v—2mv).dv. [5245a]
648 THEORY OF THE MOON; [Méc. CVl.
(10— 19/n+8m2) 3m^.{\—m) 9m^
[5245c]
—%m-.
8.(c+l— m) 2—2m-\-c 16.(1— ffl)
[5245] ^,(0, _ (— 2yl'f+34°'— 34°'
2c-{-2—2m
Reducing the formulas [5243,5245] to numbers, we get,
[5246] ^'f") = 0,00201041 ;
[5247] C'f =—0,0130618;
hence we obtain, in nt-\-s, the following inequality,
[5248] — 8%ll.sin.(2c«;+2t;— 2wt;— 2^) [5244]
Multiplying this by - = [5092c], we shall get, in dt, the term,
[52456^ dt = (2c+2-2m).C'l%^-^.cos.{2cv-\-2v-2mv).
Comparing tliis with the terms of the functions [5090^], depending on the angle
2cv-^2v — 2mv, we shall get, for (2c-\-2 — 2m) . C'^'', tlie terms of the numerator of
[5245] ; namely,
, , (10— 19m+8m2) 3m^.{l—m) 9»/2
^ ' 8.(c-|-l— 7«) 2— 2m+c 16.(1 -m)
—2AT^3Ai'^—3Af^ ; 2
as will appear by the following examination of the functions [5090j3], divided by the
common factor
[5245d] ^^'~i — •cos.(2cu+2d — 2mv).
The function [5082s line 10] contains a term, depending on the angle 2cv — 2f-j-2m»,
deduced from [4885 line 10] ; and, we find in [4885 line 11], a similar expression
[5245e] — |m^. — gTTT3'"T — ' > corresponding to the first term of [5245c]. The term neglected,
in r5090Mine71, produces the second term of r5215rl, — -^ ; and, that in
"^ -" 2— 2m-[-c
[5090&line 12], is — TTr-rj^' as in the third term of [5245c]. The term of ai)u
[5242] produces, in [5030p line 2], the term —2^'™ [5245c line 2]. The term
[5245/] jjgg^g^jgj ;^ [.5090^ line 16], gives .3./2/2) [.V24.5c hne 2]. The term — S^!,»)
[5090^ line 9], is the same as in [5045fline2]. Now, substituting in [5243, 5245], the
[5245g-] values 1-5117,5194,5157,5159], we get [5246,5247]. Lastly, we get, from [5214], the
expression [5248], by using the values [5247, 5194].
VII.i§17.] TERMS OF THE FOURTH ORDER. 649
The expression of dt ^ 1 5, gives in ni+f, the term,*
QA'-^Kee'. sin. (2v — 2mv-\-cv — c'mv — a+ra')
2 — 3/rt+c
[524'.»]
This term is sensible, on account of the magnitude of the factor A'-^'^ [5161] ; [52491
it is, therefore, useful to consider the inequality relative to the argument
2v — 2.mv-\-cc — c'mv — ^-\-ui'. The equation [4754] gives, by noticing only
the terms we have developed in § 6,t
2
dda , m 21.(2— 3m) -(4—3»! + c) , .^ ^
0 =.— +»— -• 4.(2_3;„^,) ■ ^^ • cos.(2v—2mv-^cv-c'mv—v.+..'). [5250]
We shall put
^'*''.ee'.cos.(2i' — 2mv-{-cv — c'mv — to-j-tb') [.5251 ]
for the part of a'u depending on the argument in question ; we shall have.
* (2929) This term is omitted in the product of the two quantities in [5090^1ine 20] ;
but, it is introduced in this place on account of the magnitude of Jl^'^'> [5161,5249']. rEQ4n
Having noticed this i)art of the expression, it becomes convenient to introduce the smaller
quantities, depending on the same angle, as in [5250 — 52-57].
t (2930) The equation [5250] is obtained in the same manner as [5241], by
dividing the terms of [4960e], depending upon the angle 2v — 2mv-^cv — c'mv, by the
common factor, — "
— ^.— .ce'.cos.(2î) — 2mv~\-cv — c'mv) ; [5250«1
"■/
and connecting the resulting quotients in the following manner. The term in [4870 line 7],
gives J(l — 2m); in [4879 line 4], i ; their sum is 2 — 3m; adding this to the
rionni- -^ 2(2-3».) , .
term 4892 hne / , - — - — ; — , we obtam,
2~Sm-\-c
(2— 3m) . I 1 + ^^,-V- 1 = (2-3m) j±=^!l±îl, [52506 j
Connecting this with the common factor [5250o], and adding the two terms [5241c], we
get the equation [5250] ; in which we have corrected a typographical mistake in the
original, where m- is written for — . Comparing this with [5242a], we get, [5250c]
., _2 21.(2— 3m).(4-3m+c) . ^ ^ , ^ ^ ,
H = —7)1 . , ,-, ,' , ^ ■-'— '.ee'; i = 2— 2m+c— c'm = 2— 3«+c, nearly ;
4.(2— 37rt+c) ' I ' J '
substituting these in aSu [5242i], and putting the result equal to the assumed value of
this term of aiu [5251], we get A'l'\ as in [5252], using m^ [5082ft'J. [5250rf]
VOL. HI 138
*
5
•^50 THEORY OF THE MOON ; [Méc. Cél.
[5«-2] ..., _ -2lm"-.(2-3m).(i-3m+c)
^2 — 4.(2— 3m+c).» (2— 3m+c)-—l I
Then, if we put
[5253] C'i,"ee'. sin. (2v — 2mv+cv — c'mv — n-j-ra')
for the part of nt-\-i, relative to the same argument, we shall find, by § 15
2lm^.(2—3m) , 21//i'- ^ ,„,, ^ „,
^,„ _ M2-3m+c) ^ 4.(2-3m) ^ ^ ^^'
[5'.i54] ^. — 2— 3m+c
Reducing these formulas to numbers, we find,*
15255] J'^') =—0,0134975;
[5256] C'^^ :^ 0,0534480;
which gives, in nt+:-, the inequality,
[5257] 1 0,'l 7.sin.(2i) — 2mv-{-cv — c'mv — ra+^i').
* (2931) Proceeding as in [5245rt,&c.], we find, that if nt-\-g contain a term of the
form [5253]. it will produce, in its differential ndt, the term,
[5253a] (2 — 3m-f-c).C'l"fe'.cos.(2i' — 'imv-^-cv — c'mv), nearly ;
and, by multiplying by - = -7-;- [5092f], it will produce in dt, the term,
[52536] (2 — 3m+^) • C"^"ee'- — ^.cos. (2f — 2mv-\-cv— c'mv).
Comparing this with the terms of the functions [5090p], depending on the same angle, we
shall get, for (2 — 3ot-1-c).C'^'', the terms of the numerator of [5:251]; namely,
2h>^^{^-3m) 2hn^ 2^'") + S/^'^) •
l^^^^ 4.(2-3m+c) + 4.(2-3.0 ~ ' - '
as will appear by the following examination of the functions [5090p]. The term
21mS.(2-3m) ,
[5253rf] [5082«line8] is the same as the first term of [5253c], 4_^2-3m+cJ~ ' ^^^^^
■ ^^"'L_ omitted in [50906 line 8], is the same as the second term of [5253c] ; the term
4.(2- 3m)' ■-
[5253c] ^j. ^^^^ [5251] produces, in [5090JJ line 2], the term —2A'\'\ [5253c]; lastly, the
term omitted in [5090^ line 20] produces 3^</' in [5253f].
(2932) Substituting the values of c, m [5117] in [5252], we get [5255], and
then, from [5254], we obtain [5256]. Substituting this value of C".V\ and the values
^■'^^^"'^ of e, e' [5117, 5194]; in [5253] we get [5257].
VII. i.>^M7.] TERMS OF THE FOURTH ORDER. 551
It would seem, that the inequalities depending on the angles
2cv — 2v-\-2mv±c'mv — 2^^^' [5257]
ought to be sensible, on account of the great divisors which they acquire hy
integration; it is therefore important to ascertain them carefully. By following [5257"]
the analysis before explained, noticing only quantities of the fourth order, and
representing the corresponding part of aiu, by
^ ^ [5258]
+A'^^^ey.cos.(2cv—2v+2mv—c'mv—2-.+^/);
we shall find, that the differential equation will become,*
* (-2933) We shall put, for brevity,
S = 2cv—2v+-2mv+c'mv—2zi—zy'; D = 2cv—2v-i-2mv—c'mv—2vi-\-a'; [5259a]
and the assumed value of aou [5258] will become,
aou == ^' e) e\'.cos.S+d'f1 e^c'.cos.B. f ^^^^*J
The terms of the equation [49G1], depending upon the angles »S', D, maybe found
in the functions which are enumerated in [4960c]; and, to obtain all tlie terms, we must
review the whole calculation [4835 — 4961], in order to notice the quantities which have
the factor e-e'. This great degree of accuracy is however unnecessary, on account of the
smallness of the coefficients in [5259], which are of the fourth and higher orders; we
shall, therefore, only notice the most important terms which are given by the author in
[5259]. The 6rst of the functions [4960e], which is noticed by him, is that in [4870]. [5259/1
We may deduce this selected part of the factor of e-c', from that of e^ [48701ine II],
upon similar principles to those which are used in developing a function of e, e', by Taylor's
theorem, by which the coefficient of e-e', may be derived from that of e^, &c. If we use
the value of m [4865], and put, for brevity,
M = '£-.i6+l5:n+S,n^) .^ = l.."^f .(C+15.+8™=).e^ ^5^59^]
we shall find, that the term of e^ [4870 line 11] is represented by
J\'l.cos.{2cv — 2v-\-2mv). r5259/"l
As this quantity does not contain e', it is evident, that it can be derived from the first
member of [4870] -^j^ .cos.{2v—2v'), by substituting the values of /(, u, u', v' J5259 ]
[4325,4826,4837,4838] ; then, neglecting the terms depending on c', and retaining only those
connected with e^. Now, by using merely the first terms of [4837,4838], and those
depending on the first power of e', we have,
552 THEORY OF THE MOON ; [Méc. Cél.
/ 7(2+llm-f-8m2)_7( 10+19m+8ma) \ 1
^ ddu . 3m ) Ï6 ¥.(2^=2+3H"( 3, ,00.01' o '^
[5259] V. 2c-2 + 3m ^ ' } 2
/(10+19m+8HiS) (9-|-llm-|-8m2) \ -3
3 m' ) 8.(2c— 2+ in) lie (
H .< >.e-e'.cos(2c'ii-2v+2mv-c'??n'-2ra+î3');
V 2c — 2 + m " ■ ; 4
r' = ?OT+2e'. sin.c'^nt'; m' =-. I l+c'.cos.c'?;i('|.
I Oi^Ofil I (X
If we retain these terms of e', informing the function [5259/"], it will change — — [5259f]
rrnrrv-i '"1° — -r-.( 1+3 c'.cos.c'mii) , and 2mw into 2/«D-4-4e'.sin.c'm-y. By this means the function
[52591 J u 3 ' -^
[5259/], will be increased by the terms,
r5''59A-l M.(3e'.cos.c'm«).cos.(2c!; — 2v-\-2mv) — M.{4e' .sin.c'mv).s]n.{2cv — ^v-j-'^mv) ;
the second of these quantities being obtained by means of [61] Int. by putting
z = 2cv — 2«-j-2?/u', a= 4c'.3in. t'mw [5259i].
Reducing the terms of [5259^'], by means of [17,20] Int. we get, by using the abridged
symbols [5259a],
r5259ml .M.(3e'.cos.c'mi;).cos.(2c« — 2v-\-2mv) = #ilie'.cos.S-[-M/ft'.cos.X);
— ■M.(le'.sin.c'mv).sin.(2ci-— 2D+2mi') = 2;He'.cos.S— 2.1'ie'.cos.I>.
[o259)iJ ' ^ '
Tiie sum of these two expressions gives the value of the function [5259/i] ; and, by
re-substituting the value of M [5259e], it becomes,
2
[52590] iMt'. cos.S—lMe'.cos.D= ^.{6+l5m+8m^).e'>e'.\^^.cos.S- ^^.cos.D].
A similar expression is obtained from the terms in [4S79 line 7], putting c^l, and
2
[5259/)] M=: — ^— . (l+77j).c-. Substituting this value of iVi, in the first number of [5259o],
we get the terms,
1 2
[5259?] -ï-(l+'«)-cV.^| .cos.S— è.cos.I>|=^.(— 4-4m).cV. ItV-cos.à— tV-cos.!)}.
Adding together the terms in the second members of [5259o.9], we get
2
which are the same as the first term, connected with S [5259 line 1] ; and the second term,
[5259i] connected with D, in [5259 line 3j.
VII. i.^ 17.] TERMS OF THE FOURTH ORDER. ÔÔS
therefore, we shall have, [5259"]
We may proceed in a somewhat similar manner with the term in [4892 hne 10], taking in
the first place its differential, so as to make it of a like form ; and, after reducing the "*
products, which introduce the angles S, D, again integrating, to correspond to the integral [5259i]
in the first member of [4892]. Now, if we put
2
iVi=^(10 + 19ra+8m2).e^ [5259<]
the differential of [4892 line 10] becomes,
Mdv.sin.{2cv—2v+2mv) = Mdv. cos. {2cv — 2v-\-2mv — 90''). [h25Qu]
The second of these expressions may be derived from [5259/], by decreasing the angle
2cv—2v-\-2mv by 90"; which requires that S, B [5259a] should be changed r5259„r|
into S — 90'', D — 90", respectively, and then multiplying by ih. The same changes
being made in the resulting correction, in the first member of [5259o], we obtain,
iMe'. c?i).cos.(S— 90'')— JJI/e'. dv.cos. {B—W) = JJlic'. dv.sm.S—lMe' . dv.sin.B. [52591.]
Now, integrating this second expression, according to the directions in [5259;], we get
the additional terms of [4892], as in the first member of the following equation, and, by
re-substituting the value of M [5259^'], we get its second member.
7Me' Mc'
.cos. o 4" 5~K n I . cos.B
[5259«i]
2.(2c— 2+.3m) ' 2.(2c-2+m)
= |^.(10+l9. + 8.^).eV. J-^_|^.cos.S+^-i^.cos.i)^.
The terms of this last expression are the same as the second term connected with S
[5259 line 1], and the first term connected with B, in [52591ine3].
The next terms of [5259] arise from the part of the function [4934 or 4932/t] which is
included in the table [493 Ip]. For, if we take, in the first column of this table, the term
^fee'.cos.icv-^-c'mv) [4931pline 22],
and in the second column, the term
5e . ,
— .sm. (2î> — 2mv — cv)^
which occurs also in [493 Ij? line 17], it produces, by the process used in [493InJ, the term
6m 5 c9 e' „
- V ■ 2.(2c-2+3m)-^' •^"^•(^'^''~^"+^'""+"'"^'') ' [5259x1
which is the same as the term depending on A^^'> [5259 line 2J. In like manner, by
combining the term
VOL. III. 139
[5250J-]
554 THEORY OF THE MOON ; [Mtc. Cél.
[5260]
«
AT
— 3m2
(
7.(2+1 ]H(+S?«2j
16
7.(10+19m+8m2)_
-40^i8>f '
1
1-
-#«■2-
-(2c-
-2+3,nf S
2
8.(2c— 2+3»
0 )
^i'J'ce'.cos. {cv—C7nv) [493]pline 23],
with the same term
5e
— — .sin. (2«; — 2mv — cv), in [4931^ line 17],
we get, by the method used in [4931n], the terra,
2
Gm 5 e~ e'
[5259y] — — • ;^-^H-;^247^-^i*'C0S.(2cîJ — 2v-\-2mv — c'mv), as in [5259 line 4].
The function [4î;03 line 1] contains the term —.a5u; and, by substituting the
value of 0 (5w [5259/!/], we get,
[52592] — — .eV.^^'f.cos.5+.^'f.cos.D|, as in [5259 lines 2,4].
This includes all the terms noticed by the author in [5259] ; there are other terms,
2
having the factor m.m.e~e', which he has neglected on account of their sraallness.
ddu
dv^'
Connecting these terms with t^ + w [5241c], it becomes as in [5259].
[5260ol * (2934) Taking separately into consideration the terms in the two first lines of
[5259], which depend on the angle S =2cv—2v+2mv-j^c'mv [5259o], they become
[52fi0i] of the same form as in [4990a], by putting
„ „_= C 7.(2 + ll'.+8mS) 7.(10+19m+8m^) 5^f^ -^ ^,
[5260c] /^=om . j -, 872c-2+3«.) 2c-24:3^~^^' ( " ^ ^ ^
[5260rf]
16 8.(2c— 2+3m) 2c— 2+3 m
i = 2c — 2+2/n+c'nj = 2c — 2+3w, nearly.
The corresponding term of au, or aHu, is represented by P«.cos.(w+)3) [4998c];
and, if we compare it with the assumed form of this term of aSu, in [5258 line 1],
rs^COd'l ^'■'^ ^et Pa=A'i-'e-e' ; hence [4998a] becomes, by multiplying by c, and substituting
this value of Pa,
TT
[5260e] U — (.1 I ).J1, e e-t ^.
[5260/"] Substituting in this, the value of H [5260c], rejecting the common factor cV, and
usin"- m- [5082A'], we get [5260]. Proceeding in the same way with the terms
depending on the angle D [5258 line 2, and 5259 lines 3, 4], corresponding to
rjpgQ I z;=2c — 2+2m— c'm = 2c— 2+OT, nearly,
we easily obtain the value of -4'f' [5261].
Vll.i.§17.] TERMS OF THE FOURTH ORDER. 655
3) _ — 3;»'^ C (10+19m+8wM0^f) _ (3+ 1 1 m+8m'-*) )
' ~ l—inr—{2c—2-\-m)^' I 8.{2c—2^m) To 3" ^'"^^^^^
If we denote the corresponding part of ?;/+- ^y,*
* (2935) If we take the differential of the term of ni-j-s [5262 line 1], depending
1(1^
on the ansrle 5'=2cy — 2v-\-2mv-\-c'mv, and multiply it by — = -— — r5092cl,
n ya^ "■ ■■
putting also c'=l, we get, in dt, the term,
dt = {2c-2+3m).C'[-^ e^e'.— .cos.S. [5261a]
Substituting in this, the assumed value of C'f'' [5263], we find, that the result is
represented by the fimction [526 1 r], or the numerator of the expression [5263], multiplied
by the common factor e^e'. —^ — .cos.S; and it will appear by the examination in [526l/-a?], rsaciM
that the corresponding terms of the value of dt [5090p], neglecting the same factor
[52616], agree exactly with this function [5261c] ;
21m3.(10+19/n+8m^)+120mS.^f 21»i-. (2-j-3.'rt) 63m2
16.(2c— 2+3«0 M2^3^^)~W:(2^:^3^) ^ ^^^^^^^
— 2^'f +3^i''— 3./3^«+3.^i«. j1['\ 2
By a similar process with the term depending on the angle D = 2cv — 2v-\-2mv c'mv
and the assumed value of C"f^ [5264], we find, as in [5261/-a;], that the corresponding
terms of dt [5090p], neglecting the common factor c"e'. — ^ .cos.I?, are represented [5261dl
by the function [5261e], corresponding to the numerator of [5264] ;
—3m^.{l0+l9m-\-8m^)i-l 20m^.^f^ 3m^.{2-{-m) 9m^
+ ^ /o ... „x +
16.(2c-2+«0 ■ 4.(2-7«-c) ^ ]6.{2—m)
—2A'^^^-\-3df'—3Al^'+3A,^\A['K
[r,26U]
We shall now proceed in the examination of the functions [5261c, c] in order to prove,
that they agree with those in [5090^]. The first term of [5090/? line 1] depends upon the
function [50824], which, when fully developed, contains terms of the required form, with l-'^W]
the factor eV. The terms of this function, which are retained by the author, may be
derived, in a very simple manner, from those depending on e^ [5082s line 10] ; namely,
^m^i{10+l9m+8m^). ^—.co5.{2cv-2v+2mv) ; ^,,201^]
by the process used in [5259s' — w]. For, if we substitute in the expression [5261^], the
value of the common factor g
|m2 = |m^.^ [5082r, fee], and Jlf=^.(10+197n+8m2).e2 [5259^, [52filA]
556
THEORY OF THE MOON
[Méc. Cél.
[5262]
C"-{^. eV. sm.(2cv—2v-\-2mv-\-c'mv—'2.rs—-ui')
+ C'f>. è'e'.sm.{2cv—'2.v+2mv—c'mv—2^+^'),
1
2
it becomes,
[526H3
[526U]
[526U]
(5261m]
[5261 n]
^Ma
.co%.{2cv—'2v-\-2mv).
2c—2-\-2m
Taking the differential of this expression, according to the direction in [5259s'], it
becomes,
— |Ma . dv .s\n.(2cv — 2v-\-2mv).
This is of the same form as the first member of [5259m], and may be derived from it, by-
changing M into — ^Ma ; so that, if we make the same change in the resulting
terms, in the second member of [5259m'], we shall get the corresponding terms of
[5082s], depending on cV ; namely,
7 „ 1
_2
3 771 , a
16a,
. (10+19ni4-8m2).e%
■■1:
.COS. S —
-.cos
-}
Re-substituting the value of
•y 3
2c— 2-(-3m 2c— 2+!«
[526 1/(], we find, that the coefficient of eV.cos.S, is
[5261 o]
tlie same as the first term of [5261c], which is connected with the factor 10-|-1977i+8m'';
and, the coefiicient of e-e'.cos.D, is the same as the first term of [5261e], connected with
the same factor.
.The second of the functions enumerated in [5090p], is that contained in the table
[50906]. We sliall make the following additions, so as to include those terms of eV
which were neglected in the former computation. The three columns of the table
are here marked the same as in [5090J] ; and all the terms in the third column have the
aKdv
[526];»]
common factor
(Col. 1.)
Terms of the first
factor in [5081],
between the bra-
ces,
— 2e. cos. cu
|-e^.cos.2('u
/",
21mS.(2-f-3m).ee'
(Col. 2.)
Factor Q'
[5081 or 5082s].
4.(2— 3m- c)
cos . (2r — 2m» — CV — drnv)
(Col. 3.)
Corresponding terms of [5081],
or [52Clc,f].
2]7/i2.(2+3;n).eV
3m2. (2+m).ee' .„ „ , , \
-— — — - — cos.(2v-2mv-cv-{-c7nv)
4.(2— m—c) ^ '
+
-m-
21m2e'
4.(2-3 m )
3m9e'
4.(2 -m)
cos(2y — 2mv — c'mv)
.cos.(2î) — 2mv-\-c'mv)
4.(2— 3m— c
3m9. (2+m}.e9e'
.cos. S
4.(2— )n—c)
63m2.eV
.cos.D
16.(2-3m)
9nfi. e2e
16.(2-m)
COS. S
cos.D.
r Lommnn \
I fnclor i
The terms in the third column, depending on cos.S, correspond to the two last terms of
[5261c line 1]; and, those depending on cos.D, correspond to the two last terms of
[526le line 1].
vil. i.§17.]
TERMS OF THE FOURTH ORDER.
567
We shall have, bj § 15,
'•21m-.(10-fl9ffi+8w-)+120OT°.^f Slm^. (2-f 3;n) eSm^ ■\
16. (2c— 2+3/») 4.(2— 3^7) 16.(2—3;») (
+
9)nP
2c— 2+3m
— 3m3. (10+1 9m+8»t^) + 1 20m"'. Af> , 3/«3.(2+m)
16.(2c-2+m) "^ 4.(2-Hi-f) ' 16.(2— m)
i 2
^ ; "^ [5263]
9
C?>=
2c— 2+m
Rediicins; these formulas to numbers, we find,*
1
2
[5264]
The function [4904], or aou, contains the two terms [5259i], and these produce, in
the first term of [5090^ line 2], the terms,
_2.?!li^.eV. \ A'[-\cos.S+AJ\cos.D ] ;
which are the same as the terms depending on jf^f^, A'-^'^ [5261c, c].
The next of tlie functions enumerated in [5090/;], is the function [5090^] ; and we
have, in hne 25, the neglected term 3^f\eV.cos..Ç, corresponding to the second term
of [5-261cline2]; and, in [5090o-Iine 24], the neglected term 3^f«'cV.cos.D, as in
the second term of [526 le line 2]. Again, the term — 2^î,'"e'. cos.(2i' — 2mv — c'mv), in
the first column of [5090o-], being combined with 3e-.cos.2ct;, in the second column,
gives — 3^1^'eV.cos..5; corresponding to the term depending on .4^'" [5261c]. In
like manner, the term — 2^^^'e'.cos.(2« — 2mv-\-c'mv) [50905- col. 1], being combined
with the same term 3fi^. cos.2cf, in column 2, gives — 3^^^'eV.cos.D; corresponding
to Ai^ [5261 e].
The last of the functions [5090p], is that in the table [5090i] ; and we have, in the first
column of this table, the term ^i'>e. cos.(2y — 2mv — cv) ; in the second column, the
omitted term 3.^f'ee'. cos. (ci'+c'mc) , which produce, in the third column, the term
^A['\A[^h^e'.cos.S, neglecting the common factor -1-". In like manner, we have, in
the first column, the term A[^''ee'.cos(cv+c'mvy, in the second column, 3.^',''e.cos(2tJ-2?ni'-cw);
these produce also, in the third column, an equal term |^',".^<''eV.cos.»S'. Adding this
to the preceding term, we get 3A['\A[%"e'.cos.S, corresponding to the last term of
[5261c]. In exactly the same way, we find, that the terms of ci'hi, depending on
A['^'e.cos.{2v — 2mv — cv), Af^ee'. cos.(ci' — c'mv), produce, in the third column of [5090«],
the expression 3.^-'\.^f^eV.cos.D, corresponding the last term of [5261e].
* (2936) Substituting the values [5117,5194,5157,&;c.] in [5260,5261,5363,5264],
[5261?]
[5261 r]
[5261 «]
[526 U]
we get Af^,
VOL. III.
-^ 1 J
C'f\ C"f [5265] ; and then, [5262] becomes as in [526^1,
140
[.5261u]
[52610]
[5261«)]
[526 J x]
[5265a]
[5265]
[5266]
568 THEORY OF THE MOON ; [Méc. Cél.
A'\-^ = 0,744932; I
^'(3) =—0,0153320; 2
C'f = 0,563137 ; 3
C?' =—0,0235572. 4
Hence we obtain, in nt-\-£, the two following inequalities ;
5',88. .sin.(2cv—2v-{-2mv+c'mv — 2^; — w') 1
— 0%25. .s'm.(2co—2v+2mv—c'mv—2^+^'). 2
The inequalities cle])ending on the arguments 2cv±c'mv — 2:3^=^', are very
easily found, by considering the expression of dt [5081]. This expression
gives, in that of ni+£, the inequalities,*
3/i'^'eV
2c+m ^ ^
[5267] 3^(9,^2^,
+ — ^ .sin.(2c« — c'mv — 2^+^') ;
2c — m
and it is evident, that they are the only terms of the fourth order, depending
on these arguments. By reducing them to numbers, we obtain, in nt-\-i,
the two following inequalities ;
— 3%1 6.sin.(2cD+c'wîi^— 2tr— ^')
+ 4%50.sin.(2a' — c'mv — 2ra-|-ra').
It is evident, from the expression of dt, [5081], that the inequality
depending on the argument ^v — ^mv — cv-\-ts, must be sensible.! To
* (2937) The functions [5090p], which represent the value of dt, give, in
[5090^ lines 26,27], the two following terms, which were oniiued in that table ;
[5267a] dt = tlp-.éh'. \ 3^i='>.cos.(2cu+c'm«)+3^«).cos.(2CT— c'm«) | .
\ a- , ,
Dividing this by - = -— [5092c] ; and then integrating, we get, in nt-\-e, the two
terms [5267]. A slight inspection of the functions enumerated in [5090p] shows, that there
are no other terms of this form, and of the fourth order.
f (2938) This will fully appear, by the inspection of the terms of 7i;-|-s, depending
on this argument in [5280, 5281 , 5283].
[5267']
[5268]
Vll.iU?.] TERMS OF THE FOURTH ORDER. 569
determine it, we shall represent the corresponding term of aàu, by
uiu = ^'^•"e.cos.(4î; — ^mv—cv-\-zi). [sacyj
It is evident, that there cannot be produced such terms in the differential
equation in « [4961], except by the variation of the terras of the equation [5269']
[4754], depending on the disturbing force.* We have developed these
^ÎM 7/ OH
variations in ^ 8. The first is — '- " : and it produces no term of [5270]
the fourth order, depending on cos. (4î; — 4?au — cv-\--a). The second
variation is,t
* (2939) This is evident, from the examination of the functions [4960e], which
compose the equation [4961] ; since the terms enumerated between [4866] and [4901] do
not contain the angle Av — 4my — cv. The next of these functions is that in [4908], which L5269a]
3î^t . Î/ • ou
arises from the development of — — nUTZ — ['^SOSg'] ; and we find, by inspection, that
it contains no term of the fourth order, depending on this angle. The same may be
observed of the functions [4913, 4918,4922,4928, 4942—4960]. The three remaining
functions [4911, 4925,4934], which are derived from the quantities mentioned in
[5271, 5273, 5275], produce some important terms, as will be seen in the following notes.
[52G96]
t (2940) This expression is the same as that in [4910], which is developed in
[4911, 4918]. The term depending on the second of these functions, is retained by the
author, though it produces only terms of the fifth order [5271c]. Substituting the values
[4937n, 5082Â'], in [5271], it becomes.
_2 _2
— - — .adu.cos.{2v — 2mv)-] .5î)'.sin.(2tJ — 2m«). [5271al
Now we have, in [4904 line 2, 4917], the terms of ahi, ôv', represented by
a&u= A[^'' e.cos.{2v — 2mv — cv); ôv':= — 2m..^',"e.sin.(2«j — 2mv — cv). [52716]
The first of these quantities produces, in [.5271a], the term,
4 a.
and the second, the term,
9m
J — .A[^'' e.cos.{4v — 4mv — cv) ;
..^'i".7ne.cos.(4îJ — \mv — cv) ; [5271c]
the sum of these two expressions is evidently equal to that in [5272].
560 THEORY OF THE MOON ; [Méc. Cél.
Qm' )/'3 :î m' 9/3
[5271] _ 7 ;'^ .6^.cos.(2;;-2t)') + -^^ .6tj'.sin.(2t— 2«') ;
it produces the term,
[5872] — — -.(3— 4m).J',"e.cos.(4« — 4>mv — cu+^).
The third variation is,*
[5273]
[527:3tt]
,o 4 • -T- • — .sin. C2v— 2«') — ,,yw— r. -,— .sin.(2«— 2?;')
h-.u* dv u ^ ' 2/iMt' dv ^
u
3m'.u''\6v' du
h^. u^ dv
+ /.2..4 •:7::.cos.(2i>-2t>0.
It produces the term,
3 7ft
[5274] — -^ — .(2 — 2m — c).JJ'^e.cos.(4t' — ^mv — cv+z^).
Lastly, the fourth variation is,t
* (2941) The three terms of the function [5273] are the same as those in [4924J,
which are developed in [4925]. The first of them is computed in [4923f, &c.], and
evidently contains no term of the fourth order, depending on the proposed angle. The same
is to be observed of the third term of [52 73], which is computed in [4923(7]. Tlie second
term of [5273] is, _ 3m'. «'^ dSu^ ^.^^ ^ _ ^
2^2. „4 • dv •- •V*' "J'
and it becomes, by substituting the values [4937?^, 4865],
[52736] . — = sm . (2 u — 2 m v ) .
]\ow, = — , - contams, m [49041ine2], the term,
dv dv
r 5273c] — ("2 — 2m — c).-4j"e .sin.(2« — 2mv—cv) ;
hence the preceding expression produces the term [5274], as is evident, by multiplying, and
reducing the product by means of [17] Int.
■f (2942) The expression [5275] is ths same as that in [4931], which is developed in
[5274a] [4934], by means of the functions enumerated in [4932A:] ; namely, [493 Ip, h, 4932a,/].
The first of these functions [493 Ip] contains in its second line, a term of the fifth order,
depending on .41"', which is neglected on account of its smallness. It also contains a
term depending on A'-^\ which is omitted in [4931 p line 6], but is easily found, by the
Vll.i.<^17.] TERMS OF THE FOURTH ORDER. 661
-—. { l+Îj"-.cos.(2^i— 2<)) \ ./ —i- • J - -sin. (2«-2«') +h^v'. cos.(2«-2.') [
ft < (t 1.1 f 11 ^
[5275]
it produces the term,
a, 4 — 4?« — c ^ ^
Therefore, the differential equation in ii becomes, by noticing only these
terms,*
method there used, to be, g -- j
. — — , .^'•'e.cos,(4ti— 4mw— cd): [52745]
«^ 4 — 4w — c ^ "
neglecting e-, and putting,
Ti^A^^^e; A:' = 1 ; i=2— 2m— c; {'= 2— 2m [4931/]. [52746']
This is the same as the fast term of the expression [5276]. The .next of the functions
[5274a] is [493 Iw] ; which may evidently be neglected on account of its smallness. We
then have [4932a], which contains, in its first line, a term depending on A^^, which is
omitted in the table, but is easily found to be,
2
Qlii m
- — • • -; — .^i"e.cos.(4n — A.mv—cv). r5'274cT
a, 4— 4m — c ' ^ ^
The last of the functions [5274cr] is [4932/] ; this also contains a term which is neglected
in its sixth line, and is represented by,
■iïï? r (2— 2m— c)2-l ^
~ 2^' ^ 2:(lZ:5,"^ 5 ■ ^'."^ •cos-(4r— 4mr— CT). [5274rf]
This may be reduced to another form, by observing, that, by putting c=l, we have,
very nearly, (2— 2 m— c)'^- 1 ={\—2mf—\ = —/^m; so that this term may be
represented by, -
_2 2
3 JÏI .,,, lA A \ 37/Î 3 7?l
— .m.A'^e.cos.(4«-4mt— f.), or, —. .j--^__.^me.cos(4z;-4mt;-aO, nearly. [5274e]
Adding this to the term [5274f], we get,
2
3 m, 5m
^ • 4 - 4,„_^-^'/^e-cos.(4i;— 4m^)— ct), [5274/]
as in the second term of [5276].
* (2943) Adding together the terms [5272, 5274, 5276], we obtain those connected
VOL. III. 141
662 THEORY OF THE MOON ; [Méc. Cél.
_ 2
[5277] 0 = -r^+u . < 5— 6m — c-\ — ^^ -• > . J "e.cos/4v— 4mi>— c«;+ra).
dv~ 4«, ( 4 — 4m — c ) ' ^
If we substitute in it, for a&u, tlie term,
[5378] aiu = ^'^%.cos.(4i)— 4mt;— CT+ra) [5269],
we shall (ind,*
3m2
[5279] ^'(^)
3
!L.;5-6m-c+i:fcMr^a,
[ I ' 4 — 4)n— c ) '
(4_4m— c)-— 1
Then, if we put
[5a80j C(^'e.sin.(4z)— 4mi;— cw+ra),
for the corresponding term of nt-\-s, we shall have, by ^ 15,t
with COS. (4î) — Aim — cv), in [5277]; to which we must add, as in [5241c], the two
terms -r^-\-u, to obtain [5277].
* (2944) Substituting, in [5277], the assumed value of au, or «(5m [5278] ;
[5279a] jj^^jjjj^tQf „,2 [5082A'], we easily obtain the expression of A^^'' [5279].
•f (2945) Proceeding as in [5261a, &;c.], we may take the differential of the term of
[5281a] nt-{-s [5280], and multiply it by - =: — [5092c], and we shall get, in dt, the
term, _ „2.rfi,
[52816] dt^ (4 — 47ft — c).C^^'e.-— — .cos. (4?; — Amv — cv).
Substituting the assumed value of C'^'", we find, that the result is represented by the
function [5281(/], or the numerator of the expression [5281], multiplied by the common
[5281c] e._.cos.(4.-4m.-czO;
and, we shall find, upon examination, that if we neglect the consideration of this factor, the
corresponding terms of the value of dt [5090p] will agree with the function [5281rf].
f 3m2 , 3m^.(l-»«) , 3j(0) ? A^^)_OA'W
[5281rf] [ ^^11^) + 4-4m-c ^ '^:^ J •^. ^^3 •
To prove this, we shall now compare this expression with that which is derived from the
[52816] fonctions [5090p]. The first of these functions depends on [5082«], or the value of Q'
[5082<7] ; and this last function contains, in [50825 ^'"^ ^l' ^^^'^ t^™ terms,
VII. i.<^17.] TERMS OF THE FOURTH ORDER. 563
i.(l— m) ^ 4— 4m— c S '
^,„ , 4.(1— m) ' 4— 4m— c ' ""' S'"' '
C'^) = -!^ ^ >- — ! [5281]
3 4 — 4m — c
Reducing these formulas to numbers, we obtain,
yj'(4) =—0,000799351 ; l-'5'-^82i
C'^4) = 0,00294934. t^^^^'l
Hence arises, in the expression of nt-\-i, the inequality,*
33%38.sin.(4y— 4m«— cî5+^). [5283]
The inequality depending on 4t?— 4mw — 2cw+2a, may also be sensible ;
the expression of ndt [5081, &.C.] contains the following quantity,!
— |a X function [493 Ip], — iaX function [4932a]. [5281«']
Now, the omitted term of — JaX [4931jjline6], produces the term,
_a
^^^^ . ^ .^i"e.cos.(4^ — 4mr — c«); [5281/]
a, 4— 4m— e ^ '
and, that in — ^ax[4932(dinel], gives,
— ^^H-f. . .^L . ^(ne.cos.(4y— 4Mr— ct). [sasi/-]
a, 4 — 4hi — c ^
The sum of these becomes, by using the value of ir? [5082A'],
37n".(I — in) .... ,. . .
!^ ^.yJ '* e.cos. (4î) — \mv — cv\, r5281s'l
corresponding to the second term of [5281f/]. The next of the functions [5090p] is [5090i];
it produces nothing. The term depending on [4904] produces — 2^'Ç,'", as in the last
terra of [5281 rf]. The term omitted in [5090^ hne 11] gives -— ', as in the first [528U]
term of [5281c/]. Lastly, the double combination of the terms
^^">.cos.(2u— 2mi;), A^}\.co'i.{1v—9.mv—cv) [5090/], r5281»l
gives, by a process like that in [5261m, &.C.], the term ZA^^^.A^l'^ , as in the third term of
[528 If/].
* (2946) Substituting, in [5280], the values of C'f, e [5282', 5194], it becomes
as in [5283]. ^^^^^^
t (2947) If we examine the functions contained in the expression of dt [5090p],
564 THEORY OF THE MOON ; [Méc. Cél.
[5284] i'iA^'Y- ^^' dv.cos.{^v—^mv—2cv-\-2Ta).
Hence arises, in nt-{-=-, the term,
3.( J^")-. â. sin.(4?;— 4ot«— 2ct+2^)
t^^s^J 2.(4— 4m— 2c) "
It is evident, that it is the only terra of the fourth order, depending on the
same argument, which enters in the expression of nt-\-s. Reducing it to
seconds, it becomes,
[5286] 22%26.sin. (4v— 4w«— 2cr+25,) .
We shall see, in [5578 line 10], that the tables of Mason and Burg both
[5286] agree in making the coefficient of this inequality nearly equal to 15' ;
which seems to indicate, that this coefficient is well determined by
observation ; consequently, the difference 7', between this result and the
preceding computation, must arise, in a great measure, from the quantities of
the fifth order, which we have neglected. To prove this, and to show, at the
same time, that a farther approximation will diminish the difference between
[5286'"] the theory and observation, we shall proceed to determine this coefficient, so
as to include quantities of the fifth order.
We shall denote the corresponding term of adu, by
[5287] a&u = J'f'e^cos.(4r— 4»ji;— 2ci'+2^).
It is evident, that terms of this kind are produced in the differential equation
[4961], solely, by the variation of the term of the equation [4754], arising
from the disturbing force. We have just given the four variations of these
[5967] terms [5270 — 5275]. The first variation [5270], produces no term of the
fifth order,* depending on cos.(4y — 4OTy — 2cv-{-2^). The second variation
we shall find, that the term [5284], with the factor — - , is omitted in [5090nine7],
[5284o] ^'''
and this is the only term of the fourth order, depending on the angle Av — Amv—2cv.
The integral of this expression, being divided by - [5092c], gives the corresponding
[5-2845] "
termor nt-\-i [5285]. Substituting the values [5117, 5194, 5158], we get [5286].
* (2948) The computation of the terms of the formula [5290], is made in the same
[5--.86a] j^j^jing,. as that in [5277] ; the former being multiplied by e^, and the latter by e; so
VII. i. § 17.] TERMS OF THE FOURTH ORDER. 565
[5271] produces the term
9 m'
4ii.
l2A['^—A["^.e\cos.(4>v—4imv—2cv+2i^). [5288]
The terms of the fifth order, depending on cos.(4« — 4m « — 2cv-\-2-ui),
which are produced by the third variation [5273], mutually destroy each [5288']
other, except in quantities of the sixth order.f Lastly, the fourth variation
that tlie similar terms of [5290], are of a liigher order by unity, than those of [5277] ;
and, in retaining terms of the fifth order [5286'"], we shall have to notice only the same [52866]
functions as in [5271, 5273, 5275] ; that in [5270] being, as in the former case, insensible.
* (2949) The terms of the second variation [5271] are developed in [4911,4918].
The last of these expressions produces, in [4918, 4918/], terms of the sixth order
containing e~, which may be neglected. The first term of [5271], is found as in [5288a]
[4910/], by multiplying the function [4910/.:] by 2o5i« [4904]. Now, if we combine
the term,
a
— — .cos.(2j;— 2mu) [4910;i-line 1], with 2^',"'e2.cos(2cy— 2y+2mz;) [49041inel2], [52886]
— — .^',"'e2.cos.(4u— 4mi' — 2cv), as in [5288]; [52886']
4a,
and, if we combine the term,
9ift
— .e.cos.(2u— 2mi)— ct) [4910A:line 2], with 2A-Pe.cos.{2v—2mv—cv) [49041ine2], r5288e1
we get, 9-2
--.Al''e^.co5.{Av—Amv—2cv), as in [5288]. ^5288rf]
The remaining terms of the sixth and higher orders are neglected.
t (2950) The first term of [5273] is represented, in [4923e], by the expression
—.4a(5«X function [4879] ; and the only terms of [4904], necessary to be retained, are those [5288e]
depending on A['^\ A^^\ which may produce quantities connected with e^. Now,
by retaining only the quantities which are multiplied by e^. cos.(4t) — Amv—2cv), we rsage/'l
find, that the term depending on Ai^'> [4904 line 1], combined with [4879 line 7],
produces a term of the sixth order, which may be neglected. The term,
— AA^^h . cos.(2u — 2mv—cv) [4904 line 2],
multiplied by, g-^
j-.e.cos.(2z)— 2mt; — cv) [48791ine 1], [5288g]
VOL. in 142
566 THEORY OF THE MOON ; [Méc. Cél.
[5275] produces the term,*
r53891 ^ \ 5^-g^" (l-2m).(3-2,.,).(10+19»»+8;».^) ^,o,,^^ } o /4.-4^.
^ ' 2a i 2—2m—c 4.(2c— 2+2m) ' '"^2-2»!$ * \-2CTf2n
produces qu
[5288/1] — -^■A'^e-'.cos.{4v—4mv—2cv) ;
and this is the only term of [5288e], which is of sufficient importance to be noticed. The
second term of [5273] is developed as in [4923r] ; and a little attention will show, that the
only term of whi [4904], necessary to be noticed, is ./3^''e.cos. (2 « — 2}nv — cv),
[5288i] corresponding, in [4923iy], to k^=A[''>e, f^2— 2m— c=l, nearly. Combining this
with the terra fc'.cos.v' [4923u, 4885 line 2], which is nearly equal to
— 2ê.cos. {2v — 2m V — cv) ;
making k' = — 2e, y'=z2v — 2mv — cv; we get, for the second term of [4923a;], the
expression,
[5288fc] —^.ikk'.cos.Uv—4mv—2cv) = —.A['k^.cos.{'iv—4mv—2cv).
4«, 2ffl,
This is equal to the term [5288/i], but has a different sign ; so that the two terms destroy
each other, as in [5288'] ; therefore, the whole of this function may be neglected.
[5289a]
* (2951) The fourth variation [5275 or 4931] is developed in the functions which are
enumerated in [4932/i:] ; namely, [4931p,u, 4932tt,/]. Now, the first of these functions
[493 Ijj] gives a term, which is produced, by combining, in the manner explained in [4931 n],
the term ^'j''e.cos.(2« — 2m» — cv), of the first column of [4931pline6], with the term
— fc.sin.(2u — 2mv — cv), in its second column ; which give,
__2
6m
[528911 —4A[^\ .co^.{4v—4mv — 2cv), as in the first term of [5289].
In like manner, the combination of the term A['^'^e^.cos.{2cv — 2v+2mv), in the first
column of [4931p line 25], with sm.{2v—2mv), in its second column, gives,
o
[5289c] ^ . -4j"^. .cos.(4y — \mv — 2cv), as in the second term of [5289].
«,
The function [4931m] contains nothing of the proposed form and order. The function
r4932«] contains a quantity depending on A-^^ of the sixth order, which is neglected by
[5289d] the author, on account of its smallness. The last of these functions is [4932/'] ; it contains
a term of the proposed form, which is found by combining the term .^l'".cos.(2i,' — 2mv), in
column 1 of [4932/line 1], with the term of its second column, corresponding to
[4885 line 10], _(l^±1^!^±^'^.eS.cos.(2c.-2.+2m.).
[5289c] 4.(2c— 2+2m) ^ '
This term, found by the method in [4932f'], is,
3^(')— |4""+-P^^^^i-+Ar.. ) 1
VlI.iU7.] TERMS OF THE FOURTH ORDER. 667
Therefore, the differential equation [4961] becomes, by noticing only these
terms,*
(+ 4.(2c— 2+2/«} ■ - )
Substituting J'f ^e^ cos. (4z; — 4»i« — 2c?;+2ra) for a^M, we obtain, [5291]
2— 2m— c "^2— 2ot
(I_2,„).(3_2,„).(l0+l9m+8m^j ^^^o
^,(5, _ ^J!r\l 4.(2c-2+2m) ^ ) [5292]
3 — 2 • (4_4„t_2c)2— 1
If we denote the corresponding term of nt-\-i by
C"f'e^.sin.(4r— 4m?;— 2c«+2î!), [5293]
we shall have, by ^ 15,t
_2
^ . M.(l_m)2_l ? /10+19"'+8^),^(o)g3^.os.(4t,— 4mf— 2ct) ; [5289/1
and, by using the reduction 4.(1— ?h)" — 1==(1 — 2m). (3 — 2m) [4961A], it is easily
reduced to the form of the term depending on A°'^ [5289]. Lastly, the term
./3<»ie2.cos.(2ci— 2u+2fflr) [4932/ col. 1], being combined with """"'^""^""'^ in col.2,
gives 2~'â^~ ' ^^ '" [5289]; observing, that in this case, the factor — (i^ — 1)
[4932c'] is nearly equal to unity ; since i = 2c — 2+2m = 2m, nearly. The remaining
terms of these functions are neglected by the author, on account of their smallness.
* (2952) Adding the terms [5238, 5289], and connecting the sum with the two terms
^+u [.5241c], we get [5290]. Substituting in it the assumed value of aU [5291], [5290a]
and using n? [5032^'], we get ^'^=> [5292].
t (2953) By proceeding in the same manner as in [5245a — c], we find, that the term
of nt-\-i [5293], gives, in dt, the term,
dt= (4— 4m— 2c). CT'e^. ^^ .cos.(4w— 4mj;— 2c«). [5294o]
Comparing this with the terms of at [5030p], we get, for C'l'\ the same expression
as in [5294] ; or, in other words, the terms of the functions [5030p], being divided by the
668
THEORY OF THE MOON ;
[Méc. Cél.
[5294] Cf) =
— 3m2.(5^f,'>— 2^i"')
STmi (10-fl9m+8m2)
4.(2— 2»»— c)
3m^
64.(1— m) 2c— 2+2m
•^1
T^'^^a i4.(i_,„)—i 4— 4rti
3m2.^<" 3m^(10+19m+8m3)
"" ^ 8.(2c— 2+2"/ft)r~"
(0)
4— 4m — 2c
2
3
[52945]
common
factor
, a2. dv
cos.(4t) — 4m V — 2cii),
[5294c]
[5294rf]
[5294e]
produce the terms in the three lines of the numerator of the expression [5294], as will
appear by the following examination. The first of the functions [5090p] represents the
value of Q' [50S2« or 5082q]. Now, the last of these expressions [5082<7 line 2]
contains the terms,
J.function[4885]— i.function[4889] — ^«.function [4931jj]— Aa.function [4932a];
which we shall separately examine. The mere inspection of [4885, 4839], shows, that
they produce nothing of the proposed form and order. The next of these functions is
— JaX function [4931^] ; and, as the common factor of the terms of this table is — , we
have, by using [5082A'], — |aX — = — 3m~ ; Then, by combining, as in [4931n],
the term ./3<i" e.cos.(2« — 2mv — cv), of the first column of [4931pline 6], with
— |.e.sin.(2t) — 2mv — cv), of its second column, we get,
[5294/] — 3m,^4A[^'>e^. . . ^y .cos.{Av — 4mv-2cv), as in the first term of [5294 line 1].
In like manner, the term A[^^'^e^.cos{2cv — 2!y-l-2my), of the first column of [4931/?line25],
being combined with s\n.{2v — 2mv), of its second column, gives,
[5294^] 3/«^.^'"'e^. /_^. n •cos.(4t) — imv — 2cv), as in the second term of [52941ine 1].
The last of the functions [5294c] is that depending on [4932a], which upon examination,
is found to produce no term of the required form and order. Besides these terms,
arising from the value of Q', [50S2« or 5082(7], we must add a term we have formerly
neglected, in finding the value of f 711^, which makes a part of the value of Q' [5082m].
For, it is evident, that in deducing the value of f^/K^ [5082o], from that of M,
[5082n], we have neglected the term,
^''"'' .P,.cos.(4i;— 4my+F) [5082oline2] ;
[5294A]
[5294i]
[5294i']
16.(1— m)
supposing, as in [5082n, &;c.] that P,. cos. {2v — 2mj;-f V) represents any term between
the braces in [4885]. Now, if we take this term, in [4885 line 10], we shall have,
(10-|-19OT+8rt.2)
P,=-
4. (2c— 2+2/»)
•e-;
[5294m]
VII. i. § 17.] TERMS OF THE FOURTH ORDER. 569
Reducing these formulas to numbers, we find, [5294']
and, by changing the signs of the angle in [4S851inelO], to make it conform to [5082«], we
get V = -2cv ; substituting tliese in [5'294iJ, it becomes, without noticing the factor — — ,
aim-» (10+19«+8/«s) „
- 64:0"-^ • 2c-2+2rn •^^•«o^4._4m.-2a-) ; [5294*]
as in the third term of [5294 line 1]. The next of tlie functions [5090/)], is that in the
table [50906]. This contains a quantity, which is found by combining the term -2e.cos.c«
3m-
[50906 col. 1], with the term ———— .A\''>e.cos.{4v—4mv—cv) of Q', in [50906 col. 2].
This term was omitted in [49.3I/jline6], and also in — JaX function [493 Ip], in [5294/]
computing the value of Q' [50S25'line 2]. The combination of these two terms of the
table [50906], gives — j—^-^—.A'^'^c^.cos.lév—imv—cv), corresponding to the third
term of [5294 line 2]. In the original work the divisor 4 — 4m — c is inaccurately
printed, being put equal to 2 — 2in — c.
The term depending on — [4904]x2.^^, in [5090/? line 2], gives ~2A'^=\ by
using the term of aSu [5287]; this agrees with the last term in [52941ine 1]. The [5294n]
next of the functions [5090p] is that in [5090^], whicli produces several terms. Thus, by
combining the term of —2a'hi= —2A'l,'^''e.cos.(4v—imo~cv) [5278], which would
occur in the first column of [5090j], witli the term — 3e.cos.c«, in its second column,
we get 3^V*e^cos.(4w — 4mv— 2:i)), corresponding to the first term in [5294 line 2]. t^^'*"]
In the next place, the term —2A["'>e^.cos.{iicv—2v^2nv), in column 1 [5090^1ine23],
being combined with — jrj^— r- cos.{2v—2mv), in column 2, gives,
-^jY—^.A[''^e^.cos.{4v—4mv—2cv), [5294pj
corresponding to the second term of [5294 line 2]. Again, the term — 2^,;'".cos(2y-2/««),
in tlie first column of [5090^ line 1], baing combined wltii that term of its second
column, which is contained in the first line of [5090e], bymeans of the term [5082* line 10];
namely, 3rf^^0±\9m-j-8nf') o ,„ „ , „ ,
-^^^^—^^;i^.e~.cos.{2cv-2v+2Mv),
produces the term,
— 3w^.(10-fl9ffi+8m^)
8.(2,_2^2m) -A'e ■cos.{4v-4mv-2cv), [5394,^
corresponding to the last term in [52941ine2]. Thelastof the functions [5090p] is [5090e].
This produces, in [5090nine 7], the term i.(A['^f.e^.cos.{4v—4mv—2cv), as in the [5294r]
first term of [5294 line 3]. The combination of the term AfKcos.(2v—2mv), in the
VOL. III. 143
570 THEORY OF THE MOON ; [iMéc. Cél.
[5395] *^'f = 0,00436374 ;
[5295'] Cf ' = 0,0249067 ;
which gives, in nl-j-^, the inequality,
[5296] 15',46.sin.(4«— 4wiîJ — Sczj + S^j).
The difference between this result and that of the tables is insensible ; and
[5396] we see, hy this calculation, that, to make the theory agree wholly with
observations, relative to all the lunar inequalities, it is only necessary to carry
on the approximation to quantities of the fifth order. This appears also from
the calculation of the inequality depending on sin.(« — mv), in which we
[5396"1 ^^^^^ noticed quantities of that order. For, we shall hereafter find [5589], that
the result of this analysis, compared with that which is obtained by observation,
gives nearly the same value of the sun's parallax, as that which is deduced
from the transits of Venus over the sun.
The inequality depending on the argument cv — v-if-mv — ra may be
sensible, on account of the smallness of the coefficient of v. To determine
this inequality, we shall put.
[5294«J
[5294<]
first column of [5090i], wilh 3A[^'^e-.cos.{2cv — 2v-\-2mv), in the second column,
produces # -^L"'. ./3*"V;^.cos. (4 i' — 4mv — 2cv); and the similar combination of
A\^^''c^.cos.{2cv—2v-\-'2mv), in the first column, with ^^'''.cos.(2r — 2?ot), in the second
column, gives an equal quantity, §yi„^".^<j"V2.cos.(4r — 4mv — 2cv) ; the sum of these
two terms is 3^i,'".^'/''e^. cos. (4« — 4mv — 2c?)), corresponding to the second terra of
[5294 line 3]. In exactly the same way, we find, that the double combination of the terms
A'-°\co5.{'2v — 2mv), A^^e.cos.{2v — 2mv—cv) [5090J], produces in that table, or in the
value of 3.{aiu)-, the term 3^^''^^^"e.cos.(4î)— 4wt!— cy) ; and, if we multiply this
\)y — 4e.cos.cu, which was neglected in [5090/<;], it produces the term,
f5294M] —6£'„''\^[^^e^.cos.('iv—4mv—2cv),
corresponding to the last term of [5294 line 3].
* (2954) Substituting in [5292] the values [5117, 5157 — 5167], we get for A'^^'\ a
[5295al value which is nearly equal to that in [5295]. Using the same in [5294], we get for
C'?'' a value which exceeds, by a small quantity, that in [5295]. This difference is owing
[52956] jQ ji-jg inaccurate divisor of the term mentioned in [5294»!]. Substituting, in [5293], the
values of C"-J'\ c [5295', 5194], we get [5296] ; the coefficient would be increased about
[5295c] ; , ,. . . rrr.n^ 1
P, by correcting the divisor as m [5294?»].
Vll.i.^17.] TERMS OF THE FOURTH ORDER. 671
aiu =^ A' f''.-,.e. COS. (co — v-\-mv — 3j) ; [5297]
and ^
C'^'.-,. e . sin. (cv — v+mv — w) , [5298]
i a ^
for the parts of aSii and nt+s, depending on this argument, we shall
have, by noticing the perturbations of the earth by the moon,*
* (2955) In this note we shall put, for brevity,
V = cv ■ — V -]-mv — « ; [5297a]
and we shall then examine successively the functions enumerated in [4960e], for the purpose
of collecting together the terms ofthe equation [49G1], which depend on the angle v, and
correspond to the annexed expression ;
a Su = A' f\-,.e. COS. V [52971. [52976]
'a
We shall retain the terms depending on the first power of e, neglecting the higher powers
of this quantity, and the terms depending on c', &;c. The first of the functions [4960e], r5297c-i
which containsterms depending on v, is [4872], being the development of ' ^.cos.(u — v').
Now, in retaining only the terms of the order e, we obtain from [4870A],
9m'. u'^ 9m'. a4
m:^ "^ ~ S^Ç^' ^ • ^°^-'^^- [5297rf]
Moreover, by neglecting terms of the order mc, we have cos.(t) — v') = cos.(t> — mv)
[4S37]. Multiplying this by the preceding expression [5297(7], and retaining only the terms
depending on the angle v, we get, in [4872], the expression,
9m'. a"!
-8^1?i"2<^-<=°^-^; [5297e]
and, by substituting
_3
m'.ai m'.a^ a ma nfi a r.„^, ,.„„„,„
= -. -, [4S65,5082A'], [5297e']
o^.a'"! Ui-a!^ a' «, a' a a
it becomes, by a slight reduction,
36m9
a
— . e . cos.v.
16a a' ■ [5297/-]
The second of the functions, which must be noticed, is
3m'. u'* du
. sin.(î)— t)') [4SS0]
8?t2.u5 • rf« • """^ '^> y^—i' [5297ff]
it was neglected in [4881], on account of its smallness, and not inserted in [4960e].
Substituting the values [4937n, 5297e'], it becomes successively.
672 THEORY OF THE MOON ; [Méc. Cél.
[5299]
[5297il:]
[52971]
(c— l+m).{l— (c— 1+m/— |/rt-|
3m'. a"" du . , 3m2 a du . ,
[5297ft] - 8^;:^ • « • 5^-sm.(«-»«r) = _—._.„._. s,n.(«-mr) .
Now, by puttina; c = 1, we have in r4878al the term -— = . sin.cy. If
[5297t] J I o dv a
substitute this, in the last expression [5297A], it produces the term,
m- a
A ■ — . — , . e . cos. V.
^® a a'
Adding it to the term [5297/] we find, that the sum becomes
SSm-î a
— e . cos. V.
16a a'
The third of the functions, to be noticed in [4960c], is [4S92]. This is found, as in
[4892a], by muUiplying the sum of the functions [4855,4889] by the function
[4890]. If we retain terms of the order e only, we may put the function [4890] equal
[5297/'] to - • Multiplying this by the function [4885], it produces nothing of the proposed form
a
and order; so that it is only necessary to notice the terms arising fi-om the other function,
r5297 1 or ~ X function [4S89], taking care to insert the terms depending on c, which were
neglected in the development of [4SS9].
If we substitute, in the first member of [48S9], the values h'^= a^, «' = «'"' [4937n],
we get, by dividing by a ,
[5297n] ^X function [48S9] = — ^^- . / -J . sin. (v-v').
Now, by retaining only the terms of the first order depending on e, wehave,in [4826,4837],
rcriyn-r -i « = a~^ . (l+c.cos.ci)) ; v' = Mv — 2me.sin.ct;.
From the first of these equations we get [5297c]; and from the second, we deduce [5297r];
which is easily reduced to the form [52975]. Multiplying together the two expressions
[.5897jj] r5297o,s], and then the product by dv, retaining only the terms depending on e. sin. v,
we get [52970 ;
[5297g] «-5 = a». (1— 5e.cos.cy);
[5297r] sin.(u — v') = sin.(i; — m«)-l-2me.sin.cy.cos.(t) — mv) [60] Int.
[5297s] = sin.(«— »nt))+'/ne.sin.v+ &c.;
[5297J] - . s'ln. («—«') = «^ civ. (ie+me).sin. v.
VII. i. § 17] TERMS OF THE FOURTH ORDER. 573
C"^>= IL _ i^ ^, 5300]
The integral of [5297^], being substiluted in the second member of [5297n], it becomes
asin the second member of [5297»], which is easily reduced to the form in the third member,
by using [5297c'],
K.f • r^oof.1 3/«'.a4 (5e+me) Urn^ a (10-|-4m)
-X function 48891 = . , cos.v = -— . —, . e . — —; — . cos. v. r.52Q7ui
Adding this to the sum of the terms in [5297/], we obtain,
3m^ a ( , ,{lQ+4m)} Snfi a C 21— lie— 7m )
being the same as the three first tenns, connected with e, in [5298/"].
The fourth of the functions selected from [4960e], is that in [49081inel], which, by
using [5082A'], becomes
" 20
aou = — —■ A'^^\ — . e . cos.v [5297i] ; as in the last term of [5298/]. [5297ur]
The Jifth of these functions r49C0e] is [4934], or rather, that part of it which is
contained in [4931^]. For, by combining the terms A',^''"'. - . cos.(w — Jitv) [493 Inline 31],
in its first column, with — f e . sin. (2u — 2otu — ■ cv) , in its second column, by the
method in [493 1?j], using also m^ [5082A'], we get the term,
■~ a ■ a' ■ c'l^m ' '" ' ^°^' '^ ' [5297a:]
which is the same as that depending on ^,'^' in [5298/].
The sixth of these functions [4960e] is [4946], or, it is rather the part
_2
-— . - . f a5u . dv .2 . sin. (v — v'),
'•«' « -^ [5297y]
which is contained in [4945 line 2]. Substituting, for aSu, the term,
A[^\e.cos.{2v—2mv — cv) [4904 line 2], ' ri;oo- i
and using v'= mv [4837], also m^ [5082 A'] ; it becomes, by noticing only the part
which depends upon the angle v,
VOL. HI. 144
574 THEORY OF THE MOON ; [Méc. Ce).
[5300] * From these we deduce,
_2 _2
— ■ . — .^i".e. fdv.s'm.iv — mv).cos.{2v — 'imv — cv') = — — . — ..^'".e. f dv .sln.v
4a, a' ^ "^ ^ ^ 8tt, a' ' -^
15m2 a „ 1
[5298a] = ^ .-.^,«. c — -.cos.v.
■' Da a c— I-(-m
This is the same as the term depending on Jl[^ in [5:298/].
The seventh of the functions, [4960e] is that in [4957], which is derived from [4950] or
[4882], being a term of the function - . f (— j . '— [4SSl'j. This is to be multiplied
[52986] by tiie factor — -\- u = - , nearly [5297 Z'] ; to obtain the corresponding term of [4754]
or [5298/]. Now, the variation of [4956] contains the term,
3m'|ui, ^ u"^dv
[52986'] -^
f -^ .sm.iv — v') [495Gd] ;
and by substituting h^^^a,, v! = a' ' [4937n], it becomes,
3m'. \i- f dv . ,.
-J — . sm. [v — v) .
[5298c] 2a,.(i< u^ ^
If we notice only the first term of the second member of [5297<] we easily find, that the
term depending on e, in the preceding expression, can be put under the form,
3m'. (J- /- , , , . 3m'.,a -, 1
fïaQftrfl • / a^dv.he . sm. v =: — r -. a", f e . — — — . cos.v;
[S^oaj 2a,.a'4 -^ 'ia^.a'* c—\-\-m
and, if we reduce it, by means of [5297e'], it becomes,
3m3 «a 20,a
[5298e] "~ 16a ■ c? ■ * ■ c— 1+m
[5298c'] Multiplying this by the factor - [5298&], we get the term of [5298/"], depending on f*.
Hence we find, by adding together the terras [5297«,m', a;, 5298», c'], and connecting them
with \-u [5241i;], the following equation, for the determination of this part of m:
dv"
[5298/] "— rf„2T^ ^ a "a 1 c-l+m 5 2a ' a'
Substituting in this, the assumed value of au or a&u [52976]; namely,
[5298^] «^" = -^'-'^ -l-'- '"'■ " '
we get, by reduction, the value of A'['^ [5299].
1 a'
* (2956) The differential of the term of 7U+s [5298], being multiplied bj - = —
Vll.iU?.] TERMS OF THE FOURTH ORDER. bib
^'çs) = —0,260496 ; [530 1|
C'(^' = —0,293763. IJ^soi]
[5092c], gives in dl the following term, using the abridged symbol v [5297a];
I / , . \ y^;/(-^ "' a~.dv
(It = (c— l-f-?n). C'f\- . e . — — .COS. v. [5300a]
Substituting the assumed value of C^^ [5300], we find, that this expression of cit is
represented by the function [5300c], or the numerator of the expression [5300] multiplied
by the common factor — • e . — '- — .cos.v ; and, that this is correct, will appear by the [53006]
examination in [5300f/ — s] where we shall find, that the corresponding part of the value
of dt [5090p], divided by the same common factor [5300è], is accurately representedby
the function [5300c] ;
->=^.U.(5+2m-10,.)-5^1">-|^'.-^^ 4'(^^4.3jn^)4-3^(') A'"^ I ^'"' r..nn ,
[5082. or 5082?] X "^ = Q' X ^" ; [5300c']
To prove this, we shall observe, that the Jirst of the functions [5090p line 1] is
[5082.or50829]X^= Q'X -
and, by retaining only the terms in [50827 ''"^ '^J' it becomes,
\ — ^function[4885]— àfunction[4889]— èa.function[4931p]— àa.function[4932a] i X -^ . [5300cfl
The inspection of [4885] shows that it produces nothing of the proposed form and order
in [5300c]. The next term of [5300rf] is,
-|function[4889]X^; j,3„„^^
and, by substituting [5297?<], it becomes,
3m2 a (5+Qm) a^.dv
—, — g POS V *
IC ■ a' ' c-l-\-m' ^a, ' [.5300/]
and, if we neglect the common factor [5300i], it produces the two first terms of [5300c],
depending on 5-\-2m. In the table [4931^] we find a term which is produced by connecting
the term yi"^'.-.cos.(u — mv), in the first column, with — ;Se.sin.C2u — 2mv — cv), in
a ^ ^ ^ ' [5300g]
its second column. These give, in the third column of that table, the term,
_2
6 m 1 .,,_, a
. Je . — . Ax'K —■ . cos. V.
a c — \-\-m a
Substituting this in [5300rfj, and using the value of irfi [5082A'], it produces the term,
5
|m^. — TT" "^i"'' as in the fourth term of [5300c]. [5300A]
The last of the functions [5300(/] produces nothing of importance. The second of
676 THEORY OF THE MOON ; [Méc. Cél.
[5301"] Hence we obtain, iit ni+£, the inequality,
the functions [5090/j] is [50906]; and, by combining the term — 2e.cos.cv, in
a I
its first column, with the term — fm^--, • :; .003.(1; — 7nv), in its second coUimn, or
a 1—m '
[5300i] [5082s hne 19], we get 'ÔTYZ.'^ ^ connected with the common factor [5300è], as in the
last term of [5300c]. The third of the functions [5090;;] is — function['1904]X2.— ";
and, by substituting the value of a5u [5297], and neglecting the common factor [53006],
[5300&] '^^ set —2A'f^ [5300c]. The fourth of the functions [5090pj is [5090^] ; and, by
combining the term — 2A\"\-.cos.{v — mv), in the first column, with — Sc.cos.t), in the
[53001] second column, we get 3^*"', connected with the factor [53006], as in the seventh
term of [5300c]. The last of the functions [5090p] is [5090i] ; and, if we combine
rcoAn -1 ^''^«.cos.(2y — 2mv~cv), in its first column, with 3Jl[">.-.cos.(v — 7nv), in its second
[OoUUwi ] a
column, we get §A[^''.A["\ connected with the factor [53006]. In like manner, the
combination of A[^^K — .cos.[v — mv), in the first column, with 3^',"c.cos.(2i;-27?iu-ct;),
in its second column, gives the same term ^A[^''.j1[^''\ The sum of these two terms is
3Ai^\A''^''\ as in the eighth term of [5300c]. We have yet to notice the terms of
[5081], or of [5300c], corresponding to the parts of the equation [4961], which are
contained in [5298a, e']. These terms of [5081] may be derived, in a very simple
^ °' manner, from those of [4961], by the same process of derivation which is used in computing
[5032/] from [4946/] ; namely, by dividing this last expression by — 2^-' [5082A: — /];
[530Cj)j or rather, by multiplying it by — ^a ; and annexing the common factor —^ — [5081].
The propriety of using this method of derivation is manifest from the consideration, that
the first of these terms [5298a] is derived from the function,
[5300,] (^. +u).- ./[-^) . ^„ m [4943-4945 line 2];
and this function is very nearly equal to,
1 2 /°/^/Q\ dv
Moreover, tlie second of these terms [5298e'] is derived, as in [52986, Sic], from the
function [4956 or 4882] , which is a part of the function [4881'], by multiplying it by
\-u = -, nearly [52986] ; and this last product is evidently equal to the function
[53009^], from which the first term is derived. On the other hand, the corresponding terms
of di [5300m, t)] arise from the function Q^X-j — [5300c],whosechicf term, connected
VII. i. >^n.J TERMS OF THE FOURTH ORDER. 577
* — 8',31.(l+z).sin.(cw — v-\-mv — w). [5302]
The inequality depending on the argument v — mv-\-cv-i^ is easily obtained
from ^ 15; and it is evidently expressed by,t
with Q, is 7/T ' Ifl' J \d~) • '^ [50S2m] ; and this is evidently equal to the [5300*]
product of the function [5300/-] by the factor — ia.~ — , as in [5300p]. Now, if we multiply
the expressions [5298rt, e'] by the factor — \a. — — , they become, respectively, as
in [5300«,w];
1 hm- n 1 a^.dv
+ 16 • 7/ ■•^' " c-^T^ • "7^-'°"" [5300„]
+ l!^.«.e._LOj^.^^.cos... [5300V]
16 a' c — { -\- m \/a,
Dividing these by the common factor [5300i], we obtain,
15m- 1 >/,v , 3m^ 10 u.
.Ax\ and ^
[5300w]
16 c—l+m IG ' c— 1+ m '
which correspond to the fifth and third terms of [5300c], respectively. Hence it appears,
that the value of C*'"* [5300] agrees with the preceding calculation. Substituting, in
[5299,5.300], the values [5117,5194,5158,517.3], also that of f^ = ;g^ [4320,4948'], [5300x]
we get, for Af\ C'f\ nearly the same values as in [5301,5301'].
* (2957) Substituting, in [5298], the values of C"f, e, &c. [5301', 5194, 5221], rggoa^]
we get nearly the same expression as in [5302].
f (2953) The coefficient [5303] may be computed in the same manner as that in
[5298 or 5300] ; but the change of the divisor from c — l-\-m, which is of the order
m, to c-j-1 — m, of the order 2, enables us to neglect, in [5303], all the terms which [•'>303a]
appear in [5300], except ./2<''\ The term depending on ^','" is found in the same
manner as in [5300Z], by combining the term
— 2.<4',"\- .cos. {v — mv),
in the first column of [5090^], with the term — 3e.cos.cy, in its second column ; which
gives, in the third column, the corresponding term of
dt = 3A[^'\ - . e . —.cos.{v—mv+cv). ^5303^^
. . . 1 aS
Integrating, and then dividing by -^ [5092cl, we get the expression [53031 ; and, by
n »/a, [5303c]
using the values of c, m, e, inc. [5117, 5194, 5221], it becomes as in [5304].
VOL. HI. 145
578 THEORY OF THE MOON ; [Méc. Cél.
[5303] l-m + c • â' • ^ • s'"- («— »'^+^«— ^) ;
consequently, it is equal to
[5304] — 5',01 . (l+i).sin.(«; — mv-\-cv — t^).
[5304']
Torma of
fourth
order.
[5305]
[5306]
By following the same process, we may determine the other inequalities of
the fourth order ; but, as thoy are less than the errors of our approximation,
it will be useless to investigate them by the theory, unless Vv'e wish to carry
on the approximation to quantities of the fifth order.
If we collect together the inequalities of the fourth order, which we have
just determined, they will become,*
+ n%61 .sm.(2v—2mv—2gv+cv+2è—z!) 1
— 8',ll.sin.(2cr-f2tJ— 2miJ— 2z=) 2
+ 10', 1 7 . sin.(2«— 2mvi-cv — c'mv — ^+t^') 3
pte' + 5%88.sin.(2cv— 2D+2mv+c'm«— 2^— ^') 4
— 0\2o.sin.(2cv—2vi-2mv — c'mv — 2^+ts') 5
— 3%'i6.s\n.(2cv+c'mv—2^—Ts') ' 6
4- 4%50.sin.(2cr— c'wîi'— 2îï+-n') 7
+ 33',38.sin.(4î;— 4?«t)— CD+^) 8
+ 15',46.sin.(4«— 4mu— 2cy+2!=) 9
— 8',31.(l+0-sin.(cy— ?J+J?ii"— i^) ' 10
— 6%01.(l+0.sin.(f — «u'+ci- — ïî). 11
18. We shall now consider the moon'' s motion in latitude. We have before
determined the tangent of the latitude s ; and, as the expression of the
arc, by its tangent s, is s— ^s-'+|/ — &c. [48] Int. we find, that the
latitude of the moon is very nearly represented by the following function ;t
* .'2959) If we connect together the quantities contained in [5240,5248,5257,5266,
[5305o] 5268,5283, 5296, 5302, 5304] ; the sum becomes as in [5305].
[5306a] I (2960) From s^^y.sm.gv -{- &s [48971], we get, by neglecting the second and
higher powers of 5s ; and reducing by means of [1, 2] Int.
VII.i.>^18.]
MOON'S LATITUDE.
r.( 1— i7-).sin. (gv—ù')+6s. 1 1 —jr7~+i7-'Cos.(2gv—2)) \ +-^^f.sin.(3gv—3é) ;
from which, by using the preceding value of y [5117], we get the latitude,
as in the following expression;*
679
[5307]
18542',79.sin.(5«— !))
+ 12',56.sin.(3^«— 3j)
+525\23.s\n.(2v—2mv—gv+û)
+ P,14.sin.(2f — 2mv-Jrgv—ô)
— 5\59.sin.(gv-\-cv — ù — zs)
1+ 19%85.sin.(^i" — cw— ^+to)
1+ 6'4,6.sm.(2v—2}nv—gv-ircv+ê — ra)
Moon's Latitude = / " V,39.shi.{2v—2mv+gv—cv—S+^)
\ — 2]%60.s'n\.(2v-r-2mv—gv—cv-rù-{-z!)
+ 24>%35.sin.(gv-'rcmv — i — z^')
' — 25',94.sin.(^« — c'mv — ^+ra')
— 10%20.sin.(2v—2mv—gv-irc'mv+ô — ro')
+ 22',42.sin.(2y — 2mv^gv—c''mv+ù+i!i')
+ 21%4,0.s'm.(2cv—gv—2-a+ù)
i+ 5',13.sin.(2cw4-^« — 2v+2'mv-
.—2z!—ù) I
1
2
3
"'4
5
6
7
8
9
10
11
12
13
14
16
Moon's
lattludo.
[5308]
[53066]
— ^s^ = — ^y^. s'm.^gv — Ss.y-. sin.-^u
The sum of the two expressions [5306a, J], is easily reduced to the form,
s — 153^7.(1 —iy'^).sm.gv+ôs.{i — ^y~-\-i7-.cos.2gv\-}-j'2-y^.sm.dgv. [53V6C]
Substituting this in the expression of the arc [530G], and neglecting the terms of a higher
order, it becomes as in [5307]. We may remark, that the term 5s.^7^.cos.2gv [5306c],
produces, by means of the term [4897 line 1], the expression
i J?',"' 7^. COS. 2gv . sin. (9u — 2mv — gv) ;
from which we obtain the term lB[''\7^.s\n.{2v — 2mv-\-gv) , which is of the same form
as that which is retained in [4897 hne 2] ; hence the expression of the latitude [5o06c]
becomes, very nearly,
7.(1 - if).sm.gv+.^\y^.sm.3gv-{-lB['>\7^. sin.(2«— 2mD+^D)-|-(l— Jys).^,.
* (2961) Substituting in [5o06e], the expression of 5s [4897], and then the values
[5306d]
[5306«]
580 THEORY OF THE MOON ; [Méc. Cél.
[5309] — = — , D being the eartli's radius.* Considering the smallness
19. It now remains to determine the third co-ordinate of the moon, or its
parallax. The sine of the moon's horizontal parallax is represented by
D Du
of this sine, we may take it for the expression of the parallax itself ; and,
if we substitute the value of u [53096] ; naraelj,
[5310] « = 1 . {l+e''+l7''+e.(l+e'').cos.(cv—z^)—l7''.cos.(2gv—2ù)}-\-èu ;
neglecting terms of the order — . e"* , we shall find, that this parallax
Motll'rt
purallax.
a
is represented by the folloicing formula.
[5311] -= -^^=- . (1 +e'^). 1 1 +e.n-l7''+iy^cos.(2gv-2})].cos(cv--^)-{-a&u-s6s I .
r \/ \-\-ss a ^ ' U41/ vo /j\
[5307o] of B^°\...B['^^ [5176 — 5191], also those of m, e, &,c. [5117,5194,5221], we get
[5308], nearly ; a few of the small terms being neglected.
* (2962) If we substitute the value of r [4776], in the well known expression of the
[5309o] «me of the horizontal parallax -, it becomes as in [5309], or as in the first member of
[5309/], and this may be taken for the parallax itself, by neglecting its third power. Now,
if we add to the expression of u [4826] , the part of <5u, [4904, &c.] , arising from the
perturbations, it becomes, by neglecting terms of the fourth order,
[53096] u =. ^ . j l-\-e^-\-t/+e.{l + e'').cos.cv—iy~cos.2gv+a5u j as in [5310]
[5309c] =^.(\+e~)-\l-\-{7--{l—cos.'2gv)+e.cos.cv+a<')u]^.
Developing the radical \/l+W , neglecting s" , and substituting for s, its value
[5306a], we get,
[5309<i] 77^n^= l—i^= 1 -i. j7=.sin.V+2^«-r-S'"-5-^^
= 1 — iy-.(l — C0S.2i^u) — ds.y.s'in.gv .
Multiplying together the expressions [5o09c,e], and the product by D, we get, by
neglecting 7^, &.c. of the fourth order.
[5309e]
[5309/] —^ .(l4-e-).{ l+c.(l— iy+if.cos.2^«).cos.ra+a5M— <55.7.sin.g«|.
Y l-\-ss a
This becomes as in [5311], by substituting, in its last term, the approximate value of
[5309^] y.sin.g-D = s. [5306a].
VII. 14 19.] MOON'S PARALLAX. 581
To determine —, we shall observe, that we have, in [4968],*
a
^- = -. 0,9973020 ; [53i2]
a a,
and, by [5082, 5090], t
Hence we get,î
-^ X 1 ,0003084 =. - . [5313]
l^ é ^ /n^. ( 1 ,0003084)2
a Y 0,9973020"
[5314]
Let 2 s be twice the space which the earth's attraction would make a .^gj^
particle describe in the time t , in the parallel, on which the square of the
3T
sine of the latitude is i. This attraction is — [1812, 1811/],^ the [5316]
earth M being supposed elliptical. But we have before put M-\-m = 1 [5317]
[4775"] ; m being here the moon's mass ; therefore, we have,
~ (iVi+»z).X)2 • [^318]
a 2
* (-2963) Substituting, in m [5094], the value of m [5117], we get m=0,0055796.
With this, and the values of e', m, y, A'i\ B?" [5117, 5157, 5176], we find, that t^^^^"]
the equation [4963] becomes nearly as in [5312].
t (2961) If we substitute, in the coefficient of — — ■ [5082], the values of w, A'-^\
A[^\ e [5117,5157,5158,5194], it becomes i^^.l, 0003084. This is to be put equal [5313a]
y (If
dv
to — [5090]; hence we get [5313].
t(2965) We have, in [5312], J- = ~ ■ ^=m- ; multiplying this by
a'. 1,0003084, and substituting, for the first member of this product, its value - [5313], ^5314^1
1 3 1,0003084 , ., , , r ,
we eet — = a-. , — ^==^ ; whence we easily deduce [53141.
^ n v/0,9973020 J' L J
% (29G6) Changing z into s, in [67], we get 2s =^^2; g being the force of [5316a]
gravity [54" line?]. Now, in the parallel of latitude, mentioned in [5315], we have
VOL. in. 146
582 THEORY OF THE MOON ; [Méc. Cél.
Hence we deduce,*
[5319] ^ = \ / -^^ £) "'^' (1,0003084)^
« V M-\-m ' 'as * 0,9973020
If we suppose t to be equal to a centesimal second, and T equal to the
number of centesimal seconds of time, during a sidéral revolution of the moon,
[5331] we shall have t »»^ = ™ ? "^ being the ratio of the semi-circumference to
the radius. If / be the length of a pendulum, vibrating in a centesimal
'•^ ■' second of time, upon the parallel under consideration, we shall have, as in
[5323] [86'], 2;=:^^/,t which gives,
M
^2 — ^=0 [1618'"]; Iience the expression of V [1812], becomes V=—, M
being the mass of the ellipsoidal earth, and r the distance of the attracted point from its
centre, [1G16', 1016"'"]. Now, the attraction of the earth, in the direction r, is represented
r5316c] by — ( — ^i :^ — [181 iZ, 53166] ; and, by changing r into D, to conform to the
M
present notation [5309], it gives very nearly, the expression of the gravity g = jp. [531C];
M l~
[5316d] hence [531Ca] becomes 2j = -p- . We have put, in [4775"], Jll-\-m equal to unity,
M
,,„.,, therefore, for the sake of homogeneity, we may change M into ~ — , in the preceding
[5310e] ° •' M-\-n ^ °
expression of 2; , and then it becomes as in [5318].
* (2967). Multiplying [5318] by — , and extractmg the cube root, we get,
the product of this, by the expression [5314], gives [5319].
t (2968) Noticing only the mean motions, we have nt = v [5220]. Now, when v
|.ggç,j , is equal to 2t , t becomes T [5320], and we get ?« T= 2ir, or 7i == ~ ;
whose square gives n^ [5321].
J (2969) Changing, in the formula [86'], z into e, and r into I, to conform
4
to the present notation, we get 2s ^t^; as in [53331. Multi;)lyina; it by ,-77^. we
[5323a] L J I J o J ijo
obtain J^2=:^; hence ir=-~ [5321]; and, as <=1, [5320], we have
— = A ; substituting this in [5319] and putting 4.(1,0003084)2= (2,0006168)2, we
get [5324].
VII. i. § 19.] MOON'S PARALLAX ; 583
D _ .^ / M D (2,0004108)2
.W+m / ■ 0,9973020.T2 [52,14.]
The length of a pendulum, vibrating in a centesimal second, upon the same
parallel, is equal to 0™^'-,740905 [2054],* we must increase it by its 434th ^^^^^
part, to obtain the length which could obtain independently of the centrifugal
force ; hence we have, / = 0"""-,742612. The value of D is equal to [532G]
6369374"'"- [3896, nearly] ; lastly, we have, by the phenomena of the tides, [5327]
m= —:, — [4321]; and, by observation, T = 2732 166 centesimal seconds; [5328]
hence Ave have,t
■?- = 0,01655101. f5329]
Estimating in seconds the coefficient — .(1+e^), we find it equal to [5330]
3424", 16. This being premised, we find, for the expression of the moon's
parallax, in the proposed parallel ; X
* (2970) This value corresponds with the formula [2054], putting sin.-Nj^ =sin.^]at.=^
as in [531 G]. This must be increased ^j^j part, to correct for the centrifugal force [088"],
by which means it becomes / = 0'"='-,742612, as in [5326]. This will be varied a Title
if we use the corrected value of / [2054w,or205Cp].
[53256]
t (2971) Substituting, in [5324], the values of I, m, D, T, [5326 -5328], we
get [5329], nearly. Multiplying this by 1-f-ec [5194], and then by the radius in
seconds, we get the expression [5330], nearly. This would be varied a little by correcting •'
the value of /, as in [o325J], and also by the change in the value of m [11906, 3380i].
f (2972) Substituting the value of - . (l+fc) = 3424^16 [5330] in [5311], it becomes r533Q„i
Moon's Parallax = 3424%1G.{ l-\-e.{l—ly^-\-^-)'^.cos.2gv).cos.cv-\-aSu — sSs\. [53306]
We may substitute in this }y^e.cos.2gv.cos.cv = ^j'^e.cos.{2gv — cv), neglecting the term [53301
depending on the angle 2gv-\-cv, because the term is small ; and angles of this form are
not retained in [5331]. Moreover, the chief term of Ss [5308 line 3, or 5307] is
à25',23.sm.{2v—2mv—gv); and the chief term of s [5307] is [5330rf]
7.(1 — 17^). sin. ^11 = 0,0899. sin. ^t) [5117line5]. [5330^]
Multiplying these two expressions together, we get, in sSs , the terms,
2S\6.\cos.{2gv — 2v+2mv)—cos.{2v—2mv)\ ; [ssscn
substituting this in [53303], and dividing by the radius in seconds 2062C5', it produces
the terms 0',39. j— cos.(2^i'— 2y-{-2my)-f-cos.(2i; — ^mv)\. Hence, [523 Oi] becomes,
584
THEORY OF THE MOON ;
[Méc. Cél.
Parallai.
[5331] Moon's Parallax =
3424% 16 V
1
+ 187%48.cos.(ct)— z^) \
2
+
24>%6d.cos.(2v—2mv)
3
+
38%07 . COS. (2v—2inv—cv+-^)
4
—
0^70 . cos.(2îJ — 2mv-{-cv—^)
5
0',17.cos.(2î; 2mv+c'mv ra')
6
+
1%64. cos.(2t> — 2mv — cWi+w')
7
0^33. cos. (c'm«—^')
8
0',22 . cos.{2v-2mv-cv-{-c'mv+^-^')
9
+
P,63 . cos.{2v-2mv-cv-c'mv+-^+-^)
10
0%65 .COS. (cv-\-cmv — « — to')
11
+
0'',87 . cos.(cD— c'my— w+to')
12
+
0%01.cos(2cw— 2^)
13
+
3^60 . cos . (2cî;— 2tJ+2 mv—2z,)
). 14
+
0^07. cos. (2^t;— 20) /
15
0',1 7 . COS. (2gv—2v+2mv—2ô)
16
0%0l.cos.(2c'mv—2-m')
17
(y,9o.cos.(2gv CV 2S+r,)
0',06 . cos.(2v-2mv-2gv + cv+2s-^)
18
19
0%97.(l+i).cos.(v—mv)
20
+
O'J 6 . (1 -|-«).cos.(z7 — mv-\-c'mv-y:')
21
0%04.cos(2«-2miJ+c«;-c'mw-TO+TO')
22
—
0', 1 5 . COS. (4>v — 4m«j — cv+:^)
23
+
0',05 . COS (4^—Amv — 2cz;-f 2si)
24
+
0', 1 3 . cos(2ce;-2t)+2w«+c'mz;-2a-TO')
25
+
0%02. cos.(2cu+2«— 2mzj— 2^)
26
0%\2.(l+i).cos.(cv—v+mv -) ,
27
[5330âr]
Moon's Parallax = 3424%16.p+e.(l—iy2).cos.ct)4-^72e.cos.(2^D—CT)+a(5M}
— 0^39 .COS. (2gv—2v-\-2mv)-\-0\ 39.cos. {2v—2mv).
We must now substitute the value of aSu [4904,5242,5251,5258,5269, 5287, 5297, Sic],
also [5117,5157—5175,5221], and we shall get [5331].
VII.ii§20.] EFFECT OF THE OBLATENESS OF THE EARTH. 585
CHAPTER II.
ON THE LUNAR INEaUALlTIES ARISING FROM THE OBLATENESS OF THE EARTH AND MOON.
20. We shall now consider the terms arising from the oblateness of the
earth and moon.* We have seen, in [4773], that the effect of this oblateness
is to add to the expression of Q [4756] the quantity,
( ôV ôV )
(M+m). < —H ^ = increment of Q. ' [5332]
If we put,
oLp =: the ellipticity of the earth ; [53331
a;p ;= the ratio of the centrifugal force to the gravity at the equator ; [5333']
D = the mean radius of the earth ; [5334]
(* = the sine of the moon's declination ; [5334']
we shall have, as in [1812],t
* (29T3) This subject is tieated of, in a more simple and elegant manner, in the
appendix to this volume [5937 — 5971] ; and again, in the fifth volume [12952 — 12996];
where some very small terms are noticed, which are neglected in this volume ; but they "
have but little effect on the resulting formulas. We shall, in the notes on this chapter,
restrict ourselves to the terms here investigated by the author, and shall follow the same ^ '
method of demonstration which he has used.
t (2974) We have, in [1812], for an ellipsoid of revolution,
r = f+(4.,-./o.i.itf.K-4). ,,^,
VOL. III. 147
586 THEORY OF THE MOON; [Méc. Cél.
[5:135] F =: ^ + I lap— a,, | . ^. M.(^.2— X).
If the earth vary from the elliptical form, Ave shall have, by § 32, 35, of
book iii.,*
[5336] r = ^' + ^ (ia,?_ap).CaS— a)+a/t'. (1— ,x2).COS.2^ | . M. ~ ;
[5337] ap and 0-h' being constant quantities, depending on the figure of the terrestrial
[5338] spheroid; and ■^ the angle formed by one of the two principal axes of the earth,
situated in the plane of the equator, with the terrestrial meridian, passing
through the moonh centre [1746']. It is evident, by the following analysis,
[53356] To conform to the present notation, we must put tt/(= ap [1795', 5333] ; and, in the
1 2)2
[5335c] second term, we must change - into -^, to render it homogeneous with the first term :
observing, that the radius of the earth D [5334], is supposed to be nearly equal to
unity in [1795", 1812]. Making these changes in [5335a], it becomes as in [5335].
* (2975) Neglecting the attraction of foreign bodies, and the terms depending on r"'^,
in [1811], we get, for an ellipsoid of a general form,
[533(V,] V^f^ ./„\o . d.a-+^^. r(^>./„'p.rf.«^- "3 . Z^-\
To render this homogeneous, we must multiply the two last terms by D^, as in [5335c],
and substitute, for ^itf^^.d.a^, its value M [1811', 4757] ; by which means it becomes,
[5336c] F = f + J a Y^^> -| .Z^^^ \.M. ^'.
If we substitute this value of M, in aZ<-' [1793], we get,
[53366]
[5336c'] _^.Z«)=ia9.(,2_A);
and, from [1763], we have, by changing a/i into ap, as in [5335i], also h"" into A',
to conform to the present notation,
[5336c/] ^'''' = _p.(,.3_J)+A'.(l-^^).cos.2« ;
r5336el the earth being supposed to revolve about one of its principal axes [1 762', &c.]. Substituting
these in [5336c], it becomes as in [5336]. The radius of an ellipsoid is represented by
[5336/] j_j_y^y(n, [-1775^01.15030] ; and, if we substitute the value of Y^-'^ [5336^/], we get,
„ , 1 — ap.((J.~ — ^-)-|-aA'.(l — fj,~).cos.25i = radius of the spheroid.
[5336A] At the pole, where fx=lj this becomes 1— jap; subtracting this from [5336^-], and
Vll.ii. §20.] EFFECT OF THE OBLATENESS OF THE EARTH.
687
that the term depending on cos.2ro has no sensible influence on the
lunar motions on account ol' the rapidity with which the angle w varies;
so that the value of V, which we shall here use, is the same as in the
elliptical hypothesis, with an ellipticity equal to ap ; but, in the general
case of any spheroid whatever, ap does not express the dlipticiiy [53o6/i;].
We may, therefore, suppose, in this general case, that the value of Q [4756]
is increased, on account of the oblateness of the earth, by the function,
(|aç — f-p). — .((J.2 — -i) = increment of Q [4756];
M^m being taken for the unity of mass [4775", 5336/].
We shall, in the first place, consider the variation of the orbit, or the moon's
motion in latitude, depending on this cause. If we put >- for the obliquity
of the ecliptic to the equator, and fix the origin of the angle v in the vernal
ecjuinox, at a given epoch ; tee shall have, very nearly,'*'
[5338'J
[5339]
[5339']
[5340]
[5340']
[5341]
[5342]
[5343]
putting ;j. = 0, we get the excess of tlie equatorial radius, or,
the ellipticity = ap-)-aA.cos.2^. [5336i]
Hence it appears, that the ellipticity of the different meridians varies, with the different ,roqcA.|
values of 2^, from ap — a// to af-f-aA ; instead of heing generally represented by
ap, as in [5333,5339']. From [4767,5336] we get &V, and then the first term of [5336/]
[533-2] becomes as in [5340,5340'].
* (2976) In the annexed figure, P is the pole
of the moveable equator ; P' the pole of the ecliptic ;
M the place of the moon ; so that, if the moon's
latitude be represented by /, and the declination by ^
d, we shall have PM=9Q-'—d; P'M= 90''— I; ^^
PP'=>.; PP'M= 90"— fv [5345]. Substituting
these symbols in the formula [5344f], which is the same
as [1315®], we get [5344r/], using the symbol /j- = sin.c?
[5-334']. This is reduced to the form [5344e], by means
of the expressions of sin./, and cos./ [47T6i] ;
cos.PM = sm.P'P.sm.P'M.cos.PP'M+cos.P'P.cos.P'M ;
/J. =sin.X.cos./.sin.yi)-{-cos.X.sin./;
1
J'oie o/Eriuator
Mo un.
fA= sm.X.
v/l+ss
.sin./i!-(-cos.X,
VÎ+S
[5344a]
[.53446]
[5344i'J
[5.344c]
[5344(/]
[5344e]
588 THEORY OF THE MOON; [Méc. Cél.
L5344] ft = sin.x.y^i — ss .sin.fv-\-s .cosA;
[5345] J'y being the apparent longitude of the moon, referred to the moveable vernal
equinox. We must, therefore, add to the value of Q a quantity, which we
shall represent by,*
Terms of
[5346] Q= ih'^v — ap).— .{sin.-x.(l — 5^).sin.y«-J-2s.sin.x.cos.x.sin/t)-}-s^.cos.-^ — i}.
This being premised, we shall resume the equation [4755]. We have
developed, in [5018 — 5049], the different terms of this equation, depending
on the sun's action. It is evident, that the preceding function adds to the
equation [4755] the following quantity, f
[5347] 2.(ap — lo-cp). ^ ■ .sin.x. COS. X. sin./y+ (g^ — \).H.smfv ;
[5344/]
[5346o]
[53466]
If we neglect the third and higher powers of s, we may change , into \/l—s^,
and , ^ into s ; by which means, the formula [5344e] becomes as in [5344].
* (2977) Substituting (a [5344] in [5340], and putting 2 s, for 2s.^rT^, in
1 u
the coefficient of sin./t;, we get [5346]. Now, we have — = , [4776],
which is nearly equal to ?<.\/l — s~', substituting this in [5346], neglecting s^, &ic. ,
we get, for this part of Q, the following expression ;
Q=(i[i,:p — a.'i).D^.u^. ? sin.-X.(l — s-)-.sin.^/i;-(-24-.sin.X.cos.X.sin.yD-)-s^.cos^X — ■j(l-«^)- \ .
t (2978) The substitution of the value of Q [534G6], in [4755], produces an
[5347al equation of the same kind as [5037] , in which r is composed of a series of terms, of
the form k,. s'm. (i^t-^-s^), depending on Q. When i^ is very nearly equal to unity,
r5347il ^^^'^ corresponding term of s will be very ir.uch increased by the divisors introduced by
the integration ; as in the similar case of the equation treated of in [4849], as will be seen in
r5347cl [5347r — t']. Now, /—I is of the order aiiyVocj [53475'] ; therefore, the term depending on
sin./i' must be particularly noticed ; and, in fact, it is the only one the author considers
as necessary to retain in this calculation. In making the substitution of the value of Q
[53466], in [4755], we may neglect the quantities (-j^)) ("/")' because they are
ds
r5347el multiplied by s, or —, of the order y.sm.gv, or y.cos.gv, and produce only
terms of small value, in which i, differs considerably from unity. We may also neglect
Vll.ii. ^5-20.] EFFECT OF THE OBLATENESS OF THE EARTH. 589
supposing the inequality of 6s, depending on the angle fv, to be [5347]
the term •( — I [47551, because it is multiplied by s^. Hence the equation
h-.u^ \ds J -^ ^
[4755] becomes,
^ dels , 1 /dQ\
dds ,
Now, by noticing only the terms depending on sm.fv, we get, from [5346i],
(-^] = 2.(|aç)— ap).DV.sin.X.cos.X.sin./y.
Substituting this in [5347/], it produces the term,
1 A/Q\ ^ , , , />2« .
"" hA^^' \di) "^ -•("-'— JM--^ sm.X.cos.X.sin/t. ;
which is the same ns the first term of [5347], or the first term of the function r [50371.
The other term of r is deduced from [5040 line 1], iWi.-.5s , by the successive
substitution of [5082/t',4828e] ; by which means, it becomes,
§ m^. Ss = (g- — 1 ) .OS, nearly ;
and, if we use the value of Ss [5348], it produces (g^ — l).JZ.sin./j;, as in the last
term of [5347]. Hence, the equation [5347/], by retaining only the terms depending on
the angle fv, is reduced to the following form ;
= ^a+*+2-l"-P— io-?.).— .sm.X.cos.X.sin./y4-(n-2_l).7i.sin./«.
Substituting the assumed value of 'U , or s = H. sm.fv [5348], and dividing by
sm.fv, we get,
0 = (— /^+l)-ff+2.(ap~ia?).-^\sin.X.cos.X+(^2— l).if.
Connecting together the terms depending on H, and dividing by its coefficient, we get,
-"^ ^:Zf2 -^-sm.X.cos.X.
The moon's longitude v, is counted from tlie fjced axis x, or the fxed vernal
equinox [4760'] ; and fv [534-5] is the same longitude, counted from the moveable
vernal equinox ; hence, /—I is of the same order as the ratio of the precession of the
equinoxes to the moon's mean motion. Now, the annual precession is nearly 50' [4614], and
the moon's annual motion is —-=4813'', nearly [5117, 5I17«]. These quantities
are to each other in the ratio of 1 to 340000, nearly ; hence, /—I is of the order
WcTTir; which is very small, in comparison with g-— 1 = fm^ := ^i^, nearly [5I171ine3];
VOL. III. 148
[5347/]
[53i7g]
[5347A]
[5347.]
[5347A;]
[[5347/]
[5347m]
[5347rt]
[5347n']
[5347o]
[5347pJ
[5347?]
590 THEORY OF THE MOON ; [Méc. Cél.
represented by,
[5348]
6s = H.s'm.fv.
We may, moreover, easily satisfy ourselves, that this quantity is the only
sensible one which results from the function Q [5346]. Adding it to the
differential equation [5049], and observing, that /—I is extremely small,
[5349] in comparison with g — 1, we get, by integration,
^^ — 2.(ap— iaqj) ■ D'^ .
[5350] H= h — r — • -^- sm.x.cos.x.
g^ — 1 a-
Hence we obtain in 5 , or in the moon's motion in latitude, the inequality,*
[5351]
[5351'
Is = — — ^. — r. sm.x.cos.x.sm.ji'.
Which is the only sensible inequality of the moon'' s motion in latitude, arising
from the ohlateness of the earth. This inequality is equivalent to the supposition
that the moon's orbit, instead of moving on the plane of the ecliptic, ivith a
[53521 constant inclination, moves, ivith the same condition, upon a plane passing ahmys
through the equinoxes, beticeen the equator and the ecliptic, and inclined to this
[5347g'] so that we may put /=!, in [5341n'], and we shall get,
_2.(ap-^a9) D^u
H = V— ; • — To— • sin.X . cos.X.
[5347J-] jg- — I /*
Substituting; u = -, h^z=a, = a, [4937n, 5312], it becomes as in [5350]. We
[5347«] a
may observe, that if / differ considerably from unity, it will make the corresponding
value of H, deduced from [5347n'], very small; because the divisor, in finding
JH, will be a large number of the order g^ — /^, instead of the very small one
of the order g- — 1 [5347r] ; and, for this reason, most of these terms of Q
may be neglected, considering that they are multiplied by the very small factor (o-r-aU-?)-"", ;
which can become sensible only by means of a small divisor.
[5347/]
* (2979) We have g^—\={g-\-l).{g — l)=2.{g—l), nearly [4828e],
[5351a] substituting this in [5350], and then the resulting value in ôs=H.sm.fv [5348], we
get [5351].
VII. ii. '^ 20] EFFECT OF THE OBLATENESS OF THE EARTH.
591
last plane, by an angle, loJiich may be represented as follows,*
(ap-4a(p) 1)3 . , [5353]
Angle of inclin. of the equator and fixed plane = ' '—-^ . -^ . sm.x.cos.x. ]^:^'S-,^,
° '■ g 1 « equator to
the fixed
We have found in [5329, 5117],
-=0,01655101; ^—1=0,00402175; ^5354]
a
also at the epoch, in 1750,
X = 23'' 28'" 17%9 [4353"]. ^5355]
Lastly, 019 = ^1^ [1594«,&c.]; therefore, by supposing <^p = ^i^- [2034] [5356]
the preceding inequality becomes, f
j\'l[ Moon
77
\
[53520]
J?
3^-— -
[5352!)]
■Fixed plane n
[5352cl
* (2980) The angle of inclination of the
ecliptic to the fixed plane, given in [5353],
being put, for brevity, equal to A , we shall
h ave A = — H [5353, 5350, 535 1 a] ; and
ês = — A.s'm.fv [5348]. Suppose, in the
annexed figure, that C R represents the
equator, C B the fixed plane, C L the
ecliptic, .A/the place of the moon, ML a.
circle of latitude, perpendicular to the ecliptic,
MDB the arc perpendicular to the fixed
plane ; then the difference of the arcs M L,
M B, will be veiy nearly represented by
BD=aug\eBCD x sm.CD = A .sm.fv [5352a,5345].
Hence it is evident, that if the moon's latitude, above the fixed plane, be expressed by
B M = s, its latitude, counted from the ecliptic, will be very nearly represented by
ML = MB — BB^s — A. sin. fv=s-\-hs [53526] ; as in [5352].
t (2981) The expression of A [5352a, 5353], is.
A =
_ (gp— èaç) _D2
g-l
. sin. X. cos. X.
Substituting the values [5354,5355], and that of aç [5356], we obtain,
^ = 5132',9.ap — S-'.SS ;
hence, .^-j-8^S8
"^^^ 5132%9 •
[5352(/]
[5352e]
[5357a]
[53576]
[5357«]
692 THEORY OF THE MOON ; [Méc. Cél.
Inequality
[5357] OS = — 6,487 . sin./ v .
depending
on the
[5358] It would be — 13',436.sin./i; , if the oblateness be -\^, which corresponds
°inb. *^o the supposition that the earth is homogeneous [1592a]. Therefore, if
r5358'i ^^^^ inequality be carefully observed, it will be very useful in ascertaining the
oblateness of the earth.
We shall noio consider the variations in the radius vector, and in the moon^s
longitude arising from the oblateness of the earth. We may deduce them
from the equations [4753, 4754] ; but it is more simple and accurate, to use
the formulas [919,923]. For this purpose, we shall suppose, that the
[5359] differential characteristic 6 refers to the quantity -^<i-? — a--. We shall then
observe, that the functions R, rR, [913,928'], are represented by, *
[5360]
R = — Q^\ [4774a] ; rR' = r . C^^_
dR\
Hence, the equation [919] becomes,!
[5361] 0 = -^+ -3- + 2 .fUR+,.r . ^-
[5362]
We have, in R , the term,î R = 2.(aj) — ia-p).-j . sin.x.cos.x.s.sin./y. This
contains the following term.
r5357rfl ^f '*'P =^ 3^* ' ^^® ^^^^ °^ ^^^^^ equations gives A = 6',4S7 , as in [5357] ; and, if
ap = ji!r) it becomes A=lS%4o6 , as in [5358].
* (2982) If we substitute the expression of Q [5346i], in R [5360], we shall
obtain,
[5360a] = 1_^ (ap— |a?).D2M3 5sin,2X.(l— s2)^.sin,'-yy4-2s.sinX.cosX.sin>-fs2.cos^-i(l-«')^^;
which will be used hereafter.
t (2983) The equation [5361], is the same as [919], using the expression of rR'
[5361a] [5360], and that of iJ. = M+m=l [914',5340'].
X (29S4) The term of R , retained in [5362], is the same] as that in [5360a],
[5362a] depending upon s.sinfv. Substituting in it the chief term of s; namely, s = 7.sin.^«
Vll.ii.>§.20] EFFECT OF THE OBLATENESS OF THE EARTH. ô93
éR = (a-. — iaç) . — . 7 . sin.x.cos.x.cos.(gi' — fv — .'). [5303]
This term of sR gives, in fs.dR, an expression which is exactl}' similar
and equal to iR . For the differential characteristic d [916'], refers [5363]
only to the moon's co-ordinates ; and, we have, by noticing only the preceding
term,
/ S.dR =iR .
Then we obtain,*
6.r.( — j = — 3 . (etp — lap) . — . 7. sin.x.cos.X.cos.(^T— j^ — 6).
If we substitute these values of fôAR, and 6.r.(-— j [5364,5365],
[5364]
[5305]
in the differential equation [5361], we shall find, that the expression dr
contains a term, depending on cos.(gv — -fv — &) , but it is insensible, not [5366]
having g — 1 for a divisor, which the corresponding term of 6s has.
[4897»"], it produces the term given in [53G3], which depends on the angle (g-f)-v ;
— being used instead of ti. Now, the coefficient of this angle, is of the order ^ — 1 ,
or ?n- [5347c], and the integration of dùv, in [5387], introduces g — 1 as a divisor;
and it is on this account, that the terms depending on the angle gv — fv are retained by
the author.
[53626]
[53C9c]
* ("2985) The partial differential of oR [53G3], taken relatively to /• , and
multiplied by — , gives [536.5], as is evident from the nature of the symbol & [5359]. [3366n]
If we substitute the values [5364, 5365] in [5361], tliey will produce in it an expression, ,._^.,,
[.j36bol
n .
which we shall represent by ^. Then, if we put, for a moment rSi- = u, the equation
*" [5366c]
[53611, will become 0 =-~ -{---]- -. Multiplying this by r^, and puUine; for
•- «''■ •■ '" [5366-/]
(It, its chief term -^ [5081], or 7~(Jv, nearly, it becomes, 0 = --^-\-u-\-U ;
^"' . , . . [•'536fie]
which is of the same form as [4S45], supposing jV=l. Its integral [4847] introduces
the divisor i' — JV''^ ^= P — 1 , which is nearly equal to — 1; because, in the present
case, i = g — / [4S46, 5363] is of the order m'^ [53475f]. Hence it is evident, that i^^cQfi
u =r6r , is not increased by the introduction of a small divisor in the integration. This
agrees with [5366].
VOL. III. 149
69^ THEORY OF THE MOON ; [Méc. Cél.
It is not the same with the expression of the longitude. The formula
[923J gives, in cUv, the following terms ;*
,53(.-. 3df.fs.àR+2dt^6.r.('-
[5368]
r .dv
Substituting the value of ôR [5363], we obtain, in dàv, the following
term ;t
d.ôv := — • Vt — - — - •— T '7 ' sm.x.cos.x.cos.f^w — fv — ê).
r^.dv r^ \s J /
But, this is not the only term of the same kind, in the expression of div.
[5369] The sun's action gives, in Q [4806], the term Q ^'i:^^-J\ — 2s^).
[5370] Substituting in it the value of u [4776], we obtain, in 7? «= — Q+-
[5360], the expression,!
which gives, in àR, the term,
[5372] àR=:^m'u'\r^.Sf,s;
* (29S6) Noticing only the terms depending on tlie angle gv — fv — d , or those
'• "•' which produce the factor s&s in [5373, Sic], we may neglect or, and then we obtain
from [923],
[53676] ,, Sdfi. f 6AR+'îdfi.r5R'
■■ •' dSv = 5-^ .
r^.dv
Now, we evidently have rSR'=.5.{rR') — R'&r ; and, for the same reason as in [5367a],
[5367c] we may reject R'5r ; then, using the value of rR' [53G0], we get i-5R'=ô.r.( — j;
hence, the preceding expression of dSv becomes as in [5367].
t (2937) Substituting fô.dR^SR [5.364], in [5.367]; and then using the values
[5363,5365], we obtain the expression [5368], by a slight reduction.
1 7-3
r5370al Î (2988) From u [4776], we deduce — == — ^ = r^ .{l — s^}, nearly ;
multiplying this by ^m'u'^.^l — 2s^), we get the value of the term of Q [5369];
and, by the substitution in R [5370], we obtain the term [5371], neglecting quantities
[53706] qJ- jj^p order s\ The variation of [.5371], relative to the characteristic 6, putting
5r=0 [5367ff], gives iR [5372].
vil. il. §20.] EFFECT OF THE OBLATENESS OF THE EARTH. 595
from which we easily deduce the following expression ;*
3/é.di?+26.r.r^) = 'i.m'u".r.sôs. [537.3]
We have, very nearly,t mii'\r^ = »r, also g = 1+f m^ [5117or4828e] ; ^^^^^^
hence the function [5373] becomes,
/(}R\ 14. (p- — l).s5s
3f6.dRi-26.r.(—J = r "• ^^^^^^
Substituting in it as = — ^''^~^""^\ — -sin.x.cos.x. sin.> [6351,5374a];
^~^ '■' [5376]
and s ^ 7.sïn.(gv—ê) [5050], we obtain, in [5375], the following term;|
(2989) From [5371], we get, by differentiation,
r . f'^) =-im' u'\ r\ {i-2s^). [5373al
Jr J
Its variation relative to the characteristic 5 , neglecting <5r [5367a], gives,
S.r.(^—^ = ^m'u'^.r^.sès; [53736]
and, from [5364, 5372], we have f S.dR^ §m'.u'^r^.s5s. Substituting these values in
the first member of [5373], it becomes as in its second member.
t (2990) We have nearly r = a, m'=- [4937n, &c.] ; substituting these in [5374a]
m'.u'^.i-^, we get m'.u'^.r^ ^ —— := m [4S65] ; and, from [5094], it appears that
this is equal to m^ nearly, as in [5374]. If we use the value of ^ , [5374], it becomes
m'. u 3.r3 = 4 . (^—1 ) . Substituting this in [5373], we get [5375]. '^^^''^^^
J (2991) Multiplying together the values of s and 5s, [5376]; reducing, and
retaining only the term depending on the angle gv — fv — 6 , we get,
sis = — 21"— I) ' ^ • y • sin. X . cos. X . cos. (gv —fv — ê) . [5376a]
Multiplying this by — '- , we obtain the expression of the second member of the [537661
dt-
equation [5375], as in [5377]. Multiplying this last function by — — - , we get the term
of d&v , corresponding to the second member of [5367] ; namely,
V.dv H •'/•sin.X.cos.X.cos.(5-«-/t.-<!); ^^g^^^^
adding this to the term [5368], we get [5378].
ê96 THEORY OF THE MOON ; [Méc. Cél.
[5377] 3f6.dR-\r25.r. f'l^^ = — 7.(ap— la^). — .7 . sin.x.cos.x.cos.fs-r— /r— :').
Multiplying this by -^— , we obtain, in the expression of cUv [5367],
a term which is to be added to that in [5368] ; and the sum becomes,
rco^oi -. , \Odt^.(a.n-ia.a) D^ . , r \
[5378] dij'D — — . A_l ' .—,./• sm.x . cos.x . cos.r^-i' — fv — è).
We may substitute in it, a for r, do for 7idt [603, 4828], and
^^^^^'^ n^rt^ = 1 [3709'] ; by which means, it becomes,*
[5379] d&V == — 10fl«.(ap — ia?)).-^.7.sin.X.C0S.X.C0S.(^?;— /i) — (').
(r
This value of d5v corresponds to the angle contained between the two
[5380] consecutive radii vectores r and r-{-dr, as in [923 — 925]. Now, if we put
this angle equal to dv^, dv will reiiresent its projection upon the plane of
the ecliptic, and we shall have, as in [925],t
[5381] dv = dv . -Tl -^= ^^ ;
or, very nearly,
[5382] dv = dv^ . ^ l+hs-—h-Jl, J •
(2992) Substitutina; in the factor v^r • T> ^^"^i'^'^ occurs in [5378], the values
[5379aj ^jf^±^ r = a, and n^'a^ = 1 [5378'], it becomes,
dv -D2 _ m
[53794] ^a?'l^ — '''' " a^ '
hence [5378] changes into [5379].
f (2993) The expression [5381] is nearly the same as tliat in [925], changing v into
V , and V into v in order to adapt it to the notation in [5380], which is different
from that in r923'l, observing that, on account of the smallness of s, we may change
[5381a] , J ds
— into — . Developing [5381], according to the powers and products of i, — j
dv, dv "
neglecting the fourth dimension of these quantities, it becomes as in [5382].
Vn.il S-20.] EFFECT OF THE OBLATENESS OF THE EARTH.
597
Substituting for s tlie expression,*
5 = 7. sm. {gv — c) — ^-^ — — ^ . — . sni. x . cos. x . sin.fv ; [5383]
we get,t
(h = dv, . < 1+1 . (c>-p — i«-?) . ~^. y . sin.x.cos.x.cos.(^«— ^îj — 5)-f-&c. > .
Hence we see, that to obtain the value of J5« , relative to the angle v,
formed by the projection of the radius vector r, upon the ecliptic, with a
[5384]
* (2994) Substituting in [5050], tlie values of h [5351], we get [53S3] ; wiiich
by using the value of A [5o57a], becomes as in [5383c], omitting for brevity the symbol
(3. Its differential gives [5383f/], observing that / is nearly equal to unity [5347»/].
Squaring these expressions, retaining only the products sin.^u.sin._/w, cos.^u.cos._/i,
which produce the term depending on cos.(^« — fv) , we get [53S3e,/], whose sum is as
in [53S30-] ; this is used in the following note ;
5 = y.sin.^D — A.zm.fo ;
ds
dv
= gy.cos.gv — ^.cos./« ;
is- = — ^7.3in.^y.sin./w-(-&z,c. = — ^Ay.cos.(gv — ■/i')-(-&ic.
ds^
— i-— = +gAy.cos.gv.cos.fv-{-iic. = igAY.cos.(gv—fv)-\-hc. ;
^■' -^-^^ i.(^-l).^y.cos.(^.-/.) .
[5383a]
[53836]
[5383c]
[5383rf]
[5383e]
[5383/]
[5383g]
[53;'G«]
[53866]
t (2995) Substituting [53S3o-], in [5382], we get,
civ = dv,.ll-j-i{g— I) .Ay . cos.(gv —fv}} ;
and by using the value of A [5357a], it becomes as in [5384]. Hence it appears tliat
this reduction, adds to the value of dv , the term dv^.^(^g — 1) .,;3y. cos.^^-u — J'v),
or dv.i(g — I) .Ay.cos.{gv — fv) nearly; which by the substitution of A [5357«],
becomes as in the second member of [5385]. This term of dv , is a part of that
depending on ap — laçj, which is denoted by d&v in [5359, 5379, 5385, &c.]. Adding [5386c]
together the two parts of dôv [5379,5385], we get the complete value [538G], and its
integral, putting /= 1, gives i5y [5387]. This expression is obtained, to a somewhat [5386(i]
greater degree of accuracy, in [12995] ; where small terms are computed, of the order
3m . ■ . [5386e]
— , in comparison with those which are here investigated.
VOL. til.
150
598 THEORY OF THE MOON ; [Méc. Cél.
fixed light line ; we must add to tlie preceding expression of div [5379],
the term,
[5385] (/.îî; = ic;î;.(ap— lap),— .7.sin>.cos.x.cos.(g-«;— /ij— 0) [53866] ;
which gives,
[5386] (/5t7 = — V'rft).(ap— la.?),— .7.sin>.COS.X.cos.(o-i;— /«— 0) [5386f/] ;
and, by integration,
(ap — ^as) D^ . . . r .\
[5387] ÔV = — ^. — — ^-^.-^.7.sm.x.cos.x.sm.(^i)— /i) — ").
'nequality g 1 W
This is the only sensible ineqicality in the moon^s motion in longitude, arising
from the oblateness of the earth. It may be observed, that fv — gv^é*
expresses the longitude of the ascending node of the orbit, counted from the
moveable vernal equinox ; hence it follows, that the expression of the true
longitude, in terms of the mean longitude, contains the following inequality ;
[5389] (5v=]|.-^^^^^—,—-,7,sin.x.cos.x.sin. (longitude of the ascending node).
in Ion
gituJs
depending
on tlie
oblateness
of the
earth.
[5388]
[5390]
The coefficient of this inequality isf 5^552, if p = ^i^ ; it becomes
11 ',499, if P = ^iô-
* (2996) It is evident, from [4813,4817], that gv—ê represents nearly the moon's
[5388rt] distance from the ascending node on the fixed ecliptic, counted according to the order of the
signs ; and fv [5-34.5], the moon's distance from the moveable equinox, counted in the
same order. Subtracting the first of these expressions from the second, we obtain
[53886] fv — gv-^é, which must evidently represent the distance of the node from the equinox,
or its longitude. Hence,
[5388c] —s\n.{gv—fv—ê) = s\n.(fr—gv+ê) =: sin. (longitude of the ascending node).
Substituting this in [5387], we get [5389].
t (2997) Substituting A [5357o], in [5389], it becomes,
[5390a] Sv=^'^.Ay. sin. (longitude of the ascending node).
The values of A, corresponding to the ellipticities ^ij, 2-^, have already been
computed in [5.357,5358], and found to be 6%487, 13%436, respectively. Multiplying
[53906] ^j^ggg ^,y ^.y= 0,855767 [51 17 line 5], we get the values [5390]. If we put the
coefficient of [5389] equal to A', we shall have, by comparing it with [5357a],
[5390c] A'='i.AY, or, A=—.A';
VII.ii.§20.] EFFECT OF THE OBLATENESS OF THE EARTH. 599
The ohlateness of the earth affects also the motions of the perigee and nodes
[5390'!
of the lunar orbit. For, the value of Q is, by this means, increased by
the quantity,*
Q=(ap-ioLç).(l-fs^).{i-(l-s'*).sin.^.sin.>-2s.sinx.cosx.sin/tJ-s^cos2x|.Z)V. [5391]
This produces, in the equation [4754], the following term ;t
and, by substituting for u, its approximate value,
u = - .{l + e .cos.(cv — -n)] [4826], [5393]
and observing, that h^ is very nearly equal to a [4859], we obtain, in
the differential equation [4961, or 5392a], the terms.
the equa-
tion
2.(ap_W) ^^ (l_3.sin.^X). e. C0S.(C.-.). ^53^95]
a a^
substituting this in [5357e, c], and reducing, we get the following equations, which may be
used hereafter ;
A' = 4392%C.ap— 7^6 ; [5390rf]
A'+V,6
ap =
4392%6 [5390e]
* (2998) If we change the signs of the two factors of Q [53466], which does not
alter its value ; and then vary the place of its last term, we get,
Q=(a.p — .Jtt;p).Z)"-.M^.{|(l — s^)~ — (1 — «^)-.sin.^X.sin.yt)-2«.sinX.cosX.sin./r-i^.cos2x|. [5391o]
Dividing the last factor by (1 — s^)^, and then multiplying by the equivalent expression
1 — ^s-, neglecting terms of the order s^, we get [5391]. If we neglect also the terms
depending on s, and substitute sin.-yu = i — i.cos.2fv, it becomes,
^ = (ap— ia?).D2.M3. {1— i.sin.2x+|.sin.2x.cos.2/j;| ; [53916]
which is used in the next note.
t (2999) Upon the same principles, by which we have obtained the equation [4755]
under the form [5347/], we may reduce [4754], to the following form,
600 THEORY OF THE MOON ; [Méc. Cél.
Hence we easily find, that the motion of the perigee is increased by the
following quantity nearly ;*
1^396] 5n = (ap_la?).^. v. {1 — | . sin.==X| .
Increment
of the
motions
of tlie
perigeR
and nude.
It is evident, from the equation [4755J, that the retrograde motion of the
node, ivill he increased by the same quantity. If we reduce it to numbers,
[5397] we obtain,t 0,00000026384.z) ; which is insensible.
[5392a] 0 = ^ + K--^.f^^
rfll2 " h'i \dh
[5392i]
This contains the most important part of the terms now under consideration depending on
Q ; the neglected quantities being of a different form and order from those which are
retained in [5394,5395]. Now, the expression of Q [53916], gives in [5392a], tlie
terms,
■ _ ^ . (^1?^ = _ i^lZ^^ . DhL\ \ l-|.sin.2x+i.sin.5X.cos.2> } .
If we neglect the part depending on the angle 2/î) , it becomes as in [5392]. If we
use the values [4937n], and put, for brevity,
[5392c] B = (a?— iciffi) . ~ . (1— f.sin.^X) ,
we get,
_ 1 ('B\ _ _ R 2
[5392d] Ifi' \du) ~
Substituting rt^ = — . (l-)-2e.cos.Ci;) [5393], and neglecting c^ , it becomes,
[5392e] 2 JS . - . cos.ci; , as in [5394, 5395] :
hence the equation [5392rt], is reduced to the following form,
[5392/] 0 = ^ + M — - — 2 B . - . COS.CÎ).
* (3000) Neglecting terms of the order e^ , e'^, we find that the coefficient of
[5396a] - . cos.cy , in the equation [4961], is represented by — -p [4975]; and, it is evident,
[53966] that the terms depending on B, in [5392/, or 49G1], augment the value of p by tlie
quantity &p = 2B . Now the motion of the perigee is represented, in [49846], by
[539bc] ^Y — y/yz:^).j;, which is very nearly equal to ipv ; so, that if p be augmented by
[5396rf] f5p , the motion of the perigee will be increased by ^5p.v ■=^ Bv , as in [5396,5392f].
t (3001) If we neglect terms of the order e'°, e^, &c., and also, for brevity, the
VII. ii. § 20.] EFFECT OF THE OBLATENESS OF THE EARTH. 601
We shall now make an interesting remark, upon the preceding inequality of
..... . L "^^y D 1
the moon's motion in latitude. This inequality is nothing more than the reaction "^^'J'^^y^
of the nidation of the earth^s axis, discovered by Bradley. To prove this, we 'reaction
shall put 7 for the inclination of the lunar orbit to the plane we have spoken [5398]
, .... nutation
of in [5352], which passes always through the equinoxes, and is inclined °[^'-^^_
to the ecliptic by an angle [5353], equal to — — - — - . -^ . sin. x . cos.>^. [5399]
The inclination of the lunar orbit to the ecliptic, will be,
symbol ê, we shall find, that the retrograde motion of the nodes is,
1 \/iW— 1 i • " = hp"- » , nearly [5059] ; [5397a]
observing, that p"y-sin.^u [5053], is the term of [5049, or 4755], depending on sin.^y.
The inspection of the value of Q [5391a], shows, that the quantity (y^) produces [5.39741
nothing of importance in [4755]. If we neglect s^ , and put h^=a, u=a~^, in
the other terms of [4755 line 2], we find, that this equation becomes,
Multiplying the equation [5392f/], by h-s , and substituting the preceding values of
h^, u, we get — s.(—j=^ — Bs. Again, if we take the partial differential of Q, [5397(i]
[5391a], relative to s, and multiply it by — a, putting M = a~", we shall "et [5397e]
[5397^]. Neglecting s^, putting sin.^i) = i — J.cos.2/y , and omitting the terms
depending on fv, 2/y, we get [5397 A]. Substituting cos.^X^ I — sin.^X, and [5397/]
reducing successively, using B [5392c], it becomes as in [5397 ij ;
= (ap— U?) .— .5.^I-J.sin.2x+2.cos.2x^= (ap— ^aç;) .-.«.? 3— f.sin.sx' [5397ft]
= 3Bs. [5397t]
Substituting the values [5397</, i], in [5397c], we get,
rf(/« lids „
0=-T^+«+2S«, or 0=-~^s-{-2B.v.sm.gv, nearly [5383] ; [5397ft]
hence the value of p" [5053], is increased by the quantity 2B , nearly ; consequently
the motion of the node ip"v [5397a] is augmented by the quantity Bv , being the
same as that of the perigee, [5396d], as in [5397].
Substituting, in [5396], the values [5354 — 5356], we get,
VOL. III. 151
[.5397i]
602
[5400] * (q-p-U^) D^
.s:— I
[5401]
THEORY OF THE MOON; [Méc. Cél.
— . sin.X . COS.X . COS.ÇgV — -fv — li) = inclination of orbit to the ecliptic.
Now, the area described by the moon about the earth's centre of gravity, is
\r^'.dv [372«]. This area, projected upon the ecliptic, is decreased in the
[5401] ratio of the cosine of the inclination of the moon's orbit [5400] to the radius ;
therefore, it is represented by,
[5402] ^r-.dy.cos. < } — '. -^.sm.X.COS.X.COS.(g?;— /i; — è) \ = projec. of the area èrS.rfti.
Hence, the expression of this area contains the inequality,!
[5397m] ^^ ^ ^^ ^ 0,00000026384.1', as in [5397] ;
[5397n] and, by putting ti = SOC, it becomes (5ra = 0',3, corresponding to one revolution of
the moon. This part of the motion of the perigee is insensible, in comparison with its
whole motion 0,00845199. « [5117 line 2] ; being only ^t^utt part of it.
[5397o]
[5400a]
* (3002) In the annexed figure, let
AR be the equator, AJVB the fixed plane,
AEl) the ecliptic, JVEM the moon's orbit ;
then, if we make arc A'iVi=arciVB^90'',
[5400&] and describe about N, as a pole, the
2iïc MDB, we shall have mcJ\IB=^y
[5398'], angle BAB=A [5357«].
Moreover, we have, very, nearly, in the -^^ Enuator _k
[5400c] triangle BAB, ^vcBB = A.ûn.AB= A .sm.{A]<l-\-QQ'^) = A.cos.AN ; and, as
^iV is nearly equal to AE=z fv—gv-\-d [5388], we have BB =A.cos.{fv—gv-\-è);
hence,
[5400rf] MD=^MB—BB = y—A.cos.{fv—gv-\-è) = y—A.cos.{gv—fv—è).
Now, from the extreme smallness of the arcs DB, EK, it is evident, that the arc
MB represents very neariy the value of the angle MEB, or the inclination of the moon's
[5400e] orbit to the ecliptic. This agrees with [5400]. We may moreover remark, that the angle
p-u /■« — â, or fo — gv-\-^, corresponding to the distance of the node from the equinox
varies only about 3'', in a periodical revolution of the moon ; consequently, the angle of
[5400/] inclination [5400] alters but little, during that revolution; and the factor of ^r'^.dv, in the
inequality [5403], is neariy constant in the whole of that period.
t (3003) Putting, for brevity, A'= A.cos.{gv — fv—è) [5357«], in the
"• "■' expression of the projection of the area [5402], and then developing, as in [61] Int.,
VII.ii.§90.] EFFECT OF THE OBLATENESS OF THE EARTH. 603
[5403]
[5404]
, o I (o-p— Jaa) D^ . / /• .\
ir. dv. — - . — 5-.7.Sin.X.COS.>,.COS.(g"l' — -JV — (') = a term of the projection of ^r^.dv ;
Ô
and, as Ave have, very nearly,* i".dv = ar.dt, dt denoting the moon' s mean
motion, this inequality will be represented by,
IjD". dt X-^' — ^.7.sin.X.COS.X.COS.(^T — -fv — (') = a term of the projection of ir2.rfu. [5405]
Multiplying this expression by the moon's mass, which we shall represent
by L ; then, dividing the product by \dt, Ave obtain the momentum of
the moon's force about the centre of gravity of the earth, arising from the
oblateness of the earth.f Hence we get, for this momentum, the following
expression ;
[5406]
[5406']
L.D^.— — '~ — ^•^.sin.x.cos.x.cos.f^'i; — fv — 0 = momentum of the moon, (i) [5407i
<r 1 \0 J / \ y Momen-
S
In consequence of the equality hetiveen the action and reaction, the same cause ■■;/ «""he
turn ofthe
moon cor-
respond-
ing toth
oblatenesa
ofthe
earth.
neglecting the second and higher powers and products of A', it becomes,
^r^.dv.coa.{y — A') = ^'fi.dv.\cos.y-\-A'.s\u.y\ = ^r'^.dv.cos.y-\-^r^.dv.Ay, nearly. [54035]
Re-substituting the value of A!, in the last part of this expression, we obtain the term
[5403].
* (3004) We have r^. dv = a^.ndt .\/l — e^ [1057]; and, bj' neglecting e^
changing also the mean motion ndt into dt, so as to correspond to the notation in [5404], 1^^040]
it becomes r-.dv = a^.dt, as in [5404] ; substituting this in [5403], we get [5405]. In
this process, we neglect the consideration of the perturbations of the moon's motion by the [54046]
sun's action, using the elliptical value of r^.dv [5404a]; observing, that the rejected
terms are of a different form or order, from that in [5405].
[5406a]
f (3005) The arc which the moon describes in her orbit, in the time dt, being
resolved in a direction perpendicular to the radius r, is evidently represented by i\dv ;
consequently, the velocity, in that direction, is ^--j-', and the force is proportional to it.
Multiplying this by the radius r, and by the mass L, we get the corresponding momentum
of the moon [29'], [54066]
r^.— .L, or .L, as m [5406'].
dt 2 dt
Substituting, in this last expression, for ^r^.dv, the term given in [5405], we obtain the
corresponding part of the moon's momentum, as in [5407].
[5406c]
604 THEORY OF THE MOON ; [Méc. Cél
must produce, in the particles of the earth, a momentum which is equal and
contrary to the preceding. This momentum is indicated by the nutation of
the earth's axis, and we may determine its value by means of the formulas
[5408] ij^ book V. ^ 6. For, we see, in [3101], that if we put F for the obliquity
of the ecliptic to the equator, the moon's action upon the earth produces, in
consequence of the oblate form of the earth, an increment in the angle V,
which is represented by,*
[5409] r-. r .7 .COS. (gv—fv — â) = increment of the obliquity V ;
I and >^ being the same as in that article. The element of the rotatory
[a409] jj^o|.Jq„ of the earth being supposed ndt [3015] ; the sum of the momenta
of the forces acting upon each particle of the earth, multiplied by the mass
of the particle, is equal to nC ; C being the momentum of inertia
of the earth, relative to its axis oi rotation.! io reduce this momentum to
[54096]
* (3006) Of the five terms which compose the vaUie of ê [.3101], and of ê'
[3360 or 3378], or that of V, in the notation [5408], the first is constant ; the second
" is secular ; the /bwrtA and _^/i:/i are small, and depend on the places of the sun and moon.
The third is that upon which the nutation depends ; namely,
^\''.cos.if't + pJ)
(1+x)./
c' [3086] being nearly the same as 7 [5398'] ; and,
[5409c] —f't—p' =fv —gv + é [3086', 5388],
representing the longitude of the moon's ascending node, counted from the moveable vernal
equinox. Substituting these values in [54096], it becomes,
[5409(i] 7T-,-T->,-cos.(^«— /r — a).
Now, the mean increment of v, in the time t, being represented by t [5404], it
[5409e] will follow, from the equation [5409c], that —f=f—g = 'i—g, nearly [53475r], or
f' = g — 1 ; substituting this in [5409f/], we get the increment of the inclination V
[5409/] [5409]. We may remark, that this use of the symbol V is restricted to § 20 [5408] to
[5422] ; in other parts of this chapter, V denotes the function [53-36, he.].
t (3007) The angular velocity of a particle of the earth about its axis of revolution
being n [5409'], its actual velocity, at any distance r, from the axis, is nr^.
Multiplying this by the same radius r, , and by the mass of the particle dm, we get
Vll.ii. .^20.] EFFECT OF THE OBLATENESS OF THE EARTH. 605
[5411]
the ecliptic, we must multiply it by the cosine of its obliquity, or by,*
we shall, therefore, have the following inequality, in the momentum of the
earth [541 U] ;
l-K.nC.s'm.V
7 . COS. fgV JV ÔJ =^ inequality in the earth's momentum. [5412]
We have, in [3098],
[5413]
m t denoting the mean motion of the earth [3059] ; alsof Knf = -^ ; [5414]
a being the moon's mean distance from the earth ; and, since we represent
the moon's mean motion by t [5404], and the mass of the earth by M
[4757] ; we have, very nearly, J — = 1 , which gives \.m^ = — ; [5416]
(I JYJ.
[5415]
[54105]
the momentum of this particle, equal to n.r^.dm [29]. Integrating this, relative to the
whole mass of the earth, it becomes n .fr/'.Jm; in which r/^ is represented by
z"^-\-y"^, of the formula [229], the axis of rotation being z"; consequently, this expression
becomes,
n.fr/^.(!m = n.f{x"^-^y"^) .(lm=^n.C [229], as in [5410]. [5410c]
[5411a]
* (300S) Putting the function [.5409] equal to 5V, the whole obliquity will become
V-\-ûV. Its cosine, by [61] Int. is represented by cos.V — iiV.sm.V , nearly. Multiplying
this, by the momentum nC , it produces the term, —nC.sin.F.tS?^; and, by substituting [54ni]
the value of oV [5409], it becomes as in [5412].
t (3009) This is easily deduced from X . w^ = — [.3079], changing L' into
L, and a' into a, to conform to the alterations in the notation, which is used in [5414a]
[3073,5406,5414]. We may also observe, that in deducing the value of / [5413],
from [3098], we must change h into V, to conform to [3357, 5408].
J (3010) The mean increment of v , in the time t , is very nearly represented
by nt [5095] ; consequently that of dv is mit ; and as this is put equal to dt,
[5415fl ]
in [5404], we shall have n= 1 . Substituting this and [i^M+m [4775"], in [3700],
VOL. III. 152
606 THEORY OF THE MOON ; [Méc.Cél.
thus, the preceding inequality becomes,*
[5417-1 . \Z L .y.s'm.V.COS.V.COS. (gV—fv â) = inequalitymthe earth's momentum.
4M g — 1
We have, from [2960— 2962], t
[5418] 2 C—A—B^^. ^ . (a p— i aç) . D\f 3 u . R\ d R ;
[5419] p being the oblateness of the earth; D its semi-diameter; R the radius
[5420]
[5421]
of one of its particles, whose density is n ; and * the semi-circumference,
whose radius is unity. The mass of the earth is Î M — i-^.f 3n.R-.dR ;
[5415t] we get — ^ — = I ; which, by neglecting the mass of the moon m, in comparison
M
[5415c] with that of the earth M , becomes — =1, as in [5416]. This gives a^ =^ M ,
and, by substituting it in [5414], we obtain the expression of X.m^ [5416].
* (3011) From [5416] we get "*^ = TJ* 5 hence [5413] becomes,
3i (2C-A-B) (1+X)
substituting this in [5412], we get [5417].
t (3012) Subtracting the sum of the values of A , B [2960,2961], from 2C
[2962] , we get,
[5418a] 2C-A—B^^^ . o.^.{h-l^). f^^ p .d.a^ = -.* .{^h—l^^).f^'?.3a''.da ;
in which ip [2951], is the same as in [5333'], and A = p [5335'*]. Moreover, we
[5418i] must change «, p [2947], into R, n [5419,5420], to conform to the present
notation ; hence the last expression [5418a] becomes,
[5418,] ÇIC-A—B = '^ . * . (ap-èap) . /„i m . R^ . clR.
The two members of this equation are not homogeneous ; for in the first member, ji, B,
C [2920 -2922], are of the j^/cA order in R, and the second member is only of the
[5418rf] ffiiy(] order ; we must, therefore, multiply the second member, by the square of the mean
radius of the earth D [5334], which is taken for unity in [2947'] ; and then it becomes
[.5418e] as in [5418]. In the original work, the factor 3, under the sign /, is accidentally
omitted.
X (3013) This is similar to the expression [1506a], changing the notation, as in
Vll.ii.<^2l.] EFFECT OF THE SPHEROIDAL FIGURE OF THE MOON. 607
which is to be substituted in [5418] ; and then the resulting value in [5417],
changing also the obliquity of the ecliptic V [5408], into x [5841]; [5422]
hence the inequality [5417] becomes,
L. /)-. ^^ -. 7. sin. X. COS. X. cos. (^« — -fo C) = inequality in the earth's iiioiucntuni. [5423]
[5424]
The ine-
This expression is the same as that in [5407], ivith a contrary sign. Hence
it follows, that the preceding ineqiiality of the moon's motion in latitude, is the
reaction of the nutation of the earth's axis; and, that there would be an quality
' -^ ' in the
equilibrium about the centre of gravity of the earth, by means of the forces which T^°^L\>
Ijroduce these two inequalities, supposing all the particles of the earth and moon to action
be firmly connected loith each other ; since the moon compensates for the Titil""
eartli's
smallness of the forces which act on it, by the length of the lever to which ''''''■
it is attached.
21. To notice the effect of the moon's figure, which is not exactly
spherical, we shall observe, that it introduces into Q [4756], the term,
5V' SV
(M + ni) . [4773], or more simply, ; f^^^^l
because, we have put M+m = 1 [4775"]. Now, from [1505, 1809', 4770'],
we obtain,*
[5426]
[54l8i]. Substituting the value of AI in [5418], we get,
2 C—A—B == i . (ap— >ç) . D2 M . [5421a]
and, by using this expression, and tliat of V = X [5422] ; we may reduce the inequality
[5417], to the form [5423].
* (3014) We may neglect the terms of V [1505], which are divided by r* , on
accountof their smallness ; also those depending on y<°\ F''', as is done in [1809', 1811],
and then it becomes, by accenting the letter V, so as to conform to the notation [4769],
F' = |./„V.d.«3+'^./„'p.rf.(«=r^=')= 7+'^:-/a'P-'^.(«^î^'^0 [5429]. [5425»]
Substituting this in [4770'], we get àV [5426] ; the limits of the integral being changed
from 0, 1, to 0, a. Multiplying the expression of 5V' [5426], by M-f-m^l [54256]
[4775"] ; and then dividing by m [5429], we get [5430].
608 THEORY OF THE MOON ; [Méc. CéJ.
[5427] the integral being taken from a = 0 , to a , equal to the moon'' s semi-
[5428] diameter, which ive shall denote by a , and ? being the density of the
[5429] stratum of the moon corresponding to a. We have m = -| -^ ./;' p. d.a^
[1506«] ; hence we deduce,
^ ' m 5?-'. /j,'' I' . ri . a'*
To determine f" p.d.(a^.Y'--^) , we shall observe that we have, in [1761],
for Y <-' , an expression of the following form,*
Y(2) = h'. (X — (x2)_j_/t".,a,^lZ:;i;i.sin.a+/t"'. f^.V/ï^^-cos.«
_1_ II"", (1 _,a,a) . sin. 2^+/i'. (1— ftj.) . cos.2^ .
Then, the properties of the axes of rotation [1753 — 1757], give,t
5432] 0=/;p.fL(«^/O; 0=f;?.d.{a'h"'); O ^f^ e.d.(a' h^) :
and then, from [2948—2950], we obtain,!
[5431]
* (3015) The expression of F-> [5431], is the same as in [17G1], increasing the
accents on h , by unity.
1(3016) Substituting the expression of F'^' [5431], in [1757], we get,
^^^^^"^ 4-a.(l-,j.2).sin.2a./; f.d.{a^h"")+^u{l- f-^~).cos.2^.f,'p.d.{aVi^).
Comparing this, with the value of U'"^ [1753], we get,
H=- a. r," p.rf.(«^A') ; H' = a./„« p.d.{a^h") ; H" = a, /„« p.d.{a%"') ;
r5432H
Jf"'=cL./„»p..Z.(«W"); if""=o../oV.f/.(«=AO-
Now, the properties of the principal axes give, in [1754], 12'= 0, H" = 0 , Ji"' = 0;
[•''432c] gjji^gtit^ting these in [54326], and dividing by a , we get, from the second, third and
fourth equations, the values [5432].
t (3017) Substituting the values of A, B, C [2948— 2950], in 2C—A—B,
we get the expression [54336], by putting cos.^ra + sin.^ î?= 1. This is easily
reduced to the form [543.3c], by introducing the value of F^' [5431], and neglecting the
[5433a] terms depending on h", h'", h"", on account of the integrals [54.32]. We may also
neglect the term depending on cos.2zï; because, at the limits of the integral -3=0
zi = 2*, it has the same value ; and the integral taken between these limits vanishes.
Hence we have,
Vll.ii.§21.] EFFECT OF THE SPHEROIDAL FIGURE OF THE MOON.
609
B-A = II
Thus, Ave have,*
[5433]
[5433']
= 3<x..f.p.d.{a^h').{^ -f;.2)34a.(Zw+3a./.p.(/.(a5/i^).(^— p.2).(l— a2).cos.2«4"'.ffe
=:3a./.p.f/.(a5 h' ).(^—,jP)^.diJ..d7z.
Now we have, by the usual rules of integration,
f~"'du = 2^; f_^{i—i^?y.dix = ^\ [2933i,/,or3569e];
substituting these in [5433(Z], we get [5433]. In like manner, if we substitute the values
of A, B [2948, 2949], in B — A, we get the first expression [5433^]. Substituting
in this, the value of Y"*-' [5431], and neglecting, as above, h", h'", h"", we get
[5433A] ; reducing also, by means of cos.^sf — sin.^ïï = cos.2i3; cos.^2ïï^ |-j-|.cos.4ra;
and neglecting, as in [5433a], the terras depending on cos.2ro, cos.4'W, we obtain
[5433i] ;
B— ^=a/p.£7.(o5F2)).r/(A.rfro.(l— H.2).(cos.2ra_sin.2^s)
= a/p.cZ. (o5 F2>) .(Z,x.<Zïî. ( 1 — ,;i2) .COS.2Z3
= a/p.rf.(a5A').rf,a.fZOT.(i— M.2).(l_^2).cos.2w+a/p.f7.(o5/i^).(l-,x2)2.cos.'2w.(/fji.rfw
= ia/p.rf.(a5A').(l— (j..2)2.rffii.(/w.
Substituting the integrals
y^~"rf^==2*, y!iV-f--)'-^M-- If [lî54e,/],
in this last expression, it becomes as in [5433'].
* (301S) Substituting the value of F® [.5431], in fp.d.{a^ F^»), and neglecting
the terms depending on A", h"', h"", on account of the equations [5432], we get
[5434a]. The integrals of this expression are easily obtained from [5433, 5433'], and, by
substitution, we get [5434è] ;
[54336]
[5433c]
[5433rf]
[5433e]
[5433/1
[5433g]
[5433A]
[5433i]
[5433/1]
2C—A—B B—A
= (i-O- ,e,.. +(1— ,x-).C0S.2«.-— - .
1 5 *
[5434a]
[54346]
Substituting this in [5430], and making a slight reduction, we get [5434]. Multiplying
this by the second member of [5435], and dividing by its first member C, we obtain
[5436].
VOL. III. 153
610
THEORY OF THE MOON;
[Méc. Cél.
[5434] (M+m)
[5435]
sv
m 16* r^.f^f.d.a?
We have, very nearly, in [2962],
.{(2C— ^— 5).(i-a=)+(jB-J).(I-;^.-).cos.2^]
c^\l.f:,j.a^,
therefore.
Terms of
[5436] ^M+m)._l-
m
_ri_
1 0
'f:?.d.a^' r^
[5436
terras of Q.
In this expression, th is the angle formed by the principal axis of the moon,
,,^ directed towards the earth, and the plane ivhich passes through the earthh
centre, and the axis of the moon's equator;* p- is the sine of the earth's
[5436a]
[54366]
[5436c]
[5436c']
[5436(/]
[5436e]
[5436/]
[5436^]
[54367i]
[5436i]
* (3019) Tlie notation which is here used, is similar to that for the earth [5333,5334'];
and corresponds also with [2910,3435, &ic.]. In defining the angle to, in the original
work, the words, line connecting the centres of the earth mid moon, are inadvertently used,
instead of the part printed in italics in [5436']. If we suppose the line connecting the
centres of the moon and earth to be projected upon the plane of the lunar equator, then
w will represent the angle formed by this projected line, or radius vector, and the moon's
longest axis, wliich is directed nearly towards the earth ; this axis being taken as the origin
of the angle zi ; hence we have, by supposing the angular and rotatory motion to commence
together, when -n = 0 ;
w:=: angular motion of this radius vector — moon's rotatory motion.
Now, in [3440,3433/], v represents the apparent motion of the earth in longitude, seen
from the moon ; and (p the rotatory motion of the moon ; so that, if we neglect the terms
arising from the reduction of v to the plane of the kinar equator, we may put v for the
angular motion of the radius vector, and p for the rotatory motion ; and by this means
[5436fZ] becomes,
w = V — <p.
The differential, relative to the characteristic d, affects only the moon's co-ordinates [5363'],
in its relative motion about tlie earth ; and, as (p depends on the rotatory motion, we shall
get, for the differential of the equation [5436o-], the expression dT^=dv ; therefore,
d.cos.2w=— 2cZ«.sin.23i, as in [5437,5437'].
If we substitute the expression of v—(p [5436ir], in [3447c], we get,
VII.ii.§21.] EFFECT OF THE SPHEROIDAL FIGURE OF THE MOON. 611
declination, seen from the moon, and referred to the moon's equator
[2909, 3435,&c.]. It is evident, that, bj increasing v by dv, « increases [5437]
by dv; therefore, we have d.cos.2OT = — 2(/t).sin.2t3 [5436f] ; the [5437]
differential symbol d referring only to the co-ordinates of the moon ;
moreover, we have, as in [5360],
i2__Q+i. [5438]
r
The part of dR, relative to the spheroidal form of the moon, produces the
i- II • • 1 .• .u r * [5438']
followmg expression, neglectmg the square of it- ;*
Hence we get, in to, or in the moon's true longitude, the following term of
the formula [931] ;t
a = — M + ii.sin.n -f Sic. ; [5436A]
!/ being the moon's libration in longitude [3464"] ; so that any inequahty which occurs in u,
may occur also in -n, but with a different sign, as in [5441, &c.]. If we substitute, in [5436i]
[5436A], the value of u [3456], we get,
^=-q. sin. J m i . * /'è . (^-^-) +Fl—kc.; [5436m]
and, if we change Q into K, it produces the term mentioned in [5441]. [5436n]
* (3020) The part of Q mentioned in [5425], and developed in [5436], produces
in R [5438] the following quantity ;
^ f^Yd.a^ 1 C (2C-A-B) ..AB-A) ^
-^ = — -^^y/^^- -3 • j c • (i— f^ )H Jj (^—n • C0S,2îJ J . [5439a]
If we neglect the square of (x, as in [5438'J ; then take its differential relative to d,
using the expression [5437'], we get [5439].
t (3021) We have, in Sv [931], the term -.ff'-^^^; and, if we neglect
e^, putting a = r, nearly; also (j. = 1, as in [4775"] ; it becomes 3r .ffndt.dR. [5Ai0n]
Now, nclt is nearly equal to dv [5095]; therefore, Sv contains the term 3r.ffdv.dR;
and, by substituting the value of dR [5439], it becomes as in [5440].
612 THEORY OF THE MOON ; [Méc. Cél.
The angle a is always very small [3468, 5436'] ; so that we may suppose
sin.2«=:23i. Moreover, from [3456], we find, that îj contains a term of the
[5441] form — ^.sin. \ ^'-X^^'^^^'"^ + ^\ [5436?«, «]. This term, taken
with a contrary sign, represents, in [3456, 5436Ht], the real libration of the
moon. As it increases very slowly, it would seem, that it ought to become
[5441'] sensible by double integration : this is the only term of the expression of
S which it is necessary to notice. It produces, in 6v, the term,*
[5442] 5 ^ _ ^ fjLliÉll' E sin \ V \ /sS-^r^ + F
in the
moon's
longitude
aris
from
spheroidal
form of
the moon.
he The libration K . sin. \v.l / ^S^-^ + F \ being insensible,
suppose, that it amounts to a centesimal degree. Moreover, the coefficient
[5443] y . -. ° ' ^ is extremely small. If the moon be homogeneous, it becomes f
a^ a
[5444] 4 ■ V ^ now, - is the sine of the moon's apparent semi-diameter ; hence,
* (3022) Substituting 2^ for sin.2t3, in the integral expression of ffdv^.sm.2vi,
which occurs in [5440], and then the term of « [5441], we obtain, by successive
" integrations, the expression [5442c], retaining only the most iinportant term, having the
divisor B — A, arising from the double integration ;
[54426] ffdv^.sm.2^= 2ffdvK^= — 2K.ffdv'^.sm. \ v. 1 /HB—A) _^ p)
Substituting this in [5440], we get [5442].
t (3023) The moon being supposed homogeneous, and p^l, we have,
[5443a] f^%.d.a^ = .^; f;?.d.a^ = .^; hence, j^, = -'-
Substituting this, in [5443], we get,
1544361 ^ . - /^^- =3 4 . ^ = I . sin.3(moon's semi-diameter) = |.(0,0045)« =0,000024 ;
and, if we suppose J5r=lo=54"' = 3240% we shall get 0,000024.^=0%07, for the
[5443c] > rr
coefficient of the correction [5442] ; which is msensible.
VIl.ii.§21.] EFFECT OF THE SPHEROIDAL FIGURE OF THE MOON. 613
the product of K, by this coefficient, is wholly insensible. If the moon be
not homogeneous, its density must increase from the surface to the centre;
then, this coefficient is yet less.* Hence it follows, that the preceding [5444]
inerjualiti/ of the moon'' s longitude is insensible; and, that the variation fj-om
a spherical form does not produce any sensible inequality in the motion in t^'*'*^]
lon'^itude.
As to the latitude, we must observe, that (j. is the sine of the earth's
declination, seen from the moon [5437], and referred to the lunar equator;
moreover, the ascending node of the moon's orbit always coincides with the
descending node of its equator [3433] ; therefore, we shall have,t
[5445-]
S+X.sin.(^l,' — Oi' ; [5446]
[5446']
X being here the inclination of the lunar equator to the ecliptic. Hence we
get,t
* (3024) Changing R into a, in [277'] and miiltiplj'ing by ^, we get
being less than its value a^, corresponding to p==l [5443a].
t (3025) It is found by observation, that the descending node of the lunar equator always
coincides with the ascending node of the lunar orbit [3433] • and the inclination of the lunar [5446a]
orbit to the ecliptic is nearly equal to -/ [.5400], also the inclination of the equator to the
ecliptic is X [5446'] ; therefore, the inclination of the lunar orbit to the lunar equator, is
nearly equal to v+X . Now from [5383], we find, that the moon's latitude, or the angular [544("iR]
elevation of the moon above the ecliptic, is nearly represented by s = y.s'm.f^v ê) •
hence the coiTesponding angular depression of the earth, as seen from the moon, is
—'y.sm.(gv — Ô) ; and it is evident, that by changing the inclination 7 into y+X [54404],
we get the angular depression of the earth below the lunar equator — (y+X).sin.(i?-i) 6) .
This may be put equal to its sine /a , and by using the value of s [544Gc], we "-et
l^ = — (v+>^).sïn.(gv—6)^ — s — \.sm.(gv—ê)z=— \s-{-\.sm.{gv—ô) I; [5446e]
vsrhose square is the same as [5146].
X (3026) The partial differential of fj.a [5446], relative to s, being divided by 2ds, gives
the first of the expressions [5447] ; substituting in this, the value sm.(gv—ê)=- [5446cl r...^ .
/ [.5447a]
we get the second form of that equation.
VOL. III. 154
[544Ce]
[5446^^]
614 THEORY OF THE MOON ; [Méc. Cél.
/c'a , • ^ N (>^+7)
[5447] '^ • i rf^ ] "" * + ^ • ^'"-(e"*^ — ") "= -^ • * Î
therefore, the spheroidal form of the moon, adds to the expression of
[5447'] — -i- . ( -^ ) , in the equation [4755], the term*
[5448] ./A4i;4.(-_±Z)...52_t^+^.eos.2.|.
= f^p.d.a' r^ y I C ^ C s
[5448] Now, as we have very nearly, cos.2i3 = 1, it adds to [4755], the quantity,t
[5449] A , fll^-^^ . i ^^±^ . i^ . S ^ term of [4755].
It is evident, from [5397 A;, /], that this term adds to the motion of the node,
the quantity, Î
* (3027) Substituting, in — — . \-^\ [5447'], the tenus of ^, given in [5436],
we get,
[5448a] ' /dQ\ s .'sil . u,. — ] . ( . cos. 2 m > :
[544861 substituting the last of the expressions [5447], we get [5448]; observing that A^ and
M-', are nearly equal to a , or r [4937?i,&ic.].
t (3028) Since « is very small, we have nearly cos.23J=1; hence we get,
9C—A—B B—A 2C-A—B , B—A ^ C—A
[5449a] 2L_£_Ji _^ -D_^ _ ^^^^^ = -^^ \- —^ = 2 . -^ ;
substituting this in [5448], we get [5449J.
X (3029) Substituting, in [5449], the value of s [5446c], it produces, in the equation
[4755] or in its development [5347/], the quantity,
r/p.^/.aS 1 (X+y) (C—A) . .
[5450a] A . ■'^y—. -, . ^^^^ • —-JT-^ • 7 • sm.{gv—à) .
[54506] This is similar to the term which is computed in [5397A:] ; and, by making the calculation as
in [5397fc,Z], we find, that the preceding term [5450«), produces in p", the term,
f.y.d.a^ 1 M-y C—A _
[5450c] ^P" "" ' ' f^Y^Ta' ' '^^ ' y ■ C '
„ and the corresponding motion of the node, computed as in [5397^,/], is W'-^ as in
[5450a 1
[5450].
Vll.ii.'^Si.] EFFECT OF THE SPHEROIDAL FIGURE OF THE MOON. 616
MoLioa of
tlie nodoa-
/■'p.d.«' t> C^-hy) (<^ '^) c ,. risingfiom
1 . 1^— -3 • -o • -^-^ • —-r = ^^^"^ °^ ^^ • t^^5°l
^ L^p . a. a"* r- 7 C iho sphe
-'^ "^ roidal fig-
ue u of tbo
/^ j^ niooD.
In [3545] we have * — x— = 0,000599 ; hence it is evident, that the [5451]
preceding quantity is insensible.
We find, likewise, that the spheroidal form of the moon adds to the term
— i
( j^) of the equation [4755], the term, 1 ^ [5452]
_ 4 . ''-Î-!— — - . - . ^^ -— !— ^ . s = term of [47551.
[5453]
* (3030) We have ^ = 0,000599 [3545] ; hence it follows, that C is [5451a]
nearly equal to .^; and we may, therefore, change Jî into C, in the denominator ;
by this means we shall get ^""^ = 0,000599 [5451]. Moreover, X = 1''29"' [5446']
[54516]
[3434] ; 7 1=5'' 8"" 50' [5117] ; and if we suppose the moon to be homogeneous, we
6 1 r^p d a^
shall have - . ^ . ° ' 3 = 0,000024 [54436]. Substituting these in [5450], it i[545it]
becomes, 0,00000001. d nearly. Now, in one lunar month, j; :^ 1296000'; substituting
it, we get 0-,01 , for the motion of the node in a lunar month, arising from this cause. 1 l
This is wholly insensible.
t (3031) Substituting in the term of Q [5436] the value of r=- nearly [4776] [545301
we get [54526]. Neglecting (a, on account of its smallness, and putting cos.2«=l
[5448'], we get [5452c],
j^'-^.d.a^ , C 2C—A—B , „, (B—A) , >
^ = " • %:d-:è ■''■{ —c— ■ (^ - -') + S-- ■ ( '--^)--2. j
[54526]
. u" . ^ 77-^ '-. [5452c]
//p.d.a^ 3 (C-2A+B)
This gives,
U; = " • l^Zd:a^ ■ " • r ■' [5452rf]
and by multiplying it by — — ; using also h^ =: a , u'^ = a = r nearly [4937n],
we get [5453].
616 THEORY OF THE MOON ; [Méc. Cél-
This adds lo the motion of the noae, the term,*
l^I - -.- , ei-^, . -.- . i^?#+^ = tern, of . . ;
1 0
a quantity which is wholly insensible.
..._. , * (3032) The expression [5454] may be derived from [5453], in the same manner as
[5450] is from [5449] ; namely, by changing « into |u . To estimate roughly the value
of the expression [5454], we may observe, that in the case of homogeneity, we have,
[54546] __=_; ^^=7T [3570].
Their sum is,
;,«„ 'C:^='^;=i. (_£=!> =0,001 [545,.l,neaH,;
hence it is evident, that the term [5454] is insensible, Hke the corresponding term [5450]
which is computed in [545l£/].
VII.iii.§22.] ACTION OF THE PLANETS. 617
CHAPTER III.
ON THE INEaUALITIES OF THE MOON, DEPENDING ON THE ACTION OF THE PLANETS.
22. It now remains to consider the action of the planets upon the moon.
We shall put,
P — the mass of a planet ; [5455]
X,Y,Z= the rectangular co-ordinates of the planet, referred to the centre of [5455']
the earth ;
/= the distance of the planet from the earth's centre. [5455"]
Then, it is evident, that the action of the planet P, îvill increase the value
of Q 14:1 5G], by the quantity * Terms of
_ _P.(.X+yY+.Z) P_ _^ . 3^'
or,t
[5456o]
* (30.33) The disturbing force of the planet P, upon the moon, in her relative
motion about the earth, is computed by the same differential formulas which are used for the
disturbing force of the sun. We must, in this case, change the mass m' of the sun
[4757"], into that of the planet P ; and the co-ordinates x, y', z' of the sun [4758'], [54566]
into those of the planet X, Y, Z [5455'] ; by which means, the distance r' of [5456c]
the sun from the earth [4759'], changes into / [5455"], which represents the distance of
the planet from the earth. Making these alterations in the two last terms of Q [4756], '■ ■'
we obtain the part of Q [5456], upon which the disturbing force of the planet P [5456e]
depends.
t (3034) The development of [4774] is given in [4775], and, if we multiply this by
VOL. III. 155
618 THEORY OF THE MOON ; [Mée. Cél.
[5457] Q = J j^+iP-^ ^/J ^ +^C.
Let
[5458] X', Y', Z', be the co-ordinates of the planet P, refered to the sun's centre ;
[5458'] y.'^ y'^ ~'^ ^i^g co-ordinates of the earth, referred to the sun's centre ;
then we shall have,
[5459] X=X'—x'; y^Y'— 7/'; Z := Z'—zf.
Hence, the function [5457] becomes,*
C5460] ,2 = ^ _ J^+ I p. (£'~+Y',+ Z'y. '-„'-. ^r + j,„.
[5461] ff^g shall take the ecliptic for the fixed plane., which makes 2'= 0, and, we
shall put,
Symbols. ■*■
[5462] R = the radius vector of the planet P, projected upon this plane ;
[5463] U = the angle formed by the projection of the radius, and by a fixed right
line, taken in the same plane ;
[5464] «S = the tangent of the heliocentric latitude of the planet P ;
[5465] r' = the radius vector of the earth ;
[5465'] v' =^ the angle formed by the earth's radius and the fixed line.
Then, we shall have,
[5457a] P ; changing also x', y', z , r' , into X, Y, Z, f, respectively, as in [5456è-c?],
we get,
^''"''^ vÂx^mT^^^R^^-^7+ ~p +" p ^
Substituting this in [5456] ; reducing and neglecting terms of the order Xf-^, or /~^ ;
r5457c] we get [5457] ; observing, that the terms depending on the first powerof {xX-\-yY-\-z Z),
mutually destroy each other.
* (3035) Substituting, in [5457], the values of X, Y, Z [5459], we get
f^*^°"l [5460].
VII. iii.'§,22.]
ACTION OF THE PLANETS.
619
* / =3. y'R^( I -^SS)+r'^—2Rr'. cos.(U—v'). [5466]
Hence, the part of Q, relative to the action of P upon the moon,
will be,t
* (3036) 111 the annexed figure, S
is the place of the sun ; E that of the
earth ; P the place of the planet ; and
P' its projection on the plane of the
ecliptic SMP'. Then, z'=0 gives
Z = Z' [5461, 5459] ; and the
rectangular co-ordinates of E, P,referred
to the sun, are SF = x' ; FE = y' ;
S31 = X' ; 31P'= Y' ; P'P^ Z' ;
and, by drawing EX parallel to SM,
v^ehzveEN^X; NP'=Y; P'P=Z; ^^
SE=r'; SF^R; EP=f;
angle FS£ = »'; angle F5P'= C7 ; tang. P5'F=S. Fron these symbols we easily
obtain,
X'^iJ-cos.^'; Y' = R.sm.U; Z' =RS;
.r' =r'.cos.i;'; - i/' = ?•'. sin.t;' ; 5;'= 0.
The values of the co-ordinates of the moon x, y, z, and of the radius r, referred
to the earth's centre, are given in [4776 — 4779]. Now, the distance EP=f, is
evidently equal to \/{X~-\-Y~-\-Z^) ; and. if we substitute the values [5459], we get,
by development,
/= ^{X^+Y^+Z^-) =^l{X'-xr+{Y'-^f+{Z'-zy\
= S/{{X'^+Y '^+Z'2)+(x'2+y2_)_^'a)_2(xV+ Y'y'^ Z V) \.
Substituting in this, the values of
X'2-|-F'2-]-Z'2= SP2=iJa.(l_|_Sf2); /2 = y2_^y2+V2;
X'x'+y'y'+Z'2' = JfJr'.{cos.f7.cos.i)'-fsin.f7.sin.«'| =:Rr' .cos.{U—v') ;
it becomes as in [5466].
t (3037) Substituting the values [5466/, 4776—4779], in the first members of
[5467a, è], and making the usual reductions by means of [24] Int., we get the second
[54661]
[54666]
[5466c]
[5466rf]
[546Grf']
[5466e]
[5466e']
[5466/]
[5466^-]
[5466A]
[5466t']
[5466*]
620 THEORY OF THE MOON ; [Méc. Ce).
or, by neglecting the square of S*
[54681 O = - 4- ^"-(1-2^') I 3 p \ ^'- cos{2v-2U)-j-r'"-. cos{2v-2v')-2Rr'.cos{^v- U-v') j
J- 3 P -R-s-S'.f fi.cos.(iJ — U) — i-'.cos.{v—v')l , 0^
P
As the term — does not contain either u, v, or .9, it will not enter
[5468'] /
P
into the equations [4753—4755]. The term j-Y~f3 gi^es, by its
members of these expressions ;
R R
[5467aJ X'a; + r'y4-Z'r = - . jcos.L^.cos.t)+sln.t^.sin.u + & | = -. [cos.(f7— it)4-&| ;
[54676] — xx — yy' — zz = — — .|cos.i)'.cos.i;-j-sin.t;'.sin.i)^ = .cos.(t) — ti').
l-t-ss
[5467c] Substituting these, and r"~ = -^ [4776], in [54G0], we get [5467].
* (3038) If we develop the numerator of the last term of [5467], and neglect the
square of S, we shall find, that the terms containing the first power of ^S' are the
same as in the second line of [5468]. The remaining part of this numerator of [5467] is
[546861 ^^ ™ ^^^^ ^'"^^ ™^"''^^'" °^ [5468c] ; and, by developing, using [20] Int., it becomes as in
[546Srf] ; and, by the substitution of /^ [5466], we finally obtain [5468e];
\E .CQS.{v—U)—r' .cos.{v—v')\^
[54()8c] ^JJ2_j.os_2('^_J7)_^/2.cos.2(^_,„')_Oiî,.'.coS.(y— C/).cos.(^— /)
[5468(/] =J-.{7?2+r2— 2Rr'.cos(t/-u')}+i|iî^cos(2i;-2C7)+7-'2.cos(2y-2ii')-27î/-'.cos(2f-f7-i/)|
[5468e] =^-f-+h { R^.cos.{2v—2U)+i-'^.cos.{2v—2v')—2Rr'.cos.{2v—U—v') I .
The part of this expression between the braces, being substituted in the numerator of the
last term of Q [5467], produces the third term of [5468 line 1] ; the otlier part of
è/2 SP
[5468/] [5468e]is iP ; which gives, in [5468], the term i -P- ".^T = 4^^575- Connecting
p I 2 2^2)
this with the second term of [5467], which may be put under the form '„ — , we
[5468g-] ^gj ^^, as in the second term of [.5468]. Finally, the first term j [5467], is
the same as in [5468] ; and we may observe, as in [5468'] , that this term may be neglected ;
for, / [5466] does not contain r, s, v; and its partial difierentials, relative to these
[5468/t] quantities, will vanish from the general formulas [4753—4755], which are used in this
chapter, in finding the perturbations.
Vll.iii. ^2-2.] ACTION OF THE PLANETS. 621
development, a function of this form,*
JL. = il-.[^^('')+J(i).cos.(C/— i)')+^'".cos.2(C7— tj')+^c.}=terms of Q. [5469]
Hence, the term — -r^ . ('y^j, of the equation [4754], produces the
following function ;
-^.^M"''+-4<'>.cos.(t/'-2;')+-4="-cos.2.(C/-w') + Sic.J=termsof -~.('l^\. [5470]
and it is evident, from ^9, 10, that there will result from it, in the expression
of au, the quantity, f
p
* (3039) If we substitute the value of / [5466], in the term T-^-^g, of the
expression [5468], we may develop it, in the usual manner, in a series of the form [5469].
This part of Q gives, in — T^" •(/")' ^^^'^ expression [5470]; as is evident by [54C9o]
2Ps3
differentiation. The next term of [5468] is — ^ ; and, as it is of the order s^, in
comparison with [5469], it may be neglected. The next terms of [5468] contain the angle [.54696]
2v ; but these quantities do not produce, by integration in nt-\-s [5474], any term of
importance, arising from a small divisor like i — m. The same remark may be made
relative to the terms of [5468] containing v — U, v — v' ; and, as they are also multiplied [5469c]
by the small quantity Ss, they may be neglected. Moreover, a little attention will show,
that the substitution of Q [5468], in the four first lines of [5081], will produce no terms
of the like kind, depending on angles having a small coefficient except they are multiplied
by quantities of the order of the excentricities, Sic; and, by neglecting such quantities as
in [5486', &1C.], we shall find, that the first term of importance is that in [508Iline 5],
which gives in fit the term — -^ — .2a5u. Multiplying this part of Jt by n = ~
[5092c], we get, in nflt, the term 7idt = — dv.^aSu; which will be used hereafter. [5469/]
t (3040) Substituting U=iv [.5463,5472], v'=:mv, h^^a, m = a"' [4937m], [5470a]
in [5470], and then connecting it with the two terms -X-u [47541 and with
the term of the same equation, which is developed in [4908 line 1] ; namely,
2
3m
— -ir—.ahL = —%m^M r5082A'], nearly:
2a, "• i' : ' [547051
VOL. III. 156
622 THEORY OF THE MOON ; [Méc. Cél.
TKAji^ - r. 1 ^A^'K COS. (i-m).v , A^-\cos.2(i-ni).v A^^^Kcos.SU-m^-v , „ ) „ ,
[5471] — iPa^. } i '- !^ i i L- 4-&C. V = terms of aôu ;
i beins' the ratio of the mean motion of the planet P to that of the moon.
[5472] a J J r J
Hence arises, in ndt [5081, &c.], the function,*
[5473] p^3j^ , , , , ,. \3+, 3 2 Ar N2+ 1— ^ 2 or \o+&c. [ = terms of ndt ;
conseqenily, we have, in ni+s, the following expression ;
it becomes,
[5470c] 0 = — + M +lPa^AiA^'>^+A'-^\cos.{i—m).v-irA^'''.cos.2(i—m).v-i-k.c.\—iin^Su.
Now, supposing any term of u, or Su, to be represented by,
l^470rf] ûu = B^"\ COS. n.{i — m).v,
and substituting it in [5470c], we find, by retaining only the terms depending on this angle,
and dividing by cos.?i.(i — m).v,
[_5470e] 0 = — «2.(i— »i)2.B('0-}-5('')-|-iPrt2.^u)_3,B2_5(7.).
Hence we get,
—iPoS.^i»)
[5470/] -B*"' = - — :^ --. r^ ;
and, the term of a^u [5470i], corresponding to that in [5470c], which contains
^''", is,
„ 1 -4''''.cos.7i.(i— m).v
[5470^:] ahu = —\Pa\ ,_, o_ o— r„
a
1 — §m^ — 11?. [i — ?n)
From this formula we may easily deduce any term of a&u [5471], from that which
depends on the same angle n.{i—m).v in [5470c], by multiplying the term of [5470c],
[5470ftj _^
depending on ^"'', by the factor i_|^a_,i2 (j-_„j)2-
* (3041) Substituting, in ndt = —dv.2aSu [5469/], the value of a<5M [5471], we
[5473a] get [5473], whose integral gives [5474]. Now, from [5475], we have a = -;;^ ;
substituting this in [5474], we get [5476].
VII.iii.^22.] ACTION OF THE PLANETS. 623
Pa' QA^^\sm.{i—m).v iA'^Ksm.2{i-m).v AA^^\s\n.3{i-m).v
&c. [ = terms of ?it+ f. [5474]
Now, we have — ^ = m^ [5374rt — 6] ; m' being the sun's mass. Hence, [5475j
the preceding function becomes,
.7)1 .a c^i).sin.(î"-m).i; , à^'->.sin.2(?'-?»).« , M'^'-sin 3(i-m).v }
— A Î: i !^ '- \-- ^ '- — h&c. > = terms of 7!<+«. [54761
In the case of a 'planet, inferior to the earth, ive have, by putting a for the
ratio of the mean distance of the planet from the sun, to that of the earth from ^ ^'^
the sun, and retaining the denominations of chap. vi. of the sixth book,*
a". J<" = b'^ ; a'\ J'^' = b^ ; a". A'-'^ = b'^ , &c. ; [5478]
2 2 2
which changes the function [5476] into the following ;
Terms of
~-;.m^ \ b g. sin. (^i-m).v |6„.sin.2(»-7«).i' ib ^. sin. S {i—m).v / ivomtho
^ i /. o ,• V'+. ^■^ 2 AT ^2+1 ^■^ 2 Q/^l3+^"- I = *"'"' °^ "'+- ; f5479]
I — m I l-^mr-h-m)- 1 — ^m-'-4 (i-mY 1 — mr-dn-mY )
direct ac-
tion of an
in which ive may take, for (i — m).v, the mean longitude of the planet, pllne?.'
minus that of the earth.
With respect to a superior planet, a denotes the ratio of the mean distance [5479']
* (3042) Changing a, a' [956], into R, r' [5462,5465]. respectively, in
order to conform to the present notation ; also, the angle n't — n^-f-s' — s, into U^v', [5478a]
it becomes, by neglecting, for brevity, the consideration of the excentricities,
\m-i-r'^-—<2Rr>.cos.{U—v')\-i=:^^-S..B'-^cos.i.{U—v'). [54784]
If we neglect S-, the first member of this expression becomes equal to f~^ [5466].
p
Multiplying this by rr > we get,
4^3= ^.\i^.B^'\oos.i.{V-v')\. f5478c]
Comparing this with the development in [5469,956'], we get £<''= ^'''. Substituting
[1006], and multiplying by f^'^ we obtain a^.A'^'~>==b3 , as in [5478]. Substituting [5^^^^^
these in [5476], we get [5479].
624 THEORY OF THE MOON; [Méc. Cél.
of the earth from the sun, to that of the planet ; so that we have
*
(l) (2) (3)
[5480] «'3. J(')_al6, ; o'^ ^(^' = a=.è^ ; a'\ A^^^ = ^\ h ^ ; he.
2 2-2
Terms of
f/oJr'' This changes the function [5476] into the following form,
the direct
planet?"" —■1)^0? S ^ . sin.ft— m).w 4 6, .sin.2(i— 7n).v Ih e,mâ[i—m).v } (^)
m <â .A ,* ,„>„..
These are the only sensible terms ivhich can result from the direct action of
the planet P on the moon.
But, the sun's action upon the moon may render sensible, in the motion of
that satellite, the perturbations of the radius vector of the earth'' s orbit, arising
Indirect fiom tlic ttctlon oftlic planct P upon the earth, and may produce, in the moon's
"fanets motious, incqualitics of the same order as those we have just considered. To prove
[5482] it, we shall consider the term ' ^ [4866], which is a part of the equation
h-' P
[5483] [4754]. We shall suppose — =^ ~. K. cos. Q2'n't—^n't+B), to be any
,5,-'
[5484] term of — , arising from the action of the planet P upon the earth ;t
' n"t denoting the mean motion of P, and n't that of the earth ; the
corresponding term of -7- will be,
[5485] ^= —-,.K. COS. œ'n"t—^n't+B).
urn
* (3043) The equation [5478d] holds good for a superior planet, by merely changing,
3
in the factor a'", the quantity a', corresponding to the earth's distance from the
[5480a] „,
sun, into - [5479], which represents that of the superior planet from the sun ; by
which means, it becomes,
[54806] ©'-^'"^^r °' a'^A^^^ = a?.b'^, as in [5480].
Substituting this in [5476], we get [5481].
f (3044) This form is the same as is used in various places ; as, for example, in
VII.iii.§22.] ACTION OF THE PLANETS. 625
^ '3
Hence, the term '^ .^ produces the following ;
Q p ,,'3
__ ''1:1. . K . COS. œ.'n"t—^n't+B). [Term of 4754] [5486]
Sr'
If loe consider only those inequalities of —, lohich are independent of the [5480]
excentricities of the orbit, and represent them by the series*
P or'
— .{X(''.cos(n''<-«V+s''-.'')+*^^-'-cos2(ft'V-ri7+£''-5')+£''3).cos3(n'7-n7+s''-£')+&'=|=termsof_; [54871
m a
f fO
lit tU
the term -7^—5 Avill produce, in [4754 or 4961], the function,!
— ^ •-, • i K^'\eQS.{i-m).v+K''Kcos.2{{-m).v-\-K'-^\cos.3(i-m).V'}-hc. j ; [Terras of 4754] rg^gg-,
whence results, in a^u, the function,
"l~-,7' 1 1 -|TO3_(i_„,)2+i_3„,a_4(i_,„p+j-:zp-9_c,(i_„,p+ «^c. ^ _ terms of atSw . ^g^gg-j
This gives, in nt-\-s [5095], the following terms ;
[1023, 4306, 4308, &.C.]. Now we have, very nearly, in [4777e], u'=-; and, the [5486a]
differential of its logarithm gives,
6u' or' Sr
-;7 = — — = — ^ , nearly ; [54866]
substituting this in [5483], we get [5485]. If we vary u', by the quantity (5)*', it
J . m'. !i'3
produces, in ^ , the term,
3 m'. It' 2. fa' 3m'. u'^ iu'
2A,3.u3 ~ 2^2;^ • ~^ ' [5486c]
Su'
and, by substituting the value of — ;- [5485], it becomes as in [5486].
* (3045) The form assumed in [5487], is the same as that in [4306 lines 9—1 1] ;
decreasing the accents on n'", n", i'", s", &ic. by unity, so as to conform to the notation [5487a]
here used.
t (.3046) Substituting, in [5486], the values u = cr^ u'=a'-\ h^=a[4937n],
also ^=^, [5475], it becomes, [54876]
a'3 m'
VOL. III. 157
planet.
Action of
Venui.
[5491]
(0)
2
9,992539 ;
(1)
2
8,871894;
2
: 7,386580 ;
*!"=
5,953940 .
626 THEORY OF THE MOON ; [Méc. Cél.
Tonna of
nt+e * 3ma P C K^'Vm.ii—viU , iK&'>.sm.2{i—m).v , lsKS3).sin.3{i—m).v , „ ) (^)
[54901 < ^ — -\ ^ r — — h &c. > =termsof nt+e.
illing ' i_mm' U-|m2_(i— m)a^l— am2-4(i— m)3^1— fm-2— 9(i-m)2 ^ 5
IVoni ilio
Mtimofa This function is of the same order as that which results from the direct action
of the planets upon the moon [5479,5481]. fVe shall noio compute these
several inequalities for Venus, Mars, and Jupiter.
Relatively to Venus, we have, in [4126,4132],
a = 0,72333230 ; 1
2
S
4
6
Hence we deduce, by means of [974],f
[5488a] — ï • W • K.cos.{?!i,:'t-^nt+B) ;
ir'
which is the same as the product of the assumed value of — [5483], by the quantity
[548861 ^. Therefore, if we multiply the assumed value of — - [5487] , by the same factor
2a a' ■• •'
— we shall obtain the corresponding expression, which arises from the variation of
2a
[5488c] '"'•"' , as in [5488]. This term forms a part of the equation [4754, or 4961] ; and, we
may find the corresponding part of k, or rather of ahi, as in [5470^*, A], by multiplying
a
any term of [5488], depending on K'^"-\coi.n.{i—m).v, by — j_3„j2_,j27/;_„;)2 5
hence we obtain [5489].
* (3047) Substituting the terms of a&u [5489], in ndt = — 2dv.a5u [54G9/],
[5490a] and integrating, we get the terms of nt+s [.5490]. We may remark, that i.i iLe original
work, by a typographical error, the terms of [5490], are made to depend on cos.(;'— ?«).r ;
[.54906] C0S.2. {i—m).v, &c. instead of s\n.{i—m).v; <i^2.(i— m).«', &c.
t (3048) Putting « = f, in [974], and then, successively, i = 0, i=\, we
get;
vu. iii. § 22.] ACTIOA OF THE PLANETS. 627
(0)
6^ = 85,77422 ; [5492]
b'J^=^ 83,40760.
By observations, we have z—îh = 0,0467900 ;* therefore, by supposing, [5492']
as in [4061, line 3], t
[5493]
Direct ac-
tion of
Venus.
[5494]
P J
m' ~ 3S.3130 '
we find, that the function [5479], reduced to seconds, becomes,
+ 0%577273.sin.(z— m).D 1
+ 0',241919.sin.2.(i— /«).« [Terms of nt+£] 2
+ 0%131463.sin.3.(i— m).i; 3
What we have here represented by —, is denoted by Sr'', in [4306,
line l,&c.], and we have, in that article, by means of the action of Venus,
(0) <1) (I) (2)
b, -= ; b, ^ ? ^ . [5492a]
With these formulas, we may compute the values [5492], by using the expressions [5491].
* (3049) If we use the same notation as in [4077], we shall fiod, that the mean motion
of Venus, in comparison witli that of the earth, is represented by — . Multiplying this
by m = 0,0748013 [5117], which expresses the ratio of the sun's mean motion to that
of the moon, we get the expression of i [5472], or the ratio of the mean motion of [.MgSi]
Venus to that of the moon ; consequently,
?• = 0,0748013.^ . Hence, î — ?« = 0,0748013 . ^^
1 «'■
[5493c]
and, by substituting the values of n', n" [4077], it becomes as in [5492'].
t (3050) In the present notation, P is the mass of the planet [5455], »«' that
p
of the sun [4757"] ; hence -, of the present notation, is the same as m' [4061 line 3], t^^^^a]
628 THEORY OF THE MOON ; [Méc. Cél.
rir'
-L =. 0,0000015553 1
a
— 0,0000060012.cos.(i— m).» 2
^^^^^^ 4- 0,0000171431. cos.2.(i—m).v 3
+ 0,0000027072.cos.3.(î— m).w 4
+ &c.
The function [5490], reduced to seconds, becomes,*
+ 0%448818.sin.(i — m).v 1
— 0',645333.sin.2.(i— m).v • 2
[Terms of ««+;]
— 0',068705.sin.3.(i— wî).îJ 3
&c.
If we connect this with the preceding expression [5495], toe shall have for
the lunar inequalities, depending on the direct and indirect actions of Venus,
upon the moon ;
+ 1 ',026091. sin.(i—TO).tJ 1
Indirect
action of
Venus.
[5496]
Whole ac-
tio;! uf
Vonua on
nl^-s. — 0',40S414.sin.2.(t— m).z> 2
[5197] [Terms of nt-\-B] ^
+ 0%062758.sin.3.(i— m).« 3
&c.
We must increase these inequalities in the ratio of 1,0743 to 1 [4605].
[54946]
and by putting (^'=0, it becomes as in [5493]. Substituting, in [5479], the values
[5491—5493], also that of m [5117], it becomes nearly as in [5494].
* (3051) Comparing [5487, 5495], we get,
P P
— ^0) ='0,0000015553 ;" — . iC''> = — 0,0000060012;
m! ' ' ^
^^ ^2)^0,0000171431, &c.
m'
Substituting these, r,nd m [5117], also i—m [5492'], in [5490], we get the terms of
[5496i] ,.^_|_j ^ arising from the indirect action of the planet Venus on the moon, as in [5496].
[5496a]
VII. iii.§22.]
ACTION OF THE PLANETS.
629
Relativel}' to Mars, we have from [4159, 4165],
Hence we deduce,*
a = 0,65630030 ;
(0)
6,856336
a
(1)
b^ = 5,727893
2
&''' = 4,404530
(3)
b^ = 3,255964
2
&c.
(0)
b^ = 38,00346 ;
2"
(0)
b^ = 36,20013 .
2
I Action of
AlaiB.
2
3
4
6
[5498]
[5499]
Observations give i — m = — 0,0350306 ;t therefore, by supposing, as in ^^^qq-^
[4061 line 5],
1
m' 1846082 '
[5501]
* (3052) Substituting the values [5498], in [5492a], we get [5499]. [5498o]
t (3053) Changing n' into ?i"' in [5493c], we get the value of
i— OT= 0,0748013
(«"'— n")
[5500a]
corresponding to Mars ; and, by using the values [4077], it becomes as in [5500].
Substituting these, and -' [5501], in [5481], we get [5502]. The expression [5504],
is deduced from [5490, 5503], in the same way as [5496] is obtained from [5490, 5495],
in [5496«,5]. In the original work, the coefficient of [5505 line 2], is erroneously printed,
1",201491 =0%3S9283, instead of 1",21 1491 = 0',392523 . [5500c]
VOL. III. 158
630 THEORY OF THE MOON J [Méc. Cél.
the function [5481], becomes,
Direct — 0%029177.sin.(i — m).v
action of
"""• — 0^01 1260.sin.2.(i— m).t;
t5509] _0^005584.sin.3.(^■— m).?; [Terms of nt^,-]
&c.
We have, in [4306 lines 8 — 11], from the action of Mars,
- = _ 0,0000000478 I
a
+ 0,0000G05487,cos.(«— m).« 2
[5503] ^ 0,0000080620.cos.2.(i— m).t) g
— 0,0000006475.cos.3.(z— m).ij 4
&c.
The formula [5490], reduced to seconds, becomes,
+ 0',054760.sin.(î— m).iJ 1
action of
M»- + 0',403783.sin.2.(i— m).z; 2
r.^, ., [Terms of nl-\-t]
[^5"4] _ o\021753.sin.3.(z— m).t> 3
&.C.
If we connect together the terms in [5502,5504], we shall obtain the lunar
inequalities depending on the direct and indirect actions of Mars upon the
Moon ;
Complete
action of
Mars on
n/+£.
+ 0%025583.sin.(î— m).^) 1
+ 0%392523.sin.2.(î— m).t; 2
[5505] _ 0%027337.sin.3.(/-m).. [Terms of n.+.] 3
&c.
We must decrease these inequalities, in the ratio of 0,725 tp 1 [4608].
Relatively to Jupiter, we have, as in [4167,4173],
VII. iii. >^ 22.]
ACTION OF THE PLANETS.
631
a = 0,19226461 ;
(U)
b^ = 2,176460;
3.
2
.(1)
6 = 0,619063;
(2)
6, = 0,148198;
&^'= 0,032439
2
&c.
Action of
Jupiter.
2
3
4.
5
[5506]
Hence we deduce, from [5492a],
(0)
h^ = 2,51906;
2
hl= 1,13310.
2
I
2
We have, by observation, i — m = — 0,0684952 ;* therefore, by supposing,
as in [4061 line 6],
p 1
m' ~ 1067,09 '
the function [5481] becomes,
— 0',070391.sin.(î— m).«;
1
— 0%008547.sin.2.(i— m).«
2
— 0%001275.sin.3.(i— m).t>
[Terms of n«+;]
3
&c.
[.5,507]
[5.508]
[5509]
Direct
action of
Jupiter-
[5510]
* (3054) Clianging n into n'', in [5493c], we get the value of
i-., = 0,0748013. ^:^"1, [5508a]
corresponding to Jupiter; and, by using the values [4077], it becomes as in [5508].
p
Substituting this and -, [5509], in [5481], we get [5510]. The expression [5512] is
deduced from [5490, 5511], in the same manner as [5504] is found, in the last note.
632 THEORY OF THE MOON ; [Méc. CéJ.
We have, from [4306, lines 13 — 16], by means of Jupiter's action,
- = — 0,0000011581 1
a'
[5511]
Indirect
action of
Jupiter.
[5512]
+ 0,0000159384.cos.(î— m).!) 2
— 0,0000090986.cos.2.(î— m).?; 3
— 0,0000006550.cos.3.(i— m).i' 4
&c.
Tde formula [5490], reduced to seconds, becomes,
+ 0',816336.sin.(i— ?«)-«^ 1
— 0%236377.sin.2.(i— m).« 2
— 0%0n625.sm.3.(i—m).V [Terms of nt-\-s] ^
&c.
If we connect it with the preceding [551 0], we obtain for the lunar inequalities
depending on the direct and indirect actions of Jupiter upon the moon.
Whole + 0%745945.sin.(z — m).v \
action of
^''nt^r — 0^244924.sin.2.(i— m).i' 2
— 0',012900.sin.3.(î— m).i; [Terms of «<+£] 3
&c.
[5513]
[5513'] If we take, with a contrary sign*an the inequalities resulting from the actions
of the planets upon the moon, [5497, 5505, 5513], we shall obtain the
inequalities ■produced by this action, in the expression of the moon's true
[5514] longitude; we may, therefore, reduce them to tables, observing, that {i—m).v
may be supposed equal to the mean longitude of the planet, minus that of
the earth. It would be useful to introduce these inequalities into the lunar
tables, considering the precision to which these tables have been carried.
* (3055) The inequalities of the expression of nt-\-s [5095], arising from the actions
[5515i] ^j. ^j^g planets, are given in [5497, 5505, 5513] ; and to obtain the corresponding terms of
« [5095], we must ev'dently change their signs.
VII.iii.§22.] ACTION OF THE PLANETS. ^^^
PA'.") 1 /dQ\ . . ,
The term ^3, of the expression of — -.(^-^j, gives, in the [5515]
equation [4961], the term,*
J.P«^ J"" e . COS.(CV îî). (Term of 4%1] [5516]
Hence it is evident, that the value of c is decreased by the action of an
inferior planet, by the quantity, f
P /o'
f . — . m~. 0 3 ; • (Decrement of c] [5517]
2
and, by the action of a superior planet, by the quantity,
p ^ (0)
3. . — _ in~' Oi?.h • [Decrement of c] [551S]
8 ,y^l 3
*
1 /dq\
(3056) The equation [4754, or 4961], contains the term — p'Vrfil"/' which
is developed in [5470], and contains the term _— — ^. Substituting h^=a [49377i],
^'''•" [5516a]
and M [5393], which gives — = «^.{1 — 3e.cos.(CT — w)}, nearly; we obtain the
w
term [5516], depending on e.
t (3057) Neglecting e^ c'^, and also, for brevity, the symbol •«, we have, as
in [5396a], — j^ for the coefficient of -.cos.ct, in [4961] ; hence it is evident, that
the quantity [5516] increases p, by the term Sp=:^P.a'.A^''\ The corresponding
increment of the motion of the perigee is ^Sp.v=yP.a^.A'-''Kv [5396^]. Substituting
the value of a^ [5473«], it becomes,
p
■|.— 7.m2.«'3.^(0).^_ [55166]
Now, the motion of the perigee is represented by (1 — c).v [4817] ; hence it is evident,
that the preceding expression decreases the value of c by the quantity,
4-.- .m^.a'^.A^''\ [5516c]
" m
If we substitute the value a'\A^<'-^ = b^ [5478rf], corresponding to an inferior planet,
it becomes as in [5517] ; and, if we use the value «'3.^(°'= <v>.b^ [5480&], corresponding [5516rf]
to a superior planet, the decrement of c becomes as in [5518].
VOL. III. 159
634 THEORY OF THE MOON ; [Méc. Cél.
[5519] Likewise, the term --^ [4865'] gives, in the equation [4961], the
quantity,
9
"2A2 '""a'
[5520] — ^ .«=. — . e. COS. (cîJ—n) ;
5r'
[5521] — representing the constant part of the perturbations of the radius vector
[5521'] of the earth's orbit, given in [4306]. Hence, the value of c is increased
by this means, by the quantity,!
9 m^ or'
.15522] — — . — r . [Increment of cj
i5530ol * (3058) The variation of the term [5519] is given in [5486c], namely, ^j^-^ ■ ^ :
and, by substituting,
•5"' ^'■' rr^ocn w I 3m'.u'3 6r
[55206] — = [54866], It becomes — .-•
If we use the value of u'^ [5516a], it will produce tlie term,
r5520c] -— -—.a^ — .e.cos. (ct)— «), dependmg on e.
hi the original worli it is erroneously printed,
9»i'.k'3 6r' , ,
,j..„,, ,, • . — r .e • COS. (cv — ■ss) ;
[5530rf] 2A3.M3 a' ^ •"
the sign being wrong, and «^ changed into ir^.
t (3059) Substituting «'=«'-', h~ — a [4937ft], and then --;y =m2 [-5475j^
a'
This produces, in j) [5396a], the term.
in [5520], it becomes g a ^ £ (rv-.-!.-)
[5522a] ^ ^^«-a' • ^ • C0=,. (cy— raj.
[55226] ^.P ^'"^'"^ '
and, in the motion of the perigee i&p.v [5396(Z], the term,
9
fir'
m~ . ■— . V .
[5522c] 4 '" • a'
Now, the motion of the perigee being {l—c).v [4817], it is evident, that this produces
|5522rf] an increment in the value of c, which is represented by the function [5522]. In the
original work, the word increased [5521'], is printed decreased.
VII. in. ^22.] ACTION OF THE PLANETS. 635
It is easy to prove, that all these quantities are insensible.* [5522']
ffe shall now consider the perturbations of the moon'' s motions in latitude.
The sum of the terms,
s /^\ (l + .O A!Q\ ,Te™s of 47551 [5523]
. [Terras of 4755]
K^.U \duj li^.U^ \(fs J Perturba-
tions
which make a part of the equation [4755], acquires, by the action ot the [noon^s^
planet P, the quantity, t action
planets.
3P.S , 3P.Rr'.S.cos.(v—v')—3P.R\S.cos.(v—U)
— 4- ^^ ^^ ■ ' [Terms of 4755] [55241
This function contains, relatively to an inferior planet, the term,t
* (3060) That these quantities are insensible, is evident by computing any one of the
terms; for example, that in [5517], corresponding to Venus. Substituting, in this, the
values of
P (0)
;7=3Wt^7 [5493]; m^ = 0,0055 [5117./] ; b, = 10, nearly [5491] ; ^ggag^j
we get,
I , £r . m^ . C = 0,00000006 ; [55236]
which is insensible, in comparison with the whole coefficient of the motion of the perigee
c— 1 = 0,00815199 [5117 line 2]. [5523c]
t (3061) Taking the partial differentials of Q [5468], relative to u. s, we get,
by neglecting terms of the order s^,
s
\du)~ 2u3/3 ^^•«•— yM3.y5 ; [5524a]
fdq^_ 2P.S RS.{R.cos.(v-U)-,-'.cos.{v-v')\
1 (!+««) 1
Multiplying [5524a], by — ^, and [5.524i], by — -f~, or simply, by ——.
and adding the products, we get the value of the function [5523], as in [5524], nearly ; [5524c]
neglecting the terms depending on the angles 2« — 2t7, 2« — 2«', 2v — V — v'; because
they do not produce, by the integrations, any term of s, having the small divisor g — 1 ; [5524rf]
which the other terms [5527, 5528] acquire, as will be seen in the following note.
•iP s
X (3062) We shall notice the effect of the first term of [5524] - ' in [5534«],
636 THEORY OF THE MOON; [Méc. Cél.
(,3 / (0) (1) f
[5525] f-P-^-'-Tg'S^-'^S "s (^ .Sin.(t; è) ; ITorms of 4755]
'^ t '2 '2 )
[5526] X being the inclination of the orbit of the planet P to the ecliptic, and <' the
and shall consider the rest of this function in the present note. If we divide the equation
p
[5525a] [5469], by — -^, and substitute [5473], we shall obtain, successively, the values of
f~^ [5525è,c] ; and, by using the same notation for f"^, we get its value [5525fZ] ;
[55256] ~^^A^o)j^^(iicQs.{U—v')+A^-\cos.2.(U—v')-\-hc.
[5525c] = 4-, • S i*r+ ^1"- t^os. ( U—v')-\-li'l cos.2.{U—v')+ &c. \ ;
1 1 f to) (1) (2) ■)
[5525rf] TT =-5- IK +l>,.cos.{U-v')+h^ .cos.2.(£7-^')+&c. .
J ^ a -^ i -2 2 g- )
The first of these developments is used in [55346]; the second in this note [5525Â].
[5525e] Now, as X is very small [5526, 4082], we shall have, very nearly, &' = X.sin.([^ — è)
[5526,5463,679] ; hence we get,
[5525/] S.cos.(y— ti') — iX.sin.(L^— î^'+u — â)+JX.sin.(r-j-î)'— r — è) ;
|-5505g.] S.cos.{v—U):^\'k.s\x\.{v—è)-\-^\.sm.{2U—v—è).
We shall now multiply these two last expressions by the value of f~^ [5525(7], and
reduce the product by formula [18] Int. ; neglecting the terms in which the coefficient of
the angle v differs considerably from unity ; because they are not much increased by
integration; whilst the terms depending on sin.(y — é), are considerably augmented by
[5525i] the divisor of the order g — 1, as in [53476 or 5527, 5528] ; hence we get, by making
the usual reductions ;
[5525i-] —.S.cos.^v—v) =j^^.b^.s\n.{v—è)+'kc.;
1 X ^^^
[5525/] -^..S.cos.(._l7)=-.6^.sin.(^-^)+&c.
Substituting [5525A-, l\ in the two last terms of [5524], they produce the. folloiving term
of [4755] ;
3P.P./..S.cos.(.-.0-3P.fi^S.cos.(.-f7)_ 3FR j^, ."L/?.r;.x.sin.(,,-.V
[5525m] h^.%l\p Ah^.u\a">i J ^S ^ '
Substituting, in this second member, the approximate values h^ ^ a, u^crK
[5525)t] r'=a' [5470«, &.C.] ; and, for an inferior planet, R = a.a', nearly [5462, 5477, &c.],
we get the expression [5525].
VII.iii.4.'2-2.] ACTION OF THE PLANETS. 637
lomitudeof its ascending node. This produces in s, for an inferior planet, [5526']
the term,*
* (3063) If we put, for brevity,
in the second member of [55-25ot], it becomes H'.ûn.{v — è). Tliis represents a term [55275]
of the equation [4755], or of the similar equations [5347/, m] ; and may be integrated as
in [5347/ — «']. If we su])pose the term of Ss, corresponding to [5527i], to be
represented by &'s^R".sm.{v — è), which is similar to [5348] the equation corresponding [5527c]
to [5347/h], will become,
0 =^+.-f fl-'.sin.(t»-0) + fe-2-l).//". sin.(«-â). [5527^]
Substituting, in this, the assumed vakie of s, or <h, [5527c], we find, that the two
first terms mutually destroy each other. Dividing the rest by sin.(M — è), we get the [5527e]
following equation, which is similar to that in [5347»] ;
0=fl'+(o-2-l).f/". [5527/]
Dividingby g''—\={g-^V).{g—\)=2.{g—\), nearly [5-35 la], we get H"=——^-. [5527^-]
Substituting this in as [5527c], and then resuming the value of li' [5527 a], we get,
^' = 8.{g-l).h^n\a'n^ ■ \- '''-^l S •^•^'"■(^-^) • [5527M
Substituting the values [5525ra], corresponding to an inferior planet, we get [5527«] ; and,
by using the value of a^ [5473a], it becomes as in [5527A;] ;
3Pa «3 ,- (0) (1) ^
m'
(0) (I) )
a.ôj — ftj > .X.sin.(i) — (3). [.5527A;]
'2 2" )
This agrees with [5527] ; observing, that jve have corrected this formula, for a mistake in
the original work, where it is printed ivith the prefix of a negative sign. [.5527i]
In making the calculation for « s!<periorp/aMe^, we must change the factor -7-, in the
second member of [5525f/], into -— ; and the same change must be made in [5527a, A] ;
XL-' [5527771]
by which means, this last formula becomes, for a superior planet,
VOL. III. 160
638 THEORY OF THE MOON ; [Méc.Cél.
Terms of
6s, , P ( (0) (') )
[5527] l--,-^»»'-^^-^^— &A \
6 6' = ^^ — ^.X.sin.(î) â) ; [inferior plaoetj
arising
from the
action of
an inferior
O
planet; aïià, foT o supcnor planet, this inequality becomes,
p ( (0) (1) )
[5528] 6s = ^— Y ^-^.X.Sin.(» è). [Superior planet]
and, from
tiiat of a
superior, S!" 1
planet. "^
Reducing these inequalities to numbers, by using the masses of Venus,
Mars and Jupiter [4605, 4608, 4065], we get, for Venus,*
[5529] ^s ^ — 0^276468.sin.(î;— «') ; [Action of VennsJ
[5527o]
Now, substituting as in [5525n], h^ = a, u = a ^, r'=a, and then, R^-,
get [5527J3] ; and, Idj using «^ [5473«], it becomes as in [5521q] ;
we
3Pa3 «3 ( (») (1)
p
This agrees with [5528] ; the expression being corrected as in [5527/], /or the mistake of
^ ^^ prefixing the negative sign. The terms we have here computed [5527A, «7], have the
small divisor g — 1, of the order m- [4828e] ; and, even with this divisor, they amount
only to a fraction of a second, as appears in [5529 — 5531] ; hence it is manifest, that the
terms of this kind, which have large divisors, must be wholly insensible.
[5527«]
P 1
* (30G4) Substituting, in [5527], the values of ^ = ^^ [4605], a [4126], also
("1 (0
[5529a] /j[^ h^ , deduced from [5492], g,m [5117], \=<s?' [4082], it becomes, as in [5529].
a" 5"
In like manner we obtain from, [5528], the expressions [5530,553!] ; using the mass of
Mars [4608], and that of Jupiter [4065] ; also the other elements as in [4159 — 4173,4082] ;
m , g , being as before. We have corrected the signs of the expressions [5529, 5530, 5531],
for the error [5527?,)-], ivhich is foundinthe original ivork; the numeral coefficients given
by the author being.
[55296]
[5529c]
VII.iii.§22.] ACTION OF THE PLANETS. 639
and, for Mars,
^5 = +0',005497.sin.(l'— r); [Action of Ma»J [5530]
also, for Jupiter,
o's = +0%037925.sin.(?;— ^") ; f Acion of jupHer, [5531]
è', ê'", ê'", being the longitudes of the ascending nodes of the orbits of [5532]
Venus, Mars, and Jupiter.
Finally, it is evident, that the value of g, is increased by the action of the
planet P , by the quantity,
P (0)
I . — . nî^. 63 , relative to an inferior planet ; [5533]
WÎ "2 Increment
and, by the quantity.
o( ff, by
the direct
action
of the
planets-
P (0)
I . — . m^. 0-^. 63 , relative to a superior planet* [5534]
m
+ 0",85329C = +0^276468 ; — 0",016966 = — 0%005497 ;
— 0",117051 =— 0',037925.
[5529d]
2P.S
* (3065) If we substitute, in the first term of [5524], ^' - , which was neglected rrco^ -i
in [5525a],the value of — [5525J], and retain only the part which is independent of
U — v', we obtain the expression '^ . ^^'°\ Substituting the values h^ = a , [55346]
« = a~% and a^ [5473o], it becomes successively,
This term of [4755], increases the value of p" [5397^, / ], by the quantity,
6p" = I . - . m^. a'3. ^(oi , [5534rf]
m
and the corresponding increment of the motion of the node [5397/], is,
h¥'-^=i-~-m'.a'\A^o\v. [5534,^
Now, the motion of the node is represented by (g— \).v [4817] ; hence the increment
of g is represented by,
64.0 THEORY OF THE MOON ; t^^c. Cil.
rpi 3m'.u'^,s
Ine term ^ — , which forms a part of the equation [4755], and is
t^^^^^ developed in [5021], decreases the value of g, by the quantity, ^ . ^ ;*
Decrement ^ **
—7 being the constant part of the perturbations of the radius vector of the
the mdi
reel action
or the «
planets,
[5536]
earth's orbit. Hence, the value of g is decreased by the action of the
planets, by the same quantity that c [5522] is increased by the same action.
But these quantities are insensible [5535/"].
The direct action of the planet P upon the moon, introduces in the
equation [4961], a quantity of the form,t
[5534/] * ^ m
[553%] For an inferior planet, we have a'^.A'^°^=^l>3 [5478f7] ; substituting this in [r35o4/]
fO)
[55347t] we get &g [5533]. For a superior planet a'^. A'"'' =^ a?.b^ [54806] ; hence Sg
1
[5534yj, becomes as in [5534].
* (3066) Tlie variation of the term —pcr^—r > taken relatively to u', becomes
[5535a] ^ ^ S/Ak" ^
. as in the first or second member of [5535c]. Substituting — = ; [55206], it
becomes as in its third member ; and, by successive substitutions, using the values [55346],
we finally obtain [5535cZ] ;
9m'.u'-U.s
[5535c]
[5535rf]
2/i2. ;
9m'. iP. s Su'
9 m'
m'''. 5
&r'
9 m'
«3
&r'
. s
2h^.iâ ' u' ~
2
F.M^
a'
2
■ a'3'
a'
9m^ Sr'
2 ■ a' ■ ^ ■
Now, proceeding as in [5534c, Sic] we find, that the expression [55o5<Z], produces in p"
[5535e] the term (5p" = — f m^. — ; and tiierefore in g the increment, Sg ^ — | m^. —,
as in [5535] ; being the same as that of c [5522], except in its sign. The quantities
thus computed, in [5533, 5534,5535], are of nearly the same order as that in [5522], and
must be insensible, as in [5522'].
f (3067) As an example of the manner in which terms of the form [5537], or such
as are free from the sines and cosines of the periodical angles, are introduced into [4961],
[5537a] by means of the function ^ , we may mention, those which arise from the substitution
of / [5466], in Q [5467]. For, in [669 line 1], we have, relative to the earth.
VIl.iii.§22.] ACTION OF THE PLANETS. 641
M. ^ . 7n^e'^+3î. -, . m\e'e"-{-M". -, . m^e"^+&c. ; [5537]
m m m
e" beins; the ratio of the excentricity to the semi-major axis, in the orbit of
P. Hence, there arises in the moon's mean longitude, a secular equation j",^^'","^^,^
analoîTOUS to that we have found in [5095^?], sccùiar
equation.
m"-. / (e'-^— E'^).dv. [5538]
This last expression arises from the development of the term 3
[4866 line l,5083,&c.] ; and it is incomparably superior to the former, on
p
account of the small factor — , connected with the first expression.
m *■
Thus, the indirect action of the planet P itpon the moon, transmitted by means [5539]
of the sun, is, as it regards this inequality, much more important than the direct
action, ivhich may be neglected, ivithout any sensible error.
r' = a'.H +i(i'^— e'. cos. z)'4-&c. } ; [5527b]
and for the attracting planet P ,
R = R".\ 1 +ie"^- e".cos. U+hc.]; ^5537,^
R" being its mean distance. From these values, we easily perceive, that r'^ contains
a term depending upon e'^; R^ a term, depending on e"^; Rr' a term, depending
on e'e".cos.(U — v') ; therefore, Rr'.cos.^U — v') contains a term depending on [5537^]
e'e". Substituting these in [5466], we find, that / contains such terms, free from
periodical angles, and depending on e'^, e'e", e"^ ; which are, by this means, introduced [55376]
into Q [5467], and finally into [4961]. If we proceed with the function [55.37], by the
method which is used in [5083 — .5089], it will produce terms of the form [5087], or rather rKK'i^fi
like [5095(7, or 5538] ; but they will be much less than those in [5538], by reason of the
p
small factor —, . ?n^, which attaches, as in [5476], to the terms depending on the direct
™ [5537ff]
action of the planet P.
VOL. III. 161
642 THEORY OF THE MOON ; [Méc. Cél.
CHAPTER IV.
COMPARISON OF THE PRECEDING THEORY WITH OBSERVATION.
23. In the first place, we shall consider the mean motions of the moon,
of the perigee, and of the nodes. The expression of the moon's mean
longitude, in a function of its true longitude, contains, in [5095], the
secular inequality,
[5540] I m\f(e"-E").dv.
Hence, the expression of the true longitude, in a function of the mean
Secular
of u,T'"^ longitude, contains the secular inequality
longitude ;
[5541] iv = —pn^.f(e'^—E"').ndt.
If we represent the number of Julian years elapsed since 1750, bij t, we
shall have, as in [4611],
[5542] 2e' =2E'—t.0%nil93—t^ 0 ,0000068194.
reduced to Thercforc, the inequality [5541] is represented by,t
3eC0Q(l9.
[5543] 6t,= 10%181621.î^+0%01853844.i=^; ['TtgS'™]
* (3068) From [50966] we obtain,
[5541a] v = 7it + s— |mS./(e'2— E'2) .rfy— 2 C.sin. (lî^+p) .
[55415] In the secular part of this expression —^m^.f{e'''—E"').fh, we may substitute ndt
for dv, and it will become as in [5541].
t (3069) If we put 2a, 2b, for the coefficients of t, t^ [5542], divided by
[5543a] jj^g j.^jj^jg jjj seconds 206265% to reduce them to parts of unity, we shall have,
Vll.iv.§23.] COMPARISON OF THE THEORY WITH OBSERVATION. 643
i being the number of centuries elapsed since the epoch of 1750. This [5543]
secular equation was found by observation, before I discovered the cause
of it by the theory of gravity. It is ascertained, by the comparison of a
great number of eclipses, Avhich were observed by the Chaldeans, Greeks,
and Arabs, that the moon^s mean motion has increased, from the most remote
period to the present day ; and the observed acceleration is very nearly
conformable to the preceding theory. This secular equation is placed beyond
doubt, by Mr. Bouvard, by a profound discussion of the ancient eclipses,
which were known to astronomers ; and also of those he has obtained from
an Arabian manuscript of Ibn Junis.
[5544]
[5544']
We have seen, in [5231], that the sidéral motion of the moon's perigee,
deduced from the preceding theory, differs from its true value, but by a four
hundred and forty-fifth part.* According to the theory, this motion is
subjected to a secular equation equal to — 3,00052.A; ; k being that of the [5545]
0 171793
« = ^^aôëacJ = 0,000000416438 ;
, 0,0000068194
* = 2X206265 = 0,0000000000165307.
[55436]
We also have £'=0,01681.395 [4080 line .3], corresponding to e' [5117] ; hence,
[5542] gives, by neglecting terms of the order t^,
e' = E'—at—ht"~; and, c'^ = E'^-2E'.{at-i-bfi)-\-a^f. [5543^^
Substituting this expression of e'^, in the secular equation [5541], it becomes as in
[5543/] ; whose integral is in [5543o-] ; and, by putting t==z]OO.i [5541', 5543'], we [5543e]
get [5543/-] ;
6v = §m^n .f{2E'. atdt+{2E'h—cv').tHl\ [5M2,f]
= imH.\E'. at^-^{%E' b-W)A ^5543^5
= J.IOO^ nv'n .£'a.ia+|.1003.7?!2n.(§E' h—ia^).i^. \^Uih^
This last expression is easily reduced to the form [5543], by the substitution of the values
of a, b. E' [.55436, c], also m [5117]; and, for n, the motion of the moon in
a Julian year, which is taken for the unit of time in [5541'], making
m«=129577',349 [4077 line 3, 4835]. [5543,-]
(3070) This is erroneously quoted in the original work, as a five hundred and
tbnnrf. [5544a]
sixtieth part
644 THEORY OF THE MOON ; [Mcc. Cél.
moon's mean motion [5232, 5541] ; so that the secular equation of the
[5546] anomaly [5238] is 4,00052. A;, or very nearly four times that of the mean
motion. The preceding equation was discovered by me, by means of the
theory of gravity ; and, I have found, from the theory, that the motion of the
moon^s perigee decreases from age to age ; and, that it is now less, by about
The mo- fifteen centesimal minutes in a century, than in the time of Hipparchiis*
tion of the • i /» i i
moons This result of the theory has been confirmed by the discussion of the ancient
ecieas.ng. ^^j ^j^^ modem observations.
We have seen, in [5233], that the sidéral motion of the nodes of the lunar
^ J orbit, upon the apparent ecliptic, deduced from the preceding analysis, differs
«tuauon from its true value only by a three hundred and fiftieth part. The secular
node. equation of the longitude of the node is, by the same article, equal to
[5549] o,735452.A; [5234, 5541]. This is also confirmed by the ancient eclipses.
24. We shall now consider the periodical inequalities of the moon''s motion
^ ■' in longitude. In order to compare with observation, the preceding results
inequai'i-"' of thc theory, we shall consider, as the result of observation, the coefficients
moon. of the last lunar tables of Mason, and those of the new tables of Burg.
The coefBcients of Mason's tables have been determined by the comparison
of a very great number of Bradley's observations ; and, those of Burg, by
[5550] means of more than three thousand observations of Maskelyne. These
tables have been arranged in a manner, which is quite convenient for
calculation ; so as to diminish the number of the arguments, making them
[5550] depend, the one upon the other. The following is the process for determining,
by Mason's tables, the equations of the moon's true longitude. This method
I have developed, in a series of sines of angles, increasing in proportion
to V .
* (3071) If we put successively, in [5543], { = — 20, i = — 19, we shall find,
that the difference of the two results is 6"" 16'; which represents nearly the acceleration
[5547o] p£ jjjg moon's motion, in a century, since the time corresponding to the mean of these two
values of i, or 1950 years before the epoch of 1750, which is about the time of
Hipparchus. Multiplying the preceding expression by — 3,00052 [5545], we get
[55476]
nearly 19" for the secular decrement of the motion of the perigee j instead of 15',
given by the author in [5547].
Coefficionls
Coefficients
of lîurg'a
of Mason's
Tables.
Tables.
VII.iv.§24.] COMPARISON OF THE THEORY WITH OBSERVATION. 645
We must first compute the following terms, iu which the anomalies are [5550"]
counted from the perigee ;
Tables
of Mason
und Bur^.
— 671',8.... — 66S'',6.sin.(0's mean anom.) 1
— 6%0.... — 8°",9.sin.(2.@'s mean anom.) 2
-j- 53',9. ...-}- 55%9.sin.(2.3>'s mean long. — 2.@ true long.-{-@ mean anom.) 3
-j- 76',5....-|- 75*,3.sin.(2. 3>mean long. — 2. @ true long. — (2) mean anom.) 4
— 57*,8.... — 57',8.sin.(2. Jmean long. — 2.0 true long. -|- 3) mean anom.) 5
-[-4829*,5....-(-4S28^4.sin.(2. 3)meanlong. — 2.0 true long. — Jmean anoin.) [Ejection.] 6
-j- 35',4....-j- 35',0.sin.(4. 5 mean long. — 4.0 true long. — 2. J) mean anom.) 7
-J- 124',G. ...-{- 123*,5.sin.(2. 3)mean long. — ^2.0 true long. — j) mean anom.-|-0mean anom.) 8
-j- 47',6....-)- 46%5.sin.(2. î)mean long. — 2.0 true long. — ]) mean anom. — 0 mean anom.) 9
-j- 39* ,3. ...-j- 42',0.sin.(]) mean anom. — 0 mean anom.) 10 inequali-
ties in tlio
— 21',4.... — 22'',7.sin.(]) mean long. — 0 true long. — 5 mean anom.) 11 i™n'"|'tuj<,,
— 5S^G.... — 57*,4.sin.(2. Jinean long. — 2.0 true long. — 2.]) mean anom.) 12
-|- 62',.5....+ 60%4.sin.(2.mean long, of ]) 's node — 2.0 true long.) (M) 13
-(- 1I',5....4- 17',0.sin.(l> mean long. — 0 true long.-j-0 mean anom.) 14
4- 4',9....-l- 3', I. sin. ( T) mean long. — 0 true long. — 0 mean anom.) 15
— 4*',G.... — 3',7.sin.(2.3)mean long. — 2.0 true long.-)-2. 5 mean anom.) 16
— 10',6.... — 12',4.sin.(4.5mean long. — 4.0 true long. — Draeananom.) 17
— 6',4.... — 6',3.sin.(2. 3) mean long. — 2. mean long. 3) 's node — 2. ^mean anom.) 18
— 8^,8.... — 8%3.sin.(2.raean long. 3)'s node — 2.0 true long.-}- 3) mean anom.) 19
-j- 6',9....-|- 5',3. sin. (2. mean long. 3) 's node — 2.0 true long. — ]) mean anom.) 20
-j- 6%8....-j- 7'',7.sin.(meanlong.3)'s node) 21
_[- 2',6....-j- 0',0.sin.(2. 3)mean long. — 2.0 true long. — 2.0 mean anom.) 22
— 2^,0.... — 0',0.sin.(3)mean long. — 0 true long— }- 3) mean anom.) 23
-J- 2%I. ...-]- 0%0.sin.(3.3)mean anom. — 2.3)mean long.-j-2.0 true long.) 24
-j- 2*,2....-|- 0',0.sin.(2. ]) mean long. — 2.0 true long.-|-3) mean anom.-|-0mean anom.)25
-j- l',3....-j- 0',0.sin.(2. 3)mean long. — 2.0 true long.-(-3) mean anom. — 0meananom.)26
-j- 1»,1....-|- 0',0.sin.(4. 3)mean long. — 4.0truelong. — 3. 3) mean anom.) 27
-\- l',2....-(- 0',0.sin.(2. 3) mean long. — 2.0trHe long. — 2. 3) mean anom.-(-©mean anom.)28
-|- 1*, !....-(- 0',0.sin.( 3) mean long. — 0 true long. — 3) mean anom.-|-0 mean anom.). 29
VOL. 111. 162
[5551]
646 THEORY OF THE MOON; [Méc. Ctl.
Masonand TiiG 811111 of rU tliGse tcrms must be added to the moon's mean anomaly, to
[5j52] which we must also add the function A, given hy the equation,
Coireclion
of the By Cuig. By IMasoii.
àriiy. J = — 1337^30 — 1302%0.sin.(©meananom.) 1
[5553]
— 1P,00 — 14^0.sin.(2.© mean anom.) ; 2
[5553'] ^^^^ ^^^ shall obtain the moon's corrected anomaly, bj means of whicli we
aiwraa'yl' iiiust coiiipute the following terms ;
Burç. Mason.
Equation +22692%2 +22695S3.sin.( 5 corrected anom.) 1
of the
"""'• + 776',4 + 777%0.sin. (2. 5 corrected anom.) 2
[5554] „ (JV)
4- 37%3 + 37%2.sin.(3. 3) corrected anom.) ^ ^ 3
+ 2',0 + 2%0.sin. (4.1) corrected anom.). 4
The sum of the terms in [5551, 5554] must be added to the moon's mean
Fk^f^-' longitude, and we shall obtain the moon's corrected longitude, which must be
corrected -, . i r ii •
longitude, used in computing the toUowing terms ;
Burg. Mason.
— 122 ,1 — 116%4.sin.(3) corrected long. — © true long.) 1
Variation +2141%7 +2141%1 .sin.(2. :j) corrccted long — 2.© true long.) 2
(P)
[5556] + 3%3 + 5%2.sin.(3. 3) corrected long. — 3.© true long.) ^ 3
_j- 7'',3 + 8%8.sin.(4. 3 corrected long. — 4.© true long.). 4
Second We must connect the terms [5556] with the corrected longitude of the
in°ngitude. moon [5555], and thus, form a second corrected longitude, to which we must
add the supplement of the node, or the whole ciicumterence, minus the
longitude of the node. We must also add to it the function B, determined
by the equation,
Burir. Mason.
Correction
^l^^" B = + 540',0 +552%0.sin.(© corrected anom.) ;
[5558]
[5558] and we shall obtain the moonh distance from the corrected node. We must
subtract the moon's corrected anomaly from the double of this distance, and
[5559] multiply the sine of this argument by —84^4, according to Burg ; or, by
— 84',1, according to Mason; and we shall get another inequality, which
VIT. iv.s^ai.] COMPARISON OF THE THEORY WITH OBSERVATION.
647
we must add to the inequalities [5551,5554,5556]. Lastly, we must add
the same inequality to the preceding distance of the moon from the corrected
node, in order to form the argument of latitude ; and, we must multiply the
sine of double this argument by — 406',8, according to Burg, or, by
— 407',7, according to Mason , and we shall obtain the inequality called the
reduction to the ecliptic ; which must be added to the preceding inequalities,
to obtain the longitude of the moon, counted from the mean vernal
equinox. We must here observe, that the 7nean longitudes of the moon,
of its node, and of its mean anomaly, must he corrected for the seciilar
inequalities.
From this process I have deduced the following expression of the
periodical inequalities of the moon's mean longitude, developed in terms of
the true longitude, counted upon the ecliptic. This development requires
particular attention, to prevent the omission of any sensible term.* We
Tables of
Mason and
Butg.
[55G0]
Argument
oflatitude
|.5r.Gii
[5562]
Reduction
to the
ecliptic.
[5563
[5CG4]
* (3072) We shall here point out the general principles of the method of developing
the functions [5551 — 557.3], in the forms given in [5574 — 5579], without entering into
any minute numerical details, which would be inconsistent with the limits of the present
work. In the first place, we shall show how the functions [5551, &:c.], or the expression
of the true longitude, maybe reduced, so as to depend wholly on the mean motions
nt-{-s, n't-\-i , &1C. ; noticing the secular inequalities, as in [5563], but omitting any
particular reference to them in the present note ; and then, by inverting the series, we can
obtain the expression of the mean longitude nt-{-s, in terms of the true longitude v, so as to
conform to the present theory [5095]. Several of the functions in the table [5551] do
not require any reductions; as, for example, those in [5551 lines 1, 2, 10, &ic.], which
depend on the mean motions ; but, in those inequalities which contain the sun's true
longitude, we must substitute its value, deduced from [668], by accenting the symbols [5564t/]
V, e, &.C. to conform to the notation used in this theory [4779']. Hence we have,
Sun's true longitude v'^ sun's mean longitude («'/-j-s')-}-e' ;
e' being used for brevity, to denote the periodical terms of the values of v [668], or those
which depend on coefficients, containing the excentricity e' and its powers, multiplied by
sines of the periodical angles ; and, it may be represented in the following manner ;
e'= 2a'.sin.(îVi+p').
Now, if we put a, for the coefficient of any one of the inequalities [5551] ; T', for the
part of the argument which depends on the mean motions ; and ze', for the pari of the
[5564a]
[55C46]
[5564c]
[5564e]
[5564/1
[55G4g-]
[5564A]
[5564i]
648
THEORY OF THE MOON ;
[Méc. Cél.
have neglected those inequalities which are less than a centesimal second, or
0',324. A part of the inequalities of this expression arise merely from the
[5564'] development of the formula, corresponding to the process in Mason's tables,
Tables of
Mason and
Burg.
same argument, depending on e' ; it becomes of the same form as in the first member of
[5564fe] [5564Z]. Developing this, by [21] Int., we get the second member of [5564/] ; and, by
substituting the values of s'ln.ie', cos.ie', deduced from [43,44] Int., we obtain [5564»i] ;
[5504/] o.sin.(T'+te') = a.cos.ie'.sin.T'+a.sin.ie'.cos.T'
[5564m] = a. \ l—^.r-e' S-j-^L.i^e'^— &c. | .sin. T'+a. {{e'—li^^+hc. \ .cos. T.
Substituting, in this last expression, the value of e' ^= lo.' .s\n.{i' n't~\-p') [5564A], and its
powers ; then reducing the products, by means of [17 — 20] Int., we finally get the value
[5564n] of a.sm.(T'-{-ie'), under the form of a series of terms, depending exclusively on the
mean motions ; and the whole function [5551] may be included in a general expression of
the form,
2a. sin. (p< + 7) ;
[5564o]
[5564;)]
[5564g]
[5564r]
[5564«]
[5564<]
[5564u]
[5564u']
in which the angles depend wholly on the mean motions. If we substitute this, in the
expression of the moon's corrected anomaly [5553'], we get,
3) 's corrected anomaly = 3) 's mean anomaly -J- function [5553] -|- 2 a . s\n.(pt-\-y).
The sine of this expression, or the sine of any multiple of it i, which occurs in [5554],
may be developed, as in the general formula [5564?, m], by putting,
T = i.(3)'s mean anomaly) ; ie'==i.| function [5553]+2a.sin.(p<+y)}.
By this means the function [5554] may be made to depend on the ?ne«« motions ; therefore,
the corrcc^ec? longitude of the moon [5555] will also be given in terms of the mean motions.
Substituting these in [5556], and reducing, by a similar process to that we have used, we
get, as in [5556'], the moon's longitude iivice corrected ; whence, by using B [5558],
we easily obtain the corrected distance from the node [5558'], which gives the correction
[5559]. In like manner, we get the reduction [5562] ; and, finally, obtain the true
loni-itude v, expressed in terms depending on the mcara motions; and, if we denote the
mean longitude by nt-\-s = T, the expression of the true longitude v, may be put
under the general form,
v=zT+J..B.sm.{iT-\-y);
in which the angles {iT-\-y) correspond to the mean motions.
This last formula may be inverted, by means of La Grange's theorem [629c], which, by
changing -^a; into x, then x into T, and t into v, becomes.
Vll.iv. <5-24.] COMPARISON OF THE THEORY WITH OBSERVATION. 649
which we have just explauied ; so that they cannot be considered as the
result of observation. To distinguish the different inequalities, we have
marked with an asterisk those computed by Mason, by the comparison of [5565]
Bradley's observations, and which have all been again determined by Burg
by means of a very great number of Maskelyne's observations. We shall ^^
commence ivith the great inequalilij of the first order ; and then shall give, '°"°""'
successively, the five inequalities of the second order, the fifteen inequalities of
the third order, and all the inequalities of the foxirth and of higher orders,
1
Inequali-
in tlio
mooii*a
icle.
T:=v-\-F{T); or v=T—F{T); [5564«]
Comparing together the values of v [5564m, «], we get,
2B.sin.(iT+7)=— F(r); whence, F{v) ^—SB. sin. (iv+j). [5564x]
Substituting this last expression in [5564 w], and making the necessary reductions, we
finally obtain the values of T, or nt-{-s, under the following form ;
nt-{-s = u+2 C.sm.{iv-{-p) ; [5564y]
which is the same as in [5096i], neglecting, as in [55646], the consideration of the
secular inequalities. This corresponds with the results in [5574 — 5579].
A similar process must be used, in reducing the expressions of the latitude [5595] to the
form [5596] ; or, that of the horizontal parallax [5603] to the form [5605]. There are [5564z]
no other difficulties in performing these operations, than those which arise from the great
length of the calculations, in consequence of the numerous equations, which require
attention, in order to procure accurate results.
In applying the formula [5564m] to most of the small inequalities in [5551, &ic.], we
may neglect the square and higher powers of e'. For, e' is nearly equal to -j-'jf [5565a]
[5117 line 4]; hence we have ae'^ = -—-; and, if tt<^100*, as is the case with
twenty-six out of twenty-nine of the inequalities in the table [5551], it becomes [55656]
ae'^<^0',0.3, which is insensible. Moreover, in the equations which do not exceed
12^ [5521 lines 2, 14— 29], we have «e' = j^ < 0')2 ; and the coefficient of the
r5565c1
corresponding term of ae'.cos.T' [5564m] is so small, that it may be frequently neglected ; '■ '
and then we may put simply a.sin.T', for a.sm.{T'-{-ie').
VOL. III. 163
650
THEORY OF THE MOON
[Méc.Cél
[5566']
[5567]
which have been compared ivith observations ; lastly, all the other inequalities.
We shall place, in the second column, the results of this analysis; and, in the
third column, the excess of the numbers in the second column above those in
the first. In the fourth column, Ave shall give the excess of the coefficients of
Burg's new tables, reduced to the same form as in this theory, over those of
Mason's tables in the first column. Burg retains, in his tables, the same
forms of the arguments as in Mason's tables, which had been adopted from
the tables of Mayer. It will be sufficiently accurate, in reducing Burg's
tables to the forms of the present theory, to apply to the coefficients of
Mason's tables, thus reduced, as in the first column, the difiference of the
corresponding inequalities in the two primitive tables, taken with a contrary
sign.* The functions A, B [5553,5558], difler a little in these two
tables, and we have noticed this difference. We may also remark, on this
point, that, by introducing in the primitive tables, an inequality in the
longitude, depending on
[5568] sin.(3)mean anom.+ © mean anom.) ;
and, in the latitude, an inequality, depending on
[5569] sin. (argument of lat.+ © mean anom.) ;
and, making the necessary changes in the coefficients of the inequalities,
depending on
[5567']
[5567a]
* (3073) If we suppose, that the equation [5564m] corresponds to Mason's tables ; and,
that, in Burg's tables, one of the coefficients B, is changed into B-\-<iB; it will
increase the second member of the ecpation [5564j«] by the quantity Œ.sm.[iT-\-y'),
which is very nearly equal to &B.sm.{iv-\-j). Transposing this to the first member of
'- •' the same equation, we find, that the equation [5564m], corresponding to Burg's tables,
becomes,
v—Œ.sm.{iv-Yy) =T-\-J.B.Ûn.{iT-\-y) ;
[5567c]
[5567rf]
[5567e]
which may be derived from that of Mason [5564m], by merely changing v into
D — Œ.sm.{iv-\-y) ; and, if we make the same change in [5564y], which results from
Mason's tables, we get, for Burg's tables, the following expression ;
nt-\-B = V — 5B.sm.(iv-\-'y)-\-X C.sin.(t«-|-|3).
This agrees with the remarks in [5567'].
Vll.iv4'24.] COMPARISON OF THE THEORY WITH OBSERVATION.
6Ô1
and, on
sin.(I> mean anoni. — © mean anom.);
sin. (argument of lat. — © mean anom.),
[5570]
[5571]
we can dispense with the functions A and B ; which will give to the tables r^^^y-^
a greater degree of uniformity.* Burg has introduced in his tables of the
* (3074) If we put, for a moment, the sun's mean anomaly equal to s, and the
moon's mean anomaly, corrected for the equations [5551], equal to m ; we shall have
m-\-A for the moon's corrected anomaly, which is to be used in the formulas [5554].
Now, if we put C^ 22692%2, we find, that the first, or chief term of [5554], becomes
as in the first member of [5571c]; and, by development, using [21,43,44] Int., we get,
successively, the expressions in the second members of [5571c, d] ;
Csln.(»i-f"^'^) ^^ C.cos.A.sin.m-\-C.s\n.A.cos.m
= C.\l—iA^+^\A*—&ic.l.sm.m-\-C.\A—iA^+hc.\.cos.m.
This last expression may be considerably simplified, by observing, that the chief term of
A [5553 line I], expressed in parts of the radius, gives, very nearly,
^ = — 0,006.sin.s ; hence i^2^0,000018.sin.25 = 0,000009— 0,000009.cos.2«;
and i^2c=0',2— O',2.cos.2s.
This last expression, being multiplied by sln.m, becomes insensible ; consequently, the
equation [5571(7] may be put under the form,
Csin.(w+^) = C.sin.m-f-C/J.cos.m.
If we suppose ^'=1337',3, ^"=ll',0, the expression of A [5553] becomes,
A = — yl'.sin.5 — A".s\n.'2s ;
substituting this in [5571/], and reducing by [18] Int., we obtain,
Csin.(m+^)^ C.sin.7?j — lA' C.js\n.(7n-\-s) — sin.(?K — s)^ J
—iA"C.{sin.{m-i-2s)—sm.{m—2s) ] . 2
The terms in the second line of this equation maybe neglected; for, iC [5571J],
expressed in parts of the radius, is nearly equal to xV) and A" =11' [5571^] ; hence,
^A"C = 0',6; which is nearly insensible, especially when multiplied by sin.(7n±2s) ;
therefore, the expression [557 li] becomes,
C.sin.(7n+^) = C.s'mjn—^A' C.sin.{m-]-s)-\-iA' C.sm.{m—s).
[5571a]
[5.57 li]
[5571c]
[.5571 (/]
[5571e]
[5571/]
[5571^]
[5571ft]
[5571i]
[.5571 A]
[557 J i]
[5571m]
652
THEORY OF THE MOON
[Méc. Cél,
[5572]
[5573]
Inequali-
ties in the
moon's
longitude.
motion in longitude, eight new inequalities, which are not given in the reduced
tables of Mason, except by their development. We have distinguished them
by a double asterisk. Lastly, he has compared with observation, several
inequalities, which he has found to be insensible; so that their coefficients,
given by the development of Mason's tables, may now be considered as the
results of observation ; we have distinguished these by a triple asterisk.
We may thus know, by mere inspection, the inequalities which yet remain to
be compared with observation. The differences between the two taljles
being small, enables us to deduce the development of the one from that of
the other ; and we may, by the inverse method, reduce the inequalities of
this theory to the form of Mayer's tables.
(Col.l.)
Inequalities
deduced rioin
Mason's tables.
(Col. 2.)
(Col. 3.)
(Col. 4.)
Coefficients
Kxcesg of tliuge
Excess of the
of this
coefficicn;s over
coefficients of
theory.
those of Mason's
Burg's tables over
tables.
those of Mason.
Inequality of the first order.
[.5574] — 22677%5 . sin. (cv—^)* — 22677',5 .... +0',0 . . . . + 3',1
[5571ji]
[5571o]
[5571;)]
[5571çl
The terms of this equation, depending on the arguments m±s, are as in [5568,5570].
The substitution of the values of the multiples of m-\-A, in [555-1 lines 2 — 4], produces
only some small, or insensible inequalities. The function B [5558] being small, its effects
on the equations [5559, 5560] are nearly insensible ; but, they might be noticed, in a
similar manner to that in [5571ot,&;c.].
In like manner, if we suppose the argument of the latitude to be represented by
m'-\-B, and the coefficient of the first term of the expression of the latitude by C" ; so
that the term itself becomes C'.sm.{m'-\-B) [5595 line 1]; we may develop it in the
same form as in [5571i] ; namely,
C. sin.(w!'+B) = C'.sin.m'— ABC. \ sin.(TO'+s)— sin.(w'— s) \ ;
in which the terms depending on the angles m'±5, are as in [.55G9, 5571]. The e fleet
of the rest of the terms depending on B, is so small, that they are hardly deserving of
notice.
* (.3075) The author remarks in a note upon this part of the work, that the coeflîcient
[5574o] of the inequality [5574], is one of the arbitrary terms of the theory, and he has thought
it best to adopt the result of Burg.
Vll.iv. §24.] COMPARISON OF THE THEORY WITH OBSERVATION.
653
(Col. 1.)
Inequnlilies
(lcduce-1 from
Masoii*s tables.
(CoI.i2 )
(Col. 3.)
(Col. 4.)
Coeflicionts
Excess of these
Excess of the
of this
coetriciente over
cooJficii'iits of
theory.
those of Muson's
Burg's tables over
tables.
those of Mason.
Tables
of Mason
and Burg.
Inequalities of the second order.
4- 462%5.sin.(2c»— 2ro)* + 467',4 . . .
—\903',4.sm.{2v—2mv)* — 1897%4 . . .
— 4681^5.sin.(2v— 27»!'— ci'+to)* — 4G85'',5 . . .
+ 672',5.sin.(cW— s:')* + 6S2',4 . . .
+ 407',l.sin.(2_g-u— 2Ô)* + 40G',9 . . .
Inequalities of the third order.
— 10*,7.sin.(3cD— Sîi)* — l r,4 . . .
+ 6V,l.sm.{2gv—cv—2ê-{-iz)* + 66',4 . . .
— 22',4.sin.(2^v+ct)-2â— Tï)*** _ 23',0 . . .
4-14G^0.sin.(2y— 2mu-f-cw— a)* +147%0 . . .
+ I4^5.sin.(21;— 2mi;+c'?ni'— in')* -{- 13%6 . . .
— 136',5.sin.(2« — 2mv — c'tod+ct')* — 134',5 . . .
-(- 2P,7.sin.(2u — 2mv — cv-{-c'mv-\--ui—ui')*. . . . + 24%3 . . .
— 20b',S.sm.(2v—2mv—cv — c'mv-j-u-\-a')*. . . . — 205%8 . . .
+ 68',6.sin.(r«4-c'mîJ — z-.—a')* -f 7P,0 . . .
— l\.6%8.sin.{cv—cfmv — ra-(-a')* — 117'',3 . . .
-f 178^6.sin.(2cj; — 2v+2mv—2-a)* -}-IC9%l . . .
4- 55\S.sm.{2gv—2v+2mv—2é)* -f 56',6 . . .
-\- 6',9.sm.{2dmv—2zi')* -}- 10', 1 . . .
-|-116',7.sin.(t; — mv)* +122',0.(l-f-
— 19',0.sin.(i; — mv-\-c'mv — zi')* — IS',8.(I +
+4',9 .
+G%0 .
—4^0 .
+9',9 .
—0^2 . ,
— 0',7
+ 5',3
— 0',6
+1%0
— 0^,9
+2',0
+2%6
— 0%0
+2^4
—0^5
— 9^5
+œ,8
+r,2
+0',6 1
— 0',6 2
— l',I 3
+3',2 4
— 0»-,9 5
)■
—0,1 1
+0%3 2
+0',0 3
+0',0 4
+2%0 5
—V,2 6
— l',l 7
— V,l 8
+ 1VJ 9
+0',8 10
— P,2 11
+2%! 12
— 2^9 13
+5%7 14
+5',5 15
[5575]
Inequali-
ties in
tlie moon's
longitude
reduced to
the form
of the
present
theory.
[5576]
VOL. III.
164
654 THEORY OF THE MOON ; [Méc. Cél.
Table, of (Col.l.) (Col. 2.) (Col. 3.) (Col. 4.)
Mason and , ,. .
gyp„^ Inequalities
° ili)rlii<>iii1 fm
[5577]
deduced from
Mason's tables.
Coefficients
Excess of these
Excess of the
of this
coefficients over
coefficients of
theory.
those of Mason's
Burg's tables over
tables.
those of MasoD.
Inequalities of the fourth order, and of higher orders, which have been
compared loith observations.
— 0',3.sin.(4cy— 4îï)* +0^0 1
— 2',0.sm.[2gv—2cv—2ô-\-2Ts)* 4-0'',l 2
+ l^l.sm.igv—v—ê)* + ô^G .... — 2',1 . .
— 7%0.sin.(3!; — 3mv)*
-(- 5'',7.sin.(4i; — 4mv)*
4- 0%S.sin.(ct)+2c'ff8V — -m — 2«')*
Inequali'
ties in the _ 0',8.sin.(CT— 2c'm«— «+2^')*
raoon'a
longitude
reduced to
the form
of the
present
theory.
— 0%9
3
+ 1%9
4
+P,5
5
— 0',2
6
+0',2
7
+0',9
8
+ 1'=8
9
[5578]
— S',9.s\n.{2cv-]-2v—2mv—2'a)* — S',1 .... -}-0',S . ,
+28%9.sin.(4»— 4??iî;— c«+ra)* +33%4 +4%5 . .
+15',2.sin.(4D— 4mu— 2c«+2t3)* +15',5 +0',3 — 0",4 10
—n',0.am.{cv—v-{-mv—^)* — 8',3.(14-i) +1",3 11
l',l.sin.(» — mv — c'mv+vi')* — l',S 12
-f Ç)\b.sm.{2v—2mv—2gv+cv-{-2ô—-a)* + 8',7 — 0',8 +0',5 13
-f l',2.sm.{2gv+cv~2v-}-2mv—2ê—zs)* 4-l',G 14
— '3',ô.sm.(2v—2mv—2cfmv+2i^')** — 2',6 15
_ 5%9.sin.(CT+«— 7««— ra)** — 5-,0.(l+i) -f2%G 16
+ l',0.sm.(3cv—2v+2mv—2-a)** — 2',1 17
4- 0',6.sin.(2« — 2mi>+CT-|-c'wi'— « — to')** — 2',2 18
+ 12',8.sm.{2î)— 2mv+CT— c'nu-— «+to')** +10%2 — 2',6 — 1',3 19
+ 0%S.sm.{4v—4mv—3cv—Qzs)** — 1»,I 20
4- l',0.sm.{2cv—2v-\-2mv—c'mv—2-us-{-vi')** — 0',2 .... — P,2 .... +1',2 21
-|- V,3.s\n.{cv—v-^mv—cmv — •n+a')** +!',! 22
-I- G',4.sm.{2cv—2v+2mv-\-cmv—2-!S—m')***. . . + 5',9 — 0',5 _ 23
l',2.sin.(4D— 4mt)4-ct;— w)*** 24
-j- 0',2.sin.(4c« — 4v-\-4:mv — 4ra)*** 25
— 3",9.sm.(2D— 2mt)+2^jJ— 2^)*** 26
± V,l.sm.(2gv±c'mv—2ê=P-!s')*** 27
— Û',3.sin.(2gj;+2ci)— 2D+2mi)— 2é— 2a)*** 28
± 2%0.sm.{2gv—2v-\-2mv±cmv—2ê^zi')***. 29
VII.iv.§24.] COMPARISON OF THE THEORY WITH OBSERVATION.
655
(Col. 1.)
Inoqualities
deduced from
Mason's lables.
(Col. 2.)
(^oofficiollts
ol' this
tlieory.
(Col. 3.)
Excess of these
coefficients over
ttiose of Mason's
tables.
Tables of
Mason and
Buig.
Inequalities of the fourth order, and of a higher order, deduced from Mason's
tables, which have not been compared loith observations.
-{-5',0.s\n.{2cv—cmv—2^-\-^')
—2',8.sm.{2cv+c'mv—2zi—-!s')
-|-4%7.sin.(4i; — 4mv—cv — c'mv-\-zi-{-^')
— 4',5.sin.(2« — 2mv-\-2gv—cv — 2d+w)
— 0',4.sin.(2v — 2mv — 2^i'-f-2cw+2<)— 2ra)
-|-l'',9.sin.(4w — 4mu — 2cv-{-c'mv-{-2zs — ^')
-|-l»,6.sin.(4w — 4m« — 2cv — c'mv-{-2-ss-\-zi')
— l'',2.sin.(3î)— 3mD— ci'+w)
+0',8.sm.{4gv—4è)
4-3',0.sin.(2i; — 2mv—cv-\-2c'mv-\-Ta — 2ct')
— 5',8.sin.(2t' — 2mv — cv — 2c'mv-\-ui-\-2zi')
-|-0',5.sin.(4i' — 4mv — 2gv — fy+2â-[-ra)
+4',5
-0",5
-0',4
-. igv-cv+2é+^)
0',5.sin.(4i' — 4mt) — c
imv — 3cv-\-3t!s)
■c'mv+zi')
-]-0',7.sin. (6v — 6mu — oti;-t-ora
•0',4.sin.(cD — v-\-mv-\-c!mv —
n> Q o;,, ^1,. Am„J-^'„,„ -,'
-|-0',3.sin.(4D — 4mv-\-c'mv —
-')
0-
1
2
3
4
5
G
7
8
9
10
11
12
13
14
15
16
Inequali-
ties in ihe
moon's
longitude
reduced to
the form
of the
present
theory.
[5579]
We see by this table, that the greatest difference between the coefficients
of Mason's tables and those of the theory, is 9%9 ; and, there is only 8%3 rsggoi
between the theory and Burg's tables. We might make this difference vanish
by carrying on the approximations to terms of a higher order ; but, the
preceding comparison is sufficient to establish incontestibhj, that the general [5580']
laio of gravitation is the only cause of all the moon's inequalities.
Two of these inequalities, on account of their importance, must be
determined with particular care. The first is that lohich is called the [558I]
parallactic inequality, whose argument is v — mv. It depends on the Paraiiae-
sua's parallax. It has been determined by carrying on the approximation to <i™i"j-
quantities of the fitth order inclusively ; so that we have reason to suppose.
656 THEORY OF THE MOON ; [Méc. Cél.
b
that the value whicli we have obtained, is very accurate. According
to Mason's tables, reduced to the form of the present theory, this
[5582] inequality is equal to 116',7 [5576 line 14] ; but Burg, who has determined
it by the comparison of a very great number of observations, finds it to be
[5583] greater by 5',7 [5576] ; therefore, it is equal to 122',4.* Putting this
[5584] last result equal to the coefficient (l+i)A22%0, which is given by the
theory in [5220 line 21], we obtain,t
* (3076) In ihe Monatliche Correspondenz, vol. 28, page 101, is given an extract of
a letter from Burckhardt, containing some remarks on the effect of an erroneous estimate
[5583a] of the moon's semi-diameter, in determining the value of the coefficient of the parallactic
inequality. The usual method of determining the moon's place, by observation, is, by
ascertaining the difference between the time of the transit of the moon's enlightened limb,
over the meridian, and that of some well known fixed star. In tliis method, the moon's
we«<e™ limb is observed, when the angle v—mv is less than 180'', or sin.(y — mv)
is positive; but, the eastern limb is observed, when v — mv exceeds ISO"*, or
[5583i] gjj, /^ ,^jy) ig negative. Now, it is evident, that, if there be an error in the estimated
value of the moon's semi-diameter, and, that it be taken, for example, too great by P, the
longitude of the moon's centre, resulting from this observation, will be increasedhy nearly
the same quantity, when sin.(v — mv) is positive, and decreased, when sm.{v — mv) is
[5583c] nerrative ; consequently, the error of the moon's longitude, arising from this source, will
always have the sa7ne sign as the parallactic inequality, and it will be impossible to
separate these two quantities. From this we easily perceive, that it is of great importance,
[5583rf] j^ ascertaining the coefficient of the parallactic inequality, to have the moon's semi-
diameter, to the utmost degree of accuracy. Burckhardt supposes, that it is owing, in
[5583e] g^^^^^ measure, to this circumstance, that Mayer's first estimate, given in his lunar theory,
which was published by the Commissioners of Longitude of Great Britain, in 1757,
[5583/] j^f^^gg tiiis coefficient only 115' ; being less by 7',4, than the late accurate determination
of Burg.
[5584a]
1(3077) We have, in [5584], (l+i).122%0 = 122^4. Dividing this by 122',0,
we get 1+i = 1,003 nearly, as in [5585] ; the slight difference arises from the use of
the centesimal division to two or three more places of decimals; hence, [5221] becomes.
a 1,003 . rrco/'l
[55841] â'^^ÔT' ^' '" [^^^^]-
[ 5584c]
Now, the moon's mean horizontal parallax is nearly - [5309] ; and, in like manner,
the sun's horizontal parallax is,
VII. iv.<5.24.] COxMPARISON OF THE THEORY WITH OBSERVATION.
657
therefore,
l + t= 1,002985;
a _ 1,00-2985
n' ~~ ""400"" ■
Now, the sun's parallax is —, or — .-: therefore, it may be represented
a' a a
by,
D 1,002985
= sun's parallax.
Substituting for
D
its value
0,01655101
[5329], we get 8',56 for
the sun's mean parallax upon the parallel, in which the square of the sine of
the latitude is ^ ; which is very nearly the same as has been found by
astronomers, from the last transit of Venus [5589^, A;]. Hence it appears,
that the hinar theory furnishes a very accurate method of determining the
sunh parallax.*
[5585]
[5586]
[5587]
[5588]
Uetermin-
ation of
[5589]
the sun's
parallax,
by means
of this
Junnr iri-
equatity.
[5589']
D
a''
D a
a a'
D 1,003
a 400
, as in [5588], nearly.
D
[5584rf]
Substituting in this, the value of — [5.329], and, multiplying by the radius in seconds
206265', it becomes 8',56, as in [5489].
* (.3078) We may observe, that the author, in vol.5 [12737], states the well known
fact, that this method of determining the sun's parallax, by means of the parallactic [ssggai
inequality, was given by Mayer, in page 50 of his lunar theory [5583e], almost fifty
years before the first publication of this volume of the Mécanique Céleste. According to
Mayer's calculations, from the theory, the sun's parallax lO'.S corresponds to a
parallactic coefficient of 158'',6 ; consequently, the parallax 8', 56 corresponds to
125',7, instead of 122',4, wiiich is used by La Place. Mayer supposes this coefficient
to be, by observation, only 115% corresponding to the parallax 7'',8. If he had
used the same coefficient 122", 4 as La Place, the result of his theory would make the
parallax 8',3 ; which differs but little from the truth ; and proves, that Mayer had
carried on his approximations to a considerable degree of accuracy, in computing the value
of this inequality by the theory.
Before closing this note we may remark, that Messrs. Carlini and Plana have given, in
Zach's Correspondance Astronomique, for the year 1320, page 26, a calculation of the
VOL. III. 165
[55895]
[5589c]
[5589rf]
658 THEORY OF THE MOON ; [Méc. Cél.
[5590]
The second inequality is that which depends on the longitude of the node oj
the lunar orbit, or, on the argument gv — v — ('. Its coefficient, according
to Mason, is 7,7 [6578 line 3]; but Burg, who has just determined it,
[5590'] by a very great number of observations, reduces it to 6%8 [5578 line 3].
[5591] The theory gives 5',552 [5390], if Ave suppose the earth's oblatencss to
[5592] be -^\j ; or, 1P,499 if the oblateness be -^j^-^. Hence it is evident, that
[5593] Burg's computation corresponds to the oblateness* ■j-ôi,-âT' This inequality
Oblateness is determined ivith great precision by the theory : and, we have no reason to
e°nrth! suppose, that there is, with respect to it, the same degree of uncertainty,
[5589e] sun's parallax, b)' this method, making it 8',719. On the other hand, La Place, in reviewing
his calculations, in a paper presented to the Board of Longitude oJ France, January 19,
[5589/] 1820, and printed in the Connaissance des Terns, for 1823, page 230, makes it S',65.
In his fiftl) and last edition of the Système chi Monde, page 230, he finally adopts the value
26",58 = 8',6L This differs but very little from the value 8',62 given by Burg, in a
'■ ^ late investigation, published in 1826, in vol. iv. page 24, of Schumacher's Astronomische
Nachrichten. Finally, we may remark, tliat these results difler but very little from those
[5589ft] which are obtained from the transits of Venus in 1761 and 1769. These observations
have been lately discussed with great care, by Encke, using the most approved tables ;
and, in a work entitled, "Die Entfernung der Sonne von der Erde aus dcm Venusdurchgange
[5589i] ^Q„ 1761^" page 143, he gives the parallax from 8',43 to 8',55 ; by combining, in
the best manner, the diflerent observations of the transit of 1761. The results of the
[55894] observations of the transit of 1769, are given from 8',56 to 8°,65, in vol. iv. page 25,
of Schumacher's Astronomische Nachrichten; and the final result 8', 5776 is used in
computing the Nautical Almanac for 1834. In conclusion, we may observe, that, in the
year 1763, a work was published by Doctor Matthew Stewart, entitled, " The Distance
'^"' ■' of the Sun from the Earth Determined by the Tlicory of Gravity, ^c," by means of
the observed motion of the moon's apsides. This method, though it has been approved
by Horsley, Play fair, Hutton, and others, is essentially erroneous and defective ; as is
[5589jn] shown in a jjaper presented by me to the American Academy of Arts and Sciences, and
published in the fourth volume of the first series of their Memoirs.
* (3079) This is easily deduced from the formula [5390e], by substituting ^'^6',8
[5590', 5390a— e], which gives,
6%8-|-7',6 1
L5593a] ''-P = 4392',6 = 3Ô5;Ô5' "^'"■'^-
This result is finally retained by the author, in page 230 of the fifth edition of his Système
du Monde.
VII.iv.§25.j COMPARISON OF THE THEORY WITH OBSERVATION. 659
which prevails in most of the other coefficients of the lunar theory, by reason [5593]
of the slow convergency of the approximations. As this inequality is aucfn™'""
oftlic
proportional to the oblateness of the earth, it deserves the greatest attention Jii^'^^^"""''
of astronomers. It follon-s incontestibly, from the values assigned to it by bemoan?
Mason and Burg, that the earth is less fattened, than in the case of i;;X"'''''
homogeneity. This is conformable to what has been deduced from other
phenomena, in books iii., iv., v.
longitude
[5594]
25. We shall now consider the moon'' s motion in latitude. It is found
by the tables in the following manner. If we call the moon\s corrected
longitude, the mean longitude added to all the inequalities, except the
inequality of the reduction, we shall find that the moon's latitude is represented
by the folloiving expression ;
+ 18520',8 ... +18524%5 . sin. (argument of latitude) 1
— 5%0 ... — 4',4 .sin. (3. argument of latitude) 2
+ 528',4... + 528',4. sin. (2 3) corrected long. — 2© true long. — arg. lat.) 3
4- n%6 ... + 17%6 . sin.(arg. lat. — 3) mean anom.) 5
-j- 25',1 ... -f- 2.5',1 .sin.(25mean anom.— arg.Iat.) 6
[Moon's latitude.]
-\- r,9 ... + r,9. sin.(.33)mean anom. — arg. lat.) 7
+ 9'',0 ...+ 9',0 .sin.(2Dcorr.long.-2©truelong.-arg.lat.+©niean anom.) 8
+ 3',7 ... + S",? .sin. (23)corr.long.-2©true long.-arg.lat.-©mean anom.) 9
-f 2',2 ... + 2",2 .sin. (2Dcorr.long.-2@truelong.-arg. lat. + I»mean anom. )10
+ 15'.9 . . . + 15',9 .sin.(arg.lat.-f-3)mean anom.-2l)corr.lon.-|-2©true long.)ll
-f- 5',2 . . . + 5',2.sin.(arg.lat.-|-2])meananom-23)cor-lon.-t-2©truelong.)I2
— 8',0 ... — 0',0. sin. (3) corrected longitude). 13
Reducing these formulas to sines of angles, which vary in proportion to v,
we obtain the following results ;
[5594']
Tables
of the
moon'ê
— f3,l ... — o jl .sin.(arg. lal. — © mean anoni.) 4 i^^tiiude,
' by Burg
and
Mason.
[5595]
660
THEORY OF THE MOON ;
[Méc. Cél.
(Col. 10
Inequalities
deduced from
Mason's tables.
iCol. 9.)
Coefficients
of this
theorj.
Inequali-
ties in the
moon's
latitude,
reduced to
the form
of the
present
theory.
[5596]
(Col. 3.)
Excess of tliese
coefficienis over
those of Mttson*s
tahlea.
(Col. 4.)
Excess of the
coefficients of
Burg's lahlcs over
those of Mason.
18543',9.sin.(g'«— «)* 18542%8 .
+ 13%9.sin.(3^«— 3^)* + 12%6 .
-{-52T,2.sm.{2v—2mv—gv-{-^)* +525',2
+ 0',7.sin.(2t)— 2mî;+^u— 0) + l',l
— 4%l.s\n.{gv-\-cv—ê—zr)* — 5',6
+ 19%8.sin.(gD— cu-ô+zs)* + 19',8
-f 2V,l.sm.{gv-\-cv—2v-\-2mv—ê—z:s)* + 2r,6
— 0',8.s\n.{2v—2mv-{-gv—cv—ê-}-a) — r,4
+ 6%0.sin.(2i;— 2m«— ^y+cj;+é— «)* + 6%5
+ 24\8.sm.(gv-{-c'mv—ê—zs')* + 24%3
— 2T,9.sm.{gv—c'mv—ê+zs')* — 25%9
— 9\5.s\n.(2v—2mv—gv+c'mv-\-ê—a')*. ... — 10',2
+ 22',2.sm.{2v—2mv—gv—c'7nv-^ê-\-z!')*. . . • + 22%4
-j- 25',7.sin.(2cD— g-i'— 2î3+â)* + 27',4
+ 4%3.5m-(2cv-{-gv—2v+2mv—2zi—ê)* . . . + 5',1
_ 0',9.sin.(3c«— ^v— 3ra+â)* -0',0 16
+ r,0.sm.{3gv—2v+2mv—3ê) +0%0 17
-|- 0%4.sin.(4î)— 4mï— ^t)+fl) +0%0 18
-I- (y,6.sm.{3cv—gv—2v+2mv—3-^+ê) +0',0 19
± 0', 6. sin. {cv'\-gv — 2v-{-2mv±:c'mv—zi—ê^-^) +0',0 20
T 0%6.sin.(2ci;+^« — 2v-{-2nv±:Cv—2ia—ê^-!:s) 21
+ 0',9.sm.{4v—4m,v—gv — ct+^+to) +0%0 22
— 0»,0.sin.(3)'s true longitude)** — 6',5 — 8%0 23
Here the theory agrees better with observation, than it does in the case
relative to the moon's motion in longitude. This happens, in consequence
— IM .
— r,3 .
, — 2',0
, +0',4
— 1%5
• +0',0
. — 0',1
. — 0%6
. +œ,5
. — 0',5
. +2^o
. — 0%7
• +0',2
• +1%7
. +0',8
— 3',7* 1
— 0-,6 2
-|-0^0 3
+ 0^0 4
+0'',0 5
+0',0 6
4-0^0 7
+0'',0 8
+0',0 9
— 0',5 10
+0'',5 11
— 0',0 12
-[-0',0 13
+0",0 14
-t-0%0 15
* (3080) In a note on this table, the author remarks, that the coefficient of the
inequality [5596 line 1], is one of the arbitrary quantities of the theory; and, that he
[5.596a] gj^gg jj^g preference to the result of Burg's calculation.
VII.iv.>5.-25.] COMPARISON OF THE THEORY WITH OBSERVATION. 661
[5596']
of the greater simplicity in the approximations of the motions in latitude,
which renders the results more accurate. For tliis reason, 1 have thought
it best to compute the tables of the motion in latitude, strictly by the
theory ; so as to reduce, as much as is possible, the whole science of
astronomy to the single princij)le of universal gravitation. The inequality,
— 6',487 . sin. (5 's true longitude) [5357], [5597]
is not introduced into Mason's tables, but was discovered by me, by the
theory ; and is now confirmed by observation, in an incontestible manner.
Burg found it to be equal to,
— 8',0 . sin.(3)'s true longitude), [5598]
by the comparison of a very great number of Maskeline's observations.
This coefficient is,
— 6^487 [5357], [55991
if we suppose the oblateness of the earth to be ^^^ ; it will become,
—13', 436 [5358], [5600]
if the oblateness be -jl-g, as in the case of the homogeneity of the earth.
Hence it is evident, that the coefficient — 8% which is found by Burg, [5601]
corresponds to the oblateness .* It is veni remarkable, that this [5602]
^ 304,6 "^ Doletrain-
ation
inequality gives the same oblateness as the inequality in the motion in oblateness
ofthe
longitude, depending on the sine of the longitude of the node, which toe ^^ymeans
have given in [5593]. These tivo inequalities, which, by the light they ("ne^naiity
throrv on the figure of the earth, deserve the utmost attention of observers, '[seoan
unite in the exclusion of the homogeneity of the earth.
* (3081) Substituting in [5357c], the value of ^ = 8' [5601], it becomes,
_ 8'+8%88 _ 1
'^' 5Ï32^9 ~ 304,1 ' [5602a]
which is nearly the same as in [5602] ; the slight difference arises from the use of
centesimal seconds to a greater number of decimals. The result given in [5602], is used
by the author, in page 229, of the fifth edition of his Système du Monde.
VOL. III. 166
662 THEORY OF THE MOON ; [Méc. Cél.
26. It noiv remaiiiH to consider the moon^s horizontal parallax. The
following is the expression of that parallax, at the equator, according to
the tables of Mason and Burg ;
(Col. 1.) (Cui.a )
nuig. Mason and Mayer.
342F,0 . . . 3431^4 1
— 0',3 ... — 0^3.cos.(©'s mean anom.) 2
-f- 0'',7 . . . -(- O',7.cos.(2.3) 's mean long. — 2.© true long. -|-(v) mean anom.) 3
-j- 0',8 ■■■-{- 0>',8.cos.(2. ]> mean long. — 2.© true long. — © mean anom.) 4
— 0*,1 • • • — O',l.cos.(2.3) mean long. — 2. @ true long.-)- 3) mean anom.) 5
Tables
ofihe _|_ 37s :5 . . . -J_ 37'',3.cos. (2.1) mean long. — 2.© true lon<T. — l)meananom.) [Eveciion.] 6
moon's ' o V.' o /
paiaiia.x,' -j- 0%3 ■••-!- 0%3.cos.(4. 3)mean long. — 4. ©true long. — 2. J) mean anom.) 7
by Burg,
amf"" -|- I'jfl . . . -j- l'',0.cos.(2. J)mean long. — 2,© true long. — 1) mean anom.-)-© mean anom.) 8
Mayer.
-\- 0',() . ■ • -|- 0%6.cos.(2. 3) mean long. — 2.© true long. — j) mean anom. — © mean anom.) 9
-)- 0',2...-)- O'',2.cos.(]) mean anom. — © mean anom.) 10
-)- 0',2 . . . -)- 0»',2.cos.( J) mean long. — © true long. — J) mean anom.) 11
-)- 2M) . . . -)- 2'',0.cos.(2. Dmeau long. — 2.© true long. — 2. 3) mean anom.) 12
-)- 0%4 ...-)- O',4.cos.(2.mean long, of 3) 's node — 2.© true long.) 13
-)-l87',3 . . . -)-187%7.cos.( 3) corrected anom.) ' I4
-)- 10",0 . . . -j- l0^0.cos.(2.3)corrected anom.) 15
-)- 0'',2 . . . -)- O',3.cos.(3. 3) corrected anom.) IG
-)- 26%0 ...-)- 2G^0.cos.(2. 3) corrected long. — 2.© true long.) 17
— l',0 . . . — P,0.cos.( 3) corrected long. — © true long.) IB
-)- 0'%2 . . . -)- O',2.cos.(3.corrected long. — 3.© true long.) 19
— 0%8 ... — O',8.cos.(2. 3)'s true distance from node — 3) corrected anom.). 20
To obtain the moon^s horizontal parallax for any latitude : Burg supposes
the ellipticity of the earth to be -gi-^, and Mayer uses ■^^-^. We have
supposed it to be -3!^, in conformity with the calculations in the
r.5604l preceding article ; and, we must multiply the coefficients of the table
[5603, or 5605], by unity, jniniis the product of the ellipticity by the
square of the sine of the latitude [1795"]. This being premised, we
have, for the i^ioon's equatorial horizontal parallax, expressed in terms
[5003]
Vir.iv. §26.] COMPARISON OF THE THEORY WITH OBSERVATION.
66S
depeiidiiis: on the cosines of angles, which vary in piopoition to the
longitude v ;*
(Col.l.)
Inequnlities
(ledviCP'i from
tlio tables nf
Mnsuii ami ATayer.
(Col. 5.)
Coefficients
of this
theory.
(Col. 3.)
Excess of tlicso
cnefficionts over
those of Muson'a
tables.
(Col. 4.)
Excess of the
coefficients of
Iîurg*rt tables over
those of Mason.
+3442%4 +3427',9 .... — I4',r) .... — 10',4 1
+ 1S3%5.cos.(ct— ^) + 1S7%7 .... — 0%8 .... — 0',4 2
— 0^5.cos.(2ci'— 2:3) + 0',0 . . . . + 0V5 . . . . + 0',0 3
— O',3.cos.(3cr— 3w) + 0',0 4
+ O',l.cos.(4cv— 4w) + 0'\0 5
+ 24%2.cos.(2v—2mv) + 24%7 . . . . + 0',5 . . . . + 0',0 6
+ 3S',i.cos.{2v—2mv—cv-\-zi) + 38%1 .... — 0',3 . . . . + 0',0 7
— I',2.cos.(2y — 2mv-{-cv — -n). . .
— O',2.cos.(2» — 2mv-\-dmv — ct')
+ r,7.cos.(2i'-
■M' — c'mv-\^zi') -\-
0%7 . .
■ • +
0',5 . .
•• +
0%0 8
0',2 . .
• • +
0',0 . .
•• +
0',0 9
r,6 . .
. . —
0',1 ..
•• +
O^O 10
0',3 . .
. . —
0',0 . .
•• +
0',0 11
0',2 . .
. . —
0',1 . .
•■ +
0',0 12
Vfi . .
. . —
OM ..
•• +
0',0 13
O'fi . .
. . —
0',3 . .
•• +
0',0 14
0',9 . .
• • +
0',4 . .
•• +
0',0 15
3%6 . .
. . —
0',3 . .
•• +
0',0 16
0*,2 . .
. . —
0',G . .
•• +
0',0 17
i%o.d-
^i). . .
. . +
O'.O 18
Tables of
the moon's
horizontal
parallax
reduced to
the form
of the
present
theory.
— Q',3.cos.{c'mv—-:n') —
— O',l.cos.(2t) — 2mv — cv-\-c'mv-\--!s — «'). . —
-|- I',7.cos.(2y — 2mv — cv — c'mv-}--!:s-\-ôi') . . -j- l',6
— 0%3.cos.(cv-\-c'mv — —t — -m') —
-j- O',5.cos.(cu — c^mv — ra-f-ra') -j-
4- o',9.cos.(2ct — 2v+2mv—2-a) +
4- O',4.cos.(2^« — 2v+2mv — 2&) —
— l',0.cos.(i' — mv) —
— 0-,l .cos.(4r— 4mv) _j_ 0',0 19
— 0",1.cos.(4î; — Amv — 2cv-\~2~,) -\- 0',0 . . . . -|- O', I . . . . -|- O'.O 20
— O',l.cos.(4v— 4mD— CT+ro) _ o^l . . . . -f 0^0 . . . . -f O'.O 21
— 0',2.cos.(3cw — 2v-{-2mv — 3to) r o' 0 22
— \',0.cos.{2gv—cv—2ê+zi) _ ]',o 4- 0%0 . . . . + 0',0 23
+ 0;2.cos.{2gv+cv—2ê~z,) ^ 0',0 24
— O',2.cos.(cv—v+mv—:;j) _ o-,l.(l+;) 4- 0",0 25
— O',l.cos.(2ci'4-2y— 27WI' — 2:^) 4. 0',0 .... 4- 0',1 .... 4- 0',0 26
[5605]
(3082) The expression of the parallax [5331] is, for a latitude whose sine is \/^
[5606]
664 THEORY OF THE MOON ; [Méc. Cél.
The equations of the horizontal parallax, in the tables of Mayer, Mason,
and Burg, are derived from Mayer's theory, and we see, by the preceding
table, that there is but very little difference between the coefficients of these
equations, and those of the preceding analysis. We have, however, reason
to believe, that the present analysis is the most accurate, since this theory
represents, better than Mayer's does, the moon's motion in longitude. This
is however a mere nicety in analysis ; because observations cannot be made,
with sufficient accuracy, to determine such slight differences. With respect
to tlie constant term of the parallax, it was determined by observation,
both by Mayer and Burg. This last astronomer has grounded his calculations
chiefly upon a very great number of Maskelyne's observations, and he has
[5607] found, that this constant part is less than in Mayer's tables, by 10'',4. We
have deduced this quantity, in [5330], from the experiments upon the length
of a pendulum, vibrating in a second ; and from the measures of the degrees
of the meridian : by this means, we have found, that we must still farther
|.5608| decrease, by 4',1 [56051inel], the constant part of the i)arallax given in Burg's
tables. The question then arises, whether this difference depends on the
errors of the observations, or on those of the elements which we have used
in the calculation ? This can be ascertained, by a long continued series of
observations. The only element which appears to be liable to any considerable
degree of uncertainty, is the moon's mass. We have seen, in [4628,4629'],
that to make the result of the theory coincide with the calculations of Burg,
we must decrease the moon's mass, from — - to — — . This diminution
58,6 74,2
appears rather too great, to accord with the phenomena of the tides, with
the nutation of the earth's axis, and with the inequality in the solar tables.
[5609]
[5330,5310]; and, by supposing the oblateness equal to -j^-g, we shall obtain the
[5604a] gquatorial parallax, by multiplying the function [5331], by l+iX^ri^ nearly, [1795"] ;
or by increasing their coefficients ^l^ part. This process being applied to the numbers in
[5331], gives those in [5605 col. 2], corresponding to the present theory; those in
[56046] [SGOScol. 1], being deduced from [5603col. 2], by a method of inversion similar to that
which is used in finding [5575, &ic. col. 1], from [5551, fee.].
Vll.iv. ^26.] CO-AIPARISON OF THE THEORY WITH OBSERVATION. 666
which depends on the moon's mass. Upon a full consideration of the subject,
it appears that we must still farther diminish, by two or three centesimal
seconds, the constant term of the moon's parallax, as it is given by Burg ;
who, by the comparison of a very great number of observations, had already
diminished the constant term, adopted by other astronomers, and, by
that means, obtained very nearly its true value.
VOL. III. 167
[5610]
666 THEORY OF THE MOON; [Méc Cei.
CHAPTER V.
ON AN INEQUALITY' OF A LONG PERIOD, WHICH APPEARS TO EXIST IN THE MOON'S MOTION.
27. We have remarked, in [4733 — 4736], that the moon's mean
motion, deduced from a comparison of the observations of Flamsteed and
Bradley, is sensibly greater than that which results from the observations of
[5611] Bradley, compared with those of Maskelyne ; moreover, the observations
made within fifteen or twenty years, indicate, in this motion, a still greater
diminution. This seems to prove, that there is, in the theory of the moon's
motion, one or more inequalities, of a long period ; and, it is important
to ascertain the law which regulates any such inequality.* If we examine
* (3083) The propriety of introducing an inequality of this kind, into tlie lunar theory,
has been much discussed by astronomers. It is very apparent, that the theory gives such
[5Glla] an inequality ; but, the result of the latest observations leads to the belief, that its
coefficient is insensible ; and, it is not used in Damoiseau's tables, as we have already
observed in [4746a]. This correction was proposed by D'Alembert, about sixty years
[56116] 3gQ^ ^Q account for the acceleration of the moon's motion ; before La Place had discovered
the real cause of that acceleration. To estimate the periods of the arguments of the
inequalities treated of in this chapter, we have taken, from the third edition of La Lande's
'^ astronomy; the following mean motions, in one hundred years ; supposing nt, n't, to
represent, respectively, the mean motions of the moon and sun, during that time.
[5611d] Motion of 3) 's perigee = (1—c). ni = 4069'',2 [4817];
[56116] _ Motion of 5 's node =^ {\—g).nt =—\9M\2 [4817];
[5611/] Motion of ©'s perigee = (l—f').n7= 1^7 [4817,4831];
[5611g-] Precession of the equinoxes = (/—I ).w<= 1^4 [5.347o, 4.3.59].
Hence we obtain the increments of the arguments of the first members of the following
vil. V. § 27.]
INEQUALITY OF A LONG PERIOD.
667
the lunar theory, with the most scrupulous attention, we shall find, that the
action of the planets produces nothing of this kind. This is made quite ^^^^'^^
evident, by the analysis, given in [5455 — 5539]. But, the. sunh attraction i',',T"'"^
produces, in the expression oj nt-\-s, an inequality, proportional to the sine ''"'*"'''"°
of the following angle ,**
3 1) — Qmv + 'âc'mv — 2gv — ctJ4-2ti-f si — 3;
on ilie
sun's
uciion.
[56 J 3]
expressions, in one Inindi-ecl years ; also, the times of the periodical revolutions of these
arguments, respectively, or the number of years requisite to complete the whole [561U]
circumference .360'' ;
Years
3nt—<2gnt—ai( = 200'',8
]79
[.56 1 1.-]
3nt—n't-^ c'n't—2gnt—cnt = igO^l
181
[56114]
Zni—Zn't-^-'icnt-^gnt—cnt = 195^7
184
[5611(1
2fnt-\-nt-n't-\-cn't-2gni-cnt = 20r',9
178
[5611m]
3fnt—2g7it—cnt = 205'',0
175
[561 l7j]
2 J) node-f- D perigee
2d node-)- D perigee — @ perigee
2 D node -f- J> perigee — 3 © perigee
2 J) node-}- 3) per.-© per.-|-2.precession
2 J> node-)- D perigee -)- 3. precession
The arguments [56111, m,n] correspond, respectively, to [5627,5633,5639]. The
author commenced with the use of the first of these arguments, as in [5665] ; but he
afterwards proposed to change it into the form [5611»]. Burckhardt uses the argument
23 node-f- D perigee, in his tables, published in 1812. Several papers were published by
LaPlace, Burckhardt and Burg, upon this subject, in the Connaissance des Terns, for 1813,
1 823, 1 824, Sic; and in the Monatliche Correspondenz, vols. 24, 26, 28 ; also by Carlini and
Plana, in Zach's Chrresyondance Astronomique, vol. 4, page 26, 8ic. La Place resumes
the subject in the fifth volume of this work [12755'] ; but does not there speak with much
confidence relative to the existence of this inequality. Finally, he omits it altoo-ether in
the last edition of his Système du Monde, which was published a short time before his
decease.
[5611o]
[5611;)]
[56112]
* (3084) As an example of the production of such quantities, we shall observe that
the function \~r-) [4809] contains the term,
15m'. v! 4
~ ~ë;:F~-s'"-(3''— 3i'') ;
and, we have, in u'^, a term of the form,
A. c'3. cos. (3c'm D— 3zô') [4838,&c.] ;
also, in xr"^, a term of the form.
[5613o]
[56134]
668 THEORY OF THE MOON ; (Méc.CéJ.
[5614]
The terms which compose this inequality are very small, in the differential
equations ; but, some of them acquire, by successive integrations, the
divisor (3 — 3m-\-Sc'm—2g — c)^; and this can render them sensible,
by its extreme sniallness. To determine this divisor, we shall observe, that
we have, by using the values [5117],
3— 2o— c = 0,00040849.
[5615] ^ -5
[5616] JMoreover, the annual motion of the sun's perigee is 1P,949588 [42441inel];
hence we have,*
[5617] l—c'=- 0,00000922035.
From this we get,
[5618] s—3m+3c'm—2g—c = 0,00040642 ;
consequently, we have,
A".ey^.cos. {^gv-{-cv — 25 — ro) ;
which is similar to that in [4904 line 16]. The product of these two term gives, by
reduction,
iA' A".e'^. ey^.cos.{Sc'mv—2gv—cv+2é-\-us-3-ui').
Multiplying this by the factor,
[5613c] ^.sin.(3i)— 3?;') = ^ . s\n.(3v—3mv), nearly;
r5613dl and reducing, we obtain a term of (-^\ depending on the sine of the angle mentioned
in [5613] . Substituting this in [4753, or 562U'], we find that it will suffer two integrations,
which will introduce the divisor [5614].
* (3085) The motion of the sun's perigee is {l—c').n't [5611/]; and, if we put
this equal to ll',949588 [5616], and n't = 1295977',349 [4077 line 3], we shall get,
[5616a] (1— c').1295977',349= ir,949588 ;
whence, we easily deduce [5617]. Multiplying this by 3m [51 17], we obtain,
[56161] 3 m — 3 f'w = 0,00000207;
subtracting it from [56 J 5], we get [5618]; whose square is as in [5619]. We have
[5616c] corrected the numbers [5615, 5618, 5619], for a small mistake, made by putting [5615]
equal to 0,00040859.
m being equal to — [4835] ; but, it has for a factor 3 — 2g — c, which
is very nearly equal to 3 — 3»i+3c'm — 2g — c [5615,5618]; so that it
must be considered as having only the divisor 3 — Sm-\-Sc'm — 2^ — c, which
does not appear to be small enough to render the result sensible. If the
preceding term of the expression of R depend on the square of the
[5620]
VJI. v.s^27.] INEQUALITY OF A LONG PERIOD. 669
(3— 3m+3c'm— 2j— f)' = 0,0000001651 8. [5619]
We have seen, however, in [4853', &c.], that the square of the coefticicnt of
the angle v, cannot become a divisor of the corresponding inequality, by
means of the successive integrations, when we notice only \.\\e first power of ^ ^
the disturbing force ; but this restriction does not obtain in the terms
depending on the .ignare of that force ; and, the inequality depending on
3v — 3mi'-{-3c'mv — 2gv — «)-j-2ii-t-s — 3:^', can arise only from these terms.
To prove this, we shall consider the term 3a.fJniJt.dR., of the expression [5(320']
of 6v, given by the formula [931]. This term appears to be that upon
which the inequality in question must chiefly depend. The development of
R gives some terms of the form,*
R = H.cos.(3nt—3n't+3c'n't—2gnt-—cnt+2ê-^zi—3^'). [5621]
If these terms depend only on the first power of the disturbing force, n't
and c'n't will depend on the sun's co-ordinates ; and then, the differential [5622]
dR, which only affects the moon's co-ordinates [5363'], will become,
dR = —(3—2g—c).ndt.H.sm.(37it—3n't+3c'n't—2gnt—cnt+2è+-.—3^'). ^5^33^
The double integral 3a.ffndt.dR acquires the divisor,
(S—3m+3c'm—2g—cf; [5624]
[5625]
[5625']
* (3086) This is evident, by comparing the value of R [949] with its development rsgai^-i
[957, &ic.]. It also appears, by a process similar to that in [5613rt — d], from which we
easily perceive, that — ^, or R [5360] contains a term of the form,
iZ.cos.(3«— 37nt)+3c'mj;— 2^D— CT+2â-J-ra— 3ra'). [56216]
Now, substituting nt for v, and mn = n' [4835], it becomes as in [5621]. Its [5621c]
differential, relative to d, supposing it not to affect n't, becomes as in [5623] ; but,
if we suppose it to affect the part — )ln't of the term — Znt, and put n'z^mn, as [5621(/]
above, it becomes as in [56"26].
VOL. III. 168
670 THEORY OF THE MOON; [Mtc. Cél.
disturbing force ; or, in other words, if it arise from the substitution of the
[5625"!
parts of r, v, which depend on the first power of that force ; then, the
moon's co-ordinates will contain the angles n't and c'nt. For example,
if we suppose, that the part — 2n7, of the angle — Sn't, in this term
of R, depends on the moon's co-ordinates ; we shall have,
[5G2C] djR = —{3—2m—2g—c).H.ndt.ûn.{Snt-Qn't^Scn't-'2.gnt-cnt^'2.'^-\-Tr>~S:,') ;
and, the term 3a.ffndt.dR [5620'] gives, in the expression of the
moon's longitude, the following term,
3a.(3—2m—'is:—c).n\H.sin.(3nt—3n't-\-Sc'nt—2<!nt—cnt-\-2ê-{-:z-3T^')
[5G27] oV = — ^ 7Z — 5 -^, ^ 7^ ;
*• ■' (3 — 3m-{-3cm—2g — c)-' '
which may become sensible, by the extreme smallness of its divisor. The
terms of this kind, are very numerous, and it is difficult to determine all of
[5628] them, with accuracy ; but it is sufficient for the present purpose, to prove the
possibility of such an inequality ; since we may then refer directly to
observations to determine its magnitude. This inequality must be applied
to the mean motion, and, therefore, also to the mean anomaly.
The theory also indicates an inequality, depending îipon the oblateness of
the earth, and having very nearly the same period as the preceding [5627].
We have seen, in [5340], that the expression of Q contains the term,
Q=(la? — ap).-.(,.^— X);
now, we have, as in [5344],
ix = s. cos.x4-\/i— « -sin.x. sin.fv ;
moreover, we have, in [4776],
[5628']
[5629]
[5630]
[5631] ^ — u~'
This gives in R, or in — Q, [5438], the function,^
* (3087) The square of jx [5630], or rather the square of the last term of that
[5632a] expression, produces {{-s^).sm.^'K.5m:fv = {\ — s^).sm.^\.{\ — \.cos.2fv); so that
^fi or fJ-^ — ^ contains the term — J(l— s^).sin.^X.cos.2/D . Substituting this in [5629],
we obtain in Q , the term,
VII.v.§27.] INEQUALITY OF A LONG PERIOD. 671
R = — Q^ (la?— ap).iD-. «^ (1 — |s=).sin.2x.cos.2> . [5C32]
Inequality
This function produces, by its development, some terms depending on the v ,
depending
following angle,* ^'^^^^.n
of Ihe
2fnt -\-nt — n't + c'n't — "ignt — cyj< + 2i) + ^ — 13' . °'[ 5633]
They are analogous to those produced by the function R [5626, &c.],
relative to the sun's action, which depends on the angle [5621],
37it—Qn't+ 3c' n't — 2g7it~ cnt + 2è-\-zi — 3i^ . [5634]
The coefficient of the time t , is very nearly the same in both these ^53351
angles, which differ from each other about 1 80'', in the present situation
q = — (Aap— ap) . |D3. lL-£5 . sin.2x.cos.2/i) . [5632J]
Now the expression of r [5631] gives,
I 3 1 ^2
— ^u^.([J^s^)-^=u^.{l—§s^), and —^ = u^.{l — ^s^), nearly. [^5632^]
Substituting this in the preceding value of Q, it gives in — Q, or R [543S], the
term [5632].
* (3088) This angle is produced by the development of the term m^. 4^. cos.2/t) , fsessal
which occurs in [.5632]. For the value of s, orratherof s-\-Ss , [4818,4896,4897]
contains the terms,
s = 7'.sin.(^!> — ê)-\-Bf\ ey.s'm.(gv-\-cv — ê — w) ; [56335J
whose square produces the term,
2Bf\ ey^.sm.{gv — ê) .sm.{gv-{-cv — 6 — ro) ;
or, by reduction,
s^ = — Bf\ ey^.co5.{2gv^cv—2ê—^). ^5^33^^
In like manner, the value of u, or that of «+'''* [4826, 4904], contains the following
terms,
« = ^ • J l+'^o'"- "^..e'.cos.iv — mv + c'mv — ^) | ; ^5^33^
therefore, 11? contains the term,
^^»"'* â' • ^ * ^'" ''°'*^'' ~ mv-\-c'mv — ^') . ^5g33^^
672 THEORY OF THE MOON ; [Méc. Cél.
of the sun's perigee.* All the terms of R [5632], depend entirely upon
the co-ordinates of the moon ;t so that if we represent by,
[5636] R = iT.sin. (2/n^ + nt — n't + c'n't — 'Hgnt — cn^ -f 20 + ^ — ^') ,
the term of the development of R [5632], which depends upon the
[5637] preceding angle ; we shall find, that this term acquires, in the differential àR,
the factor (2/'+l — m-{-c'm — 2g — 1 ).n ; therefore, it will have for a divisor,
in the double integral 3a.Jfndt.dR, only the ^^r^f power, and not the
square of this quantity ; hence, it is evident, that this term must be
insensible.
The term of the form Y^^^, which, as ive have seen in the third book, may
occur in the expression of the radius of the earth, can also introduce into the
[5637']
Terra of
I'.
depending
on
JMuhiplying this, by the term of s~ [5633c], and reducing, we get, in u^.s^, a terra
of the following form,
[5633e] 2K'.cos.{v — mv-\-c'mv — 2gv — cv-\-2è-{--a — ra').
Lastly, multiplying this by cos.2fv , and reducing, we obtain, m R, a term of
the form,
r5633/"l ^ ■ ^°^' (^-^^ "f" ^ — mv-\- c'mv — 2gv — cv -\-2è-^-a — ra') .
Now, changing, as in [5G21c], v into nt , and putting ?)m = 7i', it becomes as
in [5633,5636].
* (3089) Subtracting the angle [5634] from that in [5633], we get for their difference,
[5635a] 2.(/— 1) .îi<-f 2.(1 — c') .?i'/+2ot';
and, as we have, very nearly, /= 1 , c'= 1 [53475', 5617], the preceding expression
[56356] is, very nearly, equal to 2ra', which differs but little from 180'' [4081, line 3], as in
[5635].
f (3090) The variable quantities which occur in R [5632], are ti^, s", cos.2fv ;
[5636o] aZZ of which refer to the moon ; so that for this term of R, the differential d/? [5632]
changes into the complete differential dR ; and by taking the complete differential of
[5636], we get,
ggggj, dR= (2fn-^n—n'+c'n'--2gri—cn).(h.cos.{2fnt-]-nt—n't-\-c'7i't—2gnt—cnt-{-2ê-{-zi—'U!').
Substituting Ji' = mn [5621c] in the factor of this expression, it becomes,
[5636c] (2/+ 1 — m + cm — 2g—c).ndt,
corresponding to [5637].
VII. V. 4-28.] INEQUALITY OF A LONG PERIOD. 673
expression of the moon's true longitude, an inequality depending on*
sin. {Sfnt — 2gn1 — cnt + 2 ' + za) ; [5638']
[5fi39]
which is now nearly confounded with the two preceding ones [5638/]. If
this inequality become sensible, it will furnish new data on the figure of the
earth ; but some calculations, which I have made for this object, induce me
to believe, that this inequality, like the former, is insensible. The lapse of [5639]
ages, and new improvements in analysis, will throw light on this delicate and
important part of the lunar theory.
28. We shall now proceed to establish by observations, the existence of the
ineqxiality depending on the sine of the angle,
Snt — Sn't + Sc'n't — 2gnt — cnt + 2j + =i — 3^' [5627]. [5540]
* (3091) In the same manner as the term jj.^ — -j of F'-' [1528c], introduces into
V [1811 or 5336], the term,
C9
[5636a]
(lap — ap)./xa.J/. ^
„3
tlie term 1'^^' [1811, \52S(I], produces a term, whicii contains the factor — or
(x^.u^ [4776]. Now, the last term of fA [5630], gives in (j.^ tlie term,
(1 — ««)^. sin.^X .sin.^/v , or sin.^X . sin.'/"w ;
which, by means of [2] Int. gives — j . sin.^X . sin.3/y . Moreover, the complete value [56386]
of M or M-j-du [4826,4904], contains terms of the form,
- . ^ l+A,.cos.{2gv-}-cv — 2ê — zi)\; [5638c]
therefore, u* produces 4a~*. ^, . cos.(2^j;-l-c« — 2è — zs) . Mukiplying this by the
term, — 5 . sin.^ X . sin. 3/u [563Sè], and reducing by [18] Int. we obtain,
— i.fr*. A^.sm.^X.sm.{3fv — 2gv — cv -|-2â+ra) ; [5C38rf]
which, by changing v into nt, produces the angle mentioned in [5638']. The
difierence between this angle [5638'], and that in [5626], is represented by,
3.(/— 1).«^— 3.(c'— l).n'< + 3ra'; [5638e]
which, by reason of the smallness of / — 1 , <f — 1 [5635i], is now nearly equal to 3t^' ;
and as this varies slowly, the periods of the inequalities [5627, 5638'], are nearly equal to [5638/"]
each other, and to that in [5633], as in [56356,5630], or in [5611/,ot, n].
VOL. III. 169
674 THEORY OF THE MOON ; [Méc. Cél.
If we represent this angle bj E, we shall evidently have,
[5641] E = 2.1ong.Dnode + long. 3 perigee — o.long.© perigee [5611/] ;
and we shall now proceed to show, that the law of the variations of sin.jE,
is the same as that of the variations which have been observed in the moon's
mean motion.
In the lunar tables, inserted in the third edition of La Lande's astronomy, it
is supposed, that in the interval of 100 Julian years, the moon's motion relative
[5642] to the equinoxes, exceeds a whole number of circumferences, by 307'' ôS™ 12',-
[5643] and that the epoch of 1750 is IBS'* 17"'14',6. The correction of the epoch
of these tables, in 1691, has been determined by Bouvard and Burg; by
means of more than two hundred observations of La Hire and Flamsteed ;
[5644] they have both found this correction equal to — 4',4.
The correction of the epoch of the same tables, in 1756, has been
determined by Mason and Bouvard, by means of a very great number of
[5644'] Bradley's observations ; and they have found it to be 0%0. Thus, in the
interval from 1691 to 1756, the moon's mean motion was greater than the
[5645] tables, by 4',4 , which gives 6',8* for the increment of the mean motion
of the same tables in a century.
Burg has found, by a great number of Maskelyne's observations, that the
correction of the epoch of these tables is equal to — 3',0, in 1766, and
[5646] — 9',1, in 1779.
Bouvard has found, by a great number of Maskelyne's observations,
[5647] — 17',6, for the correction of the epoch of these tables, in 1789.
Lastly, by a considerable number of observations made at Greenwich,
Paris and Gotha, it has been found, that the correction of the epochs of the
[5648] same tables, in 1801, is — 28',5.
[5648']
Hence it appears, that, from 1756 to 1801, the moon's mean motion has
decreased in a sensible manner ; and, that this diminution is now increasing.
* (3092) In the interval from 1691 to 1756, which is 65 years, this correction varies
[5645a] ^,4 [5645], which is at the rate of 6',8 in a century, as in [5645].
VII.v.§28.] INEQUALITY OF A LONG PERIOD 675
For, in the interval between 1756 and 1779, which is twenty-three years,
this motion was less than by the tables, by 9',1 [5644', 5646] ; and, from [5649]
1779 to 1801, that is, in twenty-two years, it was less by 19,4.* The
epoch of 1756, compared with that of 1779, gives 39',5,t for the decrease [5650]
of the tabular motion in a century; whilst the epoch from 1756 to 1801
gives 63',3, for this diminution. Therefore, the combination of all these [5651]
observations evidently indicates the three following results. First. A mean
motion greater than that of the tables, from 1691 to 1756 [5644']. Second.
A less mean motion from 1756 to the present time [5651]. Third. A
diminution which becomes more and more rapid.
[5652]
These results are conformable to the march of the preceding inequality.
For, at the epoch of 1691, the sine of E was negative;! and, it was [5053]
positive in 1756 ; therefore, this inequality increases the moon's mean motion,
in that interval. In 1756, this sine was positive, and near its maximum; and [5654]
since that epoch, it has always been decreasing ; therefore, the inequality
decreases the moon's mean motion. Lastly, this sine was nearly equal to
* (309.3) This is the difference of tlie two corrections — 9%\, — 28*',5 [5646,5648]. [5648a]
t (3094) The difference of the numbers 0',0, — 9',1 [5644', 5646] is O',!,
corresponding to the Interval 1779 — 1756:= 23 years. This is at the rate of 39',5, in [5(550o]
100 years; as in [5650]. If, instead of — 9',1, we had used — 28',5 [5648],
corresponding to 1801, the variation would be 28',5, in 45 years ; corresponding to
63',3, in a century. These differ a httle from the results of the author in the original [56506]
work; who gives 126"=40',8, and 172",5 = 55',9, instead of 39^5 and 63V3,
respectively.
J (3095) According to the tables in La Lande's astronomy, the values of E, at the
different epochs, are nearly as follows ;
Years, 1691 1750 1756 1801 [5652a]
Values of E, 320'' 76" 87" 176". [56526]
The signs of the angles change from negative to positive, in 1750, &£c., as in [5653. &c.] ; [5652c1
and, in 1756, ûn.E attains nearly its maximum value, or sin. 90". Moreover, if we
represent, as in [5658], by y.s'm.E, the part of this correction which depends on E, [565iid]
and suppose E to increase by the quantity dE, the corresponding increment of y. sin. JÏ
becomes y.<Z£.cos.E; which has, evidently, its greatest negative value when E ^ 180'', [5652c]
or sin.£ ^ 0 ; as in [5654'J •
676 THEORY OF THE MOON ; [Méc. Cél
[5654']
nothing, in 1801 [56526]; and then, the diminution of the mean motion
was the greatest [5652e]. The decrement of the mean motion must,
therefore, be greatest about the year last mentioned.
We shall now determine the coefficient of this inequality. It is evident,
that it must produce a change, both in the epoch of the tables in 1750, and
[5655] in the mean motion of the tables in a hundred years. We shall put s
for the correction of the epoch of the tables in 1750; x for the diminution
[5656] Q^ ^j^p mean motion in a century : and, y for the coefficient of the
preceding inequality. The formula for the correction of the epochs of
the tables, will be, by putting i for the number of centuries elapsed
since 1750,
[5657]
[5658] - X .i -\- y . Sin.£. [Correction of the epoch]
To determine the three unknown quantities s, x and y ; we have
[5658'] compared this formula with the results of observation, at the three epochs
1691, 1756 and 1801 ; and, by this means, have obtained the three
following equations ;*
s+x.0,59— 3/.0,63660 =— 4',4 ;
[5659] £— .T.0,06+î/.0,99898 = 0',0 ;
£—a;.0,51 +^.0,08199 =-28^5.
These three equations give,
[Year 1691]
1
[Year 1756]
2
[Year 1801]
3
* (-3096) The coefficients of v, in the equation [5658, or 5659], are represented by
[5659al Tffff-(1^'50 — years); those of y are the values of sm.E, corresponding to the
respective years ; similar to those in [56526], but taken to a greater degree of accuracy.
Lastly, the constant terms of the second members, are the quantities computed in
[5644,5644', 5648]. The equations [5659] give the values of s, x, y [5660]; as
we can easily prove, by substituting them In [5659]. With these values, we find, that the
formula [5658] becomes,
[56596]
[5659c] — J3',46 — 31^96.^■+15•,39.sin.i: ;
from this we obtain the values [5661], using the values of E, corresponding to the
different epochs. We may observe, that the quantities — '^'fi, — 9',1, — 17',6
[5646, 5647] furnish three additional equations, of the form [5659] ; and, we can
determine the value of s, x, y, by combining all these equations, by the method of
the least squares [815e — /].
vil. V. § 23.]
INEQUALITY OF A LONG PERIOD.
677
£ =— 13^46 ;
.T = 31 ,96 ;
y= 15%39 .
1
2 [5660]
3
By means of these values, we find — 4',4 , + ()',0 , — 3',8 , — ll',3,
— 18%7 , and — 28',5 , for the corrections of the six epochs of 1691,
1756, 1766, 1779, 1789, 1801. The sum of these six corrections is —66,7;
and the sum of the six corrections determined by observations is — 62',6 ;
the whole of these corrections taken together, indicate, therefore, that we
must increase the preceding value of s by 0',7 ;* and then the formula
for correcting the tables becomes,
— i2',8 - 31%96 . i + 15S39 . sin.E .
Calculating by this formula, the corrections for the six epochs, we have.
(Col. 1.) (Col. 2.)
Corrections of the tables
by observations.
— 4',4 [5644]
(Col. 3.)
Corrrctions by the
formula.
(Col. 4.)
Excess of llipse corrections
above tlio first.
1691
1756
1766
1779
1789
1801
— 3',7
+ 0^7 1
+ 0',0 [5644'] + 0',7 + 0',7 2
— 3',0 [5646] - 3',1 _ 0,1 3
— 9',1 [5646] — 10',6 — P,5 4
— 17',6 [5647] — 18',0 — 0',4 5
— 28%5 [5648] — 27%8 + 0',7 6
The difference between the results of observation and those of the formula,
are within the limits of the errors to which these last results are liable ; they
may in part depend on the formula itself, which can be rectified by new
observations.
[5661]
[5662]
[5663]
[5664]
[5665]
[5666]
* (3097) If we suppose the expression f [5G58], to be increased by the quantity
e', it will augment each of the six numbers [5661], by the same quantity ^, [5665a]
and the sum of all of them will become — 66',7-(-6£'. Putting this equal to the
sum — 62',6 of the corrections by observation, as they are given in the second column
of the table [5666], we get — 66',7-j-6e' = — 62',6 ; whence e'=0',7; as in
[5664]. Adding this to each of the values [5661], we get the numbers in the third
column of [5666]. Subtracting the terms in the second column of this table, from those in
the third, we get the corrections in the fourth column.
VOL. III. 170
[56656]
[5665c]
678
THEORY OF THE MOON ;
[Mec. Cel
CHAPTER VI.
ON THE SECULAR VARIATIONS OF THE MOTIONS OF THE MOON ANU EARTH, WHICH CAN BE PRODUCED
BY THE RESISTANCE OF AN ETHEREAL FLUID SURROUNDING THE SUN.
29. It is possible, that there may be an extremely rare fluid surrounding
[5666] the sun, which alters the motions of the planets and satellites ;* it is,
therefore, interesting to know its influence on the motions of the moon and
earth. To determine it, we shall put,
[5667] X J 111 ^1 for the rectangular co-ordinates of the moon, referred to the centre
of gravity of the earth ;
[5668] ^'t y'l ~i for the rectangular co-cordinates of the earth, referred to the sun's
centre.
The moon's absolute velocity about the sun, will be expressed by the
following function ;t
[5667a]
[56676]
[5667c]
[5669a]
* (3098) The existenceof sucha resisting medium is now considered as highly probable,
in consequence of the observed decrease of the times of revolution of Encke's comet, in its
successire appearances between the years 17S6 and 1829. Encke has given an important
paper on this subject, in the ninth volume of Schumacher's Astronomische Nachrichten,
pag. .317 — 348 ; to which we may have occasion to refer, in treating of the perturbations
of comets. We shall here merely remark, that the extreme rarity of the mass of this
comet, makes it peculiarly well adapted to the discovery of the effects of such a resisting
ethereal fluid ; which cannot, however, produce any sensible effect on the large and dense
bodies of the planets and satellites.
t (3099) The rectangular co-ordinates of the moon, referred to the sun's centre,
are represented by x-{-x' , y-\-y', ~-|-~', as in [5667,5668]. Their differentials,
divided by dt, are,
VII.vi.§29.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 679
v_K — -r J ^ \ :f T yj -r\ 1 j__ ^ j^j^g moon's velocity.
JVe shall suppose, that the resistance which the moon suffers, is represented by
the product of the square of the velocity by a coefficient K, depending upon ^ ' ^
the density of the ether, and upon the surface and density of the moon. If "rule"""
we resolve it, in directions parallel to the axes x, y, z, we shall obtain "[^j^'»
the three following forces •*
K.(dx'+dx) ,
^^2 — • V{dx'+dxy+{dy'+dyy+{dz'+dzf; [f»™ p'-iwi lo x] 1
[5671]
■ ^H^i--^ • K/{dx'+dxf+{dy'-^dyf+{dz'+dzy ■ [fo™ I-alle: ,„ ,] 2 Exprès-
sions
of the
resistance
of the
K.idz'A-dz') , ofth,
^—r^'-\/{d3i^rdxfA-{d\j-^dyf-\-{dz'-Ydzf, [Force parallel ,o :1 3 moon
di
In the lunar theory, the earth is supposed to be at rest ; we must, therefore,
apply to the moon, in a contrary direction, the resistance which the earth [5671'
suffers. This resistance being resolved, in directions parallel to the same
dx+dx' dy+dy' dz-\-dz'
-W~ ' dt ' dt ' [5669i]
which evidently represent the velocity of the moon about the sun, resolved in directions
parallel to the axes x, y, z. The square root of the sum of the squares of the three [5669c]
partial velocities [5669è], gives the whole velocity [5669] ; as is evident from [40a b].
* (3100) Putting, for brevity,
d^^ = y/(dx'+dxr+{dy'+dyr+{dz'+dz)% ^5671 a]
we find, that the absolute velocity of the moon is -— [5669] ; consequently, the
resistance is — ^-^JI^ [^6*0], in the direction of the described arc fAv. The
[56716]
negative sign being prefixed, because the resistance tends to decrease tliis arc. To
resolve this force, in directions parallel to the axes x, y, z, we must multiply it
by the expressions,
dx'-\-dx dy'+dy dx'-\-dz
~d^ ' IbT ' ~d^ ' [5671c]
respectively ; as is apparent fi-om [40J]. Hence we obtain the expressions [5671].
Ç80 THEORY OF THE MOON ; fMéc.Cél.
axes, gives the three following forces.*
dx'
earth. df
fr,ir" — K'. % . s/dx-^+dy^+dz'^\ 1
[5672] - K- % . ^da^^+dr+d-^ ; 2
— K'. ^, . v/rf^"-^+rf2/'^+^^'^ ; ^
^' 6ez/?^ a coefficient, ivhich differs from K, and depends iipon the
resistance ivhich the earth suffers. Now having represented the forces, which
act upon the moon, parallel to the axes of x , y, and z , by,
m' (f)' a- ^''''^"•^'
we shall have, by noticing only the preceding forces,
'^^^ = K'. Jl . ^dx'^+dy'^+dz'^' 1
Relative
forces 00
the moon,
considered
as moving
^ • ^^ • K^WTWTWTW+W+W; 2
ÏSàï"" C^) =K'X. ^dx'^+dy'^+d^^ 3
real.
dy J di^
— K . —^ . \/{dx'+di:f+{dy'+dy)--j-{dz-{.dzy,
— K. ^^— • \/{dx'+dxf+{dy'+dyf+dz'+dzf
6
[5673a]
* (3101) The resistances [5671] corresponding to tiie moon, will evidently give those
relative to the earth, by taking the co-ordinates, so as to correspond to the earth, and changing
the factor K into K . This requires that we should put a?==0, y = 0 , s = 0;
in [5671]. Hence we obtain the forces relative to the earth, as in [5672]. The signs of
[56736] the forces [5672], must be changed, as in [5671'], and then they must be added to the
corresponding quantities in [5671], to obtain the forces of resistance of the ether, supposing
the moon to revolve about the earth considered as at rest. These forces are represented, in
„„ [498n'— 499a], by (— ^ j , (-j^) > (-i^j 5 hence, we easily obtain the expressions [5673].
VII. vi. ^-29.] EFFECT OF THE RESISTAACE OF AN ETHEREAL FLUID. 681
Now we have, by supposing the moon's co-ordinates only to be variable,
If we substitute the values,
COS. y sin.i' s
^- -^-^ z/ = — ; -- = - ; [5674']
which are given in [4777 — 4779], we shall obtain,*
'^«--S-l— (^?)+'>"-(f)+»-(f)l >
dv c . fdQ\ fdq\ )
.{ sm. V . [ -^] — COS. V .[-^\} 2
M I
u \ dz.
Then we have,t
drj \dyj ) [5675]
'"2-(S)-*'+(t?)-*" + (t?)-''-
* (3102) The expressions of Q [4756, 5673], may be considered as functions of
X, y, r. x', y', z' ; but if ue suppose the moon's co-ordinates x, y, z, io ^ "J
be t!ie onlv variable quantities, we shall get for t/Q the expression [5674]. Now, the
differentials of x, y, z [5674'] give,
— dv.sm.v du.cos.v dv.cos.v dus'm.v ds sdu
dx == 5 — ; dy = -— ; dz = -. [56756]
Substituting these in [5674], and connecting the terms depending on du, dv , ds , we
get [5675].
[5676a]
f (3103) Considering the co-ordinates of the moon as the only variable quanthies, we
shall have the two expressions of d(^ [5674, 5676]. In the first of these expressions,
the moon's co-ordinates are x , y , z , and in the second u , v , s; and if we
substitute, in the first, the values of dx, dy, dz [5675i], it becomes equal to the second,
and, by this substitution, produces the function [5675]. Hence it evidently follows, that [56766]
the expressions [5675,5676] must be equivalent. Now, by comparing together the coefficients
of f7«, dv , ds, in these two last expressions of t/Q, we get the equations [5677 — 5679].
Multiplying [5677], by — 1, and [5679], by — -; then, taking the sum of the [5676c]
two products, we get [5680].
VOL. III. 171
682 THEORY OF THE MOON ; [Méc. Cél.
and, by comparing these two values of Q , we shall obtain,
[5679]
\ ch ) M V f'-2
Hence we deduce,
'-»> - (f ) - : • (f) - i ■ 1 -- ■ (f ) + ^'"■" ■ (f ) I [«'='«']■
Now we have, as in [4777/ — /t],
,.„„ cos.d' , sin.v' , s'
[5681] -P' ^ — ^ ; ^' = ; .^ = - ;
v' denoting the longiiitile of the earth seen from the sun. If we take for a
[5682] fixed plane, that of the ecliptic in 1750, we may suppose 5' = 0 . We
shall represent by r'dq', the small arc which is described by the earth
in the time dt , and we shall have,
[5683] ^(dx'^ + df + dz'^) = î'dq' [40«, &c. ] .
This arc is to that which is described by the moon, in its relative motion
[5684] about the earth, very nearly in the ratio of* — to 1; therefore, it is at
[5484'] least, thirty times as great ; and we shall have, very nearly,!
* (3104) Whilst the moon describes the angle dv . with the radius a, the sun
[5684a] ^Jgg(.^.^^3es f^e angle mdv , with the radius a', nearly ; as is evident from [4837],
[4838, &;c. ] ; so that the space described by the moon is adv , and the space
described by the sun a'mdv , nearly. The ratio of the second of these expressions,
[56846] •' ,
to the first, is denoted by —, as in [5G84]. Substituting - = 400, ?« = 0,0748.
[5684c] ' ' a a
[5221, 5117], it becomes nearly equal to 30, as in [56S4'].
f (3105) Developing the terms in the first member of [5085], and substituting r'dfj'
[5683], it becomes,
^\r'Mq'^+2dx'dx-\-2dy'dy-\-2dz'dz+dx^+dy^-+dz^l
[5685a] ^
VII.vi429.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 683
If we neglect the excentricity of the earth's orbit, we shall have dq' = mdt;
the time t being represented by the moon's mean motion. Then we
have,*
[5686]
'^^■' —sin.!)'; ^ = cos.îJ'; [5687]
r dq' r' dq
consequently,
\/{dx'-{- dxY + (rf!/4- dyf + (rfz'+ dzf = ma', dt—dx.sin.v'+dy.cos.v'.
Hence, we easilv obtain, t
[5688]
Putting now s = 0 [5682], we have z' = 0 [5681] ; substituting this, and neglecting [5685J]
also di^-^dy~-{-dz^, in comparison with the other terms, we easily reduce it to the form
in the second member of [5685].
* (3106) Neglecting the excentricity of the earth's orbit, we may put 7-' = a'=- [5687o]
[4 937n], and the described arc [5683] becomes r' dq = a' dv' ; moreover, the values of
x', y' [5681], become x' = a . cos.i)', y' = a'.i'm.v' ; whose differentials are *• ■'
dx' = — ddv'. sin v', dy'= a'dv'. cos. v'. Dividing these by the above expression
r'dq' = adv', we obtain the values [5687] ; substituting these, and dq':=mdt [5686],
in [5685], we get [-5688]. The expressions [5683, 5688] may be put under the forms
[56S7e,_/'], bv merely changing, as above, r' into —, and do', or dv' into mdt.
• ' ^ =^ "' ■' [5687rf]
Lastly, the expressions of r'dq', dx', dy' [56876, c], may be put under the forms
[5687^], which will be of use hereafter ;
, mdt
Sj dx'"- -f- dy' 9 + J~' 2 = -^ ; [5687e]
, mdt 7 . , ,
v/(rfi'+rfz)2+(rfy'+f/y)a+(rf~'4-^s)2 =: — — (/a:.sin.i/+f??/.cos.i''; [5687/]
r'dd=—- dx'=—'^^—- ..^rndt^^^ [5687^^
u u' ' ^ ?/
t (3107) Multiplying [5687e] by the value of ^^ [5687§-], we get [5687A];
684 THEORY OF THE MOON ; [Méc. Cél.
/rfQ\ -(K'-K).m^.^m.v' 3K.m dx , Km dx ^ , , Km dy . ^ ,
\dy J m'2 2m' t^i ' 2it' d^ 2?<' (// '
[5691]
m
Km dz
u ' dt
Kdx Kdx
again, multiplying [5687/] successively by =-g— , — -, and neglecting, in the
last product, the terms of the order dx . dy, we gel [5687i,À:] ;
dx' , K'.
[5687A] K'. -j^^.^-dx'^J^dy"^+dz'
m^.sin.v'
rf<2 V " I y I „/2
1
[5687i]
r-r dx , ^ -K?n-. sin.t)' Km dx , ^ ,
-K. -.^[dx'+dxf+{dy'+dyf+{dz'+dzf=—^^ — -rf,-""- "
H Y .—■ .sm.y .COS. i;
u' dt
[5687/fc] -^^- -^,V{^'^'^'+<^^f+W+c'yr+{dz'+dz)^ = —Vit-
Adding together the expressions [5687/*,?',^"], we find, that the first member of the sum
L J becomes the same as the expression of f'T" ) [5673 lines 1,2]; and the second member
r5687 1 °^ ^'^'^ ^""^ '^ easily reduced to the form [5689], by substituting sin.^j;' =; A— i.cos.2y',
sin.i;'.cos.j;'= |.sin.2«', and making a slight reduction.
We may obtain [5690], from [5673 lines 3,4], by a similar process ; or, we may find
it more readily by derivation. For, if we change, reciprocally, x into y, and x' into
[5687n] y'' we shall find, that f^\ [5673 lines 1, 2] changes into (^-^^ [5673 lines 3, 4].
Now, this change in the values of o', y', is made by putting sin.i)' for cos.?)', and
cos.t)' for sin.î)', in [5681]. This does not alter the value of sin.i;'. cos. 1/= J. sin. 2i)'
[5687o] [5687i], but it changes sin.^u' [5687i] into cos.V=:^+à-cos.2w'; so that we must
ç(rrite +J.cos.2i>', instead of — J.cos.2t)', in [5687?«]. Hence it appears, that we may
[5687p] Q^jj^ij, [5690] from [5689], by writing dx for dy, reciprocally, also cos.d' for sin.v',
and changing the sign of cos.2v'.
Lastly, if we substitute ~'= 0 [5685Zi], in [5673 lines 5,6], we find, that the term
1 5687g] in [5673 line 5] vanishes, and the factor of the radical in [5673 line 6] becomes — -^ ■ ^i"
Multiplying this by the value of the radical [56S7/], and neglecting terms of the second
[5687r] p^^,.g,.i^ j^^ jy^ J,, we get [5691].
VlI.vi.^-29.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 685
If we substitute the values of x, y, and neglect the square of the
excentricitj of the moon's orbit, we shall get,*
/dq\ s /fiq\ _ [K'—K] m-.sm.{v—v') 3Km.clu
Km (Jv . .r, r-, ,s J^"* ''" ^^ ^ i\ r,
— ^,.--.sin.(2w— 2r')— -^-,- — .cos.(2î;— 2«'). 2
2u^.u dt ^ ^ )iu\u' dt ^ ^
[5692]
* (3108) Multiplying [5689] by cos.i', also [5690] by sin.r, and reducing the
sum of these products, by means of the formulas [56926, c], whicli are deduced from
[22, 24] Int., we get the equation [5692c/]. In like manner, we may obtain [5692e] ; or, [^^^Sa]
it may be more simply derived from [5692<]^, by changing v into w+90'*, where
it explicitly occurs, in both members ;
— sin.i)'. cos.!;-)-cos.î)'.sin.j; ^ sin.(« — i'') ; . [569261
cos.2K'.cos.r+sin.2i''. sin.i' = cos.(i' — 2u') ;
sin.2v'.cos.'y — cos.2[)'.sin.ii = — sm.[v — 2u') ;
[5692c]
cos.i;.
-<sin.?;.
/rfO\ , . fdOX (K'—K).m°~.sm.(v~v') 3Km idr dy )
-p +sin.r. -^ = ^ '—- ^ '- — . ^— .cos.î;+-^.sin.i>. <
\dx J \dy J M 2(t' tdt dt ^
, Km idx , n ,. dy . , ^ >
+ ^/ • J^-cos.C.-2t.')-^^sm.(.-2.') \ ;
^Q\ ('^^1 {K'—K).m"-.cos.(v-v') 3Am C dx . dy )
[56Q2d]
^ Km i dx . ^ ^,^ dy , [5^^^^]
+ 2^.' • J--^-S'n-(^-2i0-f cos.(.-2.') j .
We must substitute, in [5692f/, e], the values of dx, dy [5675i] ; and, in performing
this operation, we may use the following theorems, supposing W to be any angle
whatever ;
dx.cos.W-{-dy.s\n.W = .sin.(r — W) ^.cos.(z; fV) ;
dx.sin.W — dy.cosJV = — — .cos.(t) — JV)-\ — ^'.sin. (i; W).
^9-^"^-^^— ^''; ; [5692/]
[5692^]
The equation [5692/] may be easily proved to be correct, by substituting, in the first
member, the values of dx, dy [56756], and developing the second member, by means
of [22,24] Int. The equation [5692o-] may be found in the same manner; or, it may [5692/.]
VOL. III. 172
686 THEORY OF THE MOON ; [Méc. Cél.
/f/<?\ dv (K'—K).m^.dv , „ ^Km , dv
\dv J u' u'^.u^ ^ ^ Su'.M* dt
[5693]
2u'. lé di ^ ^ ' 2u'. n^ dt ^ ^'
dQ\ du (K'—K^.iif' du , „ 3Aw du
— ir . = -^^ 1 cos (t) 1)\ .
dv) u^.dv u'^u^ \U ^^ ^ 2m'. M« dt
[5694]
Km du
2m'. îj-* dt ^ ^
Density of
the ether,
?o'be"re-'' "^^^^ vctlue of K is not constant. If ive suppose the density of the ether
byTfùnc. to be proportional to a function of the distance from the sun, and denote
u^ce'Lm this function by,
the sun.
be more easily derived from [5692/], by changing the arbitrary angle TV into W — 90"*.
If we now put JV=v, in [5692/, g-], we shall get the two equations [5692j, 7^] ; and,
if we put TV== — (v — 21»'), or v — W^2v — 2v', we shall get [5692/,m], respectively,
making some slight reductions ;
[5692iJ dx.cos.vA-dy.s'm.v = ;
[5692/1] f/a;.sin.u — dy.cos.v = ;
qv du
[5692i] dx.cos.(v — 2v')—di/.s\n.(v — 2v') = .sin.(2v— 2d') 5.cos.(2y — 2v') ;
{11} du
[5692m] — <^a;.sin.(« — 2v') — dy.cos.{v — 2v') = .cos.(2« — 2v') -\ — ^.sin.(2i' — 2v').
Substituting the expressions [5692i, /], in [5692rf], and then, the result in [5680], we get
[5692]. In like manner, if we substitute [5692A',/«], in [5692e], and then, the result in
[5692n] [5678], we get, by multiplying by —, the expression [5693]. Lastly, multiplying
[5693] by — , we get [5694] ; observing, that the term of this expression, having the
factor — .— , may be nedected, as a quantity of the order e^. For, — is of the
dv dt ■' ° dv
du
Jt
|5692o] Q^^igj. g [4826] ; and the same may be observed of — , which is evidently of the
same order as — [5686].
dv
Vll.vi. ^29.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 687
(fÇu') := density of the ether near the earth ; [5695]
it will become, for the moon, in which u' changes into u' .cos.(î; — v'),* [5696]
(p(î/) .?'(«'). COS. (v — v') = density of the ether near the moon ; [5G97]
(f'(u') being the differential of ©(?«'), divided by du' \ so that we' [•''008 ]
may suppose,
K = H.^ («') — ^^ . <p' («') . COS. (»—?;'). f^C99]
This being premised, if we neglect those periodical inequalities, which do
* (-3109) Substituting s':=0 [5682], in the expression of the distance r' of the
earth from the sun [4777c], it becomes u'= —,. If tlie quantities ?■', u'. corresponding L^GOGa]
to the earth, be increased by ^r', 5m', for the moon ; we shall have, by taking the
variation of the preceding expression,
Su' = .ôr'= — u'^.Sr'. [56966]
r 2
The radius vector r, drawn from the earth to the moon, makes, with the continuation
of the radius r', an angle which is represented by v — v' ; and, it is evident, on
account of the great distance of the sun, in comparison with that of the moon, that the [5696c]
moon's distance from the sun must exceed that of the earth, by the quantity r.cns.[v — v'),
nearly ; hence, 5 r' = r.cos. (v — v'). Substituting this, in [5696?/], and putting [5696rf]
r = - , nearly [4776], we get,
eu' = — -' .cos. {v—v'), as in [5696]. \5696e]
Now, the function ?(«') [5695], corresponding to the earth, changes into cp(u'-\-Su'),
for the moon ; and, if we develop it, according to the powers of M, by Taylor's theorem
[617], neglecting the square and higher powers of hi', it becomes,
(P(«') + <5m'.9'(m'). [5696/ J
Substituting the value of m' [5696e], it becomes as in [5697]. Lastly, multiplying
this by the constant quantity H, we get the expression of the resistance [5699].
Encke, in making the calculation of the orbit of the comet [56676], supposed the function ^'
p{u'), or, 9(t)i to be represented by 9[~,) = i^-
688 THEORY OF THE MOON ; [Mec Cel.
not depend on the sine, or cosine, of cv — 51, we shall have,*
/■dQ\ dv H.m^.dv , , ,. 3Hm , dv
If we substitute the values,t
[5701] w = -. j 1 +e.cos.(ct) — îj) i ; f/i = dv. {1 — 2e.cos.(cv—vi) ] ;
a
we shall obtain,
* (3110) Substituting the value of K [5699], in [5693], and neglecting the terms
which depend on the sine or cosine of v—v', or its multiples, we get [5700]. For, the
first term of [5699] H.cp{u'), being combined with the second term of [5693], produces
the second of [5700] ; and, the second term of [5699],
[57006] T- • f' ( "') • ^°^- (" — ^')
u
being substituted for K, in the first term of [5693], produces,
Hnfi.dv ,, ,, „, A Hm-.dv ,, i\ ci \ -i c /■ ' \>
[5700c] -^-^-.,f'{u').cos-'(v—v') = —^--.v'{u').{i-{-l.cos.2.{v—v)l;
which gives the first term of [5700].
f (3111) If we neglect the second and higher powers of e, we get, from [4826],
[5701a] the expression of ?t [5701]. Moreover, the mean motion of the moon being represented
by t [56S6], we get, from [4828] n = 1 ; and then,
[57015] t-]-s = v~-^ .sm.(cv — ^);
whose differential is the same as the value of dt [5701]. These values of u, dt
[5701] give,
[5701c] -^ = u\\\—'\.e.cos.{cv—-a)\\ ^^ ' ft "^ <?t'.{14-2e.cos.(a'— w)}.
Substituting these, in the second member of [5700], it becomes,
/rfQ\ 1^ = ^H.m''a\dv.(f'{u').{ \—Ae.cos.{cv—zs)\—:^.a\dv.<?{u'). \ 1 — 2e.cos.(CT— ra) \
[5701d] ^ " ' ■» C3 >
^-lH.ma\dv. [ ?^— m.9'(«') \ +a«a''. J -.?(w')-2m.9'(M') j .edv.cos{cv-iz).
Integrating this, we get [5702].
vil. vi.§ 29.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 689
+ H.m.a" . \ '-. 9 ( ii')—2m.v'( «') Le. s'm.(cv — ra). 2
[5702]
Then we shall have,*
—5 . — î^ ^3= — i.H.in.a^. ^'.e .fim.(cv—z,) ; [5703]
clu J u \ds J - w ^
— ^ . — — = l.H.m.(r. { — V^ — m.cc(u) > .e. sm.(cv — u). r.5704i
\dv J xi^.dv ^ i u )
Now, if we put,
a = H.m.a^ . \ ^-^ — m.^\u') \ ; [5705]
/3 = H.m.cv^ . i ^_|m.a>'(M') I ;
[570G]
* (3112) In substituting the value of K [5699], in the second member of [5692],
and neglecting the terms depending on the sine or cosine of v — v', or its multiples
[5700a], it will be only necessary to retain the term ^-^ — '-— [5692]. Now, the fsvoSal
differential of m [5701], being divided by dt [.5701], gives, by neglecting terms of
the order e^, and observing, that c = l, nearly;
du « • / N
■^ = — --sin-lci) — ^). [57036]
Hence, the term [5703fl] becomes,
3Km e . , .
- 2;;W- „•''"•(" ''-^) 5 [5703c]
and, if we substitute — ^^a'^, nearly; also, the first term of K [5699], namely,
H.(p{u') ; we shall get the value of the first member of [5692, or 5703], as in the second
member of [570-3]. By similar substitutions, we may obtain [5704] ; but, it is more ^'^'^^^'^
easily obtained, by multiplying the differential of [5702], by
d V e . ,
-— = .sm.(cw — ^ [5701];
dv a ^ / L J ' [5703e]
and then, dividing the product by dv ; observing, that we need only notice the first line
of [5702], because the second line produces terms of the order e^.
VOL. III. 173
(390 THEORY OF THE MOON ; [Méc.Cél.
we must add to the second member of the equation [4754], or to the second
member of [4961] the following function ;*
[5707] V^ ■- • sm. (Cl' — vi).
[5708] The value of - [4968] will, by this means, be increased by the quantity!
* (3113) We have, very nearly.
^+M = - [4S90,4892f/], and A^^a, [48631
hence,
[5707a] i^^A-i^ A _ A .
Vrf«2 ~ ) ■ }fi aa'
[57076]
multiplying this by [5702], we get,
-{-H.m.a\ \ ^^-4m.ç.'(M') I .- .sm.(cv—vs).
[5707c] Now, dividing the sum of the expressions [5703,5704] by h^, or a,, and adding
the quotient to [57076], we find, that the sum becomes,
[5707rf] —H.m.a^. \ ^î^—m.<p'(u') \ -^ + H.m.a\ J ^^ — |.?n.(p'(M') ] .'-.sm.(cv—zs).
Substituting in this, the abridged symbols ci, p [5705,5706], we get [5707] ; which
represents the sum of all the terms of [4754], depending on the part of Q now under
[5707e] consideration ; as is evident by observing, that the first members of the three expressions
mentioned in [5707c] contain all the terms of Q [4754]. If we now connect the function
[5707], with the two first terms of [4754], we obtain the following equation, for the
determination of u ;
ddu , a.v . e . , .
[5707/] 0 = — +M— — + ,3 .-.sm.(ct>-«).
In which we may change n, into a [4968].
t (3114) If we put, for a moment, A= , and neglect the part of [5707/],
depending on e, the eqviation becomes as in [4963fl] ; whence, we get, as in [4963rt, è],
CL V
[5708a] u =^— A = — , for the part of u which corresponds to A. Now we have, very
[5708i] nearly, u = - [4937n], whose variation gives 5u = ~, or (5a = — a^.6ji; and,
VII. vi. ^'29.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 691
— ; consequently, the value of a will be decreased by a-a^.v. We shall [5709]
then have, as in [4973], very nearly,*
.2d.-
-1—^ + P . - ==0. [5710]
dv a
This gives,t
by substituting, for Su, the value of m [5708a] , and putting also a^=a, it becomes
.' . r^-fr>/M [5708e]
da = — o-a.v, as in [5709].
(3115.) The term p.-.sin. (cr — a) [5707/], is to be added to the second
e
a
member of [4973c] ; therefore also, to that of [4973^], which is deduced from
[4973c]. Now, it is evident, from [4973A], that the effect of this will be to add to the
e e [5710a]
second member of the equation [4973], tlie term p.-, or p.-; without altering
a, a
[4974]. To find the effect of this additional term of [4973], it is only necessary to notice
it, together with the chief term of that equation,
[57106]
dv
neglecting the other small term, which depends on - . ; observing also, that zs
[49S0], is deduced from [4978], which is not altered, by the introduction of the terms [5710c]
157071. Moreover, it follows, from r4978a,5228o1, that c — — is nearly equal to
^ A' -^ ^ [57I0rf]
unity. Substituting this in [5710&], and neglecting the terms of the order e^, it
becomes — '^'~lf~ ' '° which we must add the term p.- [5710a] ; and we shall [5710«]
obtain [5710], representing the equation [4973], adapted to the present case.
1(3116). Putting, for a moment, -^=x, and also a, = a, we find that [5710] may [57iia]
be put under the form
2(fa , ,?x , _,
~rfr + ^"^ = ^' °'' —=\?'^^; [57116]
whose integral is log.-=JP») / being a constant quantity. Now, pv being very
small, we have very nearly \^xi ■=-\o^.{\-\-\pv) [58] Int.; hence [57116']
692 THEORY OF THE MOON; [Méc. Cél.
e
[5711] - = constant. {l+i/3zj} ;
consequently,
a
[5712]
î^ecular
inequali-
ties of ilie
€ = constant. {1 — (a — ^[3).i?},
peHgV^ The ratio of the excentricity to the semi-major axis is, therefore, subjected to a
and
«cen- secular equation, arising from the resistance of the ether; but it is insensible,
[57131" ^'^ comparison ivith the corresponding acceleration of the moonh mean motion ;
because this last acceleration is, as we shall soon show [5714], multiplied
by the square of v. This resistance does not produce any secular equation in
[5713'] the motion of the perigee [5710c].*
[5711c] -=i_|_J|3y; consequently, x =z - ^ f . {\ -\-\^v) , as in [5711].
rwilrfl Moreover, we have in [5709], a = constant X {1 — a ■v\ ; substituting this in [5711],
after multiplying both members by a, we get [5712]. If we represent the increments of
l_o7Ile] ^^ ^^ arising from this cause, by &a, 5e, respectively, we shall have, as in [570Sc,5712],
the following expressions;
[5711/] Sa= — aa . u ; 6e = — e . (a — i fi) . v;
e being the constant factor of [5712]. These values will be of use in the next note.
* (3117.) Neglecting terras of the order e^ , y^, we have, in [5081^],
[5714a] dt = a^.\l—2e.cos.cv].dv,
in which cv is used for c v — w, for brevity. Supposing this quantity to vary, by
augmenting « by 5a, and e by (5 e [ 571 1/], it will be increased by
1 3
[57Hh] § a'. 5a -ll — 2e. cos.cr \ . dv — 2 a~. 6e .cos. cv . dv.
Substituting the values [5711/], it becomes, successively,
3 -3
— fa.r(^.{ l—2e.cos.cv\.vdv.-]-a''^.e.{2a. — ^).vdv.cos.cv
[5714c] 3 == , V ,
t= — 1«^. ai'.cZy4- a^. (5a — ^).e.vdv.cos.cv.
The integral of this last expression gives the corresponding increment of t -\- s, which
will, therefore, be represented by
[5714d] — 3 ^2-_ (i^^a^ a^.(5a — p) . e .^v.sm.cv-\-cos.cv.l ;
as is easily proved by differentiation, and putting c=l. We can neglect the part
depending on e.cos.cv, which may be considered as included in the elliptical motion ; then
VII.vi.§-29.J EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 693
The expression of dt [5081;^] gives, in the value of t-\-s, the terms
[ôlUdM-l,
— f .o-t?® 4- (5a — ^).ve.sm.(^CV — ro), [Incrementof HeJ [57141
Substituting ^ + s + 2e.sin.(ff — -^ for r, we shall obtain, in the [^714']
^ Secular
expression of v, the secular equation,* L'fTe'"'
moon's
(.V = f .a<' — (2a — ^).i.e.sin.(cf — ^) ; [Secular equation in v] "'[57151
therefore, the resistance of the ether produces in the moori's mean motion, a
secular equation, tvhich accelerates that mean motion, without producin<f any t^'^^^l
secular variation in the motion of the perigee.
We may prove, in the same manner, that the resistance of the ether does not (57171
produce any sensible secular equation, either in the motion of the nodes, or in equalîues"
the inclination of the lunar orbit to the ecliptic.i v'">'> ^"'^
tioii are
insensible.
putting ft- =n ^ = 1 [4S27,5701rt], it becomes as in [5714]. This contains the syware of r57i4ei
V ; but Sa, Se [571 1/J depend chiefly on its frst power ; hence it is evident, that the
secular variation of the mean motion [5714], must be much more sensible than those of [5''14/]
n and e, as in [5713, &;c.].
* (3US). Putting c==l, and £=0, in [570li], we get / = u— 2e.sin.(cf-«). [57,5^1
Transposing tlie last term, and substituting in it t for v, which may be done, if we
neglect terms of the order c^, we get d= ^ + 2£.sin.(f< — -n) [5714']. Substituting this
in the first members of [57I56,c], and neglecting e^, they become as in the second [5715a']
members of these expressions ; their sum gives the value of the function [5714], as in
[57 15 J] ;
— f.a«2 = — |-a<2 — 3o..t.e.s\n.{ct — ra) ;
(5a — ^).i'e.sin.(cr — to) == +(5a — p). i.e. sin. (c^ — ra) ;
Sum =— £a;2-j-(2a — p).<.e.sin.(rt- to). [5715^^
This last expression represents the correction of t [5714, 571 5a] ; and it is evident, that
we must change its sign, to get the corresponding correction of v, as in [5715]. [5715e]
f (3119.) In finding the secular motions of y, Ô, depending upon the resistance
of the ether, we must proceed with the equation [4755,or505lè], as we have done with
[4754, or 4973c], in finding the secular motions of e, w, [5692— 5715] ; making the [5717a]
necessary changes, to correspond to this case. In examining the reductions of the equation
vor,. III. 174
[57156]
[5715c]
694 THEORY OF THE MOON; [Méc. Cél.
Hence it follows, that the resistance of the ether can become sensible, in the
moon's mean motion only. Ancient and modern observations evidently prove,
[5718] that the mean motions of the moon's perigee and nodes, are subjected to very
sensible secular inequalities. The secular motion of the perigee, deduced
from the comparison of ancient arid 7ïiodern observations, is less by eight
or nine sexagesimal minutes, than that which results from the comparison
of the observations made in the last century. This phenomenon, of ichich no
doubt can remain, must, therefore, depend upon some other cause than the
resistance of the ether. We have seen, in [4983, &c.], that it depends on the
[5720] variation of the excentricity of the earth's orbit; and, as the secular equations
lnTih°L resulting from that variation satisfy, completely, all the ancient and modern
onthf observations, ivemay conclude, that the acceleration, produced by the resistance
of an ethereal fluid, on the moon's mean motion, is yet insensible.
[5719]
moon's
raean
iiiotiuii if
insensibli
30. The acceleration, produced by that resistance in the mean motion
rs-an ^/ ^^*^ earth, is much less than the corresponding acceleration in the moon's
mean motion. To prove this, we shall resume the formula [931] ; and, if
we apply it to the earth, we shall get, in the expression of 6v', the term,*
[5722] 6v' = — '—- .ffdv'.d'Q' ;
[5717rf]
[4755], we find, that the integral expression, in the first member of [5702], is multiplied, in
[57176] [4755], by the factor j^ + ^ ^ which is of the third order in y [5034a — b] ;
therefore it may be neglected. Now, it is on this term, that the value of a [5702, 5705]
chiefly depends ; and a produces also the part of the secular inequality of the mean
"^ motion corresponding to the square of the time, which is the most important part of the
efiect of the resistance of an ethereal fluid [5714/]. Hence it appears, that the remaining
terms of Q produce, in like manner as in [5713], only insensible secular inequalities, in
comparison with that of the mean motion.
* (3120). The chief term of [931] is,
[5723a] —.ffndt.AR;
[5722i]
which may be reduced to the form [5722], by changing mit into dv [4828] ; àR
into AO [5438]; m- into S [914', 5722'], and tlien accenting the letters to conform
to the present notation ; the mass of the earth being neglected, in comparison with that of
the sun, in estimating the value of n-
VII.vi.§30.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 695
S being the sun's mass ; supposing the sura of the masses of the earth [5722]
and moon to be equal to unity ; and, that the quantity Q', in the earth's
motion, corresponds to that which we have denoted by Q, in the moon's [5723]
theory. Moreover, the differential characteristic d' corresponds to the
sun's co-ordinates. Then we have,*
(-ri-\ (~j^)' ( / ' ) being the forces acting upon the earth, parallel [5725]
to the axes x', y, z', by means of the resistance of the ether. If we
neglect the excentricity of the earth's orbit, and represent the element of ^ „„,
° •' -^ _ [5/26]
the time dt by the differential of the moon's mean motion, we shall have,
as in [5672], for these forces, the following expressions ;t
* (3121). The equation [5724] is similar to that in [5G74] ; and, it is evident, from
[5673c], that the quantities
/dq'\ /dq'\ /rfQ'\
[l^'r W> U^'J' [5^24a]
represent the forces acting upon the earth, parallel to the axes of x', y', z', and arising
from the resistance of the ether upon the earth.
f (3122). These forces are represented by the expressions [5672] ; and we have, as
in [56876,c, 5681, 5683, &.C.], by neglecting the excentricity of the orbit,
dx ^ — a'dv'. s'm.v' ; d y' ^a'dv'.cos.v' ; dz' = a'ds' ;
^ilx'2-\-dy'^ + dz'^ = r'dq' = a'dv'.
Substituting these values in the three expressions [5672], they become respectively.
[5727a]
E.a\-^-^. sm.v ; -K.a\ — .cos.r ; -K .a\- . --. [57276]
Now we have, very nearly, dv = mdl [56S7(/] ; substituting this in [5727i], we get the [5727cl
expressions [5727] ; which represent the values of
m m &?)•
respectively. Substituting these in the second member of [5724], and also the values
dx' ^ — a'. mdt. sin. v' ; dy' = a'.mdt.cos.v' ; dz' = a'ds' ; [5727d]
which are deduced from [5727a,c], we get,
d'q= — K'.a'^m\dt. i sin.V+cos.V + ^^.|.' j . [5727e]
696 THEORY OF THE MOON ; [Méc. Cél.
J f
[5727] K'.a'^.m^. s'm.v' ; — K'. d^.m^. cos. v' ; — K'.a^.m. — ;
dt
els'
[5727] therefore, by neglecting the square of —, we shall have,
[572S] d'Q' = — K'. a' \ m\ d t ;
which gives,*
[5729] <5 V' = -^- .ffdv'. d'Q' = 1 . ^ .
[5730] We must put K' = H'.<f(u') [5699]; H' being a constant quantity,
depending on the surface, and on the mass of the earth. Hence, the secular
equation, produced by the resistance of the ether, in the mean motion of the
earth, is,
[5731] iV = • [Secular equation of tho earth]
S
The corresponding acceleration of the moon's mean motion is, hy what
precedes [57306],
Neglecting the term depending on the square of ds [5727'], and putting sin.V+cos.V=l,
it becomes as in [5728].
* (3123). Substituting, in [5722], the value of d'Q' [5728], and dv' = mdt [5727c],
it becomes, by noticing only the part depending on t^,
[5730a] l.K'.a'*.m\ffdf- = ~.K'.a'\m'U^ as in [5729] ;
substituting Z' [5730], we get [5731]. The acceleration of the moon's mean motion,
depending on l^, is | a i^ [57 1 5] ; and, by substituting a [5705], it becomes,
3
[5730i]
i.H.a\mt^[^-^-m.<p'{u')Y,
which is easily reduced to the form [5732], by using -,= «' [4937w]. Again, if we
3
[5730c]
change the sun's mass m' [4757"] into S [5722'], also w into m», nearly [5094],
we shall find that the expression [4865] becomes,
— — m, or S= — ^> as m [5733].
Vll.vi. s^30.] EFFECT OF THE RESISTANCE OF AN ETHEREAL FLUID. 697
6V =^ ^.H.(l^.a'.mt^.j3:(u') ^.•'P'(«')f' [Secular equation of Iho moon] [5732]
S a^
Moreover, we have ~ = wr \5130c] ; therefore, the acceleration of [5733]
« •' "-
the moon's mean motion, is to the corresponding acceleration of the earth's
mean motion, as unity is to,*
2H'.m. ip(tt') secular motion of the earth
TT ^r. / i\ ™ // /N> secular motion of the moon '
1Z.J3ç(h) — -.<p'(m)J
[5734]
1 • • H'.m 1-1 '" // l\ I4. rr-o-T
consequently, as unity IS to -|. ; neglecting the term ;.<»(?«;. It L^'-^^J
is evident, that,t
* (3124). Dividing the expression of &v' [STol], by that of 5v [5732], and
Hibstituting S [5730c], we get [5734]. If we neglect the term of the denominator of [o'34a]
[5734], which is multiplied by the small quantity — ; we find that the numerator and
denominator become divisible by ç(m')) and the expression changes into i-~n~- [57346]
t (3125) The resistance, which the moon suffers, must evidently be proportional to
square of the moon's seaii-diameter
mass of the moon
and that of the earth is proportional to
[5736o]
square of the earth's semi-diameter [5736a']
mass of the earth
Now, these quantities are to each other as H to H' [5699,5730]; hence we get,
H' mass of the moon square of the earth's semi-diameter
H mass of the earth square of the moon's semi-diameter ' '■ '"^" J
If we take, for the moon's semi-diameter, the angle under which it appears when viewed
from the earth, at its mean distance ; and, for the earth's semi-diameter, the angle under [5736c]
which it appears when viewed from the moon, or the moon's horizontal parallax ; we
shall find, that the expression [STSG?-] becomes as in [5736]. Substituting the values
JTt
[5737 — 5738'], we get [5739]. Substituting this, and m [5117], in f--^ [5735], [5735^]
it becomes as in [5740].
VOL. III. 175
(398 THEORY OF THE MOON; [Méc. Cél.
H' mass of the moon square of the moon's parallax
H mass of the earth square of the app. semi-diameter of the moon
From observation, we get,
[5737] The moon's apparent semi-diameter = 943' ;
[5738] The moon's parallax = 3454' ;
[5738'] and, in [4631], the moon's mass is -— of that of the earth ; therefore,
we have,
[5739] g-'=^ 0,195804.
Hence it folloivs, that the acceleration of the earth^s mean motion, produced
157401 % ^'*^ resistance of the ether, is equal to the corresponding acceleration of the
moonh mean motion, multiplied by 0,0097642 [5736ti] ; or, about one
hundredth part of the moorih acceleration.
APPENDIX.
Presented, hy the Author, to the Board of Longitude of France
August 17, 1808.*
The object of this appendix is to render more complete the theory of the
perturbations of the planets, which is given in the second and sixth books. In
striving to give to the expressions of the elements of the orbits, the most simple
forms which they can attain, we have been able to make them depend wholly
* (3126) This paper was given by the author, as an appendix to the third volume of
this work ; it was not, liowever, published, till after the appearance of the fourth volume ;
which is referred to in several places, as in [5764', 5975]. The improvement, made by
Mr. Poisson, in the demonstration of the permanency of the mean motions, which is
treated of in [5794' — .5846], was made known to the National Institute of France, in a
paper, presented June 20, 1808, and printed in the eighth volume of the JoJirnrtZ f/e rEcole
Polytechnique. This first demonstration was followed by a much more simple one, given
by La Place, in this appendix ; and some improvements were afterwards made by him,
and published in tlie fifth volume of the present work [12508, &ic.]. La Grange also gave
an elegant demonstration, founded upon the principle of the variation of the constant
quantities, in the Mémoires de l'Institut de France, for 1808, Stc. ; and in the second
edition of his Mécanique Analytique. Subsequently, the subject was resumed by RJr.
Poisson, in the same volume of the Journal, and in the Mémoires for 1816, with important
improvements; in which he extended the demonstration of his theorem on the mean
motions, so as to include terms of the third order of the disturbing masses, arising from those
of the secontZ order in the disturbed planet;and then,byinduction, he supposes this will hold
good for all powers of the masses, so far as they depend on the elements of the disturbed
planet. He also demonstrated this remarkable theorem, ' That the perturbations of the
rotatory motion of a solid body, oÇ any form, arising from forces of attraction, dejiend upon
[5741a]
[57416]
[574 1 e]
[5741d]
[5741e]
[5741/]
[5741^]
Elemcnls.
Second.
Third.
[5742]
Foiirth,
Fifth.
Sixth.
700 APPENDIX, BY THE AUTHOR ; [Méc. Cél.
on the partial differentials of a single function [913, 1195, 1258, i&c.],*
[5741'] taken relatively to these elements ; and, it is remarkable, that the coefficients
of these differentials are functions of the elements themselves. These
elements are the six arbitrary quantities of the three differential equations
of the second order [915] ; by means of which, the motion of each
planet is determined. Supposing the orbit to be an ellipsis, which is
[5741"] variable at every instant, the elements will be represented in the following
manner :
First. The semi-major axis, on which the mean motion of the planet
depends, a ;
The epoch of the mean longitude, s ;
The excentricity of the orbit, e ;
The longitude of the perihelion, to ;
The inclination of the orbit to a fixed plane, ? ;
The longitude of its node, è.
La Grange gave, a long time ago, the above-mentioned form to the
differential expression of the greater axis [5786] ; and proved, by means of
[5742'] it, in a very elegant manner, the invariableness of the mean motion, noticing
only the first power of the disturbing masses. This invariableness was first
discovered by me ; neglecting, however, the terms of the fourth and higher
[574i;i] the same equations as the perturbations of a slaghpartide of matter, attracted towards a
fixed centre;' so that, the precession of the equinoxes, and the nutation of the earth's
[5741i] axis, can be expressed by tlie same formuhis as the variations of the eUiptical elements of
the planets. We had Intended to 2,ive a particular account of these improvements of
[5741A] La Grange and Poisson, together with some notice of the papers which Mr. Lubbock has
published, on the secular and periodical inequalities of the planets, in the Transactions of
the Royal Society of London, in 1830, 1831 ; but, we'have been induced to postpone this
[5741Z] notice, by reason of the great length of the appendix to this volume. We shall, however,
resume the subject in the commentary on the fifteenth book.
* (3127) The function here spoken of is R [913]. The differentials of the
[5741m] elements a, s, e, &c., are given in [1 177, 1345, 1258, 13376, Sic] ; and they are
collected together, with improvements, in [5786 — 5791].
[5744]
VII.App.Int] INTRODUCTION. 701
powers of the excentricitics and inclinations of the orbits, which is ,,.,,,
^ [5/43]
sufficiently accurate for the purposes of astronomy. I have given, in the
second book. [1258,1337, fcc], the same forms to the differential expressions
of the excentricity of the orbit, of the inclination, and of the longitude of
its node; nothing more is required, than to give the same form to the
differential expressions of the longitudes of the epoch and of the perihelion ;
this I have now done in the present appendix.
The principal advantage of this form of the differential expressions of the
elements is, to give their finite variations, by the development of the function,
which is denoted by R, in the second book [913, &c]. If we reduce this
function into a series of cosines of angles, increasing in proportion to the
time [lOllj&c], we shall obtain, by taking the differential of each term, the
corresponding terms of the variations of the elements. We have endeavored
to satisfy this condition in the second book ; but, we can do it in a more
simple and general manner, by means of some new expressions of these
variations. These last expressions have also the advantage of proving clearly,
the beautiful theorem discovered by Mr. Poisson, on the invariableness of the
mean motions, noticing the square of the disturbing force. We have proved,
in the sixth book, by means of similar expressions, that this uniformity is
not altered by the great inequalities of Jupiter and Saturn [3906"], which is
the more important, as we have shown in the same book [3910 — 3912],
that these great inequalities have a considerable influence upon the secular
variations of the orbits of these two planets. The substitution of the new
formulas which we have just mentioned, shows, that the uniformity of the mean [5745]
motions of the planets is not troubled by any other periodical or secular
equation. These expressions give also, the most general and simple solution
of the secular variations of the elements of the planetary orbits. Lastly,
they give, in a very simple manner, the two inequalities of the moon's motion
in longitude, and in latitude [5967,5971], depending on the oblatenessof the
earth, which have been determined in the second chapter of the seventh book
[5357,5389]. This confirmation of the results, which have been obtained
relative to these inequalities, is interesting, because we can get, by comparing
them with observations, the ellipticity of the earth, in as accurate a manner,
to say the least, as by the direct measures ; with which they also agree, as [5746]
well as can be expected, considering the irregularities of the earth's surface.
VOL. 111. 176
702 APPENDIX, BY THE AUTHOR j [MécCél.
In the theory of the great inequalities of Jupiter and Saturn, which is
given in book VU, we have noticed the fifth power of the excentricities and
inclinations of the orbits. Mr. Burckhardt has calculated the terms depending
on these powers. But, it has been since found, that the inequality resulting
from these terms, is taken with a wrong sign. Therefore, we shall correct,
at the end of this appendix, the formulas of the motions of Jupiter and
[5747] Saturn, which are given in the eighth chapter of the tenth book. This
produces a small alteration in the mean motions, as well as in the epochs of
these two planets; and this change satisfies the observation of the conjunction
of these two planets, made by Ibn Junis, at Cairo, in the year 1007. This
observation varies from the formulas, by a quantity which is much less than
the error to which the observation is liable. The ancient observations, quoted
by Ptolemy, are equally well represented by these formulas. This agreement
proves, that the mean motions of the two greatest planets in the system are
now well known, and, that they have not suffered any sensible variation since
the time of Hipparchus ; it guarantees, for a long time, the accuracy of the
tables which Mr. Bouvard has constructed, by the theory, and which the Board
of Longitude has just published.
In the same meeting at which I presented these investigations to the Board
of Longitude, La Grange also communicated his learned researches on the
same subject. He has, by a very elegant analysis, expressed the partial
differential of R, taken relatively to each element, by a linear function of
the infinitely small differences of these elements; in which the coefficients of
these differences are functions only of the elements themselves. If we
determine, by means of these expressions, the differences of each element,
we may, by proper reductions, obtain the very simple exjn-essions which we
have given ; and, as they can thus be deduced from such different methods,
their accuracy will thereby be confirmed.
1. We shall resume the expression of ede, given in [1262] ; putting,
[5750] far greater simplicity, p-=l, we obtain,*
* (3128) In the equations [1262,5751], terms of the order of the square of the
[5751a] jjigjurbing forces are neglected [1253(7, &;c.] ; but it is correct in terms of the frst order of
the disturbing forces, for all powers of the excentricities and inclinations. Tiie value
_3
[5751i] fji.= l, being substituted in [541'], gives »i = a ^ which is used in [5785, &:c.].
[5754]
VII. App.'^l.] INVESTIGATION OF rf^, di, da, dc, (fe, dp, dq. 703
ede = a.ndt.^ï=7c . (^)-«-(l - «')-^'^» ^^''^^^
In this equation, / is the time ; nt the mean motion of the planet m ; [5752]
a the semi-major axis of its orbit; e the excentricity; v the true longitude 1-57531
of the planet ; R a function of the co-ordinates of the tivo planets m, m' ;
so that, by naming these co-ordinates x, y, z, x', y', z', respectively,
we shall have, as in [949, 949'],
R _ m' . (^''+H'+^^') _ ^ . [5755]
P being the distance of the two planets from each other; so that we shall have,
P = v/ ! G^' - ^y + (y- i/r+ (^'- ^r \ ; ^5756]
r' is the radius vector of the planet m' ; r that of the planet m ; lastly,
the characteristic d refers only to the co-ordinates of the planet m [916'].
We may observe, that to obtain \-j-\ we must develop R in a series
[5757]
of angles proportional to the time t ; then take its differential relative to nt, [5758]
[5759]
[57C0]
and divide it by ndt, adding to the quotient the partial differential (~\
^ being the longitude of the perihelion of the orbit of m. For, we must
not notice, in finding the partial differential of R, relative to v, the angle
nt, introduced into R, by the radius vector r of the planet m, or
by the periodical part of the elliptical expression of v, developed in a
series of sines of angles, proportional to the time. Now, in these functions
[669], the angle nt is always connected with the angle — z,, which is [5761]
introduced into R, by this means only ; therefore by adding to the partial
differential — , the partial differential Ç~\ we shall have the value* [5762]
* (3129) The two first terms of ,it-}-s [669], are not connected with — ra, but,
It IS found ill all the remaining terms; so that we have v = 7t( -{-e + cp{nt + i — vs), [5763a]
<p being the characteristic of a function. If, for a moment, we consider iï to be a function
of V, as in [3742], and represent it by R=f{v), we shall have, by the usual [57636]
notation, (-j=/'(r). Substituting ^ [5763a], in R [57636], we get, ^57^3^^
I^=f{nt + e + cf>{nt-j-s — z=)l.
704 APPENDIX, BY THE AUTHOR; [Méc. Cél.
'dR\
\d7j
[5763] of (t-)' Hence, the preceding expression of ede , will give, ^
[5764] de = ^V^-t .( I _ v/ï^^)-diî + "-^^^'.nrfL (f ) .
[5764'] Then we have, as in [7886],t
Its differentials, considering successively, nt and «, as the variable quantities, also
[5763c'] /rf.iw)\ ,, ,
putting, for brevity, nt-{-s—-a- = \v, ( ~T~ ) = <? V^^)' S'^^'
[5763i] W<""^ * + *'^"^ "^ '~^^ i •/'! "<+ « + 9 («< + s — «) i •
[5763e] (^) = — <P'("^ + s — «) ./'{n< + 6 + ?(«< + £ — w) J.
The sum of the two expressions [5î63tZ,e], being successively reduced, by using [57G3a,c']^
becomes as in [5762] ; namely,
t5^63/] ^+(^)=/'l,.+ .+ , (,.+ .-«)!=. /'W = (^).
* (3130) Substituting the value of {~-\ [5763/], in [5751], it becomes,
[5764a] cde = a.ndtVY^^. | ^ + (^) ]—-0 - «^)-d«-
Dividing this by c, and making a slight reduction, we obtain [57G4].
t (3131) The formula [5765] is the same as that which the author has demonstrated
in [7S86], in nearly the following manner. The first of the equations [606] becomes, by
[5765a] changing the origin of the time t, as in [668']; ni-\-s—u:=u—e.sm.u; and, if
we also change 7it into fndt, as in [5793], we shall get,
[57656] fndt-\-£ — w = u — e.sin.M.
In which fndt-\-s is the mean longitude of the planet m ; fndi-\-s — w its mean
[5765c] anomaly; v — « its true anomaly ; and u its excentrical anomaly [603", Sic, 668', 669].
The differential of [57656], supposing the ellipsis to be invariable, is,
|-5765^-| ndi = rfii.(l— c.cos.m) ;
and, as this must also hold good for the variable ellipsis [1168'"], we may take the general
[5765e] (jifj-grgntial of [57655], supposing all the elements to be variable ; subtracting from this,
the expression [5765fZ], we get,
[5765/] de — dia = <?«.(!— e .cos. w) — de.sm.u ;
VII. App.§l.] INVESTIGATION OF d^, ds da, de, cfe, dp, dq. 705
, , rfra.fl — e.cos.uY <Ze.sin.M.(2 — e^ — e.cos.w)
d.-d. = - -^-^^^^^^ - -^ ^-^-^^ ^. [5765]
In this formula, u is the excentrical anomaly [603" — 604], and s the [sree]
longitude of the epoch [669']. We may put the second member of [5765]
under the form,*
supposing du, in the second member, to be restricted to the variations arising from [57G5g]
s, zi ; instead of referring to the time t, as in [5765fZ]. The tliird of the equations
[606] becomes, by changing the origin of t, as in [5765a] ;
tang.è.(i)— a) = ^^l±5 .tang.jM. [g^gg^^
If we take its differential, supposing s, zs, u, to be the variable quantities ; and u,
to vary as in [5765^] ; we shall get, by multiplying by '2,
dzs du /\ _i_e 2rfe.tang.^M
~~cos.2J(r— ^) "" 7ô^u ' \/ T^^e "^ (l-e).\/III^ ' [^^"^^'J
Now we have, by using [5765A],
cos.2i(t.-^) ^ l+tang.^è-("-«) = 1+ j3^^-tang.2iw= l+tang.^è«+^.tang.=èw
1 2e l^~!:>oK\
= ■ + , .tane.^iw.
Substituting this in [5765j] ; then, multiplying by cos.*^^ n ; and reducing, by putting,
cos.^i«.tang.J« =cos.^M.sin.àu = J.sin.u ; (cos.|M.tang.^it)''= sin.^âM= J— |.cos.m ; rr.',cK^^
Multiplying this by \/ -, T^ > and reducing, we get,
rfra.(l — e.cos.w) rfe.sin.M
du = —
^/nr72 1— ea I5765n]
Substituting this value of du, in [5765/], we get the expression [5765] ; in which rg^gg ,
nothing is neglected.
* (3132) We have — (1— e.cos.M)a = — (1 — e^) -|- e.(2.cos.M— e — ccos-^m),
as is easily proved, by developing its first member. Substituting this in the numerator of [5767a]
the first term of the second member of [5765], it becomes as in [5767].
VOL. III. 177
706 APPENDIX, BY THE AUTHOR; [Méc.Cél.
[5767] — ctro.^i_e2-}- ____.(^2.cos.?t — c — c.cos.^u) — ^ _^ .(2 — c^ — e.cos.w).
[5768] The excentrical anomaly %i, is given in terms of the true anomaly v — ro,
by means of the equations [603, 606 J,
«.(1— e2) ., .
[5769] r = —. — 5^ — 7 — - — , = a.{\ — e.cos.?*.);
^ \-\-e.cos.{v — -nr) ^ ^'
whence we deduce,*
e+cos. (v — to")
[5770] COS.W = -^= ^ ^ ;
l/l — ee.smAv — ra)
[^'^'^J l+e.cos.(D-«) '
consequently,t
* (3133) Dividing the two values of r [5769] by a, we get, by successive
reductions,
(1 — e^) e.cos.(j> — «) + 6^
[5770a] e.COS.W = 1 ■
1 4" e.cos.(« — vs) 1 + Ê.cos.(î) — «) "
[5770a'] Dividing by e, we obtain [5770] ; and if we put for a moment, for brevity, cos.(« — w)=w,
it becomes,
e -|- w
[57704] COS.M = -q:^^i
whence we obtain,
sm.M = V/(l-cos.^«) = 1/ 1 -(i+i^ = ^1^-,^ = ^^
f^''^°^l _v/(l-e2)V(I-w.)
l-[-£W
[5770^1 Re-substitutingthe values of w= cos. (« — to), and \/(l — w^) = sin.(t; — to), it becomes
as in [5771].
f (3134) The value of cos.m [5770i], being substituted in the first member of
[5772nl, we get, by successive reductions, the expression in its last member. In like
manner, from sin.w [5770c], we get [5772i],
2e4-2w 2(1 — e2).w 2(1— e2).w.(l+ew)
2.C0S.M — 2e = -— ■ 2e = — — = —- -r„
l+ew 1+ew (1+ew)''
[5772al ^^_^2^
(1+ew)
,.{ 2w+2ew2} ;
VII.Api).§l.] INVESTIGATION OF d^, ih, da, de, dz,, dp, dcj. 707
r . (2.C0S.M — e — e.cos.u) ^ . (2 — e^ — e.cos.w) 1
/l— ee ^ 1 — e^
. {2.cos.(« — Ts)-\-e-\-e.cos.^v — «)! , _,
= V^l— e^.^ \, , T ^^^ ^.erf^ 2 [5772]
}l-|-e .cos.(î; — rajj''
•^ jl-(-e.cos.(î; — rajp "■
Substituting the values of e(h, de [1258], we find, that the second
member of the equation [5772] can be reduced to the following form ;*
('-'•' !e_e..|.
(14_ew)2 •» >• [577261
The sum of these two expressions, gives,
2.C0S.M — e — e.cos.^M = - — ; _ 5 2w+e4-evv^ ? . rK'>'m ^
(l+cw)2 >■ ^ ^ ^ [5772c]
Substituting this in the term which is connected with dis, in the first member of [5772],
get the term depending on dzs , in its second member [5772, line 2]. In a similar manner,
we get, from the value of cos. m [5770i],
2 f \ /" («+w)\ /eew — w\
e-' — e.cos.M = eJe — cos.m) = e . c — -^ — — - ) = e . )
^ ^ V 1+ewy \ 1+ew J
[5772cf]
l^ew
Adding to the first, and to the last members of this expression, the quantity 2.(1 — e^) ;
we obtain,
€W ( I — C^^
2 — e^— e.cos.M = 2.(1— e^) — . (1— e^) = -^^ ^ . J2.(l+ew) — ew j
1-f-ew 14-ew ' ^ ' / i
1 + ew ' ^ *
H»„„o 2 — e^— e.cos.M 2+ew ,,•,•., . . /i:=i2.sin.(t>-w)
Hence = ■— — ; multiplymg these by sm.M = ^ [K770f-\
1 — e2 1+ew fjb J l-\-c^v P'7^/]
[5771]; and substituting the result in the term depending on de [5772 line 1], we get
the corresponding term of the second member [5772 line 3].
* (3135) If we substitute f^ =1 [5750], in [1258], we shall obtain the following
expressions of ed-a, de, in which terms of the order of the square of the disturbing [5773a]
forces are neglected [1253'] ;
708 APPENDIX, BY THE AUTHOR; [Méc. Cél.
[5773] Za.ndt.r.l— ] ; [Value of the function 57721
[5774] and, as we have r.f— j = «/ — j [962], it becomes,
t^'^^^1 2a^.7ldt. •(-;—). [Value of the function 5772]
\daj
Hence, the expression of di — (Zro [5765] gives the following very simple
[57736] edra = — |^;=.sin.(D— w). { 2+ e.cos.{v—zi) } . (^~ j+a^. ndtyi-e^. cos.{v—!^}.(^~j ;
[5773c] <Ze = — ^^g.{2.cos.(j;— :;i)+e+e.cos^(t)— 5!)}.(^^j— a2.n(Z^v/i^2.sin.(i;— î:).(^— j.
These are to be substituted in [5772 lines 2,3] ; and, in performing the operation, we
may neglect the part depending on i~J~] 7 because, the terms depending on edzr
[5773d] |-5772iij,e2, 5773&], are equal to those depending on de [5772 line 3, 5773c] ; and,
they have different signs ; so that they mutually destroy each other ; as is easily seen
by the mere inspection of the formulas. The remaining part of the second member of
[5773 1 [^"'^^1' arising from the substitution of the parts of [57736, c], depending on (^ j,
becomes, v.ithout any reduction, as in [5573/] ; omitting, for the sake of brevity, the
symbol îï, which is connected with the angle 1; — sr, as in [4821/] ;
rw7Qn ( — e )•",:" ^ f ) . j (2.cos.«+e+c.cos.3j;).cos.t) + (2+e.cos.t)).sin.2«J.
lonaj i (l-|-e.cos.v)'' \dr /
The terms of the factor, between the braces, being arranged according to the powers of
e, and then successively reduced, become,
[5773^] 2.(cos.2«+sin.2i;)-fe.cos.D.^l+(cos.3u+sin.2y)} = 2+e.cos.u.|l+l|=2.|l+e.cos.r}.
Substituting this last expression in [5773/], it becomes,
2.(1- e2) „ ,^ /dR\
which is easily reduced to the form [5773], by the substitution of,
[5773i] r = ,°_;^^~''' [603].
*• ■■ 1+e.cos. « '■
Lastly, the substitution of [5774], in [5773], gives [5775], for the value of the second
member of the equation [5772].
VII. App.§l.] INVESTIGATION OF f/^, di, da, de, (fo, dj), dq. 709
equation, which was lirst discovered by Mr. Poisson ;*
ds = (h . (\—^T=Tc)+2a\ (^) .ndt. [5775']
\(l(Z J Poissoii^s
expression
If we refer, as m [1030', &c.], the motion of the planet in, to that of ds.
its primitive orbit, and put, as in [ 1 032] ,
p = tang. <p .sin. 0 ; q = tang.ç . cos.^ ; [5776]
(p being the inclination of the orbit [1030'], and ô the longitude of its
ascending node, we shall have, as in [13376,ô7516j,t ^^'^'^'^^
, df /dR\
'^P=-sya.{i-ce) • \Tq) ' [5778]
, dt /dR\
Now we have, by ^ 44, of the second book, f
0 - O ■ *' + (f ) ■"' + (")■"" + (?) ■ "■ + (?)-*+(?)-''»^ '-»'
* (3136) The expression of ds — dzi [5765] is reduced, in [576/^, to three separate
terms ; of which the Jirst is — dzi.\/ 1— e^ . The second and third terms constitute the [5775a]
first member of [5772], which is successively reduced to the form 2a^.ndt . ( i~), in
[5775] ; hence we get,
ds—d^ = — dv!. /i:rr2-}-2a2. ndt . (^) ; [57756]
and, by transposing — dzr, we obtain [5775'] ; which is correct in terms of the order m', [5775c]
as in [5773a].
t (3137) We have an = a~* [575 li] ; substituting this in [1337i], we get
[5778, 5779] ; which are exact in terms of the order m' [1337iline 3]. "■'
J (3138) R is a function of fndt [5793], and of the elements a, e, zs , s, [578031
p, q. Now, we may take its differential, relative to t, considering the elements as
constant, and the ellipsis invariable. We may also take it, supposing all the quantities to
be variable, as in [1168', &ic.]. The first of these differentials, being subtracted from the [57806]
second, gives [5780].
VOL. III. 178
710 APPENDIX, BY THE AUTHOR; [Méc. Cél.
moreover, we obtain, from [1177,5750],
[5781] da = —2a\àR ;
/'dR\ AR
(nH\ n H
—-) = — ; because the angle nt is always connected with +=
.*
5
[5783]
[5784]
therefore, by substituting the preceding values of da, de, d?, dp, and
dq; we shall have this very simple equation,!
, a.ndt.\/\ — ce f<JR \
d. = — ^_ j ;
which gives,
* (3139) We see, in [953,954, Sic], that ni is always connected with s, in the
'•' "J form of nt-\-s , or rather fndt-\-s [5793] ; so that if we suppose R to be a function
of fndt-\-î , we may represent it by R = f{fndt-\-s) ; and, by using a notation similar
to that in [5763c], we have (~\=f\fndt-\-i). and i-r)'=f'{fndi-\-E); whence
[57826] ^ '' ^ ^^
we get ( — j = [ — - j , which is equivalent to that in [5782].
t (3140) Of the six terms of which the function [5780] is composed, the fifth and
sixth destroy each other, by the substitution of the values of dp , dq [5778,5779], as
[57830] is evident by inspection. Again, the second term of ds [5775'], namely 2a^.i—\.ndt,
being substituted in [5780], produces,
fdRX , fdR\ ^ , fdR\ ,„
and this is destroyed by means of the first term of [5780], namely (-—\.da, as is
evident, by the substhution of da [5781]. Hence the function [5780], is reduced to the
i^ree terms depending on de, d-a, de; taking for ds, the first term of [5775'] only ;
[5783c] namely, ds ^ dzT.\\ — \/l—e^; hence the function [5780] becomes, by the substitution
of this value of ds, and that of (~\ [5782],
/dR\ , , ( /dR\ , àR ,^ ^ ^ ■) ,
[5783.] 0 = ^- ) . c^e + I (yj + ^^ • (1 - /l-e^) \ • d^-
VII. App.§l.] INVESTIGATION OF d^, ds, da, de, dzs, dp, dq.
711
Connecting together, in one table, these different equations, we shall have,
by observing, that n^a^ [57516], and, that the sign d, affects only 1-5785]
the co-ordinates of the body m ; *
Now, tlie value of de [5764), can be separated into two factors, so that we may put it
under the following form,
a.ndt.\/ i—e'i
( /dR\ , AR , , , )
as is easily proved, by multiplying the terms. Substituting this in [5783f/], and then
dR
dividing by the common factor ("T"' ) + "37 • ( ' — \/l— e^) , we get,
dfi
0 =
a.ndi.\/l — c'S
^f) + -^
* (3141) The equations [5786 — 5791], are the same as those which are given
in [5731,5784,5764, 5783, 1337i], respectively. The equations [5787—5791] are
correct, in terms of the Jjrst order of the disturbing masses, for all powers of the
excentricities and inclinations ; but, some terms of the order of the square of the
disturbing masses are neglected.
[5783£]
[5783/]
whence we obtain dzg [5783]. Substituting this value of d-a, in that of de [5775'],
we get [5784]. The expressions of cZra, ds [5783, 5784], are exact in terms of the [5783^-]
order m', for all powers of the excentricities and inclinations ; but some terms of the
order wi'^ are neglected.
[5786a]
[57866]
[5786c]
We may observe, that, in estimating the values of dp, dq [5790, 5791], we have
taken the primitive orbit of the disturbed planet, for the fixed plane ; so that p, q, are
considered as very small quantities, of the order of the disturbing masses; whose squares
are neglected. To avoid this restriction, the author has given other forms to these
expressions in [12528, 12529] ; by taking another fixed plane independent of the primitive
orbit. Then, if y' be the inclination of the orbit of the disturbed planet to thisneio
plane, and è' the distance of its node from a fixed point in the same plane, we shall
have, instead of p, q, dp, dq [5776,5790,5791], the system of equations [5786e — g], t^^gQji
representing the values of p, q', dp', dq' ; corresponding to this plane. From these
we easily deduce the values of dy', d6' [5786/t,iJ. The investigation of these equations
is given by the author in [12513 — 12537] ; and it is unnecessary to repeat it here.
p' = sin. 7'. sin.
q' = sin. y'. cos.
[12520]
[5786e]
712
APPENDIX, BY THE AUTHOR;
[Méc. Cél.
[5786]
[5787]
Differen-
tials of the
elements,
exact in
[5788]
terms
of the
order m'.
[5789]
[5790]
[5791]
[5792]
da = — 2a\dR ; (1)
de =
(/ra =
dip =
a.ndt
V/l— e
( ! - v/]^?).di? + ^H^TE?.n./^ (g) ; (3)
(4)
(5)
e '[Te ' '
dR
^1
a.ndt f^^\
^l_e2 * \di)J
(6)
[5793]
[5793']
Moan
motion.
[5794]
[5794']
We may substitute, ia these equations, ndt . (-f) for dR [5782], and
bj this means, reduce the preceding expressions, so as to contain only the
partial differentials of the elements ; but, it is as simple, to retain the
differential dR.
In the motion, considered as elliptical, we must substitute fndt for nt,'*
if ive wish to be rigorously correct ; now, n = a~^ [5785] ; therefore, by
putting 2, equal to the mean motion of the planet m, we shall have
[1183,5750],
^ ^ fndt = 3ffa.ndt.dR. (7)
2. From these equations, ice easily dedtice the same result, as that which
was discovered by Mr. Poisson, relative to the invariableness af the mean
[5786/]
[5786g]
[5786A]
[5786i]
dq'z=
a.ndt
v/ï:
.cos.^
dR\
dy' =
a.ndt
•'•\dp') '
d&
\/l_e2.sin.7'' \dê'J '
a.ndt /dR \
i/l_e2.sin.7 \dy/
[12528]
[12529]
[12536]
[12537]
•(3142) We have the differential of the /rsi order d^^= ndt [1180", or 5794],
[5794a] ^jjich corresponds to the variable ellipsis, and, therefore, also, to the invariable ellipsis
[1168']. In the invariable ellipsis, we have n constant, and its integral is 2^=nt-\-s;
VII.App.{-2.] TERMS, IN ^, OF THE ORDER m^ mm', m'm", he. 713
motions of the planets ; noticing the square of the disturbing force. If we
denote anj finite variation by tlie characteristic 5, and vary, in R, only sjmbui
H'/ta? relates to the planet m ; observing, tiiat \~7~) ^ ~1 [5782] ; we [5794 ']
shall have,*
Substituting, for âa, 6e, or,, &c., the integrals of the preceding values
of da, de, f?=i, &c. [5786, &c.], we shall have,t
'I'erina of
ATI ^^'
Ti Ult , /- 7.\ , of the
ôR = —- . Ô (fndt) 1 order
7ldt " m'-'
arising
from the
variation
2 of the
elemeata
of the
planet m.
+ ^«--lS->*-(T!)-(f)-/^^i
+ =^-Kf)->*-(S)-(?,)>*-(f)J *
+ .^.-K?)->-CI)-(")-/--Q|- «
but, in the variable ellipsis, n is variable, and we have ?_=fndt-{-s. Hence it is
evident, that the mean motion 7it, corresponding to the invariable ellipsis, must be
changed into fndt, in the variable ellipsis, as in [5793].
[57944]
* (3143) R IS a function of fiult, and of tlie elements s, a, e, -a, p, a ;
now, if these quantities vary by the increments 3. fndt, 6s, 6a, 6e, 6zr, 6p, on, [5795a]
respectively, we may obtain the development of R, in a series, proceeding according to
the powers and products of these increments, by means of the formulas [610 — 612, &ic.].
If we retain only the first power of these quantities, and put, for ( — V its value, forgsft]
deduced from [5782] ; namely, -- ; the increment of R will become as in [5795]. [5795c]
This equation is correct in terms of the order m'^; because, R [5755J is of the order m';
and the variations 6s, 6c, Stc, which depend on R, are also of the order 7n';
therefore, the terms of the second member of [5795] are of the order m'^; and the i^'^^"]
neglected terms of the order R6s^, R6e^, he, must be of the order m'^.
t (3144) The integral of the equation [5786] is a ;= constant — 2.fa-.dR; the [5796a]
VOL. III. 179
714
APPENDIX, BY THE AUTHOR;
[Méc. Ctl.
[5797] To obtain the value of d . ] àR -.i(fndt)i, given bjthe equation [5796],^
constant quantity being equal to the value of a, at tlie commencement of the integral.
[57966] Hence the increment of a, is represented by lîrt = — 2 fa^. dR ; so, that if we
put fdR = R', and integrate by parts, we shall have, successively,
[5796c] Sa = — 2 fa^. dR = — 2 a^. R' +fR'. 4 ada ;
as is easily proved, by taking the differentials of these expressions of (Sa, and re-substituting
rWQôc'l R' = fdR . Now, R' and da [5786], are botli of the order m' ; therefore,
fR'.4ada, is of the order m'^ ; and if we neglect terms of this order, in Sa, which
will only produce terms of the order m'^, in [5795], we shall have,
[5796rf]
[5796e]
6a
2fa^àR = — 2a^.fdR.
[5796/]
[5796êr]
[5796A]
[5796i]
[5796ft]
Hence it appears, that in finding the integral of a-. dR , we may bring the factor a^
from under the sign of integration ; neglecting terms of the order ?«'^. For similar
reasons we may bring a, e, from under the sign J^, in the integrals of the other
expressions [57SG — 5791], leaving for symmetry, « under that sign, connected with
dt , as in [5794, 5795, Sic] ; hence we get the following expressions, which represent,
respectively, the integrals of the five equations [5787 — 5791] ;
Ss = —
,, _ "V}=:îî . (i-\/i— e) .fdR + "-±^-11^
Jndt
/dR
a.\/\ — ce /• ,
(5a = . Jndt .
dR
Je
— == . fndt . T- ) ;
V/l-ee ^ \dq J
Sp :
a ^ , /dR\
[5796i] Substituting these, and also 5a [5796rf] in [5795], we get [5796], which is exact in
[5796m] terms of the order m'-. If, for brevity, we represent by R, the four lower lines of
the second member of [5796], we shall obtain, by substitution and reduction.
SR-^ .5{fndt)
ndt
R,)
[5796n]
[5796o] so that dR, represents the value of the function which is mentioned in [5797].
* (3145) If we vary in R, what relates to the planet m, as in [5794", &ic.], we
VII.App420 TERMS, IN ^ OF THE ORDER n?, mm', m'm", &ic. 715
we must take its differential, relative to the quantities corresponding to the
planet m only [5785]. To obtain the differential relative to the elements of
that planet, it tvill be sufficient to suppress the sign f, which has been
shall get the expression of &R [5796] ; or the equivalent expression [5796m] ; and the
object of the author, in [5797 — 5812], is to prove, that à.ôR contains nothing but [5797a]
periodical quantities. The value of à.SR, deduced from [5796n], is of the following
form ;
d.6R=dA~. S (fndt) J + dR. [57976]
The calculation in [5798 — 5806] is to prove, in the first place, that the second term of
this expression d/î, , produces periodical quantities only ; the process in [5807 — 5812]
serves the same purpose, relative to the other term ; namely,
In these calculations, the terms of R, are supposed to be represented by,
M.fNdt — N.fMdt [5800] ; ^^^^^^^
and, it is very easy to reduce them to this form. For, if we change fdR into fiidt. —
ndt
in [5796 lines 2.3], for the sake of symmetry, we shall find, that any one of the lines of
the function [5796 hnes 2 — 5], is composed of two terms of the form,
[5797e]
[5797/]
A.\R, .fndt . R,—R, .fndt .R^\;
A being the factor without the braces; and R^, R^, the differential coefficients,
depending on the partial difierentials of R, which occur in that line. Now, if we put
AR^ = M ; ni?3 = iV ; the preceding expression becomes,
M.fNdt-^.fndt."^-, tS^9^/']
M and iV being each of the order m' ; therefore, MN is of the order m'~ ; and,
a
if we neglect terms of the order m'^, we may introduce the factor - , of the second [5797/"]
n
term of [5797/'], under the sign /; and then, by reduction, the expression becomes,
M.fNdt — N.fMdt, as in [5800]. [5797^]
Similar processes and redactions are used, in calculating the partof d.iiî, arising from the
variation in SR, relative to the planet m', in [5813, Sic.]; and those relative to the ^^'^^'^^'
planet m'', in [5832, Sic.].
716 APPENDIX, BY THE AUTHOR; [Méc. Cél.
introduced only by the integrals of the differeiitial expressions of these
[5798'] elements [5786 — 5791] ; and then, that expression becomes identically
nothing ;* so that, if we wish to obtain the differential d of the
[5799] function ôR Mfndt), it will suffice to take its differential relative to
ndt
nt, noticing only the quantities without the sign f [5798e]. The
expression of this function is composed of terms of the form,
[5800] M./Ndt—N./Mdt [5797^]. !Fu„ct.o.«)
M and N may be developed in terms, depending on cosines of angles of
the following forms ;t
[5801] M=k.cos.(i'n't—inti-A); N=k'.cos.(i'n't—i7it+A');
[5802] i' and i being any integral numbers, positive or negative. We must
* (3146) The integrals of the expressions [57S6 — 5791], introduce the sign / in
the values of the variations of the elements of the planet m [5196f—k] ; and by this
[5798a] means they occur also in [5796]; and, as these integrals haverel'erence to the elements of
m, their differentials relative to the characteristic d [5785], must be equivalent to the
complete differentials ; so that we shall have,
[■57985] d .fJVdt == JVd t ; d . fMd t = Mdt .
Hence the differential of the function R^ [5800], relative to d, is,
d«, = AM . fJVdt — dJV . /Mdt -f MJVdt — MJYdt .
The two last terms of this expression destroy each other, as in [579S'] ; and we finally
get,
[;5798d] dR, = dM . fJVdt — dJV . fMdt .
Hence we obtain the same rule as in [5799], for finding the differential of
rgyggg, R,= M . fJVdt — JV . fMdt [5800] ]
namely, by taking the differential of R^, supposing the terms without the sign /, to
be the only variable quantities.
t (3147) The functions M, JV [5797/], depend on R, which is of the same
[5801tt] form as the assumed values of M, JV [5801] ; as appears in [957'"]. Substituting
these in [5803], we get [5804].
VII.App.>§--2.] TERMS, IN ^, OF THE ORDER w^, mm', mm", Uc. 717
[5803]
combine these two terms together, to obtain the non-periodical terms in
AAM.fNdt — N.fMdt] ; then this function becomes,
k.indt.sm.Ci'n't — int-\-A) . fk'dt.cosJi'n't — int-{-A') 1
"^ \ _ [Function .lfi,l [5804]
—k'.indt.sin.(i'7i't — int-{-A') .ykdt.cos.Çi'n't — int-j-A). 2
The integrations which are indicated in this function, being made, we find, [5305]
that the terms destroy each other, and the whole expression vanishes.*
This agrees with what we have demonstrated, in [3906'], relative to the great
inequalities of Jupiter and Saturn. The expression of
d .< iR . 6. fndt > , [5806]
is, therefore, a periodical function.
The expression of à .} —- .à .fndt >, contains only periodical
quantities ; for, we have,t
Substituting for on, its value f in^Sfan.dR, we shall have, [5808]
* (3148) The integrations, which occur in [5804], are made in the usual manner, by
changing co«. into Mw., and dividing by iW — in; and when this is done, the terms |C805a]
mutually destroy each other. We may remark, that the coefficient of t , in the values
of Jli, A* [5801], are ejMoZ to each other, being represented by i'n' — in. It is [5805i]
useless to notice other terms, in which these coefficients are unequal ; because they
produce nothing, except periodical terms, in the function [5804]; as is evident from [5805c]
[17] Int.
t (3149) The complete differential of the first member of 158071 ^—.5 fndtl
Indt ' •' y
taken relatively to the characteristic d , contains the two terms in the second member [5807aJ
of [.'>807] ; and also the additional term — „ "' . 6 . fndt ; but this term contains
the three factors àR, dn , 5. fndt; each of which is of the order m' ; jience it
is of the order ro'^, and may be neglected ; and the expression becomes as in [5807]. [58076]
J (3150) Taking the differentials of the two expressions of ^ [57 94], and dividing
VOL. III. 180
718 APPENDIX, BY THE AUTHOR; [Méc.Cél.
[5809] d A — .6 .fndt I = 3 an . -—- . ffàR .dt + Q an . — . dt -fàR .
We may unite, in one expression, all the terms of the development of R ,
which depend on the same angle i'n't — int, and it becomes of the form,
[5810] R ^ k. cos.(i'7i't — int + A) [957'"].
Substituting it for R, in the functions — r- .ffàR.dl, and — . f àR,
it LI L lid V
[5811] we find, that they are reduced to sines of double the angle* i'n't — int+A;
[5808a] them by dt , we get n = 3 fan.dR ; or, as it maybe written, 5n^=3fan.dR,
as in [5S08]. Substituting this in the development of 5. fndt, we easily obtain,
^ggjjgjj Ô . fndt =f5n .dt = 3 ffan .dR.dt.
These values being introduced into the second member of [5S07], we get,
r^Bna-i d .{ — .5. Cndi > = — p- . 3 fCan . dR . dt 4- —- . dt . 3 fan . dR .
Each of the two terms of the second member of this expression, is of the order m'^;
[5808rf] and, if we neglect terms of the order m'*, we may bring the factor an, from under
the sign /, as in [5796rf — e] ; and then the equation [5808c], becomes as in [5809].
* (31'51) From R = Tc.cos. {i'n't — int-\-A) [5810], we easily deduce the following
[5810a] expressions,
J p If 7«
[58106] — = ¥i.s\n.{i'rJt — int-\-A) ; fdR = — ., i_. . cos.(iV< — int^A) ;
[5810c] ffàR-dt = J^^2 • sin.(iW<— in<+^) ; 7^ = — ^■î^n-oos.{i'n't—int-]rA)dt
In finding these expressions, we have neglected the variations of the elements n, n', Sic,
because they produce only terms of the order m'^ in [5809]. The product of the two
expressions [5810c], being substituted in the first term of the second member of [5809],
produces a term depending on,
r5810d] sm.{i!nt—int-\-A) X cos.{i'n't—int-\-A) ^\.s\n.'2.{i'n't—int-\-A).
In like manner, the product of the two expressions [58106], being substituted in the
second term of [5809], produces another terra depending on,
jjgjO , i. sin.2.(iVî — int -\- A) , as in [581 1].
VII.App.«§,2.] TERMS, IN ^, OF THE ORDER »«^ mm, m'm", he. 719
thus, the differential d . (—- . i . J'ndtj, contains only jjeriodical quantities. [5811']
Hence it follows, that àJR, contains only periodical quantities, tvhen ive
vary in Œ, the quantities relative to the planet m only. "^ ^
To obtain the complete value of d.Œ, tve must also vary in ôR, what
relates to the planet m'. For this purpose, we shall put R', for what R [58V3]
becomes, relative to the planet m', disturbed by the action of m. We shall
then have,*
R' = "'•(^'''+yy+^--')_ ^ . [5814^
r' p
hence,
R^^ .R+m'.(xx'+yy'+zz').Ç-^^ -^) . ^^5815]
The variation of R, so far as it depends upon the variations of what
relates to the planet m', is, therefore, equal to the variation of the second
member of the equation [5815], arising from the variations of the co- [5817]
Symbol
ordinales of m'. We shall denote, by ô', the variations which correspond à'.
From what has been proved, it appears, that the two functions [5806, 58 11 'J, which compose
the value of à.5R [57967?], produce nothing except periodical terms, noticing quantities [5810/]
relative to m, as in [5812].
* (3152) Changing, reciprocally, the elements of m into those of m', we shall get,
from R [5755], the expression of R' [5814]. Multiplying this by —, we
m
obtain ,
m[ __ m'.(xx'\-yy'-\-zz') m^
m ' t3 p ' [5815a]
subtracting this from [5755] , we get.
R-^ R'^m'.^,X-+yy'+,,'y^±_^y,
[5815
from which we easily obtain R [5815]. This expression of R does not contain p,
and, on that account, it is more convenient than the expression [5755], in making the [5815c]
calculations relative to ?«', in [5813 — 5831'].
720 APPENDIX, BY THE AUTHOR ; [Méc.Cél.
to these co-ordinates. We evidently see, by the preceding analysis, that,*
is composed of terms of the form
[5819] M ./Ndt — N ./Mdt .
To obtain their differentials, relative to the characteristic d, we must
[5819] vary only the quantities w^ithout the sign of integration ; because the
quantities under that sign correspond to the elements of the planet m'.f
* (3153) If we change, in [5795, 5796], the elements ?^, e, a, e, zr, p, q,
f5818a] jj^j^ ^;^ ^,^ ^1^ ^^ ^1 ^ p/^ ^^ respectively, we shall get, by using the characteristic
[5818b] i5', as in [5817], and supposing d' to affect the co-ordinates of m' only ;
,e*, ^J^=:i.!n/«'*)+<vs<S:).av+(f).,v+(t^)...,(f;).,y+(f,).V.
i'R = ^^ .afn'dt) I
+ ^M(")->''"-(S')-(S)->''"-(f)J ,
+ •^;KI')->--(f)-(?)->--Q|- '
If we represent by /?/ the four lower lines of the second member of this last equation,
[5818e] we shall get, by substitution and transposition, the following expression, which is similar to
[5796n] ;
[5818/1 ô'R'-^^[.ô'ifn'dt)=R;;
r58l8a-l ^"^ ""^ ™^y prove, as in [5797^], that Rf is composed of terms of the forms mentioned
in [5819].
r5819a] t (3154) The terms under the sign of integration, in the second member of [5818J],
VII. App.v^2.] TERMS, IN ^, OF THE ORDER »«=, mm', mm", &ic.
721
We shall suppose these two corresponding terms of M, N, to be
represented by,*
M=k.cos.(i'n't—mti'A) ; N = k!. cos. (i' n't— ini+ A'). [5820]
Then, we must combine these terms together, to obtain the non-periodical
quantities in
A.{M.fNdt—N.fMdt) ; [582i]
and, it is evident, tliat this differential function does not contain such
quantities. We may easily prove, that
does not contain any ; by the same manner of reasoning as that which we
have used in proving that
cl.(L«.,.y„rf,) [68ir]
contains only periodical quantities ; f therefore, à.ô' R contains only [5821"]
similar quantities.
arise from the quantities o'/, à' a, h'e', (5'ra', o'p' h'q^, à' .fn'dt; which contain
terms with the sign /, like the similar expressions of àe, Sa, &;c. [5796rf — k]. rsgigj-i
Now, these quantities oV. ô'a, Sjc, depend on the co-ordinates of the planet m' ;
therefore, their differentials relative to d [5785] must vanish. On the contrary, the
factors — -, {~T-)' ( TT )' Sic, in the function [5818c], may produce, in [5818rf], r58i9e]
some terms without the sign /, containing the elements of the planet ?n, which will be
affected by the differential d.
* (.3155) The calculation in [5819—5821"], is similar to that in [5800—5812] ; and
the functions [5800, 5801, .580.3, &c.], correspond, respectively, to [.5819,5820,5821, &c.] ;
hence we obtain a result, similar to that in [5806] ; namely, that the differential d , of [53206!
the function [5818/"], does not contain non-periodical quantities.
t (3156) If we develop the function [5821'], we shall get, by observing, that n,
and fn'dt, are not affected by d ;
VOL. III. 181
722 APPENDIX, BY THE AUTHOR; [Méc. Cél.
It now remains to consider the second term of R [5815], which we
shall denote by,
[5822] p = m'. (.r.;'+^y + r=') . (^-^3) •
We have, as in [915],*
We may substitute, in the second member of this equation, for b' .[fn'dt), its value
[5821fe] fb'n'.fh; and, for ô'n', its value &'n =3ftt'n.d'R', which is similar to [5S08] ;
hence we shall have,
[5821C1 d . S ^-^'.&'.{ fn'dt) 1 = ^,- . 3//a'n'. à'R'. dt = ^.3 «'«'. ffà'R. dt ;
'- ■" ( ndt ) n'dt •'•' n'dt ' ''
observing, that we may bring an from under the sign ff, by neglecting terms of the
order m^, as in [5796e, Stc.]. Now, if we put
[5821rf] R' = 'k.cos.{i'n't — int + A),
we shall get, in like manner as in [5810c] ;
k i'n' AA'R
r5821f] rCA'R'.dt == — '- . sm.{i'n't—ini+A) ; --, = Jc.i'i7i.cos.(i'n't — int-\-A)uc-
The product of these two quantities being substituted in the last expression of [5821c],
r5821/"1 produces only a periodical term depending on s\n.2. (i'n't — int-\-A) ; in like manner as
in [5810<?, Sic.]. Having found, that the differential relative to d, of the function [5818/],
and that of [58:21'], produce only periodical quantities; their sum representing the value
[5821g-] of d.S'R', deduced from 5'R' [58 IS/], must also consist of such periodical quantities;
r5821?i] which may be neglected : therefore, we may reject the term — . R', in the value of
R [5815], and it will be reduced to R ^^ P; using the abridged symbol P [5822].
* (3157') Substitutine, in the first of the equations [915], the value of M4-m =u.
[58230] ,
^ [914'], we get,
ddx , {M-\-m).x , /dR\
[58236] ^ = -^2
{M+m).x /dR\ Mx ddx mx /dR\
multiplying this by —, we obtain [5823]. The second and third of the equations
r58''3cl [915], give smndar expressions 01 —, — . It, in these equations, we change m,
X, y, z, r, into m', x', y', z', r', respectively, and the contrary ; and, then multiply
"'' !• 1 <■ ^'■'■' ^y' "'*' mi
r5823(il ^J ~' ^^'^ ^'^^^' §^^ ^^^ corresponding values of —7^, —, — . The first of these
expressions is explicitly given, in [5825].
vu. App.§-2.] TERMS, IN ^, OF THE ORDER m^ mm', rn'm", he.
723
m'x m' (hlx mm! x m' /dR\ _ [58231
M being the sun's mass. We have also, [5824]
m'x' m' ddx' m'" v' in' /^dR'\ [58251
r'3 M ' dt^ M ' r'3 M' \dx' J
The co-ordinates y, z, y', z', ^Jroduce similar equations ; hence we
easily deduce,*
m' d.{x'dx—xdx'-\-y'dy—ydy'-\-z'dz—zdz') ^
Q being a function of x, y, z, x', y', z', of the order of the [5827]
square of the masses m, m' [5825c, c"]. The variation of the part of
P [5826], which is independent of Q, may be expressed by,
™' d.6'.{x'dx-Tdx'+y'dy-ydy'+z'dz-zdz') _
^~M' dfi
* (3158) The terms depending on x, x', in [5822], are 'iLfl. _ ^^ . The [5825a]
value of this function is found, by multiplying [5323] by — x', also [5825] by -j-'^)
and then taking the sum of the products. Hence we have,
m'.xx' m'.xx m' (x'ddx — xddx') m'.xx' C m m'
~7^ ^ ^ M ■ ~d^ ^ M ' }?~'7^
^M I \dxj \dx'J
[58256]
If we substitute, in the first term of the second member, for x'ddx — xddx', its value
d.(x'dx — xdx') ; which is easily proved to be identical, by development ; and put ^, for •'
the remaining terms of the second member, which are of the order m^; we get the
expression [5825rf]. The similar expressions in y, y' , z, z', give [58256,/] ; ^ '
Q,n Qa) being quantities similar to Q, , and of the order m^. The sum of the
equations [5825(7, e,/], being substituted in [5822], putting Q = Q,-f Q,H- Q, , becomes [S825c"]
as in [5826] ;
m' d.{x'dx-xdx')^^^^ ^5g25rf]
m'.xx'. {
f 1 1
Vr'3 r3
r 1 1
m.yy'.[
Vr" 7-3
( 1 1
■ni. rz'.l
\r^ ,.3
M df
m^ d.(y'dy—ydy')
M ' dt""
+ Q» ; [5825e]
to' d.(z'dz — zdz')
M • d? ^ ^' ' f^^^^-^]
724
APPENDIX, BY THE AUTHOR ;
[Méc. Cél
[5829]
[5830]
and, as this is an exact differential, we shall obtain the part of fà.ô'P,
which depends on the function [5828], by changing, in this function, d into
[5829'] d [5829e] ;* and then, it is evident, that it contains, in terms of the order
m^ none but periodical quantities [5829î].
The term Q will give, in fdP , the quantity fàQ . If we notice
rggoQ-, * (3159) If We neglect, for a moment, the consideration of the quantity Q, the
remaining part of the second member of [58'26] will be an exact differential, which we
'■ _ •' shall represent by dP,; so that we shall have P = dP^. Its variation, relative to i5',
[5829c] gives ô'P = d.S'P,, which corresponds to [5828]. Integrating this, we get fi'P = è'P;,
and its differential, relative to the characteristic d, gives y"d.(5'P=d.f5'P,. Comparing
'• ^ this with li'P [5829c], we easily perceive, that fà.ô'P can be deduced from this
[5829e] expression of '5'P [5S29c, or 5828], by changing, in its second member, d into d,
as in [5829'] ; hence we shall have.
[5829/]
m' à.&'.{x'dx-xdx'+y'dy-ydy'+z'dz
idz')
dt^
[5829-]
[5829A]
[5829i]
[5829A;]
[5829i]
[5829m]
[5829n]
[5829o]
If tlie function —-Ax'dx — xdx'-\-y'dy — ydy'-\-z'dz — zdz'\, by the substitution of the
at
elliptical values of r, y, z, x' , y', z', produces a term represented by A^ ; its
variation, relative to h', will become (5'^, ; which is of the first order relative to the
masses, as is evident from the import of 5' [5817] ; and, when à'A^ is multiplied,
as in [5829/], by —, it becomes of the «cconrf order. This is finally reduced to the
third order, in the second member of [5829/], by taking the differential relative to d, of
the non-periodical terms ; because it produces the differentials of the elements [5786, 8ic.],
which are of the /rs< order. Hence it appears, that, if we neglect the non-periodical
terms of the third order, relative to the masses, we may put the part of fà.à'P, which
depends on the function [5828], equal to nothing. Then, there will remain to be noticed
only the part of the function fà.i'P, depending on Q [5829o] ; which may be
represented by fd.S'P =/d.<5'Q. But, q is of the order m^ [5827], and, if we
represent it by m^A„, we shall have 6'Q, of the order m^.è'A^; which is of the
third order relative to the masses, as is evident from [5829Â] : therefore, it may be
neglected. What is proved in [5814 — 58.31], relative to the planet ?«', may also be
applied to the other planets m", in!", &ic.; but, it will still be necessary to notice the effect
of m" on m' ; m"' on m', Sic. ; m!" on m", h.c. ; in the value of R. This is done in
[5833, &tc.].
VlI.App.§-2.] TERMS, IN ^, OF THE ORDER m^, »»«', m'm", he. 72Ô
only terms of the Older m" in dQ, it will sufrice to substitute in Q,
the elliptical values of the co-ordinates, and then fdQ will contain only
periodical inequalities. Thus, J'd.oP, will contain only similar quantities. [5t3l]
Hence it follows, that fd.&R will contain, in terms of the order mr,
only periodical quantities, when we vary in R, the co-ordinates of the [5^31]
two jjlanets m and m'.
If there be a third planci m", it adds to R the function*
p' being the distance from in" to m. The part of R, relative to the
action of m upon m, then acquires a variation depending on the
action of m" upon m'. This part of R is,
„ m'.(xx'+yy'+zz') m' rrirr.
K^ ^^ —^ [OlOO\ ; [Aclionof m' on m] [5833]
the variation of the co-ordinates x', y\ z', by the action of m", produces
in [5833], some terms, multiplied by rn'm", which are functions of the
elliptical co-ordinates x, y, z, and of the angles n't, /t'7 .f But
[5834]
* (3160) The expression [5832] is the same as [Ô755]; changing the elements of
m into those of ni". It corresponds to the second terms in the expressions of R, X 15832a]
[913,914].
t (3161) The co-ordinates x', y', z'. contain the elliptical values of the orbit
of m', augmented by the terms arising from the action of the bodies ?«, m" , m'", [5834a]
&ic. When these are substituted in [5833], they produce terms of the second order of the
masses, which we shall represent by tn'm"A^; A^ containing among other terms, the
quantities x, y, z. The co-ordinates of the planets m', m", which occur in A , [58346]
introduce the angles n't, n"t [5834,950,952,953], and the co-ordinates x, y, z,
contain the angle nt . The products of the sines and cosines of such angles, produce,
in AR, some terms, which depend on the angles int-{-i'n't-\-i"n't [1214'"] ; and, as n, n', t^^'^^e]
?i" are incommensurable [1197"], these terms will be periodical. Therefore, by noticing
only the non-periodical terms, in dR , we must consider i', i" as equal to nothing ; t^^^^'']
or, in other words, we must notice in R, only those terms which are independent of
n't, n"t, as in [5835], and then it becomes of the form m'm"X [5836]; X being
a function of the co-ordinates of m, as in [5836]. Putting this equal to R, it gives t5834e]
VOL. III. 182
726 APPEiVDlX, BY THE AUTHOR; [Méc. Cél.
._„„., these angles must vanish from the non-periodical part of dR, and as they
cannot be destroyed by the angle nt , which is introduced by means of
the values of x, y, z; we need only notice, in the development of the
variation R, the terms which are indejjendent of n't and n"t . These
[5836] terms will be of the form m''m"X; X being a function of the co-ordinates
of the planet m, they introduce into fdR some terms of the form, m'm".fdX,
or m'm"X\ which produce only non-periodical quantities of the order
m'ml'\ and such quantities we have neglected in fdR.
[5838]
In like manner, the variation of the co-ordinates x, y, z, by the
action of m", can introduce in the preceding part of R [5833], only
the angles nt and n"t; therefore, we need only consider, in this part,
[5839] the terms independent of 7i't , consequently of the form m'm"X; X
being a function of the co-ordinates x, y, z, only; which, as we have
[5840] just seen, can only produce quantities that may be neglected. Thus, by
noticing only the non-periodical quantities, of the order m^ in fdR,
we may suppose, that in" is nothing, when we consider the part of R ,
relative to the action of m! upon m ; and we may suppose m' nothing,
[5841] when we consider the part of R relative to the action of m" upon m,'.
we have just seen, that in these two cases [5837, 5840], the secular
variation of fdR is nothing. This variation is, therefore, generally nothing,
[5842] when we consider the reciprocal action of three, or of any number of planets,
if we only notice as far as the squares and products of the disturbing forces,
inclusively, in the value of dR.
We shall now resume the equation [5794],
[5843] 2, = 3ffa.ndt.dR.
Its variation is,*
[5834/] àR = m'm'.dX; whence fdR = m'm" .fdX = m'm"X ; this last expression being
deduced from that which precedes it, by omitting the double sign /d, taking into
consideration, that X is a function of the co-ordinates of m only, as in [5836];
[5834ff] consequently the signs /d represent inverse operations which mutually cancel each
other. The variation of the expression fdR = m'm"X , produces in fd.SR , non-
periodical quantities, of the third order of the masses m, in', he, which are neglected.
[5844o] * (3162) Multiplying an=a-i [57516] by dR, we get an.dR=a-KdR
VII. App.s^-2.] TERMS, IN ^, OF THE ORDER m«, mm', m'm", he.
727
6P = 3(1)1. ffdtA.6R+3a-.ff(ndLdR.fdR).
We have just seen [5812, 5821", &c.], that d.o/t is nothing, noticing only the
secular quantities, as far as the order of the square of the masses of the planets,
inclusively. We have seen, likewise, that dR .fdR is nothing,* noticing
only the same quantities. Therefore, if we take into consideration only the
secular quantities, which acquire, by the double integration, a denominator of
the order of the square of the masses of the planets ; we shall find, that the
valuation 5^ vanishes. Hence it is manifest, that, if we notice the secular,
as well as the periodical quantities, this variation cannot exceed a term of the
order of the disturbing masses.f This important result ivas first obtained by
Mr. Poisso7i.
[5844]
[5845]
[5840]
Importent
ipsnll, by
M. Poisson.
[584C']
Taking its variation, and then substituting tlie value of Sa [5196d], also a~- = (ni, [5844a']
a'= a^ii, we get, successively, by neglecting terms of the order m^ ,
S.{an.dR) :^ o.(a~- . dR) = a~^. d.SR —i.a~K dR.Sa = an.d.Œ -{-a^ n.dR .fdR. [58446]
Multiplying this by 3f/^ and prefixing to the double sign ff, we obtain.
^^,
3 5.ff{an,dR)=Smi.ffc1t.d.ôR+3a^-.ff^ndt.dR.fdR) ; L^^^''^]
the terms an, a^, being placed without the signs Jf; which can be done, by
neglecting terms of the order ?«', as in [5796c, &:c.]. Now, taking the variation & [5844rf]
of ^ [5S43], and substituting [5844c], we get [5844].
* (3163) We have seen, in [5810e],that the product of the two equations [5810i],
which represents the value of —.dRfdR, produces only periodical inequalities, as [5845a]
in [5845].
t (3164) The elements of the orbit of a planet are represented in [1102, 1133], by
systems of terms of the forms,
N.sm.{gt + ^) ; N. COS. (gt+fi) ; [5846a1
in which g is of the same order as that of the disturbing masses m', m", kc. [1097c].
The double integration of a quantity, depending on an angle of this kind, in [5843 &.C.], [58466]
introduces, into i^ or «5^, the divisor g^, of the second order of the disturbing masses.
But, the terms of the Jîrsi and second orders, vanish from the expressions in [5845] ;
therefore, this divisor can operate only upon those of the third or higher orders ; and, when [5846c]
those of the third order ot'^ are divided by g^, they produce terms of the first order only;
728 APPENDIX, BY THE AUTHOR; [Méc. Ctl.
3. We shall now consider two planets m and m', i7i motion about the
[5847] sun, ivliose mass we shall take for unity. We shall put v for the angular
g.g, distance of the planet m, from the line of intersection of the tîvo orbits ;
v' for tlie angular distance of the planet m', from the same right line ; also
[5849] Y for the mutual inclination of the tivo orbits ; taking the orbit of m for the
[5850] plane of the co-ordinates, and the line of the nodes of the orbit, for the
origin of x; ice shall have,*
X = r.cos.» ; y = r.sin.iJ ; r =3 0 ; 1
[5851]
x'= r'. cos.v' ; y' = r. cos.} .sin. r' ; z' = r'. sin. 7. sin. «' ; 2
which gives, by putting,!
r5846rfl *^'"' ^f ^^* same order as the peiiodical inequalities. This agrees with [5846'] ; and, the
importance of this result of Mr. Poisson, is manifest from the consideration, that, if the
r5846 1 ^^'"'^s ^^ the second order, relative to the masses, instead of vanishing, as in [5846], were,
on the contrary, of their usual magnitude, or of the order m'^, they would produce, in
[5846/] ^ [5843], by the double integrations, terms of the order —^ , or of di finite order ;
[5846g-] which might become sensible, in the theory of the planetary motions.
* (3165) The formulas [5851], are as in [3740, 3740'], changing the value of 7,
[5851a] which represents, in [3739, Sic], the tangent of the inclination of the two orbits ; but, 7
1 y
is used for the inclination itself in [5849] ; so that, we must change / , „ , ,
[58514] I J. > 5 v/j+7- \/l+>2
[3740c], into cos. 7, sin.7, respectively ; by this means, y' , z' [3740'], become
as in [5851, line 2].
t (3166) Substituting the values of x, x', &c. [5851], in the first member ol
[5853o] [58536], it becomes as in its second member. Putting, in this, cos.7 ^ 1 — p [5852]
and successively reducing, we obtain [5853c?]. In like manner, by developing the first
[5853a'] member of [5853e], and substituting x^-\-y^-\-z^ =^ r'^ ; x'^-\-y'^-{-z'^ ^ /^ [3742d];
we get [5853e,/] ;
[58536] *^*'+yi/'-|-*'2^' = '■?"'. {cos. v.cos.î)'-|-cos. 7. sin. î>. sin. î)'|
[5853cl = rr'.5cos.2;.cos.t)'-)-sin.D.sin.T' — p.sin.r.sin.ii'l
[-5353^^ = rr'.\cos.{v'—v)—^.sin.v.sm.v'\ ;
[5853e] {x'-xY+iy'-yYM^'-^y = {x^+y'+z^)-^{œ'^+y'^+z")-^.{ix'+yy'+zz')
[5853/] = r^-\-r' ^ — 2rr'. cos.(«)' — v)-\-2^.rr'. sin.v.sin.»'.
VII.App.§3.] DEVELOPMENT OF R. 729
R
/3 = 1— COS.;. = 2.sin.''^7 ; [5852]
m'. (xx'-j-yif-^-zz') m!
'' 7^ ~~ \/(x'— xf + {y-—yf + (z'— z?
[5853]
ot'. r m'
= — r-4cos.(i''-îj)-(3.sin.u.siiw'? — / „ „ ^ , , ^ ^ ^, . -;2 r.
r'^ \/r+r' -2rr.cos(?)-t;)+2^.rr.sin«.sin.ij'
under this form, R becomes independent of the plane to which the co- rro-.n
ordinales are referred [5853o-]. Developing it, in terms of sines and cosines
of angles, increasing in proportion to the time t, by the substitution of
the elliptical values of r, r', v, v' [952, 953], it becomes a function
of the mean angular distances nt-\-s, 7i't-\-s', of the planets, from the
line of nodes ; of the distances of the perihelia from the same line ; of
the semi-axes a, a' ; of the excentricities e, e' ; and of p, or the
mutual inclination of the orbits : (3 being a very small quantity, of the
order of the square of that inclination [5852]. Under this form, R does [5856]
not contain explicitly the variable quantities p and q [1032] ; but,
we may introduce them in the following manner.
Substituting [5853(Z,/] in [5755, or 5853 line 1], we get the expression in [5853 line 2] ;
which is a function of v, v' , p [5852] ; and these quantities depend entirely on the
relative position of the two orbits, and are wholly independent of any arbitrary plane, to
which the co-ordinates can be referred ; as in [5854].
[5855]
[5857]
Instead of referring the motions of the planets to their orbits, we may
refer them to the fixed plane of the primitive orbit of m ; then z will
not vanish, and it will be represented by z^r*;* s being the sine of [5858]
the latitude of m, above that plane. If we neglect the square of the
disturbing forces, we may reject the square of s ; then we shall have,
instead of R, the following function, which we shall denote by,t
[5859]
[5853^-]
* (3167) This is similar to [3787], z being the perpendicular, and r the
hypothenuse of a right-angled plane triangle, of which the sine of the angle at the [5858a]
base is s.
t (3168) If we use z = rs [5858], and z' [5851 line 2], we get,
zz = rr'.s .sm.y.s'm.v' , instead of zz^O [5851]; [5859al
VOL. III. 183
730 APPENDIX, BY THE AUTHOR; [Méc. Cél.
[5860]
R =z — ;^.^cos.(t;' — v) — |'3.sin.?;.sin.îj'+s.sin.7.sin.î;'} 1
R. — — . 2
i/r^+Z- — 2rr'.cos.(«' — v)-\-'2.^.rr' .sm.v.s\n.v' — 2rr'.5.sin./.sin.?;'
rsSGi] In these values of R and H, we shall subtract, from v, v', the
longitude è' of the node of the orbit of m' upon m ; this longitude
being counted in the orbit of ?«. This is equivalent to a change in the
[5862] Qj-ig{ji Qj' ^ and v' ; and we shall suppose,*
[5863]
[58596]
[5859c]
[5859rf]
S = q.sm.(v — ê') — j?. COS. (27 — è') .
Then we shall have,t
hence the value of x x' -{• y y' -\- z z' [5Q53d] must be increased by that quantity ;
moreover, the value of («'— a:)^ + (y~2/)- + (2'' — z)'^ [5853/], which contains
— 2 . {x x'-{-y y'-\-zz'}, must be, for the same reason, augmented by the term
— arr'.s.sin.y. sin.î)'; and these corrections being applied to the corresponding terms of
[5853 hne 2], it becomes as in [5860]; observing, that the value of y=r.sm.v
[5851 hne 1], may be retained in this hypothesis, if we neglect quantities of the order s^.
For, the correct value of y, being similar to that of y' [5851] ; namely,
y^r. COS.7,. sin.u =?•. sln.i; — r. sin.i;.(2.sin.|7y)^ ;
7, being of the order s, we may, by neglecting s^, suppose
yz^r.s'm.v, as in [5851 line 1].
* 3169) The expression [5S63], is like that in [1335'], altering the origin of the
angles v, by writing v—ê' for v, as in [5861]. We may remark, that the angle
^ — Ô' is counted, as in [3739], from the line of nodes, or mutual intersection of the
[58626] Qj.(3-ug^ Q,j ff^g ^^jj^ of m ; and the angle v' — &' is counted from the same line of
nodes, on the orbit of m ; so that we may consider the origin of the angle v to be
on the orbit of m, at a point, which is distant, by the angle è', from the node, and
counted upon the orbit of m. In like manner, the origin of the angle v' is taken ujion
the ^;Zane of the orbit of m', and at the same distance é' from the node, but counted
on the orbit of m'. This is evident from the investigation, in [3737 — 3740'], of the
formulas [3740,3740'], which are similar to those in [5851].
t (3170) If we decrease the angles v, v by è', as in [5S6I], we shall find,
that the angle v' — v is not altered ; but the expression sin.u.sin.î)' becomes,
[5864a] sin. (.-n. sin. (.'-O;
[5862c]
[5862d]
VII.App.>^3.] DEVELOPMENT OF iî. 731
R = ^2 -\(l—hr^)-cos.(v'—v)+ir^.cos.(v'+v—2è')\ 1
2
m'
[5864]
[5865]
V//-2+7-'2— 2;t'.|(]— ip).cos.(t)'— «)+ip.cos.(î;'4-r— 2ô')[ '
^_n^r S (1— èP+è'/-sin.7).cos.(îj'— «)+(ip— i9.sin.7).cos.(î)'+tJ— 20 ) 1
'"'" ( — |p.sin.7'.sin.(i'' — v) — ^.s\n.y,s'm.(v'+v — 2t)') ) 2
m/ 3
y y/ ^■2tj.'2_2j.r' 5 (1— à(3 + à?.sin.>).cos.(«'— w)+(.lp — a?.sin.>).cos.(«'+r — 2d') ) . ^
V' (. — èp.siii.>.sin.(«' — j>) — ip.s\a.y.sm.{v'-]-v — 2^') )
now, it is evident, that we may change R into R, if we vary in R,
(3 by ^|3; v by Sv; and ô' by (5()' [5867^■],• so that we may have,* L5866]
5/3 = — ç.sin.7 ; 1
(l_i|3).5|j = cos.^ir.'5v = — ip.sin.7 ; 2 ^ggg^^
p.(5a' — 1|3.5d = — i^.sin.7. 3
which may be reduced to,
Jcos.(2;'-z;)— Jcos. («'+!> -2 r) [17] Int. ^^^^^^
Substituting these in [5853 line 2], we get [5S64]. Now if we multiply the expression
[5S63], by sin.7.sin.(v' — ê'), and reduce the product, by means of [17, 18] Int., we get,
C |7.cos.(i'' — v) — ^q .cos.(u'+ w — 2()') — ip.sin.(v' — v) >
i.sin.7.sin.(!;' — ^') = sm.y . ^ " / .
( — ^p.s'm.(v'-\- V — 2ê') \'
Substituting, in [5860], for sin.tJ.sin.t)', its value [58645] ; and for s.sin.7.sin.'i;', its
value [5S64c], we obtain [5865].
* (3171) If we put the factor of -7^ 'i [5864 line 1], equal to w; and that of
[5865] equal to w -|-i5w ; we shall have,
w = {l — lfi).cos.(v'—v)-\-y.cos.(v'-}-v—2è') ; [58676]
ow = 55r.sin.7.cos.(«;' — v) — 5p.sin.7.sin.(r' — v) — i9'.sin.7.cos.(î>'+î^ — 2^')
~ . . , , s [5867cl
— ip.sm.y.sm.[v-\-v — 2ê') .
Then 72 [5864], being considered as a function of w ; that of R [5865], will be a
[5867c']
similar function of w-{-5w. Now, if we take the variations of w [58675], considering
p, V, ê', as variable, and neglect the second and higher powers and products of 5/3,
[5864c]
1 5867a 1
732 APPENDIX, BY THE AUTHOR ; [Méc.Cél.
Thus we get,
, . /dR\
\dv J sm.y'\dà'J'
ri r. • /'<iR\ , /'dR\ V fdll\
[5868] R=R- q.sm.r . ^-) -p.Ung.h7 • (,-) - £^ • (^-^ '-
and we have, as in [5163/],
/dR\ dR , ^dR\
This being premised, the equations [5790,5791] give* the two following
5v, èê', we get,
èw = — ^5^.cos.{v' —v) -{- {] — ^^).Sv.s\n.{v'—v) -f- l&^.cos.{v'-\-v — 2è')
[5867d] + {^.èé'—U.5v).sin.{v'+v — 2ê') .
Comparing the coefficients of cos.(c' — v), sin.(D'— î;) &,c., in the expressions of i5vv
[5867c, rf], we obtain,
rcQ^- 1 — Pp = Iq.sin.y ; (1 — ip).(5« = — ip.sin.7 ; (3.5(3'— ip.5v = — \p.un.y .
\pOXHt\
These equations agree with those in [5S67] ; observing, in [5867 line 2, 5852], that we
[5867/] have 1 — Jp = 1— sin.^ly = cos.^^y. If we substitute sin.y =: 2.sin4y-cos.Jy [31]
[586~ê-] Int., in [5867nne2], and divide the result by cos.^^y, we get 5v == — jp.tang.^y.
[5867fi-'l Subtracting the equation [5867 line 2], from that in [5867 line 3], we get, ^M — èv = 0,
hence,
,,.._,, U'-^I=-P:^^^^^^ V- =_^i^ [5867^,5852].
[5867A] ^^ — ^ 2.sin.2^y a.sin.J-y.cos.^y sin.y •■ ^' ^
It is evident, by inspection, that the symbols p, 1;, è' , occur in R [5864], by
means of the quantity w only [58676] ; hence it is plain, from [5867c'], that we may
[5867i] consider jR as a function of p, «, <3'; and R as a similar function of p+5p, v-^5v,
È'^àè', as in [5866]. If we develop 5, according to the powers and products of 5p,
r5867A;] ^v , 5â', by formulas [610 — 612], and retain only the first power of these quantities, which
are of the order m', we shall have,
/dRX ^ , fdR\ , , fdR\ ^^,
Substituting in this, the expressions of à?, àv, àè' [5867line 1, 5867^, A], we get
[5868].
* (3172) The function ^ [5360], is equivalent to R, in the formulas [5790, 5791],
[5869a] where the fixed plane is supposed to be the primitive orbit of m [5775' line 2]. Therefore
we must substitute R [5868], for R, in the values of dp, dq [5790,5791],
VII.Api,..,S3.] INVESTIGATION OF dp, dg, kc. 733
expressions,
Connecting these equations with those in [5786—5789,5794], we shall
have, by taking the differential of the terms of the development of R
the corresponding terms of each of the elements of the motion of m. This '
facilitates very much the computation of these different terms. We shall
put,
R = m'.k.coiJi'n't — int + i'e — û — o-u o-v 2o-"d") [5^,72]
for one of the terms of the development of R. Then, the corresponding
terms of the semi-major axis a ; of the mean motion fndt ; of the
epoch £,• of the excentricity e; of the longitude of the perihelion ^; [5872']
and of the quantities p, q; will be represented by the following expressions,
respectively ;*
observing, that the partial differentials of [5868], relative to p, q, give the express
[58696— c/], by using [5869];
ions
dR\ . fdR\
— ^=_sm.v.^— J-, [5869ij
—\ = - tan J- (~\ L f'^\
Now we have,
, sin.^y 2.sin.^^-y ^ ]_Cos.y p
COS.J7 2.sin.iv.cosi7 ~ sin.7 ~ sin.y [^^52]. [5869e]
Substituting this in [5S69f/J, and then using the resulting value for (-—V in [57911
we get [5871]. In like manner, the substitution of [5869&], for ( — \ in [5790], gives
[5870].
* (3173) Substituting the expression R [5872], in the first member of the
vor,. Til. 184
734 APPENDIX, BY THE AUTHOR; [Méc. Cél.
[5873] 5a = • —^r~.'k-cos.(i'n't—înt-\-i's'—i!—gzi—g'-m'—2g"ô');
i'n — m
3m' iii^
[5874] 6P = L fndt = r— ^ :—.nk.sm.{tn't—int-{-i's'—ù—gz:—g'u'—2g"ê') ;
* ■' {ill — mf
[5875] J,=_.^_^^.|(lVI=72).^'p^{'^)-2«.(f;)jsin(^«^^^^^^
Periodical ^
inaquali-
ties in the . -
«''■nents. m'.an.\/l—e^ , }?+ï-(l V/l— «^)s /•/ - •.!•/,• ' ' O '/d/\
,j.j,7^T 5e= . ^- V-J^ ^^ .cos(iV«-in!;+i's'— IS— ^ra— g-^'— 2g â') ;
, , , . , -^ ) • sin.(i'«V— i»i^+i'£'— is— g-w— g-V— 2^'M') ;
e.(^n — m) \de J
[5876]
m'. an. ;/l_e2 /<?fc
[5877] i5î3 = —
[5878] ¥ = ,7î^:(îv-i) • U) • -n.(.«'^-nU + ..-.-..«-^^-2.^ 0,*
[5679] h = r>- ^";'-"^-- . ^^"+(z-+£-).sin.«è7l.cos.(zV^-m^+i'e'-ù-^--^V-2g"â') ;
'■ ■* (i?j — in).\/\—t^.s\n.y
following integral, we get,
[5869/1 fan = —-, r-v • m'k.cos.{i'7i't-int-Yi'i'—ù-g^—g'^'—2g"ê').
<- J ' " (in — tn)
Substituting this in [5796<Z], we obtain Sa [5873]. The same value of R, being
used in ^ [5794], gives [5874]. In like manner, from 5; , de, Szs [5796/— A], we
deduce [5375,5876,5877].
* (3174) Taking the partial diflerentials of R [5872], relative to |3, 6', fndt,
[.5878o] u, and using for brevity, T= i'n't — iiit + i'e' — is — gu — g'-a' — 2g"ô'; we get,
[5878c] — = m'.k.i.sm.T; \-—\z= m'.k.g.sm.T.
Substituting the first of the values [58786], in [5870], we get, by integration, p or (5p
[5S78] ; and by using the remaining three equations, we obtain from [5371], the expression
[5878(i] of Sq [5879] ; observing that we have, /3 = 2.sin.^^y. [5S52].
VII. App.-^S.] INVESTIGATION OF i.fndt, ôs, Sa, Se, (3«, Sj,, Sq.
735
These results are conformable to those in chap, viii, of the second book ;
but these new expressions have the great advantage of including all the '*'
powers of the excentricities and inclinations.*
* (3175) We may show, that the expressions of Sa, S.fndt, Se, S^ [5873,
5S74,5S76, 5S77], are simihir to those in [1197, 1286, 1294, &c.], in the following manner.
The assumed value of R in [II 95'], is R = m'.Jc.cos.^i'S,' — i^-\-A) ; so that, if we [5880a]
substitute the mean values 2,' = n't, ^ ^ jit , and A = i's' — is — g-a — g'vs' — 2g è ,
it becomes, by using T [5S7Sa], R = m'.k.cos.T, as in [5872]. Substituting these
values of A, T, and (j, = 1 [5750], in [1197] ; prefixing also the sign S before
the terms, in the first membcfs of these equations, to conform to the present notation ;
we get,
[58806]
[588Cc]
,'1\ 2m'. in , ^
■5 . ( - = — ^n — — • ^^-cos.T ;
ay m — m
3m'. ùi^ , . ^
S^ = - ahsm.T.
{i7i — in)-'
Now, by neglecting the square and liigher powers of Sa, we have S . 1-)= ;
substituting this in the first of the equations [5880t/], and then multiplying by — a"^, we
get Sa [5873]. The expression of S^ [5880f/], is the same as that in [5874].
Again, if we neglect e", as in [1283'] , we may change the factor [/l—e^ into I, in
[5876], and then it will become,
[5880d]
[58S0e]
. m .an ^
Se = . A:
.?
cos.T,
[5880/]
as in [1286,1285]. In like manner, if we change the factor ^i_ea
[5877], and multiply the expression by e, we get,
m'.an /^^A . ^
eSzs = — -:— r- • "1- • sm. 1 ;
m' — m \de J
into 1,
[5880g-]
which is the same as the integral of edzt
[1291].
[5880ft]
The expression of Si [5875], may be derived from that in [1345], neglecting terms
of the order e'. For, if we multiply edzi [1258], by — \e , and add the product
to [1345], we get, by reduction, an expression of ds — Jc^Ju, which is equivalent to
that in [5775'], using the value of r [5769] ; and, from this we easily obtain [5875].
We have thought it unnecessary to go through the details of this calculation, as it is evident rsssoil
that the result must correspond with [5775']. For similar reasons, we shall omit the
reduction of Sp, Sq , [5878,5879], to the forms [1341, Sic]
736 APPENDIX, BY THE AUTHOR ; [Méc. Cél.
[5880'! ^^ "^'^"^^ ^"*^^ ^^^ secular variations of the elements of the orbit of m,
by reducing R to its non-periodical part, which ive shall denote by,
I- J R = m' F , [.\'on-periodicalpartof R\
[5881] Then àR vanishes,* as well as da , and we shall have,
[5882] ds =- 'ÎLl^^i^. (l.^r^Ti). (f) + 2«^(f ).,«'.« J. ;
[5883] de= ^'•^V^^.n./^.Q;
[5884] d. = - —-- . ^— j ;
Secular
inequali-
[5885] dp = - ^^^, . (^^ j ;
, m'.andt fdFX
or.
[5887] ^/» = :;7f^ •«!"•>' -(^7^;;
We may here observe, that we have, as in [5755],
[5889]
„ m'.(xx'-'ryjj'+zz') m'
r-* p
r5882 1 * (3l'^6) Taking for R its non-periodical part m'F, we shall have dR = 0
[5812, 5821', 5831, &,c.]. Subtituting this in [5786], we get d'a=0 [5881']. With this
[5S826] value of dR, and f---\=m'.(-—\ [5881], we obtain, from [5788], the expression
of de [5883]. In like manner, from [5789], we get [5884] ; from [5787], we obtain
[5882] ; from [5790,5791], we deduce [5885,5886] respectively; lastly,from [5870,5871],
[5882c] ^çgggj [-5887^5883-1^ respectively. In all the equations [5882— 5888], quantities of the
order m'- are neglected ; but they are exact in terms of the order m', for all powers
[.5882(i] ° •'
and products of the excentricities and inclinations.
VII.App.p] SECULAR INEQUALITIES. 737
and hy neglecting quantities of the order m'^ it becomes,* [5889]
R= — m' .^ ^'J ' —-; [5890]
Therefore, the non-periodical terms of R depend on the non-periodical
part of ; hence we have,t
P
F = non-periodical part of — = non-periodical part of ; [5890']
this part being developed in a series of cosines of angles, increasing in proportion
to the time t; and F is the same, for both planets f5756]. U we [5891]
vary in F, the elements e, ^, p, q, of the orbit of m, and substitute for ^e,
Sw, op, oq, their values, which are given by the integrals of the preceding
* (3177) If we neglect terms of the order m'^ in [5S25] we get,
m' x" m' dilx' . m' x' ddx'
because, by neglecting quantities of the order m'^, we may put M ^ I [3709ol. [589061
In like manner, we have,
m'y' , ddy' m' 2' ddz'
7^=~'"-l^' ~^^~'^-li^- t5890c]
Multiplying these three equations by x, y, z, respectively, and taking the sum of
the products, we get,
"'■'• (^ '^'+yy'+zz') {xddx'+ yddy'+ zddz')
~ ^"^ = — m . -~ -^ . [5890rf]
Substituting this in [58S9], we obtain [5890].
t (3178) If we neglect terms of the order m'^, we may substitute the elliptical
values of x, y, z, i', y', z' [950, 952, 953, Sic], in the terms of the second ^^^^^"^
member of [5890], which are divided by d(^ ; and then we shall see, that it contains
no terms of the proposed order, except such as are periodical. For, if x' contain a ^^^^^^^
non-periodical term, its second differential ddx' will depend on the differentials of
the elements «', e', &:c., which are of the order R, or m [5786, &.c.] ; and, '■^^'^^"^
VOL. III. 185
738
APPENDIX, BY THE AUTHOR;
[Méc. Cél
[5892]
[5894]
differential equations [5883 — 5886], we shall find, that <'jF vanishes,* and
the same result is obtained with the variations of the elements of the orbit
of m'. This is demonstrated, in [3767], supposing the terms of fourth and
higher orders of the excentricities and inclinations to be neglected.
We have, as in [5867 line 1, 5867/i],
[5893] i|3 = — q.ûn.y ; 53' =
If we suppose, that (5[3 and àè' are increased by the quantities fZ(3, cW,
P
sin./
respectively, we shall have,t
[5891d]
[5891f]
[5891/1
[5892a]
[58924]
[5892e]
[5892(i]
[5893a]
when ddx' is multiplied by m'x, as in [5890], it becomes of the same order as the
neglected terms [5889']. It is unnecessary to notice the periodica! terms of ddx',
because they produce no non-periodical terms of the first order in m'. xddv' ; therefore,
tliis term may be neglected; and, for similar reasons, we may reject m'.yddtj', m.zddz'.
■m
Hence we have, by noticing only the non-periodical terms, R = — — [5890].
Substituting this in [5881], and dividing by m', we get F^ , as in [5890'].
Finally, as the value of p [5756] is symmetrical, in the co-ordinates of the two planets
X, y, z, x, y', z', respectively ; it is plain, that the non-periodical part of R,
or F, must be the same for both planets, as in [5891].
* (.3179) If we vary in F, the elements e, ra, ;?, q, of the orbit of m, we shall
get, in like manner as in [5795a — 6,5795], by noticing only the secular variations of
these elements ;
àF--
m-^'-
!
V..;-S + U>+(ï)-M
The integrals of the values of de, rfra, dp, dq [.5883, 5884, 5885, 5886], are
found, by changing, in these functions, dt into t, neglecting terms of the order m'^;
by this means, we get (5e, 53i, r5p, &q, respectively. Substituting these values of
èe, <5ra, in [5892a], we find, that the terms depending on these quantities mutually
destroy each other. In like manner, the terms which depend on 'îp, (5y, mutually
destroy each other in [5892n] ; therefore, the whole of the second member of [5892ff]
vanishes, and we have, as in [5892], i5F = 0. In a similar manner, we find, that HF
vanishes, by the substitution of the variations of the elements e', ts' , p' , q' , oftiie planet m'.
t (.3180) Taking the differentials of [589.3], and writing, as in [5894], dfi, dû',
for d.à^, d.S6', we get,
VII.App.§3.] SECULAR INEQUALITIES. 739
rf/3 = — dq.sin.r ; dt)' = — -^. [5894']
^ sin. 7
Substituting the values of dp , dq , we shall get.
*
d^ = _ '"'■""^ ('l^ . [5895]
/l_ea.sin.y ( \ ^^ / \d^J )
We have,t
dfi = — âq.s\n.j—qdr. COS. 7 ; (W = — ^ , ^-^^ _ [58936]
sin.y sin.a>
Now, y [5849] is of the same order as the greatest latitude of the planet m', above the
orbit of m ; and this varies, in consequence of the perturbations of tlie latitude, by quantities
of the order m. Moreover, p, q [5863], are of the same order as s, which is of ■'
the order m [5858] ; therefore, pt/y, qdy, are of the second order in m, m', and may
be neglected; hence the formulas [58936], become as in [5894'].
* (3181) Substituting dp [5887], in the expression of d&' [5894'], we get
[5895]; moreover, the differential of |3 = 1 — cos.y [5852], gives f/p = f/y. sin.y. ^ ^
Now, it is evident, that we may put this value of </(3 equal to that in [5894'] ; because
p would be constant, if it were not for the mutual action of the planets ; so that the ^58955]
whole of this variation of (3, arises from that of 5^ ; hence we get,
— rfj.sin.y = f7y .sin. 7; consequently, dy ^^ — dq. [5895c]
Substituting the value of dq [5888], we get [5896].
t (3182) If we put g"'^g", è = è', in the term of R [958], it becomes of
the same form as in [5872]. Making these substitutions in [959], we get,
0 = i'—i—g — g'—2g" ;
L5897a]
which must be satisfied for all the terms of R [5872]. Now, F [5881] comprises
the non-periodical terms of R, or those wliich do not contain i'n't — hit [5872] ;
and, as n, n' are incommensurable [1197'], we must necessarily have, in this case,
i'=0, 1=0. Substituting these values of i', i, in [5897a], we get,
0 = 5- + 5-' + 2 g", as in [5899] ; [5897c]
and the value of R [5872] becomes,
740 APPENDIX, BY THE AUTHOR ; [Méc.CéJ.
/dF\ /dF\ /dF\
because, if F be developed in cosines of the form,
[5898] F=^ H.COS. (g^ +^V + 2g"ô') ;
the sum g-'rg'-\-2g" of the coefficients of the angles ro, ra', ê', must
[5899] be equal to nothing, to render this term independent of the arbitrary
origin of those angles [5897c]. Therefore, we have,*
m'.amit (^ ^. /iIF\ , /dF\ }
Hence we obtain, by means of the preceding expressions of de, de',-f
[5897c'] R^m'.Ic.cos.(-gu-g'^'-2g"ê') = m'.Jc.cos.(gz^+g'^'+2g"6').
Hence we get, by means of [5890'],
[5897c;] F=l'.cos.(gvi-]-g'T^'+'2g"è'), as in [5898] ;
H being used for k. The partial differentials of F, relative to w, to', ê' give,
[5897e] by putting, for abridgement, w =^ gTs-{-g'-m'+2g"é',
/(IF
[5897/] (^
'%) = -gk.sin.^v ; (^i) = -^-'^'-sin.w ; (^,) = - 2g"k.sm.w ;
h
lence.
[5897g] 0+Q+(^) = -^•(â-+.-'+2g")-sin.w = 0 [5897c].
This last expression is equivalent to that in [5897].
[5900a] * (3183) Substituting the value of {j~\ [5897], in [5896], we get [5900].
f5901a] t (3184) The expression of dy [5900] depends upon the disturbing force of m' ;
and, if we call this part dy^, and put the other part, depending upon the disturbing
force of m upon ?n', equal to dy^, we shall have the whole value dy=dy^-\-dy^.
"■ Substituting f/y, for dy, in [5900], also 1 — |3 = cos.y [5852], then multiplying
[5901c] by ^-^, weget [5901e]. Multiplying [5883] by j-î-, we obtain [5901/]; adding
this to [5901c], we get the first of the formulas [5901^] ; and, by substituting the value of
-^ ) = ^ , which is easily deduced from [5883], by changing reciprocally
\da'J m.a'n'dt.\/[\ — t'-)
[5901c'] ,^i,e elements of m into those of m', which does not change jP [5891], we get the last
VII.App.s^3.] SECULAR INEQUALITIES. 741
f/y.sin.y eile e'dc' m.a'n'.ede
cos.y 1 — e- 1 — e'^ m .an.\/l — e^.^/l— e'a.cos. y
m', an. e' de'
m.a'n'.\/\—e'i • \/\—e"^.cos.y
Multiplying this equation by — 2.^i— e^ . ^/i— e'^.cos-r , and taking its
integral, we get,*
m.\/a g, m'.\/a'
[5901]
2.1/1— e2.v/Ti:?2.cos.7 = constant ~ •i\—e^) -^^- . (\—e'^). [5902]
m y a' - ' m-v/rt ^
expression in [590 1»-]. The similar formula, corresponding to the action of ?« on m', is
found, by changing the elements of 711 into those of m', and the contrary ; by this means, [sooirf]
we get [5901A]. Adding together the expressions [590l5-,/i], and substituting dy [.5901e],
we get [590 J] ;
dy^.s\n.y m'.andt C /dF
COS.7 \/\-
(.2
\C!l\+JL_,(^]. [5901e
ede m'.andt
1— e^ V/î=^2 * ^ V^J I '
[5901/-]
dy^.sin.y ede m'.andt 1 /'^^X in', an. c' de'
COS.7 1 — f^ \/l— e2 COS.7 \dc/y OT.a'n'.y/i_e2.y/l_e's.cos.7 '
rfy,. sin.^ , c'f/e' m.a'n'.ede
-^ + 1 72 = ; , ,^= . [590U]
cos.y 1 — €■' m.an.\/l—e^.yi—e'^.cos.y
* (3185) Multiplying the equation [5901] by — S./HT^.^/JZITa. cos.-/, we
obtain,
^ ^ ' v/(l-e2) \/(l-e'2)
[£9C2o]
m.a n ^ , , m . an , ■"
= . 2ede -{ r, .2eWe'.
m .an 7n.a a
The integral of the first member of this equation, is the same as that in [5902]. In the
second member, we must substitute an = «"*, an' = «'-= [5778a], and it becomes, t^'^^^^l
m.y/a ^ , , m'.\/a' ^ , . ,
, , , • Série -\ — . 2e'Je' ; rwnQrl
which, by integration, gives the second member of [5902]. Finally, we may observe,
that, in all the differential equations [5882—5902], we have neglected terms of the ^^^^^^^
second order in 7n, m'.
VOL. III. 186
742
APPENDIX, BY THE AUTHOR;
[Méc. Ctl.
If we put, for brevity,
[5903]
[5904]
[5904']
[5905]
[5905']
[5906]
[5907]
v/«-(i-e-) =/;
^„'.(l_e'2) =/';
we shall have,*
f3 =
[,nf+m'fY-c^
2mm'.ff'
â being an arbitrary constant quantity, independent of the elements.
The preceding value of d^ [5895], expresses the motion of the
intersection of the two orbits, produced by the action of /w', and referred to
the orbit ot m' [5862^]. We shall suppose an intermediate plane, between
these two orbits, and passing through their mutual intersection ; and shall put
9 for the inclination of the orbit of m to this plane. To obtain the differential
motion of the node of the orbit of m, upon this plane, arising from the action
of m' upon m, we must multiply the preceding value of de' byf "-^^ .
[5904o]
[59046]
[5904c]
[5907o]
[59076]
* (3186) From [5903], we obtain,
i/l_e2 = ^^; wlZTTa = 4- ; also COS.y:=l— p [5852].
Substituting these in [5902], we get,
■P
n'.f
2.(1— p).-—=^ == constant- - , — ,
\/ aa' m.\/aa m.y aa!
multiplying tliis by ?«?«'. \/^, and putting,
Hm'.\/;i^'X constant = c^ we get, 2.(1 — ^).mm'ff=^ c^ — m-.f^ — m"^.f^ ;
whence we easily deduce p [5904].
t (3187) In the annexed figure, NM, NM, represent the orbits of the planets
m, m', respectively , supposing ^'
them to be viewed from the sun,
and referred to the concave
surface of the starry heavens ;
NDM" is the intermediate plane,
or orbit ; and N the common
intersection, or node, at the ^-.
commencement of the time dt.
Orlrii cj" TTt
VII.App.'§.3.]
SECULAR INEQUALITIES.
743
Putting this motion equal to dt\ we shall have,
m'.(]t sin. y /f/jP'
dé = —
[5807']
[5908]
f sin. (J) \dp .
If we put ?' for the inclination of the orbit of m, upon the same plane, [^'-^0^]
we shall have ç+çj' = 7 ; and, [5909]
Then we shall have, as in [5905', 5909],
the angle MNM"= 9 ; the angle MNM" = <p' ; the angle MNM = ?+?)'= y ;
the arc ND = rfé ; the arc NE= cW [5907', 5905].
We shall now suppose, that the action of the body m' upon in, changes the orbit of m,
from MN to the infinitely near orbit MDE, in the time iJt ; by this means, the
node JV moves through the space ]VE=cU' [5905, 5862,c,d]. upon the orbit of m ;
or, through the space ND= dé, upon the intermediate orbit. Then, in the infinitely
small triangle NDE, we have,
sineNDE : smeNED :: NE: ND ;
and, if we neglect infinitely small quantities, we have,
angle ND E = ip ; angle iV£D = 180"— 7 ;
hence we have, in symbols,
sin. 9 : sin. 7 : : de' : de ; consequently, di) = Jd'
Substituting in this, the value of dé' [5895], we get,
m'.andt Biit.y /dF\
^1^^' sin.((>'\
sm.y
sin.(p
[5907].
d^J '
and, since an = o~' [5902&], we have,
an 1
\/l— e
^a:^) = 7 ^^««^] ■'
hence the preceding expression of de becomes as in [5908]. In like manner we
obtain the value of d'ô [5910], which represents the motion of the node of the planet m'
by the action of m; and, we can easily deduce this value of d'é [5910], from that of dé
[5908], by clianging reciprocally the elements and mass of m into those of ?«'; by which
means, / changes into /', in [5903] ; and dé [5908], changes into d'ô [5910];
F remaining unaltered [5891].
[5907c]
[5907rf]
[5907e]
[5907/-]
[5907/']
[5907/"]
[5907^:]
[5907A]
[5907t]
744 APPENDIX, BY THE AUTHOR; [Mée. Cél.
m
„ ....lit sin. y fJFx
[5910] d'& = — ■ . -. ;■ . — ) ;
[5911] (He being the motion of the orbit of m', upon this plane, produced, bj the
[5912] action of m upon ;/*'. The motions de and d'è will be equal, and the
intersection of the two orbits will remain iqyon the plane we have just
[5912'] considered, if it divides the angle of the mutual inclination of the orbits j, so
that loe may have,*
[5913] mf. sin.? = Hiy. sin.9'.
This result is the same as is found in [1164] ; ivhere ive see, that the plane
[5913'] . . , . L J ' r
m question, is that of the maximum of areas ; and, that we have,
[5914] c = mf. cos. ?+my. cos.ip'-
This equation [5914], being combined with [5913], gives the integral
corresponding to [5904] ; namely, f
[5915] P = ■
2mm'.//'
* (3188) Putting the two expressions [5908, 5910] equal to each other, and dividing
, , , , . /'dF\ m' m , • , • -,
r59]2al bv the common factor — dt.sm.yA- — , we get, - — : — = — — -, — ; which is easily
i- J ■ \dp / /■ sin.<P / . sin. 9 •'
reduced to the form [591.3]. This equation, by the substitution of the values of /, /'
[5903], becomes as in [1 164 line 1], corresponding to the equation of the maximum of the
areas ; and, by a similar reduction, we may prove the identity of the expressions of c in
[1165,5914].
[59126]
t (31S9) The equation [5913] may be put under the form,
[5915a] 0= — '«/sin.çj+m/'.sin.©'.
Adding the square of this equation to the square of c [5914], we get successively, by
using y, p [5909', 5852] ;
[59156] c^ = »î^-./^-(cos.^ip-f-sin.^^))+2mm'.//''.(cos.9'.cos.ç3-sin.ç)'.sin.ç))-|-;«'^./'2.(cos.V+sin.V)
[5915c] =m^P+2mm'.ff.cos.{<p'+<p)+nr-.f'^^7n^P-^2mm'.ff.oos.v+m'^.f^
[5915rf] = m\f^+2mm'.ff. ( I_^)+m'2./2 = [,nf-^>n'.fy-2mm'.ff'. (3.
From this last expression, we easily deduce the value of |3 [5915]; and, by an inverse
■^^^'^'^ operation, we might deduce [.5914] from [5904, 5913].
VlI.App.^S.] SECULAR INEQUALITIES. 745
These two equations, give also the following expressions ;*
[5916]
m'f'-s'm.y . , mf.sm.y
Sin.? = — ; sin.? = ;
c c
COS.Ç3 = ^-—- ^ ; C0S.9 = ,r~ri^: ' [5917]
2m/. c 2mJ .c >• J
We shall denote by -a^ and «/, the perihelion distances of m and m', from ^59191
the line of mutual intersection of the orbits. Then we shall obtain d^^ ,
by subtracting from the differential d:^, the motion of that intersection
dt\ referred to the orbit of m ;t and, it is evident, that, for this purpose, it ^
sin ©
* (3190) From [5913] we get mf=nif'.—-^ ; substituting this in [5914], we [5916(1]
obtain successively, by using y [5909'] ;
m'f , .... ,, m'f ■ , < i\ ™y ■
c = - — . cos.o.sin.ffl +sin.a).cos.ç) ? ^= .sm.((D+<p )=-. — .sin.y. [591651
sin.ip ' ^ ^ ' ^' sin. If \- ^ ^ / sjD^ç, 1. J
From iliis last value of c, we easily obtain sin.çj [5916]. Substituting this expression
of sin.? in [5913], and dividing by m'f, we get sin. 9' [5916]. Again, we have,
c— m/.cos.ip=:w7'.cos.9' [5914] ; ' [5916c]
adding the square of this to the square of [5913], and reducing, we obtain,
c2_2TO/c.cos.(p+my2=:OT'y2 ; [5916d]
whence we easily deduce the value of cos.»? [5917] ; substituting this in [5914], we
get cos.(p' [5917]. Losing the value of sin.ip [5916], we get, from [5908].
''--Jf'-Q t^^'^J' [5«16e]
and, by substituting the value of c [5915<^], we get the second form of dé [5918].
f (3191) Drawing DF perpendicular to NM, in fig. 80, page 742, we have,
JVF = ND.cos.FND = dê.cos.ç [5907c, (Z] ; [5921a]
and, if we substitute the first value of do [5918], and that of cos.tp [5917], we get,
VOL. III. 187
746 APPENDIX, BY THE AUTHOR; [Méc. Cél.
is only necessary lo multiply it by cos.;? ; now, we have,
(m/+m'/'— m'/'.p) , /dF\
[5921] dé.cos.ç=: — ^ •'^ '' -^-'.dtJ-j-y
therefore, we shall have,*
[5922] ed^^ = — m . andt.\/[—e^. (-77) + -^^ — "^-yp — - — - ^ - edt . i — j ;
[5923] ede = m'. andt.\/Y^^. (~\ .
[59216] d^ . COS., = - ~^t-X__£_i. J, . (^_ ^ .
Substituting c^ [5915rf], and dividing the numerator and denominator by 2mf, we
[5921c] get [5921]. Subtracting this quantity from the whole motion of the perihelion of the
planet m; namely, d-a, we get dw^ [.5921e] ; which represents the increment of
the distance of the perigee of the planet m from the moveable node. In the same
[5921(/] manner, we get c/ar/ [5921/] ; or, it may be more easily derived from c?w, [5921e],
by interchanging the elements of m, m', in the usual manner ;
[5921e] d., = d. + ^:^J^-^^^ . ,, . Q ;
[5.31/] .«-..'+i^^^^i'^=^)../..Q.
* (3192) Multiplying the expression of d^^ [592 le], by e, and substituting
da, [5884], we get [5922]. In like manner, multiplying the expression of d.-us'
[5921/], by £■', and substituting,
m.a'n'dt.wT^3 /dF\
[5922a] rf« = -, . [—j ;
which is deduced from [5884], by changing reciprocally, the elements of m into those
of m' ; we get [5924]. Now, we may suppose, as in [592G], that w, ra/, take the
places of -ra, w', respectively, in the function jF ; and then we shall have,
^ ' \d-aj \d-^J \dTsJ yd'^J \d-a,J xdis'J
If we neglect quantities of the order m , we shall get from [592 If,/],
VII.App.§3.] SECULAR INEQUALITIES. 747
In like manner, we have,
«*; = - „,.«V*VI=7»- ('^) + ("■/+'"/-'"/• W . ,d, . C;^) ; !=»«,
e'de' = m.a'n'diyT—r-. f^^j . [5925]
F is a function of a, a', e, e', ra, w', and f3. If we eliminate (3 [5926]
from the second members of these equations, by means of its value,
^_(^^f^)J-C^ [5915], [5927]
2mm .Jj
we shall obtain four differential equations between the four variable quantities
e, e', ^^, 3'. We may give them a still more simple form,* by putting,
h = e.sin.:^^ ; I = e.cos.^^ ; [5928]
h' = e'. sin.^/; /' = e'. cos.w'. [5929]
This renders them linear, when we neglect the higher powers of the
excentricities, and facilitates the farther integrations, by approximation, [5929']
to any powers of the excentricities. f Thus we shall have the position
so that by rejecting quantities of the order m , we shall have,
Substituting the first of these expressions in [5883], and multiplying by e , we get [5923] ,
in which terms of the order m^ are neglected. The second of the expressions [5922rf],
being substituted in the value of de', deduced from de [5883], by interchanging the '■ "^
elements of m, m', gives [5925].
* (3193) We have already seen the effect of similar substitutions, in simplifying such
results, in [1022, 1046, 10S9, &c]. "•'
t (3194) After we have obtained the values of h, h', I, /', by methods
analogous to those in [1097, &c.], we may determine e, e', w, ,w/, from [5928,5929]. [5929a]
Then a, a', being constant [5881'], we shall have /, /', from [5903]. The
constant quantity c^ is known, from the values of f, f, /3, at the epoch [5929i]
when ^ = 0, by means of [59I5(/] ; and at any other time t, the value of /3 will
be known, by substituting the corresponding values of f, f , in [5927], then from (3, [5929c]
748 APPENDIX, BY THE AUTHOR; [Méc. Cél.
of the orbits, relatively to the variable position of the line of their
[5930] mutual intersection. We shall then have the inclinations of their orbits
to each other, by means of the preceding value of [3 ; and we may
thence obtain their inclinations upon the plane of the maximum of the
areas, by means of the preceding values of f and o'. Lastly, we shall
have the motion of the intersection of the two orbits, upon this maximum
plane, by integrating the preceding expression of de [5908]. This
seems to he the most general and simple solution of the problem of the
secular variations of the planetary orbits.
[5931]
[5931']
We shall now resume the equation [5915^],
[5932] c^- (mf-^mfy — 2mm'.ff'.^.
If we neglect quantities of the fourth power of the excentricities and
[5932'] . . . .,, . „
inclinations, it will give,*
/ /- '2,1 2^771'. y/^ (3
[5933] constant = m.\/a . e'+ m'. \/«'. e'^ +
ffl.v/«+ m'. \/a'
[5929d]
we obtain y [5852]. With these values of m, m, c, f, f, 7, we deduce
cp , (p', from [59 16 or 59 1 7], and de from [5918], whose integral gives è. Thus we
shall obtain all the elements, in the same manner as in [5930,5931].
* (3195) The quantity |3 is of the second order in y [5852], and by neglecting
terms of the fourth order, we may put,
[5933a] _ 2mm:. ff. (3 = — 2mm'. ^^. (3 [5903] ;
also,
[5933a'] ff = vA7. v/r=ra.v/i^=?â = /^'. (1 — 1 e^— | e'2) .
Hence the expression [5932], becomes, without reduction,
[59336] c^= m'.a. {\ — c~)-\-2mm'.\/^.{\^ie^—^e'^)-{-m"^. a'.(l — e'^) _ 2mm'. /^. |3.
Then, by transposition, we get [5933c], and its second member is easily reduced to the
form [5933rf] ;
[5933c] — c^+TO^. r(+m'2.a'+2»im'. /^= m^. o.e^+mm', v/W. {e^+e'^)-{-m'^. a'. €'^+2mm'.^^'. (3
[5933i] " ={m.\/â-\-m'.\/a').{m.\/â.t~+m'.\/â'.e'^)-\-2mm'.\/'^:^.
If we divide this by m.^â-j- »i'- \/a' , we shall find, that the first member is a constant
quantity, and the second member becomes as in [5933].
VII.App.§4.] EFFECT OF THE OBLATEINESS OF THE EARTH. 749
D~
(ap— la?). — .(f.3_^-) . [Terra of ii]
[5034]
[5935]
and bj what has been said in [5786,5842, 5881', &c.], a and a', are
constant, noticing the square of the disturbing force ; therefore, we shall
have,*
0 := 7n.\/a . e e + m .Wa'. eoe -j -^ Li- .
my a + m'. \/a'
This equation is of the same form as that which is found in [3964], noticing
the terms dependmg upon the great inequalities of Jupiter and Saturn.
Hence it appears, that the invariable plane, determined in [1 162',&c.,5913], [5936]
remains invariable, even when we notice some terms of the order of the
square of the disturbing force [.5935c].
4. We may, by means of the differential expressions of the elements,
determine, in a very simple manner, the influence of the figure of the
earth upon the moon's motion. We have seen, in [5340,5438], that this
action produces in R, the following term ;
[5987]
ap is the oblateness of (he earth [5333]; a.:^ is the ratio of the centrifugul force [5938]
to gravity, at the equator [5333'] ; D is the mean radius of the terrestrial
spheroid [5334] ; and (^ the sine of the moon^s declination [5334'] ; which [^''39]
is represented as in [5344], by,
i^ = v/l — s-.sin.x.siuj/tj+s.cos.x ; [5940]
or, more accurately, as in [5344e],
sin.X.sin./î)-j-s.cos.X
(^ = -=:z ; [5941]
yl-\-ss
fv being the true longitude of the moon, counted from the vernal equinox
[5345]; >• the obliquity of the ecliptic [5S^:\] ; and s the tangent of the '•^^^^^
moon' s latitude [4759 "].
* (3196) We have 2|3 = 4.sin.^4-/ [5852], and, if we neglect terms of the order
y^, we get 2(3 = y^. Substituting this in [5933] ; taking its variation, dividing by 2, [5^35a]
and neglecting terms of the second order in Se , Se, Sy , v/e obtain [5935], which is [59356]
similar to that in [3964]. The equation [5935] is correct in some of the terms of the
VOL. III. 188
760 APPENDIX, BY THE AUTHOR ; [Méc.Cél.
The part of R, depending on the sun's action, is of the form* r^Q',
neglecting terms depending on the sun^s parallax, ivhich are very small
[5944c]. Then we shall have, very nearly,
[5944] i2=rQ'+(ap-iap). — .(sin.^x.sin.yi'+2s.sin.x.cos.x.sin./i'-|) [5944e,&;c.] ;
which gives,
2r .( — )=2a.( — )=4r^Q'-6.(a--ia;).— .(sJn^À.sinyi;-f2s.sinx.cosx.siii/r--i.}.
[5945] ^r.^-y = ^«.^-^ =
We shall here notice only the inequalities depending on the angle gv — fv ;
[5946] g^ being what is called the argument of latitude ; then we shall have.
[5935cl order mr' [3964', &.C.], but others of the order nî^, m^.e^.Se, Se^, Sic, are neglected,
as in [1 150', 5932', 59356, &ic.].
* (3197) Substituting the values of u, u, [4776, 4777e] in (^ [47S0], and
developing it in a series ascending according to the powers of r, we get,
[5944o]
1 m' C 7"^ r^ )
Q = - + - • 1 + -4 . - + i? • -3 + &c. ;
r r (_ r - r y
[5944a'] ^^ S, &1C., being quantities which contain v, s, v', s'. Substituting this in [5438],
we get,
[5944i] R = --.)^l+A.-+B.~ + Uc.^.
The first term of this expression of R , produces nothing, in its partial differentials,
taken relatively to the elements of the moon's orbit ; we may, therefore, neglect it; and
[5944c] ^jg^ |.j^g terms depending on r^, r^, &tc., on account of their smallness [5943]. By
this means, the expression of R, is reduced to its greatest term, depending upon r^,
which is represented by r^Q' in [5943], and is of the same order as that of the
disturbing force of the sun upon the moon ; Q' being a function of v, s , r', v', s'
[5944d] [5944a, a']. Finally, we may remark, that the symbol Q' is denoted by Q, in the
original work, but we have placed an accent upon it, in order to distinguish it from the
value of Q [5944o]. Adding this chief term of R to that in [5937], we get,
[5944e] R = r2Q'+ (ap— iaç) . — . (m.^— è) •
Substituting the value of (a [5940], and neglecting s^, it becomes as in [5944]. Its
[5944/] partial differential, relative to r, being multiplied by 2r, and then substituting [5774],
gives [5945].
VII.App.§4.] EFFECT OF THE OBLATENESS OF THE EARTH.
very nearly, s = y.s'm.go [481 8] ; 7 being the. inclination of the moon'' s
orbit to the ecliptic [4813]. Thus, we shall obtain,*
R=^ r-Q'-\-(a.p — la.:p).—.7.su\.Kcos.\.cos.(gv—fv) ;
2a^. ( — j =4^a.r^Q' — 6.(a.p — la.p),—,y.sin.-K.cos.\.cos.(gv—fv).
We have seen, in [5842], that the variation of dit is nothing,t even
when we notice the square of the disturbing force ; therelore, the
coefficient of cos.(gv—fv) must vanish from R. We shall denote by
the characteristic <5, placed before any function, the part of that fmct ion,
which depends on the oblateness of the earth ; and, we shall then have,
0 = 5. {r^Q')+(a.p —l<xp) .—.-;■ .s\n.\.cos.\.cos.(gv—fv) ;
* (3198) The value of s [5946'], is the same as in [4818^, suppos! Jig the origin of
gv to correspond to è ^ 0. From tliis expression, we get, in 2s.smfv , the term
y.cos.(gv — ■/«). Substituting this in [5944], it becomes as in [5947] ; and, from [5945],
multiplied by a, we get [5948]; observing, that in the terms whicli are connected with
ap — |a^, we may put r=«. Moreover, we have, as in [53479], /^l+TTWiJsi
g = l+?bo J nearly; so that the angle gv — fv is very small in comparison with v;
the mean increment of gv — fv in a given time, being the same as that of the longitude
of the moon's node [538Sc], and g — -/ is of the order m~ [4S28e], or of the same
order as the disturbing force of the sun upon the moon ; consequently the factor m'.{g — •/)
which occurs in dR [5949f], must be considered as of the second order, relative to the
powers and products of the disturbing forces.
751
[5946']
[5947]
[5948]
[5948']
[5949]
Symbol
0.
[5950]
[59470]
[59476]
[5947c]
[5947rf]
[5947 e]
[5947/]
t (3199) The secular variation of d.SR, or of dfi vanishes, as is shown in [5949a]
[5844 line 2, 5794", &ic.], noticing the terms of the order of the square of the disturbing
forces. Now the secular inequalities are those which are independent of the configuration of [594961
the heavenly bodies; that is to say, they depend on the variations of the elements, or on
the motions of the nodes, perihelia, inclinations, &.c., as in [4242 — 4251, &ic.] ; and as the
angle gv—fu represents the longitude of moon's node [5947rf], it partakes of the nature [5949c]
o f the secular quantities, being similar to those in [5846«], which are represented by the
angle gt-\-j2 , applied to the moon's orbit. If we notice only the terms of R [5881 1
which depend on the angle gv — fv, we may put it under the form.
R = m'.F'. cos. (gv — fv) ;
[5949d]
752 APPENDIX, BY THE AUTHOR ; [Méc. Cél.
hence we deduce,*
[5951] & . ) 2a^. [-T'j( = — 10.(ao — ^ap).— .7.sin.X.cos.X.COS.(^t)— ^).
We shall now resume the expression of d= [5784],
[5952] d^ = -^ .(l—^l-e^).{^~^+2a^^—yndt.
It is evident, that, if we neglect the excentricity of the orhit, we shall
have,t
[5953] ds = 2«^ (--\ndt ;
therefore, by noticing only the cosine of the angle gv—fv, and substituting
whose differential, relative to d , is,
[5949e] dR = — n. (g — f) . F'. sin. {gv —fv ) . dv;
[5949/] and, as the factor m'. {g — /) . F', is of the second order relative to the disturbing forces
[5947/], it must vanish from dR [5949«] ; therefore we must put F' = 0 ; and
then the expression of ït [5949rf], becomes /? = 0. Substituting this in [5947],
and retaining in r^ Q', the part ^-(''Q') [5949], corresponding to the angle
fgi} — fo^j^ vve get [5950] ; observing, that the co-ordinates of the moon produce in r^Q'^
terms depending on the angle gv — Jv , in the same manner as arguments of similar
forms appear in the expressions of the moon's mean motion and parallax in
[5220, 5331, &ic.].
* (3200) If we retain, in [594S], only those terms which depend on the angle
(gv—fv), and use the sign 6, as in [5949], we shall get,
[5951a] S .} 2a2 ^L_ j ( = 4a.8.(r^q') — C.(ap— ia(p) . — . 7.sin.X.cos.X.cos.(^y-/y) ;
Adding this to the product of [5950], by — 4a, we obtain [5951].
[5949g]
[5949;»]
[5953a]
t (3201) We have, by development, 1 — \/l—e^ = 2 «^ + ^c. ; substituting this
in the first term of [5952], we find, that it becomes of the order e; and by neglecting
terms of this order, we get [5953]. If we retain, in the second member of this last
[59536] expression, the term depending on the angle gv—fv, which is given in [5951], and
chanse ndt into dv, as in [5378'], we shall get [5954].
VII.Apj). VI.] EFFECT OF THE OBLATENESS OF THE EARTH. 753
dv for ndt [5378'], we shall get, as in [5379],
d;= — 10.(oL — \<i.^) .~^.y.dv.ûn.\.co^.\.co^.[gv—fv) [59536]. [5954]
This value of ds [5952, or 5954], is measured in the plane of the moon's
orbit ;* to refer it to the ecliptic, we must add to it the quantity \'{qdp-pdq) [5955]
[5955c]. We shall now determine p and q.
[5956]
The equation,
s =7. sin. ^t' [5946'],
may be put under the form,t
s = r.cos.(gv—fv).sin.fv-{-}.s'm.(gv—fv).cos.fv. ^^g^^^
If we compare it with the following expression, J
s = q. sm.fv—p. cos.fv, ^5958^
we shall obtain,
p = —r.sm.(gv—fv) ; q=r.cos.(gv—fv). [59593
* (3202) In computing the value of ch [5784 or 5952], from the expression [5775'],
we have taken, in [5775'line2], the primitive orbit of m, for the plane of the projection- [5955a]
so tliat the angle nt -{- s , or fncit + £ [5782, 5793], is counted on this primitive orbit.
If we represent the differential of this expression by dv = ndt -j- ds , and put dv, for rwî'ïM
its projection upon the fixed plane of the ecliptic [3778, Sic], we shall have, as in [3782],
dv, ^ dv-{-h.{qdp—pdq); so that, to obtain dv, from dv , we must add to ds the
correction \.{qdp — pdq), as in [5955]. [5955c]
t (3'20.3) We have gv =fv-{-{gv—fv) ; hence,
sm.gv = COS. (gv—fv).sm.fv+s\n.(gv—fv). cos.fv [21] Int. [5957a]
Multiplying this by 7, we get the second member of [5956], and this value of s
becomes as in [5957].
f (3204) The expression [5958] may be deduced from [1335'], by changing v into
fv ; which is the same as to count the longitudes from the moveable equinox, instead of the
fixed equinox [5345] Comparing the coefficients of sin./«, cos.fv, in [5957,5958], [5958a]
we get [5959].
VOL. lu. 189
754 APPENDIX, BY THE AUTHOR; [Méc. Cél
From these, we get,*
[5960] dp = —(g—f).q(lv ;
L5961] dq = (g—f).pdv .
The value of R contains the term,t
[5962] (a.0 — laj) . -— . sin.x. cos.x. o :
a-'
by the equations [5790, 5791], it adds to the value of dp the term,
D'' .
[5963] — (ap — laç)). — .Sin. X. COS.X. f/l) ; [TcTmofdp]
* (3505) The differentials of [5959] give,
[5960a] dp= —{g—f).y.cos.{gv-fi!).dv ; dq=r—{g—f).y.s\x\.{gv—fv).dv.
Substituting, in the second members of these equations, the values of p., q [5959], we
get [5960,5961].
t (3206) Substituting for Y.cos.(gv — •/»), its value q [5959], in the last term
of R [5947], and retaining only this part of /?, we get,
[5963a] iî = (oLp— la(p) .— .sin.X.cos.X.y [5962].
This is to be substituted in [5790,5791], as the most important part of R corresponding
to the values of dp, dq, now under discussion ; the other parts having the small factor
—, which is contained in r^Q' [5944r7, 6]. Its partial differentials relative to p, q,
give,
[5963t] (^— j =0 (;^; = (ap-^a^).— .sm.X.cos.X.
Substituting these, in [5790,5791], we get,
andt , , ^ i)3 . , „
[5963c] dp = — -;==g.(ap— Jao) . — .sm.X.cos X ; dq = 0 .
Neglecting terms of the order e^, and changing ndt into dv [5953i], we find,
that this term of dp becomes as in [5963]. Adding this part of dp to that in
[5963d] [5960], we get [5964] ; dq [.5961] is the same as in [5965], not being altered by the
term dq=0 [5963c].
VII.App.§4.] EFFECT OF THE OBLATENESS OF THE EARTH. 755
then we have the two equations,
D-
dp = — (^— /)-'y^^i' — ("^-f — 2"-^) • "i" • sill- ^ • COS. X.dv \ [5964]
dq= (g—f).pdv. ' [5965]
These equations give, in the expression of q, the constant term*
—^ ■-^. — r- .sin. X.COS.X r5965fZ] . (CoDslam pan of,] [5966]
g—f «^
From this we obtain, in the latitude s, the inequality,
* (3207) Taking the differential of [5965], supposing dv to be constant, we get,
ddq = (g—f).(Jp.dv. [5965a]
Substituting the value of dp [5964], dividing by dv^, and reducing, we obtain,
0 = ^+(S-— /f •?+(,?— /)-(<^P-W)--^'-sin.X.cos.X. [59656]
This equation is of the same form as in [865«, 870'], changing y, t, a, b, <p, &c.,
into 5', V, g — f, y, 0, &c. respectively ; by this means, we obtain from the
integral [8656,871] the following expression [5965f/], which satisfies [59656] ; as is easily
proved by substitution and reduction, by mere inspection, if we take separately into
consideration the two terms of q ;
q ^ y. COS. (gv—fv)— —- . — -.sin.X.cos.X. [5965rf]
g-f «^
The differential of this value of q, being substituted in the first member of [5965],
and then dividing by (g — f).dv, gives,
P = — y.s'm.igv—fv), as in [5959]. [5965c]
Multiplying [5965f/] by s'm.fv, and [5965e] by — cos.fv; then taking the sum of the
products, and reducing the factor of y, by means of [5957»], we obtain the value of
the second member of [5958], or the expression of « ; namely,
•5 = 7-sm.gv -— . — -.sm.X.cos.>..sin./f. \5965n
g—f «^
The term depending on a" — |o,a), being represented by ôs, is as in [5967] ; and if
we change the divisor g — ■/ into g — 1 ; / being nearly equal to unity [5947c] ; it [596.%]
becomes as in [5351].
756 APPENDIX, BY THE AUTHOR ; [MécCél.
|-5967] is = — ^-! — - — . — • sin.x.cos.x.sin./j; [596q/ J ;
which agrees with the result in [5351].
The constant part of q [5966] produces, in the function ^.(qdp — l)dq),
the following term, as in [5385] ;*
[5969] è-C^-P — 2^?) •— ^•7-sin.x.cos. A. cos.(o-« — -Jv),dv.
[59691 Putting, therefore, de^ equal to the preceding value of di, referred
to the ecliptic, we shall have,
[5970] ds^ = — ^.(o-p — la.D^.—.7-s'm.>..cos.x,cos.(gv—fv).dv ;
which gives, in s^ , and, therefore, in the moon's motion in longitude, the
inequality,
* (3208) Multiplying the expressions [5964, 5965] by J7 and — Ip, respectively,
and adding the products, we get,
[5968a] i-{Q<^V—'P^9) = —h-(g—f)-(j>^+q^}-dv—i.{o.c—o.(p).—.sin.Xcos.-k.qdv.
Taking the sum of the squares of q, p [5965f/, t], and neglecting terms of the order
(a — o-<p)^, we get,
., „ „ 2.(ap— a?)) D^ . , j- ^
[59686] V +q = T -^rZf — • -^ •y-sin.X.cos.X.cos.(g-ti— ».
Substituting this in [5968a], and retaining only the terms depending on (ap — a(p), we
get,
r5968c] h-il^^P — J>dq)^={^p — aq)).-j-.y.sin.X.cos.X.cos.(^ti — fv).dv — ^-(o-p — ttçj). — . sinX.cosX.çrftj.
r.5968rf] We may put ç = 7.co3.(^'-» — fv) [5965(Z], in the last term of [5968c], and then we
shall have, as in [5969],
[5968e] ^.(çf/p—pf/ç)^ J. (ttp—a(p).— .y.sin.X.cos.X.cos.(^«— /()).</«;.
This value of ^.{qdp—pdq) is to be added to ds [5954], as in [5955], to obtain the
[5968/] quantity which is called rfs, [5969'] ; and the sum evidently becomes as in [.5970].
Its integral gives the term of s,, or &v [597 1 ]; which agrees with that in [5387].
5973]
VlI.App. v^^ô.] GREAT INEQUALITIES OF JUPITER AND SATURN. 757
6v = — 13^. -^:i^\-i.— -.--.sin.x.cos.x.sin.C^-i; — fv). [5971]
- g-j «2
This result is wholly conformable to that in [5387].
Lastly, the luiictiou R being indeterminate, the preceding differential
expressio7is of the elements of the orbits, can also be used to determine the [5972]
variations they suffer, either by the resistance of an ethereal medium, by the
impulsion of the sun'' s light, or, by the change which the course of tim,e may
produce in the masses of the sun and planets. It is only necessary, for
this purpose, to determine the function R, which results from it, by the
considerations explained in chap, vii, of the tenth book* [8884 — 9036].
0.\ THE TWO GREAT INEQUALrPIES OF JUPFrER AND SATURN.
5. In the theory of these inequalities, given in the sixth book, we have
noticed the fifth powers of the excentricities and inclinations of the orbits.
But it has been discovered, that the values of A^^"', N^'\ &c. [3860-3860''^] [5974]
are taken with a wrong sign [386f)rt, &c.]. To correct this mistake, we
must change the signs of this part of the inequalities. This can be done, by
adding to the expression of the mean longitude, which is given in the eighth [59751
chapter of the tenth book, the double of this part, taken with a contrary sign.
This part, for Jupiter, is as in [4431,4430a] ;
6V''' = (12',536393— ï.0',00I755).sin.(5?ï^i— 2n''i+5=^— 2si') 1
— (8',l20963+t.0%004>885).cos.(5n't—2nH-\-5e''—2e") ; 2
and, for Saturn, as in [4487, 4483e line 4j ;
iv" = — (29',144591— <.0',004081).sin.(5n''^— 2w''i+5i'— 2.>) 1
+(18%879594+L0',0n356).cos.(5n7— 2n-i+53^— 2-=-). 2 ^^^'^^
The addition, to the mean longitudes of Jupiter and Saturn, of the double
[5976]
* (.3-209) This method of finding % or R [5438], has already been used
in estimating the resistance of the earth and moon, from an ethereal fluid [5672,5673].
Similar methods are used in ascertaining the values of R, in other cases, like those [^^^""o]
which are mentioned in [5973].
VOL. 111. 190
768
APPENDIX, BY THE AUTHOR;
[Méc. Cél.
of these inequalities taken with a contrary sign, can affect only the mean
motions and the epochs of these two planets. It cannot alter, except by
insensible quantities, the other elliptical elements, deduced from the
observations made between the years 1750 and 1800 ; because, during that
[5978] interval, the variations of these inequalities are very nearly proportional
to the time. We may, therefore, determine the corrections of the mean
motions, so as to make the double of these inequalities, affected tviih a
[5979] contrary sign, vanish, in 1750, when t == 0, and, in 1800, ivhen t = 50.
Thus we find, by noticing the correction of Saturn's mass, given in chap.viii,
of the tenth book [9121], that we must add to the mean longitude q" of
Jupiter, given in [9137], the function,*
[5980a]
[59806]
[5980c]
[5980rf]
[5980e]
[5980/-]
[5980fir]
[5980/1]
* (3210) We have, in [9128,9129],
n"t-\-i''- = S" 45'" 41\5+t. 30" 20"' 56',4 ;
nH-\-i'- = 231''21'"5S%9+^. jaMS" I7%1 .
Multiplying the second of these expressions by 5, and the first by — 2 ; and then putting
llie sum of these products equal to T, for brevity, we shall have,
T = 5n^<— 2w'^<+5£>— 26" = 69" 17"'5'l'',5+^24"' 32",7.
Now, if we double the expression of (hi'" [5976], and change ils sign, as in [5978] ;
then decrease the result, in the ratio of 19,232 to 20,232, on account of the change in
the estimated value of the mass of Saturn [9121], it becomes,
1 n 030
_2x^;-.(12',536393— i.0%001755).sin.r
+2 X ^^ . (8% 1 20963 +i.0',004885) .cos. T ;
the terms A" -\- B'^t, being added so as to make the expression vanish in 1750, and in
1800, when t := 0 , and ^ = 50, as in [5979]. To obtain the vakies of A'", E'\
we must first put f = 0 in [5980c], and we shall get the value of T corresponding
to this time. Substituting this, and < = 0, in [5980e], then putting the result equal to
nothing, as in [5979], we get the value of A'". Again, with t = 50, we get a new-
value of T [5980c]; substituting these expressions of ^, T, Jl"', in [59S0(?], we
obtain bOB'", from which B" may be determined. The result of this calculation
agrees very nearly with that in [5980].
In like manner, if we multiply the expression [5977] by 2, and change its signs, adding
also the terms jJ''-\-B't, we shall obtain the formula [5981]. Having computed the
VII.App.§5.] GREAT INEQUALITIES OF JUPITER AND SATURN. 759
âç'v= 16',84+^.0%1347 1
—(2S%S4.—t.0%0033).sm.(5n't—2n"t+5=''—2i") 2 [5980]
+ (15',44+^0',0093).cos.(5/i7— 2ft'''i+5-=^— 2i'^) : 3
and, to the mean longitude q" of Saturn [9138], the function,
59' == — 41% 19 — <.0',3309 1
+ (58',304— ^.0%008162).sin.(5n7— 2m''<+5=-'— 2.=") 2 [5981]
— (37',759-fï.0',022744).cos.(5n'<— 2»''^+5;'— 2-="). 3
expressions [5980, 5981], it will be easy to complete the calculations relative to the
observations of Ebn Junis [5982, &c.].
[5980i]
It is probable, that the coefficients of the function [5981], as well as those of tlie other
inequalities of the motions of Saturn, arising from the action of Jupiter, must be increased
in consequence of an augmentation of the estimated value of the mass of Jupiter by Gauss,
Nicolai, Encke, and Airy. The first estimate, made by La Place, in [40G5], is founded
on the observed elongations of the satellites, by Pound, and is — ^ — . But these [59804:1
elongations have been lately observed witii much greater accuracy, by Professor Airy, and
the result of his measures, given in vol. 10, page 404, of the Astronomische JVachrichlen
makes the mass — — . Nicolai, by the observations of the perturbations of Juno, gives [5980/1
iM^- Encke, by those of Vesta, j^^^;^ ; and by the perturbations of the comet which [5980m]
bears his name, j^^ . All these observations indicate, that the mass, assumed by La Place,
is too small by about i\ part ; and tliat the perturbations of Saturn, and several of [5980nl
the other planets, require some correction on this account. On the contrary, the calculations
of Bouvard, from numerous observations of the perturbations of Saturn and Uranus, make
the mass equal to j^^ . The cause of this difference must be ascertained by future [598O0I
observations and investigations. Some have supposed this discrepancy to arise from a
difference between the action of Jupiter upon Saturn, and upon the other planets ; but we
have nothing, analogous to this, in any known experiments or observations on the effect of [59P0p]
universal gravitation.
In closing this volume, we may remark, that the sequel of the work of Hansen, upon
the inequalities of the motions of Jupiter and Saturn ; which is mentioned in [■1458c], and ^^^^^l]
also the work on the lunar tlieory, by Plana and Carlini, [4752r(], have not been received
in this country at the time of writing this article. We must therefore defer any notice of I^'^^^^'"]
these works in the present volume.
760 APPENDIX, BY THE AUTHOR; [Méc. Cél.
These corrections have the advantage of making the formulas of the motion
of Jupiter and Saturn, given in the above-mentioned chapter, agree better
with a very important observation of Ebn Junis. This observation, reduced
to the meridian of Paris, took place the 31st of October, 1007, at 3*50'".
These formulas give 729' for the excess of the geocentric longitude of
[5983] ga,turn over that of Jupiter, at that time ; and the Arabian astronomer
found it, by observation, to be 1440' : the difference being 71 P. The
preceding corrections increase, by 388', the excess of the longitude of
Jupiter over that of Saturn; consequently, the new computation corresponds
more accurately with the observation, by that quantity ; and the difference
is reduced to nearly five sexagesimal minutes ; which is much less than
the error to which this observation is liable.
APPENDIX, BY THE TRANSLATOR.
We shall, in this appendix, point out some of the important improvements made by
Gauss, Olbers, and others, in the calculation of the orbit of a planet or comet, moving in
an ellipsis, parabola or hyperbola; with the methods of computing the place of the moving
body, at any time, by means of several auxiliary tables. For the sake of convenient [5984]
reference, we shall insert in the tables [5985,5986,5988], the most important theorems,
relative to this subject, which have been already introduced in the preceding part of the
work ; together with several new formulas, given by Gauss, in his Theoria Motus Corporum (i)
Cœ/cs^iMm, conforming, however, to the notation generally used by La Place, in this work.
In the demonstrations of the formulas included in the table [59S5lines 1 — 19], we shall
refer to any particular line of it, by including the number of the line in a parenthesis ;
thus, in referring to the value of c [5985line 1], we shall use the abridged notation (1).
(2)
From the assumed value of e ^ sin.p (1), we easily deduce the expressions (2, 3,4) ; ,3)
observing, in the formulas (3), that the development of |\/l-|-e =F \/]_e|2 becomes, by (4)
reduction, equal to 2 =F ~-\/\ — t~ = 2 ^ 2. cos. 9 ; and, that, (5)
4.sin.^Jip, 2 -j- 2. cos. (p ^ 4x03.^^9 [1,0] Int. (6)
The expression of p [378s], is the same as in (5) ; those of D , a (6), are as in
[681"]. The second and third values of p (5), are easily deduced from the first, by
using p, D (1,6). The formulas (7,8,9), are as in [606], using the second of the
expressions (4). The first of the formulas (10), is the same as in [603] ; the second and
third values are obtained by means of (5). The expression of cos. m (I !), is the same
as in [603i] ; and, from this, \\e easily obtain the value of cos.i;, in the same line. The
first expressions of sin.lw, cos.\u (J2,13), are the same as in [1, 6] Int. The second
values in these lines, are deduced from the first, by the substitution of the formulas,
(7)
(8)
(9)
(10)
, zp eos.u = ii^^Ml^i^) [603&line.5], Ot,
l-(-e.cos.r
and putting ^.(1 — cos.d) = sin.^iw , J . ( 1 +cos. «) ^ cos.^^d [1,6] Int. The third (is)
VOL. III. 191
762
APPENDIX, BY THE TRANSLATOR ;
[5964]
(13)
expressions are deduced from the
second, by the substitution of
' =-- (10);
l-|-e.cos.j; p
the fourth, or last of these values,
is deduced from third, by the
substitution of
jp=:a.(l — e^) (5).
The last of the formulas (14),
is the same as in (8); and the
second is deduced from this by
(15) using the value of tang.(45'' — |(p) (4). Multiplying together the last values of
(15) sin. Am, cos. ^u (12, 13), reducing by means of [31] Int., and using ^/l — e^ ^cos.9 (I),
we get the last expression (15) ; the second expression (15), is deduced from this, by
M7> usin2 == — - (5). The first of the formulas (16), is deduced from tiie first of
'"' " a.cos.ij] P
(14)
(15); then substituting p = t/.cos.^ç), and ^^=«.cos.<p, we get the third and
fourth expressions in that line. Multiplying together the first values of r and cos.v
(9, 1 1), we get the first expression of ?-.cos.'v (17) ; substituting e = sin.ç, or rather
_ c = cos.(90''+ç)) , we get,
cos. M — e = COS. w + cos. (90"'+ 9)
= 2 .cos.{iu 4-i<p+45'').cos. (^M — *(? —45'') [27] Int.;
whence we easily obtain the last expression (17). Multiplying the third value of cos.J?<
(13), by sin.|« , and the third value of sin.^M (12), by — cos.iu; then taking
the sum of the products, and reducing, by means of [22,31] Int., we obtain,
sin. I .(r — m) = |sin.D . 1/ L • \^i -\- e — \/l — e] ]
substituting the first of the formulas (3), we get the first of the expressions (18) ; and, by
(24) using the value of sin.r =: VP"-^'"'" (16), we obtain the second of the formulas (S).
If we repeat this last calculation, changing the factor — cos.iw, into -f cos.Ji; ,
we get,
sin.J.(i'4-w) = ism.v.X/ L .l^rfe -f ^r^e],
and by using the second expression (3), we get the first formula (l9) ; then, substituting
the preceding value of sin.v, we get the second of the formulas (19).
(18)
(19)
(20)
(21)
(22)
(23)
(23)
FORMULAS IN AN ELLIPTICAL ORBIT.
763
FORMULAS IN AN ELLIPTICAL ORBIT,
e -i^ sin. 9 ; \/(l c") = COS.O ; [ExcenHicity t]
1— c = 2.sin.2(45''-J?) = 2.cos.3(45''+à(p) ; I4-e=2.cos.2(45''4(p)=2.sin.2(45"+J<!)) ;
v/(l+e)-v/(l— e)=2.sin.*<p ; ^Çl^e)-^\/(l—e) = S.cos.^? ;
1=^= tang.2(45''— I?) ; J±^ = tang.2(45''+iç>) ;
p =; «.(1 — e^) = «.cos.^(p = (l-j-e).J) ;
D = a. (I — e) = aa. ; a. = 1 — e ;
n< = u — e.sin.M ;
tang.Jt) =r » y/f — ^Ytang.jM = tang.(45''+è'p).tang.jM ;
r = a. [I — e.cos.w) ;
n.(l — e2) a.cos.2? p
l-}-ecos.« l-}-e.cos.u l-(-f.cos.u '
COS.U =
C03.U — e
1 — e.cos.ît
COS. M
e-|-cos.«
l-[-e.cos.« '
iu=\/h.(l — cos.u)=s\n.iv.( ) =s'm.^v.[- ) =sin.it'-l — ~ — 1 :
2 ^ ^ ^ ' \l+e.cos.vJ \ P / ^ \a.(l+e)/ '
.Au=n/*^Cl+cos.M)=cos.Jy.( ,-: y=cos.|«.{ '■ -) =COS.U'.( ^- Y;
tang.iM = tang.Jj;.tang.(45''— ^<p)= iy^^^-^j.tang.iu ;
sin
cos
sin. M
r.sin.îJ
r.sin.D.cos.ç r.sin.î)
p a.cos (?
p.sin.14
cos.ip
a.cos.ip.sin.M =K/pa.s'm.u ;
[5985]
(1)
(2)
(3)
r Paramotcr 9p
(5)
rPorihelion distance D
1 (6)
[Mean anomaly ni
(7)
rTime from Perihelion (.
|_ cxpressetl in days
(8)
fExcentric anomaly \l
(3)
[Radiu^t vector r
(10)
Elliptical
formulas.
[True anomaly v
(11)
Y >• V.
(121
r.cos.i' = «.(cos. M — e) = 2a.cos.(jM+^(p-|-45'').cos.(^M — |(p — 45'') ;
sin. J.(t) — u) = I y - . sin.|?.sin.D = i / - . sin.|?).sin.M ;
sin.5.(i;-|"w) "^ I / ~ .cos.5(p.sin.«= * X - . cos.jp.sin.w.
(13)
(M)
(15)
(16)
(17)
(18)
(19)
764
APPENDIX, BY THE TRANSLATOR ;
[5986]
FORMULAS IN A PARABOLIC ORBIT.
(4)
(5)
(6)
Parabolic
formulas.
(7)
(8)
(9)
The equations of the motion in a parabola [59S61ines2 — 101, are the same as in
[691,693, &C.J ; in which 2* represents the circumference of a circle, whose radius is
(1) unity [691'" line 4J, and r=365''^J\ 25638, is the length of a sidéral year [691'", 750'].
j Parameter 2p 1
fPejihelion distance Z>1
FRadius vector r\
[True anomaly v J
(2)
p = 2D ;
(3)
D = ip ;
D
COS.Siî) l-(-cos.w '
* . V/2" '
nh'
tang.|t)-|-^.tang.»Jr}
tang.|î)+i.tang.3J«J
= j-ltang.èu+^.tang.^ét)}
" Time from the Peiihelion f,
expressed in d&^a
]
[Time from tho "1
periiielion t' daye, I
when B=J. J
(10)
3
B^
[5987] In the expressions of t [5986 lines 5, 8], we
(1) ought, in strictness, to change T into T.y/i-f-ni" ;
m" being the mass of the earth, and 1 the
mass of the sun j this is evident from [692', &ic.],
(2) where |j. = l-|-m". It is common, however, to
neglect the mass m", as we have already
observed in [692' line 4]. Instead of T, or
2ir
(3) rather T.\/l-|-m", the symbol ^=fVT^"
is used by Gauss, and by most of the
(4, German astronomers. We have already found,
in [750'],
T
T
(5)
127
= 9'i-y% 688724..., or log.-^ = 0,98626669... ;
FORMULAS IN A PARABOLIC ORBIT. 765
and, by neglecting m", we have, [59871
(6)
2x , , , St
fc=— , or log./. = log.-f^ 8,2355820...;
but, if we notice m", we shall get
to^"-)
log.fc.\/ï+^' = log.^ = 8,2355820... ;
(7)
(10)
and, since log.^l-j-m" ^= 0,0000006... , we shall obtain the corrected value of,
log.fc = 8,2.355814... ; (8)
being nearly as it is given by Gauss, in his Theoria Motus Corporum Cϔestium ; differing
from the former expression, by the very small fraction 0,0000006... We may remark,
that the mean angular motion of any planet, in the time t, is represented in [605", 605'J,
by nt = '^ ^"^ ; m being the mass of the planet ; a its mean distance from the '■^'
a*
sun ; that of the earth from the sun being taken for unity. The second member of this
expression must he multiplied by a constant quantity, which is the saine for all the planets, to
reduce it to the unit of the measures of these angles. To ascertain this quantity, we shall
observe, that the mean angular motion of the earth in a sidéral year T, is represented
by the whole circumference 2* [691''] ; and, if we change, in the second member of (U)
[5987(9)], t, m, a into T, m", 1 respectively, it becomes T.y/l^^. To
2*
reduce this to 2c [598T (1 1)], we must evidently multiply it by w, . =-, or by
the quantity Jc [5987(3)]; which therefore represents the constant quantity [5987(10)] ;
hence the mean motion [5987 (9)] becomes nt=' ^ " ; consequently n= ^ '"''. „2)
«^ ai
This value of n must be used in [5985 (7)]. If we wish to express the mean motion in
secondsjwe must multiply the expression of nt [5987(12)] by the radius in seconds 206264',67; (i3)
or, to avoid this labor, we may use the value of A." in seconds ; namely, A," ^3548', 18761
or log.A-^ 3,55000657. In estimating tlie motion of a comet, we may neglect its mass (H)
VI, on account of its smallness ; and tlien the expression of the mean motion [5987 (12)]
becomes — . This is expressed in [702'] by -^^ ; the accent on n' being (is)
omitted, to conform to the present notation. Hence it appears, that we must put y/jji = k,
to reduce the formulas of the author, in [702', &c.], to the notation of this article.
(16)
The expressions in [5986 lines 2, 3] are the same as in [807', 807"]. The first formula
in [5986 (4)], is the same as in [691 line 1] ; the second expression is easily deduced (17)
from the first, by the substitution of
cos.^iw = J -)- i . cos. V ,
VOL. III. 192
766
APPENDIX, BY THE TRANSLATOR ;
[5987] and the value of D [5986 (3)]. The expression of t [5986 (5)], is the same
as in [693]. Substituting in this, the value,
(18)
T =
[5987 (6)],
(19)
(20)
(21)
(32)
we get [5986 (6)]. This expression of t may be put equal to V^ t' [693a],
as in [5986(7)]; t' being the time from the perihelion, corresponding to the anomaly
V, in a parabolic orbit, whose perihelion distance D is equal to unity. This parabola
is usually called the parabola of 109 days ; because, it requires about 109 days to
describe an arc of 90'^ from the perihelion, in a parabola whose perihelion distance is
unity. Dividing the three expressions in [5986 (5,6, 7)], by D^, we get the formulas
[5986 (8, 9, 10)]. From that in line 8 or 9, Burckhardt has computed Table III, of this
appendix, changing v into U; and putting,
3!r.v/2
_ 27day3^ 4038..
Then, by means of this table, we can find, by inspection, the anomaly U, or v,
from log. t', or the contrary.
FORMULAS IN A HYPERBOLIC ORBIT.
[5988] The formulas for computing the motion of a body, in a hyperbolic orbit, are given in
[702] ; but, it will be convenient to alter the forms of these expressions, by writing
(1) o for a', and introducing the auxiliary quantities >f, m, proposed by Gauss j so that
(2)
e . cos.^ = 1,
and
u = tang.(45''4-i«) ;
by this means, we obtain the following system of equations, corresponding to the motion
in a hyperbolic orbit.
is
I
S
^
si
t9
FORMULAS IN A HYPERBOLIC ORBIT. 767
e_i [rm8]
e == —, = secant4-=v/i+tan!T.s4.; \/î2ZT= t&ng.^; j-p-=tang.2è4.; [ExcentncUyeJ o)
^ =a.(e^— l) = rt.tang.=»4-= (e+l).D; [Parameter 2p] (4)
D= a.{e—l) =—aa.; a=— (c— 1); [perihelion distariceC] (5)
— -.t = e.tang.a liyp.log. tang.(45''-j-2-ro) ; [Semi-transvorso axis o] (6)
^K (m 11 //icdl 1 N rTiine from tlie perihelion (,"I ,..
— J. ^ = iXe.-i ^ common log. tang. (45 +Jro) ; [ expressed in days J (7)
a* «
X = 0,4.3429448... ; log.x = 9,63778431 13... ; log.Xfc =7,8733657527... ; (S)
fc = 7 -^^ = 0,01720209895 parts of radius ; log./c=8,2355814414... ; (9)
J- -y l-\-7n"
tang, ^-a ^ I / ( — Vtang.^r = tang.^^/.tang.^u ; [Auxiliary angia a] (loj
p a.[t- — 1) ^.cos.sj^
l-j-e.cos.« l-f-e.cos.u 2.cos.fs(t) — 4').cos.à(r-^-^],) '
j Radius vector r\
(IJ)
r =: a. ( — 1 ) = ^a . ■; e . I w + - 1 — :^ > ; (i2)
.rf_l_j >, ]-f-tang. AiTT _ cos..i(v— 4)
HyperboJk
furmulas.
W = tang.(45*+i7.) = ^_^^^^l^ = eoS.4(«+4) ' [Auxil.ary ,uant..y .] (.3,
1 1 f \ ^\ l-l-cos.^.cos.v e-|-cos.t!
COS. -51 ^ V w/ 2.cos.è(i'— 4).cos.è(«-(-4) l+e.cos.u '
«2-1 2m u2 — 1
r. sin.i' ^^.cot.i/'.tang.'n :^ a.tang.4'.tang.a
= ^p.cot.4-.U— -j == |rt.tang.+ .^<- A ;
(21)
C22)
r. COS
768 APPENDIX, BY THE TRANSLATOR ;
[5989] In the demonstrations of these formulas [5988], we shall refer to any one of them, by
placing the number of the line in which it is situated, in a parenthesis, in the same
manner as in the elliptical formulas [5984 (2), fee.]. From the assumed value
e = (3), we set, by means of [1,6] Int.,
1— C0S.4 2.sin.2.H , , l+cos.X 2x03.244
(?) e — 1^ — = - — r~ ; ^+1 — ;"■= r~ '>
COS.X COS.n}/ COS.-4 COS.4
dividing the first of these expressions by the second, we get the third of the formulas (3).
In a hyperbola, the semi-axis a becomes negative, and is represented by — a [698"] ;
hence the values of j), D [5985 (5,6)] become, in the hyperbola,
p = a'.(e2— 1); D = a'.{e—l);
(3)
w
(5)
(6)
(7)
and, if we neglect the accent upon a! , for the sake of simplicity in the notation, we
shall obtain the first expressions of p, D (4, 5) ; the others are deduced from these,
by the substitution of ^l, D (•^,5). If we change, in the first equation [702], the
symbol \/fji. into k, as in [5987 (16)], and omit the accent on a, as above, it
becomes as in (6) ; using hyperbolic logarithms. We must multiply this by
X = 0,43429448... (8),
when common logarithms are used ; the quantity X being the ratio of a common
logarithm to a hyperbolic logarithm ; and then, (6) changes into (7), by the substitution of
tang.ci (15) ; this value of tang.ra being deduced from the assumed value of u (13),
(8) as in [5989(14)]. The first formula (10) is the same as that in [702 line 3] ; from this
we easily deduce the second form, by the substitution of tang-J^^ (3). The first
value of r (11) is the same as in [3796], using p (4) ; and this form is common to
the other conic sections [5985 (10), 5986 (4)]. Substituting in this, the first value of
» (4), we get the second form of r (11)- Multiplying the numerator and denominator
(10) of the first form of r (11), by cos.-^, and substituting e.cos.-^- = 1 (2), in the
denominator, we find, that this denominator becomes,
(11)
cos.vj.'+cos.v = 2.cos.|(« — 4')-cos.5(t)+4') [20] Int.,
and we obtain the third expression of r (U). The first expression of r (12) is the
same as in [702 line 2], omitting the accent upon a', as above. To obtain the
second form, we must use the auxiliary quantity u (13) ; namely,
. = tang.(45^+i-):=.;±;:-g^ [29] Int. ;
from which we get,
(.3, t«"s4- = ^ (16) ;
and then, from [30'] Int., we have, as in (15),
(12
FORMULAS IN A HYPERBOLIC ORBIT. 769
_ 2.tang.*73 __ \u+U __ 2.(m^— 1) u^—l
[5989]
(H)
From this, we get,
sec.TO = — ^ = (l+tang.^ra)i = ^^==: jYm+-Y as in (14, 15). 05)
Multiplying together the expressions of cos.ra, and tang.ra (15), we get sin.*
(15). In Uke manner, if we substitute the value of tang.Jw (16), in the expression,
cos.J=r == (l+tang.2^^)-i [34'"] Int., (I6)
we get its value (16) ; multiplying together these two expressions of cos.^w, tang.|î;i,
we get sin.|:3 (16). Substituting the first value of (14) in the first expression
of r (12), we obtain its second form. If we substitute, in the second expression of
M (13), the value,
tang.Jra == tang.|4-.tang.|y (10) ; (i8)
then, multiply the numerator and denominator by cos.J4'.co3.|«, we shall find that
the numerator becomes,
cos.i^^.cosl^J+si^.^^^.sin.l^) = cos.i(î) — 4.) ;
and, the denominator, ,|9>
cos.J-J/.cos.^t) — sin.^4/.sin.l« = cos.J(«+4/) ;
as in the last of the formulas (13). If we now substitute the last value of m (13), in
the first expression of (14), it becomes,
1 ^ 1 \ cos-i(«— 4-) , cos.j(r+4.) -i _
cos.a- ^ ' I cos.!(i;+4.) cos.|(î;— 4.) 5 '
reducing these to a common denominator,
2.cos.i(t;— 4,). cos.J('y+4.),
we find, that the numerator becomes, by using [6, 20] Int.,
cos.2|(o-+)+cos.2|(t;+4-) = li+è-cos.(y— 4.)H-{'+|.cos.(«;+v}.)J
= 1+|.cos.(d— 4)-f|.cos.(jj-f-4-) = l+cos.+.cos.D ;
as in the second formula (14). Multiplying the numerator and denominator of the
second formula (14) by e, and substituting the values [5989 (ll)J and (2), we get '^'*
VOL. III. 193
(20)
(21)
(22)
770 APPENDIX, BY THE TRANSLATOR;
[5989] the third formula (14). If we add q=l to the last of the values (14), and
(24) 1 r
substitute , , =- (11), we get,
l-f-e.cos.K p
lq=cos.3r (e=F] ;.(l=Fcos.i)) (eTI).(l^cos.D).r
(25) = — = .
cos.^ l-|-e.cos.« p
If we use the upper sign, and put
(26) 1— cos.OT = 2.sin.®ia- ; 1 — cos.î) = 2.sin.^iM [l]Int. ;
we get, by extracting the square root, the first of the formulas (17). If we use the lower
sign, and put,
(28) l_|_C0S.ar = â-COS^Ja- ; l-(-COS.l' = 2.C0S.^|« ;
we get the first of the formulas (19). The second of the formulas (17, or 19), is deduced
(29) from the first, by the substitution of p = a.{e^ — 1) (4). Substituting, in the first
of the formulas (17, or 19), the values of sin.Jji, cos.iw, cos.ct (16,15), which
give,
sin. its u — 1 cos.^ts u-\-l
^/cos.TS 2.\/u ycoi.zi 2yu
we get the first of the formulas (18,20); finally, substituting in these, the value of
p=a.(e^ — 1) , we get the last of the formulas (18,20). Multiplying by two the
product of the first of the formulas (17, 19), we get,
2sin.|-!S.cos.|'ss p
cos.w {/{e^ — 1)
and by substituting,
2.sin.iu.cos.|« = sin.w ; 2.sin.its.cos.its =: sin.-z3 = cos.ra.tang.a [3] , 34'] Int.,
(32)
also y/(e^ — 1) = tang.4. = (cotang.4')~^ (3), we get the first equation (21). The
(33; second formula (21 ), is easily deduced from the first, by the substitution of j3=a.tang.^4/ (4).
('*; Substituting in these two expressions, the value of tang.is = |.f h j (15), we get
the first and second formulas (22). Multiplying the second value of r (11), by cos.j;,
and reducing, we get, by using the last formula (14),
a.(e^ — l).cos.« (e-[-cos.v) 1
(35) r.cos.'w = — ^ — = «e — a.-- = ne — a. ;
l-J-e-cos.u l+e.cos.« cos.ra
as in the first expression (23). Substituting in tiiis, the first value of ^^^ (14), it
becomes as in the second formula (23) .
INDIRECT SOLUTION OF KEPLER'S PROBLEM. 771
From the first of the formulas (11), it appears, thiit r increases with v, and becomes [5989]
infinite, when
l+e.cos.i' = 0, or cos.r ^= = — cos. 4^ (.3) ;
(37)
e
(38)
(41)
which gives v = ISC — 4'' Now the radius ;•, corresponding to a point of tlie
hyperbola, at an infinite distance from the hcus, must evidently be parallel to the asymptote;
therefore, the angle 4^ represents the angle of inclination of the asymptote to tiie axis.
Hence it is evident, that the maximum value of v is represented by 180'' — 4^ > ''nd (39)
the greatest 77!!n«m«?n value is — (ISC' — 4-) j moreover, it follows, from the last of the
COS.A.f — 4^) (A(\'\
formulas f IS), that when v ^=r. 0 u ^ ■ ^-rr; = ' > a'la tli<it ^ increases with v, ^ '
^ '' cos.i.{-(-4)
and becomes infinite, when y = ISO''— 4-, or \-{y-\-\^ = OC . It decreases when
I' is negative, and becomes nothing at the other limit, where v = — (ISO" — 4'); or
\.(v-\) = - 90".
TO COMPUTE THE TRUE AXOMALY FROM THE TIME, OR THE CONTRARY, IN AN ELLIPTICAL ORBIT.
The true anomaly v, in an elliptical orbit, can be easily obtained from the mean anomaly (59901
«t, by means of the formula [<3C8], in cases where the excentricity e is so small, that
it is only necessary to notice two or three terms of the series ; but as the value of e '''
augments, the number of terms must be increased, so that the method finally becomes
very laborious, and it is much better to use the indirect method of solution, first o-jven bv
■' ^ a J Kepler's
Kepler, who was the original proposer of the problem. This method is very simple, and '""''''="'•
has the decided advantage of being applicable to all the varieties of the ellipsis ; but when
the excentricity is nearly equal to unity, it requires the use of a table of logarithms, to
more than seven places of decimals ; this difficulty is obviated partially in the method of (2)
Simpson, and wholly in the method of Gauss, which we shall give hereafter.
To illustrate this indirect method of solution, we shall apply it, according to the precepts
of Gauss, to the determination of the true anomaly in an elliptical orbit. We shall suppose
M, to be an approximate value of u, and x its correction; so that u = u, -{- x ,
satisfies the equation [.5985 (7)]. We must compute the value of e.sin.w^ in seconds, by
logarithms; and, while performing the operation, we must take from the tables, the variation
X of the log.sin.î/,, corresponding to r in the value of u^ ■ also the variation fj. of the (S)
logarithm e.sin.w^, corresponding to the variation of one unit in the number e.sin.w^;
the signs of X , |u, being neglected, and both the logarithms being taken to the same
number of decimals. Now when w, is nearly equal to u, or u^-j-x , the variations
of the log. sines of the arcs from ?<, to u^-\-x, will, in general, be nearly uniform ;
hence we shall have, with a considerable degree of accuracy,
(3)
(4)
(B)
(7)
■Kx
c. sin. {u^ -\- x) = e.s'm.u^ ziz — ; (8)
(14)
(15)
772 APPENDIX, BY THE TRANSLATOR;
[5990] the upper sign being used m the first and fourth quadrants ; the hiver sig7i in the second
(9) and third quadrants ; these signs being evidently the same as those of e.cos.w, [5990(13)].
Substituting this, and u = u^-\-x , in [5985 (7)], we get, by reduction,
(10) a; = . (lit — u, 4- e . sin. u,) ;
Indirect
solution of _«
Kepler's ^* '
problem.
(11) u = ti,-\- X =: nt -{- e . sin. M, ± . (nt — «, + e . sin. ?<,) ;
(J, ^ X
(12) in which we must notice the sign of the factor ± ■ ^ , according to the above
directions; and we must also have regard to the sign of the other factor (nt — !i^-|-c.sin.M,).
(13) We may remark, that the factor ± - == e.cos.w,, as is easily proved by the substitution
of sin.(M,+a?) = sin.?<,-fx cos.m, [00] Int., in the first member of [5990(8)] ; and, as
e<l, cos.M,<l, we shall have (j.>X ; tlierefore, ,7^ has the same sign as -.
If the assumed value of u, should differ considerably from «,+» , we must repeat the
operation ; using this computed value of m,+x' for a new value of «,; and this process
must be repeated, until the correct value of u is found. In most cases which occur in
practical astronomy, it will be easy to assume, in the first instance, a value of u^ which
(^5) does not difier much from v . This is particularly the case, when forming a table of the
values of u, corresponding to the regular intervals of nt , from 0"* to 360". If we
have no means of ascertaining this first value of w, , we may make the first computation
in a rough manner, using small tables of logarithms, to five places of decimals, and to
minutes of a degree. It unll tend to simplify the operation, to take for w, a quantity
''^* whose sine can be obtained frorn the tables by inspection, withont any interpolation; as, for
example, by taking the value of n^ to minutes, when the table of sines is given for every
*^^' minute ; or for tens of seconds, when tlie tables are arranged for tens of seconds ; &,c.
Useoftho jn rnakina: these calculations, and others of a similar nature, it has been found
letter II, " /• (• J 7
^fth'e convnnent to annex the small letter n to the last figure of the logarithm of any factor
*°'"°°(iO) which has a nc<ratiue value ; since, by this means, ive can very easily ascertain the sign
n"umericai of 0 quantity, which depends on the product of a number of factors, of different signs,
tion. ^^^ whose lo"-arithms are to be added together, to obtain the logarithm of the required number.
It beino- evident, that the sign of this number must be positive, if the number of the letters
n be even, but negative if the number be odd. Thus, in finding the logarithm corres-
pondin"- to the quantity —3. sin. 192'', composed of the two factors — 3 and sin. 192'*,
we may put for their logarithms the quantities 0,4771213„ and 9,317S789„ , whose
'■'^^ sum 9,7950002 corresponds to a positive quantity. We must also carefully notice the
signs of any quantities, depending on the sine, cosine or tangent of an arc ; observing that,
according to the usual rules, we have,
(20)
(21)
INDIRECT SOLUTION OF KEPLER'S PROBLEM.
773
sin. or cosec. is + in the first and second quadrants ; — in the third and fourth.
COS. or sec. is -j- in the first and fourth quadratits ; — in the second and third.
tang, or cot. is + in the first and third quadrants ; — in the second and fourth.
To show by an example the use of the formula [5990(11)], we shall suppose the mean
motion to be «< = 332''28"54',77, log. e in seconds = 4,7041513, or e = 50600'
nearly. Then, for a first operation, we shall take m, = 320'', from which we find, as
below, !(,-)-a; = 324'' I6"'20'. Taking this for u, , in a second operation, we finally
obtain!/ = 524''16"' 29',5 ; which is its true value, as will appear by the following
calculations.
[5990]
(23)
(24)
(95)
(20)
FIRST OPERATION «, = 320''.
SECOND OPER.\TION M, = 324" iG™ 20'.
11, = 326'' log.sin.9,74756i7„
e log. 4,7o4i5i3
^= 3i
u, = 324'' i6" 20' log.sin.9,7663644„ Jv = 29
e log. 4,7o4i5i3
e.sin.u, log. 4,45i7i3o„
At = i53
e.sin.M, log. 4.47o5i57ii ft = i47
e.sin.u, = — 28295' = — 7"* Siik 35' /.< -
nt = 332'' 28"' 55'
— K = 122
c.sin.M, = — 29547', 16 = — 8'' 1 2'» 27',i6 ^ — ^= 118
nt = 332" 28"' 54«,77
nt + e.sin.u, = 324" 37'" 20' = jî
u, = Ssô"* 00"' oo«
ni -\- e.sin.u, = 324'' 16" 27«,6 = .4
w, •= 324'' ifi"" 20'
{nt — u,-\- e.sin.u,) = — i** 22"' 4o' = — 4960^
multiply this by
± =-f- î^gives — ai^oo» = S nearly
(27)
(nt — M, "4" e.sin.u,) =
multiply this by
+ 7',e
=t^— r=+3[|| gives +i',g=B
A + B =u,+x
32^(f igm 20*.
.«
u = 324'' 16"" 29»,5.
Having obtained the value of u, we may compute r, v from [5985(9,11)]-
but as the method of making this calculation is sufficiently obvious, we shall not "ive an
example.
When the excentriciiy e is very nearly equal to unity, this indirect method requires the [5991]
use of tables of logarithms to more than seven places of decimals. For, if the logarithms
were correct, to the nearest unit, in the seventh decimal place, there might be an error
of 46% in computing the anomaly, in an otbit, where 1 — c=: 0,001 ; and, the error
would exceed this, by decreasing 1 — e. In this case, we may use the method of Simpson,
given by La Place in [694—698], neglecting all the powers of 1 — e = a, above the
first. This degree of accuracy is not, however, sufficient, in Halley's comet, where
1 — c = 0,03, nearly ; for, it is found to be necessary to notice the terms depending on
the second power of 1 — n ; which exceed 30% when the anomaly is 100'. If we use
the same notation as in [694', &ic.], we easily perceive, that the true anomaly v=^U-4-x
in the ellipsis, may be derived froin the value of U, corresponding to the parabola, by
an expression of the following form, in which the third and higher powers of 1 e — g
are neglected ;
VOL. III. 194
(1)
(2)
(3)
(4)
(5)
(6)
774 APPENDIX, BY THE TRANSLATOR ;
[5991]
V = V-i-S.{l—e)-i-B.i\ — ey = U+S.a. + B.o.^ ;
m
(8)
m
S being the value of the function [698], corresponding to Simpson's method, and B the
function [5991(30)], introduced by Bessel, in his tables, published in vol. 12, p. 201,
of the Monatliche Correspondenz. The same formula may be applied, ivithout any
modification, to a hyperbolic orbit, which approaches very near to a parabolic form, by
merely noticing the sign of 1 — e, which then becomes negative.
(10) In the computation of S and B, we may put, for brevity, tang.|t7=â, or,
(11) cos.^jZ7 =
1+tang.ail/ l+é^ '
Substituting these, in the expression of S [698], it becomes.
Method of
Simpson,
byBeàséi. To obtain B, we shall develop the expression [690], according to the powers of a,
neglecting terms of tlie order «■' ; hence we get the first of the following expressions ;
(13) the second form is deduced from the first, by multiplying the terms, between the braces,
by the external factor 1 +ja--t-i|"-~ j tlie third form is obtained, by arranging the terms
according to the powers of a ;
t = 5^ .(l+k-+-|,a.2).tang.i« . ^ i vj a g ^ & 2 r
/f. ^ ^^ ^- J ^- I +(-^+.'50L2).tang.ni;+>^.tang.<-i«5
= ''Z^- { (l+|a+ia2).tang.ii)+(i-ia— icL2).tang.3ir— ia..tang5J»4->2.tang.7,^|
ly— i.tang.^lt)) >
i«+|5.tangJ|K) 5 '
(14)
(15)
D~.y^2 ( tang.|!;-[-3-tang^|y-|-"--(itang-i'' — |.tang ^i" — 1 tano- si
//•'' I -fa2.(i.tang.iu— ^.tang
(16)
If a = 0, I' changes into U, and the expression of t becomes as in [691].
Putting these two values of t equal to each other, and dividing by the common factor
D^.\/2
—^, we get,
tang4C/-}-|.tang.^|l7 = tang-lv+e-tang.^iy-f-a, . | i.tang.i« — i.tang.^|y — i.tang.'|w}
+a2. {â.tang.|w — ^.tang.^ •«-|-|'5.tang.''|» J .
(17) If we put, for brevity, x = So.~\-Ba?, we shall have v=U-\-x [5991 (7)] ; and, by
neglecting x^, which is of the order a^, we shall get, by means of [29, 45] Int.,
-) tang.,«^tang.,(C^.) = ^SSl = ^ = Hi-(l+^^)+l-^^-(l+^^)-
SIMPSON'S METHOD, FOR A VERY EXCENTRICAL ORBIT. 776
Re-substituting tlie value of x [5991 (17)], and putting, for brevity, l-{-ô^ = ê^,
we get the following expression of tang.it) ; from which we easily deduce its powers
tang.'jt), &:c. ;
tang.|u = ()+U....S'<),-fa2.0,.{iJ5+iS2j} ; (20)
tang.3U'= tî3+^a..S'()2t),+a.2.(),.{|Bd2-fiS2(^â,+é3)| ; (ai)
tang.5it)= ti*-|-|a.S()<(», + 8ic. ; tang.^iv == f+hc. ''^^
If we substitute these in [5991 (IG)], the terjns independent of a will mutually destroy
each other; also those depending on the first power of a: and, if wc notice, in the
second members of the following expressions, only the terms multiplied by a-, we shall
have, by using the values [5991 (20—22)], and Sê'^ = (_|é+|é3+=()5) [5991 (12)] ; («)
[5991]
<24)
tang.Jj;+e.tang.3|v = a^J,. Jii?. (l+â9)+i5'2<3.(l+^.+ô2) | = a,a.^ ■Bâ^+jS^âé.s}
a.^.lang.ii'— i.tang.3>— i.tang.ni'5 = o?.Sè^.\\—\è^—\è'^\ = l<i?.S.\\—\è^—lè'^—t)^\; ^s)
a^^a-tang.!»— ^.tang.='i«+i\.tang.^Jf} = ^^.\gè—i,S'^i^-']. t^^'
The sum of these three formulas represents the terms depending on a^, in the second
member of [5991 (16)] ; and, as this sum is to be put equal to nothing, we shall get, by '--'''>
dividing by \'^^~^, the first of tlie following expressions. Substituting in this, the value
of 5^,2 |-599^ (23)]; also è^^ =l-\-2&^-\-èi ; and then reducing, we obtain the
second value of B.1,* ; dividing this by è*, we get the value of B ;
The values of the logarithms of S, B, in seconds, computed by Bessel, by means of
the formulas [5991 (12.30)], with their first and second differences, are given in Table IV, (^')
of this collection.
To show, by an example, the use of Table IV, we have here inserted the computation
of the true anomaly v, in an orbit which does not differ much from that of Halley's comet ;
supposing the time from the perihelion to be 60 days ;
e=z. 0,9675212; log.(l—e)= 8,5115999 ; log.peri.dist. =9,7665598. (3»)
With these data, we find,
776
APPENDIX, BY THE TRANSLATOR ;
[5991] -O
(33)
D
Its half
t = 6o days
log. CO
log.
log.
log-
log.
log-
log,
log.
log.
log.
0,2334402
0,1167201
J,778i5i3
Table III. U=Ç)T' 'O"" 58',6
2,i283ii6
Table IV. S
I — c
4,565o6
8,5ii6o
Simpson's corv. II93'',I
3,07666
Table IV. £
i — e
I — e
4,4333
8,5ii6
8,5ii6
Bessel's coir. 28',6
1,4565
From Table III, for the parabola, U^gj'' 29"' 58»,6
>impson's correction, Table IV -|- ig"" 53', i
Bessel's correclion, Table IV -|- 28',5
Sum is true anom. in the ellipsis v = 97'' 5o'" 20s, 3
(34)
(35)
In a hyperbolic orbit, in which e^ 1,0324788, we shall have,
log(e— 1) = 8,5115999 ;
and, if we suppose t = 60 days, the numerical calculation will be the same as before;
but, 1 — e being negative, the value of Simpson's correction will be negative ; and, we
shall have, in this hyperbolic orbit.
From Table III, for the parabola
Simpson's correction
Bessel's correction
U^ m-" 29"* 58',6
— 19™ 53', 1
+ 28',6
True anomaly in the hyperbolic orbit v = 97'' 10" 34',1
The inverse problem, of finding the time t, from the perihelion, when v is given, is
(36) easily solved, if 1— e be so small, that Bessel's correction, depending on B, may be
(37) neglected. For, in this case, the expression [5991(7)] becomes U =v—S.{\ — e) ;
and S may be obtained from Table IV, with the argument v instead of U. Having
found U, we easily deduce from it, the value of t, by means of Table III. Hence
it appears, that this inverse problem, in Simpson's method, merely requires a change in the
sign of the quantity S. If 1 — e should be so great, that it is necessary to notice the
term B, it will be necessary to repeat the operation, by an indirect method; or, more
conveniently, by forming a table, similar to that used in finding B, by which the
correction of Bessel may be directly obtained. But, in this case, it is better to use the
method of Gauss, which is not restricted to the first and second powers of 1 — e, but
includes also the higher powers of this quantity.
(38)
(39)
(40)
[5992]
(1)
We shall now proceed to the investigation of this method of Gauss, for the direct
solution of Kepler's problem, for computing the true anomaly v, from the time t, in
GAUSSS METHOD, FOR A VERY EXCENTRIC ELLIPSIS.
/ /7
an ellipsis or a hyperbola, luhich approaches nearly to a parabolic form ; and, in the
demonstrations, we sliail refer to any line of [5992], by merely putting the number of the
line in a parenthesis, as \\e have done in [5984 (2)], omitting, for brevity, the
number [5992]. In this solution we do not, as in the preceding method, deduce the
anomaly in the ellipsis, from that in a parabola having the same perihelion distance D ; but
we obtain it from a parabola, whose perihelion distance is increased to
[59U2J
n = D
5-3
1 — 0,9 . a
i
(41) ;
FFe shaUJirst treat of an elliptical orbit, using the same elements as before ; namely,
a the semi-transverse axis ; e the exccntricity ; 2p=2a.(l — e^) the parameter ; D the
perihelion distance ; nt the mean anomaly ; u the excentrlc anomaly ; v the true
anomaly. We shall also use the following abridged symbols, in which a, a', C, differ
from those used by Gauss ; this change is made in order to conform to the notation generally
used in this work, and to render some of the formulas more simple.
a'
=
V/ 0,1-1-0,9.6
J
CL =
= 1-e ;
P
==
.5— 5e a.
1-H9e 2a'
~2 J
/' ^
^ v/(5£)
T
=^
tang.-J« =
(S)'
,tang.
H^;
f
=
1.5.(m — sin.u)
j
9u-(-sin.«
Tt
9u-|-sin.u
\/m
20^*
+ M
or,
T =
A
C-îA
The quantities A, B, C, may be expressed in series, by the substitution of
sin.M = u — -i?<^4-yi-ôM* — &c. [43] Int. ;
which gives,
u—ûn.u = Lu^—-X^ti^J^--\-u'—hc. ;
9M-|-sin.î< = 10i<— |?/^-|-yi-ô?<^ — &c.
VOL. III. 195
(9)
(3)
B being a quantity which exceeds unity, by terms of the fourth order in n (18). By
this means, the interpolations in Table V become very easy, on account of the smallness of (4)
log. B, and C — 1, as well as the smallness of their variations ; so that we are enabled
to notice all the powers of a, with but very little additional labor. The same remarks
may be applied to the use of Table VI, relative to a hyperbolic orbit.
(5)
(6)
(')
(8)
(9)
(10)
(11)
(12)
(i:))
(14)
(15)
(16)
778 APPENDIX, BY THE TRANSLATOR ;
[5992] Substituting these in A (12), it becomes,
A = i«2— -i^M^— ^^l^^M^— &c. ;
(17) or,
\/A = iu—^^^u^—j^\-y—kc. ;
and, in liice manner, we get,
(18) B = l + _-^_-«^— &c. ;
so that A is of the stcond order, relative to u, and B differs from unity, by a
quantity of the fourth order only. We may obtain the value of A, in terms of T, by
the following process. From [48] Int., putting 2:=^m, we get u (20), and from [30''] Int.,
we have sin.u (20), by using T (II);
2.tang.iM 2.T2 i
sm.M = ^ = ^ ^ ^ 2.1 .(1 — i + i — oic.) ;
M = 2.(tang.*M— itang.siM+itang.siu— &:c.) = 2T2.(1— jT+iT^— &tc.) ;
'■*" 2.tang.iM _ 2.T*
l+tang.2iM "~ 1+T
hence we get, by substitution,
(21) 1.5.(M-sin.M) = 30.tI{|T— IT2+&C.} = 20.T*. jT— fT^+fTS— '|TH&c.| ;
^22, (9M+sin.M) = 2TJ. { V-¥T+'^T2-&c. \ = 20.T*. { i_f,T+,\T^— f^T^+kc. j .
Substituting these in Jl (12), it becomes,
_ T— fT2+9T3— '^T^-l-i5T5— &c.
(23) ~" 1- AT+|,T2— ^l-p+^T^^fcc.
T" 4 "pS _1- 2''T'3 1592'T"'1_1_236568'T'5 93097192 'T'6|Stp
™, -■- 5 ■*• I 35 J- SC25 •• "TMÔIÎS-'- 197071875-^ "poiA..
Inverting this series, we get,
A
T
(35) m &-" I TT¥-^ T^5?^-" I Î1I5S7 5-^ I 1UT1J51?5-^ T^**-^-?
as we may easily prove, by substituting in it the value of A (24), and reducing, by
which means we shall find, that the terms mutually destroy each other.
If we substitute this value of - in C (14), it becomes,
^^^ C = 1 +Tf r^-^+^ lî-^M-âlMf t^^+t/t¥5VV^-45+S^c.
Hence it appears, <A«< C differs from unity by terms of the secoiid order in A, or
(^'^ of the fourth order in u (17). The quantities Jl, B, C, are functions of T,
(28) which have been computed by the preceding formulas, and inserted in Table V. By
(99) means of this table we can easily find, by inspection, the values of A, C, log..B, for
GAUSS'S METHOD, FOR A VERY EXCENTRIC ELLIPSIS. 779
(30)
any given value of T, or the contrary; and, as the quantities C, log.B, vary so
slowly, in the most useful part of the table, it is very easy to take out the corresponding
numbers, wliich we shall hereafter find to be one of the great advantages of the method
of Gauss. After this digression on the method of computing Table V, we shall proceed t^^)
to the investigation of its uses in the direct solution of Kcjpler^s problem, of finding r, v
from t, in a very excentric ellipsis.
Substituting the value of nt [5987 (12)], in [5985 (7)], neglecting m on account
D . (^^'
of its smallness, and then putting n= [5985 (C)], we get (34). From this we
easily deduce (35), since by multiplying together the two factors of (35), and reducing, it
becomes identical with the second member of (34). Now, the value of B (13), gives,
9m + sin. M = 20 A^.B ; 03)
substituting this in (35), in the factor without the braces, also the value of A (12), we
get (36); whence we easily deduce the expression (37) ;
,'i — e\z
k.t .[ — =— ) = u — c.sm.w (3<)
[.'iQSa]
,^ , . . ( 1 — e , l4-9c u — sin.n }
(9w + sin.?0 . < . }
^ ' ^ ^ 10 ^ 10 9«+sin.u 3
i „ C 1 — e I + 9e A )
20A'. B . { + — . — {
^ 10 ^ 10 15 3
(35)
(36)
C il ? )
= 2B . ] (1 — e).^-+ — . (l+9c).^- [ ; (37)
in which we must substitute the value of log.t = log-y/f^ = 8,2355814... [5987 (8,16)].
If we now suppose,
A^'= r\ ~lj ■ tang.JM> , or A = ^.tang.^w (10), (38)
and substitute it in the preceding expression, every term will have the factor (1 — e)^ ;
then dividing by this quantity, we get,
t . = 2iî . . Uane.Àw 4- J.tang.^J^wK _
Multiplying this by,
7 • Bk.^^- ^ ''
Bk^ \ 5 y ■ Bk.^2 '-' ' <'''
we finally obtain.
780 APPENDIX, BY THE TRANSLATOR ;
i'^l^ -^.^ = f.{tans.à-+^-.tang.3J.|.
(40) J5J5^ k
Now, from the construction of Table III [5987 (22)], it appears, that the tabular number,
corresponding to the anomaly w, represents the logarithm of the second member of this
expression ; so that, if we put.
0,l+0,9.c/ VI— 0,9.a/
and then substitute D^ and a' (9), in (40), we shall get, by making successive
reductions in its first member, the following expressions ;
a' (0,l+0,9.e)H t , . , , • -n ,, ,„ ,■ , ' ,
/4g\ f —^^ ! — — :^ number of Uie log., Ill lablc in, corresponding to Ine anomaly t£j ;
so that, if B, and, therefore, J),, be known, we can determine the relation of w and t,
by means of Table III. Hence it appears, that, in the direct solution of Kepler'' s problem,
in a very excentrical orbit ; where t is given, to find r, v; vie can obtain iv from t,
by means of (42) ; and then, from w, we get A, by means of formula (38); namely,
(43)
A= (3.tang.2i«'= 'çr^ ■ tang.^iif = —.tang.^Ji
l+9.e
Now, B differs so little from unity (18), that we may, in a first rough calculation,
(«) suppose J9=l ; and, upon this supposition, we can compute the approximate values
(45) of w and A (42,43). With this value of A, we find, from Table V, the
expression of log.B ; and, by repeating the calculation, with this value, we get the
corrected expressions of w, A. In general, this second operation will be sufficiently
accurate, except u be very great. It frequently happens, when several observations
(46) are computed, for successive days, that the value of log.i? is very nearly known at the
commencement of the operation ; in this case, we must use this approximate value of B,
in the first operation ; and, it will generally happen, that one operation, in such cases, will
be sufficient to obtain the correct value of w.
(47)
(48)
(49)
(50)
Having obtained the value of A, we find, from Table V, the corresponding value
of C ; from which we get,
T = tang.'éM = c^TJ (H. 14),
vifith more accuracy and less labor, than it could be directly obtained from Table V.
Substituting this value of tang.^gW, in (1 1), we get the first expression of tang.|j) (51).
Substituting in this, the second value of A (43), rejecting the factor (1 — e)^, which
occurs in the numerator and denominator, then introducing the first value of y (10),
we get the second expression (51);
GAUSS'S METHOD, FOR A VERY EXCENTRIC ELLIPSIS. 781
Having found u, v, we may compute r from either of the formulas [5985 (9,10)],
or from the following ;
_ J0.cos.2|« _ D _ {C—^A).D
cos.^it' ~ (l+T).cos.2^ ~ {C+iAycosJiV
[5'J92J
(SI)
(52)
The first of these expressions is easily deduced from the last formula [5985 (13)], by
substituting a.(l — e)^D [5985(6)], then squaring and reducing. The second is (53)
obtained from the first, by putting,
and the third is deduced from tlie second, by the substitution of tiie value of T (14).
The inverse •problem of finding the time t,from the true anomaly v, is also solved by means
of Table V. In this case, we must first compute T, from i;, by the formula (11) ;
(55)
1 e
T = .tang.^iv. (56)
1+e " ^
With the argument T, we must enter Table V, and take out the number A, and
the log.-B; or, what is more convenient, and, at the same time, more accurate, the
(57)
number C, and the log.i? ; then compute A, by the formula (14),
CT
A = q:^; (58)
lastly, we must find t, by means of the formula (37). This expression, being divided
by the factor of t, gives,
t = l.D^.AKB.(\-e)-^. I l+,V^.(l+9e).(l-e)-^ | ;
and, if we put,
t, = l.DlAKB.(l-e)-^; t._ = t,.,\.A.{l+9eUl-e)-^ ;
we shall have,
(59)
(60)
t = t, + t, ; (61)
and, it is under this form, that the value of t is computed in the introduction to Table V,
observing, that we have,
log.^- = 2,0654486 [5987(8)], and log. J, = 8,8239087 . ^^^
VOL. III. 196
782 APPENDIX, BY THE TRANSLATOR;
[5993] yy-g i^jgy ^]g^j compute t, from v. by means of Table III ; but, this table does
(^3) not facilitate the operation, as it does when finding r from t. In using Table III, for
this purpose, it will not be necessary to compute A. For, we bave, in (43,56),
,.„,.i.,.=^..(S:)s ,»,.... = (sy.T.
Dividing the first of these expressions by the second, we get the first of the equations {66);
substituting j~ (10), we get the second expression {66); and, by using A (58),
we get the last of the formulas {66) ; from which we easily obtain the first value of
tang.|w (67). The second formula (67) is derived from the first, by the substitution
of the second value of y (10).
tang.iw _ fA\h p-l-9A4_ ( ^\- _ . / C .
e) - VTvV V 7^\+%^) '
(65)
(66)
tanç.Ji) \Ty A5+5e
or,
(67) l'l"g-2
tang.ittJ =
(68)
Havin°- found, in Table III, tlie time corresponding to this anomaly iv, we must
3
multiply it by — — , to obtain the time t from the perihelion ; as is evident from
the first of the formulas (42).
Table V is given for every thousandth part of a unit, from A = 0,000 to
t09) yl = 0,300. It was thought to be unnecessary to extend it any farther ; because
A = 0,3 corresponds to T= 0,392.374 = tang-^Jw (1 1), or w = 64" 7" ; and,
with such large values of ti, the indirect method of solution is the shortest, as we have
already observed. This table is arranged so as to make it most convenient for use in
finding B, C, with the argument A, in the first problem, where t is given to
find V, which is by far the most frequently required. In this case, the number T is
not used. In the second problem, the argument T is used to find B and C, which
are small and easily computed ; and then A is found directly, by means of the
formula (58).
We shall apply this method to the compulation of the same example, as in [5991 (33)].
EXAMPLE 1.
Given, e = 0,9675212, t = m'^y\
be. perih. dist. D = 9,7665598, a = 1 — e = 0,0324788.
(73) '^ '
(,'2 __ 0,1 -f 0,9. f = 0,9707691 ; to find t, v, in an elliptical orbit.
(70)
(71)
(72)
GAUSS'S METHOD, FOR A VERY EXCENTRIC ELLIPSIS.
FIRST OPERATION TO FIND I).
0,'2 log. 9,9871159
a'
D
t= 60
log.
log. CO.
Its half
log.
9,9935579
2334402
1 167201
1,7781 5i3
U = gô^SS" Table III.
log.i'
2,1218695
a
è
log.
log. CO.
log.
8,5 1 1 5999
128841
9,^989700
/3
èC= 48'' 29"
log.
taog.
same
8,2234540
o,o52g4
o,o52g4
Approx. A = 0,02 1 347
log.
8,32933
TO FIND THE RADIUS /".
C4-o,2.jî = i,oo42gi2 log. CO. g,g98i4o3
C — o,8.^î = 0,9829401 log. 9,9925270
D log. g,76655g8
4» = 48'' 55'» io',47 sec 0,1823567
same 0,1823567
log.
o,i2ig4o5
SECOND OPERATION TO FIND V.
783
[5992]
(74)
— BLox 34, found by the first operation 2,1218661.
Hence w = go"* 58™ 2i«,6, by means of Table III.
(75)
Jio = 48'' 29*" ios,8
same 8,223454o
tang. 0,0529828
same 0,0529828
A = o,02i35ii log. 8,3394196
C = 1,0000210
o,8.-î = 0,0170809
C — 0,8.^ = 0,9829401 log. CO. 0,0074730
1 -|-e = 1,9675212 log. 0,3939194
I e = (j|_ log. CO. 1 ,4884001
tang.Sà" 'og- 2^o,II92I2l
il) = 48'' 55™ io',47 tang. 0,0596060
» = 97'' So" 2o',94 (51).
(76;
(77)
This value of v differs 0',64 from that found in [5991 (33)], by noticing only the
corrections of Simpson and Basse).
EXAMPLE II.
In the inverse problem, with the same elements, we have given,
the anomaly t> =: 97" 50"" 20',94 ,
to find t, in the following manner, by means of the formula (61).
(78)
I — 6 = 0,0324788 log. 8,5115999
I ■\- e = 1,9675212 log. CO. 9,7060806
4w = 48'' SS"" io»,47 lang. 0,0596060
same 0,0596060
T = 0,02173163 log. 8,3368925
Hence C = 1,0000210 log. 91
I -|-o,8.T = 1,0173773 lug. CO. 9,9925180
A = o,02i35ii log. 8,3294196
Corresponding log. B in Table V o,ooooo34
Constant log. 3,o654486
|. log. D 9,6498397
è log. A 9,1647098 (79)
log. B 34
è log. (i — e) arith. co. 0,7442000
. , days
< = 42 ,09
log. 1,6243015
('onstant 8,8239087
A log. 8,3394196
. I + 9e =9,707691 log. 0,9871159 (SO)
(i — e) log. CO. 1,4884001
.7''"",9o8
log. i,353o458
t = 60
days
= '. + '■
784
APPENDIX, BY THE TRANSLATOR:
[5992] We shall now compute the same example by means of Table III ; by which means it
^^'^ will evidently appear, that the preceding form is the shortest and most simple.
(82)
(83)
(84)
è » = 48<* 55"" 4o',47
I — e
log-
log. CO,
tang,
same
T =3 0,02172163
C = 1,0000210
i-|-o,8.T = 1,0173773
A = o,02i35ii
Corresponding log. B, Table V
8,5115999
9,7060806
0,0596060
0,0596060
8,3368925
9'
log-
log,
log. CO. 9,9925180
log. 8,3294196
0,0000034
a'2 = 0,9707691
log.
2 log-
sum
half
lana.
è» = 48''55"> io',47
hw = 48''29"'io»,8
to = 96*58" 3i',6 Table III log. ('
Table V log. B
D log.
D^ log.
d' log. CO,
9,9871159
9,7060806
o,3oio3oo
91
9,9926180
!)9,9867536
9,9933768
0,05960(10
tang. 0,0529828
2,1218662
M
9,7665598
9,8832799
64421
( = 60
dnys
log.
I,778i5i4
[5993]
We shall now proceed to the explanation of the method of computation in a hyperbolic
orbit ; in which the elements are ; « the semi-transverse axis ; e the excentricity ;
2p ==ia.(e^ — 1) the parameter; D the perihelion distance. We shall also use the
following abridged symbols, which are similar to those in [5992 (9 — 14)], corresponding
to an elliptical orbit. In the demonstrations in this article, we shall refer to any line of
(5) [5993], by merely putting the number of the line in a parenthesis, as in [5984 (2),&c.].
(1)
(2)
(3)
(4)
(6)
C)
(8)
(9)
(10)
(11)
(12)
c-?;^ .
"+1/
B =
V/ 0,1+0,9.6 ;
5e-5 _ e-1 _ y/5{e-\-})\ ^ /{'±}\
l+'Je ~~ 20.'= ' ^ \X U+ye/ y/ V2-^'V
e 1
tang.^Jis = ^qT^-tang.^èt' =
tang.(45''+|^) ;
î'o •(« — -) +is-log.M
,v(m— ^)+iyog.w
2^ A
; — M
T =
A
C+tA
GAUSS'S METHOD, FOR A HYPERBOLIC ORBIT. 785
We may observe, that the expression of u (9) is the same as in [5989(12)] ; and [5993]
the last expression of tang-.i«', or T (8), is deduced from it, in the same manner
as in [5989(13)]. This last value of T (8) gives,
1+Ti
i '
1— T*
hence, ''^'
1 (1+T*) (1— Tn 4T1
"- u = I^tT - -Ï+TT = ^z-]^=4T^l+T+T^+T'+&c.) ; .a,
and, from [58] Int., we have,
log.«=log.(i±5) = log.(l+T*)-log.(l-T') (u,
= (t*+|T^+^T^ + &c.)— (— Ti+|T^_>T^+&ic.)
= 2Ti.(I+JT+iT2-KT3+&c.); hence, ^,^^
§-[i •(« - J) -log.« I =3Ti.{iT+|T2+?T3+&ic}=2T^.{T+|T2+|T='+^'T^+&c}; ^^^^
^^ •(" - 1) +«-log-«=2T^^ (i]i+,VT+,'5T='+,'oT3+&ic)+,|(l+,^T+iT=^+|T3+&c) j ^^^^
=2Ti. f l+BT+l.T^+I^T^+^c. S .
(IB)
Substituting the expression (16, 18) in the value of ^ (10). and rejecting 2T* from
the numerator and denominator, we get,
T+fT^|-T3 + ^°T^-}-&C. -p I ^rpa , 24'r3 I .59--T4-l_ &•,.
^ I+6JT+/3T3+I5T3+&C. •^ + =■■1 +3li i-5S55l +ûiC. ^^^^
this may be derived from the expressions [5992(23, 24)], by changing the signs of ^4, T;
and, if we make these changes in [5992 (25,26)], we shall get, for an hyperbolic orbit,
/J
Extracting the square root of the expression of A (19), we get,
\/A = Ti. 1 1+ ft T + r*r% T^+ &c.} ; ^^
substituting this, and (18), in B (H), we get.
(94,
VOL. HI. 197
786 APPENDIX, BY THE TRANSLATOR ;
[5993] Now, if we consider ■us as a small quantity of the first order, we shall have T (8)
^^" of the second order, and A (19) of the second order. Hence C, B (22,24) differ
(26) from imity, by quantities of the fourth order. These values of A, B, C, T have
relations to eacli other, which are very similar to those in the ellipsis [5992 (15 — 31)].
These quantities, for the hyperbola, are given in Table VI ; which is arranged in the same
way as Table V, for the ellipsis ; and is used in the same manner as in [5992 (29), &ic.].
(38) The numbers in Table VI are computed for values of A, from 0 to 0,300 ; which is
sufficiently extensive for practical purposes.
We have in [5983 (6)],
Jc
(29) — . i = e . tan^. zi — log. tang.(45''-J- J-ra).
a^
Substituting in this the values of tang.w, tang.(45''-j- ^ w) [5988(15,13)];
D
(30) also, a = [5988 (4)],
we get (34) ; and, as we have identically,
(31) |e=^V•(e-l) + ^(TV + T^^e) ; -1 =A-(«-l)-(TV + T\e);
as is easily proved by reduction ; we may substitute these factors of %i — -, andoflog.w,
in the second member of (34), and it will become as in (35). This may be still farther
reduced, by observing, that the product of the expression (11), by 2A^, gives,
(32) 2-V • ("—7,) + T^-log-« = 2J5.^*;
substituting this, in the denominator of the value of A (10), and then multiplying it by
(33)
X 2B.^A , we get,
|i-("-^) -log.M|=|B.^
Using these two last expressions, we find, that the function (35), is reduced to the form
(36), or the equivalent expression (37) ; which is very similar to that corresponding to
the eUlpsisin [5992(37)].
e— IM / 1
(34) A;.( -^j .< = 4e. (« — -)— log.M
(35)
(36)
(37)
= (e-l).2B.^*+(J^+Ae).f.B.^-
= 2B. \ [e-[).A^+^\.{\+9e).A^ | .
\.[ u ) — los;.t«;
u.
GAUSS'S METHOD, FOR A HYPERBOLIC ORBIT. 787
If we suppose, [WDS]
^ = (-[ ~lj • tang.î'jM) = ^ . tang.2^i« (7), (M)
3
and then substitute this first value of A in (37), we shall get, by dividing by (e — 1)^,
—^.t = 2B. (-q^ V- {tang.4t« + J.tang.'iwj . (=9)
Multiplying this by.
/l+9e\é 1
we finally obtain,
-^ .< = ^-.{tang.^îo+J.tang.^àw;}. Ho)
(•II)
This is of exactly the same form as [5992 (40)], in an ellipsis ; and, if we put, as in
[5992(41)],
we shall get, as in [5992 (42)],
a' (0,l+0,9.e) t
.t = . t ^= — =■ number of the log., in Table III, corresponding to the anomaly w ; (42)
Boi BOi D,i
so that, if B be known, we can determine the value of iv, by means of Table III.
Therefore, in the direct solution of Kepler's problem, in a hyperbolic orbit; where t is
given, to find r and v ; we can find w from /, by means of (42) ; and then, from w,
we get A, by the following expression, which is th€ same as in (38);
A= f3.tang.^Jî«= ' ^^ . tang.'' J w = ^.tang.^Jw.
Now, B differs so Utile from unity (24). that we may, at first, suppose B^l;
and, with this assumed value, we can find the approximate values of w, A (42,4-3).
With this value of A, we obtain, from Table VI, the expression of log.J9; and,
by repeating the calculation, with this value, we get the corrected expressions of
w, A. In general, this second operation will be sufficiently accurate, as we have (46)
observed in the similar calculation for an elliptical orbit [5992 (46)].
Having obtained the value of A, we find, from Table VI, the corresponding value
of C ; from which we get,
(43)
(44)
(45)
788 APPENDIX, BY THE TRANSLATOR j
[5993] ^
(«) T = tang.^J^ = Qj^.j^ (8, 12),
with greater accuracy and with less labor, than it could be directly obtained from Table VI.
Substituting this value of tang-^l^r, in (8), we get the first expression of tang.|« (49).
Substituting in this, the second value of A (43), rejecting the factor (e — 1)=, which
occurs in the numerator and denominator, then introducing the first value of y (7),
we get the second expression (49) ;
(48)
(49)
tang.è. = ^î±} . \/
A ^ y.tang.^t<;
C+M \/C+fA '
Having found «, v, we may compute r, from either of the formulas [5988(11,12)],
or from the following expressions ; which are similar to those in an ellipsis [5992 (52)] ;
D (C+^A).D
(50) r = _^ \ ^ t J
(1— T).cos.2^y (C—iA). cos.^iv
The first of these formulas is deduced fiom the last expression in [5988 (20)], which
gives, by squaring and reducing,
_ (m+1 f o.(g— 1)
(51) 4u ■ C0S.2 4 i; '
Now, from the value of T (8), we have,
^-^ = '-(^ù'=(^^' ^'^° «.(e-l) = l> (30).
Substituting these in the preceding value of r, it becomes like the first expression (50) ;
and the second expression is deduced from the first, by using the second value of T (47).
(53)
Theinverse problem of finding the time t,from the true anomaly v, may be solved by means
of Table VI. To effect this, we must first compute T, from v, by the formula (8) ;
e 1
(54) T = — . tang.2|i; .
With the argument T, we must enter Table VI, and take out the number A, and
the log.i?; or, what is more convenient, and, at the same time, more accurate, the
number C, and the log.S ; then compute A, by the following formula; which is
easily deduced from (47) ;
CT
(55)
A =
(56)
1— tT'
lastly, we must find t, by the formula (37). This expression, being divided by the
factor of t, gives.
GAUSS'S METHOD, FOR A HYPERBOLIC ORBIT. 789
[5993]
observing, that we have, as in [5992 (62)],
log.^= 2,0654486 ; logji = 8,8239087. (s?)
Ihen, if we put,
t,^\.D^.A^.B.{e—\)-^; t, = t,.i,.A.{l+9e).{c—\)-^ ; (58)
we shall have the following expression of /, which is exactly similar to that for an (59)
ellipsis [5992(61)];
t = t^ + 1„ ; (60) .
and, it is under this form, that the value of t is computed in the introduction to Table VI.
If we wish to use Table III, which does not, however, facilitate the operation, it will
not be necessary to compute A. Then, we shall have,
tang. > = AK Q-g)* (38) ; tang.J. = ^^.(^J (54). (6»
If we divide the first of these expressions by the second, then substitute the values of
y (7), also that of A (56), we shall get, by successive reductions,
tang-l^. _ (A\h /l+9eY_ ( A\^= t / ^ • (ca)
tang.J. Vt; • V5e+5; VTvV V 7^.(1— |T) '
tang4^« = \/ :,,:^^' '^^S-hv = \/ ^^^^~::^y tang-è^^-
Having computed the value of w, from (63), we may then find, in Table III, the time
3
corresponding to the anomaly «•. We must multiply this time by — ;— (42), to (64)
obtain the time t from the perihelion. The remarks made in [5992 (69 — 72)], relative
to the construction of Table V, will apply, with the proper modifications, to Table VI.
To illustrate this method of computation we shall give the following examples.
EXAMPLE I.
Given, e = 1,2500000 ; log. perih.dist. 0,0200000 ; r = GO""^)", to find t, v. (^s)
VOL. III. 198
790
APPENDIX, BY THE TRANSLATOR ;
[5993]
APPROXIMATE OPERATION TO FIND V.
a'2
(66)
(67)
(68)
= 0,1 +o,9.e= 1,225
log.
log-
0,088 1 36 1
a'
0,0440680
D
log. CO.
g,g8ooooo
its half
g,ggooooo
t = 6o'î''y'
log.
I,778i5i3
Approx.
log. t'
1,7922793
; Cr=66''44'", nearly, in
Table III
e — I = o,25
log.
g,3g794oo
a'a
log. CO.
9,9118639
Constant
log.
9,6989700
Sum given /2
log.
9,0087739
iU==33'' 22m
tang.
9,8 1 85849
same
9,8i85849
Approx. JÎ = 0,044253
log.
8,6459437
iponding log. j5 = o,ooooi45 Table VI.
TO FIND THE RADIUS r.
C — o,2.jî = o,ggi236o
log. CO.
o,oo3823o
C -1- o,8..4 = 1 ,035496g
log.
o,oi5i488
D
log.
0,0200000
è» =. 3i''48™3i%3
sec.
0,0706768
same
0,0706768
log.
o,i8o3254
CORRECTED OPERATION TO FIND V.
Subtract log.B = o,ooooi45 gives correc. log.*' i ,7922048
Hence U or iv = 6&^ AA™ iG^g in Table III.
same
4 tc = 33"^ 22™ o8=,4 tang,
same
log.
Constant A = 0,044260g
0,8.^ = 0,0354087
C = 1 ,0000882
9,0087739
9,8186233
9,8186233
8,646o2o5
C-j-o,8..^ = 1,0354969
e-[~ ' = 2,25
e — I = 0,25
Sum is 2 log. tang \ v
à u = 3i<i4Sm3t«,3
o = 63'' 37"" 02«,6
log. CO. 9,9848512
log. 0,3521825
log. CO. 0,6020600
9,585ii42
tang. 9,7925571
(69)
EXAMPLE II.
In the inverse problem, with the same elements, we have given,
e = 1,2500000 ; log. perlh. dist. 0,0200000 ; and y =:= 03" 37™ 02^G
to find t, in the following manner, by means of formula (60).
e — I =0,25 log. 9,3979400
e+ I = 2,25 log. CO. 9,6478175
èî) = 3i<'48"'3i«,3 tang. 9,7925572
same 9,7925572
(70)
T = 0,0427437 log. 8,5308719
Hence C = i ,0000882 Tab. VI log. 383
I — o,8.T = o,9658o5o log. co. o,oi5iio6
A = 0,0442609 log. 8,6460208
Constant log.
2,0654486
J log. D
o,o3ooooo
à log. A
9,323oio4
log. B
i45
4 log. (e — i) arith. co.
3oro3oo
^=52''^^=,42I log.
i,7ig5o35
Constant
8,8239087
A log.
8,646o2ù8
. I -j-ÇÊ^ 12,25 log.
i,oS8i36i
e — 1 log. CO.
G020600
'j= 7'^''^'.579 log.
0,8796291
60'
days
= t
t .
COMPUTATION OF THE ORBIT OF A COMET.
791
We shall now compute the same example by means of Table III; by which means it
will appear, that the preceding method is the most simple.
C I ^ 0. 2^)00000
e -j- I = 2,25ooooo
et) = 3i''48"'3i»,3
T = 0,0427437
■ 0,8. T = 0,9608050
C = 1,0000882
.4 = 0,0442609
log. 9,3979400
log. CO. 9,64781-5
tang. 9,7925572
same 9,7925572
log. 8,630871g
log. CO. o,oi5iio6
383
log.
log.
Corresponding log. B, Table VI, is
8,6460208
0,0000145
a'2 = 1,225
log.
2 log.
o,o88i36i
9,6478175
o,3oio3oo
O,oi5iio6
383
sum 2^o,o52i325
half 0,0260662
J» = 3i''48"'3i',3 tang. 9,7925572
•iio = 33'' 22"> o8»,4 tang. 9,8186234
to = 66"* 44" i6%8 Table III log. t' 1,7922044
Table VI log. B i45
■D log. 0,0200000
i)* log. 0,0100000
a' log. CO. 9,9559320
i= 60 ^' neai-ly. log. 1,7781509
ON THE METHOD OF COMPUTING THE ORBIT OF A COJIET.
A shon time before the publication of the first volume of the Mécanique Céleste,
containing La Place's method of computing tlie orbit of a comet [754 — S49], Dr. Olbers
gave a much shorter process for solving the same problem, in a work published at Gotha,
in 1797, entitled Abhandlu.ng ilber die leichleste und hequemste Méthode die Balm eines
Cometen aus einigen Beobachtungen zu bcrechncn ; and as this method is but little
known in our country, we shall here give a full explanation of it, and shall simplify in some
respects, the calculation by means of Tables I, II, of this collection ; which have been
computed and examined with particular care, in order to render them correct, to the nearest
unit, in the last decimal place. We have used Table II, in an abridged form, for several
years, and have found it convenient and sufficiently accurate, as it regards the number of
decimal places. We shall first explain the method of Dr. Olbers, by the geometrical
process, which he used, and shall afterwards, in (262 &z;c.), show how his results can be
obtained by an analytical process; noticing the smah terms that he has neglected, and
which require attention in some particular cases.
In finding the orbit of a comet, we have given, by observation, three geocentric longitudes
and latitudes, together with the times of observation ; and from the solar tables we have
the Sun's longitudes and the radii vectores. We shall use the symbols in the following table
(9 — 29) ; most of them being like those which are given by La Place, [7Gl"',820'''&c.].
The unaccented letters being taken for thefirst observation ; the same letters with one accent
for the middle observation; and with two accents for the third observation. We have
inserted in the same table (30 — 45), several theorems which are useful in these calculations
with the demonstrations in (40 — 130). In treating of this subject we shall refer to any
line of [5994], by placing the number of the line in a parenthesis, as in [5984 (2), &;c.].
[5993]
(71)
(72)
[5994]
(1)
(2)
OiberSij
metliod of I
computing
the orbit of
a Comet.
(3)
(1)
(5)
CO)
(7)
(8)
792
APPENDIX, BY THE TRANSLATOR;
[5994]
(9)
(10)
(")
(12)
(13)
(14)
(15)
(16)
■ (17)
(18)
Symbols.
(19)
(20)
(21)
(29)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Fundaniea-
tal equa-
tions.
(33)
(34)
(35)
(36)
(37)
(38)
(38')
(39)
(40)
(11)
(42)
(44)
t, t', I" , The times of observation;
©J ^1 ©") Longitudes of the Sun, differing i8o'' from those of the earth Jl, A', A", respectively ;
a, cl', a". Geocentric Longitudes of the comet ;
6, 8', 6", Geocentric latitudes of the comet ; southern latitude being considered as negative ;
Distances of the earth from the sun ;
Distances of the comet from the earth ;
Curtate distances of the comet from the earth;
Radii vectores of the orbit of the comet ;
Heliocentric longitudes of the comet ;
Heliocentric latitudes of the comet ; southern latitude being considered as negative ;
The difTerences of the heliocentric longitudes of the comet and the earth ;
Longitude of the ascending node of the comet;
Inclination of the comet's orbit to the ecliptic ;
Arguments of latitude of the comet, or distances from the ascending node counted on the orbit ;
Arguments of latitude of the comet reduced to the ecliptic and counted from the ascending node ;
R, R', R".
îi> Çi'> î"
h f'' P"'
r, r', r" ,
n,
f,
u, u', u",
w, w', w",
p^ =u — l
V, V, !)",
c
tang.6'
The true anomalies of the comet ;
The perihelion distance of the comet ;
The chord of the path of the comet betvpeen the first and third observations.
m
sin.(©'-a')
p"= M.f ;
[Approximnte"!
value of JIf.J
"tang.6"— m.sin.(©'— CL") t'—t
rS = /}2— 2fl.f.co3.(©— a)+f9.sec2.â ;
r"2 = Ri'-i—2R".Mf,.cos.{Q"—o,'i)-\-M2.f^.sec''.6'i ;
c2 = »-2-|-)-"2— 2iifl".cos.(Q"— ©)
4-^2iJ".cos.(©"— a)+2Jl!fiî.cos.(i3— a")}-p
+ 5— 2>/.cos.(n,"— a)— 2JV/.tang.fl.tang.6"|.f2
(A)
(O
sin.-w =: — . tang.9 ;
p"
sin.^" = !-77. tang.S" ;
r"
f.sin.(0— a) .
sin.Ê = ^ -^ ;
r.cos.-ra-
7-".C03..nr"
cot.tu = tang.m".cot.Œ-.cosec.(/S"— ;S) — cot.(/3"— yg) ;
sin.(!o"-|-'=')
SU3l.
or,
tan]
/ g^-gx sin.(.-4-^ _ (l!^)
\ 2 / Sin.(OT"— ot) * V 2 /
n = 0 — V) = longitude of the ascending node ;
tang.? = tang.ra-.cosec.u) .
cos.u = cos.-ET.cos.îi! = cos /sr.cos.(^ — n) ;
cos.u"= cos.ot".cos.(/3"— n) ;
Xj = longitude of the descending node = i8o<^ -|- fl ;
[? = Inclination]
1
(43) tang.
tang.Jt) = cot.ix— (-Ttj • cosec.i;^ ;
(44-, tang.èu" = — cot
ta+fr)'.
cosec.à;^; ;
(44")
(45)
tang.(èt)+Jx) = tang.(45''~J).cot.ix ;
D = r.cos.^iu = Perihelion distance.
COMPUTATION OF THE ORBIT OF A COMET. 793
[5994]
We shall suppose, in the annexed figure 84, that s is the place of the sun ; a, h, c,
the places of the comet in the first, second, and third observations ; a', b', c', the i'^)
corresponding places of the earth. Draw the chords ahc, a'h'c, intersecting the
radii sb, sb', in the points A, A', respectively; then by Kepler's first law [3G5'], we
have,
(' — t : i" — t' : : sector sab : sector sbc . (*')
Now if we consider tlie chord « c as a very small quantity of the first order, in
comparison with the radius sb, the segment hb will be of the scco/k/ order. In this (■is)
Geometri
case, tlie triangle sac will be of the ^rsi order, and the elliptic, hyperbolic or parabolic JfiauoTof
segment a be ha of the third order; so, that the sector s abc will difier but very (^g""*'
little from the triangle sake. In like manner if we suppose the chords ab, cb to be
drawn, we shall find that tbe sectors sab, scb, differ respectively from the corresponding
plane triangles sab, scb, by quantities of the //»«/ order; therefore the error will be
but very small, if we substitute in (47), the ratios of the areas of these plane triangles, (^"^
instead of the ratio of the areas of the sectors. Now these plane triangles have the same
common base sb, and the perpendiculars let fall upon it from the points a,c, are
evidently proportional to ah, ch, therefore their areas must be in the same proportion;
hence, we shall have, very nearly,
(51)
(5S)
(53)
t' — t : t'' — t' : triangle s«i : triangle sc 5 :: ah : ch.
The same reasoning may be applied to the segments of the chord aV, described by the
earth ; therefore, we shall have, very nearly,
t'—t : t"—t' :: ah : ch : : a'h' : c'h' ;
which is equivalent to the supposition, that if the two chords a c, a'c', be described with
uniform velocities, in the time t" — t, by a fictitious comet and j)ianet, the fictitious bodies
will be at the points h, h', when the real bodies are at b, b', respectively ; and it is
upon this hypothesis that the method of Dr Olbers essentially depends.
We shall now take the point h' , as a centre, and shall suppose the line h's, to be
continued infinitely, till it meets the concave surface of the starry heavens, in the point
S, figure 65, representing the geocentric place of the sun at the second observation.
Moreover, we shall suppose three lines to be drawn through A' figure 84, page 797, (54)
parallel to the lines a'a, b'b, c'c, in the same directions, and continued infinitely to the
heavens in the points A, B, C, figure S5, representing respectively the geocentric places
of the comet, in the first, second, and third observations. Through the extreme points
A, C, we shall draw the great circle CHAJV, intersecting the ecliptic SJV'm the point .A'';
also the great circle SB, intersecting the arc AC in H. To avoid the confusion of
having many lines on the same figure, we have not actually drawn these three lines through ,^^
the point A', but have merely marked, in figure 84, page 797, the point a' of the line
h'd'A, and the point c" of the line h'd'C; supposing h'a" z= a'a, h'd' = dc. Then it is
VOL. III. 199
(55)
794 APPENDIX, BY THE TRANSLATOR;
[5994]
evident from this construction, and from the proportions between tlio lines ah, ch, a'h', c'h'
(52), that the right line, connecting the points «", c", will pass through the point h ; and
'^^' this line will be divided by the point h into the segments ha", he", which have the same
ratio to each other, as ha, he ; as will be more particularly explained in a similar case in
(70, &c.). Then as the line shb, when viewed from h', is projected on the concave
surface of the heavens, in the great circle SB ; and the line a" he', when viewed
from the same point h', is projected in the great circle AC ; it follows, that the point h,
which is the intersection of these two lines sb, a"c", must be projected in the heavens in
the point H where the two great circles AHC, SHB intersect each other. Therefore,
H will be the geocentric place of the comet in the heavens, in the middle observation, if
the bodies were to move uniformly in the chords ae, a'c' and the comet be at the point h,
when the earth is at h'.
(38)
(59)
(60)
(61)
(02) Now if we suppose P, figure 85, to represent the pole of the ecliptic ; °f the first point
of Aries; PAA', PBB', PHH, FCC circles of latitude ; we shall have,
(G3)
J A' = è, BB'= ê', CC'= 6" ; TA' = a, rB' = a', T C" = a", °fS= ©',
and we sliall put for the geocentric longitude and latitude of the point H,
cfW = a, ; HH' = L ;
,54, also, the angles, ASA! = b ; BSE' = b' ; CSC = b" ;
and the arcs, SA^©'—^; SB = 0—a.' ; .S'C'= ©'— a".
Then, in the rectangular spherical triangle ASA', we have,
(65, tang.^S^'='^^ [1345='^];
which, in symbols, is the same as the first of the equations (66) ; the second and third of
these equations, are found in the same manner, from the rectangular spherical triangles,
BSB', CSC ; the second of these expressions, is evidently equal to the assumed value
of m (28);
tangJ ,, tang.â' tang.4"
(^6) tang.è = -r— T^.:^ r; tang.i'== -^— -~ — = m ; tang.i"= °
(67)
(68)
(69)
sin.(©'-a)' °' sin.(©'— a') ' ^^ sin.(©'-a")-
We shall suppose in figure 86, that the paths of the earth and comet are projeeted
orthographically vjion the plane of (hat circle of latitude which is perpendicular to the
radius, drawn from the sun to the earth at the time of the middle observation ; or in other
words, that the plane of projection is perpendicular to the line b'h's figure 84 ; so that
the point h^, of figure 86, is the projection of this line, or of the three points b', h' , s,
upon the plane of this figure. AVe shall suppose, that the points «, , c, , .^, , C, ,
represent, respectively, the projections of the places of the earth a', c', and of the comet
a, c, at the times of the first and third observations ; also I-l„ the projection of the
point h; so that the points S, A, H, C, in figure 85, correspond respectively to
the points s, A^, H,, C, in figure 86. Then it is evident from the principles of the
I
COMPUTATION OF THE ORBIT OF A COMET.
PPolc of theEcUpnc
orthographic projection, that the lines rt,f,,
.4, C, , are divided in the points /(,, //.;, in
tlie same proportion as the lines n't', «c, in
figure 84, are by the points /(', h. Therefore,
if we draw tlie line ^,.î, , parallel and equal
to (7, /îj; C, C^ , parallel and equal to e,/i,;
then join //„ C., H,A„ , we shall have, as in
(52), very nearly,
t'—l : t"—t' ::A,H,: C,H, : : A,.^„ : C,C„.
IN'ow by construction the angles C,C^H, A^A^H,, are equal, and as the sides about
these angles are proportional, in the triangles C^C^H^, A^A^M^ , therefore these triangles
are similar, and the angle A^H„A^ :^ the angle C^H„C„; consequently the three
points A„,H„,C„, are situated in a straight line ; \^hich is divided by the point H„,'m
the same proportion as the line A,C^ is divided by the same point; so that we shall
have, as in (52),
t'~t: t"—t' ■.-.AJL : CM..
795
[5994]
(70)
(71)
(T2)
(73)
(74)
From the above construction, it is also evident, that the line h^A.. is equal and parallel
to a, .^, ; and h^C„, equal and parallel to c^C^; so that if the lines ^j ./2, , A, C, be ('S)
continued infinitely, they will represent the projections of the lines liA, h'C (55), drawn
in figure 85, from the centre of the sphere h' to the geocentric places A,B,C, of the comet, P^)
in the starry heavens, at the first and third observations. In like manner h^ J/, represents
the projection of the line h'hH figure 84, drawn from the centre A', through the point h,
towards the point H in the starry heavens. The line «,Ci, figure 86, continued to A,, <■"'''
represents the projection of the ecliptic, upon which we shall let fall the perpendiculars A„A^,
A^a^,HJi^, C^c,, C„Cy Then it is evident, from the constmction, that a^a^^k^A, (78)
is the projection of the curtate distance p at the time of the first observation ; and as the
geocentric longitude of the comet is then equal to a., and the longitude of the axis (79)
796 APPENDIX, BY THE TRANSLATOR;
[5994]
drawn through a, , perpendicular to the plane of the figure is ©', the inclination of the
line p to this axis is 0' — a ; consequently this projection of p is represented by,
(80) a,a, = h. A, = p.sin.(©'— a).
In like manner the projection of p" is,
(81) qC3 = A.C3 = p".sin.(©'-a").
Again, as the line .S'A' figure 85, is perpendicular to the plane of projection in figure 86,
the angles formed about the point »Si figure 85, will be projected about the point h ,
(82) figure 86, without any alterations in their magnitudes or relative positions ; so that the
angle ASA' figure 85, will be projected into AJi^A' in figure 86, and so on for the other
angles ; hence we shall have, by using the same symbols as in (64) ;
■ (83) AJi^A„_ == b ; HJi,H^ = V ; CJi, C._ = h" ; AJi.H, = b'— h ; C.h^H, = h"— b'.
Now in the rectangular plane triangles A^h^A^, CJi^C^, we have by using the values
(80,31,83),
(84) h^^=_Ml_ _ P-sin.(©'-a). ^^ ^ _ KC _ p".sin.(©'-a") ^
' " cos-A^h^A„ cos. 6 ' ' ' c os.C^h^Cr, cos. 6"
In the plane triangle A,HJi^ we have,
(85) sm.A„HJi^ : sm.AJi.H., :: h,A„ : A^_H„;
hence we get the first expression (86), using for brevity,
(85) sin. jH":= sin.^^iTjAj = sin.CjH,A, ;
and by substituting the symbols (83,84) in its second member we get the second expression
(86). In like manner from the triangle C^H^h^ we get the expression (87) ;
(86) A„H,.sm.H=h,A„.sm.A„h,H = ^'^'"'^®,~-^ . s\n.(b'—b) ;
cos. 6
(87) CM. . sin.H = h, C„ .sin. CJi.H^ = j!:^"-(©'— "■') _ si„. (i"_ m
cos. 6
Dividing the equation (87) by (86), then substituting in the first member, the expression,
CM. t"—t'
A„ H„ i'
(-^4);
putting also p" = J\Lp, as in (29), we get the approximate values of M (92). Developing
the first members of (89,90), by [22,34'] Int., and substituting the values (66), we get
successively,
co:mputation of the orbit of a comet.
sin.(i' — b) =sin.6'.cos.i — cos.ô'.sin.i = cos.i.cos.t'. (tang.6' — fang. 6)
= COS. J. COS. J'.
tang.^ \
s.n.(©'.
sin.(i" — b) =^ sin.i". cos.i' — cos. 6". sin. 6'= cos. 6'. cos. 6". (tang. 6" — tang. J')
,, ,,, / tang.â"
= cos.6'.cos.6". -^—, — . — t,
Vsni.(©'-a")
Substituting the values (S9,90) in M (92), and rejecting the factor cos.b.cos.b'.cos.b'',
which occurs in the numerator and denominator, we finally obtain the approximate value of
J\I (93) ; which is of tlie same form as in (30) ;
t"—i' cos.b".sm.{b'—b) sin.((v)'_cL)
M
I'—t cos.b.s\n.{b"—b') sin.(©'— a")
t" — t' m.sin.f©' — a) — tang.é
[Approximate"!
ViiiUL' of .V.J
f t
ig.ô" — 7n.sin.(@' — a")
We shall show hereafter, in (306, &c.), iiow this approximate value of JH may be corrected
for the error of the hypothesis (50), where the ratio of the areas of tiie triangles, is used
instead of that of the sectors. Again, we have, in the right angled spherical triangle
BBS, figure 85, page 795,
C0S.SJ5 = cos.SB'. cos.BB' == cos.(©'— a').cos.ô' [IS^S^'J ;
and this evidently represents the cosine of the angular distance of the sun and comet
sb'b = SB figures S4, 85, in the second observation. In like manner, by decreasing by
unity, the accents of the symbols, so as to make them correspond to the first observation,
we get,
cos. s a' a =: cos. (© — a) . cos. Ô .
Now in this plane triangle sa'a, we have, sa = r,
sa' ■== R, aa' = p. seed, and, by using [62] Int.
we obtain the expression (9T), which is easily
reduced to the form (98), being the same as the
first equation of La Place's method [806] ;
r^^.^^-2R.(p.secJ).\eos.{Çî)-a.).cosJl+{p.sec.êf
=iJ2_2iJ.p.cos.(©— a)4-,A sec^.^.
This last expression is the same as the value of
r^, (31), corresponding to the Jirst observation.
If we add two accents to the symbols of this
expression we get,
r"^=R"^—2 R"J'.co5. (©"—a") +p"2.sec2.«3" ; ^"'^
VOL. III. 200
797
[5994]
(90)
(9))
(93)
(93)
(93')
(94)
(95)
(9C)
(9C-)
(97)
(98)
(90)
798
APPENDIX, BY THE TRANSLATOR;
'-^ ' which, by substituting p" = iVi.p (29), becomes as in (32) ; corresponding to the third
observation.
We shall now suppose, fora moment, that the place of the comet at the first observation,
is determined by three rectangular co-ordinates x, y, z, whose origin is the centre of
the sun. Tiie axis of x is drawn in the plane of the ecliptic, towards the first point of
Aries ; the axis of y, is drawn in the same place, towards the first point of Cancer ; the axis
of z, is perpendicular to the ecliptic, and directed towards its northern pole. In like
manner, we shall suppose, that x', y', z', represent the co-ordinates of the comet, at
the second observation ; also x", y" z", those at the third observation ; then it is
evident, from the principles of the orthographic projection [118], that if c represent the
line or chord, between the places of the comet at the first and third observations, we shall have
the first of the expressions of c^ (106) and by developing and substituting,
r2=.ï2^y2_j_,a. r""~ = x""" -^ y"^ + z"^ ;
it becomes as in (107) ;
c^=(x''_a:)2-f(y"-y)2+(^"-=)2=(x"2-fy'S+~~"^)+(^^+/+^=)-'2.(x^^"+y2/"+r^")
(100)
(101)
(102)
(103)
(104)
(105)
(106)
(107)
(108)
(109)
(110)
(HI)
(112)
(113)
(114)
Now we have, as in [762,768],
X = R.cos.A-\- p.cos.a ;
a;" = JÎ".cos.^" -f p". cos.a";
•■"■^+r^-2.{xx"+yy"+zz").
y = R.sm.A -\- p. sin. a, ;
y"=.ii".sin.^"-f p".sin.a".
p.tang.é.
■■ p". tang.â".
(115)
Substituting these values in the first member of (110), we get its second member, and by
successive reductions, it becomes as in (112) ; using the values of A, A" (10);
xa;"+y/ = iî.iî".(cos.y4.cos.„2"-f-sin.^.sin../2")4-iî".r.(cos.^".cos.a-fsin.^".sin.a)
-\- i?.p".(cos.^.cos.a"-|-sin.^.sin.a") -f-p.p".(cos.a.cos.a"-f-sin.a.sin.a")
= iï.iî".cos.(^"— ^)-|-jR".p.cos.(^"— a.)-Liî.p".cos.(.4— a")+p.p".cos.(a"— a)
=/î.iî".cos.(©"— ©)— iî".p.cos.(©"— a)— JJ.p".cos.(©-a")-(-p.p".cos.(a"— a).
CoTiiet tPi
^ Substituting this and i;;" = p.r".tang.ltang.ô" (108, 109), in (107);
and then putting p" = ^lp. (29), it becomes as in (33), the terms
81 being arranged according to the powers of p. We have as in
[5858] c = r.sin.lat. or c = r.sin.ra (18); putting this equal
to the value of z (108), we get sin.w (34), corresponding to the
first observation, and in like manner we obtain for the third
observation sin.jr" (35). In fig. 87, if S be the place of the sun,
C, that of the comet,and Tthat of the earth, at the first obervation ;
g^^^ C,C a line drawn from the comet Cj, perpendicular to the plane
COMPUTATION OF THE ORBIT OF A COMET.
799
[5994]
(116)
of the ecliptic ; we shall have in the plane triangle, STC,
TC:^?, SC^ r.cos.zi, CST = e, STC = © — a ;
and since,
SC : TC :: sin.STC : sm. CST ; (in)
we shall have, in symbols,
r.cos.-a : p :: sin. (© — a) : sin.s; (U3)
whence we obtain sin. s (3G), corresponding to the first observation; and by putting two
accents upon the symbols, we obtain the similar expression of sin.s" (37), corresponding
to the third observation.
In figure 88, NA'C represents the ecliptic, A,C, the heliocentric places of the comet at
the first and third observations ; A', C, these places reduced to the ecliptic ; N the ascending
node of the comet's heliocentric orbk NAC. Then in the rectangular spherical triangle,
NJ'J,v:e have cot.AJS^A' = cot.^^'.sin.iV^' [1345 31J ; or, in symbols,
cot.(p = cot. w. sin.w, which is easily reduced to the form (40). In like manner, in the
triangle NCC, we have, by putting for a moment, p" — (3 = 2p„ and NC = tv -\- 2(3i ;
cot.<p = cot.i3".sin.(w -\- 2|3i) = cot.w".{sin.w.cos.2(3,-fcos.w.sin.2j3i.}. Putting these two
expressions of cot.ç (120', 122) equal to each other, and dividing by sin.w.sin.2(3,.cot.ra",
we get, tang. •!3".cot.w.cosec.2p. = cot.2p, -f cot. w, whence we get cot.w (38). This
expression may be reduced to the form (38') ; which is rather more convenient, in using
logarithms. For we have, in the triangles, iV,^^', NCC;
tang. -53
tang.çi = -r^— ;
s\n.io
88
(119)
(ISO)
(120')
(121)
(122)
(1220
tang. (?
tans.w
sin.(i(;4-p"— /3) '
(122-)
and by putting these two expressions equal to 27hac^
each other, we get the first equation (123'). ^
Putting for brevity, m'j=?= w + I- (p'' — (3) = "'+P,, we get the second form in (123') (123)
and, by development we obtain the third form. From this last expression, we easily
. cos. ta". COS. Î3
deduce (124), and by multiplying it by tang./3, . t, ; we get the first expression
cos.ra . cos.w
(124'), which is easily reduced to the second form, which is the same as (38'),
tang. -a" sin.(w + p" — p) sin.(w, -|- p^) sin.Wj.cos.Pj + cos.Wj.sin.p,
IV. sin.(îfj — Pj)
2sin.M, . cos.pj tang.M'i
tang.w
tang.sr" + tang.ra
tang.ra" — tang.CT 2cos.Jfi .sin.Pi tang.^
sin.-n''.cos.« -\- cos.-n".sin.ra
sin.t«,.cos./3j — cos.iVi.sin.p,
tang.jfj =
sin.ra .cos.nr — cos.w .sm.ar.
sin.('n" 4- w)
^^"S-P' = sin.(."-^]-'^"S-^'"
(123')
(124)
(124)
800 APPENDIX, BY THE TRANSLATOR; 1
[59941
•^ Again in tlie triangles NAA', NCC, fig. 88. we have
(las) cos.iV^ = cos.^^'.cos.iV.f ; cos.jYC = cos.CC\co3.iYC' [1345^'],
which in symbols, becomes as in (41,42). These values of it, m", give,
(126) X ^= "" — " = ^'^^ AC ;
adding this to v we gel v" = v -\- x- Then the formvda (45), which is the same as
[5986(4)], gives,
(137) D = ?-.cos.^Ju = r". cos.^i.(v-|- x) 5
hence,
/r\4 cos.i.(w4-v) cos . il) . cos. J y — sin. Jy . sin. fv , , . .
(1-28) [-j;Y = ^^ . = -^ = ^ = cos.ix — tang.iu.sin.ix-
V / cos.^y cos.Ji;
(128) Dividing this by sin.Jx» we get the value of tang.iy (44). In the same way, we may
obtain the expression of tang.|w" (44') ; or more simply, by changing r, v, u,
corresponding to the first observation, into r", v", ii", which corresponds to the third ;
by which means % (24), changes into — x . The expression (44), may be reduced
to the form (44"), by putting,
(129) tang.l = iy^ 4 (4-'^) ;
by which means it becomes,
tang.?
tang.^w == cot.lx r-^- , or, tang.| = cos.ix — sm-sX-tang.ii,' ;
(129') sm.J-x
hence we get, by successive reductions, and using [1,6,31,29] Int., the following expressions,
1 — tang. I 1 — cos.i-x + sin.i-x.tang.^^u 2sin.^2~X + Ssin.ix-^os.^x-'ang.v^y
(130)
1 -|- tang.l 1 -|- cos.^x — sin-^x-tang-J^u 2cos.^,[x — ^sin.^x-cos.^x-'ang.iv
tang.J^v 4- tang. il)
C130') =tang.ix- ', T . 1 =tang.ix-tang.(^i;4-ix)-
1 — tang.J^x-tang.^w
Substituting 1 = tang.4.5'', in the first member of (130), and then reducing, by means of
(30) Int. it becomes,
tan£!;.45'' — tang.| , ,
(130") ~ _ '^^ = tang.(45'' — |) ;
1 4- tang.45'' tang.l °^ ^
hence the expression (130') becomes as in (44").
We shall now proceed to illustrate these formulas by an example in (1T3, Sic). The
(131) data being as in (174 — 17-5). With these, we can compute in (176 — 181), the
coefficientsofthe fundamental equations (31,32,33), as in (182 — 186). From these equations ,
whence we find, by inspection, in Table II, T=^27<'^J", instead of the real value by
observation T = 8^"^^ ; and as this is three times toj great, we may decrease p in that
ratio, and take for a second value p ^ +• This gives in (184,186),
7-2 _|. /'2 = 1^747 . c2 = 0,0374,
VOL. III. 201
(135)
COMPUTATION OF THE ORBIT OF A COMET. 801
[5994]
by llie process, which is explained in (134 — 163), we may compute the values of p, ;i32)
r, r". in successive approximations with the help of Tables I, II, as in (167 — 191) ; the
argumenlstobe used in Table II, being the sum of the radii r -|- r" at the top of the page,
and the chord c at the side. Having obtained these approximate values of p, r, r", we can ,,33,
deduce from tlicni the approximate elements of the orbit, as in (193 — 20.5). The chief
difficulty in this solution, is in finding the value of p, which will satisfy the equations
(lS-2, 183,186), or, as they are called, {A), (B), (C) ; to which we may also annex the (134)
equation (X)), or the sum of tlie equations (A), (B), which represents the value of
,.2_j_ ^"2_ The method of operation, to find the valu 3 of p, is explained in the precepts,
in the four first pages of Table II, to which we may refer, observing particularly the directions
at the bottom of the fourth page, to vary p in the successive operations by some aliquot (135)
part of its last value, represented by -.p; p being an integral number, positive or
negative; by this means any term A, depending on the first power of p is augmented
1 2 1 /2 \
by -.A. and if it depend on p^, it is augmented by the quantity --A-] .(-.A).
P P ^p\p J
We may also observe, that in making the first rough estimate of the value of p, we can
use with advantage the two equations (C), (D), or the values of r^ -\- r''~, c^ ; found "^^'
to one or two places of decimals. In this process we must enter Table II with the argument
,.2 _j_ ,."2 at ti^g bottom, and t^ at the right hand side column. In this case we have ''^"'
only two equations, {C),{D), to satisfy ; instead of the t/iree equations (A), (B),(C) ,
required in the general and more accurate process. Most commonly, we may, for a first
hypothesis, take p = 1 ; and if the resulting time T, deduced from Table II, be too great. C^b)
we must, in general, decrease proportionally the value of p ; and in one or two trials,
without the trouble of taking any proportional parts, and with a very few minutes labor, we
can get a pretty close approximation to the value of p. When this is obtained, we can use
it with the equations [A), (B), (C), in getting the correct value of p, by the process
explained in page 2 of Table II, or by the similar calculation in (153 — 163). In the
examples which we shall give in (207 — 242), for finding p, we have neglected the
consideration of the equation (D), but it may not be amiss to show the advantage of
using it, by applying it to these examples. Taking therefore the first example, and usino-
the equations (D), (C), (184,186), we find, that if we put p ^ 1, and use two
places of decimals we shall get
,.a _|_ ^/ 3 ^2^02— 1,50 + 2,01 = 2,53 ; £2 = 0,02 — 0,11 + 0,50=0,41 ;
(139)
(HO)
(141)
(142)
(143)
B02 APPENDIX, BY THE TRANSLATOR ;
r 59941
whence T = "''"-'^G nearly. This must be increased a little, because the time is too
(144) small, as we have clone in (189). Again if we put p = 1, in the second example,
(207, 8ic.) we shall get, from (210, 209),
,•2+ r"2 = 4,71 ; c2 = 0,6S ;
which gives, in Table 11, T = 42'i"Js instead of ll''-'')%9734. We may, therefore,
C'*^' for a second supposition, put P = 4» because these two values are nearly in that ratio.
Substituting p = i in (210,209) we get,
7-2 4- r"2 = 2,42 ; c^^ 0,104;
hence we get in Table II, J"^ I4<iay3 nearly; so that p must be still further
(146) decreased ; and the value assumed in (212) is |. In Example III, (216, &,c.), we have
by putting P = 1, in (219,218),
r3 + r"2 = 4,98; c^ = 0,91 ;
whence T = 49''"^" instead of lO''"^" ; so that for a second value we may take
f = -1, which gives,
r2 + r"2 = 1,79 ; c^ = 0,009 ;
,j47) whence T = S'^"-",'- This is much loo small, therefore we may take p = i ; hence,
r3 + ,-"s = 1,89 ; 0^ = 0,050;
whence T =1 9'^"^% which must be increased a little as in (222). In Example IV,
(226, &tc) we have, by putting p = 1, in (229,228),
r^^- ?-"2=l,9S; c2 = 0,12;
whence r= 14''''>' ; which is nearly four times too great, therefore we may take for
(148) the next operation p = j, as in (231). In Example V, (235, Sic), we have, by putting
p=l, in (230,235),
j-s + r"2 = 4,39 ; c^ = 0,]6;
whence T=20'^''>'% which is more than double the actual value, we may therefore
(H9) assume p = I as in (237) for the next operation. In Example VI, we have by
putting p=l, in (240,239),
,•24- r"2 = 1,27 ; 0^=0,45 ;
whence T= 21'''^^^; which is twice the actual value; we may therefore take for the
next operation p = à) as in (241). What we have here stated will serve to show the
method of using the equation (D). We shall now proceed to the explanation of the
process with the equations (A), (B), (C) ; and it will suffice, for this purpose, to explain
(153) part'cularly, the calculations in the first example in (173 — 206).
COMPUTATION OF THE ORBIT OF A COMET. 803
(IÔ3)
(154)
(155)
(156)
(157)
In making the calculation of p, from the equations (A), (B), (C), or (182, 183, 18G)
of the first example, we have placed, in the first column of the table (187—191), the
successive values which are assumed for p. The second column contains the corresponding
terms of /", deduced from the equation (A), in the third, the value of r"'^, deduced
from (Z?; ; tiie fourth the value of c-, deduced from (C). In the fifth column are
the corresponding values of 7-, »■", c, deduced from r~, ?■"-, t^, by means of Table I ; and
in the sixth column is the resulting value of T, deduced from Table II. Thus by putting
p^l in the equation (.2), we find that the terms become 7--^l,014—O,2S8-|-l ,103=1,829,
as in column 2; and with this value of i~, we get r= 1,35, in Table I. In the
same way, we get, from (5), (C), the expressions ?" = 0,84.; c = 0,64 ; then with
r-\-r" ^= 2,19 and c = 0,G1, we obtain, by the mere inspection of Table II, r=27'^">%
nearly. This time being about three times as great as the actual value by observation,
T = B'''"'', we may take for a second hypothesis p^ ^ ; and by repeating the operation get
T^ 7'''>%6GO. The calculation of the coefficients of p, p^ in these equations is made in ^
columns S, 9, 10, conformably to the precepts in pages I, 2, 3, 4, of Table II; and the
results are transferred tocolumns 2,3, 4. In going through these calculations, we have always
varied p by an aliquot part - . p of its last value, according to the precepts in the table
and in (135). Thus we have, in the first instance, taken p=l (187), then p=^ (188) ; to
this second value J^j part is added, for the next operation; and as this is found to be too (IM)
great, it is decreased by ^^^ part ; finally this last value is increased by ^J^^ part or
0,0004, multiplied by its last value ; and then the resulting expression of T becomes (JC2)
gjajs^ agreeing with the observations. Similar processes are used in the other examples,
as may be seen by inspection of the calculations, without any particular explanation.
(159)
(ICO)
(163)
(165)
(106)
In the first example (173 — 206), we have gonethrough the whole calculation (176 — 181)
for finding the coefficients of the equations {A), (B), (C), (18-2 — 186) ; and deducing from
them the values of p, r, r' (187 — 192). From these last quantities we have finally (i64)
deduced the elements of the orbit, as in (193 — 205). This one example will suffice for
the illustration of the method of calculating the coefficients (176 — 181), and the computation
of the elements (193 — 205) ; but for the sake of explaining more particularly the uses of
Tables I, II , we shall insert several examples of the computation of p, r, r", similar
to (187 — 192), from the fundamental equations (A), {B), (C), corresponding lo difierent
comets and shall select, for this purpose, some which have been already calculated by Olbers, (la?)
Delambre, Ivory, Sic. We may remark, that if any one of the coefficients of the erjualions
(A), {B), (C), be negative, we may add its arithmetical coniplement to I OfiQOOO, and (les)
then reject this last quantity. Thus, in finding the first value of r-. in the following table
(187); instead of using 1,014—0,288+1,103 we may take, (leo)
1,014 + 9,712 + 1,103 — 10,000 ;
and as each figure of the arithmetical complew eut can be taken separately, tvhilc performing ^^^^^
the process of the addition of these quantities, toilhout the trouble of actually writing down
the figures of the arithmetical complement, u-e can make this addition, by one operation^
notwithstanding the difference of the signs : by this means the calculation is somewhat (i72)
abridged.
804
[5994]
(173)
(171)
(175)
(1750
APPENDIX, BY THE TRANSLATOR ;
EXAMPLE Ï.
This example is the same as that of Dr Olbers, in page 54 of his Mhamllung, &c., in which he gives the
computation of the orbit of the comet of 1769, from the observations of September 4'''', 8"', 12"', 1769.
© long. Comet's geo. long. Com. geo. lat. South. log. © dist. earth.
1769 Sep. 4'" 14''; © = i62''42"'o5» ; a = So-'Se'»!!'; 9 = — 17* 5im 3g. ; jog.R =o,oo3i32;
8'' 14''; 0' =i66''35"'3i'; a' = loi'^oo'" 54^ 6' = — 22'^ oS" 02» ; log.B' = o,oo2665 ;
12'' 14''; ©" = I70''29"'20s; a" = 124'' 19™ 22' ; Ô" = — 23''43'" 55' ; log.iî" = 0,002184 .
Hence we deduce,
© — ci. = 8i<f45»' 54"; © — a"= 38'' 22™ 43» ;
©' — a :^ SS-J 39» 20» ; 4g)' — a' = 65* 34™ 37' ; ©' — a"= 42* le^cg»;
©"— a = 89'' 33m ogS ©"— a"= 46'' 09" 58» ; ©"— © = 7'' 47"" 1 5» ; y." — cl = 43-' 23" 11».
log.flâ = 0,006264 ; log./J"a = o,oo4368 ; log.2iJ =o,3o4i62 ; log.2iJ" = o,3o32i4 .
In this example the ffieeii's geocentric latitudes being smith are considered as nc^a<i«e ; and the rules fur
the signs of t lie angles ISggo (23,24, 25)] are to he ohseived in finding the coefficients uf ail the terms of
the fundamental equations (28 — 33).
I. CALCULATION OF THE THREE FUlSfD.lMENTA L EQUATIONS (3[,32, 33).
(176)
(177)
To find m, M (28, 3o).
6' tang. 9,608237,,
©' — a' cosec. ,040712
m log. 9,648949,,
— sin.(©'— a") log. 9,827766"
0,29972 log. 9,476715
(ang 6"=— 0,43963
(178)
— o,i3ç9i^Denomin. of Jil
m log. 9,6489491
©'—a, sin. 9,998750
—0,44432 log. 9,647699»
— tang. 6^=0, 32 224
(179)
i\um.of3/=-0, 12308 log. 9,086644
Den. ot M log.co.ar.o,854i5i"
t" — t' log. 0,602060
(' — t log.co.ar.9,397940
M log. 9,940795
To find r2 (3i).
Ri
i,oi453
log.
— 2R
log.
©-a
cos.
coeff. of p
=—0,28854 log.
9
sec.
scc2.9=
=i,io3S4
log.
To find
j-rtz>
Rii-2 1,01011
log.
—2R"
log.
©"—a"
cos.
M
log.
coeffi.of |)=
=— I,2l47
I log.
6"
sec.
M
log-
M.sec.6"
log.
Jia.sec2 a
"=0,9085
2 log.
0,006264
o,3o4i62,
9,i56o45
9,46o207„
0,02l452
0,042904
(32).
o, 004368
o,3o32i4,
9,84o464
9,940795
o,o84473„
o,o3837i
9,940795
9,979166
9,958332
vieO)
(181)
To find c2 (33).
—2R
R"
log-
log,
cos.
o,3o4i62„
0,002184
9.995976
— 2,00596
log-
log.
COS.
0,302322„
272"
o,3o32i4
7,892666
i^' term^o,0i57o
8,195880
2R .
log.
CO;,
log.
o,3o4i62
9,894275
9,940795
2"''term=i,37795
log-
los;.
log.
COS.
0,139232
coefr. of f=i, 39365
— 2
M
a"— a
o,3oio3o„
9.940795
9,861378
i-''term=— 1,26825
log-
log,
tang
tang
o,io32o3„
— 2 .
M .
S .
9" .
o,3oio3o„
9.940795
9.508 1 75„
9,643092,,
2"''term= — 0,2472;
log.
9,393092,,
COMPUTATION OF THE ORBIT OF A COMET.
Three Fundamental Equations.
7-2 = 1,01453 — o,28854.f 4- i,io384 f^ (Ji)
r"~ =: 1,01011 — i,2i47i-/i -(- o,go852.f2 (JJ)
Sum r2 ^ >-"2 = 2,02404 — i,5o335./i -j-2,oi236f2 (/))
Add the other terms of c-, — 2,oo5g6 + i,39365.f — I,5i548.f3
Sum is c2 = 0,01868 — 0,10960./! + 0,49688.^ (C)
Coinpuliition of f from the equations (.?),(£), (C).
805
[591)4]
(182)
(183)
(18-1)
(185)
(186)
Col. 1.
Col. 2. 1 Col. 3. 1 Col. 4.
Col. 5.
Col. 6.
Assumed Viilues
of J.
Corresponding terms of the
equations {A), (B), (C).
Values of
>■, >■", c.
T
r-2 j r"2 1 c9
Hypothesis I.
f = 1,0
as in (i4i).
i,oi4
—,288
i,io3
1,010
-I,2l4
0,908
0,018
— 0,109
o,4g6
r= 1,35
r" = 0,84
3.^doys
r-{-'"= 2,19
1,829
0,704
o,4o5
c = 0,64
Hypothesis 11.
t=J=o,33333
1,01453
—,09618
,12265
1,0101 1
—,40490
,10095
0,01868
— ,o3653
,o552i
r = 1,0202g
r"=o,84o33
r+r"= 1,86062
7,53g
1
,i3o
i,o4ioo
0,70616
o,o3736
c = 0,19339
7,660
Hypothesis HI.
Add 1 or
0,01667 makes
f = 0,35
1 ,01453
— ,ioogg
,13522
1,01011
-,425i4
0,11129
0,01868
— ,o3836
,06087
r = I ,o24oy
»■"= 0,83443
r-\-r" = 1,85852
7,903
18
,1 17
1,04876
0,69626
0,04119
c = 0,20395
8,o38
Hypothesis IV.
Less j; or
0,00175 makes
f = 0,34825
i,oi453
— ,10048
,13387
1,01011
— ,423oa
,11018
0,01868
— ,o38i6
,06025
r = 1 ,02368
r" = o,835o3
7,9^3
18
7,99'"'
r 4- r"= 1,85871
1 ,04792
0,69727
0,04077
c = 0,20191
Hypothesis V.
Add o,ooo4-/> or
0,000 1 4 makes
f = 0,34839
1,01453
— ,ioo52
,13398
1,01011
-o,423i9
0,11027
0,01868
— ,o38i8
,060 3 1
r = 1,02372
r" = o,834g9
r + r"= 1,85871
7,903
18
79
1,04799
0,69719
o,o4o8 1
c = 0,20201
8,000
Col. 7. Col. 8. Col. 9. Col. 10.
Coefficients of
i
I
JiJ
2ffcr
,ooo4
0,38854
O,og6i8o
48og
o,ioog8g
— 5o4
o,4o4go3 o,o36533
20245| 1826
o, 100435
4o
1,21471 o,iog6o
o,425i48'o,o38359
3135! 191
o,423o23^o,o38i68
169I 1 5
0,1005250,423193 o.o3Si83
Coefficients of
f-
1 0
1
1 uo
+1
,0008
i,io384 |0,go852
0,1226490,100947
1 3365! lOogS
306; 3 52
o,i3522o 0,1 I13g4
— i352 — iii3
3 3
o,i3387i 0,110184
107 88
0,1 33978 0,1 10372
o,4g688
o,o552o8
552
i38
0,060867
—609
0,060259
48
o,o6o3o7
With these last found values of f = 0,34839, r= 1,02372, r" = 0,83499, we shall now compute the
elements of the orbit, by means of the formulas (34—45) ; observing, that as r > r", the comet must be
nearer the perihelion at the third observation than at the first.
(187)
(188
(189)
(ISO)
(191)
1,152)
Computation of the elements of the orbit.
f = 0,34839 log. 9,542066
M log. 9,940795
t" — 1/ '
log. 9,482861
j" 0,83499 '"§• <^o- ,078319
6" tang. 9,643o92„
•IS-" = — cf 1 2™ 37'
9,304a72n
/> log. 9,543066
r 1,02372 log. CO. g.gSgSig
6 tang. 9,5ù8i75„
«^ = — 6<'i7m45' sin. g,o4oo6o„
VOL. III.
202
©-a
see. 0,002627
log. CO. 9,g8g8i9
log. 9,542066
sin. 9,995499
i= 19" 48'» 35» sin. 9,53ooii
A = 342'' 42°' o5' = © -I- iS'S^ "
/3 = 2'' So"» 3o» = .4 -f 6
/3" = S"* 55'" 06' found in (196)
yS" — ^ = id 24" 36»
w = -;'' II'" 35' found in (197)
/S" — tJ = 10'' 36'" oi' = y5" — ;S -|- to
U = 355"* igtn o5' = /3 — u>
(193)
(m)
(195)
806
[5994]
APPENDIX, BY THE TRANSLATOR;
(196)
(197)
(197')
(198)
■ a"
sec. o,oo5636
log. CO. 0,078319
log. 9,482861
sin. 9,858147
('/ = i5''25"'46' sin. 9,424963
All = SSo"* 29"" 20» = ©" -Jf- 180^
;5" = S"* 55"< 06' = Jill + s"
— ^ = 3ii 24™ 36» cosec. 1,225625
OT- cot. 0,957317»
V tang. 9,209914»
24,7090 '°g-
-cot.(/5" — ^) — 16,7824
1,392856
cot.jc = 7,9366 log. 0,899087
u) = 7'* II'" 25" cosec. 0,902518
OT tang. 9,042683,1
Inclination f = — 41'' 23™ 4i' tang. 9,g4520i„
- «" = '35" 54"' oo» > f^^^j i„ „^^ ^d ,„,„^„
m" = i4<' 00"' 29» )
Perihel. — u= i49''54"' 29»
Long, u = 355'' 19™ o5*' found '" ('95)
Long. Perih. = i45'' i3"' 34» on the orbit.
Time of
OT COS. 9,997373
w COS. 9,996571
u= 9*32"' 46» COS. 9,993944
COS. 9,994364
COS. 9,992525
COS. 9,986889
(193) /3"_u
«" = 14*^ oowi 29»
^ = u" — u= ifi 27"' 43«
4;(^ := 2<f i3'» 52» cosec. 1,409711
è log-r
arith.
co.9,9g49og
è log.r"
g.960840
Number 23,ig85
log.
i,36546o
— cot.J;^^ ^ — 25,6674
tang.4i)" = — 2,4689
\v'i = — 67'' 57'" 00
COS.
9,574512
same
COS.
9,5745 1 2
)-"
log.
9,921681
log.
Per. Dist. B= 0,11768
half
t)" = — 135'' 54*" 00» tab. Ill log. 2,789133
Time from Per. 24'''»»,8422
Third obs. Sept. 1 2""»», 5833
Perihelion, October 7''''!;%4255.
log.
9,070705
9,535352
i.BgSigo
(198')
(199)
(200)
(201)
(302)
(203)
(204)
(205)
(206)
The value of v" being negative, indicates, that the comet was approaching towards
the perihelion at the time of the third observation. The heliocentric latitudes,
« = _C''17'"45'; tn" =— 9"12'"37%
being south and increasing, it is evident, that the comet had passed the descending node ?3,
a short time before the first observation ; and we have therefore calculated the longitude
of that node 355'' 19'" 05"-, to which corresponds 9=: — 41''23"'41', which is the
same as to put ^ = 175'' 19"'05', and <? = 41''23'"4I''. Hence the approximate
elements of the orbit are,
Longitude of the ascending node 175'' 19'" 05';
Inclination 41" 23-" 41'
Longitude of the Perihelion 145"13'"34'
Perihelion distance 0,11768
Time of passing the Perihelion 1769, Oct. 7''''J'%4255.
To illustrate the process of finding P, r, r", from the fundamental equations (31,32,33),
we shall give the following additional examples.
COMPUTATION OF THE ORBIT OF A COMET.
EXAMPLE II.
Thd equations in lliis cxnmple, correspond to those of the comet of i8o5, as given by Mr Ivory in the
Transactions of the Royal Society for i8i4, page 170.
r2 = 0,973662 -[- 1,408969.^ -\- i.oooooo.f'î ; (.1)
r"2 = 0,969967 -(- o,23oo47.f -)- o,i3i45o.f2 ; (g)
c2 = o,o435o5 + o,ii52oo.f + o,5i8768./i9 ; (C)
ra _[- r"-2 = 1,943629 -\- i,6390i6.f -f- i,i3i45o.f3. (iJ)
Interval between the extreme observations T =: H'>"y<^,f)j34.
Col. 1.
Col. 2. Col. 3. 1 Col. 4.
Col. 5.
Col. ti.
f
r2 1 r"a
c2
r, r", c
T
Hypothesis 1.
f = § or 0,16667
as in (146).
0,97366 0,96997
,23483 3834
2778 365
0,043 5o5
19200
i44io
r= 1,11188
r" = 1,00596
r-\-r"= 2,11784
c — 0,27769
ii,3g2
21
,324
1,23627 1,01196
0,077115
11,73-
ii,84i
i5
87
Hypothesis II.
Add 5Ô or o,oo833
f = 0,175
0,97366 0,96997
,24657 4026
3062 402
o,o435o5
20160
15887
r= i,ii84i
r" = 1 ,007 1 0
r+r" = 2, i255i
i,25o85 i,oi425
0,079552
c = 0,28205
1 1 ,943
Hypothesis III.
Add IÏÛ = 0,00125
f = 0,17625
0,97366 0,96997
,24833. 4o55
3io61 408
1
o,o435o5
2o3o4
i6ii5
r = 1,1193g
r" = 1,00727
n,84i
19
,ii4
r4-»-" = 2,12666
i,253o5| i,oi46o
o,07gg34
c = 0,28271
1 1 .974
Col. 7.| Col. 8. 1 Col. 9. Col. 10.
Coefficients of f.
X
1
150
1,408969
o,23oo47
0,11 5200
0,234828
I1741
o,o3834i
1917
0,019200
960
o,24656g
1761
o,o4o258
288
0,020160
1 44
n.24833i
L.,o4o546
o.o?o3o4
Coefficients of /i'2. 1
1
6
1
==1
1
280
1 ,000000
o,i3i45o
0,518768
0,166667
0,02 1 go8
o,o8646i
0,027778
2778
69
o,oo365i
365
9
o,or44io
144 1
36
o,o3o62 5
437
2
o,oo4o25
57
0,015887
227
o,o3 1 064
0,004082
o,oi6ii5
807
[5994]
(207)
(208)
(209)
(210)
(211)
Henco f^ 0,17625, r^i,iig3g,
Mr Ivory makes, /i =.0,17620, r = 1,11936,
r = 1,00727.
r" = 1,00727.
(212)
(213)
(214)
(215)
EXAMPLE III.
These equations are similar to those given by Mr Ivory in the Transactions of the Royal Society for i8i4,
page 160 ; and correspond to the comet of 1781..
r2 = 0,976625 — o,3o3724.f + i.oooooo.fa ; (jÎ)
r"2= 0,972873 — i,457243.f -\- 3,788i66.f2 ; (S)
£2 = 0,080278 — 0,353719., + i,2378i8.,2 ; ( C)
r-2 -j- r"3 = i,94g4g8 — 1,760967-, + 4,7881 66., 3. (D)
Interval between the extreme observations T" = 10 °''.
(216)
(217)
(2!8)
(219)
(220)
808
[5994]
APPENDIX, BY THE TRANSLATOR;
(221)
(222)
(223)
(234)
(235)
Col. I.
Col. 2.
Col. 3.
Col. 4.
Col. 5.
Col. 6.
f
/■2
r"2
cS
r, t", c
T
Hypothesis 1.
P = o,33333
as in (i47).
0,97663
— ,10124
,11111
0.97287
-,48575
,42091
0,030278
— ,1 17906
,137535
r = 0,99323
r" = 0,95291
8,902
14
1 37
r-\-r" = 1,94614
0,98650
0,90803
0,049907
c = 0,2234
9,o53
Hypothesis H.
Add 50 malces
f =0,35
0,97663
— ,io63o
,1225o
0,97287
— ,5ioo3
,464o5
0,030278
— ,i238oi
,i5i633
r = 0, 99640
?■'' = 0,96275
9,735
23
43
r-j- r" = i,95gi5
0,99283
0,92689
o,o58tio
c = 0,24106
9,801
Hypothesis HI.
.Add 100 malies
f = 0,3535
0,97663
— .10737
, 1 2496
0,97287
— ,5i5i3
,47338
0,030278
— ,i25o3g
, 1 54681
r = 0,99711
r" = 0,96494
9,760
195
r 4- r" = 1 ,96205
0,99422
0,93113
0,059920
c = 0,24479
9,960
Hypothesis IV.
.Add 500 or o,ooo88
f = 0,35438
0,97663
— ,10763
,.2559
0,97287
— 5 1642
■47575
0,030278
—,125352
,155455
r = o,9973o
r" = 0,96550
9,760
233
r -(- !■'' ^ 1 ,96280
0,99459
0,93220
o,o6o38i
c = 0,24573
10,000
Col. 7. Col. S. Col. 9. 1 Col. 10.
Coefficients of f .
h
1
55
I
lou
1
4110
o,3o3724
1,457243
0,35371g
0,101241
5062
o,io63o3
io63
o,48574S
24287
0,117906
5895
o,5ioo35
5 1 00
0,i238ûi
1238
0,107366
268
o,5i5i35
128S
o,i25o39
3i3
0,107634
n.5i64.,3
o,i25352
Coefficients of ^2. |
1
I
10
J
1,0000003,788166
1,237818
0,11111 1
11 III
278
0,420907
42091
1052
0,137535
13754
344
1
ill
1
sou
1
SfJO
8ÙU
0,122500
2450
12
o,464o5o
9281
46
o,i5i633
3o33
i5
0,124962
625
1
0,473377
2367
3
0,1 54681
773
I
o.i25588'o,475747
0,T 55455
(226)
(227)
(228)
(229)
(230)
(231)
(232)
(233)
(234)
Hence ;i = o, 35438, r = 0,99730, r" = 0,96530 ; which agree with Mr Ivory's calculation, excepting
a unit in the last decimal place.
EXAMPLE IV.
These equations are equivalent to those given by Mr Ivory, in the Transactions of the Royal Society for
i8i4, page i65 ; and refer to the comet of 176g.
r2 = T ,017347 — 0,778609.^1 -(- I,O00O0O./l2 ; (^)
r"~ = 1,010107 — 1,297813.^ -|- 1,033677.^2 ; (£')
c3 =0,004678 — o,0275i8.f -l-o,i396i9.()2 ; (C)
j-2 _|_ r"2 = 2,022454 — 2,076422.^ -1- 2,o33677.f2. (JD)
Interval between the extreme observations T" = 4 "^'^
f
r2
r"-2
t2
r, r'l, c
T
Hypothesis I.
f = i = 0'25
as in (i48).
1,012
—,194
62
1,010
-,324
64
o,oo46
—,0068
87
r = 0,94
r" = 0,86
3,119
r + r"= 1,80
0,880
o,75o
o,oo65
C =: 0,080
Hypothesis II.
p = J = 0,33333
1,01235
-25954
,11111
1 ,0 1 0 II
—,43260
,11485
0,004678
— 9'73
i55i3
r = 0,92947
r" = o,832og
3,856
3
,191
/• + )■"= 1,761 56
0,86392
0,69236
0,011018
c = 0, 10497
4,049
Hypothesis III.
Sub. j'o or o,oo4i7
p = 0,32916
1,01235
—,25629
,io835
1,01011
—,42720
,11200
0,004678
— 9o58
i5i27
r = 0,92974
r" = 0,83362
3,856
3
,i4i
r 4- r"= 1,76336
o,8644i
0,69491
0,010747
c = 0,10367
4,000
Coefficients of f. 1
i
0,778609
1,297813
0,027518
0,194
0,324
0,0068
I
0,259536
— 3244
0,432604
— 5408
0,009178
— ii5
10,256292
0,427196
0,009058
Coefficients of f-. i
1
15
1 ,000000
1,033677
0,13961g
0,062
0,064
0,0087
5
1
0,1 11111
— 2778
17
o,ii4853
— 2871
18
o,oi55i3
— 388
2
o,io835o
0.1 120no
o,ni5l27
Hence
The true values being
(5 = 0,32916, r = 0,92974, »•" = 0,83362.
f;— 0,32911, r ^ 0,92974, r"=o,8336i.
COMPUTATION OF THE ORBIT OF A COMET.
EXAMPLE V.
809
[5994]
The foUouins equations were comi)utcil liy Dr Olliers in liis Abhandlung, &c., page Sg ; they correspond to
the comet of i6Si, and were computed from Halley'^ elements, and not deduced from actual observations.
r2 = 0,96734 — 0,59992 f, 4- i,24328./>3 (.;?)
r"3 = 0,96941 — o,4oi85.f -f 2,2oo87,f3 (B)
c2 = 0,019726—0,122756.^-1-0,265982.^2 (C)
rS + i-"a = 1,93695 - o,gg477./>+3,444i5.f2 (D)
Interval between the extreme observations T = 8''^''*,o47.
Col. 1.
Col. 2. Col. 3. 1
Col. 4.
Col. 5.
Col. G
T
P
r-2
,-"2
c3
r, »■", c
Hypothesis I.
f = 0,5
as in (i49)-
0,96754 0,96941
— ,2g646 — ,20092
,31082 ,55022
,oig726
—,061378
,066495
r = 0,99091
r'' = 1,1 4835
J--|-r" = 2,i3g26
6,363
1 4
,32 3
6,699
OjgSigo
1,31871
,024843
c = 0,15763
Hypothesis 11.
Add 15 or ,o5
p := 0,55
o,g6754
— ,32611
,3760g
0,96941
— ,22102
,66576
,019726
— ,067516
,080459
r = 1,00872
r"= 1,18918
7;74o
i4
33
r + r"= 2,19790
1,01752
i,4i4i5
,o3s669
c ^= 0,18074
7,786
Hypothesis III.
Add 50 or 0,01 1
f = o,56i
o,g6754
—,33263
,39129
0,96941
—,22544
,69266
,019726
—,068866
,083709
r = i,oi3o2
r"= i,ig85g
7,776
3
,256
r-t^r" = 2,21161
1 ,02620
1,43663
,034569
c = 0,18593
8,o35
Hypothesis IV.
Add 1 =o,ooo56i
lUUO
f = o,56i56i
0,9675/
-,33296
,39207
0,96941
—,22566
,69404
,019726
— ,o6Sg35
,083876
r = I, 01 324
r" = 1,19908
7,776
4
,267
r-\-ir" = 2,21332
1, 02665
1^43779
,034667
e ^0,18619
8,047
Col. 7.
Col. S.
Col. 9. Col. lO.J
CoelTicients of p. i
t
tV
r
o,5g2g2 o,4oi85
0,122756
0,296460
3g646
o,20og25
20093
0,061378
6i38
0,326106
6522
0,221017
4420
0,067516
i35o
0,332628
33?
0,225437
225
0,068866
69
0,332960
0,2 2 5663
0,068935
Coefficients of fi. 1
TDWcT
r,34,J28
3,20087
0,265982
o,3io82o
62164
3io8
0,550217
',110043
55o2
o,o664g5
1329g
665
0,376093
i5o44
i5o
0,665762
2663 1
266
o,o8o45g
3218
32
o,3gi386
782
o,6g2659
i385
0,08370g
167
o,3g2o68
o,694o44iO,o83876|
(235)
(236)
(237)
Hence f = o,56i56i, r=i,oi334, r"=i,ig9o8. The actual values, according to Halley's theory, upon ,333
which the proposed equations are founded, are t = i,oi44, r" = 1,2000 ; which agree, very nearly, with the
preceding result.
EXAMPLE VI.
These equations correspond to the comet of i8o5, in the calculation of Mr. Ivory in the Transactions of the
Royal Society for i8i4, page 175.
r2 = o,988ig2 — 1,271721.^-1-1 ,oooooo.f 2 ;
t"-2 = o,g8ig87 — 2,3ii644.f -f- i,88i447f'- ;
e2 = 0,043371 — o,07448g./> -|- o,485838 p9 ; (239)
r2 -\- r"2 = 1,970179 — 3,583365.f -f- 2,88i447-f2. (3«)
Interval between the extreme observations 7"= 12 *^',o36.
VOL. III.
203
810
APPENDIX, BY THE TRANSLATOR;
[5994]
CS-40
Col .1.
Col. 2.
Col. 3.
Col. 4.
Col 5.
Col. 6.
f
,a
r"i
ca
r , r", c
T
Hypothesis I.
f= I
as in (240-
0,98
— 1,27
-j-i,oo
0,98
— 2,3l
-1-1,88
o,o43
—,074
+.485
r = 0,84
r" = 0,74
r-\-r" =1 ,58
c = 0,67
24days
0,71
0.55
0,454
Hypothesis II.
f = è
0,98819
—,63586
,25oOO
0,98199
-i,i5582
,47o36
0,043371
- 37244
,121459
r = 0,77610
r" = 0,54455
11,653
3
,238
r-i-r"= i,32o65
0,60233
0,29653
0,127586
f, = 0,35719
11,894
Hypothesis HI.
Add jij or o,o5
f = o,5o5
0,9881g
—,64222
,25502
0,98199
-1,16738
,47982
0,043371
— 37616
,i23goo
r = 0,77523
r" = 0,54263
1 1 ,938
36
3
r + r"= i,3i785
c = 0, 36008
0,60099
0,29443
0,129655
1 1 >977
Hypothesis IV.
Add jijj or,o o33
p = o,5o83
0,98819
— ,6465o
,25844
0,98199
-I, 17516
,48623
0,043371
— 37867
,125557
r = 0,77468
r" = o,54i35
11,938
28
67
r-f->-" = i,3i6o3
o,6ooi3
0,29306
o,i3io6i
c = 0,36202
i2,o33
11,938
28
70
Hypothesis V.
Add 5^0 = o,ooi7
f = o,5o847
0,98819
— ,64673
,2586i
0,98199
-1,17555
,48656
0,043371
— 37880
,12 5641
r = o,77465
r" = 0,54129
r-\-r"= i,3i594
0,60008
0,29300
o,i3ii32
c = 0,36212
1 2 ,o36
Col. 7.
Col. 8. Col. 9. Col. 10.1
Coefficients of p. 1
è
I
I,27i73i|2.3i 16440,074489
o,63586o 1,1558220,037244
6359 ii558| 372
1
0,642219
4281
1,1673800,037616
7783| 35i
1
(i,6465oo 1,1751630,037867
3i5j 3g2 i3
o.6407i5!i, 1755550,037880
Coefficients of f 2 1
i
s'ô
1
200
2
ISO
ah
T5ÛÔ
1 ,000(100
1,881447 o,485838|
O,2 5o000
5ooo
25
0,470362
9407
47
0,121459
2429
12
o,255o25
3400
II
0,479816
6398
21
o,i23goo
i652
5
o,258436
172
0,486235 o,i25557J
324 84|
0, 258608 •0.486559 0,1 2 aUi j
(242) Hence
Mr Ivory makes
f = o,5o847, r = 0,77465, r" = 0,54129.
f = o,5o8i, r =: 0,77472, r'' == o,54I44•
(242')
From these examples we see that the interval of time T, between the extreme
observations, is found in Table II, with a sufficient degree of accuracy, and that the results
agree with the calculations by logarithms of other astronomers, although the table is only
carried to the nearest unit in the third decimal place. While treating upon this subject, it
may not be amiss to recall to mind the remarks of La Lande, in the third volume, page 259,
of the third edition of his astronomy, relative
to the degree of accuracy in the cometary
calculations. He has there given a table
of the elements of the orbits of those comets
which had been previously computed,
giving the longitudes and angles to seconds,
and thelogarithmsof the perihelion distances
to five or six decimals ; but at the same time
observing, that though he has inserted the
seconds, no confidence could be placed in
them; neither could we depend on the
correctness of the logarithms of the
Col. 1.
Col. 2.
Col. 3.
Col. 4.
Examples.
Time T by
Observation.
Time T by
Table JI.
Errors.
I.
8''''y%ooo
8'"'y',o2o
-f o''^y%020
II.
I.''^^^973
ii'"'^",98i
-(- o'''■'>•^oo8
HI.
days
10 ■' ,000
9''''^^998
- 0''">'^002
- o"='^-',ooi
IV.
4'""",ooo
3"^^^999
V.
s-^'^'Mi
8"'=>'^o6o
+ o'"'^",oi3
VI.
i2'^''",o36
days
12 ,129
+ o'^''^^093
COMPUTATION OF THE ORBIT OF A COMET. 811
[5994]
perihelion distances in the fourth decimal place, as is abundantly manifest, by comparing the
results of the calculations of différent astronomers. To estimate the degree of accuracy with
which the time T can be ascertained, by entering Table II, with the values of r~^r"^ at the
bottom of the table and c^ at the right hand side ; we have computed the value of T, for (243.,
the six preceding examples; as in the third column of the annexed table; the times by
observation being given in the second column, and their differences or errors respectively, in
the fourth column. These errors being very small, it is evident, that the method of
combining the equations (C), (Dj, or the values of r- -f- r"^, c^ ; by means of Table II,
must generally give a very close approximation to the value of p.
Gauss varied the forms of the equations (31,32,33), by the introduction of several
auxiliary numbers A, B, B' b, b", c, c", he. which are deduced from the co-efficients of
the terms in the original equations ; changing also the unknown quantity p into u ; so
as to reduce the expression of c^ (33), to the form in (244). The object of these
transformations is to render the calculations more convenient for computation by logarithms,
by putting them under the following forms;
(243)
(244)
When the equations are given in this form, we may determine u, by means of Tables I, II,
or by successive apjiroximations, in the same manner as we have found p in the preceding
examples ; using in Table II the arguments, r -f- r" at the top, with c at the side ; and
it is evident, on account of the decrease of the number of terms in the expression of c^
(244), that the calculation of u is more simple than that of finding p in the former (S")
examjiles ; but the ?aving of labor is nowise sufficient for the trouble of reducing the equations
to the forms (244), when the time is deduced from Table II, in the manner we have here
pointed out. We may also use the equations (C), (D), or the values of r^ -j- r''^ and
c^, in finding the first rough estimate of u ; in like manner as we have proceeded with the
similar expressions in terms of p in (136 — 150). This process may be illustrated,
by the two following examples. Thus if we put ?« = 0, in (248,247), we shall have
r= -|- /'^ = 2,49, c- =0,028, whence we obtain, by inspection in Table II, T^7''*>'^3 (2«')
nearly ; which is less than the time by observation 14''^>%0493. We also observe by
inspecting the same vertical column, corresponding to r^ -f- r"^ = 2,49 : that this last
mentioned time corresponds very nearly in the margin to c- = 0,11 ; substituting this
in (247) we get 0,11 =: 0,028 -f- u^, whence we obtain 7i = 0,28, or nearly 1/ = |, (2«)
which is assumed in (249). In like manner, in Example VIII, we have, by putting
„ = 0, in (2.54,2.53) r^ + r"^ ^ 12,53, c^ = 0,051 ; which correspond in Table II,
to 14'^^''', S . If we suppose 7L = 3-V, we get r- -f ?-"2 = 23,2, c^ = 0,062 ; corresponding
in Table II to 18''"5'%9. As the actual time by observation falls nearly midway between
■' ■' (246)
these two times, we may assume, for an approximate value, t<- = rrj, as in (255).
812
[5994]
(247)
(248)
APPENDIX, BY THE TRANSLATOR ;
EXAMPLE VII.
The following equations correspond to the second comet of i8i3. They are equivalent to those given by
Gauss ill vol. 28, page Sog, of the Monatliche Correspondenz ; or by Encke, in the Jahrbuch, for i833, page 284.
r2 = 1,2441 5 -\- i,92565.u -{- ZfiGgjS.u^ ; (Jl)
j-"a = 1,24837 4- i,5i429.M -+- o,-jg33i.u^ ; (£)
c2 =0,028219-]- «3; (O
ra -\- r"-2 = 2,49252 -f- 3,43994.m + 3,863o4.w2 ; (D)
Interval between the extreme oliservations T = i4 ^^',o493.
(249)
Col. 7. Col. S. Col. 9. |Col. 10.
Coelficients of u .
i
65
1
1
4000
1,92565 |i, 51429
o,48i4i2 0,378572
— 9628 — 7571
0,4717840,371001
— 2358| — i855
0,4694260,369146
117', 93
0.469543 0,369238
Coefficients of u^. 1
i
1
100
2
200
1
400
3
40ÔÔ
3,06973 ]o,7933i
1 ,00000
,7674320,198328
,25oooo
,191858
-7674
77
0,049582
—1983
20
0,062500
— 25oo
25
0,184261
— 1843
5
0,047619
—476
I
0,060025
— 600
I
O,i82423|o,o47i44
91 24
0,059426
3o
o,i835j4 0.04-168 0,059456]
Hence we have « = o,243836, r = 1,37702, r" = 1,29026 ;
According to Gauss, u = 0,24388, r = 1,37708, r" = 1,29027 ;
(250) According to Encke, r = 1,37705, r"= 1,29027.
We may observe, that the last, or fourth hypothesis, may be dispensed with, by interpolating between the
values of f, r, r", given in the second and third hypothesis, so as to make T correspond to the proposed
interval 1 4''°^'',o493.
(251)
(252)
(253)
(254)
EXAMPLE VIII.
The following equations correspond to the comet of 1825, calculated by Nicolai in the tenth volume of the
Jlstronomische JVachrichten, page 238.
r2 = 6,2o536 -\- 43,23445.M -{- 8o,07556.m2 ; (^)
r"2 = 6,332i3-f-46,4i4ii.M+ 93,5o6io.«2; (B)
c2 = o,o5i58-|-«2; (C)
r2 _^ r"a = 12,53749 + 89,64856.u + i73,58i66.m2 (Z>)
Interval between the extreme observations T = i6iiay3,782i.
COMPUTATION OF THE ORBIT OF A COMET.
Col. 1.
Col. 2.
Col. 3. 1 Col. 4.
Col. 5.
Col. 6.
u
r-3
,"2 c3
r, r", c
T
Hypothesis 1.
u = 5ij ^ o,o5
as in (246).
6,2o536
2,16173
,20019
6,332x3 o,o5i58
2,32071
,23376! ,0025o
r = 2,927
r" = 3,g8i
16,237
II
,180
/■4-r"= 5,908
8,56727
8,88660' o,o54o8
c = 0,23355
16,438
Hypothesis II.
Adil i makes
u = o,o58333
6,3o536
2,52201
,2724s
6,332:3' o,o5i58
2,70749
,3 18 1 7 ,oo34o
J- ^ 3,0000
»■" = 3,0590
16,374
83
,319
r+r" = 6,0590
8.99985
9,35779 0,05498
c = 0,23448
16,775
Hypothesis III.
Add ôj,fi or
0,0003916 makes
t( ^ n,n",869 5
6,3o536
2,53462
,37520
6,332 1 3 o,o5i58
2,72103
,32135 343
r = 3,00353
r" = 3,061-9
16,374
8q
r-\-r" = 6,06432
c = 0,23454
,323
9.ci5i8
9,3745 1 j o,o55oi
16,786
Col. 7.| Col. 8. 1 Col. 9.
Col. lO.j
Coellicienls of u. |
B
500
43,23445
46,4i4ii
3,16172
,36029
2,32071
38678
2,52201
1367
3,70749
1 354
3.534(»
3,73103
Coefficients of u-. \
I
400
1
IS
2
JO(J
80,07556
93,50610
1 ,00000
0,30019
667J
556
0,33376
7792
649
O,0035o
83
7
0,27248
272
o,3i8i7
3i8
o,oo34o
0,27520
o.33i35
o,oo343
813
[5994J
1255)
The Talue of T by observation, falls between the results of these two last hypotheses, and by taking
parts of the corresponding variations of the values of p, r, r", we get the final values corresponding to the
actual value of T ;
f = o,o5852 ; r = 3,ooi63 ; r" = 3,o6o8o.
This manner of finding the orbit of a comet has an imperfection, which obtains in several
other methods; namely, that it fails in accuracy in the particular case where the value of
M (30, or 9-2) appears under the form 3/ = g ; which happens when the apparent j^alh of
the comet is in the ecliptic, or in any other great circle passing through the sun. For in this
case, as the points A, B, C, figure 85, page 795, are situated in the same great circle, passing
through S, we shall have all three of the angles h, h', h" (64), equal to each other, and
then the expression (92) becomes M = %. Hence it is evident that this method can be
most successfully applied, in cases where the arc B H, is considerable, in comparison with
the arc ..3 C. When the ratio of these arcs, B H, j1 C, is small, there may be instances
in which the method, without actually failing, becomes somewhat uncertain, on account of
the inaccuracy in the estimated value o( M, in consequence of the neglected terms (93'),
which have a more important influence than usual, and it is an object of interest, to obtain
a more correct estimate of the value of M. We shall therefore proceed to investigate the
complete value, by the analytical methods, used by Gauss, Ivory, Encke, &c., without
neglecting any terms, and we shall obtain in (306, &:c.), the correction to be made to the
approximate value, which is given in (30). Finally we shall give, in (355, &c.), the process
to be used in the excepted case mentioned in (251).
(256)
Analytical
investiga-
tion of
Olbers's
method.
(357)
(258)
(259)
(260;
(Cei)
(2C2)
VOL. III.
204
814
[5994]
1263)
(264)
(265)
(366)
(2CC')
(267)
(268)
(269)
(270)
APPENDIX, BY THE TRANSLATOR ;
Using the same notation as in (100 — 104), we have, identically,
0 = [x'li" - x"y').x + {x"y - xy").x' + (x,/ - x'y).x".
For the Jirst term is balanced by the fovrth, the second, by the fifth, and the third by the
sixih ; so that the second member is identically equal to nothing. We shall now represent
the double of the area of any one of the plane triangles sah, sbc, sac, figure 84, page 792,
by including the corresponding radii in brackets ; so that we shall have,
[rr'] = 2. area of the triangle sab ; ['"''■"] = 2. area of the triangle sbc ;
[;•;•"] = 2. area of the triangle sac.
The plane of the comet's orbit being inclined to the ecliptic by the angle ç (21) ; it is
evident, by the principles of the orthographic projection, that the double of the projections
of the areas of these triangles, upon the plane of the ecliptic, will be obtained by multiplying
the expressions (266) by cos.?, so that we shall have,
[«•']. cos. 9= 2. projection oi sab ; [rV"]. cos. ç = 2. projection of sbc ;
[r-/'].cos.ip = 2.projection o(sac.
We shall represent the co-ordinates, of the projection of the point a, hy x, y ; those of the
point b, by x' , y' ; and tliose of the point c, by x" y" (100, Sic.) ; as in figure 89, where
a, |3, 7, represent respectively the projection of the points «, b, c, of figure 84, upon the
plane of the ecliptic. Now we evidently have,
area sa.p^
= i<
3, X o-aj = ix'y ; area (SttjBj = |^Pi X a^Pj = iy'.{x' — x) ;
(271)
(272)
(273)
area sjBp^ = J«(3, X |3/3, = ix'y' ;
subtracting the sum of the two first expressions from
the third, we evidently get the value of the triangle,
so.p = Ix'y' — ix'y — iy'. (x' — x) ;
and if we neglect the terms ix'y' — i^'y', which
mutually destroy each other, it becomes as in the first
of the expressions (273). If we change the accents
on xy, so as to correspond to the other triangles
s^y, so-y, we shall obtain their values, as in (27.3).
Triangle «ap = i.(xy'— x'y) ; triangle s^y = ^.{x'jj" — x" ij') ;
triangle so-y == ^.{xy" — x" y).
Substituting these in (268), we get the following system of equations depending on the
COMPUTATION OF THE ORBIT OF A COMET.
815
principle tiial tlie three observed places of the comet a, b, c, figure 84, are in the same
plane passing through the sun ; this plane being inclined to the ecliptic by the angle <p ;
xi/'— I'l/ = [rr'].cos.q, ; {r',j" — x",/) = [r'r"].cos.<p ; {nj" - ^''y) = [rr"].cos.(p.
Introducing these values into the equation (2(i3), and then dividing by cos.cp, we get
the equation (277). This equation must be satisfied, whatever be the position of the axis of
X ; and if we change this axis into that of i/, we shall find that the values x, x', x", will
become y,y',y", respectively, without altering ['■'/■"],[/•/•"],[(•/■']; hence we get (278).
In like manner, by changing the axis of x into that of z, we get (279).
0 = [r'r"] . X — [n-"] . x' + [rr'] . x" ;
0 = ['■''■"]. y -K']. y +[/T']. y ;
Q=[r'r"].z-irr"].z' + [rr'].z";
We may remark, that the whole number of accents on each of the terms of these equations, is
three; and this symvietry obtains in many other of the equations of this article. The
recollection of this eireumstance u'ill sometimes assist in distinguishing the symbols from each
other. If we substitute .^=: 180''-)-©, (10) in (108), we shall obtain, for the co-ordinates
X, y, z, at the first observation, tlie expressions (281), and, by accenting the letters, we get
the values corresponding to the other observations as in (282,283) ;
a; = p.cos.a — iî.cos.@ ; y = p.sin.u. — jR.sin.© ; z = p.tang.é ;
a;' ^ p'.cos.a' — jR'.cos.©'; / = p'.sin.a' — /î'.sin.©' : s' ^ p'.tang.â' ;
x"= p".cos.a"— 7î".cos.©" ; y" = p",sin.a"— R".sm.(^"; z" = p".tang.()".
Substituting these in (277 — 279), we obtain,
0 = [;■'/■"] . \ p.cos.a — i^.cos.© } — [rr"] . I o'.cos.a' — Tî'.cos.©'}
-I- [rr'] .{!>". COS. i)."—R".cos.Q"] ;
0 = [rV'J.f^sin.a — iî.sin.©! — [rr"] .{/.sin.a' — iî'.sin.©'i
+ [r/] . J/'.sin.a" — /?".sin.©"i ;
0 = [r'r"] . ^tang.a — [rr"] ./.tang.ô' -f- [rr'] . /'.tang.â".
If we divide (284,285,286), by any one of the areas [r'r"], [rr"], [rr'], we shall find,
that these three equations contain fve unknown quantities ; namely, the two ratios of the
areas, and the three radVi p, p', p" ; any two of which, may be eliminated. In doing this,
we may observe, that the equations (284, 285), are wholly independent of each other ; and
we may, in either of them, change at pleasure the direction of the axis of x. If we decrease
the angles in (284), by the quantity ©', we shall get (292) ; if we decrease the angles in
(285) by a, and then change the signs of all the terms, we shall get (293) ; lastly, if we
decrease the angles in (285), by ©', we shall get (294). The same results may also be
obtained by combining the equations (284, 285) by the usual methods ; thus, if we multiply
[5994]
(2730
(274)
(275)
(276)
(277)
(278)
(279)
(279')
(280)
(281)
(282)
(283)
(284)
(285)
(286)
(287)
(288)
(289)
816 APPENDIX, BY THE TRANSLATOR;
[5994]
(390) (284) by cos.©', and (285) by sin.©', then take tbe sum of (he products, reducing
them by [24], Int. we shall get (292). Again, multiplying (284), by sin.a', also (285) by
— cos. a', then adding the products, we get (293) by reduction, and using [22], Int. Lastly,
<29i) multiplying (285), by cos.©', and (284), by — sin.©', then adding tlie products, we
get (294). The equation (295) is the same as (286).
0^[r';-"].{p.cos.(a-©')-iî.cos.(©-©')|-[,T"].lp'.cos.(a'-©')_/î'|
+[rr'].5p".cos.(a"_©')-iî".cos.(©"-©')};
0= [r'r"] .{p.sin.(a'— a)+iî.sin.(© — a')^— [n-"].7?'.sin.(©'— a')
— [n-'].{p".sin.(a"— a')— iî".sin.(©"— a')|:
0 = [r'r"] .{p.sin.(a-©')+J?.sin.(©'-©)|-[n-"].p'.sin.(a'-©')
+[,r'l\ p".sin.(a"-©')-fi".sin.(©"-©') , .
(995) 0 := [r'r"] .p. tang. ^ — [rr"].p'.tang.â' -f [rr'].p". tang.t)".
Multiplying (294) by tang.ô', and (295) by — sin.(u.' — ©') ; then taking the sum of
the two products, we find that the terms multiplied by p' vanish, and we get,
(392)
(393)
(294)
(396)
(297)
0 = [;•'/■"] .^itang.â'.sin.(a — ©') — tang.â.sln.(a'— ©') j + [/r'I.iî.tang.ô'.sin. (©' — ©)
+ [rr'].p". \ tang.â'.sin.(a"— ©') — tang.ô".sin.(a.'— ©') | — [;T']./Î".tang.ô'.sin.(©"— ©').
Dividing by the coefficient of f", we finally obtain,
„ [rV'J |tang.ô'.sin.(a — ©')— tang.lsin.(a' — ©')}
'' "^ [77] ■ |tang.â".sin.(a'— ©') — tang.â'.sin.(a"-(^| ' ''
tang.^' I r>V'].7?.sin. (©' — ©) — [rr'].J^".sin.(©"— ©') \
"^ [rr] ' tang.â".sin.(a' — ©') — tang.()'.sin.(a"— ©')
In like manner, the plane triangles sn'li', sb'c', sac', figure 84, page 792, corresponding to the
earth's orbit, give by using a notation like that in (266),
(298) \RR''\ = 2.area of the triangle sab' ; \R'R"\ = 2.area of the triangle sh'c' ;
[IiR"'\ = 2. area of the triangle sa'd.
The area of any one of these triangles, as sa'I), is found by multiplying its base sa' = R,
by half the perpendicular let fall upon it from its vertex b', or by i R'.sin.a'sù' ; therefore,
this area is represented by ^RR'.sm.a'sb' ; and as the angle a'sb'= A' — A=0 — ©,
'■^^^ the area becomes i E/?'.sin.(©' — ©). Substituting this in the first expression (298),
we get the first of the equations (300) ; in like manner, the second and third of the formulas
(298), become like those in (300). Inexactly the same way, we get the expression [300'] ;
C299') observing, that the angle asb = v' — v ; the angle csb = v" — v' ; the angle asc ^v" — v ;
[RR] = RR'. sm.{Q' - Q) ; [R'R"] = R'R".sm.{Q" -Q') ;
^3»°) [RR"] = RR".sm.{&'—&) ;
(300) [rr'] = rr'.sin.(i-'— 1») ; [rr"] = r'r" .s\n.{;v" — v') ; \>r"] = rr".sm.{v' — v).
COMPUTATION OF THE ORBIT OF A COMET. 817
[5994]
The second of the equations (300), gives the first expression (301) ; multiplying its
numerator and denominator by R.sm.{(;?)' — ©), we get its second expression ; substituting
in its denominator the value, [Rli'] (300), we get the last of the formulas (301) ;
R" ,in (B"-m = ^^'^"] = L^'i^n ■7?sin.(C^'-G) _ [R'R"].R.s\n.{Q' -Q)
'^ '^ '^ R' liRf.s\n.{& — &) [iuf] ' (3"^
substituting this last expression, in the numerator of the second line of the second member
of (297), we get,
" — ['"''""] tang.â'.sin.(a — fy) — tangJ.sin.(a' — ©')
['•'•'J ' tang.d".sin.(a'— f^') — tang.ô'. sin.(a" — ©') "''
+ \ I^ _ [^'^"] ] 7?.tang.é'.sin.(g/ — Q) ''""''
I [r/J lRR']y tang.d".sin.(a'— ©')— tang.é'.tang.(a"— ©') "
Now putting for brevity,
^^^ tang.é^sin■( g — Q') — tang.é.sin. ( a' — €i') .
tang.r.sin.(a'— ©') — tang.â'.sin.(a"— ©-) ' (303)
_ tang.O'.sin. (©' — ©)
tang.a". sin.(a' — ©') — tang.O'. sin.(a" — ©') ' (304)
the preceding expression of p" (302), or M.p (29), becomes of the following form ; in
which nothing is neglected;
'—[^•-.■'+1[^] -[IS I •«=•«•
(305)
(30G)
Dividing this last expression by p, we get the correct value of M. If we suppose, as in
Olbers's hypothesis (53), that.
[//■"] [R'R"] t"—t'
[n'] [RR'] t'—t '
the term depending on M^ will vanish from (306), and we shall have, very nearly,
(307)
hence,
f f
I' If I "^ ' (308)
^ i"—f ^j ^ t"—t' tang.^'.sin.(a — ©')— tangJ.sin.(a'— ©')
t' — t' ^ ' ~ t' — t ' tang.a".sin.(a'— ©') — tang.é'.sin. (a"— ©') " (309)
This expression of M is the same as the approximate value, assumed by Dr. Olbers (30) ;
as is evident, by substituting in it the value of m (28), and making a slight reduction. To
estimate the value of the neglected terms in the value of M, we may proceed in the
VOL. III. 205
818 APPENDIX, BY THE TRANSLATOR;
[5994]
(310)
(311)
(312)
(313)
(314)
following manner. Taking the rectangular co-ordinates of the comet, in the plane of its
orhit, and representing them in the three observations, by x, y, x', y', x", y"; putting
|ji=l,or neglecting the mass of the comet, in comparison with that of the sun, as in
[760^'"], we obtain from [^61], by accenting the symbols, the following equations ,
d'^K.' , x' _ dY y'
"rf^ + /3 — 0 ; d? + ^2 = ^-
Now if we take, for the origin of the time t, tlie moment of the second observation, when
the co-ordinates are x', y' ; and suppose that at the end of the time t, these co-ordinates
become x", y", respectively ; we shall have by Taylor's or Maclaurin's theorem [G07a]
the expression (315). Substituting in this the value of fPx', and of its differentials, deduced
from the first of the equations (312), we shall get (316) ; which is easily reduced to the
form (317);
(317)
(315) x"=x'+^.^ + i. —.fi+i.---^ .t^+fcc.
, „.. , - , C f/x' 1 rfr' 3x' ) „
' -" ^ -'•* ^ I dt r 3 dt ?•'* 5
d\.' J X
= -l'-*-; + *V^-S + -| + l'l'-*7-+-l
In like manner, we can obtain the similar expression of y". The intervals of the times
(318) between the observations, namely, t' — i, i" — t', t" — t, are to be reduced to parts of the
radius, by multiplying them by k [5937(8)] ; and we shall, for brevity, express these products
by r, t', t" ; as in (319) ; observing that these symbols have the same symmetry as in
(318) (279^) namely, that the number of accents in each of the equations (319) is three. We
shall also use the abridged expressions (320—323).
(319) t"=^'.(/' — 0; 'r = k.{i" —t'); r' = l: {t" — t); r'=r+'r";
t"2 .r"3 dr'
(320) «^,= 1— i. -^— J ,— , . — +&C.;
(321) w„=r"— i.^ — hc.;
(323) W"= T — -^ . ^ + &C.
While the body moves from the second point b, to the third point c, figure 84, the time
increases from t' to «", the increment being t"—i', or r (319), expressed in parts of the
(324) radius. Substituting this for t in (317), we get the expression of x", (328), using the
symbols (322, 323) ; in like^ manner we get the similar expression of y" (329). If we
COMPUTATION OF THE ORBIT OF A COMET.: 819
[5994]
change, in this calculation, t" into t, the quantity t will change into — t" (319); by <''-=)
which means w' (322), changes into u\ (320), and w" (323) into — w^^ (321); making
these changes in x", y" (328, 329), we get x, y (326,327). Finally as the jilane
of the orbit is taken for the plane of projection (310), we shall have z = 0, z"= 0, as
in (32T',329').
It
dy'
X = w, • X' — «'„ --^ ; (=«6)
y == u', . Y — w,, . ~ ; (32-)
z = 0 ; (32-)
x" = w' . X' + rv". ~ ; (328)
at
II I I s 11 ^h' (329)
Z" = 0 (329')
Multiplying (326) by y', and (327), by — x', then taking the sum of the products,
we get the first expression (331). Again, multiplying (329) by x', and (328) by — y';
then taking the sum of the products, we get the first expression (332). Lastly, multiplying (330)
(326) by (329), also, (327) by (328), and subtracting the last product from the preceding,
we get the first expression (333). The second form of either of these expressions, is easily
deduced from the first, by the substitution of
(330)
s.' dy' — y' d x'
rf^ = ^a-{y - e^)=^p,
which is easily deduced from [366,596c], using (311), and [5985(5)].
(x'f/y' — y'd\')
xy' — x'y = w„ . -^ = w„ . \/p ; (33i)
U'dy' — y'dx')
x'y"— x"y' = 10" . ^ -^ ' = w". s/p ; (332)
(x'f/y'— y'(/x') (x'Jy' — y'dx'\
xy"_ x"y = w,Ao" . ' '-^ ■' + lo'.tv,, .i l—J. L = {iv,.io" + w'.w,,) . x/p. (333,
Now the expressions (320 — 323), give successively, by using t' = t -\- t", (319),
,.,.«/' = T-i.--i.r._-f&c.; w'.W„='r"-l— _i..".-+&c.; ,,,,,
u;ac"-{-w'.w„= t+t"— —3. |t3+3t2.t"+3t.t"2-[-t"3|+&:c.
"'" (335)
T'3
= ^'—i- ^+^''-' (336,
820 APPENDIX BY THE TRANSLATOR ;
[5994]
Substituting in the first members of (331 — 333), the following expressions, which are
deduced from (274), by putting <p ^ 0, as in (310).
(337) xy' — x'y = [rr'] ; x'y"— x"y' = [r'r"] ; xy" — x"y = [rr"] ;
and in their last members, the values (321,323,336), we get,
(338) [r/] = \r" — -X3.r"^—k.c.lyp = r".\l—~ . r"^ — ^A Yp ;
(339) [,V']=^T —-TTs' 'r^ + Sic. >.v/p = '^•jl ~7^ • 1-2+ &c.>.v/p ;
Dividing these expressions, the one by the other, we obtain,
(340)
(341) ^-^ = ^ J 1 _ _L . (^2 _ ^"3) + &c. ];
[rr] r" I 6r'3 ^ ^^ <, '
[r'r"]
b
[r
(343) Ï;:^ - r'r Qr' • "^ M i- ^C. j .
As these formulas may be used for any of the heavenly bodies, we shall obtain the expressions
(344—346), corresponding to the earth's orbit, by merely changing r, r', r", into R, R, R ,
respecti vely,
[RR"]
1
(344) ltL!±J = _ . M — -5-3 . (
.a
6R
T
'-) + &c. I ;
[RR
(345) <■
[RR]
^ = - . 5 1 — ^ • (^" - ^"-) + ^c. \;
[RR] r" I 6R3 5
i^E] -^.\ 1 î- . (r'^- r.) + &c. ? .
C3«) If the intervals between the three observations be equal, or t" = r, we shall have
t3 t"2 = 0, and then the expressions (341,344), will give, hij neglecting terms of the
jourth order in r, r', (333— 340), or of the ^Aùy/ order in the factors of —, (341,344);
(348) [^ [Rm^^^^:^ (319),
[rr] [RR] t" i'— < ^ ^
which agrees with the supposition of Dr. Olbers (307). Hence ive see the great advantage
^^^' of having the intervals of time between the observations equal to each other, in computing the
C03IPUTATI0N OF THE ORBIT OF A COMET.
821
orbit of a comet, ly this method; because it makes the factor of M„ R, (306), nearly
insensible; and gives a more accurate value of the expression M.p, than it would if the
intervals u-ere unequal. If observations cannot be obtained, in which the intervals r, t"
are equal to each other, we must select those which are nearly equal ; in order to diminish
as much as possible the effect of the factor t^ — t"-. If R'=r', the expressions
(341,344), become equal; hence it is evident, that if r' be nearly equal to /?', and
the intervals t, t" differ considerably ; it will be rather more accurate to compute
equal to it, than to put each of these
• , from the solar tables, and put f— -:
[RR] '^ [rr']
quantities equal to — (34S). Finally, we may observe, that after we have computed,
by a first approximation, the values of p, r, r", we may, by interpolation, find an
approximate value of r, by supposing the values to increase uniformly ; by which means
we shall have,
t' — t
r" = r -\ . (r" — r).
[5994]
Grout ad-
vantage of
having the
(319)
intervals of
time equal
between
the obser
vations.
(350)
(351)
(352)
(353)
With these we may obtain the corrected value of the function (341), to be substituted
in (306), to get a more accurate value of M; with which the calculation can be repeated,
in any extreme case, where it shall be found necessary.
In the case where the value of M (309), appears under the form of M = ^ , we
may deduce the value of p" = M.p from the equation (293), instead of (294, 295),
which are used in finding (297). Then as radius p' does not occur in (293), we shall
have,
[r'r"] sin. (a' — a)
'' ~ [rr'] ' sin. (a."— a') * **
+
[r'r"].R.sm.(f£)—a.')—[rr"].R'. sm.{0'—o.')+ [rr'].sm.(0"—a.').V.~
[rr']. sin. (a" — a')
If we divide the expression (341) by (344), we get, by a slight reduction, the expression
(357) ; in like manner, from (342, 345), we get (358) , lastly, from (343, 346), we obtain
(359). The equation (360), is evidently identical;
[r'
r"]
[rr']
[r
n
['
-r']
[;■
■r"]
['"
'r"]
[rr']
[RR]
^ '' \R'^ r'
+ &c.
+ &c. ] ;
'2 _ ^3) . ( J-
I
(334)
(355)
«56)
[rr']
VOL. III.
[RR']
(357)
(358)
(339)
(.3011)
206
822 APPENDIX, BY THE TRANSLATOR ;
[5994]
Taking, as in (269), the ecliptic for the plane of projection ; we shall represent the
(361) rectangular co-ordinates of the earth, by X, Y, at the first observation ; X', Y', at
the second observation ; X", Y" at the third observation ; hence the identical equations
in the earth's orbit, corresponding to (277,278), In the comet's orbit ; becomes,
(362) 0 = [R'R"] . X — [RR"] . X' + [RR'] . X" ;
(363) 0 = [R'R"] . Y— [RR"] . Y + [RR] . Y" .
If we take for the axis of X, the line whose longitude is lSO''+o-', we shall evidently have,
(364) Y = iî.sin.(© — a') ; Y' = R'. sin.(©'— a') ; Y" = R". sln-(©"— a ) .
Substituting these in the numerator of the second line of (356), it becomes,
(365) [>■'>■"] . Y — [n"] . Y + [rr'] . Y".
If we substitute, in this expression, the values of [»•';"], [rr'], [rr'], (357, 358, 360)
and neglect, for a moment, the terms depending on the factor ^ -j ^^ ^vill become,
(366, ^^ . I [R'R"] . Y - [RR"] . Y + [RR] . F' j ;
and as this vanishes, by means of the equation (363), it will be only necessary to retain the
(367) terms of (357, 358), which are multiplied by that factor — — —^. In the case now
under consideration, this factor is very small, because when the apparent motion of the
comet is in a great circle, we shall have r' = R' [780"] ; and if the intervals i' — t
i" — tf, or r", T, be nearly equal, we shall have ■f^ — t"^ = 0 ; and we may
therefore neglect the product of this quantity, by the preceding factor in (357) ; putting also
T '= 2t" in the factor r'^ — t"2 (35S), by which means we get -|.(t'2 — t"2)=:1t"2;
hence the term of (358), depending on this factor, becomes,
(368)
(369)
(370)
[RR
/]'^ \r'^ ?-'V ^ \r"> T'y ' \R'^ r'y'
(371)
(372)
[RR
nearly; as is evident by using only the first term of the second member of (345).
Substituting this in the numerator of the second line of (356,or365,&;c.), and putting in its
first line,
p^ =^ = '-P^ (341,369,319),
we finally obtain the following value of p", which can be used in the case now under
consideration, ivhcn the geocentric longitudes a, a', a", vary from each other much
more than the geocentric latitudes é , è', ô" ;
t" — t'' sin.(a' — a) , , iî'.sin.(a'— ©') /I 1 \
^ — t'—t • sin.(a"— a') ^ ^ sin.(a"-a') \R^ r ' J
COMPUTATION OF THE ORBIT OF A CO]\IET. 823
[5994]
We may obtain another form of the expression of p" by eliminating p' from (292,295) ;
this is done by multiplying (292) by — tangJ', and (295) by cos.(a' — © ), and
taking the sum of the products, by which means we get,
( — p.tang.â'.cos.(a. — ©') +iî.tang.ô'.cos.(0 — ©')
0 = [rr] . j ^ \ —[rr"].R'. tang.â'
' -j- p. tang. â. COS. (o
5-(©-©'))
(a'-©') S
( — p '. tang.é'.cos.(a"— ©') + R". tang.ô'.cos.( ©" — © ) )
+ [n-].j [
( 4- p"- tang.d".cos. (a' — ©') )
(373)
Dividing this by the coefficient of P , we obtain,
, [rV ] C tang.^'.cos.(a — ©') — tang.é.cos.(tt.' — ©') ,
p' , .
[rr'] ^tang.^'.cos.(a-' — ©') — tang.()'.cos.(a" — ©')i
(37-1)
_ ^ [>■;•]. JZ.tang.^'.cos.(© — ©') — [rr"]. R. tang.â' -fJVr'jJ?^ tang J'. cos.(©"— ©') \
( [rr']. \ tang.d". cos.(a' — ©') — tang.â'.cos.(a"— T©^)] ~ "^
The second line of this expression may be reduced, by a process similar to that in (364 &c.).
Taking for the axis of X the line whose longitude is 180''+©', we shall have
in like manner as in (364),
X = fi.cos.(©-©'), X'=./î'.cos.(©'-©')=i2'; X=i?".co3.(©"-©') ; (375)
and then the numerator of the expression in the second line of (374), becomes.
j [r'r"lX-[rr"\.X + [rf].X' [ . tang.ô'. ,376)
If we substitute in this, the parts of \r'r"], [rr"], [rr'], (357,358,360), which depend
on the first term of the second members, it becomes,
[W] • ^ ^^'^"^- ^ ~ [^«' ]-^' + [Ri?'].X" S ; (377,
which vanishes, by means of (362). Hence we obtain the same result as in (367); namely,
that it is only necessary to notice the terms depending on the factor — — — • and by <^'*>
supposing the intervals i' — t, t"— t' to be nearly equal, we shall find as in (370), that
the only part of this numerator, which it is necessary to notice, arises from that part of
-^^, which is denoted by -\- ^.rr'.f--^ — ^\ (370). Substituting this in the (379)
second line of (374), and puttiiig, in the first line, the value (371), we finally obtain the
following expression of p", which can he used, in this excepted case, when the geocentric (3790
latitudes ê,ê',è", vary from each other more than the geocentric longitudes a a' a"-
824
APPENDIX, BY THE TRANSLATOR;
[5994]
t" — t \ tang.«'.cos.(a — ©') — tang.â.cos.(a' — ©')
^ ~ V — t' ^tang.â".cos.(ciL' — ©')— tang.ô'.cos.(a" — ©')
(380)
^, , ^ iî'.tang.â'
' ^ * ^ tang.ô". cos.(a'— ©') — tang.ô'.cos.(a" — @')
(381)
(382)
(383)
(334)
(385)
(386)
(337)
(388)
(389)
(390)
(391)
(392)
(393)
(394)
(395)
For convenience in the calculations we have arranged the formulas (372, 380), as in the
table (387 — 392). If we neglect the term of p'' (372), depending on tt', and use
the symbol M' (387), it becomes p" = JkZ'.p, so that M' represents an approximate
value of M, (29). With this we may compute the equations (31 — 33), and from thence
deduce, as in (192), the approximate values ?', r", p. This value of p we shall represent
by (p) ; and from r, r", we may find the approximate value of r (353), to be
used in computing the term of the order "'-l^g — ~A-, vvhich occurs in (372).
) (319), in the second term of (372), and then dividing
the whole of the second member, by the expression of M' (387) ; we find that the quotient
becomes equal to F' (388) ; consequently, this expression of p", will become as in
(389). In like manner, by using the abridged values of M", F" (390,391), we find
that the expression of p" (380), becomes as in (392) ; (p) being as before, the value
of p, deduced from the first approximation, in which .
unity.
is supposed to be equal to
M' =
t" — t' sin. (a' — a,)
t' — t ' sin.(a" — ^0,') '
F' = l+ir'
sin.(a'— ©0
sin. (a' — a)
(p) ■ \R'' '
p" = M.p = M.F'. p ; or M=M'. F:
t"—t' tang.é'.cos.(a— ©')— tang.é.cos.(a'— ©')
"^^"^ fZTt ' tang.â".cos.(a'-©')— tang.â'.cos.(a"-©')
F"=l+h^
tang.ô'
tang.ô'. cos.(a-
p"=^.p=iM".F".p;
R
. e>')-tang.â.cos.(a'— 0) '(p)'
M=M". F".
1
m'
or
To Le used when
the longitudes
a, a', a",
vary faster than
the latitudes
To he used when
the longitudes
0,, a', 01",
vary slower than
the latitudes
6, 6', 6".
P"
If we compare the correct value of M=- (306), with its approximate values
(309, 387, 390), we shall find, that the first, or general form, is by far the most accurate ;
especially when the intervals of the observations are nearly equal, or r^— r"2 = 0;
since in this case, the value of M (309), is correct in terms of the second^ order, in
r r", inclusively (306,347, 8ic.) On the contrary, the values of M (389,392),
aie found by multiplying the assumed values M', M" (387, 390), by the factors F',F"
(388, 391), which contains terms of the second order in r, r" ; so that these expressions
COMPUTATION OF THE ORBIT OF A COMET.
825
[5994]
(395)
(H90;
(397)
of M may be considered as less accurate than that in (30) or (309), by at least, terms
of one order, in t, t'. Now from the mere inspection of the approximate values of M,
given in (309, 387, 390), it is evident, that when the apparent path of the comet is near the
ecliptic, and the latitudes è, 6', é" differ but little from each other, the expressions (309, 390),
will have very small numerators and denominators ; therefore the resulting value of M
or M' may be considerably affected by the imperfections of the observations ; but this
would not be the case with the expression (387), supposing the longitudes of the comet to
vary rapidly. On the other hand, when these longitudes vary slowly, the expression
sin. (a' — a), sin.(o.' — a'), are small ; consequently, the numerator and denominator of
(SSI), may be so small that the errors of the observations can have an important
influence on the resulting value of M'. Hence it follows, that when the expression (309)
becomes uncertain, on account of the smallness of its numerator and denominator, we can
use the expressions (387 — 3S9), if the longitudes of the comet vary more rapidly than
the latitudes ; or the expressions(390 — 392), if these longitudes vary slowly in comparison
with the latitudes. The method of using the formulas (387 — 392), is so similar to that in
the preceding examples (173, &c.), that it is unnecessary to give any examples for illustration.
We shall, therefore, close our remarks on this method, by observing, that after the approximate
values of the elements have been obtained, we may correct them by taking more distant
observations, as we have already observed in [820"', Sic, 849a, &c.].
Since the preceding article was prepared for this appendix, a new method of computin''
the orbit of a comet has been proposed by Mr Lubbock, and published in the fourth volume
of the Memoirs of the Astronomical Society of London, and in a separate pamphlet "On
the determination of the distance of a comet, &ic.;" in which he has reduced the question
to the solution of a quadratic equation. As we have not made any numerical computations
by this process ; v.e shall restrict ourselves to the explanation of the principles of the method,
with such illustrations as may be necessary.
If we suppose the intervals of time t' — /, t'' — t', between the observations to be
equal, we shall have r" = r (319), and by neglecting terms of the order ^^. we '^"''
shall have, as in (320—323),
Substituting these in (326 — 329'), we get, by taking the differences of the resulting
expressions.
f/x'
X X
Y" — \
dy'
rfz'
0.
(308)
(399)
(400)
(40 J)
(402)
(403)
LuI)bock*a
method of
computing
the orbit oi
a comet.
(404)
(400;
(^f ' '' ' (It ' cli mi)
The sum of the squares of these iliree equations, produres the first îiik! scccnd of the
VOL. III. 207
826 APPENDIX, BY THE TRANSLATOR ;
[5994]
following expressions of c^ ; from the second we easily deduce the third by means of the
(407) formula [572, line 5], putting (j. = I, as in (311) ;
(408)
The values of r^, r"^, may be deduced from ;'^ and its differentials, by Maclaurin's
theorem [G07«], in the same manner as we have obtained x" from x' in (315,&c.) ;
and we shall have,
(«9) r^ = r"2 — r . -V^ + J r2. ^^ — &;c.
dt dr
Subtracting 2r'^ from the sum of these values of r^, r"^ ; neglecting the terms depending
on T^, and the higher powers of -r, we get,
(411) ^2 _ 2/2 +,."2 = ^2.
dfi
The second member of this equation may be reduced, by means of [595], For if we put
(■»H'> for a moment r = r^, the expression [595], becomes, by supposing as in (407') ti-^l,
1 fh-2
412)) 2 r* . r — ^^-„ ;= h^.
a
Adf
Taking its differential, and dividing by di- , we get,
1 ddï
(412) r-* — 7-„ = 0 .
a 2dt^
Re-substituting the value of r, and making a slight transposition in the order of the terms,
we get,
cP. (,-2) 1 1
(413) !^ — : = •
2dt^ r n'
hence, the equation (411), becomes,
(414) ,2_2,.'2+r"2=2r2/l _ 1
\r a
Mr Lubbock's method is grounded on the two equations (408,414); by substituting the
values of c, r, r' , r", in terms of p', and assuming the following expressions of ,", p",
COMPUTATION OF THE ORBIT OF A COMET. 827
[5'J94]
p = ^1 • p' ; (-115)
p" = X^ , p' . (4)6)
Tlie values of X, , X„ , may be deduced from the equations (294,295), by the ehmination
of p". For if we multiply (294) by tang.d", also, (295) by — sin.(a"— ©'), and d")
take the sum of the products, we shall find that the terms depending on p" will vanish,
and we shall have,
0 = [rV"].p.{tang.â".sin.(cL — ©') — tang.().sin.(a"— ©')}+ [r'r"]./î.tang.ô". sin.(0— ©)
_[_[,-r"].p'.|_tang.â".sin.(a'— ©')+tang.ô'.sin.(a"— ©')}— [rr'].i?".tang.é".sin.(©"— ©').
Dividing by the co-efficient of p, we obtain (419). In like manner, if we multiply (294)
by tang.^, also (295) by — sin. (a — 0'), then take the sum of the products, and
divide by the co-efficient of p", we shall get (420) ;
[rr"] C tang.é'.sin.(a" — ©'
f— [r')-"] ■ ( tang.â.sin.(a"— ©';
— ©') — tang.é".sin.(a'— ©')
P
+
p" =
+
') — tang.d".sin.(a— ©')
tang.a" \ [r'r"].R.sm.{©' — ©) — [ r r'] .R". sin. (©"— ©') \
[r'r"] ' tang.â.sin.(a"— ©') — tang.d". sin. (a — ©')
[rr"J C tang.é.sin.(oL' — ©') — tang.ô'. sin. (a — ©')
5') — tang.ô'. sin. (a — ©') ) ,
^')— tang.()".sin.(a — ©') \ '^
[rr'J I tang.â . sin. (a" — ©']
tang.ô {[?y].R".sin.(©" — ©') — [rV'].i?.sin.(©' — ©)|
[rr] ' tang.ô.sin.(tt." — ©') — tang.é".sin.(a — ©')
Substituting in the last term of each of these expressions, the value of R". sin.(©" — @')
(301), we get,
_ [ rr'q C tang.<!'.sin.(a."— ©') — tang.é".sin.(a^— ©Q )
P == |- ,./,-'/] • I tang.â.sin.(a" — ©') — tang.â".sin.(a — ©') 5 ' ^
( [r'r"] _ [R'R"] ) [rrq i^.tang.y^sin.((2)' — ©)
'^ i Trr^ [^^'] > ['■''^'l' tang.lsin.(a" — ©')— tang.â".sin.(a — ©') '
[rr"] <; tang.a.sin. (a' — ©Q — tang.é^sin. (g — 0')
''"" [rr^ ' ( tang.ô.sin.(a"— ©') — tang.ô". sin. (a— ©') " ^'
-1
[r'r"] _ [R'R"] } ■R.tang.lsin.(©^ — ©)
Irr'] [RR'] ) * tang.é.sin. (a"— ©') — tang.ô". sin.faT^^^©^'
If we neglect those terms of the second members of these equations, which are multiplied
by the extremely small quantity -j— rr — [RR] ^'^*^^)' ^^^ ^'^^^^ '^a^^'
(418)
(419)
(420)
(421)
(422)
828 APPENDIX, BY THE TRANSLATOR ;
[5994]
^^^ ^ [rr"] ^ tang.é'.sin. (a^' - ©Q — tang.é". sin.(a' — ©Q ^
'' [r'r"] ■^tang.â.sin.(a"— ©') — tans.â".sin.(tt — ©')^'
(424)
(435)
(434)
p ;
[rr"] CtangAsln.(a' — ©') — tang.é'.sin.(a — ©') ^
[rr] ' 5 tang. Ô. sin. (a" — ©') — tang.â".sin.(a — ©') C '
Comparing these with (415,416), we get,
_[rr"] ( tang.ô'. sin. (a" — ©') — tang.ô". sin.(a — ©')
'""[77] ' I tang.ô.sin.(a"— ©') —tang.ô", sin. (a — ©')
(426) ^ ^lîZÎ] ^ tang.lsin.(a' — ©') — tang.ô'. sin.(a — ©') ) _
° [""'] * ( tang.ô.sin.(a" — @') — tang.ô".sin.(a — ©') ^ '
ïrr"] îr)-"1
in which we must substitute the value of tlie factors - — ^ and - — -. Now if we use
[)■)•'] [rr']
the abridged symbols A, y^, y^ (428,429,430), and suppose the intervals i! — t, t" — t'
to be equal, or, r' = 2r = 2r" (319) ; we shall find from (342, 343), that both these factors
become equal to 2A (431), and the values of \, \, p, /' (425,426,415,416) become,
as in (432,433);
(428) ^ = 1 — — 3 ;
(429) _2 \ tang.fl'.sin.(a" - ©') - tang.ô". sin.(a' - ©') ^ _
^' ■ \ tang.ô.sin.(a" — ©') — tang.ô". sin. (a — ©') 3 '
(430) y =2 \ tang.ô.sin.(a' — ©') — tang.ô'. sin.(a — ©') ^
' ^ tang.ô.sin.(a" — ©') — tang.ô". sin. (a — ©')>'
(431)
[n-"] [rr"]
[r'r"] ~" [rr'j"
' =2A;
1
(432)
\=A.y,;
whence.
p = '^■7i ■ p' ;
(433)
K = ^.7, ;
whence,
p"=.3.y,.p'.
Hence it appears that each of the values of \, \ (432,433) contains the unknown factor
A=l — ~- ; which is an inconvenience that Olbers's method does not suffer ; since his
value of M, deduced from p"=M.i> (29), by the substitution of p, p" (432,433),
does not contain this factor; for by using the value of p, p" (432,433), we have
P" >=
M = -=^ — . Substituting this last value of M, also, p=^.7, ./ (432), in (31 ,32)
we get (4.36,438). The expression of r'- (437), is similar to (31). The same values
<-^^^' of M, p, being substituted in (33), give the first expression of c^ (439), and the second
expression is the same as in (403). Lastly, substituting the values of r^, r'-, r"a
(436 — 438) in (414), we get (440) ; observing that terms of the order t^ are neglected
1
COMPUTATION OF THE ORBIT OF A COMET. 829
[59941
in the second member of (439, 440) ; but may be introduced, by noticing the terms of a
higher order, wliich are neglected in (406 k.c.) ;
r^ = R^ — 2.y. .l?.^.p'.cos.(© - a) + y,^A^ p'^.sec^.é ; («oj
r'^ = R'^ — 2.R'. p'. cos.(©' - a') + /a. sec^.d' ; («t)
r"2 = R"^ — 2.y, .R". A.p'.co5.{©" — a") + 7/. A^.p'-\ sec^ô" ; («e)
' ,-2 4- r"3 _ 2.RR". COS. (©" — ©) A
J + 1 2.y,.R".cos.{©"— a) + 2.y„.R.cos.{(^ —a") I Jl.p' V==4.ra. ^ 4 — ' ( ; P^P'^'^f'™ °'1 ''='°'
. + j^— 2.7,.7,.cos.(a"— a)— 2.7,.7„.tang.é.tang.â"|.^2_/a)
R^ — -2.R'^ + R"^ \
r-2.v,.R.cos.(©-a)+^./î'.cos.(©'-a')) / ,, ,,r -,
+ { '^ ) .A.f' \_g 2)' ^? I Expression Of I
( — 2.7v/î".cos. (©"-a") j ( ~ ~'^ 7 r~a \ ' U-2r'2+,-"2j
2
is'
+ {y?. sec^.d— --.sec2.4'+7,2_sec2.é"|.^a./a
(440)
(441)
(442)
Multiplying the equation (440) by — 4, and adding the product to (439) ; after substituting
the values of r^, r"^ (436, 438), we get the fundamental equation of Mr. Lubbock's
method,
In this equation, «3', B', C are functions of the given quantities R, R', R', ©, ©', ©",
a, a', ol", ê,6',è"; and of the unknown quantity A (428). If we put A=^l, in the first
operation, we shall obtain the approximate values of A', B", C ; and then putting
- = 0, to correspond to a parabolic orbit, we shall finally obtain the quadratic equation,
^' + 5'./+C'.p'2 = 0; (443)
for the determination of an approximate value of p', or A.f. With this value of p', we
may find an approximate value of r' by means of (437), and this is to be used in finding jl
(428). This last value of A must be substituted in (439,440), in order to get a more
accurate expression of the equation (441, or 443) ; and thence a corrected value of .3.p'. '^^"'^
The same process is to be repeated till the true value of Ap is found ; and ihen from
(436 &1C.) we get r, r', r",k,c. What we have said, will serve to explain the principle of
this method, which is illustrated by examples, in the works of Mr. Lubbock, mentioned '■'^*^'>
at the commencement of this article.
VOL. III. 208
830 APPENDIX, BY THE TRANSLATOR ;
[5994]
(446)
(147)
(449)
(450)
(451)
(402)
If we compare these two methods together, we shall see that the peculiar advantage of
Mr. Lubbock's method is, that the determination of p' is reduced to the solution of a
quadratic equation (443) ; but the accuracy of this equation, is considerably impaired, in
the first operation, by putting A= I (442) ; and this defect can be remedied only by
successive operations, with repeated solutions of the quadratic equations after correcting the
(448) coefficients, which increases the labor considerably, and sometimes alters very essentially
the coefficients of the equations, so that it changes materially the successively approximating
values of p'. This is evident by the inspection of the coefficient of Jî-p', in the second
and third lines of the first member of (440) ; where we see that when the interval of time
is small, the term which is to be divided by A is nearly equal to the sum of the other two
terms of this coefficient, and has a different sign ; so that the resulting coefficient, arising from
the difference of these expressions, is frequently so small as to be materially afl"ected by the
divisor A, which affects the largest term of this coefficient. Similar rermaks may be made
relative to the three terms of the coefficient of A^. p'~, In the fourth line of the first member
of the equation (440). Moreover the intervals between the observations are required to be
equal in the equation (414) ; and the peculiar form of the second member of this equation is
founded upon this circumstance ; so that this method could not be applied, without some
(453) modification, when the intervals are unequal. Neither of these objections apply to the method
of Dr. Olbers, because the fundamental equations (31,32,33), contain only the known
coefficients of f>, p^, and the equations may be used whether the intervals be equal or
'''^''* unequal; the equal intervals being however the best. Finally, in consequence of introducing
the three radii r, r', /', into the equation (414), we are under the necessity of computing
the coefficient of the equation (437), in Mr. Lubbock's method, as well as the value of A,
neither of which are wanted in Dr. Olbers's method, or in the similar method of Mr. Ivory.
Thus, we see, that these methods, which are the best now known by astronomers, have each
(457) their peculiar advantages and disadvantages. They are short and simple in their application ;
taking into view the difficulties of the problem ; and, by either of them, an astronomer can
obtain the elements of the orbit, in a few hours. Instead of being employed several days, or
'*^^' weeks, as in the early calculations of the orbits of comets.
[5995] METHOD OF COMPUTING THE ELEMENTS OF THE ORBIT OF ANY HEAVENLY BODY; THERE BEING GIVEN
THE TWO RADII rtVi, THE INCLUDED ANGLE v'—v = 2f, AND THE TIME t'—t
OF DESCRIBING THE ANGLE 2/.
This is a very Important problem, in the computation of the elements of the orbits of the
planetary bodies ; and the method of Gauss, which we shall give in [5999] depends
(1) essentially upon it. He has given two different solutions ; the one by the process of
quadratures; the other, by developing the quantities in series, and reducing them to tables,
2) as in Tables VIII, IX, X. We shall restrict ourselves to this last method ; which has
different forms in the ellipsis, parabola, and hyperbola; and it is therefore necessary to
^'' consider each of them separately.
(455)
(456)
ELLIPTIC ORBIT COMPUTED FROM r, /, v' — v,t' — t.
TO FIND THE ELEMENTS OF AN ELLIPTICAL ORBIT.
In the first place we shall suppose the orbit to be elliptical and shall use the following
symbols (6 — 16) which are similar to those in [5985]. For convenience of reference we
shall also insert in the table (17 — 67), most of the formulas which are used in this method;
and shall afterwards give the demonstrations in (68&C.);
r, r' the radii vectores ;
t', v' the mean anomalies ;
u, u' the excentric anomalies ;
the semi-parameter jJ = «.(1 — e^) = a.cos^.p = b.cos.cp ;
a = the mean distance ; that of the sun from the earth being unity ;
6 = the semi-conjugate axis = a.\/i—c^ = a.cos.<p = = \/(ip [5985 (5),.378??2] ;
e = the excentricity = sin.ip ; \/i—e^ = cos.(p ;
-2f^v'-v; v = F-f;
2F==v'+v; v'=F+f;
2g=u~u; u = G—g;
2G = u'-\-u; u'=G-'s-g.
l.sin.g = s'm f.\/Tr' ;
b.sin.G = sin.F.y/î'y;
sin.y.sin.G = sin.o-.sin.F
p.cos.^ = (cos./-|-e.cos.F).\/7T';
p.cos.G^ |cos.i^-|" e.cosf\.^rr' ;
cos f.\/rr' = {cos.^ — e.cos.G|.a ;
cos.F.^r7= jcos.G — e.cos.^}.a;
: tang. (45"' — w);
r' — r ^ 2afi.sin.o-.sin. G
4.tang.2w
• V^y ;
cos.2w
r" -\-r = 2a — 2ae.cos.^.cos. G = 2a.sin^.g- -f- S.cos./.cos.^.^/jy .
^ (2 + 4.tang^ 2^') .^= 2.cos./ .(1+21).^^';
v/:^+v/p_,
2.C0S/
+ 21;
\/'-+ \/'-, = (2 + 4.tang^.2«;) ;
/;• 4.tang.2?c
V r cos.2k'
. %\x?.\ f tan2,-.2i<'
cos.y COS. y
2.(/+ sin2.iir).cos./:v/;7
« = ■ ^"i '-^^■,
s\a. .g
r Assumed I
value of Mj.J
[AssumecJ "I
value of /.J
831
[5995]
(*)
(5)
(C)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
When
COS./ is
positive.
(30)
(31)
(35)
832
APPENDIX BY THE TRANSLATOR ;
[5995] ,_
^\2.{l + smKls:).cos.fVrr'\ ,
(33) i/a = ± -. >
kt
,,,, — = u' — e.sin.w' — u-\- e.sm.u ;
(o^J 3
C Upper sign, if sin.ff be positive."!
Lower sign, if sin.^ be neg;itive. J
: 2g — Se.sin.^.cos.G ;
: 2g — s\n.2g + 2.cos./.sin.^
kt
s/rr'
(35)
(36)
(37) m =
2icos.y.(rr')*
(38) log.m^ = 5,5680729 + 2.1og.« - .S.log.cos/— §.log.(7T') ;
^ , ,7 , ^-^.^x? /2^ — sin.2^
[Assumed "I
value of ÏH. J
(39) ±m=(Z + sin^|^f +(Z + sin^|^)
sin.^^
(40) m
(41) a; = s\n.^g = J.( I — cos.^) = J versed sin.^ ;
2g — sin.2^
(42) X== ■
CLTpper sign, if sin.^ be positive. "I
Lower sign, if sin.^" be negative. J
[Used when sine g and "!
cosine / are positive. J
[Assumed "I
value of I.J
Assumed
sin.^jg'
(43) I = a; — f -t- 7-^ =
(44) y — I J^
(45) A =
(46) h ^
■m sin.^g — ^.(2g--sin2g).(l— |.sin.%)
9X
ft.(2^— sin.2^)
m
-Ma;— I) v/i + x'
}»•-=
i + Hl'
(47) X = -5 / .
yi
\/r+l/i^
(48)
When
COS./ is
negative.
(49)
2. COS. y
= 1— 2L;
_ sin. 2*/ tang.22ît)
(50) -'" 3
COS./"
kt
COS./ '
2^. (— cos./)*.(rr')*
— 2.(1, — sin.%).cos./:i/;y
(51) a = -
sin.^^
(52)
± M=-(i^sin.%f +(^sin.2ig)
„ ,H- / 2p- — sin.2p-
sin.^^
[Assumed T
value of X.J
C Assumed "I
value of à. J
[Assumed "1
value of y. J
C Assumed "1
value oi /i.J
[Assumed l
value of L.\
[Assumed "1
value of J/. J
ELLIPTIC ORBIT COMPUTED FROM r, /, v' — v, f — t.
M=-iL -,r )*+ ,!^JSL. = "^VL^^ ;
r = - 1 +
4 i5-('» 1)
L — x M
i-?û-(*-|) K^L-x'
H =
H =
M2
(Y+l).Y^
T ^^'
X = Li ;
m^ cos./i/;y
!/- sia-'.sr
a =-2.
3P CQS.fYrr' .
sin-.^-
/)/.rr'.sin.2/"\^
27 =
Y.rr'.s\n.2f\^
log./.- = 8,2355814 .... [5987 (8)] ;
log.fc in seconds = 3,55000657 .... [5987 (14)] ;
with a,p, we get cos.p = \/l — ea= \// -; (H);
^ co3.^ \/7P.cos.f v/;7
COS. u = • = cos.fi".cosec.(p cos.f.cosec.ffl ;
e ae '^ ^ a •' -
. sin./sin.G .
sin.!* :=: -. = sin./.sin.Cr.cosec.e' ;
sin.o-
o
mean daily motion ^ ka ''; or,
log. mean daily motion in seconds = 3,55000657 — f log. a.
Other formulas of a similarnature may be deduced from these, particularly the expressions of
sin.(|/q=^o-); cos.(J/zfJ-); sin.QFqrJG); co%.{\F ^\G) ;
which may be conveniently used in logarithmic computations. In general, however, the use
of these auxiliary angles requires more labor than the common processes of spherical
trigonometry ; and the formulas we have given are all that are necessary. We shall now
proceed to tlie demonstration of these formulas (17 — 67).
If we select the last values of sin.iw, cos.iw [5935(12,13)], and then accent the symbols
r, », M, we shall get the corresponding values of sin.|w', cos.|m' ; substituting these in
the first member of (69), it becomes as in its second member ;
VOL. III. 209
833
[5995]
(53)
r Assumed "1
Lvalue of r.J
(54)
r Assumed 1
Lvalue of //.J
(55)
(50)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(6t)
(65)
(66)
(66')
(67)
(07)
(66)
834 APPENDIX, BY THE TRANSLATOR ;
[5995]
\ TT §
(69) sin4w'-cos.jM =Fcos.iM'-sln.iM= \ -^r; ^f ^' ^sin.iw'-cos.JrTcos.lD'.sm.J^K
(«''.(1 — e-)) ^
Multiplying this by h = «.(1 — (?)'" (H), and reducing, by means of [21,22] Int. we
get,
(TO) J.sin.J.(H' =F m) = (?•;•')*. sin.|.(i;' ^ v) ;
substituting the values (13 — 16) we get (17,18); the upper sign giving (17), the lower,
(18). Multiplying crosswise ihe two equations (17, 18), and dividing by h.\/^', we get
(19). In like manner, if we substitute the third values of [5985(12,13)], in the first member
of (71), we obtain its second form, and by connecting together the teims depending on e,
and reducing, by means of [23,24] Int. we get (72),
('1) jj.{cos.|m'. Cos.^M±sin.|M'.sin.jM| = {(l -\- e).ç,o=,.^vi' .co?,.\xi ^^{X — t).%\a.\v .•sm.\v\.\y'Çp
C (cos.|«'.cos.J« ± sin.it)'. sin. Jîj) ^
C -\-t.{co'S,.^X!' .CQ?,.\v:::ç.'^m.\v' ,%\x\.\v)' )
C'a) p.cos.(^M' q: ^m) = {cos.(jî)'q= J«) -|- e.cos.(J.w'± l'v)\. \ tt-'
Substituting (13 — 16), we find that the upper sign of this last expression gives (20), the
?ower (21). Multiplying (21) by — e, and adding the product to (20), we get,
(73) p.{cos.^ — c.cos.G} = \/r7.(l — e^).cos./;
substituting p=ia.(l — e^) (9), and dividing by 1 — c^ we get (22). In like manner,
if we multiply (20) by — c, and add the product to (21), we get,
(74) p. I cos. G — e.cos.^l =^,y". (1 — e2).cos.F;
substituting the same value of p, and dividing by 1 — e^ we obtain (23).
(75)
(76)
(77)
(78)
(79)
We have,in [5985(9)], j-=a.(l — e.cos.tj), r' = a.(l — c.cos.m') ; taking the sum,
and the difl^erence of these quantities, we get, by means of [27,28] Int.,
r — r= ae.^cos.tt — cos.?(' j = 2ae.sin.^.(2/4"")'S'n'2-("' — ") '■>
r' -|- r=2a — ae.^cos.?t'-j- cos.î<}=2a — 2ae.cos.^.(u' -\-u).cos.i.{u' — u) ;
substituting the values (15,16), we obtain the first forms of the values of r' — r, r'-{-r
(25,26). The second expression (26), is deduced from the first, by changing the term 2ft
into 2rt.(sin^.^-l-cos^.^), by which means we obtain,
r'-\-r= 2a.sin^.^ 4-^cos.^ — e.cos.G j .«.2. cos. g=^ 2fif.sin2.^-|-^cos./.;/^| .2.cos.^ (22).
These admit of further reductions, by the introduction of the symbol w (24) ; and
if we put for a moment 45''-[- w = w, we shall have \ /^ '- ^ tang-, w ; substituting
this in the first member of (SO), and successively reducing, by means of [34', 32, 31] Int.,
we finally get the expression (81), which is the same as (29),
ELLIPTIC ORBIT COMPUTED FROM r, r', v' — v, t' — t. 835
[5995]
\ /^^-f \/-= tang^.w -f cotan^w = 2 + {tang.w — cotan.wp
Csin.w cos.w)^ , Csin^.w — cos^.w) ^, ( — cos.2vv)2
= 2+< ; [ =2+]—^ [=2+K • o (
^cos.w sin.w) ( sin.w.cos.w ) ( t.sin.aw )
= 2+{— 2.cotan.2w|2 = 2+4.tang^2w.
(80)
(81)
Multiplying this last expression by \/'n', we obtain tlie first value in (27); finally, if we
multiply the assumed value of 1+2/ (28), by 2.v/r7. cos./, we shall get the second
expression in (27) ; and, we may incidentally observe, that the comparison of (28) with (81) (Si)
evidently shows that I is positive. The same expression (79), gives.
/ yr sin-.w cos^.w
-, = tan2:-.w — cot-.w = — ~^—i — ■
(82)
r \/ r cos-.w sin'.w
the numerator of this expression is easily reduced to the form,
(sin^.w -(-cos^.w).(sin-.w — cos^.w) = sin^.w — cos^.w = — cos.2w = sin.2w; (gg.j
and the denominator is,
(sin.w.cos.w)^ = (^.sin.2w)^ = (j.cos.2w)^ = J.cos^.2«« ; (82,,
hence we easily deduce the expression (30). Multiplying this by \/r7, we obtain the
second form of (25). From the assumed value of 1 +2/ (28), we get, by substituting
(81), the first expression of I (83) ; reducing by means of [1] Int., we get the last form
in (83), which is the same as (31), and is composed of the given quantities /, w,
2+4.tang^.2w 1 — cos/ tang^.2i« sin^.j/ tang^.2(f
4.CO3/ ^ 2.C0s/ ' cos./ cos/ COS./ (^3)
Transposing the last term of the second expression (26), and dividing by 2.sin^.^, we
get successively, by using the last of the formulas (27) ;
r-}-/ — 2. COS./. COS. fC-\/rP 2.cos./.(l +2Z).\/^ — 2.cos./.cos.^.\/r7
" ^ 2.sin2.^ "^ 2.sin2.^ <^^'
_ J2?+l— cos.g}.2.cos.//,~ I 2?4-2.sin^.ig } .^.cos.f.s/^'
'H.sm^.g ~ a.sin^.^- " (85)
This last expression is easily reduced to the form (32) ; and its square root is as in (33) ;
to which the double sign ± is prefixed, so that ^' + sm'J.lg yl + x u\\ JJ^f^y
siu.g sin.g- » ■" •?
be considered as a positive quantity.
(85')
Substituting n [5987(12)] in [5985(7)], and neglecting the mass m, on account of its
smallness, we get the first formula (87) ; the second is deduced from the first, by accenting (sg)
836 APPENDIX, BY THE TRANSLATOR ;
[5995]
ht . kl'
3
(87) — = M — e.sin.M ; — = ii — e.sin.M'.
(88)
(89)
Subtracting the fii-st of these expressions from the second, and for t' — i, which represents
the interval of time between the observations, putting simply t, we get the expression (34).
This is easily reduced to the form (35) by substituting u' — ii^2g (15), and,
sin.?*' — sin.M = 2.sin.(2-M' — ^u).CQS.[iu' -\- iu) = S.sin.^.cos. fr (15, 16) ;
but from (22), we have,
„ ^ yrr'
(90) e.cos. Cr = cos.^ — cos./ . ;
substituting this in (35), and putting 2.sin.^. cos.^ = sin.2^, it becomes as in (36). The
symbol m (37), is used for brevity, and when cos./ is positive, the expression of m,
(90') will be a real and positive quantity ; being a function of the given quantities r, r f, t, k;
and its equivalent logarithmic expression is given in (38) ; using the value of log.fc
[5987(8)]. Multiplying (37) by the denominator of its second member, we get,
3 3 ?.
(91) kt = m.2 - . cos. ^/. ( rr') * ;
substituting this in (36), and then multiplying by a^, we obtain,
3 3 3 3 X i
(92) m.2"^\ cos.V- {rr'f — (2o' — sm.'2g).cfl + 2.cos./. sin.^. ( rr'y.a^.
Using the value of i/a (33), we find that each term of the expression contains the factor
3. 3
cos.^/.(rr')*, and by rejecting it, we get,
(93) m.22 = -i- (2^ — sin.2g) . ^ '^ . ^ "'^' ' ± 2. j 2.(; + sin^.iç) | ^ ;
sin .g
3
dividing this by ±2^, we get the expression of ± m (39) ; the order of the terms
being changed. This equation contains the known quantities I, m ; and from it we may
determine the unknoion quantity g. In the case which most frequently occurs, g is so
y p» ^-^ sin 2 o*
(94) small that the common tables of logarithms do not give the factor . g ' °= X (42),
with a sufficient degree of accuracy. In this case, we must develop it, in a series, ascending
according to the powers of sin.-Jg' ; and then the value of the factor, which is represented
(95) by the assumed symbol X, can be obtained with accuracy, in the following manner.
Changing y into sin.|^^ in [4G] Int. we get the value of the arc ^g, in terms of
(96) sin.lg- ; multiplying this by 4, we get the expression of 2g (98). Moreover,
(97) sin. 2^ = 2.sin.^.cos.^ ; sin.^ = 2.s'm.ig.cos.ig ; cos.g =1 — 2.sm.'^ig ; hence,
ELLIPTIC ORBIT COMPUTED FROM r, r', v'—v, i—t. 837
r5995 1
sin.2^ = 4.sin4^.(l — 2. sin.^ |^).cos.^g- ; ^^^.^
and since,
cos%= (1 — sin.2|o-)i = 1 _ l.sin.21^ — |.sin.''i^ — &c.,
we find, that sin. 2^ becomes as in (99) ; subtracting this from (98), we get 2g — sin. 2^
(100), being tlie numerator of the value of X (94),
2g = 4.sin.è^ + î.sm?^g + i'o.sin.^^^ + &c. ; (»8)
sin.2^ = 4.sin.^^ — lO.sin.^io- -}- i-sm.^g — &c. ; ''^'
2§- — sin.2^ = f .sin.3^^ — f§.siu.% — 8ic. = f .sin.'Jg .{ 1 — ?^-^m?ig — &c.|. (loo)
The denominator of X (94) is,
sin.'^ = (2.sin.^^.cos. Jo-)^ = S.sin.^^^ .\l — ^sm.^^g — Sic { ; (loi)
dividing the expression of the numerator (100), by that of the denominator (101), we get,
X= ê-M+f-sin.^i5-+ &ic.p (102)
expressed in a series ascending according to the powers of sin.^^^ = a; (41). To obtain
the law of this series, we shall resume the expression of X (94), which gives,
X. sin.^^ = 2^'- — sin. 2^. (io3)
Taking its differential, and dividing by ilg , we obtain,
— .sin.'o- -{- SX.sin.'^o-.cos.^ = 2 — 2.cos.2g=: 4.sin.^^. (lo-i)
The differential of x = sin.^^^ (41), gives,
dx = dg .%m,^g COS. ^g = \dg.5ia. g , or dg = -. — ; (ws)
substituting this in (104), and dividing by ^.sin.^g, we obtain,
dX _ & — G X. cos.^ (105')
dx sin.*^
but, from (41), we get,
cos.g-= 1 — 2x; sin.^^- = 1 — cos.^^ = I — (1 — 2a?)^:=4a; — 4x^; ^loe)
substituting these in (IDS'), and multiplying by 2x — 2xx, we finally obtain,
(I— x).2x.~ = 4— 3.(1— 2a;). X.
Now if we assume for X, an expression of the form (108), c, , f, , &ic., being
constant; we shall find, that its differential, divided by dx, will become as in (109).
Substituting these in (107), we get (HO) ;
VOL. 111. 210
(107)
838 APPENDIX, BY THE TRANSLATOR ;
[5995]
(10
(109)
(110)
^^=^.\c,-\-2c,.x+3c,.x"-+4c,.x^+^c.\;
= (8 — 4c,)..x + {8c,—4c,).x'^+ {Sc, — 4c,).x''+ &tc.
Putting the coefficients of the different powers of a; equal to nothing, we get, successively,
(111) c, = f; r,= f.c,; c^^'^.f^; c^ = ^.c,k.c.;
the law of continuation being manifest ; substituting these in (108), we finally obtain,
4.6 4.6.8 „ 4.6.8.10 ,, 4.6.8.10.12
(112) X = 4 + .X + ■ • . x^4- ■ . x^-\ . x'^ 4- Sic.
""^3.5 ^3.5.7 ^3.5.7.9 ^3.5.7.9.11 ^
This value of X may be computed by means of a table, with the argument x; but it is
much more convenient to find and use the small quantity | (43), of the order x^ (115);
or of the fourth order in g, instead of X (112), which contains terms of the order
X. If we divide the fraction 'I , by the expression of X (112), we shall get,
(113)
(114) •— . = f — a: + |3.a;2 + ,%2j.a;3 4- Sic.;
9 X
substituting this in the assumed form of | (43), namely, | = x — t ~t" tT^ 5 we get,
(115) | = #5.a;2_j-A2__3;3_j_gjc.
With this formula we may compute the values of | , as in table IX, for the small values
of X, when the usual tables would not be sufficiently accurate. The numbers in this
(116) table are given for the values of x, from x = 0,001, to a; == 0,300. This last
(iiT) value corresponds to ^=66'' 25'"; and for greater values, if any should occur in
practice, we may use the indirect method of solving the equation (39), in its present
form without making any reduction ; assuming a value of g, and repeating the process,
till we obtain an expression which will satisfy that equation. From the first expression of
I (43), we easily deduce the second value of X (42). Finally, if we substitute the
assumed value of X (94), in the first value of | (43), it becomes successively, by
using « (41),
sin. 2^
(118)
(119)
^'-i + i"- ^ -^Z = si"-' J5— * +
'''°^ ^~ ^'^^■2g-sm.2g- '^ « ' ,|.(2^-sin.2g)
and this last expression is easily reduced to the second form in (43).
(121)
In the case now under consideration, sin.^ is positive ; so that we must use the upper
sign of the value of m (39) ; and by substituting sin.^^^ = a? (41); also the second
ELLIPTIC ORBIT COMPUTED FROM r, r', v'—v, t'—t. 839
[5995]
value of X (42), it becomes as in the first expression of m (40) ; the second form
is deduced from the first, by the substitution of tlie first assumed vahie of y (44). The *'"'
second form y (44), is easily deduced from the second expression of m (40). Squaring
this, we get,
/ + 1- = — ; whence, a; = - — Z as in (47) ; (,23,
and if we use the assumed value of h (45), which gives,
f + ^ + I = -7- ; <i23)
we shall get successively,
4
- ,V(^- 1) = ,V(| - ^ + 1) = ,« . (1 + ? + 1-'^) = ,? .^^'_^J
(124)
(125)
_ 9m^ (î/2 ^
"" 10^2 • ^ X — 1 ^ •
Substituting this, and / + x (123), in the first expression of y (44), we obtain,
y=l+-y^; or, (y_l).?^_(y_])^V^
whence we easily deduce the expression of k (46).
When the heliocentric motion is between ISC' and 360'' ; or generally when cos./ is
negative, the value of m deduced from (37) becomes imaginary, and I (31) is negative.
To avoid this we must change,
I into — L; m into — I\l.\/ — 1 or M.{ — 1)- ; y into — 1, and h into H;
by this means, we find that (28) changes into (43) ; (31) into (49) ; (37) into (50), after
dividing by ( — 1)^; (32) into (51) ; (39) into (52), after dividing by ( — I )'^ ; (40)
into (53), divided in the same manner ; (44) into (54), after dividing by — 1 ; (45) into
(55), changing the signs of the numerator and denominator ; (46) into (56), with the same
changes of the signs ; lastly, (47) into (57).
To determine the value of y, or rather of hg.yy, from the cubic equation (46), a
table was computed by Gauss, being the same as Table VIII, of the present collection. (120)
This table answers also for computing log. FF from H, as is evident from the
consideration, that if we change y into — F, yy changes into YY, the equation (46) for finding
y, changes into that in (56) for finding F, and log.yy changes into log. YY. This table
is calculated from A = 0, to h = 0,6. From 0 to 0,04 the intervals in the values of h are „.,,
taken equal to 0,0001, which do not require the use of second differences : and this is by
(120)
(127)
(128)
(134)
(135)
S40 APPENDIX, BY THE TRANSLATOR ;
[5995]
far the most important part of the table ; from 0,04 to 0,60 the intervals are 0,001, and then
(132) it is necessary to notice the second differences, if we wish to have the logarithms correct in
the last figure of the decimals. If /( exceed the limit of the table, we may obtain the
(133) solution of the cubic equation (46 or 56), by any indirect process, or by some one of the
well known methods of solution.
The values of /, ?n, h (31,37,45) are positive ; and as it is supposed in the equations
(49,50) that cos./ is negative, (126), we shall also have Zr and Jl/ positive. We have,
by [32] Int. — sin^.^= cos./— cos^.i/; substituting this in the value of L (49) it
, r , , cosS.l/ , tançr2.2j« , , ... , „ ,
becomes L, = I -\- -f- - — ^ — - ; and as each term is positive, we shall have
( — cos/) (—cos/) '
L]>1; therefore H (55) is also positive, | being small (1 15 &.c.) ; moreover as
(136) "*> v'i+i (90', 85') are positive, we shall have y =: —t=^ (44) positive; and for similar
(137) reasons Y^ . — (54) is positive. If we now trace the successive values of h,
Y L — X
while y decreases from co positive, to 0, we shall see, by the mere inspection of the
(138) second member of the formula (46), that h decreases with y ; becomes 0, when
y = 1 ; and is negative, when y falls between 1 and 0 ; so that there is always one
positive value of y, which exceeds 1, and will satisfy the equation (46), for any positive
value of /(, from h = 0, to h = œ. In like manner, by the inspection of the equation
(56), we find that while Y decreases from co positive to Y = ^, H will remain
'"^' positive ; and that it will become negative when Y falls between 0 and | ; so that we
(142) have always one positive value of Y, which exceeds i, and satisfies the equntion (56),
for all positive values of H. from II z= co, to its least limit. After tliis digression on the
nature of the roots of the equations (46,56), we shall now proceed to the explanation of the
'^■*^' manner in which these roots are obtained by approximation.
If I be known we shall have the value of h (45) or H (55) ; and then from the cubic
(145) equation (46 or 56) we can obtain y, or Y; and finally, from (47 or 57), the value of
(140) X. Now as I is a very small quantity of the fourth order in g (113), wc may at first
(147) neglect it in the values of h or jH (45 or 55), putting h= g-^r, , or 11= ' 5 .
With this value of h or II, we find, from Table VIII, the corresponding value ijf log. yy,
or log.YY; whence we obtain, from (47 or 57) the value of x, and with this we get,
in Table IX, the corresponding value of |. Having obtained |, we may repeat the
(]49) calculation, using (45 or 55), to obtain a corrected value of x ; and generally, one operation
will be sufficient to get the true result. Having found x, we get g from the equation
(41), T = sin^.50', or versed sine g=2x. Wc may here remark, that both of the angles
u' — u = 2g, and v' — v=2f, (13,15) fall between 0'' and 360''; or between the
(151) same multiples of 360'' ; consequently the angles g,f, fall between the same multiples
of 180".
(139)
(140)
ELLIPTIC ORBIT COMPUTED FROM ,-, / v'—v, t' — t. 841
. [5995]
Now considering g as a known quantity, loe shall •proceed in the investigation of the
formulas (58 — 61), for the determination of the elements of the orbit. We have, from the (152)
equations (-10,41) / -|- r = ? -(- sui". Jo- = — ; substituting this expression of /-j-sin^.|^,
in (32), we get the value of a (58). In hive manner, from (53,41), we have,
L-X=L- S.n^. k = yâ ; (154,
substituting this in (51), we get the value of a (59). Dividing the square of the equation
(17) by the expression of a (58), and rejecting the factor sln^. ^, which occurs in both
members of the equation, we get the first expression (155). Substituting the value of m^
(37), we get its second form ; and the third form is easily deduced from this, by using
2. sin./, cos,/ = sin. 2/;
(155)
J2 _ >/. s\n^. f(rr')i y=^.(n-')3.(2.sin./.cos/)2 _ C y.>T'.sin.2/ ) 2
a ~~ 2m9.cos./ ~" kH^ ~ ( kt \ '
52
now we have - =jj (H) ; hence we get the expression of p (60). In like manner,
fit ( 1 5(i )
by squaring the equation (17), then dividing by the expression of a (59), and substituting
JIP (50), we get (61). Now if a planet revolve about the sun, in a circular orbit, at the
distance a; the angular motion in the time t will be represented by nt =^ — [5987(12)1,
neglecting the mass of the planet, on account of its smallness. Multiplying this by ^a^,
we get the area of the circular sector ^.\/a.kt, described by the radius vector, in the time (isg)
t, in this circular orbit, whose mean distance, or semi-parameter is a. If we retain the
same mean distance, and suppose the orbit to be an ellipsis, whose semi-parameter is p (9), ''^^'
the area described by the radius vector, will be decreased in the ratio of the square roots of
the parameters of s/p to \/a [383"], and it will therefore become l-s/p.kt (158) ;
which may represent in figure 84, page 792, the area of the sector sab; included between
the radii Sa=^r,Sb = r', and the elliptic arc a?i. On the other hand, the area of the '"^''
triangle Snb, included between the radii .S'a =: r, Sb = r", and the chord ah
represented, in [5994(300')], by
(ICO)
(162)
(163)
^.[jr] = ir/. sin.(î;' — ■!;) = ^rr' . sin.2/ (13).
Dividing the area of the sector (160), by that of the triangle (163), we obtain the ratio of
these two areas as in the first of the following expressions ; and by comparing it with the
value of y, deduced from (60), or that of Y from (61); we find that they are equal C^^')
to each other, as in the third and fourth expressions (164) ;
area of the sector sah ^.\/p.kt
area of the triangle 506 ^r7-'.s\n.2f Ci64)
VOL. III. 211
842 APPENDIX, BY THE TRANSLATOR ;
[5995]
(165)
(165')
()66)
(168)
(172)
(173)
(174)
(175)
Hence it appears that y or Y represents the ratio of the area of the elliptical sector
sail, to that of the triangle sab. If we substitute,
m
V// + s,n2.àff = /r+T=- (41,40),
and X (42) in (39), we get the expression of m (168), corresponding to figure 84,
page 792 ; sin.^ being supposed positive. In like manner, if we substitute,
M
(167) \/i,— sin^.Ag- =: \/L — ,r = — (53),
in (52), we get the value of M (169), corresponding to sin.^ positive,
m = --{- —.X .
y y^ ■
(169) M = .X.
Y '^ y3
Now if we suppose the quantity ?«, which is proportional to the time t (37), to represent
the area of the sector sab ; the quantity — (164), will represent the area of the triangle
(170) ^
sab (164) ; and their difference, which is — ;• X (168), will therefore represent the area
of the segment, included betW'Cen the chord «6, and the elliptic arc ab. Similar remarks
"^'' may be made relative to M (169), observing that when the angle bsa exceeds ISO'',
we have the sector equal to the difference between the segments and the triangle. Hence
1 3 X
it is manifest that the quantities m, (Z -f xy, (I -\- x)" .—, in the equation (39 or 40) ;
1 ? X .
and the quantities M, {L — x)''^, (L — *)"'T^ in (52or 53), are respectively proportional
to the sector, the triangle, and the segment ; and these geometrical considerations serve very
much to illustrate this part of the calculation. We shall now show the use of these formulas,
by the following examples, given by Gauss.
EXAMPLE I.
Given, log.r = 0,1394892, log.r' = 0,3978794, «' — « = 5/= 224»', i = 206 ^^',80919; to find the
elements of the orbit a, p, e ; the true anomalies v, v' ; and the excentric anomalies u, u'. In this example,
the value of Y exceeds the limits of Table VIII ; we must, therefore, in this case, deduce V from the
original cubic equation (56), instead of using that table. We have computed G, in (i8i), by the formula (65) ;
we may also determine sin.G by (25); and we find, from these formulas, that sin. G and cos.G, are
positive, therefore G (182), falls in the first quadrant of the circle, [5990, (23, 24)]. In like manner, we have
computed sin. F (iS3) from (66) ; we may also compute cos.i^ from (23), and as both expressions are positive,
F must also fall in the first quadrant.
ELLIPTIC ORBIT COMPUTED FROM r, r', v' — v,i' — t.
To find X.
t' log. 0,3978794
r log. 0,1394892
^ tangl.(45''-|-!t') log. 0,2583902
45<q.„-_4g'']4'"43;-8 tang. 0,0645975
w! = 4''i4'"43',78
0,3978794
0,1394892
sum 0,5373686
half 0,2686843
2«j=8'' 29™ 27^,56
tang9.2i/j
cos.f
f= II2<f
è/= 56*
sin2. if
COS./
= 0,0594959
= 1,8347335
(ir)^log 0,8060529
ar.co. 9,1939471
tang. 9,i74o3i4
same 9,17403 14
ar.co.cos. 0,4264246»
log. 8,7744874»
ar.co.cos. 0,4264246»
sine 9,9185742
same 9,9185742
log. 0,2635730»
sum is L = 1,8942294
^ = 0.8333333
L — |- = 1,0608961
MM (176)
Approx. H
Hence from the cubic > , • , xr I TT'
equation (56), we get J APP"^"^="« ^ = '.591432
log. 0,0256728
log. 0,6724334
log. 0,6467606
VV log. 0,4035762
MJU log. 0,6724334
MM ^
— — = 1,8571935 log. 0,2688572
L = 1,8942294
Approximate x =: 0,0370359
Corresponding | = 0,0000801 in Table IX.
L — 4 = 1,0608961
L — ^ — f = 1,0608160 log. o,02564oi
MM log. 0,6724334
Corrected H log. 0,6467933
Hence we get from (56), corrected V = i,59i5ii
MM
= 1,8370008
YY ^
L = 1,8942294
Corrected x = 0,0372196
yy log. o,4o36i92
MM log. 0,6724334
log. 0,2688142
Corresponding | = 0,0000809 in Table IX.
L — |- ^ 1 ,0608961
L — |. — ^ = 1,06081 52 log. 0,0256397
MM log. 0,6734334
Corrected H log. 0,6467937
Hence we get from (56), corrected V = 1,5915124
VV log. o,4o362oo
MM log. 0,6724334
MM „, ,, .
. = 1,8570064 log. 0,2688134
L = 1,8942294
X = sin2.Jg = 0,03722 3o
To find MM. (5o)
constant log. 5,5680729
? = 2o6''''y', 80919 log. 2,3i55698
same 2,3i55698
arith. co. log. ( — cos./) x 3 1,2792738
f log. r /
X = sin2. j g
is
g
MM
arith. comp. 9,1939471
MM log. 0,6724334
To find a. (59)
11^07'» 2C«,3
22'' l4"» 525,6
VV
— COS./
a
To find p, and e = sin. j. (61, 64)
k
t
rr'
2/
-V
[/a
f, = 75<'23wo7",3
To find F, G, », »', u, u'. (65, 66)
g
cos. g. cosec. f = o,g565oi8
/
y-r
cosec. 0,0142840
cos. 9,9664018
log. 9,9806858
cosec. 0,0142840
ar. CO. log. 8,7442935
cos. 9,5735754»
log. 0,2686843»
u^G — g^= — 17'' 22™ 40*
u'=G-|-g= 27'' 07™ oG"
843
[5995]
(170)
(177)
log. 8,5708114
sin. 9,2854057
cosec. 0,4218017
same 0,4218017
log. 0,2688134
log. o,3oio3oo
log. 9,5735754
log. 0,2686843
log. 1,2557065 (l"8)
ar. CO. log. 1,7644186
ar. CO. log. 7,6844302
log. 0,5373686
sin. 9,84i77i3»
log. 0,2018100»
log. 0,0297987
log. 0,6278532
COS. 9,4019455 (179)
(180)
- '^'"'■'coa./. cosec
a
? = 0.0398875
log.
8,6008372
COS.
G = 0,9963893
log.
9,9984291
(181)
G
f
g
4''52mi3»
sin.
sin.
cosec.
sin.
8,9289080
9,9671659
0,4218017
(162)
F =
9.3178756
a83)
/=
v = F—f=-
v' = F-\-f=
112''
- lOOl*
124<i
G
g
= 4'' 52'» 1 3*
= 22'' 14" 53»
(184)
844
[5995]
APPENDIX, BY THE TRANSLATOR;
EXAMPLE II.
(185) Given log. r = 0,3307640, log. )•'= 0,3222239, »'— « = 2/= 7''34™ 53',73, t = ^i^^^^gSSgi ; to find
the elements of the orbit a, p, e = sin.? ; the true anomalies v, v' ; and the excentric anomalies u, u'.
(186) A considerable part of the calculation of this example, is given in the introduction to tables VIII, IX ; and
it is unneecessary to repeat it here ; we shall merely give some of the results of this part of the process ; namely,
(187)
i = o,ooii2o5685 ; log. — - = 7,27i5i33; log.™ = o,oo2i633 ;
w = — bm 275 ;
log.mS = 7,2736766 ; log.y/rr' = 0,3264940 ; x = sin2. ig = 0,0007480186.
With these we shall compute a by the formula (58) ; p from (60) ; «f or f from (64) ; G from (65) :
F from ((56) ; then v, v', u, u', from (i3— 16).
(188>
(189)
(190)
(191)
(192)
(193)
To find a.
^g I"* 34»' 02S,o3
g- y o8"> o4^",o6
2
/
a
log. 6,8739124
sin. 8,4369562
cosec. 1,2621764
same 1,2621764
log. 7,27i5i33
log.
COS.
o,3oio3oo
9,9990488
log. 0,3264940
log. 0,4224389
To find p, and e = sin. ?.
k
ar.co.log.
1,7644186
t
ar.co.log.
8,658884o
rri
log.
0,6529879
=sf
sm.
9,1203696
y
log.
log.
0,0010816
Vv
0,1977417
Va
log.
0,2112194
<» =
l4'i 12m 02^,0 COS.
9,9865223
cos.gf.cosec. ?
To find V , !)',«, u^
COS. 9,9993498
cosec. 0,6102727
4,0702635 log. 0,6096225
Vr.
\/r
a
f
cosec.
ar.co.log.
cos.
log.
0,6102727
9,5775611
9,9990488
o,3264g4on
.co3./.cosec.f=— 3,261 1940 log. 0,5133766»,
cos.G? = 0,8090695 log. 9,9079858
log.e = log.sin.j 9,8897273
G — 324<'oo"' i8«,4 sin.
/=. 3<'47™26»,865 sin.
g cosec.
F = 3 14'' 42" 54',95 sin.
/ = 3"* 47™ 26^86
V =F —f = 310'' 55m 28'
d' = ^"4-/= 3i8<'3o"'22'
G = 324'' 00"' i8«,4
g- = 3'' o8mo4«,i
u = G — g = 320'' 52» i4«
«' = G 4- g = 327'' 08» 23'.
9,769 i653„
8,8202909
1,2621764
9,85i(5326„
In this example, cos. G is positive (189) ; but sin. G (25) is negative, because r' — r is negative ; therefore
G must fall in the fourth quadrant [5990, (23, 24)]. Again, sin. F (190) is negative, and cos. F, deduced from
(22), is positive ; therefore F falls in the fourth quadrant.
These examples will suffice for illustrating the calculations in an elliptic orbit ; we shall now proceed to
explain the similar calculations in a parabolic orbit.
PARABOLIC ORBIT, COMPUTED FROM r, r', «' — « = 2/.
TO FIND THE ELEMENTS OF A PARABOLIC ORBIT, THERE BEING GIVEN r,r', u— r=2/.
In a parabolic orbit, we shall use the symbols (2 — 10), most of them being similar to those
in an ellipsis [5995(6, &ic.)]- We shall also insert in the same table (11 — 25), several
formulas which are useful in these calculations ; and shall afterwards give the demonstration
in (26—60).
r, r, the radii vectores ;
V, v , the mean anomalies ;
p=2D, the semi-parameter ; [5986(2)].
D = ^p, the perihelion distance ;
2f = v'~v; v = F—f;
2F=y'+v; v'=F-\-f;
r' = r.tang-.sr ;
cos.y = cos.yisin.2z ;
Cfc = 1 — f.sins.iy ; log.fe = 8,2355814 . . . [5987(8)] ;
^ = co3.(|F— i/) = cos.iy ;
|-, = cos.(iF+i/) = cos.iy';
845
[5996]
(1)
(2)
Symbols.
(3)
(4)
(5)
(6)
•(7)
(8)
(9)
(10)
Fonnulas
in a para-
bolic orbit.
(12)
\/r?
J-(r+r')
2rr'
= cos.i^-{- cos.f;
1 + cos.J'.cos./;
(13)
(14)
p =
2rr'. sin^./
r -f r' — 2.cos.f.^/r7
sm.z.sm.f\2
. sin.|y J
(15)
2.sm.f.cos. f.rr' , 4.sm^. f.(rr')^ l/2 , . . „ , ,
V/p
r
2.C03./
3p^
= 1+2/;
= CA; . < -^^— > . sin.iw.
( cos.z ) "^
fcf
m
2^.{cos.ff.{rr'Y
log.m2 = 5,5680729 -|- 2.1og.< - 3.1og.cos./— |.log.(rr') ;
VOL. III. 212
[Assumed "I
value of Z.J
[Assumed l
value of m- J
(16)
(16)
(17J
(18)
(19)
846 APPENDIX, BY THE TRANSLATOR ;
[5996]
(20)
sin2./^,T' _
^^ 2/.C0S./ '
(21)
,«=Z2_J_^p.
\/? + v^^_, ,
2.C0S./
M- ^'
(23)
^"3 s a '
2^. (-cos./f.(rr')*
[Assumed 1
value of L.J
[Assumed "I
value of Jl/.J
sin^./l/r/
(24) p= — ~Y^ — -;
— 2L.C0S.J
(25) M = —L^^%Û.
The formulas in the prececiing table are easily demonstrated in the following manner.
(26) Substituting D=lp (5), in the first expression of »• [5986(4)], we get '>'==- — — - ;
whence,
\^£ = cos4v = cos.{hF-if) (6);
and in like manner,
^|, = cos.|«' = cos.aF+i/-) (7);
these agree with (11,12). Multiplying the product of the two formulas (11,12), by 2, and
then reducing the second member, by means of [20] Int., we get (13). Taking the sum of
the squares of the two expressions (11,12), and reducing, by means of [6,27] Int., we get,
as in (14) ;
^:^^±^Ucos^(iF^/j+cos^.(|F+è/)=l+|.cos.(F-/)+|.cos.(i^4/)=l+cos.
Multiplying (13) by — cos/, and adding the product to (14), we eliminate cos.F,
and obtain,
(26')
(27)
(28)
]). (?• + r') — 2p. COS. f.\/rr'
2^y
(29) —^r:j ■ — = 1 — cosV=sin-./;
which is easily reduced to the first form (15). If we substitute, in this, the value of / (8),
we get the first of the following formulas, and by successive reductions, using y (9), we
finally reduce it to the second of the forms (15) ;
PARABOLIC ORBIT, COMPUTED FROM r, r', v' —v = 2/. 847
(29')
(30)
(31)
2.tanf.z.sm^.f ^.sin^.z.sm^.f 2.sm^.z.sm^.f
p ^ )• . — :^ r. '■ = r.
l+tang-.z — S.cos./.tang.z 1 — 2.cos./.sin.«.co3.2 1 — cosyisin.S^
â.sin^.z.sin-./" 2.sin-.s.sin-.f /sin. «.sin.
= r. ^ = r.— — r— ^ =r.[ '
1 — cos.ij 2.sm'~.iy \ sm.ly
Substituting D=lp (5) in [5986(6)], we get,
p ^
i! =— . {tang.|y + ^.tang». 'j;] ;
and by accenting the letters,
f =g.{tang.|«' + e.tang3.|t;'*. ^3,,,
Subtracting the first of these expressions from the second, and changing t' — t into t,
in conformity with the notation of this article, we shall get, by multiplying by k, the
expression (32). The second member is easily reduced into two factors, as in (33 or 34)»
3
kt = If. j (tang.iu' — tang.iu) -f- i.(tang3.i«' — tang^i?>) } Oa,
3^
= \'p'^ . \ tang.iu' — tang.iu M 1 + i-tang^.|u' + itang.iw'. tang.|« + |.tang^.|y \ (33)
3
= Ijj^.^tang.iy' — tang.U'^.p +tang.!w'. tang.iu + iV.(tang.|u' — tang.|«)^}.
Now we have,
sin.iy' siniy sin.|«'.cos.iv — cos.lu'.sin.iu sin.(àî)' — \v)
tang.|«' — tang.Ju = — —, ~ = - — ^— ^^ = — ^ ^-^
cos.il! cos.|u cos.^D .cos.iu cos.|j;.cos.|w
sin./
[599(5]
(34)
COS.iîj'.COS.jU
and the product of the expressions (11,12), gives £5-^^;= = cos.|u'. cos.jU ; hence the (ss)
preceding expression becomes,
, , 2.sin./.i/P7
tang.iu' — tang.iv = . ^3^,
By similar substitutions, we obtain,
. , , • cos.iîj'. cos.iî) 4- sin.|u'.sin.i« cos.(W — Iv) cos.f
I + tang.iy'. tang.iu = = ~ —=: ^^^ ~=^ ^
cos.èD.cos.|y cos.jW • cos.jW cos.iy .cos.jj; O')
2.C0S./ \/rr'
P
Substituting (36,37) in (34), we get,
{37')
848 APPENDIX, BY THE TRANSLATOR
[5996]
1 ( 2.C0S.f.[/rr'
(38) Kt = p^. Sin./ ^rr' ■ < '- 1 i
3
2.sin./.cos./.rr' 4.sin^/()T')^
2.sin./.v/;7\2
(39)
This last expression is the same as the first of the formulas (16). If we multiply the last
term of the second member of (39), by p, and divide it by the first value of jj (15)-
we get,
2.sin./'.cos./!rr' 2.sm.f.^^.ir-\-i-' — S.cos./.^/iy } 2.sm f.\/rp.\ r-\-j'-\- cos. f.\/^'\
Substituting in this last expression, the first value of i/p (15), we get the second expression
(16). These two forms of Gauss, are reduced to the form (16'), by Burckhardt, in the
*^'' following manner. Substituting the assumed value / ^ r.tang^.z (8), in the second
expression (16), we get (42) ; and by successive reductions, using the symbols z, y, C
(8,9,10), we finally obtain the expression (43), which is the same as (16'),
i/2 ^
(42) kt = ^—-- r^.{l -\- tang^.2: -j- cos.f.tang.z] ■ f 1 + tang^.5: — 2.cos./.tang.z}^
o
l/2 ^
= - — r^sec.'z.{l +cos./.sin.2;.cos.2;].| 1 — 2,cos./.sin.z.cos.z]*
i/2 '
= î— -. r^. sec.^z.U + J.cos./.sin.2x|.|l — cos./.sin.25:|*
=^. A sec.^z4l+l.cos.yl.\l—cos.yf=^.Asec.^z.\l+i.(l-2.smUy)]4^-sirr'.lyl'^
_V/2
1 3
^.r^.sec.^z.{| — sin^.ly} . 2 ^.sin.Jy ==?•'. sec.^^.Jl — f.sin^.|y}.sin.|2/
(43)
= r^ sec.^r.CA:.sin.|y=Cfc. f ^— ) .sin.ij/.
To facihtate the use of this last formula, Burckhardt computed Table VII of this collection,
which contains the values of the logarithms of C = ^- — '— , for intervals often
minutes in the value of y, from y = 0'' to y =20''; and by means of it, we can
(**)
very easily compute the time t, corresponding to the radii r, i\ and the included arc
2/"= v' — V ; as may be seen in (J3), or in the example which is given on the same page
with the table. The assumed values of l,m, L, M (17,18,22, 23), are precisely the
('5) same as in the ellipsis [5995(28,37,48,50)]. Multiplying (17) by 2.cos.f.\/T?, we get,
r' -{- r = 2.C0S. f.\/r7 + Al. cos. f .\/^ ,
PARABOLIC ORBIT, COMPUTED FROM
?-, V-
v=2f.
hence the denominator of tlie first expression in (15), becomes 4Lcos.f.i/'iy ; and tlic
value of p is reduced to the form (20). Again, since (17) is reduced to llie form (-22),
by changing I into — L, we may, m the same way, get (:24) from (20). Substituting the
value of p (~0)} J" t'l^ first expression of Jet (IG), we get.
849
[599G]
(■IB)
kt = P.(2.cos/)^(^■')' + i-l--{2-cos.f)-.{rry = 21 cos.y.( rry.{l^ + |./^- j . m
Substituting this in the value of ?» (IS)) it becomes of the very simple form (21). In a
similar manner, the substitution of the value of p (21), in let (16), and then in M
(23), gives (25); and this maybe derived from (21), by changing, as in [5995(127)]
/ into — L, and m into J\I.{ — 1)=. If we compare the equations (21, 25) with the (43,
similar ones in an ellipsis, [5995(40, 53)], we shall find that they agree, if we suppose
x = 0, or sin^.|5- = 0; whichmakes |=0 [5995(115)]. Henceit is evident, that in
calculating an orbit, upon the supposition that it is an ellipsis; if we obtain x = 0, that is
to say — — Z = 0, or — — L=0, [5995(47, 57)], we may immediately conclude (49)
that the orbit is a parabola, and we can then calculate the elements of the orbit, by any of
the formulas in the preceding table (11 — 25). Thus we may find jj from (15 or 20), (50)
also, D=^\p, and then we may obtain F from (13 or 14). We shall illustrate these
formulas by the following example.
EXAMPLE.
Given in a parnbolic orbit log. j- = 0,2476368, log. 7^=0,2929648, and k' — ï = 2/ = 3o'' 18'" 43», to find
the elements D, p ; the anomalies v, v' ; and the time of describing the arc t.
To find t. To find p, D, v.
4 log. r' o,t464S34
k log. r o,i23Si84
z =r 46* 29"' 39» ,6
2Z = 92'' 59"" 195,2
J^ 15"* 09"' 21«
y=^ i^i 26"' 27»,2
v/r . sec. 2
43/= 7^43" i3>,6
* = 5â''=5'%6222
tang. 0,0226640
sine 9,9994089
COS. 9.9846256
COS. 9,9840345
J log. r o,i238i84
COS. 9,8378575
log. 0,2859609
Multiplied by 3 0,8578827
Table VII. log. C 1,7591607
sine 9,1282047
log. 1,7452481
/
z
iy
r
P
D = ip
i/J? =cosa.4t'
r 2r
iv = 3'' 3o"' 53"
sine
sine
ar. CO. sin.
9,4173807
9,86o52i4
0,8717953
sum
doubled
log-
log.
0,1496974
0,2993948
0,2476368
o,547o3i6
o,3oio3oo
log.
log.
P,346ooi6
0,2476368
log.
9,9983548
COS.
9,9991824
ble III.
log.
0,7037928
0,3690024
(51)
(52)
{ô:î/
Time from tlie perilielion cori-espoading to r, f, 1 1 ,83 48 log. i ,0727052
VOL. III.
213
850
APPENDIX, BY THE TRANSLATOR ;
[5997] TO FIND THE ELEMENTS OF A HYPERBOLIC ORBIT ; THERE BEING GIVEN THE RADII r, r , THE ANGLE
v' — v=2f, AND THE TIME t OF DESCRIBING THE ANGLE 2/.
We shall here use the same symbols as in the elliptical orbit [5995(G, &:c.)], changing
C
'■' M into —, and u' into Cc; using also the auxiliary angle -^^ [5988(3)].
For convenience of reference, we shall insert these symbols in the following table
(3 — 9, &,c.), together with the formulas which are used in this method (9 — 59), and
(2)
their demonstrations in (60 — 172).
Symbols-
(3) r, r' the radii vectores ;
(1) V, v the mean anomalies ;
(5j «= the semi-transverse axis = 6. cot. 4-;
7 1 • • • // o 1 \ s'm. f.\/r?
(0) 0= the semi-coniugate axis = rt.i/(e 1) = — ;
^ *' ^ V V / tang.2w '
(7) p =: « (e^ — 1) = i>-\/e^ — i = «.tang.-4^ = i.tang.-^]^ = semi-parameter ;
(8) e =
Formulas COS.-i.
fur a hy- "
perbolic
; secant 4- =^ excentricity ;
urbit.
. tang./", tang. 2?j tang./'.tang.2n
(0, ^/ea_i=tang.+ =-^^'' - - ^./ ^ .
2.{_l-z)
2.{L+z)
(10) i«=-;
(11) u' =1 C c;
(12) c = tang. (45" + ») ?
(13, Z=\. ^^^C-^jY',
(14) C = tang.(45'' + W);
c^ 5- — 4.1og.c
(15) z = T — ;
(16) tang.2?i = 2.\/(7+l2) ;
2.sin.4'.tang.2w
(!') taiig.2JV=— ^ ;
^ sin._/.cos.2i«
(18, 2f=v'-v; v^F—f;
(19) 2F=i''-f-i;; v'=F+J;
[Corresponding to r, v.]
[Corresponding to r', v'. \
(20) sin.gï^- ?
(21) cos.^y=è
, ^
^
^+
il
^ _ C — c sin.(JV— «) .
(22) tang.^w— ( c+c). tang.H" cos.(yV -f- ?i).tang.l+ '
HYPERBOLIC ORBIT, COMPUTED FROM r,r',v'—v,t!—t. 851
[5997]
\).a )
COS
sin
Cc—V _ sin.(JV + «)
(Cc+ l).tans.i4. cos.(JV — ?i).tang.^4.
Ce
a c ^ } i. c )
/-j-j'
,2\2
y/'^ = tans.(45'^ + w) ; )çX'- = tang.(45'^- iv) ; [v^of t]
'" 1 I / _ [-When cos./l r Assumed T
"1~ \/ / L 13 positive. J Lvalueof I. S
r V r
s.cos.y
sin^.J f , tang^Sw
^+ — ^^
COS./ cos.y
2*.(cos.y)^.(/T')^
= 1+2?;
(24)
tang.ij; =______-__ = _ ^^/ ^^ ^_ ;^ ^ , ; (25)
(20)
cosf^la . I r.{ C + - )-( c- 4- i) I .(^-^J; (27)
(28)
,i„.F=J..|c-I|.|^|";
c».F=i«.|,(.+l)-(c+i)j.(l,)i
r ( C c }
- =ke. < — +y, ? —1 ; (30)
r' C 1 )
- = le . ■? Cc+-p;- ^ — 1 ; (31)
(3S)
(33)
2 _ (1 + ^z).{z + r^)^ - log.{v/i— . + y/z i ^3^j
2.(,- + c^)
(36)
(37)
(38)
r .^S3ume.l 1 (39J
l_vaIi)o of m. J ^ '
852
[5997]
(40) m
(41) y '■
(42) h =
(43) h =
(44)
APPENDIX, BY THE TRANSLATOR ;
irC'
(y-i)y
y+i
o
TO- _
y
(45)
r
m L -
(47) Jll :
(48) 31
(49) 5^.
(50) H
(51) j;^
(59) 2
(53) T'
(53')
^ = 1 — 2L ;
2. COS./
sin.^i/ tang^.2ît'
COS./ COS./
t«
= _(L+c)*+(L + cy*.z= r.(L+~)';
= -^ + (^ + ^~)-^ = -(L+^*'
M2
(Y+1).Y
1
y— T- '
= ya"" ^ '
= — . ? ^ liyp.log.tang.(45'' + ^) \
a" CXe.tang.2A'
= . — . < ^ comm. log. tang
Xk ( cos.2)j
,(45"+^)|:
(54) i^ = —
rt' C e.tang.2;i , , fArd,\}.
^ --^- hyp. log. tang. (45* + n) >
COS.!
(54')
(54 ) loi
[Assumed I
value of y.]
[Assumed ~|
value ol h.j
r When COS. /"[ r Assumed "1
L is negative, J Lvalue ot /-.J
[Assumed 1
value of JI.}
C Assumed I
value of rj
[Assumed 1
vplue of H. J
3 -^
-= -_ ) ^^^^ comm. log. tang. (45 '+ n) > ;
Xk I C0S.2A )
5.fe= 8,2355814 ... : log.x = 9,6377843 ... ; log.- = 2,1266342 ... ;
HYPERBOLIC ORBIT, COMPUTED FROM r, r', v' — v, t' — t. 853
[5997]
■r
' + '• - ('+ ]y°^-f-\^' __ ^ • 1 ^ - ^ W' - ^cj \ • "^"^^v/^
- 8 . ^ L + f (v/c - ^)' I • cos./V^
2.(1 — z) .cos.f.\/T? 2m^. cos./.v/;y fc^_5^
tang^Srt y-. tang^. 2» ~ 4 j^. ?•/. cos.y.tang-. 2?j
— 2.(.L + =:) .cos/.y/^y _ — 2.J\P. COS./. y/^ _
k^fi
P
tang'-. 2« r^. tang2. 2»i 4 Y^. rr'. cos.y. tang2.2n '
siii.y;tang./.\/JT' «/^. sin./.tang./.\/Jy /i/.rr'.sm.2f.^
■2.(1— z) 2m^
/i/.rr'.sm.2f.\^
= V ^f )
— sin/.tang./!^/jT' — Y^. sin./.tang/.v/ir' / V.rr'. sin.2f N^-
2.(i + c) "^' 2jiï^ ^\ kt
(55)
(55)
(50
(57)
(58)
<59)
We shall now give the explanations and demonstrations of the formulas in this table,
taking them generally, in the order in which they occur. The symbols (a — 9) are '^''')
similar to those in the table, page 767, or like those for the ellipsis, [5995(6 — U)], page
831, changing as usual I — e^ into «^ — 1, &c. ; the formulas in (6, 9, 17), cei)
depending on / will be noticed in (149,150). We have iu [5988(13)],
M := tang. (45" + ^ w), («i-,
and in like manner,
u' = tang. (45" + i zi). jsa.
When the quantities w, -u^ have been obtained, from the times t, l', by means of
[5988(6 or 7)], we can easily deduce u, u'. Instead of the symbols w, zs', Gauss
uses the quantities c, C, putting,
' = { '3m?) i *=&' '= !.a«,.(«'+,.).,.„,(«-+;.0!*=(...O^; .«
these values give,
-=tang.(45"+J^)=u (61'); Cc = tang.(45"+ ^^') =«' (62); (64,
being the same as in (10,11). In the course of the calculations, the new symbols
VOL. III. 214
854 APPENDIX, BY THE TRANSLATOR ;
[5997]
n, JV, z, Z, are introduced, depending on c, C These assumed values are given in
(12— 15), in terms of c, C. Ifweput, in [5989(12,14)],
(70)
(71)
(72)
the first of these expressions will become as in (12) ; and the last form of [5989(14)]
will give,
c^— 1 / 1
(65)
(66) tang.2n = -— = i[c — -
Novv the assumed form of z (13) gives,
(67) y/r=i.\à—c-i i ; ^r+-z=i.ià-\-c-i ] ; /rfi-f- v/r=ci ;
(68) \/z.\/ï+-z = ^ï+T^=i.{c-c-^); z=i.(c-2 + c-'); l+2z=h{c + c-^)-
Substituting the first of the expressions (68) in tang.2« (66), we get (16). Dividing
the numerator and denominator of (15) by 8, it becomes,
|.(c2_c-)_log.c*
<691 ^ :=:
è-(c-c-)'
Now the product of the first and third of the equations (68) gives,
l.{c^-c-^) = {\4-2z).{z + z^f;
moreover the third power of the first of the equations (68), being multiplied by 2,
produces,
i..{c-c-^f = 2.{z + z^f;
substituting these and the value of â, given by the third of the equation (67), in (69),
we get (34) ; which is reduced to the form (35) in (119 &;c.) The assumed values of
/, F (18, 19) are similar to those in the eUipsis [5995(13, 14)]. If we divide the
last of the expressions of sin.J?;, cos.|j; [5988(18, 20)], by i/r, and substitute
the corresponding values of i« = -(10), we shall get (20, 21). The similar values
of sin.|»', cos.il)' (23, 24) are found in the same manner, by merely accenting the
(73) letters r',v', and using u' = C c (11), instead of the value of ii (10). Dividing
(20) by (21), we get, without any reduction.
Oc-i—C-ià /e + l\^
tOrtfT ■*-'i1 r I
(74)
Multiplying the numerator and denominator, of the first factor of the second member of
this expression, by O à, it becomes ^^ ; and we have as in [5988(3)],
HYPERBOLIC ORBIT, COMPUTED FROM r,r',v' — v,i' — t. 856
, V [59971
(74';
e — ly tang. ^ ^ '
hence we get the first of the expressions of tang.^u (22). The second expression can
be deduced from the first, by substituting the values of c, C (12, 14). For if we put,
for a moment, 45<'-|-n=n', Ai»" -\- N = JV' , the expressions (12,14) become (75)
c = tang.?i' ; C = tang.JV' ; hence we get,
„ ,,, , sin.JV' sin.w'
C q= c := tang.^-' q: tang.n' = — zp ■
cos.JV' cos.n'
__ sin. N'. COS. w' ::p cos. N'. sin.?t' sin.(JV" zp n')
cos.JY'.cosm' cos.JV'.cos.w' '
and if we divide this expression of C — c, by that of C -{- c, we obtain,
C—c_sm.(JV' — n') sin. (.A''— n) _ sin.(JV— n)
C+c ~sm.{JV'+n') ""siu.(90<'-|- JV+n) "^ cos.(JV+ n) '
(76)
(76')
(77)
substituting this in the first expression (22), we get its second form. In like manner, by ,g
dividing (23) by (24 J, we get the first expression (25) ; hence w-e may obtain its second
form, by substituting the values of c, C (12, 14). It is, however, easier to derive (25)
from (22) ; observing that if we change c into c~^ , in (20, 21), we shall obtain the
formulas (23, 24) respectively ; moreover the change of c (12), into c~' , requires that
(79)
we should change tang.(45''-l~ ?;) into r~;77w'77~; > or tang. (45'' — n) ; which is
equivalent to a change in the sign of n; making these changes in (22), we obtain (25)
by a slight reduction. Multiplying (21) by (23) we get (SO) ; also (20) by (24) gives
(81) ; (21) by (24), gives (82) ; and (20) by (23) gives (S3),
sin.lv'.cos.iv = ia. \ C 7-, +c i . \ ;- ^ '; (80)
,i.'.cos.|« = ia.[c--^,+c- -i^ . Y^]^;
, , • ^ ^ 1 1 > (c^— 1 ) i
cos.Ju.sm.iy ^ ia. < C — "ri — c + — ,"• . < ;— } -;
i ^ c ) (_ rr 'i
cos.4y'.cos.4« = ia.^ C+ -^+c + -| • '^^ 5.
f 1 1^6+1
sin.V. sin.^«; = |a. ^ "^ C"~'^"~75 ' 777)i '
(81)
(82)
(Kl)
irr')
Subtracting (81) from (80), and substituting in the first member for,
sin.Ju'. cos.^D — cos.Jt;'. sin.i«, (84)
its value, sin.(|i;' — ^v) =: s'm.f (18), we get (26). In like manner, the sum of
856 APPENDIX, BY THE TRANSLATOR ;
[5997]
(82,83), gives by substituting for the first member, its value cos-dw' — ^v) =cos.f,
the expression (27). The sum of (80,81), substituting sm.F ^= sm.{^v' -\-iv) (19),
(85) gives (28) ; lastly, by subtracting (83) from (82), and substituting cos.F=^cos.{^v'-}-iv),
we get (29). Dividing the last expression of r [5988(12)] by a, and substituting the
value of u (10), we get (30); accenting r, u, and substituting u' (11), we get
W (31). Subtracting (30) from (31), we get (32); and the sum of (30,31) gives (33).
(86) The assumed values of w, I, m (36, 37, 39), corresponding to the case of cos./ positive,
are the same as in the ellipsis [5995(24,28,37)] respectively ; and the resulting value of /
[5995(31)], is the same as in (38). The similarvalues of L, M, (45 — 47), corresponding
to the case of cos./ negative, are also the same as in the ellipsis [5995(48 — 50)]. If
we substitute the value of tang.« [5939(14)], and tang. (45'' -|- Aw) [5989(12)] in
[5988(6)], it becomes,
(88)
(91)
(93)
k
(87) -.t = ie.]u — -[ — hyp. log.w,
iu-l\-
a2
r'-«5
and by accenting t, u, we get,
— .t' = he .) u' ; ( — hyp. log.w'.
Subtracting the first of these expressions from the second, then changing t' — t into t,
to conform to the notation in this article, we get (90) ; which is easily reduced to the form
(91), by the substitution of the values of u,u' (10,11); eliminating e by means of
(27), which gives,
C 1 1 ) , m'
= |e.|c+^|.^c-^|-2,log.c
Jc— -^cos./.v/;y <; 1)
(89)
(90) -.1
we get (92),
h
a'
Eliminating e, from (33) by means of (89), we get, by making a slight reduction.
(93) := i C -\-
a
HYPERBOLIC ORBIT, COMPUTED FROM r, r', v' — v, f — t. 857
whence we easily deduce the first value of a (55). Multiplying (37) by 2.cos. /.\/r7, [5997]
we get, r' -}- r ^ (2 -j- Al). cos .f.\/rr' ', substituting this in the preceding value of a (94)
(55), we obtain,
(2 + 4/).cos.//r7 —(c + -Ycos././ï^ 8 . ^ / — i. ("c - 2 + ^-) I cos.f.^
(95)
which is easily reduced to the second form (55). The third form is easily found, from (45)
by a similar process ; or it may be easily derived from the second form, by changing
/ into — Z,, as in (37, 45). If we substitute the value of z (13), in the second and
third forms of (55), we get,
9.{l—z).cos.f.{rr'f _ —8.{L -\- z).cos.f.{rr'y-
Multiplying (15) by ^.(c j we get,
substituting this in the second member of (92), and then multiplying by a^, we get (100). m)
Now the square root of the first expression of a (97), being multiplied by c ,
gives,
(c - ^) . a* = 2Ï. (I - z)i. (cos./)i (rrf ; ^99,
substituting this and its cube in (100), it becomes as in (101),
let = ^c — - j . a*, cos./, {rr')^ + ^-{^~\) ' "*■ ^ (WJ
=2l (cos./)Mrr')^. | (/- zf -^{l-zf. Z\ ; <"»'^
hence the value of m (39) becomes as in the first form of (40) ; and by substituting in it,
the assumed value of yz=\-\-Q, — z).Z (41), it becomes m=-y.{l — «)*, as in the (io2)
second expressions (40, 41). Squaring this value of m, and dividing by if, we obtain (io2)
z (44). By a similar process, usmg the second value of a (97), we may reduce the
value of M (47) to the first form in (4S) ; and by substituting the assumed value of
y= _ 1 + (L + z).Z (49), we get the second forms of M, Y (48, 49) ; finally, fi-om ("«)
VOL. III. 215
858 APPENDIX, BY THE TRANSLATOR ;
[5997]
these we easily deduce z (52). We may also obtain (48) from (40), by the same
process of derivation which is used in [5995(127)], namely, by changing,
(104) I into — Z, ; m into M\ — l)i; y into — Y; and li into iî.
By developing in series, we obtain,
(104.) \/{z + z') =z^-^ Iz^- \zi + &c. ;
multiplying this by 1 + 2s, we get,
(105) ( I + 2z)y (s + z^) ^è^ f J + |s* + &c.
Moreover,
(106) ^iqi7+^5:=l +si-t-is — ^22_j_gjc. ;
whose hyp. log., by (58) Int. is,
(107) hyp. log.|v/r+7 + \/z\ = (sl+ \z - iz^ + &c.) - U^- + ~2Z- iz^ + ^c.f
+ i.(zi + is — &c.)3 — J-.(si + is— Sic.)* + &c.
= {zi + is - iz-- + &c.) - i.(s +s* + Iz^ - is* + &c.)
_j_ a . (J + |,2 + 3^* _|_ &c.) - I .(S^ + |S^ + &iC.)+iS^+&C.
1, 3. Sl
(108) — ■' — s'^ -Titr^ — ^^■
Subtracting (108) from (105), we get,
(109) (I + 2^)-/(- + z^) — hyp. log. {/Î+; + i/s}= I s^+ is* + &c.,
moreover, the cube of (104') is,
(s + s2)* = #+fs* + &c.;
substituting these expressions in (34), we get,
(110) z = ll±A^+^- = t + f^ + &^- _._,,, &e.
2.(si + iJ+&.c.) l+i~' + ^c. ^ ^-
To obtain the law of this progression, we shall multiply the value of Z (-34), by
2.{z-^z^f, which gives,
(HI) 2.(s+s2)i,_z^ (1 _f. 2s).(^+ s7— log.{/r+i + /s|.
The differential of this expression, being divided by dz, gives, without any reduction.
HYPERBOLIC ORBIT, COMPUTED FROM r, r', v' -v,t'- t. 859
[5997]
3.{z + s^. ( 1 + 2^) . Z + 2.(^ + z^)k ^ (112)
= 2.(^ + z^y + 1.(1 + 2zy. {z + z^)-^ ^J:-i+-_^^- ^ . cnao
The last term of the second member being reduced, by rejecting the factor ^1+2 + \/^
which occurs in its numerator and denominator, becomes,
hence that second member may be put under the following form, by taking the terms in
the same order as in (112'), and bringing the factor ^.{z + s^)^ , without the braces ;
i.{z + z'^)-i.\4.(z + z^) + {l+2zr-l}=:i.{z + z"^)-i{S.{z-\-z^)\=A.{z + z"-)K (IM)
Substituting this in (112'), and then dividing the whole equation by {z-\-z^)-, we get,
by transposing the term depending on Z,
(2z + 2z^).'^ = 4 — {3-\-6z).Z. d'S)
If we compare this equation with that in [5995(107)], we find that the former may be
derived from the latter, by changing X into Z, and x into — z ; making the
same changes in [5995(112)] which is deduced from [5995(107)], we get,
^ ^ 4. G ,4.6.8- 4. 6. 8. 10, 4. 6. 8. 10. 12 ^
^= *- 3—5 ^ + 3TTT^^-3.5.7.9^ + .3.5.7.9.11 " ^ - ^^- ""''
Making the same changes in [5995(114,116)], and writing Ç for |, v/e obtain (117,118);
substituting the second of these expressions, in the first, we get (119), ^
10 5 2 52
9Z=6+^+35-~'-i575-"'+^'^-' '"'^
2 52
^=35-^'- 1575 •^' + ^''-' '"'>
10 5
From the last equation we obtain the value of Z (35), In Table X, are given the (120)
values of Ç (118), corresponding to z ; from 2 = 0,001 to « = 0,300; which
are to be used in solving the equation (40 or 48), as we shall see hereafter, (130 — 134).
The comparative magnitudes of z, ?, in Table X, have a striking analogy with those of '
<125)
(126)
(127)
860 APPENDIX, BY THE TRANSLATOR ;
^^^^^'^ a;, I, in Table IX ; as is easily seen by the inspection of the tables ; moreover, in
(122) consequence of the smallness of ?, in comparison with z, we may in the firsi
approximation towards the values of z neglect Ç, as we have neglected | in
''^^ [5995(146, &;c.)]. If we now assume for h the value (42), we shall get,
, , , m2
(124) 5 _]_ / 4- ^ = _ .
Substituting this, and the value of z (44), in the expression of Z"' (35), we get by
successive reductions,
Z- = i + «.(^ + O = ft.(t + ^ + 0 = ft-(n-' + ?-5) = -ft{?-5)
=ft-e^')-"i=.v(^')-('-)^
whence we obtain,
(^— )-^ = -'^-(,'^,);
and by substitution in the assumed value of y (102 or 41), we get,
(128) y_i^io,(jL_^ or (?/ — i).(f-A) = -^^.A;
whence we easily deduce the value of h (43). In like manner we may obtain, from
the assumed values of Y, H, (49, 50), the expression (51). This may also be very
(129) easily deduced from (43). by the principle of derivation (104) ; observing that if we
chan<^e the signs of the numerator and denominator of (42), and then make the changes,
which are indicated in (104), it becomes as in (50).
We may deduce the value of z, from the cubic equation (43 or 51), in the same
(130) manner as x is obtained from [5995(46 or 56)], in [5995(145, &tc.)] ; by first neglecting
(131) I, on account of its smallness, and putting, ^'' = j-j-j (42), or H=£— — ^ (50).
With this value of h or H, we find in Table VIII, the corresponding value of
(132) \q„ y y or log. Y Y; and then from (44 or 52), the approximate value of z ; also from
Table X, the con-esponding value of Ç. Tliis opération is to be repeated till the assumed
and computed values of Ç agree, and in general, it will be found that one single operation
' is sufficient to give a very close approximation to the true value. Hence we see that the
calculation for finding z, in a hyperbolic orbit, is nearly the same as that for finding x
(134) in the ellipsis ; and we may observe that the cwaji^fes -^ — ^ ^~'W [^995(47,57)],
which are positive in the ellipsis [5995(47,57,41)], becotne negative in the hyperbola,
(44, 52, 13), and vanish in the parabola [5996(49)] ; so that the sign of these functions,
determines the nature of the conic section.
(135)
(135)
HYPERBOLIC ORBIT, COMPUTED FROM ,-, r', v'—v,t'—t. 861
[59971
Having this computed the value of z, tve may noio consi<hr it as one of the data of the
problem, to be used iufnding the éléments of the orbit. The value of c may be found
from the formula,
f- I +2- + 2.^(2 +=2); (130
which is easily deduced from the first and third of the equations (6S) ; by multiplying the
fu-st of these equations by 2, and adding the product to the third equation. Wemay also
obtain c, from the formulas (16, 12), namely,
tang.Sn = 2.[/{z -\- z~) ; c= tang.(45'' + n). (is?)
The remarks in [5995(131 — 144)], relative to the roots of the cubic equation in y or Y,
corresponding to the ellipsis, may be applied also, with proper modifications, to the hyperbola,
as is evident by considering that the formulas, [5995(46,56)], in the ellipsis, are of the same
forms as those in the hyperbola (43, 51). Finally, we may observe, that if z exceed
the limits of Table X, we may use the indirect methods of solution, without changing the
form of the equation (40 or 4S). In this last case, if we suppose the elements of the orbit
to be known approximatively, we may determine very nearly, the value of n, by means
of the formula.
tan2;.2ji =
sin/V»"
cos. 2ft
which is easily deduced from (16) ; for if we square (16), and add 1 to both members of
the resulting equation, we get,
1 + tang-. 2n = l-\-4z -{- Az^ or sec^. 2« = ( 1 + 2^)2 ;
whence sec.2rt = 1 -|- 2z, and,
sec.2?i — 1 1 — cos.2ft 'i.sm^.n sin^.w
2 2.cos.2« 2.cos.2« cos.2« '
This value of z, is to be used in finding K '" Tabic X; and then a corrected value of
h or H (42, 50), may be obtained, which must be substituted in (43 or 51), to obtain
VOL. III. 216
(138)
(139)
which is easily deduced from (26), by the substitution of,
i.u — - ) = tang.2?j {my <"'>
Then z may be deduced from n, by the following expression of its value,
(142)
(143)
862 APPENDIX, BY THE TRANSLATOR;
[5997]
(144)
a more accurate value of y or Y. These operations arc to be repeated till we obtain a
value of y or Y, which will satisfy the equation (43 or 51) ; and then from (44 or 52),
'"^' we get the true value of z, to be used in Computing the elements of the orbit. We shall
now give the demonstrations of the remaining formulas in the preceding table, which are
used in this part of the computation.
Comparing the first of the equations (68) with (16), we get
1
(146)
(151)
(154)
4Y(s + s^) = 2.tang.2« :
and we have, in (13),
substituting these in the second and third of the formulas (55, 55'), we get the first of the
(147) formulas (56,57) respectively. Substituting in these, the value I — z = — (44), and
(148) L-\-z^y^ (52), we get the second expressions in (56,57). Substituting the value of
m^, (39), in the second form of (56), we get its third form ; and in like manner, by using
JVP (47), we may reduce the second form of (57) to its third form. The value of
(149) «v/e2 — I, deduced from (140), is the same as the last of the formulas (6). Dividing this
(150) by the first of the expressions of a (.56), we get the second form of \/e^—\ -(9); and
in like manner, by using the first value of a (57), we get the third, or last of the formulas
(9). Multiplying the equations (26,32) together crosswise, and dividing the product by
^•{c j, which occurs in both members, we get,
.(c-i).si„/= (,.'-. ).(^v-y-
(152) If we change, in (12), c into C, and n into N, it becomes as in (14), and by
making the same changes in (66), which is derived from (12), we get,
(153) tang.2iV = ^.(C — —\
We have also, as in [5995(30)],
/ — r /r /r 4. tans;
{n-y ~vV~ V r' ~ C0S.2
2îw
'iw''
Substituting these and
(155) e =: sec.4' ; (e^ — 1)^ = tang.-l (8,9),
in (151), it becomes,
4. tang. 214)
(155) 2.sec.+.tang. 2iV.sin./ = -^^^^ ■ tang.+ ;
HYPERBOLIC ORBIT, COMPUTED FROM r, r' v' — v, t' — t. 863
[5997]
dividing this by 2.sec.4..sin./, we obtain the expression of tang.2A' (17). The third
expressions of p (53,59), are the same as those in the ellipsis [5995(60,61)] ; they can (i56)
be easily deduced from (6), by squaring it, and then dividing by a, by which means we
get, for «.(e- — 1), or p (7) , the following expression ;
SYn?f.rr'
P = ^- ; ''*'>
«.tang^.2n
substituting in this, the last of the values of « (56,57), and using 2.sin./.cos./= sin.2/,
we get the last of the values of p (53,59). From these we easily obtain the second
forms (53,59), by putting sin.2/ = S.sin.yicos/, and using, in (53), (i58)
{]ctf = S.{cos.f )^(rrY.m^ (39); (i58)
and in (59),
{]dy' = 8.{—cos.fy.{rr')^.J\P (47). dss",'
,.2 1
Lastly, substituting in the second form (58), the value ~ z= (44), we get its first (iso)
form; in like manner, by using — = -— ; (52), we reduce the second form of (59) ;,60)
to its first form. Instead of representing the times from the perihelion of the first and second
observations by t,i', as in (87,83), we shall now represent them by T — ^t, and
T-^-^t, and then the two expressions (87,38) will become,
aoi)
:^. (T - 10 = è c/u -^-)-hyp. log.M ; -^.{T+it}=ie/u'- V\-hypAog.u'.
3
The half sum and the half difference of these two expressions, being multiplied by ff
k
give (163,165), and by the substitution of the values of u,u' (10, 11), we get their
second forms (164, 16G) ;
0
«^ < / ^ C 1 c \ )
Now if we use the values,
(162)
(103)
(1G4)
(165)
(I6f.)
864 APPENDIX, BY THE TRANSLATOR ;
[5997]
(167) c= tang. (45" + n) = tang, n' ; C= tang. (45'' + JV) = tang. N',
(12, 14, 75), we shall have as in (66, 153),
(168) c =2.tang.2n; C — ^= 2.tang.2JV;
and by using [30"] Int. we get,
2 c 2.tang.?i' . „ ,
(169) — .^ g = -^2_^=sin.2n =cos. 27i ;
1 -\- c^ l-[-tang."'n
(169') , , „„ = , i' r^' 2 A/-' = sin . 2 JV' = COS. 2 JV ;
1 _|_ C^ I -j- tang. ''.A'
substituting these in the first members of (170, 171), and making successive reductions, we
finally obtain,
^ , C I- c l+c2 /^ 1\ 2 ^ ^^, 4.tang.2JV
(170) Cc-i 7^ -^=-J—.{C— ^)= r-.2.tang.2JV= ^ :
' ' ' c C c C c V C/ COS. 2» ° cos.2?i
Cl,cl + C2/1\ 2„ „ 4.tang. 2n
(171) Cc— -^4--—= — '-^ — .(c 1= ^;-^».2.tang.2n= ^r^^- .
''"' c Cc ^ C C \ c] C0S.2JV ° C0S.2JV
Substituting the last expressions (170, 171), and also c, C (167) in (164,166), we get the
(172) first values of T,\t (53, 54), adapted to the use of hyperbolic logarithms ; the second
forms (53', 54'), are adapted to common logarithms, by using the factors X, Xit, (54").
which are the same as in [5988(8, 9)].
To illustrate the preceding formulas, we shall give the following example, from Gauss.
EXAMPLE
(173)
Given log. »- = o,o333585, log. j-' = o,2oo854i, «' — «= 2/=48<i 12", ( = 5 1 '■^^■',49788. To find the
elements of the orbit a, p, e, and the true anomalies v, v'.
We have given the calculation of z, in the introduction to Table X, and it is not necessary to repeat it
(174) here. The results of this calculation are 10 = 2"* 45"' 28',47 / = OjoSygôoSSS ; log. -5 = 8,7030725,
(175) log. ^i/ ^ O,o56o846, log. m2 = 8,7591571 log. y'^rs 0,1171063, z = o,00748583. With these we shall
compute n from (16), 4 ^''o^ (9)» t> from the last of the formulas (6). From this we shall deduce
,j75> a and p, by means of (5,7) ; ^ is deduced from (17), «, »' from the last of the formulas (22, 26) ;
lastly, T, t from the formulas (53', 54')- The computation of t, is made merely for the purpose of verification,
(l"7) as it is one of the data of the problem.
HYPERBOLIC ORBIT, COMPUTED FROM r, r', v' — v, t' — t.
To find n. (i6).
z = 0,00748583
I + Ï = 1 ,00748583
V^ (: +• 22) = è-tang- 2«
2
3n = 9*5i"'ii",8i6
»i = 4<'55"'35»,9o8
log. 7,8742399
log. 0,0032389
log. 7.8774788
log. 8,9387394
log. o,3oio3oo
tang. 9,2397694
To find 4. (9).
i= 0,057960388
z = 0,00748583
Z_s = o,o5o474558 log.co. 1,2969275
/ = 24<'6'" tang. 9,6506199
è log- 9.6989700
tang. 2n (as above) 9,2397694
4 = 37<'34"59«,77 tang. 9,8862868
To find V. (22).
n= 4*55'"35',9o8
JV= 8''o/i"'53«,i27
JV — n= 3*09'" I7',2i9 sin. 8,7406274
jV-j-ji= t3''oo"' 29',o35 ar.co.cos. 0,0112902
4 4 = 18'' 47"» 29^,885 cot. 0,4681829
4»= 9'' 25" 29s ,97 tiing. 9,22oioo5
i)=i8''5o'"59',94
To find T. (53').
{x ft)-! constant log. 2,1266342
a (in column 2) log. 0,6020619
ai log. o,3oio3o9
Factor log. 3,0297270
\e (in column 2) log. 9,7388027
2rt sec. 0,0064539
2J\'' (in column 2) tang. 9,4621341
First term of r= I72''''^^63o56 log. i.,iZ-!w;i
Factor (above) log. 3,0297270
45<H-A' log.tang. 0,1241703 log. 9,0940177
Second term of r=—i32''"y^96725 log. 2,1237447
r= 39''='y^6633I
è<= 25<'^y^7489I
a = 6. cot. 4 (5)
p=6.tang.4
To find b, a, p. (6, 5, 7)-
/ sin. 9,6110118
y;^. log. 0,1171063
arith. co. 0,7602306
6 log. 0,4883487
4 (in the first column) tang. 9,8862868
log. 0,6020619
log. 0,3746355
To find jY. (17)-
/ arith. co. sin. 0,3889882
2 log. o,3oio3oo
4
2tO
5<! 3o« 56',94
2j\r i6<' 09'" 46',253
sin. 9,7852685
sec. 2oi56
tang. 8,9848318
tang. g,462i34i
To find V'. (25).
ar. CO. COS. 0,0006587
sin. 9,3523527
same 0,4681829
Jb' = 33"^ 3iin 295,93 tang. 9,8211943
v' = 67'' 02"' 59»,86
To find it, (54'); Jor verificatioti.
\ constant log. 9,6377843
e = sec. 4 '°g- 0,1010184
A e log. 9,7388027
3,0297270
2« (as in column i) tang. 9,2397694
2j\r
First term of ^t = Io6''''J^I2393
4i^-\-n]og.t3ng. o,075o575
Second term of Ji = 8o*''5'^375o2
sec.
I75I42
log.
2,0258i33
log.
log.
3,0297270
8,8753941
log.
i,9o5i2ii
it= 25'^'^J■^7489I
it = 25''''J'S 74894 by observation.
o''*J'^,oooo3 difference.
ît — i3 .91440 = time from the perihelion of the first observation.
r+4<= 65 ''^^4I222 = time from the perihelion of the second observation
866
[5997]
(178)
(179)
(180)
(181)
(182)
(183)
VOL, III.
217
866 APPENDIX BY THE TRANSLATOR ;
[5998]
'- ■' GAUSS'S METHOD OF CORRECTING FOR THE EFFECT OF THE PARALLAX AND ABERRATION OF ANY NEWLY
DISCOVERED PLANET OR COMET, IN COMPUTING ITS ORBIT, BY MEANS OF THREE GEOCENTRIC
OBSERVATIONS, WITH THE INTERVALS OF TIME BETWEEN THEM.
In the computation of the orbit of a newly discovered planet, by the method in [5999], it
(1) becomes important to avoid the trouble of repeating, with much labor, the preliminary
(2) calculations, similar to those in [5999(300 — 379)], to correct for the effect of the planet's
parallax, which at the commencement of the calculation is wholly unknown. This is
(•''> effected in a very elegant manner by Gauss, by applying an equivalent correction to the
(4) place of the earth in the ecliptic ; supposing at each observation, a fictions or second observer
(5) to make the observation of the planet. The place of this second observer being in theplcme
of the ecliptic, at the point where the line drawn from the planet, through the actual place of
(^' observation on the surface of the earth, and continued beyond, intersects the plane of the
0) ecliptic. It being evident that the geocentric latitude and longitude of the planet is the
same in both places of observation ; but the distances of the planet from the two observers
will be varied, by the distance of tiie two places of observation. In consequence of this
change of place, we must apply a small correction to the distance of the earth's centre from
the sun ; and also to the longitude and latitude of the earth, so as to reduce them to the
assumed situation of the second observer. After these reductions have been made, the rest
'^' of the calculation must be continued ; supposing that the second observer is situated at the
times of the three observations, in the three points of the ecliptic, deduced in the
abovementioned manner, from the actual places of observation ; since it is a matter of
indifference, from what places the planet is observed, provided we carefully ascertain the
assumed positions of the places of observation, which are used in the calculations.
(8)
(10)
(11)
We shall put, at the time of any observation,
(12) A = ISO' + © = the heliocentric longitude of the earth's centre ;
,13) L = the heliocentric latitude of the earth's centre ;
R = the distance of the centres of the earth and sun.
(15) In like manner ./3„ L^, R^, represent the heliocentric longitude, latitude, and distance from
the sun's centre, of the place of the first, or actual observer, upon the surface of the earth.
(16) Also, Jl„, L„, R^, the corresponding heliocentric longitude, latitude and distance of the second
or fictions observer.
(17) a, Ù, the geocentric longitude and latitude respectively of the planet ; being the same for
both observers ;
(18) pi the distance of the planet from the first observer ; p, + P„ its distance from the second
observer ; p^ the distance of the first and second observers from each other.
Z the longitude, and z the latitude referred to the ecliptic, of the first, or actual observer,
(19) O ' i
as seen from the centre of the earth ; r, the distance of the first observer from the centre
of the earth.
TO CORRECT FOR THE PARALLAX AND ABERRATION.
/
First Observer yy
We shall suppose that the plane of
the annexed figure 90, is the plane of
the ecliptic ; S, the place of the sun ;
(St, the line drawn from the sun towards
the first point of aries ; C, the centre
of the earth ; O', the actual place of
the first observer ; F, the corresponding
place of the fictious or second observer ;
CC, 010', perpendiculars let fail,
upon the ecliptic, from the points C, O',
respectively ; C A, F B, O E,
perpendiculars let fall upon Sf; also, ■Son
FH, C G, perpendiculars let fall upon
OE; lastly, C I is drawn parallel to CO. Then by the preceding notation we have,
SC' = R; SO'=^R^; SF = R,; C'0'=-t;
°{'SC=A; rSO=^jlr, 'Y'SF=A_;
OCG = Z; 0'C'I=z; OFH^o-; 0'FO = ê;
and by the usual rules of plane trigonometry we have,
SC=,SC'.cos.L = iî.cos.Z,; CC = IO=R.sm.L;
SA = SC.cos.CSA = SC.cos.A = R.cos.L.cos.A ;
CA=GE=R.cos.L.sm.A; C'I= CO = C O.sm.O'C I=t.cos.z ;
TO' = C'0'.sin.O'CI=T.s\n.z ; CG=AE=CO.cos.OCG= CO.cos.Z=t.cos.z.cos.Z.
OG = r.cos.s.sin.Z ; SB = SF.cos.FSB = R„.cos.A„ ; FB = EH=^ R^.s\n.A^ ;
FO = FO'.cos.O'FO = FO'. cosJ = p,.cos./) ; 00' = FO'.sm.aFO = p,.sm.ô.
FH = BE= F0.C05. OFH= FO.cos.o. ^ p,.cos.0.cos.a ;
OH=FO.sm.OFII = p^.cosJ.sin.a.
Now, by referring to the figure, we evidently have,
SB + BE=SA + AE; EH + 0H= GE-{- OG; 00' =10 + 10'.
Substituting the values (27 — 31') in (3:2), we obtain the three following equations,
R„.cos.A„ -\- pj.cosi.cos.a = iî.cos.Zy.cos.,^ + r.cos.z.cos.^;
R^.sm.A„ -\- pj.cos.ô.sin.a := iJ.cos.L.sin..^ + r.cos.z.sin.^ ;
p,.sin.é = iî.sinX -j- r.sin.z.
867
[5998J
(20)
(31)
(22)
(23)
(S4)
(25)
(25)
(26)
(27)
(270
(27")
(28)
(29)
(30)
(31)
(31)
(32)
(33)
(34)
(35)
868 APPENDIX, BY THE TRANSLATOR;
[5998]
(41)
(43)
(43)
If we assume the value of m (37), we shall get from (35), ?„.sin.â = m.tang.â ;
(36) whence we easily deduce p„ (40) ; substituting this in the second terms of the equations
(33,34), we obtain (38,39) ;
(37) m = (R.s'm.L + r.sin.r).cotang.â ;
(38) R„.cos.A„ = i?.cos.L.cos.-4 + r.cos-z.cos.Z — m.cos.a ;
(39) R„.sm.A„ = R.cos.L.sm.A + r.cos.s.sin.Z — m.sin.o- ;
(40) i>„=m.sec.ê.
The equations (37 — '40) are perfectly accurate, and they give the values of R„, A„, p„.
Tills value of p,, is used in (116, 1 17), in finding a corresponding correction of the time t,
depending on the aberration. Multiplying the equation (38) by cos. ^2, and (39) by
sin..^2 ; then taking the sums of the products, and reducing by means of [24] Int., we get
(44). In like manner, if we multiply (33) by — sin../3, and (39) by cos..'2, we find
that the sum of the products, reduced by [22] Int., becomes as in (45).
(44) R^=: R.cos.L.cos.{^„ — A) -\- r.cos.z.cos.{Z — A^) — m.cos.(a — A„);
(45) R„.s\n.(A^ — A) ::=r.cos.z.sin.(Z — A) — m.sin.(a — A).
On account of the smallness of L and A^ — .1, we may put,
(46) cos.L=l, cos. (^2 — A) = l, s\n.{A., — A) = A„ — A;
also in (45), we may change R„ into R ; hence we finally obtain from (37,44,45,40),
the expressions (47 — 50) ;
(47) m = {RL -j- r.sin.2).cotang.é ;
(48) R^ = R +r.cos.r.cos.(Z — A) — m.cos.(a — A) ;
r.cos.z.sin. (Z — A) — m.sin. (a — A)
(49)
(50)
(51)
(52)
(53)
(54)
Pj = m.sec.f).
If r, m, are given in seconds, we must divide them by 206265*, or multiply them by
sin.P, nearly. With these formulas, (47 — 50), we may compute the corrections of the
place of the earth, for each of the three observations.
In making these calculations we must compute the longitude and latitude of the zenith ;
or as it is very commonly called, the longitude and the complement of the altitude of the
nonagesimal degree of the ecliptic, for each of the three observations. The data in each
of the observations being the obliquity of the ecliptic, the latitude of the place of observation
reduced to the centre of the earth, on account of its elliptical figure, and the right ascension
(55) of the meridian. Various methods have been given for this purpose in books of astronomy
(5g, and navigation ; but that which is derived from Napier's formulas [1345 "'^'^^J, is as simple
and short as any ; it was published by me several years since, in a work on navigation, in
TO CORRECT FOR THE PARALLAX AND ABERRATION.
nearly the following form. In tlie annexed figure, W°f S
is the equator, E its pole, P the pole of the ecliptic, Z
the zenith of the observer. Then we have given, the
side PE equal to the obliquity of the ecliptic, the side
EZ equal to the complement of the reduced latitude of
wm
the place of the observer, and the angle PEZ equal to -^^
tlie difference between the right ascension of the meridian
and 270'', or the right ascension of the arch EP W ; so that we have the sides P E, EZ,
and the included angle P£^, to find the angle £ P .Z, and the side PZ. Having computed
this angle and side, we shall then have, by noticing the signs.
longitude of the zenith = 90"
We shall now put, for brevity,
<iS=EZ+PE;
angle PZE=Z;
EPZ; latitude of the zenith = QO"* — PZ.
A =
cos.D
COS. S '
2D= EZ—PE;
angle EPZ= P ;
angle PEZ =
= E,
B ■'■"«•"; C =
— tang.S.
Then from Napier's formulas [1345 ^'^S'^"], we have, by changing the letters JÎ, B, C, into
P, Z, E ; and the arcs a, b, c, into EZ, PE, PZ, respectively ;
cos.x?
tang4(P + Z) = — ^ . cotang.JE = -4.cotang.JJE ;
COS. O
tang.KP— ■2) = -^-^ • cotang4i:= B.^.cotang.|E=£.tang.i(P + Z);
sm.o
^„^ cos. I- (P+Z) „ ^, cos.i(P+Z)
tang.JPZ= V,r, J\- tang.S = C. ■= ^ ' '
cos.i(P-
cos. i(P—Z)
The values of D, S, do not vary sensibly, during the interval between the extreme
observations, and we may put the preceding expressions under the following logarithmic forms ;
2.S' = Polar Distance of the observer -|- Obliquity of the Ecliptic;
■2D = Polar Distance of the observer — Obliquity of the Ecliptic ;
log. .4 = log.cos.D — log.cos..S' ; log.B = log.tang.D — log. tang.S ; log. C = log tang.S.
log. tang.J(P+Z) = log.^ + log.cot.J£ ;
log.tang.i(P - Z) = log .5 + log.tang.KP + Z) ;
log.tang.^PZ=log.C + log.cos.i(P + Z) — log.cos.è(P— Z).
This method is peculiarly well adapted to this calculation, because it is short, simple, and
VOL. III. 218
869
[5998]
{.■i7)
08)
(59)
(60)
(CI)
(61')
(tii2)
(W)
«4)
((i51
iUI.)
(67;
(67)
(t*)
(119)
('«)
on
870
APPENDIX, BY THE TRANSLATOR ;
[5998]
requires only four openings of the table of logarithms for each observation ; moreover the
numbers A, B, C, do not sensibly vary in the time included between the extreme observations,
so that the same numbers are used in all three of the observations. Thus in the example
[5999(277 — 279)], the obliquity of the ecliptic varies only 0'',42, in the interval between
the extreme observations. To illustrate these formulas, we shall apply them to the three
observations in the example [5999(277 — 285)] ; and as the altitudes and longitudes of the
zenith are not required to any great degree of accuracy, we shall only use five places of
decimals in the logarithms. Then the co-latitude of the place of observation, [5999(281)],
gives, rZ = 3S''31"'2r; the obliquity of the ecliptic, PE = 23" 2T" 59' , [5999(277)].
''•*> Their half sum, and half difference gives S= 30''59"'40' ; D^l''3l'"4V. Then we
m have, from (68),
(72)
(73)
(74)
(75)
(78)
(70)
(80)
-D= 7''3i»"4i' COS. 9,99624
.S=: 30'' 59" 40' COS. 9,93309
A log. o,o63i5
D tang. 9,12107
S tang. 9,77868:= log. C
B log. 9,34239
Subtracting 270'' from the observed right ascensions of Juno [5999(274 — 276)], which was
observed on the meridian [5999(282)], we get the resulting values of E (84), corresponding
to the three observations of the following table. Then by means of the formulas (69 — 71),
we obtain the values of the angle P , and the side PZ. Subtracting the angle P, from
90* we get the longitude of the zenith (88) 5 and subtracting the side PZ from 90'',
(83) vve get the latitude of the zenith (89). The calculations for all three of these observations
(81)
(82)
are as in the following table.
First obsenation.
(84) R. A.Merid, — 270^= £ = 87'' lo"' 22". (274).
.S.log. o,o63i5
èE=43''35'"i ij cot. o,o2i44
(85) .i(P+Z)=5o''32"'42^ tan. 0,08459
B. . . log. 9,34239
(30) 4(P_Z)=i4'*57'»53» tan. 9,42698
(87) Sum =65''3o'»35»=£PZ
(68) Comp.=24''29"'25*=long. zen.
(.89)
Second observation.
JE = 85" 43» 46». (275).
jî.log. o,o63i5
4JS:=42"'5i"'53' cot. o,o324o
cos. 9,8o3io
a log. 9,77868
sec. 0,01498
è(P-fZ)=5i''i5"'09» tan. 0,09555
B. . . log. 9,34239
^PZ tan. 9,59676
èPZ=2i''33"'4i'
/'Z=43''07"'22«
Latitude = 46''52'"38'
è(P— Z)=i5''i9»46nan. 9,43794
COS. 9,79650
C log. 9,77868
sec. 0,01 573
liPZ tan. 9,59091
Sum=66<'34»'55»=£PZ
Comp.=23<'25"'o5>= long. zen. ^pz^ 2i<'i-jm56.
PZ=42<'35"'52«
Latitude = 47''24'"o
Third observation.
£=85''ii''' 10^ (276).
A. log. o,o63i5
4£=42''35"'35^ cot. o,o3653
è(P+Z)=5i"3i'"o6« tan. 0,0
B. . . log. 9,34239
.i(P-Z)=i5<f28'"o8« tan. 9,44207
Sum=66<'59"'i4'=£PZ
Comp.^23''6o"'46'^long. zen
cos. 9,7939
C. log. 9,7786
sec. 0,0160
èPZ tan. 9,5886
.\PZ= 2i''n"'5';
PZ= 42"23"'5/
Latitude = 47'i36'"oe
These results are the same as in [5999(283 — 285)].
(90) ^^ ^^^^ ^^^^ '° 0^)> ^^^ ^^™^ latitude of Greenwich as that given by Gauss, 51''28"'39';
but it would be rather more accurate to reduce it, on account of the oblateness of the earth;
(91) the difference is, however, of no importance, in the present example, on account of the
smallness of the parallax. In calculating the parallaxes in longitude and latitude, in a total
(92) or annulai- eclipse of the sun, the longitude and latitude of the zenith may be required at the
times of the four contacts of the limbs of the sun and moon ; and during this interval the
value of A, B, C, remain unchanged. In fact, the numbers vary but very little in several
(93) years, so that we may compute a table for the obliquity 23'' 27™ 40% like that in (96), with
TO CORRECT FOR THE PARALLAX AND ABERRATION.
871
die variations corresponding to a change of 100* in the obliquity, or in the latitude, and by
this means we can obtain, by inspection, for any places inserted in the table, the values of
log. A, B, C ; and can make any allowance for a small variation in the latitude of the
place of observation, arising from any correction in the observations, or in the reduction
for the ellipticity.
[5998]
(94)
(95)
Table computed for the
obliquity 23''
27"'
4o«.
Reduced
Var.
OS..Î
Var.l..-../i
Var. log. CI
Places.
latitudes.
log. Jl
+
100»
I0-. B
-)- 100'
log. C
+ iuo« 1
Lat.
Obi.
Lat.
Obl.
Lat.
Obl.
+
d m 1
—
+
—
+
Albany,
42,27,13
0,079670
63
97
9,475733
293
739
9,853328
223
223
Berlin,
52,20,24
0,061608
49
75
9,3241 35
618
1099
9'77'i97
240
240
Cambridge, (E.)
•52,01,28
0,062166
49
76
9,33 ro54
600
1080
9,773925
24o
240
Cambridge, (A.)
42,12,03
0,0801 5o
52
97
9,478383
288
733
9,855355
222
222
Dublin, (Obs.)
53,12,07
0,060090
48
a
9,3o4i66
670
ii55
9,763705
242
242
Edinburgh,
55,46,0-;.
o,o556i8
47
67
9,a334oi
878
1376
9,74ioii
249
249
Greenwich, (Obs.)
51,17,28
0,063466
49
77
9,346396
562
1038
9,780232
238
238
Havanna,
23,03,34
0,120000
64
1 48
9,597658
96
5i6
io,oo3o45
210
210
Leon, I. (Obs.)
36,i6,52
0,091680
55
112
9,529940
202
634
9.902005
2l6
216
London,
51,19,29
o,c634o6
49
77
9,345714
564
io4o
9^779944
2 38
238
Oxford, (Obs.)
51,34,28
0,062963
5o
77
9,34o586
576
io54
9,777800
239
239
Paris,
48,38,5i
0,068207
5o
83
9,394413
452
9i8
9,802627
233
233
Philadelphia,
3q,45,44
0.084828
53
io4
9,501872
248
687
9,874738
219
319
(9C)
We may observe that the same rules of Napier (63 — 65) may be used in finding the
apparent longitude and latitude of a planet from its right ascension and declination, as in the
observations which are computed in [5999(277 — 285)] ; supposing in the preceding
figure 91, page 869, that the point .Z represents the place of the planet; and using its right
ascension, instead of the right ascension of the meridian ; and its distance PZ from the north
pole of the equator, instead of the co-latitude of the place of observation. To illustrate this
by an example, we shall take the first observation of Juno [5999(274,277)], namely, right
ascension 357'^ 10™ 22',35 ; declination 6" 40'" 08' south; obliquity of the echptic
23''27'" 59%4S. Hence we have,
(97)
(98)
(99;
(100;
angle FEZ = 357" 10™ 22 ',35 — 270'' = 87'' 10"' 22^,35 = E,
Pi; = 23''27™59',48,
S = J {EZ+ PE) = 60'' 04"' 03%74 ;
(loi;
EZ= 96''40'"08^ (102)
D = i{EZ — PE)= 36'' ae™ 04^26. (102 )
D sin. 9,7754222
S arith. co. sin. 0,0621735
è£=43<^5'"i i»,i8 cotan. 0,0214379
.i(P—Z)=35''5i -"375,34 tan. 9,8590336
è(P4.Z)=59''23"'28«,44
Sumï^^gb-fi 5"'o5',78=.Ei'Z
90
COS. 9,9046102
arith. co. cos. 0,3019201
cotang. 0,0214379
i(P-f-Z)=59'^23"'28M4 Ian. 0,2279683
4(P — Z) (io3) ar.co.cos. 0,0912754
iiP+Z) (io3) COS. 9,7068655
S ( 1 02') tan. 0,2397466
iPZ=47''29'"45«,79 tan. 0,0378875 (103)
PZ=94''59'"3i',58
90
Latitude 4''59"'3i'',58 south.
354''44'"54»,22=longitude of Juno.
(104)
(105)
(106)
872 APPENDIX, BY THE TRANSLATOR ;
[5998]
These results agree with those in [5999(283)]. After we have found the angle EPZ, we
may compute PZ, by means of the formula [1345'='], which gives,
sm.PEZ.sm.EZ
(107) sin. PZ =
sm.EPZ
but ic is rather more accurate to determine P Z by means of the tangents, as in tlie
formula (65).
The effect of the aberration of the planet cannot be so completely determined as that of
(103) the parallax in the preliminary part of the calculation of the orbit. Gauss adopts the usual
()09) method of correcting the observed places for the effect of that part of the aberration which
is common to the fixed stars ; namely, by adding 20', 25 to the longitude of the sun, which is
given hy ike solar tables, neglecting the small correction from the inequality of the motion of
' the earth, and applying to the observed places of the planet, the same corrections for the
aberrationin longitude and latitude, as if it were a fixed star. These corrected values are to
be used throughout the whole calculation of the orbit. Moreover, when the distance of the
(ji-i) comet from the earth has been nearly determined, by the first approximation, as in the
example [5999('i26)], we must apply a correction for the remaining part of the aberration
(113) of the planet ; by decreasing the time of observation, by the time t^, which is required by
the light, in passing from the planet to the earth, supposing it to take 493 seconds, or
(114) 0''^'% 005706, 1.1 passing from the sun to the earth, when at its mean distance. It being
evident that this corrected time corresponds to the actual place of the planet, in its orbit, at
the time that the particle of light quits the planet, which after the interval of time t, ,
strikes the eye of the observer. Moreover, we may remark, that these reduced times
corresponding to the orbit of the planet, are those which enter into the calculation of the
orbit in [5999], and not the actual times at the place of the observer. Finally, the
correction 3<" the distance i>. = m.sec.ê (40), requires a corresponding correction in the
aberration, which upon the same principles is represented by,
(115)
(116)
d") 493^p„= 493'.m.sec.ô = 0'"^y',005706.m.sec.ô ; log.0,005706 = 7,75633 ;
but this correction is generally insensible, as in (121), and may be neglected.
EXAMPLE.
Given the geocentric longitude of the planet «, = 354ii 44"' 54' ; its geocentric latitude 6 = — 4'' Sg" 32*
(io6); longitude of the zenith Z= 24'' 29™ (88) ; latitude of the zenith z = 46'' 53" (89) ; heliocentric
longitude of the earth .d =i2'i aS" 54' [5999(277)] ; heliocentric latitude of the earth X = -f- o'!49 [5999(277)] ;
distance of the earth from the sun R = 0,9988839 [5999(277)] ; distance of the observer from the centre of
the earth r^8',6o, being put equal to the sun's mean horizontal parallax, the mean distance of the earth
from the sun being supposed 206265*.
(118)
(119)
( 120) From the above data we get Z— A = iï'^ oo'" ; a, — Az= 342'' 16"" ;
COMPUTATION OF THE ORBIT OF A PLANET.
To find the correction, of the time for the aberration.
m (col. i), log. i,889i3„
i' sin. 4,68557
493' log. 2,69285
H sec. o,ooi65
To find
m.
M
log-
9.9995"
L
log.
9,69020
«£ = 0,48945
9,68971
r
log.
o,g345o
z
sm.
log.
9,8633o
r sin. z = 6,27769
0,79780
Sum = 6,76714
log.
o,83o4o
e
cot.ing.
i,o5873„
m
To find
R..
log-
1,8891 3„
r
log.
0,93450
z
COS.
9,83473
Z — A
COS.
9,99040
I»
sin.
4,68557
-f- 0,0000279
log
5,44520
— m .
log.
1,88913
a— JÎ .
COS.
9,97886
I« .
sin.
4,68557
■\- 0,0003577
log.
6,55356
Sum = o,ooo3856
= correction.
Add R = 0,9988839
gives,
-«2 = 0,9992695
Correction of lime = — o«, 1 86
log. 9,26920»
As this correction of the lime — o',i86 is so very
small it may be neglected.
To find jîj .
same o,g345o
same 9,83473
Z—A sin. 9,31788
Ri ar. CO. log. o,ooo32
+ ''.2' log. 0,08743
log. 1,88913
a — -9 ■ ■ sin. 9,4837i„
-^2 . ■ ar. CO. log. o,ooo32
— 23',6i log. 1,37316,,
Sum = — 22',39 = ./«j— JÎ; hence,
.«2=.;? — 22',39
873
[5998]
(121)
(122;
(123)
(124)
(125)
(126)
GAUSS'S METHOD OF DETERMINING THE ORiSlT OF A PLANET OR COMET, MOVING IN ANY fONlC SECTION
BY MEANS OF THREE OBSERVED GEOCENTRIC LONGITUDES AND LATITUDES, TOGETHER WITH THE
TIMES OF OBSERVATION.
We shall here give the excellent method, published by Gauss, in his Theoria Motus
Corporum Cœlestium, by which he determined the orbits of the newly discovered planets
Ceres, Juno, Pallas, and Vesta; by means of three geocentric observations, with the times
of observations ; the intervals between the observations being small, corresponding to an
arc of a few degrees in the motion of the body. The importance of this method was
exemplified several times in the computations of the orbits of these four planets,
particularly Ceres, which was discovered by Piazzi, a kw days before its conjunction
with the sun. It remained obscured in the sun's rays above ten months ; and
after the conjunction, was sought for, in vain, during several weeks, by many
European astronomers. It was feared by some that they would be unable to find it
again, and that it might be considered as wholly lost. But when Gauss furnished the
elements of its motion, they were able easily to distinguish this very small planet from the
numerous little stars which appear so much like it ; and on this account, he may be
[5999]
(1)
VOL. III.
219
874
APPENDIX, BY THE TRANSLATOR ;
[5999]
(!■)
(!■')
(2)
(2/)
First
(3)
m
Second
excepted
case.
(5)
considered as its second discoverer. The great simplicity of this method, as well as the
rapidity with which Gauss performs such laborious calculations, was shown in the very
remarkable instance, of his computing to a considerable degree of accuracy, in the period of
about ten hours, the orbit of the planet Vesta, by observations embracing a period of
nineteen days, with a geocentric motion of the planet of only four degrees.
figure
portion
surface
"/-ftV
riaccs ef the earth
The annexed
92, represents a
of the concave
of the starry heavens, the
sun S, being the centre
of this surface ; ^AA'A"G
the ecliptic; a CC'C'G'
the heliocentric orbit of the
planet or comet, whose
elements are to be com-
puted. ^, A', A", the
heliocentric places of the
earth, at the times of the
three observations ; C, C,
C ", the corresponding
heliocentric places of the planet ; B, B', B", the geocentric places ; the arcs AB, A'B', A"B",
being always less than 180'^. Then as the sun, earth, and planet are situated in a plane,
which is projected in the heavens, in a great circle, it is evident that the arcs A CB, A' CB',
A" C" B", are portions of great circles, and we shall suppose them to be continued, till they
intersect each other, in the points E, E', E". Lastly, we shall suppose the points B", B,
to be connected, by a great circle, which intersects A'B' in the point B*, and the orbit
£l G', in the point M. From this construction, it is manifest, that the situation of the point
B*, will be indeterminate, if the arcs BB", A'B" coincide ; or, in other words, if the points
A', B, B',B", fall in the same great circle. This case we shall exclude from our calculations,
cep'ted with the remark, that we must select such observations as vary considerably from this
situation ; so that the slight errors of the observations may not materially effect the position
of the point B*, which is an object of importance in these calculations. Moreover, the situation
of the point B*, or of the arc B B" is indeterminate, when the points B,B", coincide ; or
are in opposite parts of the spherical surface ; we must therefore, for the same reason, avoid
the use of observations, where the geocentric positions, in the first and last observations, are
very near to each other, or are very nearly in opposite parts of the heavens. We shall also
exclude this case from our calculations. It is important to observe that the geocentric and
heliocentric places of the comet, in any particular observatioii, fall on the same side of the
COMPUTATION OF THE ORBIT OF A PLANET. 875
[5999]
ecliptic; the latitudes being cither both north, or both south; moreover, the heliocentric
place of the planet, is always situated in a point of that part of the arc of the great circle,
which is included between the geocentric place of the planet and the heliocentric place of the noiiocen-
trie dI&cc
earth. Thus, in the first observation, the hehocentric place of the planet C, is situated «( the
' ' planet.
between the heliocentric place of the earth A, and the geocentric place of the planet B (2) .
This will be evident from the following considerations. If the planet be at an infinite
distance from the earth, the point C will evidently fall infinitely near to B ; and if that (vj
distance be infinitely small, the point C will fall infinitely near to A. Moreover, it is plain,
that if we suppose the situations of the sun and earth to remain unaltered, while the distance
of the planet from the earth a a, figure 84, page 792, increases in the direction of the line
«a', from nothing to infinite, without altering the geocentric position of the planet in the (7,^
heavens, or the position of the line a a! ; the heliocentric place C, figure 92, will gradually
move from A towards B ; which are the two extreme points or limits corresponding to an
infinitely small, or an infinitely great distance of the planet from the earth ; therefore, the
point C will always fall between A and B. Hence we shall have,
CB^ABKISO"; C'^'<^'JS'< 180'^ or 2<5', (30, 24) ; C"5"<^"jB"< 180". (sj
In the calculations of this article, we shall use the following symbols, which are similar to
those in [5995—5997].
Sytiibols.
t,t',t", Times of observation ; (9)
©. ©'. ©". Longitudes of the Sun ; (lOj
A, A', A", Longitudes of the earth, differing 180'* from ©, ©', ©", respectively ; (11)
O., a.', O.", Geocentric longitudes of the planet ; ('2)
6, 6', 6", Geocentric latitudes of the planet ; southern latitudes heing considered as negative; (13)
a*, 6*, Geocentric longitude and latitude of the point B*; southern values of the latitude 6* being negative; 03',
R, R', R", Distances of the earth from the sun ; (M)
Pi'f/'ti"' Distances of the planet from the earth ; (15)
f, />', f", Curtate distances of the planet from the earth ; (16)
r,r',r", Radii vectores of the orbit of the planet ; (17)
y8, yS', /2", Heliocentric longitudes of the planet ; (18)
OT, Œ-', isf", Heliocentric latitudes of the planet ; southern latitudes being considered as negative ; (19)
V, v', v", True anomalies of the planet ; (ao)
u, u', u", Arguments of latitude of the planet, or distances from the ascending node, counted on the orbit ; (21)
CC'C", C=angle G'CS; C = angle G'C'B' ; C"= angle G'C"£"; (22)
w,w', w". Arguments of latitude of the planet, reduced to the ecliptic, and counted from the ascending node ; (23)
i,î',i", i = -i.\-cAB; i-i = 3rcA'B'; J- " = arc A" B" ; (24)
i* i* = ATC B'B*; (25)
n, Longitude of the ascending node of the orbit of the planet ; u = 180"* -|- n ; (2(5)
J, Inclination of the orbit of the planet to the ecliptic ; (27)
E,E',E'i, E =ans\eA'EA" ; £' = angle .«JEVÎ" ; E" = angle AE"A' ; (28)
7f,2f',2f", 2/=arc C'C" = »"—»'; 2/' = arc CC" = »"— o ; of "= arc CC = v' — v : (29)
z,z' 2 = arcC'B'; z' = arc C'fi* = arc C'2î' — arc £'£• = 2 — .f», (25) ; (30)
^, f " f = arc CE' ; C = arc CE' ;
(31 J
876
[5099]
(32)
Formulas
uaedin
(33)
these cal-
culations.
APPENDIX, BY THE TRANSLATOR;
6 =
R.sin.fsin.jAiiE'—J-i') .
RiiMn.<f".sin.{JlE'—S} '
i{'.sin.J'.sin.(.^"£— J")
C34) c =
(35) d =
(36)
(37)
(38)
(39)
(40)
(40-)
(41)
(41')
(41")
(4J"'/
(42)
(43)
(44)
(45) x('_
(46) X =
2iî'3.siu3.<f'.sin.J-«
b.sec.S'' — a _
i^sic.J"*— I '
tang-iT*
t Assumed 1
value of a. J
r Assumed "I
Lvalue of b.J
[Assumed 1
value of C.J
[Assumed "1
value of d. J
t Assumed 'I
value of C.J
"— b.sec.'f—i '
[rr'], [r' r"], [r »"], represent as in [5994(266)], the double ol the areas of the plane triangles sab, she, sac,
in figure 84, page 792, respectively. The radii, corresponding to any particular triangle,
being included between the brackets ;
^ I [»■ r"J '5
tang, w
sin.J" "
{P4-a.).e , , ,,
„ , — =-^^ ■ — — ; ('30')
(;-±-:)_eos... ^ + ^
6/^+'
+
Q' = c Q.sin. w ;
Q'.sin4.z = sin. (2— w — <r") ; or, ([26)
o = log. Q' + 4. log. sin. 2 — log sin. (z — w — <f *) ;
['•'-"] ,_(P + a).g.sin.J-'
[r';-"J ■ b.sm.{z — i*) ^ '
I [,' r"] ■ 5 " P ■■
[First unknown"!
quantity P. J
t Second unknown"]
quantity Q. J
t Assumed "I
value of W.J
[Assumed 1
value of Q'.J
[>• '']
(■69)
_ Jî.sin.J. sin. (./?'£" — J-' -f J *)_ a .
' jR'. sin.cf'.sin.(^-E" — cf) ~ 6" '
6,. sin.J-»
if.sin.J"
(i33')
sia.(.^£' — <r) '
B".sin.<f "
'sin.(.^"£' — J"')
cos.(.^£'— J-)
- ; whence a=^; (32,44,45)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
i?.sin.<f
cos.(.^"£'_J")
~ ii". sin.J'"
C [r r"] , } sin.E
= < =—. — 7TZ . r
J [»■''■"]■ S
sin.£'
sin.(î + ^'£ — ,r'); (182)
•^ llrrn 5 sin.B' ^ ^
[Assumed "|
value of bf.j
[Assumed T
value of Wj.J
[Assumed "1
value of «.J
[Assumed "I
value of x". J
[Assumed '1
value of x-J
[Assumed "i
value of X". J
t Assumed ~|
value of p. J
[Assumed "1
value of p". J
[Assumed "1
value of q. J
[Assumed ~l
value of q", J
.[rr']
q = x..(\p— I); (igS)
q" = «.". (x"p"— I); (198)
p=r.sin.,>-; q = i-.co3.f; (182,195); tang.f=^; r = p.cosec.f =q.sec.f ;
P"
p"=r".sin.^"; q" = r". cos.f " ; (181,198); tang.f"=^; T" = p". cosec.^" = q".sec.^" ;
log. k = 8,23558:4 ; [5987(8)]-
COMPUTATION OF THE ORBIT OF A PLANET. 877
[5999]
The points B,B',B", are given by observation, also tlie points^,./!?',.//", by tiie solar
tables j and when they are connected by 2;reat circles, as in figure 92, we shall have several '^*''
spherical triangles, whose sides and angles can be computed, by the common processes of
spherical trigonometry ; frequently using, with much advantage, the formulas of Napier
[1345,^^'^^''"'^']. Gauss has given many other similar formulas, but it is not necessary to
repeat them here, because the computations, by the usual methods, are in general more
simple, short, and accurate than those in whicii many auxiliary angles are introduced ; since
the small fractional parts which are neglected in these auxiliary angles, may have a tendency
to produce small errors in the results. We shall now give the enumeration of the triangles
which are to be computed, inserting some of the formulas, to which we may have occasion
to refer.
(56)
(57)
First. From the point B, draw the arc Bb, perpendicular to the arc A G ; tlien in the
rectangular triangle AbB, we have the perpendicular Bb=ô= the geocentric latitude of .^g,
the planet at the first observation ; and the base Ab=o. — A=^ the difference of longitudes
of the points B,A; whence we find the angle BM^j, as in the first of the formulas (62), '*''
which is the same as [1345"]; and the hypothenuse &, asin the first of the formulas (63), which ^^yj
corresponds to the second of [1345 -']. In like manner, by letting fall perpendicular arcs.
First
from the points B', B", upon the arc AG; we may form similar triangles, corresponding to ?'""«•
the second and third observations ; from which we may deduce the values of ■)-', y", S', è",
(62,63) ; or they may be more simply derived from the expressions of y, 6, by merely
accenting the letters, to correspond to the particular observations. The values ô, à', ô",
are always considered as positive.
(Cr)')
(01)
tang.d tang.ô' tang.â"
tang.7 = ■ , .- ; tang.y = ^-— — — ; tang.7 = -r~-j — — ; (ga,
sm.(a — .^) sm.(o.' — A) sm.(a — A)
tang.(a— ^) tang. (a' — A') „, tang. (a"— ^")
taBZ.S= -^-^ '; tang.(5' = — ^ ; -' ; tang.5"== — ^-^^ '. (63)
COS.7 cos .7 COS.7
We shall suppose, that neither of the expressions of tang.7, '«^ng-y'» ••^"g->") appear
under the form §, in the observations which have been selected for computing the orbit. (^'')
Second. In the triangle A A'E", we have the angles A'AE"= y, AA' E" =180"— y' ,
and the side AA'=A'—A ; to find by Napier's formulas [1345 «.^i], the sides AE", A'E";
and then the angle E", by [1345'*], or ^E" by [1345 '«or 1345"]. In like manner, in
the triangle AA"E', we have the angles A"AE'=7, AA"E'= 180"— >", and the side
AA!'=A!' — A; to find by the same formulas, the sides AE', A"E', and the angle
AE' A" = E'. Lastly, in the triangle A' A'E, we have the angles A"A'E=y,
A'A"E=\80"—'}", and the side ^'^"= ^"— ^' ; to find, by the same formulas, the
sides A'E, A'E, and the angle E.
VOL. III. 220
Second
process.
(65)
(66)
(67)
878
APPENDIX, BY THE TRANSLATOR ;
[5999]
Thud
process.
(68)
(68')
(69)
(70)
(71)
(72)
(74)
(73)
(76)
(79)
Fourth
proccsa.
(80)
(81)
(89)
Third. To find the point B* ; we have given, in the triangle i?£'B", the side
BE'=AE'—AB = AE'—5, the side B"E'=A"E'—A"B"=A"E'-o", and the
angle BE'B'= E' ; to find the angles EBB", E'B'B, by Napier's formulas [1345«.«],
and the side BB", by [1315'*]. Then in the triangle BE"B*, we have given, the angle
BE"B"=E", the angle E"BB*=E'BB", and the side BE"=AE"—AB=AE"-Ô;
to find the sides 55», B*E", by Napier's formulas [1345™'5i], and the angle BB*E",hy
formulas [1345'=']. Finally, we have B*B"^BB"— BB* ;
B*B'= B*E"— E" B'= B*E"— A'E" + A'B'= B'^E"— Jl!E"-\- Ô'.
In the plane triangle STC^, figure 87, page 793, the sides TC,, ST, S C, , or the
corresponding symbols p,, R, r, are respectively proportional to the sines of the opposite
angles TSC^, SC^T, STC^ ; and these angles are represented in figure 92, page 874, by
the arcs AC, CB, 180'' — AB ; as will evidently appear, if we suppose in figure 87, page
t") 798, a line SB to be drawn through S, parallel to TC,, and continued infinitely, in the
heavens, towards this point which is marked B, in figure 92 ; so that we shall have, in
figure 87, the angle j9;S'C^= angle SC^T; and the lines SC^, ST, being continued infinitely ,
fall iu the points C, A, figure 92. Hence we have.
sin.^C sin.CB s'm.AB
P,
R
From these we obtain the expressions of r, p^ (77, 78) ; and by accenting the letters we
get the similar quantities corresponding to the second and third observations, using the
symbols (24,30) ;
r= R
s'm.AB R.sia.5
r'=z R'.
sm.A'B'
R'.sm.ô' R'.sm.ô'
(77)
(78) P,
sm.CB sin. CB' ' " 'sm. CB' sm.C'B'
,. R". sm.A'B" R". sin. 5"
sjn.s
sm.CB"
sm.CB"-
R
sm. AC
sm.CB'
,_ sin.^'C _ R'.sm.{ô'—z)_
sm.Czï sm.z
R".
s\n.A"C"
sm.CB"'
Hence it is manifest, that when the situations of the points C, C, C" are known, we can
determine the values of r, r' , r" ; p^, />/, p/'.
Fourth. We shall now show these points C, C, C", can he determined by means of the
quantities P, Q (38, 39). We shall suppose M to be the point of intersection of tlie great
circles B"B*B, C'C'C, and for brevity we shall put,
2f= arc C'C'^MC "— M C = v"— v' ; 2/' = arc C C"= M C"— M C ^ v"— v ;
2f"= arc CC =MC' — MC = v'— v ;
(82) observing that <Ae«c symbols have the same symmetry relative to the number of accents, in
COMPUTATION OF THE ORBIT OF A PLANET. 879
[5999]
f,f\f''; C'C", CC", CC; as in the similar expressions [5994(279')] ; moreover, the («a")
values of /,/',/", in terms of v,v',v", are the same as in [5995 (13 &.c.)].
Tlie equation [5994(278)] is founded upon the supposition that the three places of the planet '^^ '
or comet a, b, c, figure 84, page 792, are situated in the same plane, passing through the sun,
which is the origin of the rectangular co-ordinates x,y, z; but the plane of xy, and the
direction of the axis x, are wholly arbitrary. Now if we take the plane of the orbit for that
of xy, it will be represented in figure 92, page 874, by the plane drawn through the centre
of the sphere S, and the great circle M C C'C" ; and if we take the line SM for the axis
of X, and the line perpendicular to it, in the same plane, for the axis of y, we shall find that
the radii r, r, r", form with the axis of cr, three angles which are represented by the arcs
MC, AW, MC", respectively ; therefore, by the usual rules of trigonometry, we shall
have,
x = r.cos.MC; x' ^r'.cos.MC ; x" =.r" .cos.MC" ;
y=r.sm.MC; y' ^r'.sm.MC ; y" =r".sm.MC".
(83)
(84)
(85)
(80)
(86')
(87)
(88)
Substituting these values of y, y', y" in [5994(278)], we get,
0 = [/ ;"].r.sin.MC — {rr"\r'.sm.MC' + [r/J.r". sm.MC" ; m
and by comparing [5994(300')], with (S2), we obtain the following expressions, which
have the same symmetry, in the accents as in (82') ;
[rr'] =3 r r'.sin.2/" ; [r'/'] = r'r". sin.2/; [rr"] = 7-)".sin.2/' . m)
Substituting these last expressions in (89), and dividing by rr'r", we get (91), which is
the same as (92), using the values of 2/, 2/', 2/" (82) ;
0 = sin.2/.sin.;tf C - sin.2/'. sm.MC + sin.2/". sm.MC" ;
0 = sin.C'C".sin.J»/C — sin.CC'.sin.MC'+sin.CC'.sin.^WC".
This may be considered as a theorem in spherics, signifying that the points C,C',C",
are situated in the same great circle MCC'C" ; M being any point whatever of the
circumference of this great circle. If we suppose the point M to be placed on the
continuation of the arc CM, of the great circle, so as to increase the distance CM,
by the quantity 90^ the term sm.MC, will change into sin. (MC+ 90''), or cos. MC,
and the other terms of the equations (91, 92), being changed in the same manner, we get,
0 = sin.2/.cos.MC — sin.2/'. cos.MC + sin .2/". cos.iliC" ; (93)
0 ^ sin. C'C". cos.AJC — sin. CC". cos.MC + sin. CC. cos.JiC", (93)
(91)
m)
Theorem
m
sphericit.
(92')
(92)
tvhich is merely another form of the theorem in spherics (92). We shall now suppose that
perpendicular arcs of great circles are let fall from the points C, C, C'', £, E', E", upon '^'''
the great circle J/2?B''E/ ; the arcs C^C, l!^/-Ë', are the only ones, which are actually drawn (95i
in the figure ; the others being omitted, to avoid confusion. We shall represents these arcs,
Theorem.
880 APPENDIX BY THE TRANSLATOR ;
r 59991
(96) by the Roman capital letters C, C, C", E, E', E", respectively. Then in the rectangular
spherical triangle MCC, we have, as in [1345='], the first of the following equations, or
(97) the value of sin. C,C or sin.C ; the second and third of these equations correspond to
the points C, C", and are easily derived from the first, by increasing the number of
accents ;
(98)
sin.C = sin. CMC,.sm.MC ; sin.C'= sin. C;iiC^.sin.;V/C" ; sin.C"=sin. CMC^.sm.MC"
Substituting these in (89), after multiplying it by sin. CMC,, we get,
(99) 0=[rV"].r.sin.C— [rr"].7-'.sin.C'+ [rr'].r". sin.C".
In the right angled spherical triangles E'E^'B, CCB, we have, by [1345 ='],
(100) sm.E'E;= sm.E'BE;. sm.BE' ; sin. CC,= sin. CSC .sin. CB.
Dividing the first of these expressions, by the second, and observing that,
sm.E'BE; = sïn.CBC,,
we get.
(101)
whence we obtain,
(loi'i sin. CC,=
sin.E'E; sin.-BE' _
sin. CC, ~ sin.CB '
sin.E'E,'. sin. CJ5
sin.BE'
substituting,
(102) BE'=^AE' — AB = AE'—Ô: E'E;=E'; CC, = C,
(103) W6 g^*^ the first of the equations (105) ; and by adding another accent to the letters E',E',
we get the second expression (105), corresponding to the point E". In exactly the same
(104) way, we obtain the values of sin.C (106), and sin.C" (107).
. ^ sin.E'sin.Ci5 sin.E". sin.Cl^
(105) sin.C=
(106)
.C'=
sin.(^E'-^) sin.(^E"— (5) '
■ sin.E.sin. C'B* sin.E". sin.C'B*
(107) sin.C"=
sin.{A'E-&'-\-o*) sm.{A'E"—à'-\-è*) '
sin.E.sin. C"B" sm.E'.sm. C"B"
sm.{A"E—ô") sm.(A"E'—ô")
Dividing the first of the equations (105), by the second of (107), we get the first equation
(108) ; in like manner, by dividing the fii-st of the equations (106), by the first of (107),
we get the second equation (108) ;
COMPUTATION OF THE ORBIT OF A PLANET. 881
sin.C _ sin.C^ sin.(^"-E'— ^") sin.C'_sin.C".B* sm.{A"E—è")
sm.C"~~ sm.C'B"' sm.{AE'—&)'' ^i^'~sin.C"B" " sm.{A'E—5'+ Ô*) ' '"*'
Dividing the equation (99) by ?■". sin.C", and substituting (108), we obtain,
^ ^ , „. r.sm.CB s\n.{A"E'—ô") „ r'.sin.C'B* sm.{A"E — ô") , , ,
'=f'' '" ^-.".sin.C'ii" • sln.iAE'-ô) " t''' ^V'.sin.C'i?" " sin. (^'£-.5'+'^*) + ^" ^^ '""
Substituting tlie values,
r.sin.C5 = iî.sin.â; r'.sin.C'B' = /J'.sin.â' ; r". sin.C"B"= jR". sin.^" (77); (ho)
observing also that sin.C'5* may be put under the form,
sin C'B* sin ''
sin.C'B*=sin.C'£'. ^ o> = sin. C'B'. -^^ ; (30), (ni)
sm.C'^' sin.z ^ '
we get,
n—r/ •"! ^-^'"-^ sin.(^".E' — j")_ il', sin.6' sin.(.^"i: - Ô") sm.z'
^-^ ' -'■iî".sin.o"- sin.(^£'-<5) ^''' -"-ii". sin.o" ' sm.{A'E-ê'+ô*) ' iii^ + '^'"'"^ ' '"'^
and if we use the assumed values of a, b, (32,33), it becomes,
sin.js/
0 = a . [/)•"] — [7 r"] . b . 1- \rr'\ . (ii3)
sin.s
From the assumed value of P (38), we easily deduce.
[5999]
substituting these in (113), we obtain.
hence we get,
r-\- 1 sm.z
[rr<]-\-[fr"] P+ 1 sin.s'
[,T"] P-\- a sin.z
g this, and the i
we obtain.
(114)
(115)
(116)
Substituting this, and the value of r'= — 'r—^ (77), in the assumed value of Q (39); ("^)
C P+1 sin.z' ) /î'3. sin3.<5'
^ ^ P + a sm.2 5
(118)
sin^.z
Multiplying this by ^^^-^^„ we get,
VOL. III. 221
882 APPENDIX, BY THE TRANSLATOR ;
[5999]
q.sm\z P+l
(119) — rr^- ~ — = b . — . Sin. S — sin.z.
2iî'3.siiA6' P + a
Now, from [21] Int., we have, by using z = z'-{- &* (30),
(120) sin.z = sin. (z'4- <5*) = sin.s'.cos.(5*-|- cos.z'. sin.é* ;
substituting this in the last term of (119), we obtain,
Q.sm'^.z C P+ 1 )
(•*" ^^^- — = < b.— C0S.5* > .sin.z' — sin.i5*.cos-.z'.
2iî'3. sin3. S' I P+ a 3
The assumed value of the first expression of tang.w (40), gives,
/P-(-l\ sin. (5* sin.rP.cos.w
(122) b . -fr-, — ) — cos.(5* = = : .
\P-f-a/ tang.w sin.w
Substituting this in (121), we get (123) ; thence by successive reductions, and the
(123) re-substitution of z' =z—S* (30), we obtain (125) ;
©.sin^.z
(123) ^
s . ^ ( COS.W . )
-- =sm.o*. < .sin.z' — cos.z' >
. i5' I sin. w )
2/î'3.sin3
sin.5* , . . ,, sin.6»
(124) =-: .jcos.w.sin.z' — sin.w.cos.z'{= -: .sin.fz — w)
sin.w sin.w
sin (5*
C^' =--- .sin.(z— w— 5*).
sin.w
Multiplying this last expression by — ^ , and substituting, in its first member, the assumed
value of c (34), we get,
('25') c Q.sin.w.sin^. z = sin.(z — w — i5*) ;
Funda-
mental
equation and by using Q' (40'), it becomes,
(126) Q'.sin^.z = sin.(z — w — Ô*) ;
or by using logarithms,
(126') log. Q' + 4.1og.sin.z — log. sin. (z — w — 6*) =0;
from which we must find the value of the unknown quantity z. We may observe that the
assumed value of,
sin.o
(127) tang.w = — T^XK ~ ('**^)'
cos. 5*
\P+aJ
COMPUTATION OF THE ORBIT OF A PLANET. 883
[5999]
may be rendered more convenient for calculation, in the following manner. Multiplying the
P-L. a
numerator and denominator by — , it becomes,
(f +a).tang.6* (P-j-a).tang.3*
tang.w— J 3ec ,s# (p_)_j)_(p_(_3) — p (i.sec.<5*— 1) + (6.sec.<5* — a)' '"*'
Substituting in the numerator, the expression,
tang. 5* ^= e. (b.sec.6* — 1), ("a»)
depending on the assumed value of c (36) ; and in the denominator,
b.secA* — a ^ (h.secà* — l).c/,
(130)
depending on the assumed value of d (35), we find that the whole numerator and
denominator becomes divisible by b.sec.i*— 1, and we finally obtain the second expression
of tang.w (40), namely,
(P+a).e
tang.w = ^ J ' . (130')
F-\-a
The calculation of the quantities a, J, c, d, e (32 — 36), which depends on known quantities,
constitutes the fourth operation. The actual values of b, c, e, are not required, but
merely their logarithms. If we put -3=6,, and substitute the values of a, i (32,33),
we find that the factor i?".sin.5" occurs in the numerator and denominator, and by
rejecting it, we get the following expression,
■R.sin.^ sin.(v4"-E'— d") sin.(.^'-E— ^'+ 6*)
'~'R'.sm.ù''sm.{A"E-&")' s\n.{AE'—lj ' "^''*
Now from the second and third forms of the equation (107), we get (132) ; from the second
and third forms of (106), we get (132') ; and from the second and third forms of (105), we
get (132') ;
sm.(A"E'—S") sin.E' _
sm.{A"E—ô") sin.E
sin.E
(132)
sin.(A'E — r/+ 6*)=-:— -,.sin.(./2'£"— o'+o*); (m)
sm.E
1 _ sin.E" 1
sm.{AE'—ôy^ sin.E' " s\n.{AE"—S) ' "^"'
Multiplying these three expressions together, and rejecting the factor sin.E. sin.E'. sin.E",
which occurs in the numerator and denominator of the second member, we get,
sin.(^"£'— é") sh.(A'E — è'+ 6*) s\n.(A'E"—o+ 6*)
sm.(A"E—Ô"y s\n.{AE'—5) ~ sm.{AE"—ô) ' "^"
B84
[5999]
APPENDIX, BY THE TRANSLATOR;
Substituting this in the second member of (131'), we obtain the value of h, (42), satisfying
(133') the equation h, = T-, or a =M, (131).
(133"i
Special
cases.
(134)
(135)
(136)
(137)
(138)
(138')
(139)
(W»")
(140)
(141)
(142)
There are two special cases, where some modification must be made in this calculation.
The Jirst is when the great circles BB", A"B",
coincide ; as in the annexed figure 93 ; in which the
point B coincides with E', and B* with E. In this
case, the quantities a, b (32,33), become infinite,
because the factors,
sin.(^ E'— Ô), sm.{A'E—S'+ 6*),
which occur in the denominators of these values of
a, b, vanish. When this happens we must divide
the equation (113), by b, and substitute the assumed
value of r=^/ (133'), and 2=2 — <i* (30), A
also —^ = 0. Hence we get,
0 = b,.[r'r"]-''''-^'-^*\[rr"].
sin.;!
Multiplying the numerator and denominator of tang.w (40), by b,, it becomes, by putting
as in (133') bb, = a,
tang.w =
b, .sin.^*
{P -{-!)— b,.cos.ô*
P + a
and as a is infinite, the denominator is equal to P+l — è,.cos.5*; consequently this
value of tang.w, becomes the same as the expression of tang.w, (43).
The second case is where ô* = 0. Then the expression c (34), is infinite ; and
w=;0 (40); hence it would seem that the factor c.sin.w (125'), becomes indeterminate.
But if we multiply together the expressions of c, tang.w (34,40), and the product by
cos.w=l, we get, by rejecting the factor sin.5*, which occurs in the numerator and
denominator,
1
c.sin.w :
2iî'3. sm\ô' . U . ( J^) — cos.S* \
Multiplying the numerator and denominator by P-{-a, and substituting cos. 5* = 1 ,
we get,
COMPUTATION OF THE ORBIT OF A PLANET. 886
[5999]
P+a
2jK'^ sin"*.» . I o.{F-\-l) — r — a {
l a
Now when cos.(S*=l, the expression of d (35) becomes d=j — r,or6 — a=tZ.(6 — 1); ci44)
consequently,
è.(P+l)-P— a=(è— l).P+(i— a)=(6— l).P-}-f/.(è— l)=(i— l).(P+rZ). (hs)
Substituting this in (143), and then multiplying by Q, we get for Q' (40'), the
following definite expression ;
„, „ . (P+a).Q
(e=c(e.sm.w= 2R^sinW. {b—l).{P+d) ' ("")
Lastly, substituting â* = 0, and w=iO (140), in the second member of (126), we
find that the whole equation becomes divisible by sin. s; then substituting the expression <'■"'
of Q (140), and extracting the cube root, we get, in this second case.
Fifth. When P, ^, arelcnown, we can obtain \\ from {AO), and then z from the equation (ns)
(41 or 41'). In a first approximation we may assume for P, Q the values P', Q' rmh
process»
(259) ; and by repeated processes, in the manner explained in (259 — 267) we can
compute the true values of P, Q ; from which we finally deduce the required value of (150)
z. If we develop the second member of (41), by [22] Int. ; it may be put under the form,
Q'. sin^2 — COS. (w -\- â*).sin.r = — sin.(w + 5*).cos.z ; (isi)
squaring this equation and substituting cos^.2 =1 — sin^.z, it produces an equation of the
eïg-^iA degree in sin. s ; which according to the general theory of equations may have eigfAf (iss)
roots, real or imaginary. Several of these roots must necessarily be real, and they may all
be very quickly found, by supposing sin.~ to increase gradually from 0 to 1, and ('53)
selecting, by inspection, those values which nearly correspond to this equation ; and then
by a few operations, correcting these first assumed quantities, so as to get the precise values
of z which satisfy it. We may reject all the negative values of sin. z, because they (154)
would make r' (77) negative, ^' being supposed positive (61) ; we must also reject
those in which z exceeds 5' as is evident from (8) ; and also from the consideration (155)
that if sin. ((5' — z) were negative, it would render p', (78) negative. When the intervals
of the times are moderate, it is generally found that there are four values of sin. z, which
satisfy the equation, of which one is most commonly negative, and can be rejected ; (i56)
sometimes there are three negative values, and only one positive value, consequently there
is then no ambiguity as to that which is to be used. In case of having three positive values,
vor,. III. 222
886 APPENDIX, BY THE TRANSLATOR ;
[5999]
it commonly happens that one of them is very nearly equal to (5'. This value satisfies
(157) the analytical conditions of the problem, but not the physical conditions. The analytical
conditions require that the planet should be situated, at the times of the three observations,
(158) somewhere on the lines a'a, b'b, c'c, fig. 84, page 792, respectively, and that the selected
points a, h, c, should be situated in the same plane and at such distances as to make the
(159) areas of the sectors sab, she, proportional to the times. Now all these analytical conditions
are completely satisfied by supposing the planet, at the times of the three observations, to
(160) he in the same places as the earth, so that the points C, C, C" may coincide with A, A\A",
(161) respectively, in fig. 92, page 874 ; in this case, we shall have C'B' = A'B' = 6'. This
result is evidently incompatible with the physical conditions of the problem, which require
(162) that the light in coming from the planet to the earth, should proceed from points a, b, c,
fig. 84 ; wiiich are at some distance from the eye of the observer at a', b', c', respectively.
(163) In most cases it will be found, that where there are three positive values of sin. z, we
can neglect one of them because it is nearly equal to à' (161), another because z
(164) exceeds 5' ; and then the remaining one can be used. If it should however happen that
the equation admits of two solutions, which satisfy the proposed conditions of the problem,
(165) we shall thence obtain two different orbits. In this case the true orbit is to be determined,
by comparing it with observations taken at greater intervals of time.
As soon as we have ascertained the value of z, we can find r', from the equation,
, R'.sm.S'
(167)
(168)
(iBC) •- sm.z ^''^^- ^°'' ""'^ ''""'' '" (^^^^' [rr']+[rV'] = [r'r"].(P+l);
substituting this in the first member of (116), and dividing by P-j-1, we get, by
re-substituting z' = z — «5* (30),
[r'r"] _ b s\n.{z — 5*)
[fl^ ~ P + a ■ simz
Dividing the value of r' (166), by the preceding expression, we get (168). The
equation (169) is easily proved to be correct, by the substitution of the value of P (33) ;
these expressions are the same as (41" 41"') ;
[r r"] ^_ (P + a).R'.sin.^' _
[i-'r"] '^ ~ b.sm.{z — (5*) '
(169) [rr] ' [r'r"] • ' p '
(170) We shall suppose the arcs C'c, C"c", fig. 92, page S74, to be let fall from the points
(171) C, C" respectively, upon the great circle ABE' ; then in the right angled spherical triangle
(172) C'dE", we shall have, by [1345^»] and (24,. 30),
(
COMPUTATION OF THE ORBIT OF A PLANET. 887
[59991
sm.C'c=sm.E'.sm.C'E'==sm.E''.sm.{C'B'+B'E")==sm.E''.s\n.{C'R+A'E''~A'B') U")
=sin.JE",sin.(2+^'£"— <S') ; (.74)
in like manner, in the right angled spherical triangle C"c"E', we have, by using (31) ;
sin. C"c" = sin.£'.sin. C"£' = sin.E'.sin.i^". {i75)
Now in the two right angled spherical triangles C'c'C, €"c"C, we have, by using C (22),
the fii'st of the four following expressions of sin. C'c', sin.C'c"; from these we deduce (i^^)
the second and third forms, by using (29, 90) ; the last forms are the same as those in
(174, 17Ô) ;
sin.C'c'=sin.C.sin.CC =sin.C.sin.2/'=sin.C. ^^=s\n.E".sm.{z^d'E"—&) ; (n?)
[rr"]
sin. C V=sin. C.sin. CC ^=sin. C.sin.2/ :=sin. C. — ~ :=sin.JG'.sin.^''. (178)
Dividing the two last of the expressions (HS), by the corresponding ones in (177), we
eliminate sin.C, and, by a slight reduction, obtain the value of r''.sin.^" (181); and
this value is for brevity, put equal to p", in (49, 53). In like manner, by supposing (i?d)
perpendiculars Cc,, C'c'^ to be let fall from the points C, C, upon the great circle ./2"J5",
so as to form the right angled triangles Cc,C", C'c\C", we get the expression of ?•. sin.C
(182) ; which may also be derived from (IS I), by changing the quantities, relative to liie
point C, into those of the point C", and the contrary. This value of r.sin.i^, is, for
abridgment, put equal to p in (48, 52),
CI80)
(180')
r
\rr"^ sin.JS ' . , , ^ „
ir r"l sin E
In the last place, we shall suppose the arcs Cc„, C"c"„, to be let fall perpendicularly upon
the great circle A'B^ ; though we have not actually marked these arcs, in the figure, to
avoid confusion ; then, from the right angled spherical triangle CE"c„,v,e obtain (184);
and from the triangle C"Ec"^, we obtain (185),
sin.Gr„=:sin.jE".sin.C£"=sin.£".sin.(C£'4-^£'— ^£')=sin.£".sin.(C4-./4£"— ^£');
sin.C"c",=sin.£.sin.C"jG=sin.£.sin.(C"£'4-^"jG— ^"jB')=sin.£;.sin.(r+^"-B-^"£'). (,85,
Now by proceeding, as in (177, 178), we get, in the right angled spherical triangles CC'i\_,
C"C'c"„, the first of the expressions (187, 188) ; from these we deduce the second and (,8^,,
third forms, by using (29, 90) ; the last forms are the same as in (181, 182) ;
(183)
(184)
sin. Cc,=sin. C '.sin. CC'=sia. C'.s-in.2/"=sin. C. tlj =sm.E".sm.{^-\-^E"—AE') ;
rr
(187)
888 APPENDIX, BY THE TRANSLATOR ;
[5999]
(188) sin. C "c",=sin. C'.sin. C'C"=.sin. C'.sin.2/=sin. C ^Ç^=sm.E.s\n.{i"-\-A"E—JÎ"E').
Dividing the two last expressions of (187), by those in (188), and substituting P (38),
we get, by a slight reduction,
(189) r.sm.{^+JlE"—AE') = r".P. ^^„ . sm.{i"+A"E—A"E') ;
sin.ii>
(190) Substituting CB=^—AE'-^5, C" B"=?"—A"E'+Ô".{31, 24), in the values of r, ?•"
(77), we get,
(191) r.sin.(^— ^E'+(5)=iî.sin.i5 ;
(192) r".sm.{C'—A"E'+&")=R".sm.5".
Developing the first member of (191), by [22] Int. ; and then dividing by R.sm.6, we get
(193) ; substituting, in this, the assumed values of X, x, (46, 44,), we get (194),
. cos.(AE'—S) sm.(AE—S)
(193) r.sin.^. D-^—x~ r.cos.^. - — ^—. — j — = 1 ;
it.sm.d /t.sin.ô '
]
(194) r.sin..*'.X — r.cos.f.— =1.
x.
Substituting in (194), the expression r.sin.<'=p (182), and then multiplying by x, we
(195) get, by using the symbol q (50); r.cos.^=x.(Xp — l)=q ; as in (50, 52). Again if
we develop, in the same manner, the expression (192), and divide by R".s\n.&", we
shall obtain (196) ; and by substituting (47, 45), we get (197) ;
, cos.{j1"E'—ô") s\n.{A"E'—5")
(196) r".sm.<".- 577-; — ^, — — ?-".cos.f". — 5^-1 — j;; — = 1 ;
^ ' it".sin.ù" il".sin.û"
I
(197) r".sm.i".X"—r".cos.i".-i:^ 1.
Substituting r''.sin.i"^= p" (181) ; then multiplying by x.", we get,
(198, r".cos.i" = x". (X"p " — 1) = q" (51,53).
Hence it appears, that we may deduce r, Ç, fiom the expressions of p, q, as in (52) ;
and r", ?", from p", q", as in (53). There can be no ambiguity in the values of
(199) ^j ^") because r, r", must necessarily be positive. The accuracy of the calculation can
be verified by substituting these values in (189), to ascertain whether this equation is
satisfied, by the results we have obtained. There are two cases in which other methods
are to be followed. In the first place, when the point B coincides with E', or with its
(200) opposite point, in the spherical surface ; or in other words, when AE'—ô is C, or 180";
because then the equations (182, 191) are identical; x (44) becomes infinite, and,
COMPUTATION OF THE ORBIT OF A PLANET. 889
[5999]
Xp — 1 = 3- = 0 (50) ; (201)
so that q (50) is indeterminate, (n this case we must find r' , ^" from (53), as in the (202)
former method; then r, Ç, from the combination of (189), with (182 or 191); by
methods similar to the preceding, and which require no particular explanation. We may (aoa)
also observe that when AE — 15 is very nearly equal to 0'', or 180'', the same method (204)
must be used, because the former is deficient in accuracy ; adopting that combination of
(189) with (182), or with (191), which will give the best form, to the resulting equation, (sos)
for the determination of r, Ç.
r^^"i J.
(206)
The second case which requires modification is where the point B", very nearly coincides
with E',oï with its opposite point; in this case the determination of »•", Ç", by the
preceding method would be impossible or inaccurate, on account of the smallness of
sin.(^"£' — Ù"), in the value of x" (45). Then r, ^, must be determined by the (207)
former method ; but r',^, must be found by combining (189) with (181), or with (192), isosj
upon similar principles to those adopted in the preceding case, in (205). The case where (aoo)
the points B, B , coincide with E', or with its opposite point, is excluded in (4).
Having found the arcs 4", i" ; the points C, C", together with the point C", will be given (Sio)
in position ; and the arc CC" = 2f', can be determined by means of the given arcs,
Ç=CE'; ?"=^C"E', (.31), can)
and the angle CE'C'=E' (28); using Napier's formulas [1345''®'''"], to find the angles
C'CE' ,CC'E', and [1345'*] to obtain the included side CC". Moreover in the triangle
C'L'C", we have the angles CEC\ C'CE, and the side CE, to find C'C' = 2f;
also in the triangle C'E"C, we have the angles C'E"C, C'CE", and the side CE", to find
CC =2f". These values of 2f, 2/", are however much more easily obtained by the (2i3)
[rV] I [rr'] 1
following formulas ; observing that the logarithms of , — — . —, r-— • . -, , have been
obtained by a previous calculation in (168,169) ;
(912)
(214)
sin.2/"= Hth • -T • sin.2/'. 'S'^)
These formulas are easily proved to be correct, by the substitution of the values of, [rr'], aiti)
[r'r"], [rr"], (90), in the second members, and making a slight reduction. Hence we have
a new confirmation of the previous calculations ; because we ought to liave 2/+ 2/""=2/'; (Si?)
and if any difference be found, we must re-examine the calculations. If the difference be
small, we may apportion it between 2/, 2/", so that their log. sines may be equally (si?')
increased or diminished, by which means the equation,
VOL. III. 223
89Q APPENDIX, BY THE TRANSLATOR ;
[59991
(319)
(220)
(225)
(232)
[rr'] r.sin.2/"
will be satisfied. If /,/", differ but little, the error may be equally divided between
2/ and 2f".
Alter we have obtained, in this manner, the position of the body in its orbit,
we may compute the elements in two different ways ; the one by combining the first
observation with the second ; the other by combining the second observation with the third ;
using the intervals corresponding to the times of observation ; by the method given in
[5995 &ic.]. Before these operations are commenced, we must correct the observed times,
(221) for the effect of aberration, by subtracting from the times of observations, the number of
seconds represented by, «„ <„, t^, respectively, and computed by the following formulas,
(222) ^ = 493^p,; t, = 493>.p;; t.^Am^^;';
observing that 493 seconds is the time required for the light to pass from the sun to the
^*^'' earth, when at the mean distance, which is taken for unity. This, expressed in parts of a
(223) day, is O"!"' ,005706 [5998(114)], whose logarithm is 7,75633. The values of p,, p/, p/'„
are found, as in (78,77,1 90), to be,
_ R.sn^.{AE'—l) _ ?-.sm.(^E'— 0
'^^' '''~sin.«-^£'+<5) ~ sin.<5 '
R.sm.{à'— z) _ ?-'.sin.(^'— z)
sin.2 sin.'J'
„ ■R".sin.(^".E'-r) _ r".sin.(^"£'— ^")
'^^ ^'~~sin.(C"-^"£'-f-«5'') "~ sin.*'
If the situations of the body, at the times of the three observations, be nearly known, by
(227) any previous calculations, we may immediately correct the observations for the effect of
aberration, and suppress this part of the calculation. Using tliese corrected times of
(228) observation t, t', t", and the value of k (54), we shall put, as in [5994(319)] ;
(229, r"^k.{t'—t); r = k.(t"-t'); r'^k.{t"—t); r'==r + r".
When we have gone through the calculation, as far as to find the value of y, or Y
(23b) [5995(129&;c.)], which expresses the ratio of the area of the elliptical sector sab
[5995(164)], to that of the corresponding triangle sab ; we can use this value of y or T,
to compute more correct values of P, Q, by the formulas (235,256) ; and then a corrected
''^'' value of z from (41,40',40). This part of the calculation is to be repeated till the
assumed and computed values of P, Q, agree. As the values of y or Y, differ
according as we use the different triangles or sectors, sbc, sac, sab, we shall denote them by
y, y', y", respectively ; so that we shall have by using the same notation as in (37 or 90) ;
(333) sector sbc = Jy.[/r"] ; sector sac = |y'.[''''"l ; sector sab = iy''. [rr'] ;
COMPUTATION OF THE ORBIT OF A PLANET. 891
[5999]
in which the accents have the same symmetry as in (82'). Now by Kepler's first law, the
sectors sbc, sab [5994(47)], are proportional to the intervals of time t" — t', t' — t or t, t"
(229) ; hence we have,
sector sab ^ "[^'^ W- [t'] ^ y^' p ,gg. ,
sector sbc t hy. [rV] y
consequently,
(234)
(«35)
Correct
value of P.
and as y, y" are very .nearly equal to unity [5995(44,31 Sic.)], we shall have P = — , (236)
for a very near approximation to the value of P ; to be used in a first operation, as we
shall see in (259&ic.). When the intervals r", r, are nearly equal, the expressions y, y")
will commonly not differ much from each other, and then the assumed value of P (236), (236)
is very near its true value. We shall now investigate the value of Q ; putting it under
such a form as will enable us to assume, at the commencement of the operation, a quantity, (23'')
which is very nearly equal to it. We have in [5935(10)], the following system of equations,
in which the anomalies are counted from the perihelion, (238;
p = r.(l 4" e.cos.v) ; p =r'.[\ -\- e.cos.y') ; p = r".(l -|- e.cos.t)''). (239)
Multiplying these three equations, by the values of ['V'], — [rr"], [rr'\ (90), respectively,
and adding together the products, we get,
p. I [/,•"]— [rr"]+[r/] I = rrV". I sin.2/— sin.2/'+sin.2/" J (240)
-(- rr'r".e.{sin.2y.cos.i) — sin. 2/'. cos.u'-(-sin.2/". cos.i)"}. (2<"
The coefficient of e (241 ), vanishes by means of the formula (93) ; the arbitrary position
of the point M being taken so as to correspond to the position of the perihelion, from which (242)
the angles v, v', v", are counted (938) ; hence we have,
p, I [rV']—[rr"] + [?•/] \ = rr'r". \ sin.2/— sin.2/'+ sin.2/" \ . (243)
Now by [31, 26] Int., we have, by observing that f'=f-\-f" (29), <2«)
sin.2/= 2.sin./cos./; sin.2/'— s in. 2/'= 2.sin.(/"—/').cos. (/"+/') (245)
= — 2.sin./cos. (/"+/'). (245')
Adding these two equations together, and reducing, by means of [28] Int. and (244), we
get successively,
siD.2/—sin.2/'+sin.2/-"=2.sin/.{cos./— cos. (/'+/')!
= 2.sin./.i2.sin.è.(/-f/'+/").sin.è.(/"+/'-/j=4.sin./.sin./.sin./". ''*''
892 APPENDIX BY THE TRANSLATOR ;
[5999]
Substituting this in (243), and dividing by the coefficient of p, we get,
4.r/?-". sin j/". sin ._/''. sin._/"
^''" P ^ [r'r"] — [rr"] + [rr'] "
(248) If we substitute the value of [rr'] (90) in [5995(60)], we shall get ^p =^ '^^■', using
(249) y",f",f—thc., for y,/, f &tc. as in (232 &c.), also t" for k.{t'—t), as m (229).
In like manner, in the triangle or sector corresponding to the radii r', r", we have
V \r'r"\
(250) \/P = — ■ The product of the two expressions of y'p (248, 250), gives,
y y". [>•»•']■[.'/']
(251) p — — ;; -■
TT
Putting this expression of p equal to that in (247), we get,
r , „-, r ,n , r n 4rr". nV. sin/.sin./"'. sin.f
(252) [r'r"]— [»T"]+[rr'J = „ . n r ' 'n ■
Multiplying the numerator and denominator of this expression by 2rr'r". cos.f.cos.f'.cos.f" ;
we find that the numerator becomes,
(253) „"_ (2rr'. sin./", cos./") . (2r'r". sin./.cos/).(2rr". sin/'.cos/)=rr".[rr'] .[»^r"].[rr '],
(354)
as is evident from (90), observing that 2rr'.sin./".cos./"=rr'.sin.2/'^[rr'], &c. Using
this reduced value of the numerator, and rejecting the factor [rr'], [rr"], which is common to
the numerator and denominator, we obtain the first of the following expressions: the second
is derived from the assumed value of Q (39) ;
rr
(255)
.r".[rr"] Q-hr"]
[rV']— [rr"]+[rr']=^ „ ,„ ^ f fô = -^^hr* •
<- J L J 1 L J 2yy .rrr .cos.j.cos.J .cos-J 2r^
Dividing these two last expressions, by the coefficient of Q, we get,
(256) Q==tt".
Correct >r" * cos./cos./'. COS./" * yy" •
valueof Q.
Now the angles / /', /', being generally small ; their cosines do not vary much from unity ;
r
,'2
(257) moreover as the radius r' falls between r, r", we shall have —77 , nearly equal to
unity, in most cases, in practice. Hence it is evident, that we may take, at the
(258) commencement of the operation, Q=tt", for a very near approximation to the value of
Q ; it is not however so close an approximation as the assumed value of P (236), on
account of the magnitude of the factor cos./.cos./'.cos./". The success of Gauss's
method essentially depends on this happy selection of the unknown quantities P, Q, whose
(258) values are so nearly known by means of the times t, f ; F being nearly proportional t<i
the ratio of their times, and Q proportional to their products.
COMPUTATION OF THE ORBIT OF A PLANET. 893
[5999]
We shall now show how, by means of the approximate values of P, Q (236,258), Approii-
mato val-
namely, "os «fp,
t"
-P=— ; Q=""; (259;
we may compute the elements of the orbit. The preliminary calculations for finding
a, b, c, d, e, 5, Ô', Ô", k, k", X, X" (32— 3G, 62, 63, Sic, 44—47) being made ; we may (2co)
substitute in (40) the assumed value of P (259), and we shall get the value of w ; then
from (41') we may obtain by a few trials the value of ~ ; substituting this in ( 166) we get (acij
/; also,
frr"! [rr"]
L;,]./. (168), LJ.,'. (1G9); (202)
hence we deduce p, p" (48,49); q, q" (50,51); Ç, r (52); Ç", r" (53);
then we obtain the arcs /,/', /", as in (211 — 215). With these values of r,r',r", f,f',f", (203)
we may compute the corresponding values of ['■'■'], [r'/'J, [rr"] (90) ; and with these we
can obtain new values of -P, Q (38,39). If these last expressions are equal, respectively, (264)
to the assumed values (259), we may conclude that we have obtained the true expressions
of r,i-',r'', f,f' ,f', &,c. But if the assumed and computed values of P, Q, differ (205)
from each other, we must repeat the calculation, in the same manner as in (260 — 265) ;
and the same process is to be continued, by assuming the last found values of P, Q, for (aoo)
a new operation ; and when the assumed and computed values of P, Q agree, they
must be taken for the correct expression of P, Q, to be used in the rest of the (267)
calculation, in finding the elements of the orbit.
(908)
Taking the extreme observations, for this purpose, we have, by the preceding calculations
the values of r, r", 2/"'= v" — v, and the corrected interval of time t"—t. With these
we can find, by the precepts in [5995] for an elliptical orbit, the elements corresponding to
the plane of the orbit ; namely, the semi-major axis, and the excentricity e ; also, the
time and place of the perihelion in its orbit. If the orbit be a parabola we can use [5996], (269)
and if it be a hyperbola we must use [5997]. The place of the node and inclination of
the orbit, to the ecliptic, may be obtained, by means of the triangle nAC, or aA"C", (270)
figure 92, page 874; and it may be useful, for the purpose of verification, to make the
calculation hi both triangles ; and take the mean of the results, if there should be any slight
difference. In the triangle nAC, we have given, the angle Q.CA^=^C, the angle <?^^)
a.AC=z\SO'^ — y, and the included side ACz=AE' — ?, to find the sides aA, nC,
by Napier's formulas [1345^"'^^], and the angle Aa.C=cp, by [1345'*]. If we use the (272)
triangle V-.TC", we have the angles ciC"A'= 6'", n^"C"= 180"— 7", and the side
A"C"^A!'E'—l:"; to find, as above, the sides Q.A",nC", and the angle ^"i2C"=(p. (273)
VOL. III. 224
894
APPENDIX, BY THE TRANSLATOR;
[5999]
EXAMPLE.
We shall take, for an example of this method of calculation, the following observation3 of the planet JunO,
made by Dr. Maskelyue at Greenwich. The times of observation may be reduced to the meridian of Paris, by
adding the difference of meridians, which Gauss puts equal to 9" 20" ,9 = o'^^'' ,00649 2.
Data.
(274)
C275)
(27G)
(277)
(278)
(279)
(S80)
(281)
(283)
(284)
(285)
(286)
Observation.
I.
II.
III.
Mean time at Greenwich.
October 5<i lo'i 5i"' 06» or 5'^''J',453.i52
17 09 58 10 or 17 ,415393
27 09 16 4i or 27 ,386585
App. Right Ascen-
sion.
357* 10" 22» ,35
355 43 45 3o
355 II 10 95
App. Declina-
tion south.
(i<i- 40'" 08s
8 47 25
10 02 28
At these times we have, from the solar tables, the following results.
Observation.
I.
II.
III.
^'s longitude from app.
Equinox.
192'' 28"» 53» ,71
2o4 20 21 54
2l4 16 52 21
Nutation of
equin. point-
+ I5^43
-I- i5 5i
-f-i5 60
©'s distance
from Earth.
0,9988839
0,9953968
0,9928340
Q's latitude.
— o'Ag
+ 0 79
— o i5
App. Obliquity of
the ecliptic.
23"* 27» 59»,48
23 27 59 26
23 27 59 06
With these data we obtain the apparent longitudes and latitudes of Juno, at the times of observation, as in the
following table ; the latitudes being south are marked negative. Also the longitudes and latitudes of the zenith,
which are equivalent to the longitudes and complements of the altitudes of the nonagesimal degree of the
ecliptic ; the latitude of the place of observation being 5i''28"'39' ; and the right ascensions of the meridian
being the same as the right ascensions of Juno, because the planet was observed in the meridian. This method
(283) of making these calculations is given in [5998(88, 89, 106,)].
Observation.
App. longitude of
App. latitude of
Longitude of
Latitude ot
Juno.
Juno.
the Zenith.
the Zenith.
1.
354* A^"' 54^,^7
— 4'' 59"' 3 1 «,59
24'' 29™
46'' 53™
II.
352 34 44 5i
— 6 21 56 25
23 25
47 24
III.
35i 34 5i 57
— 7 17 52 70
23 01
47 36
The parallax of Juno being unknown, we must use the method explained in [5998] ; by applying a correction
to the sun's place, as in [5998(121 — 126)], where we have computed the corrections corresponding to the first
(287) observation, as in the first line of the following table ; in which we have given the con-ections for all three ot
the observations ; the corrections of the time in the third column are so small that they may be neglected.
(289)
(390)
(291)
(292)
(293)
(294)
(295)
(296)
Observation.
II.
III.
! Reduction "of ©'s
longitude.
I — 22',39
! — 27 21
: —35 82
Reduction of ©'s
distance.
-|- o,ooo3856
-j- 0,0002329
-\- o,ooo2o85
Reduction of
the time.
— o»,i9
— 012
— 012
These longitudes are reduced to the epoch of the mean vernal equinox, corresponding to the beginning of the
year i8o5, by adding the corrections for the precession as in the following table (3oi — 3i2). We must also
correct the longitudes and latitudes for the aberration, as in [5998(110, iii)]; by applying to the planet's
longitudes and latitudes the same corrections as if it were a fixed star ; these quantities being also contained in
the same table. The correction for the aberration of the sun in longitude is made in (277 — 279), where the
tabular numbers have been increased 20^,25.
(29Î)
(298)
(299)
Observation.
Reduction'of precession
Juno's aberration
Juno's aberration
to January i, i8o5.
in longitude.
in latitude.
I.
n',87
— i9»,ii
+ 0^53
II.
10 23
— 17 II
+ I 18
Ill
8 86
— i4 .82
+ 1 75
COMPUTATION OF THE ORBIT OF A PLANET.
895
Wc shall now apply these corrections to the lono;itucles and latitudes, in order to obtain the values of A, A', A" ;
a, a', a" ; 9,6', 8", R, R', R" ; observing that the signs of the nutation of the equinoctial points (277 — 279),
are such as are used in finding the apparent place from the mean; and these must be changed, in [3oi', 3o3']
in finding the longitudes from the mean equinox.
Observation I. Observation II. Observation III.
i2<'28"'53',7i 24''2o'«2i.,54 34''i6»"52',2i
— i5 43 — i5 5i — i5 60
— 22 39 — 27 21 — 35 82
+ II 87 -)- 10 23 -I- 8 86
Q's longitudes — 180'',
Nutation of Equinoctial points.
Correction for parallax of Juuo,
Precession to Jan. i, i8o5,
Juno's longitude,
Nutation of the equinoctial points,
Precession to Jan. i, i8o5.
Aberration as a fixed star,
Juno's latitude,
Aberration as a fixed star.
Sun's distance.
Correction for Juno's parallax.
Corrected distances R, R', R",
Logaaithms of these distances,
Mean times of observation at Paris, found by
adding o'''''^',oo6492 to the times at Greenwich,
From (3o2, 3o4) we get.
.^=I2''28"'27S,76
354''44"'54',27
— i5 43
4- II 87
— 19 II
a=354''44'"3i',6o
— 4''59'"3i«,5g
+ 53
ô=— 4''59"'3i',o6
0,9988839
-(- o,ooo3856
R = 0,9992695
log. i? = 9,9996826
(=Oct. 5"''>'',458644
^— a=i7''43'n56',i6
A' — A=ii 5i 21 29
.4'=24''i9"'49'>o5
352''34'"44»,5i
_ i5 5i
+ 10 23
— 17 II
().'=352''34"'22',I2
— 6''2i™56»,25
+ I 18
-6<'2i'"55",07
0,9953968
-(- 0,0002329
iî' = 0,9966297
log. R' = 9,9980979
i' = Oct. i7''"5",42,885
.<?'— a'=3i''45"'26",93
A"—A'= 9 56 20 60
.;«"=34''i6'"09»,65
35i<'34"'5i',57
— i5 60
+ 8 86
— i4 82
(,_"=35!''34'"3os,oi
— 7<'ly'»52»,70
+ 1 75
6"= — jii 7™5o»,95
0,9928340
4- o.ooo2o85
R" = 0,9930425
log. i{"= 9,9969678
<"=Oct. 27''^>',393o77
.4"— a"=43<'4i"'39»,64
A'i—A=7i 47 4 1 89
[5999]
(300)
Dau.
(301)
(301')
(302)
(303)
(303')
(304)
(305)
(306)
(307)
(308)
(309)
(310;
(3U)
(312)
As all the latitudes have the same sign, we have considered them as positive, in the following calculations
(3i2'— 319, &c.), and have drawn the figure 94, page 8g4, to conform to this supposition, making the points
B, B', B", C, C, C", Stc., fall below AA", instead ol above, as in figure 92, page 874. The change of the
directions in the lines AB, A'B', A''B", of the figure, are indicated by the signs. Thus if we had supposed
a to be negative, in finding y (3i4), we should have tang.â and tang.j, ricgadVe; but this negative
value of y merely indicates that the arc AB falls below AA", as in figure 94, instead of above, as in figure 92,
page 874. Hence we see that by a careful attention to the actual situations of the points of the figure, we
may avoid, in a great degree, the trouble of noticing the signs in these preliminary calculations; and by referring
to the figure, are less liable to mistakes, than we should be, if we restricted ourselves exclusively to the analytical
method of computation.
To find y, yi, y". (62).
6 (3o6) tang. 8,9412495
A — a (.3ii) subtract sin. 9,4836865
y = i6<'bo'"o8',38 tang. 9,4575630
6' (3o6) tang. 9,0474879
.1' — a' (3ii) subtract sin. 9,7212540
>.' = ii<'58"'oo»,33 tang. 9,3263339
6" (3o6) tang. 9,1074080
A" — a.'i (3ii) subtract sin. 9,83 12855
>" = io''4i'"4o',;7
tang. 9,2761225
To find <f, ef', J". (63).
A — ()_ (3ii) tang. 9,5048260
y (3i4) subtract cos. 9,9828366
<f=jîB = i8<'23™59',2o
A' — a.' (3ii)
y (3 16)
<f ' = A'B' = 32"*i9m24',93
.4" — a" (3n) tang. 9,9650091
y" (3i9) subtract cos. 9,9923903
J" = .î"£" = 43''ii'"42>5 tang. 9,9736188
(312')
(312')
Preiimin-
iiry calcu-
lations.
(319'")
(313)
tang. 9,5219894 (314)
tang. 9,7916902 (314')
subtract cos. 9,9904579 ,3,5,
tang. 9,8012323 (316)
(317)
(318)
(319)
896
[5999]
APPENDIX, BY THE TRANSLATOR ;
To find E, A'E, A"E, in the triangle EA'A".
(320) £jj/^//— i8o<i— 5,'=i68<'oi'»59',67 (3i6). Using Napier's Rules [i345'
(321) E£l'J}'= y"= 10 4i 4o 17 (3l9)
(323) ■
'■]•
(323)
Prelimiu-
ary calcu-
lations.
(324)
(325)
l326)
(329)
(330)
(330')
(331)
Sum=2Si=i78 43 39 84 ; S, =89''2i">49',92
Difference=2Z)i =i57 20 19 5o ; Dj =78 4o og 75
Z)i sin. 9,9914519
,S'j aiith. CO. siQ. 0,0000268
4(4"—^')= 4'*58'"io',3o tang. 8,g3g2834
t{A"E—A'E)= 4 52 24 38
li(AiiE+AiE)=56 58 5o 78
Difference is J1'E=^2 06 26 4o
Sumis j2"£=6i 5i i5 16
tang. 8,9307621
i)j COS.
Si arith. co. cos.
i(^"— ^') (3 1 2) tang.
9,2932968
1,9545834
8,9392834
à(^"-E+^'£)=56''58'"50S78 tang.
A«—Âi (3 1 2) sin.
0,1871636
9,2370422
S'E (325) aiith. co. sin.
y (3i9) sin.
0,1028335
9,2685128
£=2i«i9'"34»,oo sin.
8,6o83885
To find El, AE', A"E', in the triangle E'AA'K
(327) J5;/^^//=i8o— >^=i63''59™5i«,62 (3i4)
(328) E'A"A=y"= 10 4i 40 17 (3i9)
Sum ^2S; =174 4i 3i 79;
Diffei'ence^2Z)o:=i53 18 11 45 ;
Sj ==87''2om45',9o
Dj =76 3g o5 73
JD, sin. 9,9881058
S» arith. co. sin. o,ooo466o
J(.4"— ^)=io''53'"5o»,95 tang. 9,a844852
4(jî"£' — «£')=io 37 1 5 55
i.(^"£'+^-E')=43 49 45 33
Difference is AE'=33 12 29 78
Sum is A"E'=54 27 00 88
tang. 9,2730570
-D2 COS. 9,3633710
S'a arith. co. cos. 1,3343907
i{A"~A) (3i2) tang. 9,2844852
^{A"E'+AE')=43''49'"4S^,33 tang. 9,9822469
A'' — 2 (3i2) sin. 9,5697089
AE' (33o) arith. co. sin. 0,2614699
y" (326) sin. 9,2685128
E'— 7'*x3"'37',70 sin. 9,0996916
To find E", AE", A'E", in the triangle E''AA'.
(»2) £"^^'=180"— }.=i63''59"'5is,62 (3i4)
t333)
(334)
1335)
(336)
E"A'A=y'= II 58 00 33 (3i6)
S3 =87''58"'55«,97
D3 =76 00 55 64
Sum 2^3=175 57 5i 95 ;
Difference 21)3^15201 5i 29;
n^ sin. 9,9869333
S, arith. co. sin. 0,0002694
4(jî'— jî)= 5''55"'4o%64 tang. 9,oi63358
ll{A'E"—AE")= 5 45 25 19
i(A'E"+AE")=^35 28 32 49
Difference is AE"^2g 43 07 3o
Sum is A'E"=4i 1 3 57 68
tang. 9,oo35385
-O3
(3l2)
è(^'£"+^-E'')=35''28'»32',49
A'— A (3 1 2)
AE" (335)
y' (3i6)
£"= 4'^55'"46',22
cos. 9,3832o5i
arith. CO, COS. 1,4533373
tang. 9,oi63358
tang. 9,8528782
sin. 9,3127087
arith co. sin. 0,3047442
sin. 9,3166918
sin. 8,934144?
COMPUTATION OF THE ORBIT OF A PLANET.
897
To find the angles B, B", in the triangle EiBB",by [i345*''"].
.îE'=33<'i2'»29<,78 (33o) jî"£'=54''27moo»,88 (33o') BjB'B"=jE'=7''i3'"37',70 (33i)
^B=i8 23 59 20 (3i4) A'iB"==4^ 11 42 o5 (Sig)
[5999]
E'B=iA 48 3o 58=^£'— <
E'B''=ii i5 18 83 (339)
EiBii=ii 1 5 18 83=^"JS'— cf";
Sum 2S'4 =26 o3 49 4i ; «4 =i3''oi"'54»,7i
Uiff.iiD, = 3 33 II 75 ; Z)^ = i 46 35 88
J)^ sin. 8,49i4o56
S^ arith. co. sin. 0,6468671
^BE'Bii= 3<56'"48»,85 (337)cotan. 1,1996098
è(B"— JS)= 65 19 46 66
k{B"+B)= 86 28 3g 26
Sum is B"=i5i 48 25 92
Diff. is B=^ 21 08 52 60
tang. 0,3378825
(337)
(338)
(340)
Pcelimin-
«ry calcu-
latione.
£=2i<io8'"52',6o (343)
E"= 4 55 46 22 (336)
Sum 2 «5 =2604 38 82
Diff 2i>5=i6 i3 06 38
^5 =l3''o2mi9',4l
Us = 8 06 33 19
D^ sin. 9,i494o55
S5 arith. co. sin. 0,6466425
4BE"=5<'39'»34«,o5 (345') tang. 8,996067g
i(E''B"— £'B*)=3 32 43 98
à(E"B'4-B'S*)=5 45 01 93
tang. 8,7921 1 5g
-D4
«4 arith.
iBE'B" (337)
COS.
CO. CO.S.
cotang.
9.99979' 2
o,oii33ig
1,1996098
(341)
i(B"-)-.B)=86<i28m39.,26
tang.
1,210732g
(342)
(343)
riangle Ei'BB', by [i345so>*'].
^£"=29<'43'"o7»,3o (335)
^B=i8 23 5g 20 (3i4)
(344)
(345)
BE"=ii ig 08 10
(3450
S,
hBE' (345')
è(£;".B*-fB'B*)=5''45'«oi',93
COS. 9,9g56356
arith. co. cos. o,oii343g
tang. 8,gg6o679
tang. 9,oo3o474
To find i »,
.î'£"=(336) 4i''i3"=57»,6
AE'—S, AEi'—f, .l'E—J-'+f*, AiEi'—J'-Jri*,ii.c.
J-'— <f"=^î'B*=3i 56 II 77
<r'=.4'B'=(3i6) 32 19 24 93
i*=B'B*=z o 23 i3 16
.î£'(33o)=33''i2"'29',78
J- (3i4)=i8 23 5g 20
AE'-i=i4 48 3o 58
AE'—i sin. g,4o75423
COS. 9,9853302
^£"(335)=29''43'»07',3o
<r (3i4)=i8 23 5g 20
AEi'—f=\i 19 08 10
AE'i—i sin. g,2928537
.4'£(325)=52<'o6m26',4o
.r'-J*(349)=3i 56 II 77
A'E-i '-fr=2o 10 14 63
■t'E—i '-I-J » sin. g,5375gog
(346)
Sum E''B*=Q 17 45 gi. The sum is talsen because £"B* is opposite to the greatest of the two angles S E" :3-iî)
(348)
(349)
(350)
(351)
(351')
j}'E(325)=52<'o6">26',4o A<E" (336)=4i''i3"'57«,68
J'(3i6)=32 19 24 g3J i< (3i6)=32 ig 24 g
AiE—î'=\g 47 01 47
AiEii—t'= S 54 32 75
^'E"(336)=4i''i3"'57',68
J~'— #"(34g)=3i 56 II 77
WEii—S'+i*= 9 17 45 gi
A'S'i—i-JTS' sin. 9,2082704
./Î"£(326)=6i<'5i'«i5',i6
<f"(3i9)=43 II 42 o5
^"-B— <r"=i8 39 33 II
A"E—ii' sin. 9,5o5o663
!l''E' (33o')=54''27'"oo',88 (s.yj)
<f" (3i9)=43 IT 42 o5,353)
A'iE'—i''==i\ i5 18 83(354)
A"E'—i " sin. 9,2904350 (355)
COS. 9.ggi566[ (356)
To find iî.sin.J
R (3o9) log. 9,9996826
f (3i4) sin. 9,499'994
iJ.sin.J~ log. 9,4988820
J?'.sin.<f'
iî".sin J~".
R' (3o9) log. g,g98o97g
f' (3i6) sin. 9,7281105
iî'.sin.if' log. 9,7262084
R" (309) log. 9,9969678 ,357)
J"" (3ig) sin. 9,835363i (358)
i}'',sin.<f'" log. 9,83233o9 (gjm
VOL. 111.
225
898
APPENDIX,
BY THE TRANSLATOR
)
[5999]
To find a, b, c, d, e. (32—36).
(3ti0)
(3G1)
(302)
(3C3)
A"E'
AE<-
iî.sin
fl"sin
.y
' (355)
(35i) arith.
(359)
(359) !"■'*•
sin.
CO. sin.
log.
CO. log.
9,2904350
0,5924577
9,4988820
0,1676691
Ai'E—S" (355)
A'E-S'+S-* (35i)
W.smJ-i (359)
Rii.sin.e-" (359)
sin.
arith. co. sin.
log.
arith. co. log.
9,5o5o663
0,4624091
9,7262084
0,1676691
(364)
3=0,3543593 (32)
2
log-
log.
9,5494438
6 (33) log.
J-* (35 1 ) secant
9,8613529
(365)
o,3oio3oo
0,0000099
(366)
(367)
3.1og.(iJ'.sia.J'') (359)
i* (35i)
c-i (34)
c
sin.
log-
log.
9,1786252
7,8295726
ft.sec.cf "=0,72671 28
1^0,3543593
ft.secf*— 3^0,3723535
J.sec.J* — 1= —0,2732872
d=— 1,3624994
log.
(364)
log-
log, subtract
(35) log.
9,8613628
(368)
(369)
7,3092278
2,6907722
9,5709555
9,4366i92„
(370)
o,i343363„
(371)
(372)
J« (35i)
t.sec J »— 1 (369)
tang,
log. subtract
7,8295825
9,4366 1 92„
(373)
e (36) log.
8,3929633n
(374)
/{.sin.tT (359)
(375)
AEi—S (36i)
(376)
■X=L\
,2340696
(377)
JlE'—f (35i')
(378)
R.s\n.t (359)
!
(379)
To find K, «.", X, k". (44—47)-
R'l.sin.J-" (359)
M'lE'—J-" (355)
log. 9,4988820
arith. co. sin. 0,5924577
log. 0,0913397
cos. 9,9853302
arith.co.log. o,5oiii8o
(46)
log. o,,
log. 9,83233o9
arith. co. sin. 0,7095650
»"=3,4825384 (45) log- 0,5418959
Mi'E'—f" (356) COS. 9,991 566i
Ri'.sin.S^" (359) arith. co. log. 0,1676691
\" (47) log- 0,1592352
First
approiT-
ointion.
To find the j
(380)
«'— f=ii,96324i (3io)
k (54)
(381)
t" (229)
T (38 1) su
(382)
i'=— = 1,1997804 (259)
T
a= 0,3543593 (364)
(i = — 1,3624994 (370)
(383)
(384)
(385)
(386)
(387)
First Apphoximation to P, Q.
To find the first values ofP, Q, w, Q' and the equation in z. (4i')-
log. 1 ,0778489
log. 8,23558:4
log. 9,3i343o3
subtract log. 9,2343285
log. 0,0791018
P-|-a= 1,5541397
P+d^ — 0,1627190
e (373)
w=I3*I6"'54^77 (4o)
J-»= 23 i3 16 (35i)
log. 0,1914901
log. CO. 0,788561 8n
log. 8,3929633n
tang. 9,373oi52
("— ('=9,971192 (3 10)
k (54)
log. 0,9987471
log. 8,23558i4
T (229)
t" (38i)
c (369)
w (386)
log. 9,2343285
log. 9,3i343o3
Q = tt" (259) log. 8,5477588
log. 2,6907722
sin. 9,36i24o4
Q' (4o')
log. 0,5997714
Hence the equation (4i') becomes,
o,59977i44-4.1og.sin2;— log.sin.(2;— i3''4o"'o7'',93)=o.
w+if* =i3 4o 07 93
To find z by approximation from the preceding equation, (386.)
By a slight inspection of the table of log. sines, we find that « = i4'' may be assumed for a first process,
in the following table ; and î= i5'' for a second process. The errors of these assumed values leads to a third
value i4''45"', and so on, by repeated operations as in the following table, till we get the correct value of z.
In the same way we may find the other values of z, which satisfy this equation; as in the second example of
the table.
COMPUTATION OF THE ORBIT OF A PLANET.
899
[599U]
Assumed value ot z,
Its lo;:;. sine.
1 4-'
9.384
9,4 1 3
i4''45™
9,406
7,624
0,600
i4'^3o"'
9,3986
9,4oi52
i4''35'"
9,4oio3
i4''35'"09>
9,401 II
32<i2m
9,72461
32"3"'
9.72481
32''2'"26''
9,72470
Muhipliod by 4,
A.M lo^.Q',
7,536
o,6oo
7,652
0,600
7,5944
0,5997
8.1941
8,i6i5
7,60608
0,59977
8,2o585
8,21086
7,6o4i2
0,59977
8,2o38g
8,203o2
7,60444
0,59977
8,89844
0,59977
8,89924
0,59977
8,89880
0,59977
9,49857
9,49856
0,00001
v; — i3''4o"' 7',g3)leg sine,
Uifiereiice,
8,i36
7,762
8.252
8,366
8,224
8,276
8,20421
8,20421
9,49821
9,49839
— 0,00018
9,49901
9-49877
4-0,3^4
—0,1 14
— o,o52
-|-o,o336
— o,oo5oi
-{-0,00087
0,00000
-|-0,O0024
(.TSS)
Hence we find tliat the value of z, corresponding to this equation is z =i4'^35'"09S; the other value
z =32''2'"26» is nearly equal to J" '=32''r9"'24',93 (3i6), and is to be neglected, as in (i57, &c.)
To find r< (jj), and the factors (4i", 4i"')-
iî'.sin.if' (359) log. 9,7262084
z=i4''35"'09» (390) sub. sin. 9,4011076
log. o,325ioo8
r> (77)
.f • (35i)
z—i* =i4''ii'"55',84
23"" 1 3', 1 6
.4'£-J'(354)=i9''47"oi',47
z=i4 35 09 00
z-f-^'£-J''=34 22 10 4:
Its log. sine =9,7516861
j'£'/_j>'(354)=8''54"'32',75
(390) z^i4 35 09 00
z-|-jî'£"— <f '=23 29 4i 75
Its log. sin. =9,60061 13
I Ir'r"] " S
iî'.sin.,r' (359)
P+a (383)
log. 9,7262084
log. 0,1914901
b (364) log. CO. 0,1 38647 1
z— if* (394) arith. co. sin. o,6io324o
(4i") log. 0,6666696
P (382) subtract log. 0,0791018
(4i"') log. 0,5875678
(.TOO)
First
Approxi-
matluii.
(;»1)
1 392)
(394)
(395)
(39(5)
(397)
(398)
To find p, (p"), (48,49) ; q, q" (5o, 5i) ; ^, ^n, r, r" (52, 53).
(394)
I :r'T"] s
z-\-^'E—i' (398)
E (326)
E' (33i)
P (48)
'^ (379)
X. = 1,2340696 (376)
X px =3,1977206
q = xpjt — X = 1 ,96365io (5o)
P (4o3)
— = tang.f (52); f=23'^i7"'33',38
e'= CE' (409)
q (407)
r (52)
log. 0,6666696
sin. 9,75i686i
sin. 8,6o83885
arith. co. sin. 0,9003084
log. 9,9270526
log. 0,4864482
log. 0,0913397
o,5o484o5
log. 0,2930643
log. 9,9270526
■"I
(396)
['■'■']■
z-^^iE"-J-' (398)
E" (336)
E' (33 1)
log. 0,5875678 <399)
sin.
sin.
arith. co. sin.
9,6006113
8,9341447
o,9oo3o84
P" (49) log.
(379) log.
x"= 3,4825384 (376) log.
>"p"»"= 5,2937488 log. 0,7237633
0,0226322
0,1592352
0,5418959
tang. 9,633g883
sec. 0,0369220
log. 0,2930643
log. 0,3299863
--k"p"k" — H." = 1,8112104 (5i) log.
(4o3) log.
0,2579689
0,0226322
T, = tang.,>-"(53) ; ^" = 3o'*i i'"o4',25 tang. 9,7646633
?"=C"E' (409)
q" (407)
r" (53)
sec.
log.
0,0632798
0,2579689
(400)
(401)
(403)
(403)
(404)
(405)
(406)
(407)
(408,
(409)
(410)
(411)
log. 0,3212487 (412)
900
[5999]
First
Approxi-
mation.
(413)
(413')
APPENDIX, BY THE TRANSLATOR ;
Tofijld the arc CC" = 2f', in the triangle CE'C", by [i345'"''"].
Ç=CE' = 53<'i7'n33»,38 (409)
}ii=aiEi = 3o n o4 25 (409)
Sum 2S6 = 53 28 37 63 ;
Diff. 2D^ = 6 53 3o 87 ;
Ss =26'*44"'i8',82
J5e = 3 26 45 M
(414)
(415)
1416)
(417)
-De
hE'
sin. 8,7789252
arith. co. sin. 0,3468646
3''36"'48«,85 (34i) cotan. 1,1996098
è(C'— 0")= 64 4i 56 92
è(C-f C")= 86 45 58 08
SuraisE'CC"=i5i 27 55 00
Diflf.is£'C"C= 2204 01 16
tang. 0,3253996
Se
à-B' (340
COS.
arith. co. cos.
cotan.
9,9992141
0,0491 i5i
1,1996098
è (C-{-C") =86''45"'58',o8
tang.
1,2479390
£'C"C (4 16)
CE' (4 1 3)
E' (33i)
arith. co. sin.
sin.
sin.
0,4261700
9,5970663
9,0996916
2f=CC" = 7''36"'32'.42
sin. 9,1219279
(418)
(419)
(420)
To find the arcs CC'=7f" , C'C"=2f, (214, 2i5).
r (4i2) log. 0,3399863
5 ILLJ .r' ( (394) arith. co. log. 9,33333o4
jf'=ca' {All) «'°- 9>i"9'79
2/=C'C" =3''29'"47',5o sin. 8,7862446
2f'=CC' =4 06 44 95 (4i9)
(421) Sumis2/'=CC" =7 36 32 45
Computed CC"=7 36 32 42 (4i7)
r" (4 12) log. 0,3212487
< -7 — Pi •'■' ( (396) arilh.co.log. 9,4124322
2/ (417) sin. 9,1219279
3/i'=CC'=4''o6'n44',95 sin. 8,8556o88
(423)
(4231
(424)
(425)
(42G)
(427)
(428)
To find f, , f/, f," ■ in
To find fi and f, (224).
.4£'=33<ii2"'29»,78 (33o)
,>-=23 17 33 38 (409)
AE'—^= 9 54 56 40 sin. 9,236o3
r (4i2)
i (358)
f/ (224)
log. 0,32999
arith. co. sin. o, 60080
order to correct t, V, t", t, t", for the aberration, (223)
To find f/ andt„ (226).
■T' =32<'i9m24',93 (3 1 6)
z =i4 35 09 (390)
sin. 9,48382
log. o,325io
<r'— 2=17 44 i5 93
log. 0,06682
Constant log. of aberration 7,76633
Correction £î=o,oo6665 log. 7,82316
Observ. Oct. 5,458644 (3io)
(429) Corrected Oct. 5,461989=^.
(4301
(431)
r' (392)
J-' (358) arith. co. .sin. 0,271?
(223) Constant
log. 0,
log. 7,75633
To find f/' and t^ (226).
^"£'=54''27»oo',88 (33o')
^"=3o II o4 26 (409)
^"£'—^"=74 1 5 66 63 sin.9,6i38i
r" (412) log. 0,32125
j" (358) arith. co. sin. 0,16464
log. 0,09970
log. 7,76633
(223) «2 = 0,006873 log. 7,837i4
Oct. 17,421885 (3io)
Oct. i7,4i5oi2=(', corrected.
Oct. 5,45ig89^i, corrected.
Int. r—<=t 1,963023 log. 1,0778409
Constant ft (64) log. 8,23568i4
Corrected t" (229) log. 9,3i34223
f," (226)
Constant
(3= 0,007178 log. 7,856o3
Oct. 27,393077 (3io)
Oct. 27,385899=(", corrected.
Oct. i7,4t5oi2=:<', corrected.
Int. t"—t'= 9,970887 log. 0,9987338
Constant ft (54) log. 8,23558i4
Corrected t (22g) log. g,2343i52
COMPUTATION OF THE ORBIT OF A PLANET.
To find y" from r,r', ij"s V—t. Li'A-e [5995(187)].
r' (39.)
r (4i2)
log. o,325ioo8
log. 0,3399863
(■'
=t,Ulg4.(j5'Ll-!t') log. 9,9951145
r
45*fio=44''55"'o9'.957 tang. 9,9987786
u;= — 4"5o»,o43
iw= — 9'"4o',o86
(39.)
(412)
o,325ioo8
0,3599863
sum 0,6550871
half 0.3275436
(it')* log. 0,9826307
arith. co. 9,0173693
tang. 7,44907,,
same 7,44907,1
y":=2<io3'"22',475 (420) sec. 0,00028
tang2.2t/j.sec/""=o,ooooo79i log. 4,89842
constant log. 5,5680729
V — t 11,963023 (43o) log. 1,0778409
same i ,0778409
3Xlog.sec./" (439) 0,0008391
^.log. {rr') arith. co. (433) 9,0173693
log. 6,7419631
/"^2''o3"'22*,475 (435) sec. 0,0002797
4/"=i 01 4i 2375 sine 8,2538985
same 8,2538g85
?in2.4/".sec/"=o,ooo322i6 log. 6,5060767
tang2.2i/;.sec/"=o,ooooo7gi (436)
/=o,ooo33oo7
1=0,83333333
i-|- 1=0,83366340 subtract log. 9,9209908
mm (438) log. 6,7419631
A = 0,0006621 7 log. 6,8209723
Corresponds in Table VIII, toapp. log. y"y"=o,ooo6383
log. y"^o,ooo3i92
To find P.
arith co. log. 9,9996808
log. 0,0002285
log. 9,3i34223
arith. co. log. 0,7656848
y"
(447)
y
(447)
t"
(43i)
T
(43i)
Corrected P= ?^ (235)
y'V
Assumed value of P (382)
log. 0,0790164
log. 0,0791018
Difference — o,oooo854
JCa
We may remark that the value of ft (445) does
not require, in this example, any correction for the
quantity | [5995(14-)], which is wholly insensible.
To find y from r',r", 5/, ("—('.
r" (4i2) log. 0,3212487(412) . . . 0,3212487
r' (3g2) log. o,325ioo8 (392) . . . o,325ioo8
^ =tang'i. (45''-|-u>) log. 9 996 1 479
45''-4-io=44''56'"ii',3o2 tang. 9,9990369
w= — 3"'48',698
2t«= — 7'"37',396
sum 0,6463495
half o,323i748
(r'r''ys log. 0,9695243
arith.co.log. 9,0304757
tang. 7,34587„
same 7,34587»
/=i''4.4»'53',75 (419) sec. 0,00020
tang2.2tc.sec./=o,ooooo492 log. 4,69194
[5995(38)]
I'l—V
|.Iog. {r'r")
constant log. 5,5680729
9,970887 (43o) log. 0,9987338
same 0,9987338
3xlog.sec./ (439) 0,0006066
arith. co. log. (433') 9,o3o4757
901
[.5999]
(433)
Firsl
Approxi-
malloil.
(433)
(433')
(434)
(435)
(4311)
(430
(437)
/=i''44"'53s,75
4/=o52 26 875
7)1 m
(435)
log. 6,5966228 (436)
sec. 0,0003022
sine 8,1834375
same 8, 1834375
sin2.4/.sec./=o,ooo23285 log. 6,3670772
tang2.2«i,sec./^rf3,ooooo492 (436)
I ^0,00033777
I =0,83333333
Z-|-|- = 0,833571 10 subtract log. 9,9309427
mm (438) log. 6,5966228
[5995(147)] ft =0,00047389 log. 6,6756801
Corresponds in Table VIII, to app. log. yy =0,0004570
' log. y ^0,0002285
To find q.
T (43i)
t" (43 1)
2.log. r' (392)
r (4i2)
r" (4i2)
y (447)
y" (447')
/ (439)
y (4i7)
/" (439)
Corrected Q (256)
Assumed Q (382)
log. g,3343i52
log. 9,3 134223
o,65o3oi6
arith co. log. g,6700i37
arith. co. log. g,67875i3
arith. co. log. 9,9997715
arith. co. log. 9,9996808
secant 0,0003022
secant 0,0009581
secant 0,0002797
log. 8,5475964
log. 8,5477588
Difference — 0,0001624 ('*53)
(430)
(410)
(441)
(443)
(443)
(444)
(44i)
(44fi)
(417)
(447')
(447 )
(447 )
(448)
(449)
(450;
(151)
(45S;
(453)
VOL. III.
226
902
APPENDIX, BY THE TRANSLATOR;
[5999]
Spcolld
Approxi-
mation.
(454)
(455)
Second Approximation to P, Q.
With the corrected values ol t, t", (43i), and the computed vahies of P, Q (448, 45i), we must repeat
that part of the calculation, which is contained in (382 — 453), in order to obtain a nearer approximation to the
values of P, Q. We shall give this calculation at full length, and in the same form as in the first process
(382^453); but the part (422 — 43i) relative to the aberration, is given with sufficient accuracy ; and it is
not necessary to make any correction in it. The labor of this re-computation is much decreased from the
circumstance that the same form of calculation is retained, and the results are not much varied.
(456)
(457)
(458)
(450)
(460)
P= 1,1995445 (448)
a= 0,3543593 (364)
d= — 1 ,3624994 (370)
log'. 0,0790164
P-\-a= 1,5539038 log. 0,1914342
i'-(-d=— 0,1629549 log. CO. 0,78793 26„
e (373) log. 8,3929633,.
w=i3'^i5'"4i',oo tang. 9,3733201
J-'= 23 i3 j6 (35i)
w-|-cr*=i3 38 54 16
Q (45i)
c (369)
w (458)
Q' (4o')
log. 8,5475964
log. 2,6907722
sin. 9,36o58i8
log. 0,5989504
Hence ihe equation (4i') becomes,
o,59895o4-|-4.log.sin.s— log.sin.(z— i3<'38"'54',i6)=o.
To find z hy approximatiun from the equation, (459).
(461)
(402)
(403)
Assumed value of-,
Its log. sine.
Multiplied by 4,
Add log. Q',
i4''35"'
9,40103
i4''33»'
9,40006
i4"33'"23»
9,4oo25
7, 604 1 2
0,59895
7,60024
0,59895
7,(5oioo
0,59895
Sum,
(j— i3<i38™54',i6)sine,
8,2o3o7
8,21265
8,19919
8,19688
8,19995
8,19995
0,00000
Difference,
— 0,00958
-|-0,002 3l
This operation is much abridged, because we are able
to assume, in the first operation, the value of z,
computed in (3go), which varies but very little from
the result here found, namely z = i4'' 33™ 23'.
To find T> {-j-j), and the factors (4i", 4i"')-
i?'.sin.<r' (359)
(464)
(465)
(466)
(407) "f * (35i) o''23"'i3',i6
(468) z— <f* =i4''io"'o9»,84
(469) AiE—Si{Zg^)=i^''ATo\'Ai
(470) 2 (462)=i4 33 23 00
log. 9,7362084
i4 33'"23' (462) sub. sin. 9,4002490
r' log. 0,3259594
(471) J+-«'-E-<r'=34 20 24 A
,,,^3j Its log. Bine =9,7513596
[T'r"-\- S
,4'£//_j'/(395)=;8<'54'"32»,75
r=i4 33 23 00 f [rr''^ j
z-\-Ji'E^'—i'=73 37 55 75 \ ['■'■'] '^ !
Its log. sin. ^9,6ooo9"5
JÎ'.sin.J'
P+a
6 (392)
Z-Î'
(4i")
(359)
(457)
log. 9,7262084
log. 0,1914242
log. CO. o,i38647i
(468) arith.co.sin. 0,6112070
log. 0,6674867
P (456) log. subtract 0,0790164
(4i"') log. o,58847o3
COMPUTATION OF THE ORBIT OF A PLANET.
903
To^ndp.p", (48,49); q, q"(5o, 5i); f , <:" ; r, r" (52, 53).
[5999]
Second
Approxi-
z-\-.rE—f'
E
E'
(472)
(4oi)
(402)
aril)
X (4o4)
» =
P
1,2340696
(4o5)
xpx =
3,2oi3348
log. 0,6674867
I
(470)
log. o,5684703 (47,-))
sin. 9,75i3596z_(-.4'£"_^' (472)
sin. 8,6o83885
arilh. co. sin. O|goo3o84
'og- 9>9275432
log. 0,4864482
log. 0,0913397
log. o,5o533ii
q = xpx — ic = 1,9672652 log. 0,2938639
p (4-7) log- 9.9275432
^ = tang.^(52); ^=23''i6'"4o',26 tang. 9,63368o3
^^CE' sec. 0,0368739
q (48i) tang. 0,2938629
E"
El
(4oi)
(402)
sin.
sin.
aritli. CO. sin.
(4o4)
log.
log.
9,6000975
8,9341447
o,9oo3o84
0,0230209
0,1592352
0,541895g
log. o, 3307368
«"=3,4825384 (4o5) log
>." p"x''= 5,2984890 log. 0,724l520
q" = >."p"K" — x" = 1,8 159506 log. 0,2591040
P" (477) *"''• '°g- 0,0230209
1)"
!-jy = tang.^"(53); f "=3o<io8'"3o»,24 tang. 9,7639169
ÇI = O'E'
q" (48 1)
log-
log.
0,0630914
0,2591040
0,3221954
(474)
(475)
(470)
(477)
(478,
(479)
(480)
(481)
(483)
(483)
(484)
(485)
(486)
To find the arc CO' = 2/', in the triangle CE'C".
^=CE' = 23''i6'n4o',26 (483)
Çii—C'iEi = 3o 08 3o 24 (483)
Slim 28-, = 53 25 10 5o ;
Diff. 2J>7 = 6 5i 49 98 ;
s,
iE' (4i4)
Sj =26''42'"35',25
Dt= 3 25 54 99
sin. 8,7771576
arith. co. sin. 0,347297
cotan. 1,1996098
^(C— C") 64''37'"5i',8o
1(0-)- C") 86 45 55 3i
SumisE'CC''=i5i 28 47 "
Did. is£'C"C= 2208 o3 5i
tang. o,324o65i
-O7
■St
àE' (4i4)
.J (C-\-C") =86<'45'n55',3i
E'Ci'C (490)
CE' (487)
E' (4i6)
of=CC" = 7''34'»56',36
(487)
COS. 9,9992204
arith. co. cos. 0,0490053
cotan. 1,1996098 (488)
tang. 1,2478355
arith. compl. sin. 0,4239133
sin. 9,5968064 (^83)
sin. 9,0996916 (490)
sin. 9,i2o4ii3 (451)
To find the arcs CC'=2f" , C'C"='2f, (214, 21 5).
r (486) log. o,33o7368
jl^.r'^ (473) arith.co.log. 9,3325i33
ifi=CC" (491; ein. 9,12041
2f=C'C" =3'^29'»oi',64 sin. 8,7836614
2/"=CC'=4 o5 54 75 (493)
Sum is 2/= CC" =734 56 39
Computed CC"=7 34 56 36 (491)
r" (486)
2f (491)
2/"= CC'=4''o5"'54',75
log. 0,3221954
arith. co. log. 9,4115297
sin. 9,i3o4ii3 (493^
sin. 8,854i364 .493)
(494)
(495)
904
APPENDIX, BY THE TRANSLATOR ;
[5999]
(490)
Hpcoml
Approxi-
mation.
(497)
(497')
(493)
(499)
(500)
(501)
(502)
(503)
(504)
(503)
(506)
(507)
(508)
(5091
(510)
(511)
(512)
C513)
(514)
(515)
(516)
(517)
To find y" frotn r, r', if", t'—t, (432—44?)
log.=o,3259594
Iog.=o,33o7368
(466) .
(486)
;o56
-=tang4. (45<'-(-u>) log. 9,9952226
r
45i*_|_îo=44'*55'"i6%37 tang. 9
«.= — 4"'43',63
2t0= 9"'27',26
/"= 2<'o2™57',375 (493)
tang2.2tt).sec/""^o,ooooo757
0,3359594
0,3307368
r"
sum 0,6566962 _=tang4.(45<'4-«)) log. 9,9962360
half 0,3283481 " — .. ■
{rr'y^ log. 0,9850443
arith. co. 9,0149557
45*+i«=44''56"'i6',55 tang. 9,9990590
w= — 3'"43
tang. 7,43936„
same 7,43936„
sec. 0,00028
log. 4,87900
(436') constant log. 5,568072g
t'—t (437) log. 1,0778409
same 1,0778409
3X>og sec./" (5o3) o,ooo8334
J. log. {rr') arith. co. (497') 9!Oi49557
m m log. 6,7395438
f''=2do2"'5-j',3-j5 (499) sec. 0,000277
è/"=i 01 28 688 sine 8,2524236
same 8,2524236
sin3.jy"'^sec.y '=0,00031998
tang2.2M).sec.y '=0,00000757
Z^o,ooo32755
s=o,83333333
log. 6,5o5i25o
(5oo)
Z-f-|- = °>83366o88 log. sub. 9,9209896
m m (5o2)
h = o,ooo6585o
log. 6,7395438
log. 6,8185543
Corresponds in Table VIII, to log. y" y" =^ 0,0006347
log. y" = o,ooo3i73
To find y from r', r", rf, ("—<'.
-'' log.= o,332i954 (486) ... 0,3221954
log.= 0,3269594
(466)
■II
To find P.
y" (5ii) arith.co.log. 9,999682
log. 0,0002271
log. 9,3i34223
arith. co. log. 0,7656848
y (5n)
t" (447")
T (447'")
Vt"
Corrected P=^,— (235)
y"T
Assumed value of P (456)
log. 0,0790169
log. 0,0790164
Différence -|"°>°o°°°o5
. . . 0,3269594
sum 0,648 1 548
half 0,3240774
(r'r")- log. 0,9722322
arith. co. 9,0277678
tang. 7,33578„
same 7,33578h
sec. 0,00020
log. 4,67176
2W^= — 7'"26^',90
/=i''44"'3o',82 (493)
tang2.2M). sec./ =0,00000470
(436') constant log. 5,6680729
f'—t' (437) log. 0,9987338
saine 0,9987338
3X log. sec./ (6o3) 0,0006021
^.log. {r'r") arith. co. (497') 9,0277678
m m log. 6,6939104
/= I ''44"" 3os,82 (499) sec. 0,0002007
4/=o"'62'"i6»,4i «'°^ 8,1818625
same 8,1818625
secS.J /.sec./^o,ooo23ii6 log. 6,3639067
tang2.2Ui.sec./=o,ooooo470 (600)
I:
5 .
= 0,00023586
= 0,87333333
Z-)-|= °.833569i9 log, sub. 9,9209417
7/1 711 (602) log. 6,6939104
h = 0,00047094 log.
6,6729687
orrespond
i in Table VIII, to log. yy
log. y
To find Q.
0,0004541
0,0002271
1
(447')
log-
9,2343i52
t"
(447")
log.
9,3 1 34223
2.log. r'
(466)
0,6619188
r
(486)
arith. co. log.
9,6692632
r"
(486)
arith. co. log.
9,6778046
y
(6.1)
arith. co. log.
9.9997729
y"
(6.1)
arith. co. log.
99996827
/
(5o3)
sec.
0,0002007
/
(49 >)
sec.
0,0009614
f
(6o3)
sec.
0,0002778
Q
Assumed value of Q (456)
Difference
log. 8,54''6o96
log. 8,54-i6964
-f- 0,0Û00l32
COMPUTATION OF THE ORBIT OF A PLANET.
905
Third Approximation to P, Q.
[5999]
Third
Approxi-
umtioD.
With the computed values of P, Q (5i2, 5i5), we must ap;ain repeat the operation, as in (456— 5i7) to (518)
obtain the final vahics of P, Q. The form of calculation is the same as in the last process, anil the numbers
vary but very little, so that the calculation is repeated with great facility ; and it serves as a verification of
the process.
P= 1,1995459 (5l2)
a= o,35435g3 (364)
d^ — 1,3624994 (370)
loff. 0,0-90169
P-(-a= 1,5539052 log. 0,1914546
P-|-d= — 0,1629535 log. CO. 0,7879363,,
e (373) log. 8,3g2g633n
w=i3<fi5'"4i ',44 tang. 9,3723242
J"= 23 i3 16 (35i)
w-)-<f '=i3 38 54 60
Q (5i5)
c (36o)
W (522)
Qi (4o')
log-
log.
8,5476096
2,6907722
9,36o5857
log. 0,5989675
Hence the equation (4i') becomes,
0,598967 5-|-4.1og.sin.ï — log.sin.(« — i3''38"'54',6o)=o.
(519)
(520)
(521)
tS22)
(523)
(524)
To find z hy approximation from the equation, (523).
Assumed value of r.
Its log. sine,
i4''33"'23^
9.40025
l4''33"'23^8
9,4002555
i4''33'»23»,72
9,4002548
Multiplied by 4,
Add los. Q',
7,60100
0,59897
8,19997
8,19989
-|-o,oooo8
7,t3oiO220
0,5989675
8,1999895
8,1999982
— 0.0000087
7,6010192
0,5989675
Sum,
(:— i3<'38"'54'.6o)sine.
8,1999867
8,1999875
—0,0000008
Difference,
(525)
The value of z, obtained in (462), is here
assumed as the first operation, and by a very easy
calculation we find 2:=i4''33"'23^,72 nearly. (50g)
(527)
To find r' {n), and the factors (4i", 4i"')-
R'.im.i' (464) log. 9,7262084
Ï i4''33"'23»,72 (525) sub. sin. 9,4002548
rl
log. 0,3259536
t* (35i) o''23"'i3',i6
2— <f*=i4''io'"io«,56
.«'£— J' '(469)=i 9''47'"o I ',47
z (526)=i4 33 23 72
z-\-A'E—i'=M 20 2 5 19
Its log. Bine =g,75i36i8
.4'£//_j/(469)=8''54°'32',75
2=i4 33 23 72
z-(-.4'£"— ^'=23 27 56 47
Its log. sin. =9,6001010
P'.sin.J' (464) log. 9,7262084 (528)
P-(-a (520) log. 0,1914246 (529)
b (466) arith.co.log. 0,1 386471 (530)
z—i* (532) arith.co.sin. 0,6112011 (531)
^rr7!l-'4 (4'") 'og- 0.6674812 (532)
P (519) log. subtract 0,0790169 (533)
H-^.r'^ (4im) log. 0,5884643 (534,
(535)
(S3e)
VOL. III.
227
906
APPENDIX BY THE TRANSLATOR ;
[5999]
Third
Approxi-
mation.
(537)
(538)
(539)
(540)
To find p, p", (48, 49) ; q, q" (5o, 5i) ; ^, ^" ; r, r" (52, 53).
I [rV] S
z-\-A'E—i'
E
E'
(532)
(536)
(475)
(476)
log. 0,6674812
sin. 9,75 1 36 1 8
sin. 8,6o83885
cosec. o,9oo3o84
(541)
(542)
(543)
P
X (478)
x = 1,2340696
xpx =3,2oi3io4
q = xpx — « = i ,9672408
P (54i)
^ = tang.^=23''i6»4o',62
^=CE'
q (545)
r
(479)
log
log.
log.
log-
log.
. sub.
log.
tang.
sec.
tang.
log.
9,9275399
0,4864482
0,0913397
(544)
(545)
(546)
0,5053278
0,2938575
9,9275399
(547)
9,6336824
(548)
(549)
0,0368743
0,2938575
(550)
o,33o73i8
(534)
log. 0,5884643
x-\-AiE'i-S-> (536)
E'l (475)
E' (476)
sin.
sin.
cosec.
9,6001010
8,9341447
o,goo3o84
P" log.
X" (478) log.
x"= 3,4825384 (479) log.
o,o23oi84
0,1592352
0,5418959
^"p"K"= 5,2984585
log-
0,7241495
q" = x"p" x" — x."= 1,8159201
p" (54 I)
p"
!-j5 = tang.4-"=3o''oS'"3i»,23
f " = C"Ei
q" (545)
log, sub.
log-
tang.
sec.
log.
0,2590967
o,033oi84
9,7639217
0,0630926
0,2590967
r"
log.
0,3221893
(551)
(552)
(553)
(554)
(555)
(55C)
(557)
To find the arc CC" = if', in the triangle CE'C".
^=CE' = 23'^i6'"4o',62 (547)
Çii=C"E' = 3o 08 3 1 23 (547)
Sum
Diff.
2Z»8 =
53 25
6 5i
II 85;
5o 61 ;
iE'
(488)
e
4(C-c")=
64'f37'»53'
8645 55
,3r
33
S^ =26'^42'"35s,93
Xio= 3 25 55 3i
sin. 8,777 It
arith. co. sin. 0,3472948
cotan. 1,1996098
tang. 0,3240734
(558) SumisE'CC"=i5i 23 48 64
(559) Difi.is£'C"C= 2208 02 02
(560)
-Da
«8
è-E' (488)
i(,C-\-C") =86<'45">55',33
cos. 9,9992204
arilh. co. cos. 0,0490060
cotan. 1,1996098
tang. 1,2478362
E'C'C (559)
CE' (55 1)
E' (490)
arith. compl. sin. 0,4239210
sin. 9,5968081
sin. 9,0996916
2f=Ca' = 7''34"'56»,95
sin. 9,1204207
To find the arcs CC'=7fii , OC"=-if, (214, 2i5).
(561)
(562)
(563)
(564)
r (55o) log. 0,33073 1 8
jfrin-'''^ (537) arith.co.log. 9,3325x88
•ifi=CC" (56oJ sin. 9,1204207
2/=3'^29'»oi',92 sin. 8,7836713
2/"=4 o5 55 07 (56i)
Sum is ■if=^ 34 56 99
Computed above =-j 34 56 96 (56o)
r" (55o)
(56o)
2y"=4''o5'"55>,07
log. 0,3221893
arith. co. log. 9,4ii5357
sin. 9,1204207
COMPUTATION OF THE ORBIT OF A PLANET.
907
To find y" from r, r', nf", t'—t. (496—511).
log. 0,3259536
log. 0,33073 18
r'
-=!tang4.(45ii-|-u") log. 9,9952218
45*f io=44''55'"i6',32 tang. 9,99880545
tc= — 4°'43',68
2tc= — 9'"27',36
(53o) . . . 0,3259536
(55o) . . . o,33o73i8
sum 0,6566854
half 0,328342-
{rr')i log. 0,9850281
arilh. co. 9,014971g
tang. 7,43943,1
same 7,43943n
/"=2''o2"'57',535 (562) sec. 0,00028
log. 4,87914
tang2.2u?.secy "^0,00000757
(436')
t'—t
constant log. 5,5680729
(437) log. 1,0778409
same i ,0778409
SX'og-sec.y" (572) 0,0008337
â.log. (rr') arith. co. (566') 9,0149719
log. 6,7395603
/"=3(f02'"57*,535 (568) sec. 0,0002779
^/"=i 01 28 768 -sine 8,252433:
same 8,252433i
siQ2.4/".sec/"^o,ooo32ooo log. 6,5o5i44i
tang2.2ic.sec/"=o,ooooo757 (569)
/=o,ooo32757
5=0,83333333
6
J-|- ^=0,83366090 subtract log. 9,9209895
mm (571)
7j=o,ooo65S52
log. 6,7395603
log. 6,8185708
Corresponds in Table VIII, to app. log. y"y"^o,ooo6348
log. y"=o,ooo3i74
To find y from r',r", 2/, t"—t'.
log. o,32 2i8g3
log. 0,3259536
=tang4.(45H-«') log- 9.9962357
45''-}-u'^44''56"ii6»,53 tang, 9,99905892
w= — 3'"43',47
2W=: — 7'»26',94
To
find
P.
y"
y
(580)
(580)
ari
t"
T
(447")
(447'")
arith co. log. 9,9996826
log. 0,0002271
log. 9,3i34223
log. 0,7656848
Corrected P= ^ (235)
y"r
Assumed value of P (5i9)
log. 0,0790168
log. 0,0790169
Difference :
-0,0000001
(55o) . . . 0,3221893
(53o) . . . 0,3259536
sum 0,6481429
half 0,3240714
(/r'')^ log. 0,9722143
'arith.co.log. 9,0277857
tang. 7,33582„
same 7,33582^
f = id44m3o',g6 (56i) sec. 0,00020
tang2.2tc.sec./=o,ooooo470 log. 4,67184
(436') constant log. 5,5680729
l"—t' (437) log. 0,9987338
same 0,9987338
3xlog.sec./ (572) 0,0006021
arith. co. (566') 9,0277857
[5999]
(5C5)
Third
Approsi-
matiuil.
(566)
(566')
(507)
(568)
(569)
(570)
g.log. (r'r")
/=i'*44"3os,96
4/=o52 i5 48
m m
(568)
log. 6,5939283 (571)
sec. 0,0002007
sine 8,1818622
same 8,1818622
sin2.4/.sec./=o,ooo23ii7 log. 6,363925i
tang2.2W),sec./=o,ooooo470 (569)
I =0,00023587
I =0,83333333
Z-|-| = 0,83356920 subtract log. 9,9209417
7nm (571) log- 6,5939283
(572)
(573)
(574)
(575)
(576)
(577)
h ^0,00047096
1 Table VIII, to app.
log.
log- yy =
6,6729866
(578)
Corresponds ii
=0,0004541
(579)
To find
Q-
log. y =
= 0,0002271
(580)
T
(447')
log.
9,2343 1 52
,"
(447";
log.
9,3i34223
2.log. r'
(530)
0,6519072
r
(55o)
arith.
CO. log.
9,6692682
r"
(55o)
arith.
CO. log.
9,6778107
(581)
y
(580)
arith
CO. log.
9,9997729
(582)
y"
(58o)
arith
CO. log.
9,9996826
(583)
/
(572)
secant
0,0002007
/'
(56o)
secant
O,ooog5i4
/"
(572)
secant
0,0002779
Corrected Q
log.
8,5476091
(584)
Assumed value of Q
(5i9)
log.
8,5476096
(585)
Difference= — o,oooooo5 (586)
The differences between the assumed and computed values of P, Q (583,586J, are so very small that it will
not be necessary to repeat the operation ; and we may suppose the expressions of r, ri'J ', deduced from (his
last calculaUon to be their true values ; from which we may deduce the elements ot the orbit, in the following
(557)
908
[5999]
(388)
(589)
APPENDIX, BY THE TRANSLATOR ;
To compute the elements of the orbit.
We have for this purpose log. r=o,33o73i8 (55o) ; log. r"=o,322i8g3 (55o) ; 2/'=7''34m56,.96 (56o) ;
t=Oct. 5''''^',45i989 (429) ; «"=Oct. 27''*^',385899 (429), or i"-(=2i''^'",9339io. With these data we may
(601)
(602)
(603)
(604)
(605)
(606)
(607)
(608)
(609)
(610)
(611)
(612)
(613)
(614)
(615)
(616)
(617)
(618)
Computa-
tion of the
eleiaents.
(590)
(591)
déterra
r"
r — '=^"8
45<'-|-ic=
w=
310=
Ine the elements, by the meth
To find ï = sin2.i^. [5ç
r'l log.=o,322i893
r Iog.=o,33o73i8
«.(45''-H«) log. 9,9914575
44''5i'»32',85 tang. 9,99786438
od explam(
95(187)].
(588) . . .
(588) . . .
sum
half
{rr"^ log
arith. co.
tang.
same
) sec.
log.
)
) log-
same
) arith. CO.
log.
sec.
sine
same
;d in [0993]
0,3221893
o,33o73i8
•
Tc
2/'=
a
find
=■,-34''
\/—e
To find a, [5995(58)].
g (612) arith. compl.log.sin.
same
?n2
— (608) log.
2 log.
/' (595) cos.
VS^ (591") log.
a log.
p and e = sin. », [5g95(6o,
7c (54) arith. co. log.
t'l—t (5y6) arith. co. log.
>-r" (5gi') log.
'56s,g6 (588) sin.
yi (606) log.
S/P [5995(60)] log.
Va (594) log.
î=cos.? ; f=i4''i2ino5',3 cos.
1,2621295
(591')
(591")
0,6529211
0,3264606
1,2621295
7,27i6i36
(59S)
(592)
= _ 8"'27',i5
= _I6"■54^3o
/'=3-i47«.28',48 (58f
tang2.2t/).sec/'=o,oooo2423
constant log. (436
<i'_i=2i,9339io (589
3Xlog sec./' (595
f.Iog. (rr") (592
TO m
/' (595
èy'=i<'53'"44',24 (595
0,9793817
9,0206183
o,3oio3oo
9,9990486
(593)
(594)
(595)
7,6917447
7,6917447
0,0009514
0,3264606
0,4224118
(595)
5,38444o8
12.9)]-
(596)
5,5680729
i,34iii6o
i,34iii6o
0,0028542
9,02o6i83
1,7644186
8,658884o
0,6529211
9,1204208
(597)
7.2737774
0,0010819
(598)
(599)
(600)
0,0009514
8,5195500
8,5195500
0,1977264
0,21 12059
9,9865205
sIn2.i/'.sec./'=o,ooi0966i log. 7,o4oo5i4
tang2.2M).sec.y^o,oooo2423 (595')
Z=0,OOII2084
1=0,83333333
Z-}-§ = o,834454i7 log. sub. 9,9214025
S
m m (597)
h =O,00225l0
log. 7,2737774
log. 7,3523749
Corresponds in Table VIII, to y' y'= o,0O2i638
mm (597) log. 7,2737774
To find G, F, v, «", u, u", [5995(65,66, &c.)].
f (600) sub. sine 9,3897547
(612) cos. 9,9993497
cos.g'.cosec.?=4,0700o56 log. 0,6095950
— •;?' (591")
a (594)
/' (592')
(f (60 1 )
■V—cosf'.cosec.f^ — 3,2609391 log. o,5i33427»
ma
— =0,00186902 log. 7,2716136
Z ^0,001 12084 (602)
7772
X =—jr—l=sin^.ig ^o,ooo-j48i8 log. 6,8740061
àg=i'*34'"02*,64 sin. 8,4370o3o
g=3'io8'"o5',28
After finding a in the second column (594), we may
find the mean daily motion in seconds from [5995(67)].
a (594) log. ar. co. 9,5775882
its half 9,7887941
constant log. 3,55coo66
Daily motion 824',877 log. 2,91638891
log. 0,3264606»
ar.co.log. 9,577588a
cos. 9,9990486
cosec. 0,6102453
[5g95(65)] cos. G^ 0,8090665 log. 9,9079842
[5995(41,47)] G= 324''oo'"i7.,4 sin. 9,7691682»
/' (595) sin. 8,820342a
g (612,591) 3''o8"'o5',3 cosec. 1,2621295
F= 3i4''42"'5i'',4 sin. 9,8516399»
/'= 347 28 5 (595)
V =F—fi= 3io55 22 9 [5995(i3)]
i,"=F-(-/'= 3i8 3o 19 9 [5995(1 4)]
H=G — g= 320 52 12 I [5995(i5)]
u"=G+g= 32708 22 7 [59g5(i5)]
COMPUTATION OF THE ORBIT OF A PLANET.
909
We may remark that the expression of cos. G = 0,8090665 (608), corresponds to G=: 3î4''oo"'i7M or to
G= SS'^SgiTiîiî'.e. The first of these expressions is to be used as in (610), because the corresponding values of
v,v" (6i5,6i6), are in the fourth quadrant of the true anomaly, where the radii r, r" are decreasing, as in (588);
but tlie other value of G, gives v, v", in the first quadrant of the true anomaly, when the radii r, r" are
increasing. The mean anomalies nt=u—e.sin.u, »i("=u"— e.sin.u" [5985(7)], corresponding to the first
and third observations, may be found in the following manner.
To find nt, [5985(7)]. To find nV, [6985(7)].
e (601)
radius in seconds,
— sin.u (617) 32o''52"'ia',i
log. 9,3897547
log. 5,3i4425i
sin. 9,8000855
— e.sin.« = 3i934»,9^ 8 52 149 log. 4,5o42653
Mean anom. nf =329.44 27 o
e (601)
radius in seconds,
— sin.a" (618) 327''o8"'22«,7
log- 9.3897547
log. 5,3i4425i
sin. 9,7344746
-e.sin.M" = 27457^,1= 73737 I log. 4,4386544
Mean anom. nt" =334 45 Sg 8
To find Ui ?.
We may find the longitude of the node u, and the inclination ?, of the orbit to the ecliptic ; by means of the
triangle 'oJlC, in which we have given,
the angle u.4C=}.=i6''oo'"oS',38 (3i4) ; the angle uC.5=£'CC"=i5i''23"'48",64 (558);
.;3C=.4£'— C£'=33''I2»"29^78— 23''i6'"4o',62=9''55'"49',i6 (33o, 55i) ;
to find the angle u, and the sides uC, u.3, by Napier's formulas [i345'°'"J.
■uJiC= i6<'oo'»o8%38
uCjî=r5i 23 48 64
Sumis 285=167 23 57 02
Diff.is2Z)9=i35 23 4o 26
S,= 83''4i'"58',5i
X)9= 67 4i 5o i3
-D9
4./ÎC = 457 54 58
è(U.^— uC)= 4 37 23 46
è(u.«+t5C)=i6 43 18 36
Diff. is uC= 12 o5 54 90
sin. 9,96623 16
arith. co. sin. o,oo263io
(628) tang.
tang. 8,9077624
Sum is u-5= 21 20 4i
Lon.^(3o2)= 12 28 27 76
Diflt. is u=35i 07 45 9
uC^ i2''o5'"54',9
u (635) is ^35 1 07 45 9
Sum is long. ofC= 3 i3 4» 8
Sub.» (6i5)=3io 55 23 g
Long, perihelion
COS. 9,5792122
arith. co. cos. 0,9596291
tang
8,g388gg8 (631)
tang. 9,4777411
S,
à./ÎC(628)
4(u.«+uC)=i6''43'"i8',36
vC (632) sin. arith. co. 0,6786204
uv3 0(629) sin. 9,4403996
AC (628) sin. 9,2366642
^uC=i3''o6'"36',o sin. 9,3556842
52 18 17 9
From the time of the last observation, corrected for aberration as in (429), October 27, 385899 to the epoch
January I, i8o5, the interval is 64 ^'^,6i4ioi. Multiplying this by the daily motion 824^,877 (618), we get the
mean motion in that interval, i4''48'"i8s,6. Adding this to ni" = 334''45"'59s,8 (625); we get the mean
anomaly at the epoch, equal to 349''34"'i8*,4. This last expression being added to the longitude of the
perihelion 52''i8"'i7',g (636), gives the mean longitude at the epoch 4i''52"'36',3. Hence we have the following
elements of the orbit.
Elements of the orbit of Juno.
Log. of the mean distance a= 0,4224118 (5g4).
Log. of the semiparameter p = 0,3964528 (5g8).
Log. of the excentricity e= 9,3897547 (601).
Daily runlion 824',877 (618).
Inclination of the orbit to the ecliptic = i3''o6"'36',o (634).
Long, of the ascending node I7id07'n45«,9 (635).
Long, of the perihelion in the orbit 52''i8'"i7',g (636).
Mean longitude at the epoch 4i''52"'36',3 (639').
With the daily motion 824' ,877(618), the planet would describe the whole circumference 360* in about i57i
days, which represents the time of revolution of the planet. If we compare these elements of the apparent orbit,
corresponding to the epoch i8o5, with those in [40791], corresponding to the year i83i ; we shall find that they
VOL. III. 228
ji'a.
[5991']
(010)
(620)
(621)
(622)
(623,
(624)
(025)
(626)
(627;
(62«)
(629)
(63«;
(63S)
(633)
(63H)
(635)
(636)
(637)
(638)
(639)
(639-)
(640)
(64))
(642)
(643)
(044)
(645)
(646)
(647)
(648)
910 APPENDIX, BY THE TRANSLATOR.
[5999]
(649)
agree as well as could be expected, taking into consideration tiiat all the calculations in this article are deduced
"(650) from the motion of the planet in a geocentric arc of less than four degrees. These elements were sufficiently
accurate to trace the path of the planet for several days, until other more distant observations could be obtained,
for correcting them.
This method, like all others of a similar nature, requires some modification in particular cases. First.
(651) When any one of the three geocentric places of the planet coincides with the heliocentric place of the
earth, or with its opposite point at that time ; because then the arc, connecting this geocentric place of the
(652) planet, and the corresponding heliocentric place of the earth becomes indeterminate. Second, When the
(553) geocentric places ot the planet in the first and third observations coincide. Third. When the three geocentric
(654) places of the planet are situated in a great circle, passing through the heliocentric place of the earth in the
second observation. In the first of these cases the situation of one of the great circles AB, A'B', Jl'' B" , remains
(655) indeterminate ; in the second and third cases, the situation of the point B* is indeterminate ; and in these two
last cases, the defect is inherent in the problem itself, and cannot be rectified. We must, therefore, in selecting
(656) the observations, which are to be used, avoid those which are at the same time near the node, and near the
conjunction or opposition with the sun ; we must also avoid those observations in wliich the geocentric place of
(657) (]jg planet, in the third observation, is near to that in the first observation ; finally, we must reject those in which
all three of the observed places of the planet lie nearly in a great circle passing through the heliocentric place
(658) of the earth, in the middle observation. We may easily rectify the rules in the first case (65i),by supposing the
points E, E', E", figure 92, page 874, to coincide, and then finding this point of coincidence by means of the two of
(659) t]jg three arcs AB, A'B' or A''B", which are given in position and magnitude ; supposing the other arc to be
(660) infinitely small, but taking it in the direction towards the common point E, For example, if the points JÎ, B,
coincide, we may suppose the arc AB to be infinitely small, and that it is taken in the direction of the great circle
(661) ABE. It being evident that this small change in the place of the planet, at the time of the first observation,
^662) can produce no sensible effect in the result of the calculation. In this ease the factor -^ — , ^ ' , r , which
' sm.{AE'—J-)
occurs in the expression of a (32) becomes, - — —j- , which may be put equal to nothing, on account of
(663) the extreme smallness of sin J"; hence we have a^o (32). This value of a is to be substituted in (35,4o),
and we shall get the value of w, to be substituted in (4i') ; then the calculation is to be completed in the usual
manner. The method of proceeding is nearly the same, when the points A", B", coincide in the third observation ;
(664) and as a, 6, (32,33), become infinite, because sin.<r":=o, we must put as in (42) a=66, ; and tang.w
(4o) changes into tang.w, (43); also the factor, 7~ (4i")) changes into - = 6,. When the points
(665) A', B', coincide, we have 6=0 (33); hence (4o) becomes, tang.w= — ' — F»^ — ^^^S-^*) <"" w^— tf»;
and so on for the other quantities. It is unnecessary to enter more minutely into the consideration of these
(566) uncommon cases, as the method of proceeding is sufficiently obvious.
In all the preceding calculations, we have supposed the orbit to be wholly unknown, at the commencement
(lOT) of the calculations; but it is evident that the same method can be observed for correcting the approximate
(668) elements, in a manner similar to that in [825 — 829]. Taking P and Q for the unknown quantities; and then
separately varying each of them, by a small quantity, in two successive operations, so as to obtain two equations,
similar to [82g], for correcling the assumed values of P, Q. This method is so plain, that it requires no
particular illustration. We may however remark that when the arcs 2/, 2/', 2_/ " are large, the assumed values
(669) of P, Q (269) may not be sufficiently accurate for the first operation, and then we may use the expressions
(670), computing the values roughly, by means of the approximate elements, which have been previously found.
r.sin 2 / " 4r'*. sin. /'.sin./'"
r".sm/ p.cos.J'
This value of P is easily deduced from (38, go) ; and if we multiply the expression of Q (3g), by that of p (247),
(CTl) and the product by [c 1'], we get p.Q [rr"]=S rr'i r". sin. /.sin./', sin./". Substituting in the first member,
the v.ilue of [r r"] (go), and then dividing by 2 pre' .sin./', cos. /', we get Q (670).
TABLE I. — OF SQUARE ROOTS.
The proposeil number is to be found, as far as Oic scconrl decimal jilaco, in tlie side cobimn of the (able, and the third
decimal at the top of one ot the vertical co]\i]iMis; ilie number oorrespoiidins is tlio jeciuired root.
o.oo
o.oi
o.o:!
o.o3
0.04
o.o5
o.oG
O.OT
0.08
0.09
O.IO
O.I
0.12
O.I 3
0.14
O.I 5
0.16
0.17
0.18
019
0.20
0.21
0.22
0.23
0.24
0.25
0.26
C.27
0.28
C.29
o.3o
0.3
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0-46
0.47
0.48
0.49
0
1
o.oooovl o3i62
II. 1 0000 10488
0.14142 14491
0.1702! 17607
0.20000 20246
0.25361
0.2449")
0.26458
0.28284
o.3oooo
0.31623
o.33i66
0.34641
o.36o56
0.37417 37550
o.5o
o.5i
0.52
0.53
0.54
0.55
o56
0.57
0.5S
0.59
22583
24(598
26646
28460
3oi66
31780
333i7
34785
36194
0.38-30
0.40000
o.4i23i
0.42426
0.43589
0.44721
0.45826
0.46904
o 47958
048990
o.Soooo
0.50990
0.51962
o. 5291 5
o 53852
0.54-72
0.55678
o. 56569
0.57446
o.583io
0.591 61
0.60000
0.60S28
0.6164.
0.62450
0.63246
o.64o3i
0.64807
0.65574
0.66332
0.67082
0.67823
0.6855
0.69282
0.70000
38859
4oi25
41352
42544
43704
44833
45935
4701 1
48062
49092
5oroo
5io88
52o58
53009
04472
10954
14832
1788g
20494
22804
24900
26833
28636
3o332
31937
33466
34928
36332
3-683
53944 54037
0.7071 1
071414
0.72111
0.72801
0.73485
0.7416:-
0.74833
0.75498
0.761 58
0.76811
54863
55-6-
56657
57533
58395
59245
6oo83
6091 o
61725
62530
6332
64i og
64885
6565i
66408
67157
67897
6862g
69354
70071
70781
71484
72180
72870
73553
74229
74900
75565
76223
7687-
1
4o24g
41473
42661
438i8
44944
46043
47117
48166
49193
50200
5ii86
52i54
53io4
55857
56745
57619
5848i
59330
60166
60992
61806
62610
634o3
64187
64962
6572T
66483
67231
67971
68702
69426
70143
70852
71554
72250
72938
73621
74297
74967
7563i
76289
76942
o5477
ii4o2
i5i66
1 81 66
20736
23o22
25lOO
27019
288
30496
32094
336i5
35071
36469
37815
3gii5
4o373
41593
42778
43932
45o56
461 52
47223
48270
49295
50299
51284
52249
53198
o6325
ii832
1 5492
i843g
20976
23238
25298
27203
28983
O.
3o65g 3o822
54129 54222
54955 55o45 55i36
55946
56833
57706
58566
59414
60249
61074
6188
62690
63482
64265
65o38
658o3
66558
67305
68044
68775
69498
70214
70922
71624
72319
7300-7
73689
74364
75o33
7569-
76354
770C:6
32249
33764
35214
366o6
37947
39243
4o49'
41713
42895
44o45
45i66
46260
47329
48374
49396
50398
5i38i
52345
53292
07071
12247
i58ii
18708
2I2l3
23452
25495
27386
29155
G
56o36
56921
57793
58652
59498
60332
61 1 56
61968
62769
6356
64343
65ii5
65879
66633
67380
681 18
68848
69570
70285
70993
71694
72388
73075
73756
7443i
75 ICO
75763
76420
77071
32404
33912
35355
36742
38079
39370
40620
4i833
43oi2
44i5g
45277
46368
47434
48477
49497
50498
51478
5244c
53385
543 14
55227
56i25
57009
57879
58737
59582
6o4i5
6123-
62048
62849
63640
64420
65ig2
65955
66708
67454
681 91
68920
69642
7o356
71063
71764
7245'
73i4-
73824
74498
75i66
75829
76485
771 36
07746
1 264g
nil 25
i8g74
21448
23664
256t)o
27568
2g326
30984
3a558
34o5g
354g6
36878
38210
39497
40743
41952
43128
44272
45387
46476
47539
48580
49598
50596
5i575
52536
53479
54406
I
08367
I 3o3K
16432
■923:'
21679
23875
25884
2774g
29496
3ii45
3271 1
34205
35637
3701 4
3834i
39623
40866
42071
43243
553 1 7
56214
57096
57966
58822
59666
60498
6i3ig
6212g
62929
637
64498
65269
66o3o
66783
67528
68264
68993
69714
70427
71 1 33
71833
72526
73212
73892
74565
75233
75895
76551
44385 44497
45497
46583
47645
48683
49699
50695
51672
5263i
53572
5^98
55408
563o3
57184
58o52
58907
59749
6o58
61400
62209
63oo8
63797
64576
65345
66106
66858
67602
68337
69065
69785
0894^
i34i6
16733
19494
21909
24o83
26077
27928
29665
3i3o5
32863
34351
35777
37148
38471
39749
40988
42190
43359
9
45607
46690
47749
4878;
71204
71903
72595
73280
73959
74632
75299
75gôi
-76616
7726(3
5o7g4
5176g
52726
53666
54589
554g8
56391
57271
58i38
58992
59833
6o663
61482
62290
63087
63875
64653
65422
66182
66933
67676
684ii
69138
69857
70569
71274
71972
72664
73348
74027
74699
75366
76026
76681
7733o
094s-
13784
1702g
19748
22l36
242go
26268
2810
2g833
3i464
33oi
344g6
35917
37283
386oi
39875
41110
423o8
43474
44609
45717
46797
47854
48888
49900
5x865
52820
5375g
54681
55588
5648o
5735g
58224
59076
59917
60745
61 563
62370
63i66
63953
64730
65498
(16257
67007
67750
6g2io
6gg29
70640
71 344
72042
72732
73417
74095
74766
75432
760g 2
76746
773g5
WUrn tho quantity j: wliose root is to I)c found consists
of sevL'ml plact'ô of deciumls anil is less than 0, ], it
will 1)0 convuniunt to lind thfi root of 100:); and divide
till; result by 10, which is done l)y merely changing tho
deciinjil point two ligures in finding lOOx, and one tiguro
in dividing by tO.
150
15
30
45
(10
75
90
105
131)
135
210
205
200
195
190
185
180
175
170
165
ice
155
21
21
20
20
19
19
18
18
17
17
10
10
42
41
411
3il
3h
37
30
35
34
33
32
31
u:t
112
CO
511
.57
.51 i
54
.53
51
,50
48
47
HI
82
8(1
7,s
70
74
72
70
(W
Oli
04
02
lll.S
103
mil
98
95
93
90
88
85
83
80
78
12li
12:1
120
117
in
111
los
105
102
99
90
93
117
Ml
1411
137
133
130
1211
123
no
IK.
112
109
IKS
104
11)0
1.5li
152
14-
141
1411
I3i;
132
12.^
121
lo'J
185
180
17li
171
107
1U2
156
153
149
144
14U
1.13
14B
14-1
142
140
138
130
134
133
130
13?
126
15
15
14
14
14
M
14
13
13
13
13
13
30
29
29
28
28
28
27
27
20
20
20
2.5
44
44
43
4:1
42
41
41
40
40
39
;t8
38
59
58
.58
.57
.50
.55
54
.54
53
.52
51
50
74
73
72
71
70
m
08
67
Rfi
05
M
63
89
88
80
8.5
84
83
82
HO
79
78
77
76
104
102
101
99
98
97
95
94
92
91
90
88
118
117
115
114
112
110
109
10-
100
104
109
101
133
131
130
128
120
124
122
121
119
117
115
113
134
12
25
37
50
62
74
87
99
112
123
122
121
120
119
118
117
iir
115
114
113
112
111
1
12
)2
12
12
12
12
12
12
12
II
11
II
11
2
2.5
24
24
24
24
21
23
2:1
23
23
2:!
22
22
3
37
37
30
30
30
35
35
35
35
31
.34
34
.33
4
49
49
48
48
48
47
47
40
41,
■II.
45
45
44
5
62
61
01
60
60
59
59
.58
.58
57
57
50
.56
0
74
V3
73
72
71
71
70
70
09
08
l»i
67
67
'/
80
85
85
84
83
83
82
81
81
80
79
78
78
8
98
98
97
90
95
94
94
93
92
91
90
90
89
9
111
110
lOU
108
107
106
105
104
104
103
102
101
100
110
109
108
107
too
105
104
103
102
101
100
99
98
11
11
11
11
11
11
III
III
10
10
10
10
10
22
0.-)
22
21
21
21
21
21
20
20
20
90
20
33
33
32
32
32
32
31
31
31
30
30
30
29
44
44
43
43
42
42
42
41
41
40
40
40
39
55
55
.54
.54
.53
53
.52
.52
51
51
.50
.50
49
00
65
65
64
64
63
62
62
61
01
60
.59
59
77
70
76
75
74
74
73
72
71
71
70
69
69
88
87
80
81;
85
84
83
82
82
81
80
79
78
99
98
97
90
95
95
94
93
92
91
90
89
88
96
95
94
93
92
91
90
89
88
87
ec
85
34
83
82
81
10
10
(I
9
9
!l
9
0
0
9
fi
9
8
8
8
R
19
19
19
19
18
Is
IS
Is
Is
17
r,
n
17
17
16
l(i
29
•29
•28
28
■2-
27
•Si
■j-
2i.
2t
21:
20
'-Î.5
25
25
94
38
38
38
37
37
36
30
36
35
35
3 1
34
3-1
33
33
:«
48
48
47
47
40
46
1.-,
15 1-1
41
4:1
■13
-!■'
4^'
41
41
58
.57
56
56
.55
55
.51
5.'.
53
.52
.52
51
51)
,511
49
49
ta
67
liO
65
64
64
63
62
62
61
60
60
.59
5S
.57
77
7li
75
74
74
73
72
71
70
70
69
lï<
67
60
66
65
86
86
85
M
83
82
81
SO
79
78
"7
77
70
75
74
73
73
72
TABLE I.— OF SQUARE ROOTS.
Tlie proposed number is to he fourni, as for as the second decimal place, in the side column of the table, and the third
decimal at the top of one of (he vertical columns; the number coi-responiliny; is the re([uircd root.
0.60
o.Ci
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.71
0.73
0.73
7/4
0.75
0.76
0.77
0.7S
0.79
0.80
0.81
0.S2
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
O.yl
0.92
0.93
0.94
eg*;
0.96
0.97
0.98
0-99
1. 00
1 .01
1.03
i.o3
1.04
i.o5
1.06
1.07
1.08
1.09
I. in
I.I I
I.I 2
I.i3
1. 14
i.i5
i.iG
1. 17
1. 18
1,19
0
0.77460
0.7SIO3
0.78740
0.79373
0.80000
0.80623
0.81240
0.81854
0.S2462
o.83o66
0.83666
0.84853
0.85440
0.86023
o.866o3
0.87178
0.87750
o.883i8
0.88882
0.89443
0.90000
o.go554
0.91 1 o4
o.gi652
0.92195
0.92736
0.93274
0.93S08
0.94340
0.95394
0.95917
0.96437
0.96954
0.97468
0.97980
0.98489
0.98995
0.99499
1 .00000
I .oo4g9
I .ooggS
1. 01 489
1 .01 980
I .02470
1.02956
i.o344i
I .03923
I .o44o3
I. 04881
1.05357
I .o583o
I .o63oi
I. 06771
1.07238
1.07703
I .08167
I .08628
1 .09087
0
1
77524
78166
78804
79436
80062
80685
8i3o2
8igi5
82523
83i26
83726
84321
84912
85499
86081
86660
87235
S7807
88374
88938
90056
90609
91 1 59
9170b
gasSo
92790
93327
93862
94393
94921
95446
95969
96488
97005
97519
98031
9853g
99045
99549
ooo5o
00549
01045
01 538
02029
025i8
o3oo5
03489
03971
o445i
04929
o54o4
05877
o6348
06818
07285
07750
08213
08674
091 33
]
2
3
4
5
6
7
8
9
77589
77653
77717
77782
77846
77910
77974
78038
78230
78294
78358
78422
78486
78549
78613
78677
78867
78930
78994
79057
79120
79183
79246
79310
79498
79561
79624
79687
79750
79812
79875
79937
80125
80187
8o25o
8o3i2
80374
8o436
80498
8o56i
80747
80808
80870
80932
80994
8io56
81117
81179
81 363
81425
81486
81548
81609
81670
81731
81792
8ig76
82037
82098
82i58
82219
82280
82341
82401
82583
82644
82704
82765
82825
82885
82g46
83oo6
83x87
83247
833o7
83367
83427
83487
83546
836o6
83785
83845
83905
83g64
84024
84o83
84i43
84202
8438o
84439
84499
84558
84617
846-76
84735
84794
84071
85o29
85ob8
85i47
852o6
85264
85323
85381
85557
856i5
85674
85732
85790
8584g
85907
85965
86139
86197
86255
863 1 3
86371
86429
86487
86545
86718
86776
86833
86891
86q48
87006
87063
87121
87293
87350
87407
87464
87521
8757g
87636
87693
87864
87920
87g77
88034
88091
88148
88204
88201
88431
88487
88544
88600
88657
88713
8876g
88826
88994
89051
89107
89163
89219
89275
8g33i
89387
89554
89610
89666
89722
89778
8g833
89889
8gg44
goi 1 1
90167
90222
90277
90333
9o388
90443
9049g
90664
90719
90774
go83o
go885
90940
90995
9' 049
91214
91269
91324
91378
91433
91488
91 542
9' 597
91761
gi8i5
91869
91924
91978
92033
92087
92141
92304
92358
92412
92466
92520
92574
92628
92682
92844
92898
92952
g3oo5
93o5g
93ii3
93167
g3220
93381
93434
93488
g354i
93595
g3648
93702
g3755
93915
93968
94021
94074
94128
94181
94234
94287
94446
94499
94552
g46o4
94657
94710
94763
9481 5
94974
95026
9507g
95i3i
g5i84
95237
95289
95341
95499
95551
95603
95656
95708
95760
95812
95864
96021
96073
96125
96177
96229
g628i
96333
96385
96540
96592
96644
96695
96747
96799
96850
96902
97057
97108
97160
97211
97263
97314
97365
97417
97570
97622
97673
97724
97775
g7826
97877
97929
98082
98133
98184
9S234
g8285
g8336
983S7
98438
98590
98641
986gi
98742
98793
98843
98S94
98944
99096
99146
99197
99=47
99298
99348
99398
99448
99599
99649
99700
99750
99800
99860
99900
99950
001 00
001 5o
00200
002 5o
oo3oo
oo34g
oo3gg
00449
ooSgS
00648
00698
00747
00797
00846
oo8g6
oog46
oiog4
01143
oiig3
01242
01292
01 341
01390
oi44o
01 587
01637
01686
01735
01784
01 833
01882
oig3i
02078
02127
02176
02225
02274
02323
02372
02421
02567
02616
02665
02713
02762
0281 1
02859
02908
o3o53
03l02
o3i5o
o3i99
o3247
03296
03344
03392
03537
03586
03634
o3682
o373o
03779
08827
03875
04019
04067
o4ii5
o4i63
04211
042 5g
o43o7
04355
04499
04547
o45g4
04642
04690
04738
04785
04833
04976
o5o24
o5o7i
o5ii9
05167
o52i4
05262
o53o9
o545i
0549g
o5546
o55q4
o564i
o5688
o5736
05783
o5g25
o5g72
0601 g
06066
061 1 3
06160
06207
06254
o63q5
06442
06489
06536
06583
o663o
06677
06724
06864
0691 1
06958
07005
07o5i
07098
07145
07191
07331
07378
07424
07471
07517
07564
07610
07667
07796
07842
078S9
07935
07981
08028
08074
08120
08259
o83o5
o835i
08397
08444
08490
o8536
08582
08720
08766
08812
08858
08904
08950
08995
09041
09179
09225
09270
09316
09362
09407
09453
09499
2
3
4
5
6
7
8
9
65
64
7
6
i3
i3
20
IQ
26
2h
33
32
3q
38
46
45
52
5i
59
58
63
6
i3
19
25
32
38
44
5o
57
59
58
6
6
12
12
18
17
24
23
3o
2Q
35
35
4i
4i
47
46
53
52
6
II
17
23
29
34
4o
46
5i
53
52
5
5
II
10
16
16
21
21
27
26
32
3i
37
36
42
42
48
47
5
10
i5
20
26
3i
36
4i
46
47
46
5
5
9
9
14
i4
IQ
18
24
23
28
28
33
32
38
37
42
4i
62
61
6
6
12
12
19
18
23
24
3i
3i
37
37
43
4J
5o
49
56
55
60
55154
6 5
II II
17' 16
22! 22
28 27
33i32
39 [38
44 43
49I49
5o
49
5
5
10
10
i5
i5
20
30
25
25
3o
29
35
34
40
39
45
44
48
5
10
i4
19
34
?9
34
38
43
TABLE I. — OF SQUARE ROOTS.
The proposed number \s to be fourni, as f.\r as the second decimal place, in tiie siilc column of the table, and the third
(lecimoi at the top of one of the vortical coluuuis ; the number cor respond iu;i;- is tlio required root.
0
1
■2
3
4
5
6
7
S
9
I.2tl
1.09544
09590
09636
og68i
09727
09772
09818
09864
09909
09955
46
45
1.31
r. 10000
10045
loogi
101 36
10182
10327
10272
io3i8
io363
io4o8
—
1. 22
1. 1 0454
10499
10544
10589
io635
10680
10725
10770
10816
10860
I
6
5
1.53
I.KK)o5
I ogSo
10995
no4i
1 1 086
ui3i
U176
11221
11265
11810
2
9
9
1-24
I.I j355
1 1 400
11445
ii4go
11 535
ii58o
11624
ii66g
11714
1176g
3
4
M
18
i4
18
1.23
1.11S03
11848
Ii8g3
iiq37
11982
12027
1 2071
121 16
1 2161
I2305
5
28
23
1.26
I.I2250
12294
I233q
12383
12428
13472
125l7
I256i
1 3606
13660
6
28
27
1.27
1. 1 2694
12739
12783
12827
12872
12916
12960
i3oo4
i3o4g
i3og3
7
82
82
1.28
I..3I37
i3i8i
l3225
13270
i33i4
13358
1 3402
1 3446
13490
13634
8
37
36
1.30
1. 1 3578
1 3622
i3666
1 3710
13754
13798
i3842
1 3886
18930
i3g74
9
4i
4i
i.3o
1.14018
1 4061
i4io5
i4i4g
14193
14237
14280
14324
1 4368
i44i3
i.3i
1. 14455
14499
14543
14586
i463o
14673
14717
1 4761
i48o4
14848
1.32
1.14891
14935
14978
l5022
i5o65
i5io9
i5i52
15195
i523g
16282
1.33
I.I 5326
15369
1 541 2
1 5456
15499
i5542
15585
16629
16672
1571 5
44
43
1.34
I.I5758
i5So2
1 5845
1 5888
i5g3i
15974
16017
16060
i6io3
16146
I
2
4
4
9
1.35
i.i6igo
16233
16276
i63ig
16362
i64o4
16447
16490
1 6533
16676
1.36
1.16619
16662
16705
16748
16790
16833
16876
16919
16962
17004
3
i3
i3
1.37
1.17047
17090
17132
17175
17218
17260
i73o3
17346
17388
17481
4
18
17
1.38
1. 1 7473
17516
17558
1 7601
17644
1-686
17739
17771
17813
17866
5
22
22
1-39
I. .7898
1 7941
17983
18025
18068
18110
18162
18196
18237
1827g
6
26
3i
26
80
1.40
I.I8322
1 8364
i84o6
18448
i84gi
18533
18575
18617
18669
18701
7
8
85
84
1.41
1. 18743
18786
18828
18870
18912
i8q54
18996
igo38
19080
19122
9
40
39
1.42
1 . 1 91 64
19206
19248
19290
19331
19373
19415
19457
19409
19541
1.43
1. 1 9583
19624
19666
19708
19750
1 9791
ig833
19875
1 991 7
19958
1.44
1.20000
20042
20o83
20125
20167
20208
20230
20291
20333
20374
1.45
i.2o4i6
20457
2049g
2o54o
20582
20623
20665
20706
20748
2078g
42
4i
1.46
i.2o83o
308 -73
2ogi3
2og55
20996
21037
31078
31120
21161
21 203
—
—
1.4-
I. 21 244
21285
21326
21367
21408
2i45o
21491
2l632
21673
21614
I
4
4
1.4s
I. 21655
21696
21737
21778
21820
21861
21902
2ig43
21984
22025
3
8
8
1.49
1 .22066
22107
23147
22188
2222g
22370
223ll
32352
22393
22434
8
4
5
i3
12
16
31
i.5o
1.224-4
225l5
22556
225g7
22638
22678
22719
22760
23801
33841
17
21
i.5i
1.228S2
23933
22g63
23oo4
23o45
23o85
23l26
23167
28207
28248
6
25
35
1.52
1.23288
23339
2336g
23410
23450
23491
2353i
23572
28612
33653
7
29
29
1.53
1.23693
33734
23774
238i4
23855
23895
2 3g35
23g76
24016
24o56
8
34
33
1.54
1.24097
24137
24177
24218
24258
24298
24338
24378
24418
3445g
9
38
37
1.55
1.24499
24539
2457g
2461 g
24660
24700
24740
24780
24820
24860
1.56
1.24900
24940
24g8o
25020
25o6o
25lOO
35i4o
25i8o
26330
36260
1.57
i.253oo
25340
2537g
254ig
2545g
25499
35539
25579
36618
26668
1.58
1.25698
25738
25778
25817
25857
25897
35g36
26976
2601 6
26066
4o
39
1-59
1.26095
26135
26174
26214
26254
26293
26333
26373
26412
26452
I
2
4
8
4
8
1.60
i.264oi
26531
26570
26610
3664g
26689
26728
26768
26807
26846
1. 61
1.26886
26925
26964
27004
27043
27083
27122
27161
27201
27240
3
12
13
1.62
1.2727g
27318
27358
27397
27436
27475
275i5
27554
27698
27682
4
t6
16
1.63
1.27671
27711
27750
27789
2-828
27867
27906
27945
37984
28028
5
20
20
1.64
1.28062
28102
a8i4i
28180
2821g
28258
28297
38335
28874
28418
6
24
28
32
23
1.65
1.28453
28491
28530
28569
28608
28647
28686
28726
28768
28802
7
8
27
3i
1.66
1.2S841
28880
28919
28957
28996
29035
29074
29112
29161
29190
9
36
85
1.67
1.29228
29267
29306
29345
29383
29422
29460
29499
2g538
29676
1.68
I. 2961 5
29653
29692
29730
2976g
29808
39846
29886
2g923
29962
1.69
i.3oooo
3oo38
30077
3oii5
3oi54
30192
3o23i
3o36g
8o3o7
8o346
1.70
i.3o384
3o422
3o46i
3o499
3o537
30576
3o6i4
3o652
80690
80729
88
37
1.71
I .30767
3o8o5
3o843
30882
30920
30958
30996
3io34
81072
3iiii
1.72
i.3ii49
31187
3l225
3i 263
3i3oi
3i33g
3i377
3i4i5
81453
^'f9'
I
4
4
1.73
i.3i529
3i567
3i6o5
3i643
3i68i
31719
31757
31796
3i833
81871
2
8
7
1.74
1 -31909
31947
31985
32033
32061
32098
32136
32174
82212
32260
3
11
i5
19
23
II
1.75
1.32288
32325
32363
83401
32439
32476
325i4
32552
32690
82627
4
5
i5
19
22
1.76
1.32665
32703
32740
32778
32816
33853
32891
33939
82966
83oo4
6
1-77
i.33o4i
330-9
33ii6
33i54
33192
33229
33267
333o4
83342
33379
7
27
26
I. -8
1. 3341 7
33454
33492
33529
33566
336o4
33641
336-9
88716
33764
8
3o
80
'•"9
1. 33791
33828
33866
33903
33940
33978
3401 5
34062
84090
34127
9
34
83
0
1
-2
•3
4
ij
6
7
,c.'
9
TABLE I.— OF SQUARE ROOTS.
The proposed number is to be found, as far as the second decimal place, in the side column of the table, and the third
decimal at the top of one of the vertical columns ; the number correspoudinf; is the required root.
0
1
2
3
4
5
6
7
8
9
1.80
1.34:64
34201
34239
34276
343:3
34350
34387
34435
34462
34499
38
1. 81
1.34536
34573
34611
34648
34685
34722
34759
34796
34833
34870
—
1.82
1.34907
34944
34981
35019
35o56
35093
35: 3o
35167
35204
3524:
I
4
1.83
1.35277
353i4
3535i
35388
35425
35462
354gg
35536
35573
356io
2
8
1.84
1.35647
35683
35720
35757
35794
3583i
35868
35904
35941
35978
3
4
5
i:
i5
1.85
i.36oi5
36o5i
36o88
36:25
36:62
36198
36235
36272
363o8
36345
19
1.86
1.36382
364i8
36455
364g2
36528
36565
36602
36638
36675
367:1
6
23
1.87
1.36748
36785
36821
36858
368g4
36931
36g67
37004
37040
37077
7
27
1.88
1. 371 1 3
371 5o
37186
37222
37259
37295
37332
37368
374o5
3744:
8
l" 1
37
1.89
1.37477
37514
37550
37586
37623
37659
37695
37732
37768
37804
9
34
1.90
1.37840
37877
37913
37949
37985
38022
38o58
38094
38: 3o
38:67
:
4
i.gi
I.38203
38239
38275
383: 1
38347
38384
38420
38456
384g2
38528
2
3
4
5
6
7
1.02
1.38564
38600
38636
38672
38708
38744
38780
388:6
38852
38888
II
i5
1.93
1.38924
38960
38996
39032
39068
3g:o4
3gi4o
39:76
3g3i2
39248
1.94
1.39284
39330
39356
3g3g2
39427
39463
39499
39535
3957:
39607
'9
22
1.95
1.39642
39678
39714
39750
39786
39821
39857
3g8g3
39929
3gg64
7
g
26
3o
1.96
1 .40000
4oo36
40071
40:07
40143
40:78
402:4
402 So
40385
4o32i
33
1.97
1.40357
40392
40428
40464
4o4gg
4o535
4o57o
40606
4o64:
40677
36 y
1.98
I. 4071 2
40748
40784
40819
4o855
40890
40936
4og6;
40996
4:o32
—
1-99
I. 41067
4iio3
4n38
4:: 74
4:209
4:244
4:280
41 3: 5
4:35:
4:386
I
4
2.00
1.41431
41457
41492
4:527
4:563
4:598
4:633
41669
4:704
4:739
2
3
7
1:
2.01
1. 41774
4i8io
4:845
41880
4:9:5
4:g5:
4: 986
42021
43o56
42og2
4
i4
2.02
1. 421 27
42162
42197
43233
42267
42302
42338
43373
42408
42443
5
18
2.o3
1.42478
425i3
42548
42583
436:8
42653
42688
42724
42759
42794
6
22
2.o4
1.42839
42864
42899
43934
42969
43oo3
43o38
43073
43: 08
43:43
7
8
25
2.o5
1.43178
43213
43348
43283
433:8
43353
43388
43422
43457
43492
9
=9
32
35
2.06
1.43527
43563
43597
4363:
43666
43701
43786
4377:
438o5
43840
4
2.07
1.43S75
43gio
43944
43979
4401 4
44o4g
44o83
44: : 8
44:53
44:87
1
2.08
1.44222
44357
4429:
44326
4436:
44895
44430
44465
44499
44534
2
3
4
7
2.09
1.44568
446o3
44637
44672
44707
44741
44776
44810
44845
44879
: I
i4
2.10
I. 4491 4
44948
44983
450:7
45o52
45o86
45:2:
45:55
45:90
45224
5
6
18
2. II
1.45258
45293
45327
45362
45396
45430
45465
45499
45534
45568
21
25
28
32
2.12
I .45602
45637
4567:
45705
4573g
45774
45808
45843
45877
45gi:
7
8
2.l3
1.45945
45979
460:4
46o48
46082
461:6
46:5:
46:85
462:9
46253
2.l4
1.46287
46322
46356
46390
46434
46458
46492
46536
4656:
46595
34 y
2.l5
1 .46639
46663
46697
4673:
46765
467gg
46833
46867
4690:
46935
I
3
3.16
1 .46969
47003
47037
47071
47:05
47:39
47173
47207
4724:
47275
2
7
2.17
I .47309
47343
47377
474::
47445
47479
475i3
47547
47580
476:4
3
:o
2.18
1 .47648
47682
477:6
47750
47784
478:7
4785:
47885
479' 9
47953
4
:4
2.19
1.47986
48020
48o54
48088
48122
48:55
48189
48223
48257
48290
5
6
7
17
2.20
I.4S324
48358
4839:
48425
48459
48492
48526
48560
48593
48627
20
24
2.21
I. 48661
48694
48728
48762
487g5
4882g
48862
48896
48y3o
48963
8
27
33
2.22
1 .48997
49o3o
49064
49097
4g:3:
4g:64
49:98
4g23:
49265
49298
9
3:
2.23
1 .4g332
49365
493g9
49432
4g466
4g499
49533
49566
49599
49633
3
7
2.24
1 .49666
49700
49733
49766
4g8oo
49833
49867
49900
49933
4gg67
I
2
2.25
i.Soooo
5oo33
50067
5oioo
5oi33
50167
50200
5o233
50266
5o3oo
3
10
i3
2.26
i.5o333
5o366
5o399
5o433
5o466
5o499
5o532
5o566
5o5g9
5o632
4
5
6
2.27
i.5o665
50698
50733
50765
50798
5o83:
5o864
50897
50930
5og64
17
2.28
1.50997
5io3o
5:o63
51096
5:: 29
5i:63
5:195
5:228
51261
5:294
20
23
26
2.29
i.5i327
5i36o
5:394
5:427
5: 460
5:493
5:526
5:559
5:592
5:625
7
8
2.3o
i.5i658
51690
5:723
5:756
5:78g
5:832
5i855
5:888
51921
5:954
32 9
3o
2.3l
1.51987
5303O
53o53
53o85
52:18
52:5:
52:84
533:7
5225o
52283
—
2.32
i.523i5
52348
5238:
534:4
52447
53480
525:2
53545
53578
526:1
I
3
2.33
1.52643
53676
52709
52742
52774
52807
52840
52872
53go5
52938
2
6
2.34
I. 52971
53oo3
53o36
53069
53ioi
53:34
53:67
5319g
53232
53264
3
4
5
:o
i3
2.35
1.53397
53330
53362
53395
53428
53460
53493
53525
53558
53590
:6
2.36
1.53623
53655
53688
5372:
53753
53786
538:8
5385:
53883
^^/J'.^
6
19
2.37
1.53948
53981
540: 3
54045
54078
54: 10
54:43
54:75
54208
54340
n
22
2.38
1.54272
543o5
54337
54370
54402
54434
54467
544g9
54532
54564
8
26
2.39
1.54596
5463g
54661
54693
54726
54758
54790
54822
54855
54887
9
29
0
1
2
3
4
5
6
7
R
9
_
TABLE I. — OF SQUARE ROOTS.
Tlie
proposed
numbe
■ is to bi
found.
as lar as the second t
eciiiKil
])Uice, in the sii
e column of the table, and the third
ilcciim
I at llie top of one of the vertical ooluniiis; the luiniber con-e^pondins is t!ie rc()uiro(l root.
0
1
0
3
1
5
6
7
8
y
a.4o
I. 54919
54952
54984
55oi6
55o48
55o8i
55ii3
55i45
55177
55310
33
2.41
1.55242
55274
553o6
55338
55371
554o3
55435
55467
55499
55531
__
2.42
1.55563
55596
5562S
55660
55692
55734
55756
55788
55820
55853
,
3
2.43
I .55885
55917
55949
55981
56oi3
56o45
56077
56109
56i4i
56173
2
7
2.44
I. 50205
56237
562(i9
56301
56333
56365
56397
56429
5646i
56493
3
4
5
10
i3
17
2.45
1.56525
56557
56589
56621
56652
56684
56716
56748
56780
56812
2.46
1.56844
56876
56908
56939
56971
57003
57035
57067
57000
57i3i
6
30
2.47
I. 57162
57194
57226
57258
57290
57321
57353
57385
574/7
57448
7
23
2.48
1.57480
5-5i2
57544
57575
57607
57639
57671
57702
57734
57766
8
26
2.49
1-57797
57829
57861
57892
57924
57956
57987
58019
58o5i
58082
9
3o
2.5o
i.58ii4
58i46
58177
58209
58240
58272
583o4
58335
58367
583g8
32
2.5[
I.58430
5846i
58493
58524
58556
58588
58619
5865i
58682
58714
I
3
2.52
1.58745
58777
588o8
5884o
58871
58902
58934
58g65
589g7
59028
3
6
2.53
1.59060
59091
59123
59154
59185
59217
59248
59280
59311
59342
3
10
2.54
1.59374
59405
59437
59468
59499
59531
59562
59593
59625
59656
4
i3
2.55
1.59687
59719
59750
59781
59812
59844
59875
59906
59937
59969
5
6
16
19
22
2.56
1.60000
6oo3i
60065
60094
60125
6o!56
60187
60319
6o25o
60281
7
8
2.57
i.6o3i2
60343
60375
6o4o6
60437
60468
60499
6o53o
6o562
60593
26
2.58
1 .60624
60655
60686
60717
60748
60779
60810
60842
60873
6ogo4
9
29
2.59
1 .60935
60966
60997
61028
61059
61&90
61121
61 1 53
611 83
61214
2.60
1. 61 245
61276
6i3o7
61 338
61369
6i4oo
6i43i
61463
61493
61 524
01
2.61
1. 61 555
61 586
61617
61648
61679
61710
61741
61771
61803
61833
1
3
2.62
1.61864
61895
61926
61957
61988
62019
62049
62080
631 11
63142
3
6
2.63
1. 621 73
62204
62234
62265
63396
62327
62358
62388
63419
62450
3
9
12
2.64
1.624S1
62512
62542
62573
62604
63635
62665
62696
62727
62757
4
2.65
1.62788
62819
62850
62880
6291 1
62942
62972
63oo3
63o34
63o64
5
6
16
19
22
2.66
1.63095
63126
63i56
63187
63218
63248
63379
63310
63340
63371
7
2.67
I.6340I
63432
63463
63493
63524
63554
63585
636i5
63646
63677
8
25
2.68
I .63707
63738
63768
63799
63839
63860
63890
63921
63951
63g82
9
38
2.69
1 .64oi 2
64043
64073
64io4
64i34
64i65
64195
64335
64256
64286
3o
2.-0
1.64317
64347
64378
64408
64438
64469
64499
64530
6456o
645go
2.71
1. 64621
6465i
64682
64713
64742
64773
648o3
64833
64864
648g4
I
3
2.72
1 .64924
64955
64985
65oi5
65o45
65076
65 1 06
65i36
65i67
65 197
2
6
2.-3
1.65227
65237
65288
653i8
65348
65378
65409
65439
65469
65499
3
9
2.74
1.65529
6556o
655go
65620
6565o
6568o
6571 1
65741
65771
658oi
4
r
12
i5
18
2.75
1.65831
6586i
65892
65922
65952
65982
6601 3
66043
66072
66103
6
2.76
1.66:32
66163
66193
66233
66353
66383
663i3
66343
66373
664o3
7
21
2-77
1.66433
66463
66493
66523
66553
66583
6661 3
66643
66673
66703
8
24
2.78
1.66733
66763
66793
66823
66853
66883
66913
66943
66973
67003
9
27
2.7g
1.67033
67063
67093
67123
67153
67183
67212
67242
67272
67302
29
2.80
1.67332
67362
67392
67422
67451
67481
67511
67541
67571
67601
I
3
2.81
1.67631
67660
67720
67750
67780
67809
67839
67869
67899
2
6
2.82
1 .67929
6-958
6798S
68018
68048
68077
68107
68137
68167
68196
3
9
2.83
1.68226
68b 56
68385
683i5
68345
68375
684o4
68434
68464
68493
4
5
2.84
1.68523
68553
68582
68612
68642
68671
68701
68731
68760
68790
11
2.85
t. 6881 9
68849
68879
68908
68938
68967
68997
69027
6go56
6go86
6
7
8
17
20
2.86
1.691 1 5
69145
69174
69204
69234
69363
69293
6g322
6g352
69881
23
2. 87
I .f)94i I
6g44o
69470
69499
69529
69558
69588
69617
69647
69676
9
26
2.88
1 .69706
69735
69765
69794
69823
69853
69882
69912
69941
69871
2.Sy
1.70000
70029
70059
70088
701 1 8
70147
70176
70206
70235
70265
28
2.90
1.70294
7o323
70353
7o382
70411
70441
70470
70499
70529
7o558
2.yl
1.70587
70617
70646
70675
70704
70734
70763
70792
70822
7o85i
I
3
2.92
1.70880
70909
70939
70968
70997
71026
7io56
71085
71114
71 143
2
6
2.93
1.71172
71202
7i23i
71260
71289
7i3i8
71348
71377
7i4o6
71435
3
8
2.94
I. 71464
71493
7i523
71552
7i58i
71610
71639
71668
71697
71727
4
5
II
i4
17
2.95
1.71756
71785
71814
71843
71873
71 901
71930
71959
71988
72017
6
2.96
1 .72047
72076
72105
72134
73163
72193
72221
7225o
72279
723o8
7
20
2.97
1.72337
72 366
73395
72424
72453
72482
72511
72540
73569
73598
8
22
3.98
1.72627
72656
72685
73714
72743
72772
72800
7282g
72858
72887
9
25
2 4(9
1.72916
72945
72974
73oo3
73o33
73061
73090
73118
73i47
73176
0
1
2
3
4
5
6
7
8
9
2a
TABLE I.— OF SQUARE ROOTS.
The proposed number is to be found, as far as the second decimal place, in the side column of the table, and the third
decimal at the top of one of the vertical columns ; the number corresponding is the required root.
0
1
2
3
4
5
6
7
8
9
3.O0
1.73205
73234
73263
73292
73321
73349
73378
73407
73436
73466
29
3.01
I .73494
73522
7355i
73580
73609
73638
73666
73696
73724
73753
—
3.02
I. 73781
738ro
73839
73868
73897
73926
73954
73983
740:1
74o4o
I
3
3.o3
I .74069
74098
74126
74:55
74:84
7421 3
74241
74270
74299
74327
2
6
3.o4
1.74356
74385
7441 3
74442
7447:
74499
74528
74557
74585
746:4
3
4
9
12
3.o5
1.74642
74671
74700
74728
74757
74786
748:4
74843
74871
74900
5
i5
3.06
1.74929
74957
74986
76014
76043
76071
76:00
76129
76167
76186
6
17
3.07
1.75214
75243
76271
75300
75328
75357
75385
75414
76442
7547:
7
20
3.08
I .75499
75528
75556
75686
756:3
76642
76670
75699
76727
76766
8
23
3.09
1.75784
75812
75841
76869
76898
76926
76966
75983
7(3011
76040
9
26
3.10
1 .76068
76097
76126
76163
76182
76210
76238
76267
76295
76324
3.11
1.76352
76380
76409
76437
76465
76494
76622
76660
76679
76607
3.12
1.76635
76664
76692
76720
76748
76777
76806
76833
76862
76890
3.i3
I. 7691 8
76946
76976
77003
77o3i
77069
77088
77116
77144
77172
3.t4
1.77200
77229
77267
77286
773:3
77341
77370
77398
77426
77454
3.i5
1.77482
775il
77539
77667
77595
77623
7765:
77679
77708
77736
3.16
I .77764
77792
77820
77848
77876
77904
77933
7796:
77989
78017
3.17
I .78045
78073
78101
78:29
78:67
78185
78213
7824:
78269
78298
2»
3.18
1.78326
78354
78382
784:0
78438
78466
78494
78622
78660
78678
—
3.19
I .78606
78634
78662
78690
78718
78746
78774
78802
78830
78867
I
2
3
6
3.20
1 .78885
78913
78941
78969
78997
79026
79063
7908:
^9',2§
79:37
3
8
3.21
I. 791 65
79193
79221
79248
79276
793o4
79332
79^60
79388
79416
4
II
3.22
I .79444
79471
79499
79627
79555
79583
796::
79639
79666
79694
5
i4
3.23
1.79722
79750
79778
79806
79833
79861
79889
79917
79944
79972
6
17
3.24
1 .80000
80028
80066
8oo83
80111
80139
80:67
80194
80222
80260
7
6
20
22
3.25
1.80278
8o3o5
8o333
8o36i
8o388
8o4i6
80444
80472
80499
80627
9
25
3.26
i.8o555
8o582
80610
8o638
80665
80693
80721
80748
80776
80804
3.27
i.8o83i
80869
80887
80914
80942
80970
80997
81026
81062
81080
3.28
1. 81 1 08
8ii35
8ii63
8:191
81218
81246
81273
8i3oi
8i328
81 356
3.29
I. 81 384
8i4ii
81439
81466
81494
81621
81 54g
81676
81604
8i63i
3.3o
I. 81659
81687
81714
81742
81769
81797
81824
8i852
81879
81907
3.3i
I. 81934
81962
81989
82016
82044
82071
82099
82126
82164
82181
3.32
1.82209
82236
82264
82291
823:8
82346
82373
82401
82428
82455
3.33
1.82483
82510
82538
82666
82692
82620
82647
82676
82702
82729
3.34
1.82757
82784
8281 1
82839
828(36
82893
82921
82948
82976
83oo3
27
3.35
1 .83o3o
83o57
83o85
83ii2
83 1 39
83:67
83i94
83221
83248
83276
I
3
3.36
i.833o3
8333o
83358
83385
834:2
83439
83467
83494
83521
83548
2
5
3.37
1.83576
836o3
83630
83657
83686
837:2
83739
83766
83793
83821
3
8
3.38
1.83848
83875
83902
83929
83967
83984
84oii
84o38
84066
84092
4
II
3.39
1. 84120
84:47
84174
84201
84228
84265
84282
843io
84337
84364
5
A
i4
16
3.40
I .84391
844 1 8
84445
84472
84499
84626
84554
84581
84608
84635
7
19
3.41
1 .846(52
84689
84716
84743
84770
84797
84824
84851
84878
84906
8
22
3.42
1 .84932
84959
84986
85oi 4
86o4:
86068
86096
86122
85i49
86176
9
24
3.43
1.85203
8523o
86267
86284
863i:
85338
86365
85391
854i8
85445
3.44
1.85472
85499
86626
85553
86680
86607
85634
86661
86688
86716
3.45
1.85742
86769
86796
85822
85849
86876
86903
86930
I'P'^J
86984
3.46
1 .8601 1
86o38
86066
86091
861 1 8
86:45
86172
86199
86226
86263
3.47
1 .86279
863o6
86333
86360
86387
864:4
86440
86467
8fi494
86621
3.48
1 .86548
86574
86601
86628
86655
86682
86708
86736
86762
86789
3.49
I. 8681 5
86842
86869
86896
86922
86949
86976
87003
87029
87066
3.5o
I .87083
87110
87x36
87:63
87190
872:6
87243
87270
87297
87323
26
3.5i
1.87350
87377
87403
87430
87457
87483
87610
87537
87663
87690
—
3.52
1.87617
87643
87670
87697
87723
87750
87776
87803
87830
87866
I
3
3.53
1 .87883
87910
87936
87963
87989
880:6
88043
88o6()
88096
88122
3
5
3.54
I. 881 49
88175
88202
88229
88256
88282
883o8
88335
8836:
88388
3
4
8
10
3.55
I. 8841 4
88441
88468
88494
8862:
88647
88674
88600
88627
88653
5
i3
3.56
I .88680
88706
88733
88769
88786
88812
88839
88866
88892
88918
6
16
3.57
1.8S944
88971
88997
89024
8go5o
89077
89103
8gi3o
89166
89182
7
18
3.58
1 .89209
89235
89262
89288
893.5
89341
89367
89394
89420
89447
8
21
3.59
1.89473
89499
89626
89662
89678
89605
89631
89668
89684
89710
9
23
0
1
2
.3
4
.5
6
7
8
9
TABLE I.— OF SQUARE ROOTS.
The proposed number is to be found, as far as the second decimal place, i
decimal at tlie top of one of the vertical columns ; the number conespouilins;
the side column of the table, and the third
1 tlic required root.
3.60
0
1
2
3
4
5
G
7
8
9
■.89-37
89763
89789
89816
89842
89868
89895
89921
89947
89974
27
3.6!
1.90000
90026
90053
90079
90106
90182
90168
90184
90310
90287
—
3.62
1 .90263
90289
903 16
90342
90868
9o8g4
90431
90447
90478
90499
I
3
3.63
1 .90326
90552
90578
90604
90681
90637
90688
90709
90735
90762
2
5
3.64
1.90788
90814
90840
90866
90898
90919
90945
90971
90997
91024
3
4
8
3.6-5
1.91050
91076
91102
91128
91154
91181
91207
91288
91269
91285
5
i4
3.66
i.giSii
91337
91364
91390
91416
91442
91468
91494
91620
91646
6
16
3.67
I. 91 572
91599
91625
91631
91677
91708
91729
91766
91781
91807
7
'9
3.68
I. 91 833
91859
91885
91911
91937
91964
9' 990
92016
92042
92068
8
22
3.6y
1.92094
92120
92146
92172
92198
92224
92260
92276
92802
92828
9
24
3.-0
1.92354
92380
92406
92432
92468
92484
92610
92686
92662
92688
3.-I
I. 92614
92640
93666
93691
92717
93743
92769
92795
92821
92847
3.72
,.92873
92899
92925
92951
92977
98008
98028
98064
98080
98106
3.-3
1.93132
931 58
93184
98210
98286
98261
98287
98818
93339
93865
3.-4
I. 93391
93417
93442
93468
98494
98620
93540
93572
98698
98628
3.-5
1 .93649
93675
93-01
93727
93752
93778
98804
98880
98856
98881
3.-6
1 .93y07
93933
93959
93985
94010
94o36
94062
94088
941 1 3
94189
3.—
1.94163
9419'
94216
94242
94268
94294
94819
94345
94371
94397
26
3.-S
1.94422
94448
94474
94499
94625
94551
94676
94602
94628
94654
—
3.-9
1.94679
94705
94731
94756
94782
94808
94888
94869
94885
94910
I
3
5
8
3.80
1.94936
94962
94987
9601 3
96088
96064
95090
96116
96141
96167
2
3
3.81
1.96192
95218
95243
96269
96396
96820
95346
96871
96897
96428
4
10
3.82
1.95448
95474
95499
95525
96661
96676
96602
96627
96668
96678
5
i3
3.83
1 .95704
95729
95755
96780
96806
95832
95857
96888
96908
96934
6
16
3.84
1 -95959
95985
96010
96086
96&61
96087
96112
96188
96163
96189
7
8
9
i8
3.85
1.96214
96240
96265
96291
96816
96342
96867
96892
96418
96443
21
23
3.86
I .Q6469
96494
96520
96645
96671
96696
96621
96647
96672
96698
3.87
1 .96723
96749
96774
96799
96826
96860
96876
96901
96926
96962
3-88
1 .96977
97000
97028
97068
97079
97104
97129
97166
97180
97206
3.S9
1.97231
97256
97282
97307
97882
97358
97383
97408
97434
97469
3.90
1.9-484
97509
97535
97660
97585
97611
97686
97661
97687
97712
3.91
1.97-37
97762
97788
97818
97888
97S64
97889
97914
97939
97965
3.Q2
i-9"99o
98015
98040
98066
98091
981 16
98141
98167
98192
98217
3.93
1.98242
98267
98293
98318
98848
98868
98894
98419
98444
98469
3.94
1.98494
98520
98545
98670
98696
98620
98645
98671
98696
98721
25
3.Q5
1.98746
9877;
98796
9S822
98847
98872
98897
98922
98947
98972
I
3
3.96
1-98997
99023
99048
99078
99098
99133
99148
99178
99198
99228
2
5
3.97
1.99249
99274
99299
99824
99349
99374
99399
99434
99449
99474
3
8
3.98
1-99499
99524
99549
99376
99600
99626
99660
99675
99700
99725
4
10
3.09
1.99750
99775
99800
99826
99850
99875
99900
99926
99950
99975
5
6
7
i3
i5
18
4.00
2.00000
00025
ooo5o
00076
001 00
00126
00160
00175
00200
00225
4.01
2.00230
00275
oo3oo
00826
oo35o
00876
oo4oo
00425
00449
00474
8
20
4.02
2.00499
00624
00549
00674
00699
00634
00649
00674
00699
00724
9
23
4.o3
2.00749
00774
00798
00828
00848
00878
00898
00928
00948
00978
•
4.04
2.00998
01022
01047
01072
01097
01122
01147
01172
01196
01221
4.o5
2.01246
01271
01296
01821
01 345
01870
01896
01420
OI446
01470
4.06
2.01494
oi5i9
01 544
01669
01694
01 61 8
01643
01668
01698
01718
4.07
2.01742
01767
01792
01817
01842
01866
01891
01916
01941
01966
4.08
2.01990
0201 5
02040
02064
02089
02114
02189
02168
02188
022l3
4.09
2.02237
02262
02287
02812
02336
02861
02386
02410
02486
02460
4.IO
2.02485
02509
02534
02669
02688
02608
02688
02667
02682
02707
24
4.11
2.02731
02736
02781
02806
02880
02866
028-9
03904
03929
02968
4.12
2.02978
o3oo2
03027
o3o63
08076
08101
08126
o8i5o
08176
08199
I
2
4.1 3
2.03224
03349
03273
08298
08822
o8347
08872
08896
03431
03445
2
5
4.i4
2.03470
03494
o33i9
03544
03568
03593
c36i7
o3642
03666
08691
3
4
5
7
4.i5
2.0371 5
08740
03765
08789
o38i4
08888
08868
08887
08912
08936
10
12
4.16
2.03961
03985
o4oio
o4o34
04069
o4o83
04108
04182
04167
o4i8i
6
i4
4.17
2.04206
o43 3o
04255
04279
o43o4
04828
04363
04377
o44o2
04426
7
17
4.18
2.044 5o
04475
04499
04524
04548
04578
04597
04633
o4646
04670
8
19
4.19
2.04695
04719
04744
04768
04793
04817
o484i
04866
04890 0491 5
9
22
0
1
•2
3
4
5
6
7
8 9
TABLE I.— OF SQUARE ROOTS.
The proposed number is to be found, as for as the first decimal place, in the side column of the table, and the second
decimal at the top of one of the vertical cobimns ; the number correspondiiig- is the required root.
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7-1
7-2
7.3
7-4
7-5
7.6
0
2.04939
2.07364
3.09762
2.12l32
2.14476
2.16795
2.19089
2.21359
2.23607
2.25832
2.28035
2.30217
2.32379
2.34521
2.36643
2.38747
2.4o832
2.42899
2.44949
2.46982
2.48998
2.50998
2.52982
2.54951
2.56905
2.58844
2.60768
2.62679
2.64575
2.66458
2.68328
2.70185
2.72029
2.-386I
2.75681
1
?7489
7.8 2.79285
2.81069
.82843
7-9
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9-'
9.2
^.3
9.4
9
9.6
9■■^
9.8
9-9
1 0.0
I o. I
o5i83
07605
1 0000
12368
14709
17025
19317
2i585
2383o
26053
28254
3o434
32594
3473. ■
36854
38956
4io3q
43io5
45i53
47184
49199
51197
53i8o
55i47
57099
59037
60960
62869
64764
66646
685i.'i
70370
72213
74o44
75862
77669
79464
81247
83019
2.846o5 84781
2.86356
2.88097
2.91548
2.93258
2.94958
2.96648
2.g8329
3.00000
3.01662
3.o33i5
3.04959
3.06594
3.08221
3.09839
3.11448
3.i3o5o
3.14643
3.16228
3.17805
0
8653i
05426
07846
10238
12603
14942
17256
19545
21811
24o54
26274
28473
3o65i
32809
34947
37065
39165
41247
433ti
45357
47386
49399
51396
53377
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59230
6ii5
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64953
66833
68701
70555
72397
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83196
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86705
oSôyo
08087
1 0476
12838
1 5x74
17486
19773
22036
24277
26495
28693
3o868
33o24
35i6o
37276
39374
41454
43516
45561
47588
49600
51595
5357'
55539
57488
59422
61 343
63249
65i4i
67021
05913
08327
10713
i3o73
i54o7
17715
20000
22261
24499
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28910
3io84
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35372
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5
061 55
08567
10950
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1 5639
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■ "" 41868
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93428
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96816
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11609
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14803
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1
90172
91 890
9359S
95396
96985
9866/
oo333
01993
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05287
06920
08545
10161
11769
i336g
14960
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18119
70740
72580
74408
76225
78039
79821
8i6o3
83373
85i32
86880
88617
90345
93062
93769
95466
97153
oo5oo
o3i59
o38o9
o545o
07083
08707
10322
41661
43721
45764
47790
49800
51794
53772
55734
57682
59615
61 534
63439
6533o
67208
69072
70924
72764
74591
76405
78209
80000
81780
83549
85307
87054
88791
905 17
92233
93939
g5635
97321
9899S
00666
02324
03974
o56i4
07246
0S869
10483
I
5oooo
51992
5396g
55930
57S76
59808
61725
6363g
655i8
67395
69258
71109
72947
-4773
76586
78388
801
8ig57
83735
85482
06398
08806
11187
i3542
15870
18174
20454
22711
24944
27156
29347
3i5i7
33666
35797
37908
40000
42074
44i3i
46171
48193
5o200
53190
54i65
56i35
58070
60000
61916
638i8
65707
67582
6g444
71293
73i3o
74955
76767
78568
80357
82135
83901
85657
06640
09045
11434
13776
1 61 02
i84o3
20681
22935
35167
27376
29565
31733
3388o
36oo8
38ii
40208
43381
44336
4637,
48395
5o4oo
52389
54363
56320
58263
60193
63107
64008
65895
6776g
6962g
71477
733i3
75i36
76948
78747
8o535
823l2
84077
85832
9
iig2g i2ogo
13528 i3688
15278
16860
i5ii9
16703
18377
18434
8733:
88g64
90689
92404
94109
95804
9748g
99166
00833
02490
o4i38
05778
07409
ogoSi
10644
I2250
1 3847
1 5436
17017
1859,
5
06882
09284
11660
i4oog
i6333
1 8633
20907
23i5g
25389
27596
29783
3ig48
34094
36220
38328
4o4i6
42487
44540
46577
48596
5o5gg
52587
54558
565 1 5
58457
6o384
62298
64197
66o83
67955
69815
71662
73496
753i8
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og523
1 1 896
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1 6564
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3ii33
33383
2 56io
37816
3oooo
32164
34307
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38537
40624
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50799
52784
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56710
5865o
60576
62488
64386
66371
68143
70000
71846
73679
755o(
77128 77308
87402
8gi37
90861
92575
94279
9^973
97658
99333
00998
02655
o43o3
o5g4i
07571
09193
io8o5
1 2410
i4oo6
i55g5
17175
18748
6
87576
89310
gio33
92746
94449
96142
97825
99500
01164
02820
04467
061 o5
07734
09354
10966
12570
i4i66
15753
17333
18904
78g37
80713
8348g
84353
86007
87750
gi204
92916
g46i8
96311
97993
99666
oi33i
02985
o463i
06268
07896
09516
11127
12730
14325
iSgii
17490
19061
8
79106
80891
82666
84429
86182
87924
89655
91376
93087
94788
9647g
98161
99833
ùi4g6
o3i5o
04795
0643 1
080 58
09677
11288
1 2890
14484
16070
17648
19218
246
215
344
243
242
241
240
23D
238
237
236
235
234
1
2.5
i'l
24
24
24
24
21
24
24
24
24
24
23
i>
4;i
4H
4H
4!l
4,s
4K
4rt
4K
4H
47
47
47
47
:t
74
74
7:t
73
7;!
72
72
72
71
71
71
71
70
•1
iW
».s
!)H
i)7
97
(III
»(i
«m
il.'i
11.5
ill
iH
!)4
;",
l-i;l
12:!
122
122
121
121
1211
1211
11:1
lUI
IIH
11.^
117
(
I4K
147
mi;
14«
14.5
14.5
144
1 1:1
I 1:1
112
142
141
140
7
172
172
171
1711
llW
l(i!
ir„s
11,7
1117
ICI,
1115
1115
1114
h
1!)7
HI;.
111.",
i:m
I'.ll
Hi:'
|.|.,
l!l!
I'.lii
I'.ln
i.-<;i
MH
1H7
S)
231
221
22U
21»
21b
217
211)
2i;.
214
21a
212
212
211
233
232
231
230
229
*28
227
226
22.5
224
•^3
2-Î2
221
2:h
23
&
23
23
23
23
23
23
00
22
-»
22
47
411
4li
411
411
4il
4.5
45
45
45
45
44
44
711
70
m
(iil
(iil
(1.-^
68
118
118
117
67
67
m
H:l
!I3
m
(12
H2
HI
111
illl
00
!I0
8«
8!l
88
11-
iir,
116
115
115
114
M 1
11:1
113
112
112
111
HI
1 111
1:1' 1
IH!1
ins.
v.r,
i:!7
lllll
13,1
135
131
134
133
133
II ill
111"
1(1"
1111
111(1
Kill
I5;i
15s
15K
157
1.51.
1.55
1.55
ISd
IKi,
1K5
184
IKli
1R2
182
181
180
179
178
178
177
210
209
208
207
20U
205
204
203
203
202
20J
200
199
220
219
218
217
216
215
214
213
212
211
210
209
1
22
22
22
22
22
22
21
21
21
21
21
21
"
44
44
44
43
43
43
43
43
42
42
42
42
3
(16
66
65
65
65
65
64
64
64
63
(13
63
4
88
R8
87
87
86
86
86
85
85
84
84
84
5
110
nil
109
109
108
108
1117
107
106
106
105
lO.-i
6
132
131
131
130
130
129
12^
128
127
127
126
12^5
7
1.54
1.53
1.53
1,52
151
151
1511
149
148
148
147
141.
8
176
175
174
17^
173
172
171
1711
170
U19
168
167
9
198
197
196
195
194
194
193
192
191
190
189
las
21
42
ca
83
104
125
146
166
187
207
206
205
204
203
202
201
200
199
198
197
196
195
1
21
21
21
20
20
20
20
20
20
211
2,1
20
20
2
41
41
41
41
41
40
40
40
40
40
39
39
39
3
62
62
62
fil
61
61
60
60
110
.59
59
.59
59
4
83
82
82
82
81
SI
RO
8(1
80
79
79
78
78
5
104
103
103
102
102
101
101
100
mil
99
99
9s
98
6
124
124
123
19,2
122
1"1
121
120
119
119
118
lit
117
145
144
144
143
141
1 11
111
MO
139
139
Kt
137
137
8
166
165
164
163
162
1112
161
lllll
159
158
15,-
l.)V
1.56
9
186
185
185
184
183
182
181
180
179
178
177
176
17d
180
185
184
183
Ifl
19
18
18
37
37
37
37
.56
.56
55
55
74
74
74
73
93
93
92
92
112
111
no
no
130
130
129
128
149
148
147
141.
107
107
166
165
182
13
36
55
73
91
109
127
146
164
181
180
179
178
177
176
175
174
173
172
171
170
109
18
18
18
18
IS
18
18
17
17
17
17
17
17
36
.311
36
36
35
.35
35
35
.35
34
31
34
34
54
54
54
53
.53
.53
.53
.52
.52
.52
.51
51
bl
7.)
72
75
71
71
70
70
70
69
69
68
68
68
91
90
90
89
89
88
88
87
87
86
86
85
85
109
108
107
107
106
106
105
101
104
103
103
102
101
107
126
12.5
12.5
124
123
123
122
121
120
12(1
119
118
145
144
143
142
142
141
11(1
1119
13S
13S
1:1,
136
135
163
102
161
160
159
108
158
1;.,
I06
15j
lu4
1j3
152
108
167
166
165
164
163
162
161
160
1,59
1.58
157
156
17
17
17
17
16
16
16
16
16
16
16
16
16
34
33
33
33
33
33
32
IfJ
32
32
32
31
31
50
50
.50
50
49
49
49
48
48
48
47
47
47
67
67
66
66
66
65
65
64
IH
64
113
113
62
84
84
83
83
82
82
81
81
80
80
711
79
78
nil
lllll
IOO| 99
98
98
97
97
91.
95
95
94
94
IIS
117
nil
116
115
114
113
113
112
111
1 1 1
nil
109
134
134
1.33
132
131
130
130
129
12s
127
12 1
126
126
151
150
149
149
148
147
146
145
144
143
142
141
140
TABLE II.
This gives ilie time 7" of describing a parabolic arc by a comet, tlie sum of tlie extreme radii r -\- r' being found at llji
top, anil the chord c at the left side of tlie pace.
Sum .)! thu ra.lll r ■- r . |
Chord
C.
0,01
0,0-2
0,03
0,04
0,05
0,00
0,07
0,08
0,09
0,10
0,11
Days lilif.
Days III if.
Days ](lil'.
Days Idif.
Days |(lif.
Days |dif.
Days |iiil
Days Idil".
Days l.lif.
Days lijif.
Days |dif.
0,00
0,01
0,02
o,o3
0,04
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,00c
0,027
i4
0,000
o,o4l
o,t>-8
9
11
o,oiio
o,o5o
0,099
0,143
8
1->,0(X
o,o58
0,11 5
0,!~0
0,31 g
i4
33
0,000
o,o65
0,129
0,193
0,232
o,3o6
6
i3
■9
37
38
OjOOt
o,t)7i
0,143
0,211
0,279
0,344
o,4o3
6
1 1
18
24
3i
4i
(>,000
0,077
0,1 53
0,339
o,3o3
0,375
0,444
o,5o8
5
ti
16
22
30
36
45
0,000
0,082
0,164
0,345
o,3a5
o,4o4
o,4So
0,553
0,620
5
10
i5
31
26
33
4o
49
0,000
0,08"
0,174
0,360
0,346
o,43o
o,5i3
0,593
o,6()9
0,740
5
to
i5
'9
25
3o
36
AA
53
0,OOCJ
0,093
0,184
0,275
o,365
0.455
o;543
0,629
0,71 3
0,793
0,867
4
â
18
23
28
33
39
46
56
0,000
0,096
o,lg3
0,288
0,383
0,478
0,571
0,662
0,752
o,83g
0,923
1 ,000
5
8
i3
18
22
27
32
37
43
59
0,0000
0,0001
o,oot.v^
0,00(KJ
0,001''
0,00 2 ^
o,oo3(')
0,0049
0,0064
0,008 1
0,0100
0,0121
0,0001 1
0,0002
0,0004
0,0008
0,0012
0,0018|
0,0024 0,0032|
0,0040
0,0050j
0,0061
0
\ • { r -\- r' )- or r^ -|- r"* nearly. |
TABLES rOR COMPUTING THE ORBIT OF A COMET.
Table I. This is a table of square roots, adapted to the calciilalion of the orbit of a Comet, by methods similar to tha'
proposed by Dr. Olbers. AVe have by inspection, in this table, the root of any number, from 0,001 to 10,19 ; and by
using the small tables of proportional parts, given in the margin, the root may he obtained Jrom the number, or the
number from the root, to five places of decimals. This requires no particular explanation, since the arrangement is the
same as that of a common table of logarithms. We may also observe that when the quantity .r, whose root is to be found,
is less than 0,102, it is convenient to find the root of lOOx, and then divide the result by 10 ; which is done by merely
transposing the decimal point. Thus if x^ 0,0961, we may find the root of 9,61 =3,1, and transpose the point one figure,
and we shall obtain 4/ 0,0961 =0,31. In like manner, if we have c3 = 0,080087, we get by the table v/ 8,0087 = 2,82996,
whence c = 0,282996 ; by this means the proportional parts are more easily obtained.
Table II. The argument at the top of the table is the sum of the two radii vectores of the comet r, r" ; the mean
distance of the earth from the sun being taken for unity. On the left side column of the table, is the length c of the
chord, connecting the extreme parts of these radii. The corresponding number represents the time T, given by
Lambert's formula [750, 750'] ; supposing the comet to move in a parabolic orbit,
T=9''^'",688724.f(r+r"+c)t — (r+r"— e)f|.
Thus if r4-r" = 2,20, and c = 0,20, we shall have r=8''''''^619. The proportional parts for the fractions of r -f r"
beyond two places of decimals, are placed at the right hand side of the page, those for c, in the column at the bottom of
the table, nearly below the corresponding tabular time T. In using Table II. we must enter it with the values of r-{-r"
and c ; taking them to t%vo places of decimals ; and find, by inspection, the corresponding chief term of T. The variation
of T, corresponding to the successive tabular values of r -\- r", is given in the same horizontal line with the chief term
of T; and we must find also the variation, corresponding to the successive tabular values of c, in the vertical column,
immediately below the chief term of T". The increments of T, corresponding to the fractional parts of r + r" and c.
beyond the second decimal place, are to be found and added to the chief term T, to obtain the true value' of T.
In general it will be sufficiently accurate to use for the argument of the proportional parts in the table in the side
column, the tabular number in the column of differences corresponding to the chief term of T; but when very great
accuracy is required, we may find it for the exact value of c ; by taking a proportional part of the difference of the two
nearest numbers in the table.
To show, by an example, the use of this table, we shall suppose r-(-r"= 1,96280, c = 0,21.573. Then we shall have for
the chief term of T, corresponding to 1,96 and 0,24 the value 9 ^ ,760 ; the differences between this and the next
numbers being 25, and 406, respectively. The proportional parts corresponding to the decimals ,00280 and ,00573 are 7
and 233 ; the sum of these three quantities is 9,760 -\- 0,007 + 0,233 = 10''*^^ the value of T required.
In the right hand column of the table is given the value of c~. At the bottom of the table is given the values of
.J . (r -)- r") 2, which maybe used, instead of r3 -[-r"2, in the first approximation to the value of ç. In this case the
calculation is made merely by inspection ; using the nearest numbers in the table, and taking them to one or two places
of decimals ; without using the tables of proportional parts, which are exclusively adapted to the values of r -^ r" and c.
These two tables are designed to facilitate the computation of the value of g, from the three equations {A), (B), { C),
which are similar to these in the following system ; in which r,r" represent the radii vectores at the first and third
observations; c the intercepted chord ; ^ the curtate distance 0/ the comet from the earth; the interval bettueen the
observations, expressed in days, being given and represented by T. The equation ( /) ) which is the sum of the
equation (.4), (B), may be used in the first approximation to the value of §. It is not absolutely necessary, to use
the equation ( Z) ), but it will frequently be found to have a tendency to abridge the calculations.
3a
TABLE II.
This gives the time Tof describing a paraboUc arc by a comet, the sum of the extreme radii r -
top, and" the chord c at the left side of the pace.
;•" bein^ found at the
Sum of the liadii r -r r", |
Chord
C.
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,20
0,2
1
iiT?.
0,22
Days !dif.
Days |dif.
Days |ilir.
Days Idil'.
Days |dil.
Days |dil'.
Days lilil.
Days |dir.
Days |dif.
Days
Days [liif.
0,00
0,000
0,000
0,000
11,000
0,000
0,000
0,000
u,ooo
0,000
0,000
0,000
O.ljOOO
0,01
0,701
4
0,103
4
o,Ioq
4
0,11 3
3
0,116
4
0,120
3
0,1 23
4
0,127
3
0,1 3o
3
0,1 33
3
0,1 36
3
0,0001
0,02
0,201
8
o,2oq
8
0,217
S
0,225
7
0,232
8
0,240
7
0,247
6
0,2b3
7
0,260
b
0,266
7
0,273
6
o,ooo4
o,o3
o,3oi
1 3
o,3r4
12
0,326
1 1
0,337
1 1
0,348
1 1
0,359
I I
0,370
10
o,38o
10
0,390
9
o,3qq
10
o,4og
9
O.ÛOWJ
o,o4
0,401
17
0,418
16
0,434
i5
0,449
i5
o,464
i4
0,478
i4
0,492
i4
o,5o6
i3
0,519
i3
0,532
i3
0,545
12
0,0010
o,o5
0,300
21
0,521
20
o,54i
iq
0,560
IQ
o,57q
18
0,597
18
0,61 5
17
0,632
16
0,648
16
0,664
16
0,680
16
0.0025
0,06
o,5q8
25
0,623
24
0,647
24
0,671
22
0,693
22
0,71 5
21
o,736
21
0,757
20
0,777
IQ
0,790
IQ
0,81 5
iq
o,oo35
0,07
o,6q4
3o
0,724
29
0,753
28
0,781
36
0,807
26
0,833
25
0,858
24
0,883
23
o,9o5
23
0,928
22
0,950
23
0,0049
0,08
0,789
35
0,824
33
o,857
32
0,889
3i
0,920
29
o,q4q
29
0,978
28
1,006
27
i,o33
2b
i,o5q
25
1,084
25
0,0064
0,09
0,882
40
0,922
38
0,960
37
0.997
35
I,032
33
i,o65
33
1,098
3i
1,1 2g
3i
1,160
29
1,189
29
1,218
28
o,ùo8 1
0,10
0,972
46
1,018
A4
1,062
4i
i,io3
3q
I,l42
38
1,180
36
1,216
35
I,25l
35
1,286
33
i,3i9
32
1, 35 1
32
0,0100
0,11
1,069
53
1,112
4q
1,161
46
1,207
44
1,25:
42
1,293
4i
1,334
39
1,373
38
1, 41 1
36
1,447
36
1,483
35
0,0121
0,12
1, 1 39
62
1,201
56
1,257
32
1 ,3f!9
4q
1,358
47
r,4o5
45
1 ,45o
43
1,493
43
1,535
40
1,575
3q
1, 61 4
3q
0,01 44
o,i3
1 ,284
66
1,330
58
i,4oS
55
1.463
32
i,5i5
4q
1,564
47
1 ,61 1
46
1,657
45
1,702
4i
i,74:>
4i
0,0169
o,i4
1,435
68
i,5o3
62
1,565
57
1,623
54
1,676
52
1,728
5o
1,77a
49
1,827
4l
1,874
45
0,01 96
0,1 5
i,5q2
71
T,fi63
64
1,727
5q
1,786
57
1,843
55
i,8q8
32
1,950
5i
2,001
49
0,0225
0,16
1.754
73
1,827
66
I,8q3
63
1,956
59
2,01 5
:)7
2,072
53
2,13-
3û
0,0356
0,17
1,921
75
1,996
6q
2,o65
65
3,1 3o
62
2,192
59
2,25l
38
0,0289
0,18
2,093
78
2,171
71
2,242
67
2,30Q
63
2,374
61
o,o324
0,19
2,270
So
3,35o
74
2,434
69
3,493
67
o,o36 1
0,20
2,45 1
83
2.534
-6
2,6tO
72
o,o4.x>
0,21
2,637
85
2,722
78
0,044 1
0,22
2,82s
87
o,o484
,0072
,0085
,0098
,0113
,0128
,0145
,0162
,0181 1 ,0200
,0221 1 ,0242
'^
4 . ( r + J-" )= or r'' + r'" nearly. |
80
83
86
89
92
95
98
8
8
9
9
9
10
10
16
17
17
18
18
IQ
20
24
25
26
27
28
29
2q
32
33
M
36
37
38
3q
4o
42
43
45
46
48
4q
48
5o
53
53
55
57
5q
56
58
60
62
64
67
69
64
66
6q
71
74
76
78
72
75
77
80
83
86
88
Proporliona
parts for the Chord.
lOI
104
107
no
ii3
116
119
123
125
128
i3i
i34
1 37
—
_.
10
10
1 1
1 1
1 1
12
12
12
i3
i3
i3
i3
14
20
21
21
32
23
23
24
24
25
26
26
27
27
3o
3i
32
33
34
35
36
37
38
38
39
4û
4i
40
42
43
A4
45
46
48
4q
5o
5i
52
54
65
5t
53
54
55
57
58
60
61
63
64
66
67
69
61
62
64
66
68
70
71
73
75
77
79
80
82
71
73
75
77
79
81
83
85
88
90
95
94
96
Si
83
86
8S
90
q3
q5
q8
100
102
io5
107
no
91
94
96
99
102
io4
107
no
1x3
ii5
118
121
123
E X A M P L E I .
To show the use of these tables we shall apply them to the <letermination of the value of g from tlie three followina
equations, correspondinç to observations of the Comet of 1779 ; as in page xiii. of Dr. Olbers' Jlhhaiidlung, &,c.
r2=0,9S240 4-0,87363.ç-l-2,33263.§2; (A)
r"2 = 0,98361 4- 2,llSC9.ç + 2,88041.52 ; (B)
(;2 — 0,04133 4- 0,006845. § + 0,208501. §2; ( C)
r3 + r"-3 = 1,97101 -f-2,99232.g + 5,21304.ç2 . ( Z) )
Time T= ll''='y^834 .
COMPUTATION OF p FROM THE ABOVE EQUATIONS,
(./!), (E),(C).
r2.
r'la.
C2.
r, r", c.
T.
Hypolhesisl.
g = 0,3
0.98240
0,26209
0,20994
0,98861
0,63560
0,25924
0,04168
,00205
,01876
r =1,20599
r"= 1,37239
r+r" = 2,57838
c =0,2.5038
11,645
19
18
11,682
1,45443
1,88345
0,06269
Hypothesis II.
Add -^^ makCB
§=0,3075
0,98240
,26864
,'^057
0,98861
,65149
,27236
0.041830
.002104
,019716
0,063700
r =1,21311
r''= 1,3829]
I- -j- I-' =2,59602
c =0,25239
11,690
13
111
1,47161
1,91216
11,814
Hypothesis III.
Add ^-i-y or ,00123
5 = 0,30873
0,98240
0,26971
,22233
0,98801
,65410
,27454
0,041880
,002112
,019374
r =1,21427
r'= 1,38405
r+r"=2,,59892
c =0,25271
11,090
20
127
1,47444
1,91725
0,063866
11,837
Hypothesis IV.
^"Wtju " .MOIS
J = 0,30858
0,98240
,26958
,22211
0,98861
,05377
,27427
0,041880
,002111
,019854
r =1,21412
r"= 1,38413
r+j-" = a,59a55
c =0,25207
11,090
19
125
1,47409 1 1,91 61.5
0,003845
11,834
Coetlicients of Ç.*
0,S73C3
0,20209
655
0,26864
107
-âTTUtr
0,20971
—13
0,26958
2,llSi,9
0,63500
1589
0,65149
2G1
0,05410
—33
0,6537'
,002053
51
,002104
8
,002112
,002111
CoefBcienlsof g^
2,33203
2,8804110,208501
A
0,20994
1050
13
0,2.592410,018766
1296 938
16 12
Ï5ÎT
"bUt!'
-TSUTT
0,22057
170
0,9-^33
—22
0,27230
218
0,019716
158
0,274,54 0,019874
—27 1 _20
0,274271 ,1119854
TABLE II.
This gives tlie time IT of dcscrihiiii; a parabolic arc by a comet, the sura of tlie extreme radii r -{->■'
top. and the chord i- at tlio lelt siile of the paiie.
liciiiT luiiml at tin
t^iimi'l'tho Kailii r~\-r".
Cboril
c.
0,00
0,01
0,02
o,o3
o,o4
o,o5
o,c6
0,07
0,08
0,09
o,io
0,1 1
0,13
0,1 3
o,i4
0,13
0,16
0,17
0,18
0,19
0,20
0,21
0,3.2
0,23
0,24
0,25
0,26
0,27
0,28
0,39
0.-23
Days jdil"
0.000
0,139
0,2
0,41 S
0,557
0,696
0,834
0,972
1,109
1,246
1,383
i,5i8
1 ,653
1 ,786
1,919
2,o5o
2,180
3,309
2,435
2,56o
3,683
3,800
2,915
3,033
.0-26.=
0.-24
Days |.lil'.
o,oou
0,l42
o,2S5
0,437
0,569
0,711
o,85
0,993
1, 1 34
1,274
i,4i3
1,553
i,6go
1,837
1,963
2,098
3,332
2,364
2,495
2,634
3,75o
2,874
3,995
3,112
3,322
0,25
Days lilif.
,0288
<.),00(
0,145
0,291
0,436
o,58i
0,725
0,870
I, or 4
1,1 58
i,3oi
1,443
1,585
1,736
1,867
3,006
3,1 45
3,383
2,4i8
3,552
2,685
3,816
3,945
3,073
3,195
3,3i4
3,435
,0313
0,26
Days |(lir
o,uoo
0,1 48
0,396
0,444
0,592
o,74o
0,887
i,o34
1,181
1,327
1,473
1,618
1,762
1 ,906
2,048
3,190
2,33i
3,470
3,601
2,74
2,880
3,01
3,145
3,273
3,398
3,5i9
3,633
0
,0338
0,-27
Days |<iif.
0,000
0,1 5 1
o,3o2
0,453
0,604
0,754
0,904
i,o54
1, 204
1,353
i,5o
1,649
1,797
1,944
2,089
2,334
3,378
3,521
2,663
2,804
3,943
3,080
3,3i5
3,348
3,478
3,606
3,739
3,845
83
,036c
0,28
Days I dil'.
0,00(
0,1 54
o,3o8
0,461
o,Ci 5
0,768
0,931
1,074
1,236
1,378
i,53o
1,681
1, 83 1
1,981
3,1 3o
3,378
2,425
2,571
2,716
2,860
3,oo3
3,i44
3,283
3,420
3,556
3,688
3,817
3,942
4,060
,0392
0,29
0,782
0,937
1,093
1,348
i,4o3
1,557
1,71 1
1,864
2,017
2,169
,0421
0,0000
0,000 1
0,0004
0,0009
0,001 ()
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,01 2 1
0,01 44
0,0169
0,0196
0,0225
o,o256
0,0389
o,o324
o,o36i
o,o4oo
o,o44 1
o,o484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,084 1
c"
[t -\- r")^ or r- ■
r"- ticarly.
Proportiooal parts for the Chord.
113
116
120
134
128
I 32
1 36
i4o
144
1 48
l52
1 56
,
II
13
12
13
i3
i3
i4
i4
i4
i5
i5
16
3
22
33
24
25
26
26
27
28
29
3o
3o
3i
3
34
35
36
37
38
4o
4i
42
43
44
4b
47
4
45
46
48
5o
5i
53
54
56
58
bq
61
63
5
56
58
60
62
64
66
68
70
72
74
7b
78
6
67
70
72
74
77
79
83
84
86
89
^1
94
7
78
81
84
87
90
92
qi
9a
1 01
104
106
109
8
qo
q3
96
99
102
106
109
112
n5
118
122
125
9
1 01
io4
108
112
ii5
"9
133
126
i3o
1 33
1 37
i4o
lu the first approximation we shall use the equations (C)(jD), computing
the numbers to one or two places of decimals. Now if we suppose ç = 1. these
equations become (;2 = 0,26, r2 -)-)-"2 = 10,2. The first of these numbers is to be
found ou the right hand side of Table II., and the second at the bottom, the
corresponding value of T is nearly 31 days. This being nearly three times the
actual value of T, we may take for g one third part of the value first assumed, or
g = J ; then repeating the preceding calculation, with one more decimal in c-,
we get c2 = 0,067 ; r2 + r"2 = 3,53, and the corresponding value of T is about 124
days ; so that we must decrease e a little more. We shall therefore take for the first
hypothesis of the preceding table ç = 0,3, and use the equations ( jî ), (B), ( C ),
making the c^ilculation to a greater degree of accuracy, and we get 7"= 11,632.
This time being rather too small, the value of § is increased j-Vi '° Hypothesis
II., and the resulting value of T becomes 11,81-1. Increasing ç by 2-50, we obtain
in a third hypothesis 7-= 11,837, which is rather too large. Finally decreasing
this last value of § by ^.jn part, we obtain ç = 0,3085S, and T= U''''^',S34, and
this value of T agrees with that by observation. We may use the values obtained
by this last o()eration, as being very near the true values so that we shall have
g = 0,30S58, r=l,21-J12, r"=l,38443, c = 0,23267 ; which are almost identically
the same with those obtained in the above mentioned work of Dr. Olbers.
Trop, parts for lliu sum of llio Itailii
■ I 3 I 3 I 4 I 5 I 6 I 7 I 8 I 9
I
1
I
3
3
3
3
3
4
4
4
5
5
6
6
6
6
7
7
8
8
'.
8
lO
9
10
10
1 1
1 1
I 3
1 1
i3
13
i4
i3
i4
i3
i5
i4
16
i5
17
i5
i8
iG
iS
17
19
18
20
18
31
19
22
20
23
20
23
TABLE II.
This gives the time T of describing a parabolic arc by a comet, the sum of the extreme radii ;■ -f- 1
top, and the cliord c at the left side of the page.
being found at the
Sum r,r th" liiuhi r H- r". 1
Chord
C.
0,30
0,31
0,3-2
0,33
0,34
0,35
0,36
0,37
0,38
0,39
Days |dif.
Day» |,lir.
Davs lilit:
Days |ilif.
Days |i!ir.
Days lilit'.
Days Idif.
Days |,lif.
Days Idif.
Day» |dif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0«i
_
0,0000
0,01
0,169
3
0,162
2
0,164
3
0,167
2
0,169
3
0,172
2
0,174
3
0,177
2
0,179
3
0,182
2
0,0001
0,02
o,3l8
6
0,324
5
0,329
5
0,334
5
0,339
5
0,344
5
0,349
5
0,354
4
0,358
5
0,363
5
o,ooo4
o,o3
0,477
S
0,485
8
0,493
8
o,5oi
■ 7
o,5o8
8
o,5i6
7
o,523
7
o,53o
7
0,537
7
0,544
7
0,000g
o,o4
0,636
1 1
0,647
10
0,657
10
0,667
II
0,678
9
0,687
10
0,697
10
0,707
9
0,716
:o
0,726
9
0,0016
o,o5
o,7o5
i3
0,808
i3
0,821
i3
0,834
i3
0,847
12
0,859
12
0,871
12
0,883
12
0,895
12
0,907
12
0,0025
o,o6
0,954
i5
0,969
16
0,985
i5
1,000
16
1,016
i4
i,o3('
i5
1 ,045
i5
1.060
i4
1,074
i4
1 ,oSS
i4
0,00 36
0,07
1,112
iS
i,i3o
19
i,i4g
18
1,167
17
1, 1 84
18
1,202
17
i,2ig
17
1,236
16
1,252
17
1 ,260
16
0,0049
0,08
1,270
21
1, 29 1
21
I,3l2
20
1,332
21
1 ,353
30
1,373
19
1,392
20
I,4l2
19
i,43i
19
1 ,45"
iS
0,0064
0,09
1,427
24
1, 45 1
24
1,475
23
1,498
23
1,521
22
1,543
22
1,565
22
1,587
22
i,6og
21
i,63o
21
0,0081
0,10
1,584
27
1 ,61 1
26
1,637
26
1,663
26
1,689
25
1,714
24
1,738
25
1,763
24
1,787
23
1,810
23
0,0100
0,1 1
1, 74 1
3o
1,771
29
1 ,800
28
1,828
28
1,856
26
1,884
27
1,91 1
27
1,938
26
1,964
26
1.990
26
0,0121
0,12
1,897
33
1,930
3i
1,961
3i
1,992
3i
2,023
3o
2,o53
3o
2,o83
29
2,112
29
2,l4l
28
2,169
29
0,0144
0,1 3
2,o53
35
2,088
34
2,122
U
2,1 56
33
2,189
33
2,222
33
2,255
3i
2,286
32
2,3i8
3i
2,34g
3o
0,0169
0,1 4
2,208
38
2,246
37
2,283
36
2,319
36
0,355
36
2,391
35
2,426
M
2,460
34
2,494
33
2,527
33
0,0196
0,1 5
2,362
4i
2,4o3
40
2,443
39
2,482
39
2,52 1
38
2,559
37
2,596
37
2,633
37
2,670
35
2,7o5
36
0,0225
0,16
2,5i5
Âà
2,559
43
2,602
42
2,644
42
2,686
40
2,726
40
2,766
40
2,806
39
2,845
38
2,883
38
o,0256
0,17
2,668
4C
2,714
46
2,760
45
2,8o5
45
2,85o
43
2,893
43
2,936
42
2,978
42
3,020
4o
3,06c
4i
0,0289
0,18
2,819
5o
2,860
49
2,918
48
2.966
47
3,01 3
46
3,059
46
3,io5
45
3,1 5o
M
3,194
43
3,237
43
o,o324
0,19
2,97c
53
3,023
52
3,075
5i
3,126
49
3,175
5o
3,225
48
3,273
47
3,320
Al
3,367
46
3,41 3
45
o,o36i
0,20
3,119
56
3,175
55
3,a3o
54
3,284
53
3,337
52
3,389
5i
3,440
5o
3,49"
5o
3,540
48
3,588
48
o,o4oo
0,21
3,26"
60
3,327
58
3,385
57
3,442
56
3,4g8
55
3,553
54
3,607
53
3,660
52
3,712
5i
3,763
5i
o,o44i
0,22
3,4i4
63
3,477
61
3,538
60
3,598
59
3.657
58
3,71 5
57
3,772
56
3,828
55
3,883
54
3,937
53
0,0484
0,23
3,559
66
3,625
65
3,690
64
3,754
62
3,816
61
3,877
60
3,937
58
3,995
58
4,o53
57
4,110
56
0,0629
0,24
3,702
70
3,772
69
3,84 1
67
3,908
65
3,970
64
4,o37
63
4,100
62
4,162
60
4,222
60
4,282
59
0,0576
0,25
3,844
74
3,918
72
3,990
70
4,060
69
4,129
68
4,197
66
4,263
64
4,327
64
4,3gi
62
4,453
61
0,0620
0,26
3,983
78
4,061
76
4,1 37
74
4,21 1
73
4,284
70
4,354
70
4,424
67
4.491
67
4.558
65
4,623
64
0,0676
0,27
4.11Q
83
4,202
80
4,28?
79
4,36 1
76
4,437
74
4,5ii
72
4,583
71
4.654
70
4,734
68
4,792
67
0,0729
0,28
4.252
88
4,340
85
4.425
83
4,5o8
80
4,588
77
4,665
76
4,74 1
75
4,816
73
4.S8g
71
4,960
70
0,0784
0,29
4,38 1
95
4,476
90
4,566
86
4,652
84
4,736
82
4,818
80
4,898
78
4,976
76
5,05;
75
5,127
73
0,0841
o,3o
4,5o3
io3
4,606
97
4,7o3
91
4,794
89
4,883
86
4,969
84
5,o53
81
5,1 34
80
5,2i4
78
5,2g2
76
0,0900
0,3 1
4,73o
io5
4,835
98
4,933
94
5,027
90
5,117
88
5,2o5
85
5,290
84
5,374
81
5,455
80
0,0961
0,32
4,961
107
5,068
100
5,1 68
95
5,263
92
5,355
90
5,445
87
5,532
85
5,617
83
0,1024
0,33
5,195
109
5.3o4
102
5,406
97
5.5o3
94
5,59-^
91
5,688
89
5,777
87
0,1 o8g
0,34
5,433
no
5,543
io4
5,647
99
5,746
96
5,842
93
5,g35
90
0,1 1 56
0,35
5,674
ii3
5,787
io5
5.892
101
5,Qg3
97
6,090
95
0,1225
o,36
5,919
ii4
6,o33
107
6,1 4o
io3
6,243
99
0,1296
0,37
6,168
ii5
6,283
log
6,392
104
0,1 36g
0,38
6,419
118
6,537
no
0,1 444
0,39
6,674
i'9
0,1 52 1
,0450
,0481
,0512
,0545
,0578
,0613
,0648
,0685
,0722
,0761
C2
4 . (r + r"i!i or T- + r"a n.-iirly. |
p
roport
oiial parts for the Chord.
122
125
128
i3i
1 34
1 37
i4o
143
146
149
l52
i55
i58
161
164
167
170
173
176
17g
182
I
12
i3
i3
i3
i3
14
i4
i4
i5
l5
x5
16
16
16
16
17
17
17
18
18
18
2
24
25
26
26
27
27
28
29
2g
3o
3o
3i
32
32
33
33
34
35
35
36
36
3
37
38
38
3g
40
4i
42
43
AA
45
46
47
47
48
49
5o
5i
52
53
54
55
4
4g
5o
5i
52
54
55
56
57
68
60
61
62
63
64
66
67
68
6g
70
72
73
5
61
63
64
66
67
tig
70
72
73
75
76
78
79
81
82
84
85
87
88
go
gi
6
73
75
77
79
80
82
84
86
88
8g
91
93
95
97
»?
100
102
io4
106
107
109
7
85
88
90
92
94
96
g8
100
102
1 04
106
109
n 1
n3
ii5
117
ng
121
123
125
127
8
08
100
102
io5
107
no
112
ii4
117
ng
122
124
126
129
i3i
1 34
1 36
I 38
i4i
i43
1 46
9
no
ii3
ii5
118
121
123
126
12g
i3i
i34
1 37
i4o
142
145
i48
i5o
1 53
i56
i58
161
164
In making these successive operations, it is convenient to vary §, by some aliquot part of its value, represented by ,'.ç ;
f being an intec:ral number ; since by this means we are enabled to deduce any one of the coefficients of §, in the successive
operations, from tliat which immediately precedes it ; as in the small tables of the preceding examj>lc. Thus if we
represent by A^ , ./7, , ./Î, , A^ , the successive vahiesof the term 0,87.3fi3ç of (he equation {A'), we shall have, when
1 = 0,3, .à, := 0,8736.3. ç= 0,26209, in the first operation. In the second hypothesis, this is to be increased by
~. .^, = 0,00655 ; by which means it becomes A, = 0,26*64. In the third hypothesis this is increased,!- . jî, = 0,00107;
making -5, = 0,26971. In the fourtli hypothesis, it is decreased 5555 . A^ = 0,00013, making the final value A^ = 0,26958 ;
as in the preceding table. In like manner if the coefficient of g'-, in any operation be represented hy A^ and we increase
5 in the next operation by the quantity ^.ç, the value of .^ will become A . {\ -\-%-\-};i)^=A-\-l .A ■\-Jy.B\ using
for brevity B = l.A. From this formula we obtain the successive values of JÎ, as in the third table of the preceding
e.\aniple. In this way we obtain the values, in the successive operations, with very little additional labor.
TABLE II.
This gives tlie time T of ilesci-ihins; a par;ibolic arc by a comet, (lie sura of the extreme radii r-j-r" being fomiJ at t)ie top,
nml tlie oboril e at tin U-lt side of the |iai;e.
Sutii ol" the llaiiii r -f- ?'.
Piop. Jjarla for lliu sum ol* tlie Kadii.
T 1 3 1 ,3 1 /i 1 5 1 6 1 7 1 8 1 0
Cliord
c.
0,40
0,41
0,4-^
0,43
0,44
0,45
0,46
I
2
3
0 0
0 0
0 I
[
C
I
I
0
I
1
I
2
I 1 ' 1 1 -
1 I 1 I
1 I 2 3
2 2 3 3
Duys |.lif.
Days |.lif.
Days |,lif.
Uays |ilil'.
Days Iclil'.
Days |.lir.
Days |dif.
0,00
0,000
0,000
0,000
0,000
0,000
o,ocx>
0,000
CJ,000(l
0,01
0,184
2
0,186
2
0,188
3
o,iqi
2
o,ig3
2
0,195
2
0,197
2
0,000 1
4
0 I
I
2
2
3334
0,02
o,368
4
0,372
5
0,37-
4
o,38i
5
o,386
4
0,390
4
0,394
4
0,0004
o,o3
0,55 1
7
0,558
7
o,565
7
0,572
6
0,578
7
o,585
6
o,5gi
7
o,ooog
5
1 I
2
2
3
3445
o,o4
0,735
9
0,744
9
0,753
9
0,762
9
0,771
9
0,780
8
0,788
9
0,0016
6
7
I I
I I
3
2
2
3
3
4
4455
4 5 6 6
o,o5
0,919
II
0,930
II
0,941
II
0,952
1 1
0,963
1 1
o,g74
1 1
o,g85
II
0,0035
8
J 2
2
3
4
5 (i 6 -
0,06
1,102
i4
1,116
1 3
i,i2g
i4
i,i43
i3
i,i56
i3
I, leg
l3
1,182
i3
o,oo36
9
I 2
3
4
5
5678
0,07
! .385
Hi
i,3oi
Hi
1,3 1 -
16
1,333
i5
1,348
i5
1,363
16
1,37g
i5
o,oo4g
5
6
6
6789
o,u8
l,40h
'9
1,48-
18
i,5o5
18
1,533
17
1,540
18
1,558
17
1,575
17
0,0064
10
I 2
3
3
4
4
o,og
1,65 1
21
1 ,672
20
1,692
20
1,712
20
1,732
20
1,753
19
1,771
20
0,008 1
1 1
1 2
I 2
I 2
4
5
789 10
7 8 10 I 1
0,10
1,833
23
1,856
23
1,879
23
1,903
33
1,934
22
1,946
21
1,967
22
0,0100
i3
1 3
1 3
4
5
7
8 9 10 13
8 10 11 i3
0,11
2,0 [(>
25
2,o4l
25
2,o6(j
25
3,091
24
2,Il5
24
2,i3g
24
2,i63
24
0,0121
i-i
4
6
7
0,12
o,i3
2 , 1 gS
2,379
27
3o
2,225
2,4og
28
3o
2,253
2,43g
27
29
3,380
3,468
26
29
2,3o6
2,4g-
27
=9
2,333
2,526
26
28
2,359
2,554
26
28
0,0 1 44
0,0 i6g
J 5
i 6
3 3
2 3
3 3
5
5
5
6
6
7
8
9 11 12 i4
10 1 1 i3 i4
10 12 i4 i5
0,14
2,56<>
33
2,593
32
2,625
3.
2,656
32
2,688
3i
2,719
3o
2,749
3o
0,0 1 g6
17
9
o,i5
2.741
35
2,7-6
34
2,810
34
2,844
34
2,8-8
33
2,911
33
2,g44
33
0,0225
18
2 4
2 4
5
6
7
8
9
II i3 i4 16
11 i3 i5 17
0,16
3,921
37
3,958
37
3,gg5
37
3,o32
35
3,067
36
3,io3
35
3,i38
35
0,02 56
19
10
0,17
3,101
3y
3,i4o
4o
3,180
38
3,218
3g
3,257
37
3,2g4
38
3,332
37
0,0289
20
2 4
2 4
2 4
6
8
8
10
12 14 16 18
i3 i5 17 19
i3 i5 18 20
0,18
3,280
42
3,323
42
3,364
4i
3,4o5
4o
3,445
4i
3,486
39
3,535
39
o,o334
2 1
6
1 1
0,19
3,458
45
3,5o3
44
3,547
44
3,591
43
3,634
42
3,676
42
3,718
4i
o,o36i
22
7
9
1 1
0,20
3,636
48
3,684
46
3,73o
46
3,776
45
3,821
45
3,866
44
3,gio
44
o,o4oo
23
24
3 5
2 6
7
7
9
10
1 2
1 2
i4 16 18 21
l4 17 19 23
0,21
3,8 1 4
49
3,863
49
3,gi2
49
3,961
48
4,009
47
4,o56
46
4,102
46
o,o44i
0,22
3,ggo
52
4,042
52
4,094
5i
4,i45
5o
4,ic)5
5o
4,2.45
49
4,2g4
48
o,o484
2 5
3 5
8
10
i3
i5 18 20 23
0,23
4,166
55
4,221
54
4,275
53
4,328
53
4,38 1
53
4,433
5i
4,484
5i
o,o53g
26
3 5
8
10
i3
16 18 21 23
0,24
4,341
5-
4,3g8
57
4,455
56
4,5ii
55
4,566
55
4,631
54
4,675
53
0,0576
27
38
3 5
3 6
8
8
1 1
1 1
i4
14
16 ig 22 24
17 20 22 25
0,25
4,5i4
61
4,575
5g
4,634
59
4,693
58
4,75 1
57
4,So8
56
4,864
55
0,0635
29
3 6
9
12
i5
17 30 23 26
0,26
4.(i87
64
4,75 1
63
4,81 3
6i
4,874
60
4,934
60
4,gg4
59
5,o53
58
0,0676
0,27
4,859
66
4,g25
65
4,ggo
64
5,o54
63
5,117
62
5,17g
62
5,341
60
0,0739
3o
3 6
9
9
10
12
i5
18 21 24 27
0,28
5,o3o
69
5,09g
68
5,167
66
5,233
66
5,399
65
5,364
64
5,428
63
0,0784
3i
3 6
12
16
19 32 25 28
0,2g
5,200
72
5,273
70
5,342
70
5,4i2
68
5,48o
67
5,547
67
5,6i4
65
0,084 1
32
3 6
i3
16
19 22 26 29
33
3 7
10
i3
17
20 23 26 3o
o,3o
5.368
75
5,443
73
5,5i6
73
5,589
71
5,660
70
5,73o
6g
5,7gg
68
0,0900
34
3 7
10
i4
17
20 24 27 3 1
o,3i
5,535
78
5,6i3
77
5,690
75
5,765
74
5,839
73
5,gi3
72
5,984
70
0,0961
0,32
5,700
81
5,781
80
5.861
79
5,940
77
6,017
75
6,092
75
6,167
73
0,1024
35
4 7
1 1
i4
18
2 1 25 28 32
0,33
5,864
84
5,948
83
6,o3i
82
6,ii3
80
6,193
79
6,373
77
6,349
76
o,io8g
36
4 7
1 1
i4
18
32 25 39 32
0,34
6,025
89
6,1 14
86
6,200
85
6,285
83
6,368
83
6,45o
80
6,53o
79
0,11 56
37
38
4 7
4 8
1 1
1 1
i5
i5
19
19
22 26 3o 33
33 37 3o 34
0,35
6,i85
9'
6,277
9°
6,367
88
6,455
87
6,542
84
6,636
84
6,710
82
0,1225
39
4 8
12
16
20
23 27 3 1 35
o,36
6,343
96
6,438
94
6,533
92
6,624
89
6,7 1 3
88
6,801
87
6,888
85
0, 1 296
24 28 32 36
25 3g 33 37
25 2g 34 38
26 3o 34 3g
0,3-7
6,496
lOI
6,5g7
98
6,695
95
6,790
94
6,884
9"
6,975
90
7,o65
88
0,1369
40
4 8
13
16
30
o,38
6,647
106
6,753
102
6,855
100
6,g55
97
7,o53
95
7,1 47
93
7,240
91
o,i444
41
4 8
1 2
16
31
0,39
6,793
112
6,905
108
7,0 1 3
1 04
7,117
lOI
7,218
99
7,3i7
96
7,4 1 3
95
0,l52I
42
43
4 8
4 9
i3
i3
17
17
21
33
o,4o
6,933
120
7,o53
ii4
7, '67
,09
7,376
106
7,389
102
7,484
101
7,585
9&
0,1600
44
4 9
i3
18
22
26 3 1 35 4o
0,4 1
7,194
123
7,3i7
ii5
7,433
no
7,542
108
7.65o
io4
7,754
102
0,1681
I '■>
18
18
23
23
27 32 36 4i
38 33 37 4i
28 33 38 42
2g 34 38 43
0,42
7,45g
124
7,583
1 17
7,700
112
7,8 13
log
7,921
106
0,1764
45
1 4
0,43
7,727
126
7,853
u8
7,971
ii4
8,o85
no
0,1849
46
5 9
i4
0,44
7,998
127
8,125
120
8,245
ii5
0,1936
47
48
? 9
J ru
14
14
19
19
24
34
0,45
8,273
12g
8,4oi
121
0,2025
49
5o
5i
52
5 10
5 10
J 10
J 10
i5
i5
i5
16
20
20
20
21
25
25
26
26
29 34 3g 44
30 35 40 45
3 1 36 4i 46
3i 36 42 47
,0800 1
,0841
,0882
,0925 1
,0968 1
,101:3
,1058 1
C^
' 1
' ]
' 1
' 1
53
J 1 1
16
31
27
32 37 42 48
J
J . (r -f- r')3 or r^ + r ^ nenily. |
54
55 (
J 1 1
5 1 1
16
17
22
22
27
28
32 38 43 4g
33 39 A4 5o
Proporlional parts fur tlje Ciionl.
184
i85
i8(
) 18-
188
i8g
190
19
'9
i 193
ig4
ig5
'9
^ 197
56 f
3 I 1
17
23
28
34 3g 45 5o
—
—
—
—
—
—
—
57 (
Î I I
■7
23
29
34 4o 46 5 1
I 18
19
IC
■Ç
) 19
19
19
It
) i<
) 19
19
20
2(
3 20
58 (
) 12
17
23
29
35 4i 46 52
2 37
37
3-
3-
38
38
38
3{
i 3i
S 39
39
39
3(
i 39
59 f
i 12
18
24
3o
35 4i 47 53
3 55
56
se
se
) 56
57
57
5-
7 5f
i 58
58
59
5(
) 59
4 74
74
74
7-
75
76
76
7(
' 7'
J 77
78
78
7i
i 79
5 92
93
93
9^
94
95
95
9f
> 9^
' 97
97
98
9!
5 99
60 (
J 12
18
24
3o
36 43 48 54
6 no
III
112
1 15
Ii3
ii3
:i4
11;
II
) 116
116
117
11!
i 118
70 -
1 14
31
28
35
42 49 56 63
7 "29
i3o
1 3c
i3i
1 32
l32
1 33
i3<
i3i
i i35
1 36
i37
i3
7 i38
80 i
. 16
34
32
40
48 56 64 72
8 i47
1 48
i4ç
1 5c
i5o
i5i
1 52
i5:
1 5.
i54
1 55
1 56
i5
7 t58
90 (
i 18
27 36
45
54 63 73 81
Q 1 66
16-
16-
iW
16c,
I-r
i-i
I—.
I-
J 174
175
176
1-'
i 177 1 C)0 1 1 (
1 30
3<, 4o
5o'6ol7.il8ol9o|
A4
TABLE
II.-
-To find the lime T
; the
sum
of the
rad
ir+r
", and the chord c bein
ge;iven.
Sum .if tlH> RaJii r -|- r". 1
Chord
0,47
0,48
0,49
0,50
0,51
0,52
0,53
0,54
0,55
0,56
Days |dir.
Days |iUr.
Days |dil'.
Days |d>f.
Days |dir.
Days |dif.
Days |dir.
Days |dll'.
Day» [dif.
llavs \Aii.
0,00
0,000
0,000
0,000
0,000
0,000
O.OOO
0,000
0,000
0,000
0,00C)
0,0000
0,01
o>i99
2
0,201
2
o,3o3
3
0,206
3
0,208
2
0,210
3
0,312
3
0,3l4
2
0,3l6
3
0,3l8
I
0,0001
0,02
o,3g8
5
o,4o3
4
0,407
4
0,4 1 I
4
0,4 1 5
4
0,419
4
0,423
4
0,437
4
0,43 I
4
0,435
4
0,0004
o,o3
0,598
6
0,604
6
0,610
6
0,616
7
0,623
6
0,62g
6
o,635
6
0,64 1
6
0,647
5
0,652
6
o,ooog
0,04
0.797
8
o,8o5
9
0,8 1 4
8
0,822
8
o,83o
8
0,838
8
0,846
8
o,854
8
0,862
8
0,870
8
0,0016
o,o5
0,996
10
1,006
II
1,017
10
1,027
10
I,o37
II
i,o48
10
i,o58
10
1,068
9
1 ,077
10
1.087
10
0,0035 1
0,06
1,195
12
1,207
i3
1,230
13
1,233
i3
1,345
13
1,357
13
1,269
13
1,381
13
1,293
11
i,3o4
13
o,oo36
0,07
1,394
i4
i,4o8
i5
1,433
i5
1,438
i4
1,452
i4
1,466
i4
i,48o
i4
1,494
14
i,5o8
i4
1,522
i3
o,oo4g
0,08
1,592
17
i,6og
17
1 ,636
16
1,643
17
1,659
16
1,675
16
i,6gi
16
1,707
16
1,723
i5
1,738
16
0,0064
o,og
1=791
'9
1,810
19
1,83g
18
1,847
19
1,866
18
1,884
18
1,902
18
1,920
18
1,938
17
1,955
iS
0,0081
0,10
1.989
21
2,010
31
3,o3i
21
2,o52
20
2,072
21
3,og3
30
2,Il3
30
2,i33
20
2,i53
19
2,173
20
0,0100
0,1 1
2,187
33
2,210
33
2,233
23
3,356
33
2.279
22
2,3oi
33
2,333
33
2,345
23
3,367
22
3,38g
31
0,01 3 I
0,12
2,385
25
2,4lO
25
2,435
25
2,460
25
3,485
25
3,5l0
24
3.534
34
3,558
34
2,583
33
3,6o5
33
0,01 44
o,i3
2,582
28
2, 610
27
2,637
27
3,664
27
2,691
27
3,718
26
3,744
26
2,770
36
2,796
35
3,821
26
0,01 6g
o,i4
2,779
3o
2,809
3o
2,839
29
2,868
29
2.897
28
2,g25
29
2,954
28
3,983
38
3,010
27
3,037
27
0,01 g6
0,1 5
3,976
33
3,008
33
3,o4o
3i
3,071
3i
3,102
3i
3,i33
3o
3,i63
3o
3,193
3o
3,333
3o
3,253
29
0,0335
0,16
3,173
34
3,207
34
3,241
33
3,374
33
3,3o7
33
3,340
33
3,373
32
3,4o5
33
3,437
3i
3,468
3,
o,o256
0,17
3,369
36
3,4o5
36
3,44i
36
3,477
35
3,5i2
35
3,547
35
3,582
34
3,616
34
3,65o
33
3,683
33
0,028g
0,18
3,564
39
3.6o3
38
3,64i
38
3,679
37
3,716
38
3,754
36
3,7QO
36
3,836
36
3,863
36
3,898
37
o,o324
0,19
3,759
4i
3,800
4i
3,841
40
3,881
39
3.920
40
3,g6o
38
3.998
39
4.037
38
4,075
37
4,112
38
o,o36i
0,20
3,954
43
3,gg7
43
4,o4o
42
4,083
42
4,124
4i
4,i65
4i
4.206
4i
4.347
4o
4,287
4o
4,337
39
o,o4oo
0,21
4,i48
46
4,ig4
45
4,339
44
4,283
44
4,337
44
4,37!
43
4,4 1 4
42
4.456
42
4,498
42
4,540
42
0,044 1
0,22
4,342
48
4,3go
47
4,437
4l
4,484
46
4,53o
46
4,576
45
4,621
44
4,665
45
4,710
44
4,754
43
o,o484
0,23
4,535
5o
4,585
5o
4.635
49
4,684
48
4,733
48
4,780
47
4,827
4i
4,8-4
46
4.920
46
4,966
46
o,o52g
0,24
4,728
52
4,780
53
4,832
5i
4,883
5i
4,934
5o
4.984
49
5,o33
49
5,083
49
5,1 3i
48
5,179
47
0,0576
0,25
4,919
55
4,974
54
5,028
54
5,082
53
5,i35
52
5,187
52
5,33g
5i
5,290
5i
5,341
5o
5,391
49
0,0625
0,26
5,111
57
5,168
56
5.234
56
5,280
55
5,335
55
5,390
54
5,444
53
5,4g7
53
5,55o
53
5,603
52
0,0676
0,27
5,3oi
60
5,36i
59
5,420
58
5,478
57
5,535
57
5,592
56
5,648
56
5,704
55
5,759
54
5,8i3
54
0,072g
0,28
5,491
62
5,553
61
5,614
61
5,675
59
5,734
60
5,794
58
5,852
58
5,gio
57
5,967
57
6,034
56
0,0784
0,29
5,679
65
5,744
64
5,808
63
5,871
62
5,933
61
5,994
61
6,o55
60
6,ii5
60
6,175
58
6,233
59
0,084 1
o,3o
5,867
67
5,g34
67
6,001
65
6,066
65
6,i3i
64
6.195
63
6,258
62
6,320
62
6,382
61
6,443
60
o,ogoo
o,3i
6,o54
70
6,124
69
6,ig3
68
6,261
67
6,328
66
6,394
65
6,45g
65
6,524
64
6,588
63
6,65 1
63
o,og6i
0,32
6,240
73
6,3 1 3
71
6,384
70
6,454
70
6,534
69
6,593
67
6,660
67
6,727
67
6,794
65
6,859
65
0,1024
0,33
6,425
75
6,5oo
74
6,574
73
6,647
73
6,719
71
6,790
70
6,860
70
6,g3o
68
6,gg8
68
7,o6(5
67
0,1 o8g
0,34
6,6og
78
6,687
76
6,763
7fi
6,8 3g
75
6,914
73
6,987
73
7,060
72
7,i3s
71
7,2o3
70
7,373
69
0,1 1 56
0,35
6,792
80
6,872
80
6,952
78
7,o3o
77
7,107
76
7.183
75
7,258
74
7,332
74
7.406
72
7.478
72
0,1225
o,36
6,973
83
7,o56
83
7,139
81
7,220
79
7.299
79
7,378
78
7.456
76
7.532
76
7,608
75
7.683
74
0,1296
0,37
7.153
86
7,23g
85
7,334
84
7,408
83
7.491
81
7,572
80
7,652
80
7.733
78
7,810
77
7,887
76
0,1369
0,38
7,33i
9"
7.421
88
7,5og
86
7,595
86
7.681
84
7,765
83
7,848
82
7,930
80
8,010
80
8,ogo
79
0.1444
o,3g
7,5o8
93
7.601
91
7,693
89
7.781
89
7,870
87
7.957
85
8,042
85
8,127
83
8,310
83
8,2g2
81
0,1 521
o,4o
7,683
96
7.779
94
7,873
93
7.966
91
8,o57
90
8.147
88
8,335
87
8,333
86
8,4o8
85
8,493
84
0,1600
o,4i
7.856
99
7,955
98
8,o53
96
8,149
94
8,243
g3
8,336
9'
8,427
90
8,5i7
89
8,606
87
8,693
86
0,1681
0,42
8,027
io3
8,i3o
lOI
8,33i
99
8,33o
98
8,428
96
8,534
94
8,618
93
8,711
91
8,803
90
8,893
89
0,1764
0,43
8,.g5
107
8,3o3
io5
8,407
io3
8,5 10
lOI
8,611
99
8,710
97
8,807
06
8,903
94
8,9g7
g3
9.090
91
o,i84g
0,44
8,36o
112
8,472
log
8,58i
107
8,688
io4
8,792
103
8,8g4
100
8,994
99
9.093
97
9.190
96
9,386
94
o,ig36
0,45
8,522
117
8,63g
Ii3
8,753
III
8,863
108
8.971
106
9.077
io3
g,i8o
102
9.283
100
g,382
99
9,481
97
0,2025
o,5o
9,689
1 36
9.825
13g
9.954
124
10,078
120
io,ig8
118
io,3i6
ii5
10,43 1
ii3
0,3 5oo
0,55
11,178
143
11,331
136
o,3o35
,1105
,1152
,1201
,1250
,1301
,1352
,1405
,1458
,1513 1 ,1568
r^
i . (r + r'\'- or r~- + r ^ nearly. |
163
164
166
I
16
16
17
2
32
33
33
3
4P
49
5o
4
65
66
66
5
81
82
83
6
97
q8
■ 100
7
ii3
ii5
116
8
i3o
i3i
1 33
9
1 46
1 48
i4g
168
170
172
174
176
178
180
182
184
186
1 88
190
192
194
1 96
,98
200
303
17
'7
17
17
18
18
18
18
18
19
ig
ig
19
19
20
20
20
20
34
34
34
35
35
36
36
36
37
37
38
38
38
3q
39
40
40
40
5o
5i
52
52
53
53
54
55
55
56
56
57
58
58
5q
5q
60
61
67
68
69
70
70
71
72
73
74
74
75
76
77
78
78
79
80
81
84
85
86
87
88
89
90
9'
92
93
94
95
96
97
9«
99
100
lOI
101
102
io3
104
106
107
108
109
no
112
ii3
ii4
Hi
116
118
iiq
120
121
118
iiq
120
122
123
125
136
137
12g
i3o
l32
i33
i34
1 36
l37
.3q
i4o
l4l
1 34
1 36
i38
i3q
i4i
l42
i44
1 46
147
i4g
i5o
l52
1 54
i55
.57
1 58
160
1 6a
i5i
1 53
i55
1 57
1 58
160
162
1 64
166
167
169
171
173
175
176
178
180
183
TABLE II. — To find tlie time T\ the sum of the radii r-\-r\ and Ihc choi-d <• being given.
Siinioftlie Kadii r-\-r".
1
'rup
pa,
ij II
r til
, Ml,
1 ol
llir
{ml,
~1
Chonl
c.
0,57
0,58
0,59
0,60
0,61
0,62
'|2|J|4|5|6|7|8|9|
1
3
3
0
0
0
0
0
I
0
I
I
0
I
1
I
I
2
I
I
2
I
I
2
I
2
2
I
2
3
Diijs |.lil'.
jiiiys i.i.r.
Uiiys |.lir.
U.iys Idif.
Uiiys |.lir.
Days IlIIi".
0,00
0,O^KT
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,2 iq
2
0,221
2
0,223
2
0,225
2
0,227
3
0,239
2
0,000 1
4
0
1
1
2
2
2
3
3
4
0,05
o,43cj
4
0,443
4
0,447
3
o,45o
4
0,454
4
o,458
3
0,0004
<),o3
o,65b
6
o,6()4
6
0,670
5
0,675
6
0,681
6
0,687
5
O.OOOC)
5
1
1
2
3
3
3
4
4
5
0,04
0,878
7
0,885
8
0,893
'
0,900
8
0,908
7
0,915
8
0,0016
()
1
1
1
1
2
3
2
3
3
4
4
4
4
5
5
6
5
6
o,o5
1.097
9
1,106
10
1,116
9
1,125
10
i,i35
9
1.144
9
0,0025
8
1
3
2
3
4
5
6
6
7
0,06
1,3 16
12
1,328
II
1,339
II
i,35o
12
1,362
1 1
1,373
1 1
o,oo36
9
1
3
3
4
5
5
6
7
8
0,07
1,535
i4
1, 54c.
i3
1 ,562
i3
1,575
i3
1,588
i3
1 ,60 1
i3
0,0049
0,0064
0,08
■ .-54
i5
1 ,769
16
1,785
i5
1,800
i5
i,8i5
i5
i,83u
i4
10
1
2
3
3
4
4
5
6
7
8
9
0,09
1.97J
17
1,990
17
2,007
17
2,024
17
2,o4l
17
2,o58
17
0,008 1
1 1
12
1
I
2
3
4
5
6
6
7
7
8
8
9
10
10
II
0,10
2,192
'9
2,211
19
2,23o
19
2,249
19
2,268
18
2,386
'9
0,0100
i3
1
3
3
4
4
5
6
7
8
8
9
10
12
i3
0,1 I
2,410
21
2,43i
21
2,452
21
2,473
21
2,4cj4
20
2,5i4
21
0,01 3 1
1 4
I
7
10
II
0,12
2,628
24
2,652
23
2,674
23
2,697
23
2,720
22
2,743
22
0,01 44
i5
2
3
5
6
g
9
10
1 1
12
i3
i4
0,1 3
2,847
25
2,872
24
2,896
25
2,921
25
2,946
34
2,970
24
0,0169
16
2
3
5
6
8
II
i4
o,i4
3,064
27
3,091
27
3,118
27
3,i45
26
3,171
36
3,197
26
0,0196
17
2
3
5
7
9
9
10
10
12
14
i5
0,1 5
3,282
'9
3,3 11
'9
3,340
28
3,368
=9
3,397
28
3,425
27
0,0225
18
'9
2
2
4
4
5
6
7
8
II
i3
i3
i4
i5
16
17
0,16
3,490
3i
3,53o
3i
3,56i
3i
3,593
3o
3,622
3o
3,652
'9
o,o256
0,17
3,716
33
3,749
33
3,782
32
3,814
33
3,847
3i
3,878
32
0,0289
o,o324
20
2
4
6
8
10
12
i4
16
18
0,18
3,933
35
3,968
35
4,oo3
34
4,o37
34
4,071
34
4,io5
33
21
2
4
6
8
1 1
1 3
i5
17
■9
0,19
4,1 5o
37
4,"i87
36
4,223
37
4,260
36
4,296
35
4,33i
36
o,o36i
23
23
2
2
4
5
7
7
9
9
II
12
i3
i4
i5
16
18
18
20
21
0,20
4.366
39
4,4o5
36
4,443
39
4,482
38
4,520
37
4,557
37
o,o4oo
24
2
5
7
10
13
i4
17
19
22
0,21
4,582
4i
4,623
4o
4,663
40
4,7o3
4o
4,743
4o
4,783
39
0,044 1
0,23
4,797
43
4,84o
43
4,883
42
4,925
42
4,967
4i
5,008
4i
0,0484
25
3
5
8
10
i3
i5
18
20
23
0,23
5,012
45
5,o57
45
5,102
ÂA
5,i46
A4
5,190
43
5,233
43
0,0529
0,0576
36
3
5
8
10
i3
16
18
21
23
0,24
5,226
47
5,273
47
5,320
46
5,366
46
5,4 1 2
46
5,458
AA
27
28
3
3
5
6
8
8
II
II
i4
i4
i6
17
19
20
22
23
24
25
0,25
5,440
5o
5,490
48
5,538
49
5,587
47
5,634
48
5,682
47
0,0625
29
3
6
9
12
i5
17
20
33
26
0,26
5,654
5i
5,7o5
5i
5,756
5o
5,806
5o
5,856
49
5,905
4q
0,0676
0,0729
0,0784
0,084 1
3o
3i
32
33
3
3
3
3
6
6
6
7
i5
16
18
0,27
5,867
53
5,920
53
5,973
53
6,026
5i
6,077
5?
6,129
5i
9
13
21
34
35
27
28
0,28
6,080
55
6,1 35
55
6,190
54
6,244
54
6,298
54
6,352
52
9
13
19
22
0,29
6,393
57
6,349
57
6,4o6
57
6,463
56
6,519
55
6,574
55
10
lO
l3
i3
16
17
19
20
22
23
36
36
'9
3o
o,3o
6,5o3
60
6,563
59
6,623
58
6,680
58
6,738
58
6,796
57
0,0900
0,0961
34
3
7
10
i4
17
20
24
27
3i
0,3 1
6,714
62
6,776
61
6,837
61
6,898
60
6,958
59
7,017
59
35
4
7
7
7
1 1
i4
18
21
25
28
32
0,32
6,924
64
6,988
63
7,o5i
63
7,ii4
62
7,176
62
7,238
61
0,1024
36
4
I [
i4
i5
18
23
25
Jo
32
0,33
7,i33
67
7,200
65
7,365
65
7,33o
64
7,394
64
7,458
63
0,1089
37
38
4
1 1
19
ig
32
26
33
0,34
7,342
68
7,4io
68
7,478
67
7,545
67
7,612
66
7,678
65
0,1 1 56
4
1
8
11
i5
23
27
3o
34
0,35
7.55o
71
7,621
70
7,691
69
7,760
69
7,829
68
7,897
67
0,1235
39
4
8
12
16
30
23
27
3i
35
o,36
7,757
73
7,83o
73
7,903
71
7,974
71
8,045
70
8,ii5
69
0,1396
4o
4
8
12
16
20
24
28
32
36
0,37
7.963
76
8,039
74
8,ii3
74
8,187
73
8,260
73
8,333
71
0,1 369
4i
4
8
12
16
21
25
29
33
37
o,38
8,i6()
78
8,247
77
8,334
76
8,400
75
8,475
75
8,55o
73
o,i444
42
4
8
i3
17
21
25
29
34
38
0,39
8,373
81
8,45^
79
8,533
78
8,611
78
8,68g
77
8,766
76
0,l52I
43
AA
4
4
9
9
i3
i3
17
18
22
22
26
26
3o
3i
34
35
39
4o
o,4o
8,577
83
8,660
81
8,74 1
81
8,823
80
8,902
79
8,981
78
0,1600
0,41
8,779
86
8,865
84
8,949
83
g,o33
83
9,114
83
9,196
80
0,1681
45
5
9
i4
18
33
2-7
32
36
4i
0,42
8,981
88
9,069
86
9, 1 55
86
g,24i
85
9,326
83
9,409
83
0,1764
46
5
9
i4
18
23
28
32
37
4i
0,43
9,181
9'
9,272
89
9,36i
88
9,449
87
9,536
86
9,633
85
0,1849
47
5
9
i4
■9
24
28
33
38
42
0,44
9,38o
93
9.473
92
9,565
91
9,656
89
9,745
89
9,834
87
0,1936
48
49
5
5
10
10
i4
i5
19
30
24
3 5
29
29
34
34
38
39
43
AA
0,45
9,578
96
9,674
94
9,768
94
9,863
92
9,954
91
10,045
89
0,2025
i5
3o
45
46
47
48
o,5o
to,544
III
,o,655
109
10,764
107
10,871
106
0,977
io4
1 1 ,08 1
io3
o,25oo
5o
5
10
20
25
35
40
0,55
■1,45-
i3i
,1,588
127
Ii,7i5
125
ii,84o
122
1 ,96.1
120
12,082
118
o,3o35
5i
5
10
i5
20
26
3i
36
4i
0,60
13,736
i5o
2,886
i43
1 3,029
137
o,36oo
52
53
5
5
10
16
16
31
21
36
27
3i
32
36
37
42
42
54
5
II
16
22
27
32
38
43
49
55
6
II
17
22
28
33
39
AA
5o
56
6
1 1
17
2 3
28
34
39
45
5o
57
6
u
17
33
29
34
4o
46
5i
58
6
13
■7
3 3
29
35
4i
46
52
59
6
13
18
34
3o
35
4i
47
53
60
61
6
6
13
12
18
18
24
24
3o
3i
36
42
43
48
49
54
55
,16-25
.1682 1
.1741 1
,1800 1
,1861
.1922
â
63
63
64
(i
■9
19
'9
25
3i
37
38
38
43
AA
45
5o
5o
5i
56
^ . {r -t- r")^ or T^ -f- r'"^ nearly.
6
6
l3
i3
25
26
32
32
^8^
204
206
208
210
212
214
2lt
1 2I«
22
0 22
2 224
22
6 228
2J0
—
—
—
—
_ —
. —
_
65
7
i3
20
26
33
39
46
53
%
I 20
21
21
21
21
21
2
! 22
2
2 2
2 22
2
3 23
23
66
7
i3
20
26
33
4o
Aft
53
59
2 4i
4i
42
42
42
43
A-
i ÂA
4
4 4
i 45
4
5 46
46
67
7
i3
20
27
34
4o
Ai
54
66
3 61
62
62
63
64
64
6;
) 65
6
6 6
7 67
6
8 68
69
68
7
i4
20
27
34
4i
48
54
61
4 82
82
83
84
85
86
8f
' 87
8
8 8(
J 90
9
0 91
92
('9
7
i4
21
28
35
4i
48
55
62
5 102
io3
104
io5
1 06
107
I of
109
II
0 11
112
II
3 ii4
ii5
6 122
124
125
126
127
128
1 3c
i3i
i3
2 i3.
i i34
i3
fi i37
1 38
70
7
i4
21
28
35
42
49
56
63
7 i43
1 44
1 46
1 47
1 48
i5o
i5i
1 53
i5
4 15;
) l57
i5
8 160
161
80
8
16
24
32
4o
48
56
64
72
8 i63
1 65
166
168
170
171
17;
174
17
5 17?
179
18
1 182
1 84
90
9
18
27
36
45
54
63
72
81
9 i84
i85
18-
189
191
193
19^
196
'g'
S 20c
203
20
3 2o5
307
100
10
30
3o
4o
5o
60
70
80
90
TABLE II.
— To fiml (he time T
the sum 0
fttie i-i
A\\ J
+ '■",
and
the cliord c
beinp:
given.
Sum ol' tlic milii r-^r", 1
Chord
c.
0,63
0,64
0,65
0,66
0,67
0,68
0,69
0,70
0,71
0,72
Days |.lif.
Diiys |dir.
Days |.liC.
Da)S lilif.
Days |rlil'.
Days |ilil'.
Dasy |dif.
Days Idir.
Days |dil'.
Days l<lif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,0 1
o,23l
2
0,233
I
0,234
2
o,336
2
o,238
2
0,240
1
0,241
2
0,243
2
0,245
2
0,247
1
0,0001
0,03
0,46 1
4
0,465
4
0,469
3
0,472
4
0,476
3
0,479
4
0,483
3
0,486
4
o,4go
3
0,493
4
0,0004
o,o3
0,692
6
0,698
5
0,703
5
0,708
6
0,714
5
0,719
5
0,724
6
o,73o
5
0,735
5
0,740
5
o,ooog
o,o4
0,923
7
0,930
7
0,937
7
0,944
8
0,952
7
0,959
7
0,966
7
0,973
7
0,980
6
0,986
7
0,0016
o,o5
i,i53
9
1,162
9
1,171
9
1,18c.
9
1,189
9
1,198
9
1,207
9
1,216
8
1,224
9
1,233
8
0,0025
0,06
1,384
II
1,395
1 1
1, 406
10
i,4i6
11
1,427
11
1,438
10
1,448
1 1
1,459
10
1,469
10
1,479
1 1
o,oo36
0,07
1, 61 4
i3
1,627
i3
1 ,64(
12
1,652
i3
1,665
12
1,677
12
1,689
i3
1,702
12
1,714
12
1,72(1
12
0,0049
o,oS
1,844
i5
1,869
i5
1,874
i4
1,888
14
1,902
i4
1.916
i4
i,g3t
i4
1,944
14
1,958
i4
1,973
i4
0,0064
0,09
2,075
16
2 ,09 i
16
2,107
17
2,124
16
2,i4o
16
2,i56
i5
2,171
16
2,187
16
2,203
i5
2,218
16
0,0081
0,10
3,3o5
18
2,323
18
2,341
18
2,359
18
2,377
18
2,395
17
2,4l2
18
2,43o
17
2,447
17
2,464
17
0,0100
0,1 1
2,535
20
2,555
20
2,575
19
2,5g4
20
2,614
20
2,634
19
2,653
19
2,672
19
2.6gi
19
2,710
'9
0,01 2 1
0,12
2,764
22
2,786
22
2,80b
22
2,83o
21
2,85i
21
2,872
22
2,894
21
2,915
20
2,935
21
2,956
21
o,oi44
0,1 3
2.oq4
24
3,018
23
3,04 1
24
3,o65
23
3,088
23
3,111
23
3,i34
23
3,1 57
22
3,179
23
3,202
22
(J ,01 69
o,i4
3,223
26
3,249
25
3,274
26
o,3oo
25
3,325
25
3,35g
24
3,37/
25
3,399
24
3,423
24
3,447
24
0,0196
0,1 5
3,452
28
3,480
27
3,5o7
27
3,534
27
3,56i
27
3,588
26
3,6i4
27
3,64i
26
3,667
26
3,693
26
0,0225
0,16
3,681
3o
3,71 1
29
3,740
^9
3,769
29
3,798
28
3,826
28
3,854
28
3,882
28
3,gio
28
3,938
27
o,0256
0,17
3,910
3i
3,941
3i
3,972
3i
4,oo3
3i
4,o34
3o
4,064
3o
4,094
3o
4,124
3o
4,1 54
29
4,i83
29
0,0289
0,18
4,i38
34
4,172
32
4,204
33
4,237
32
4,269
33
4,3o2
3i
4,333
32
4,365
32
4,3g7
3i
4,428
3i
o,o324
0,19
4,367
35
4,402
34
4,436
35
4,471
34
4,5o5
34
4,539
34
4,573
33
4,606
33
4,639
33
4,672
33
o,o36i
0,20
4,594
37
4,63 1
3?
4,668
36
4,704
36
4,74o
36
4,776
36
4,812
35
4,847
35
4,882
35
4,917
34
c3,o4oo
0,2 1
4,822
39
4,861
38
4,899
39
4,938
37
4,975
38
5,oi3
37
5,o5c
37
5,087
37
5,124
37
5,161
36
0,044 1
0,22
5,049
4i
5,ogo
40
5,i3i
40
5,170
40
5,210
40
5,25c.
39
5,28g
39
5,328
38
5,366
38
5,4o4
38
0,0484
0,23
5,276
43
5,319
42
5,36 1
42
5,4o3
42
5,445
4i
5,486
41
5,527
4i
5,568
40
5,608
4o
5,648
40
0,0529
0,24
5,5o2
45
5,547
44
5,591
44
5,635
AA
5,679
43
5,722
43
5,765
42
5,807
42
5,849
42
5,891
42
0,0576
0,25
5,729
46
5,775
/fi
5,821
46
5,867
45
5,912
45
5.957
45
6,002
45
6,047
43
6,090
A4
6,1 34
iÂ
0,0625
0,26
5,954
49
6,00 3
48
6,o5i
48
6,099
47
6,i46
47
6,193
46
6,239
46
6,285
46
6,33i
46
6,377
45
0,0676
0,27
6,180
5o
6,23o
5o
6,280
5o
6,33o
49
6,379
49
6,428
48
6,476
48
6,524
48
6,572
47
6,6 1 g
47
0,0729
0,28
6,4o4
53
6,457
52
6,509
5i
6,56o
5i
6,611
5i
6.662
5o
6,712
5o
6,762
5o
6,812
49
6,861
49
0,0784
0,29
6,629
54
6,683
54
6,737
54
6,791
53
6,844
52
6,896
52
6,948
52
7,000
52
7>o52
5o
7,102
5i
0,084 1
o,3o
6,853
56
6,909
56
6.965
55
7,o2r
55
7,075
55
7,1 3o
54
7,184
54
7,238
53
7,291
53
7,344
52
0,0900
0,3 1
7,076
59
7,i35
57
7,19=
58
7,25o
j7
7,307
56
7,363
56
7,419
56
7,475
55
7,53o
54
7,584
55
o,og6i
0,32
7,299
60
7,359
60
7,4 1 9
60
7,479
59
7,538
58
7,596
58
7,654
57
7,711
57
7,768
57
7,825
56
0,1024
0,33
7,521
63
7,584
62
7,64(>
61
7,707
61
7,768
60
7,828
60
7,888
59
7,947
59
8,006
59
8,o65
58
0,1 o8g
0,34
7,743
64
7,807
64
7,871
64
7,935
63
7,998
62
8,060
62
8,122
61
8,i83
61
8,244
60
8,3o4
60
0, 1 1 56
0,35
7.964
67
8,o3i
66
8,097
65
8,162
65
8,227
64
8,291
64
8,355
63
8,4i8
63
8,48i
62
8,543
62
0,1225
o,36
8,184
69
8,253
68
8,321
68
8,389
67
8,456
66
8,522
66
8,588
65
8,653
64
8,717
65
8,782
63
0,1296
0,37
8,4o4
71
8,475
70
8,545
70
8,6i5
G9
8,684
68
8,752
68
8,820
67
8,887
67
8,954
66
g,020
65
0,1 36g
o,38
8,623
73
8,6g6
73
8,769
71
8,84o
71
8,911
71
8,982
69
9,o5 1
G9
9,120
69
9,-89
68
9,257
67
0,1 444
0,39
8,842
75
8,917
74
8,991
74
9,o65
73
9,1 36
73
9,211
71
9,282
71
9,353
71
9,424
70
9,494
69
0,l52I
o,4o
9>o59
78
9,i37
76
9,2i3
76
9,289
75
9,364
75
9 -430
74
9,5i3
73
9,586
72
9,658
72
9,73o
71
0,1600
0,4 1
9,276
80
9,356
79
9,435
78
9,5i3
77
9,590
■76
9,666
76
9,742
75
9,817
75
9,892
74
9,966
73
0,1681
0,42
9'492
82
9,574
81
9.655
8u
9,735
80
9,8 1 5
78
9,893
78
9.971
78
10,049
76
10,125
76
10,201
75
0,1764
0,43
9,101
84
9,791
84
9,875
82
9,957
82
10,039
80
10,1 19
81
I0,20'
79
10,27g
78
10,357
78
10,435
78
0,1849
0,44
9«2i
87
10,008
85
10,093
85
10,178
84
10,262
83
10,345
82
10,427
82
io,5og
80
io,58g
80
10,669
79
0,1936
0,45
1 0,1 34
S9
10,223
88
io,3ii
87
10,398
86
10,484
85
10,569
85
10,654
83
10,737
83
10,820
82
10,902
83
0,2025
o,5o
II, 184
lOI
11,985
101
11,386
99
11,485
97
1 1 ,582
97
11,679
9O
11,775
95
1 1 ,870
93
1 1 ,g63
^l
i2,o56
92
o,25oo
0,55
12,200
116
I2,3i6
ii4
i2,43o
112
12,542
II I
12,653
no
12,763
108
12,871
107
I2,g78
io5
i3,o83
io5
i3,i88
io3
o,3o25
0,60
1 3, 166
1 35
i3,3oi
i3i
1 3,432
129
i3,56i
126
13,687
125
i3,8i2
122
13,934
121
i4,o55
118
14,173
118
14,291
116
o,36oo
0,65
1 4,36 1
i56
i4,5i7
i4g
i4,666
i45
14,811
i4o
i4,95i
i38
15,089
i35
l5,224
1 33
1 5,357
i3o
0,4225
0,70
16,049
1 63
16,212
i55
16,367
i5i
0,4900
,1985
,2048
,2113
,2178
,2245 1
,2312 1
.2381
,2450
,2521
.2592
c"-
Ji . (r + r")» or r= -)- r"= nearly. |
2l3
2l5
217
2ig
221
21
22
22
22
22
43
43
43
AA
AA
64
65
65
66
66
85
86
87
88
88
107
108
loq
no
111
12,8
129
i3o
i3i
1 33
1 49
i5i
l52
i53
i55
170
172
174
175
177
192
194
193
197
199
45
67
89
112
1 34
1 56
178
201
225
227
229
23
23
23
45
45
46
68
68
69
90
91
92
ii3
114
ii5
i35
1 36
1 37
1 58
i5q
160
180
182
1 83
2o3
2o4
206
23l
2 32
233
2 34
235
236
237
238
239
—
23
23
23
23
=4
24
24
24
24
46
46
47
47
47
47
47
48
48
6q
70
70
70
71
71
71
71
72
92
93
93
94
94
94
95
95
96
iih
n6
117
117
118
lib
119
iiq
120
i3q
i3q
1 40
i4o
i4i
142
142
i43
143
162
162
i63
1(34
i65
i65
166
167
167
i85
186
186
187
188
189
190
190
19;
208
209
210
211
212
212
2l3
214
2l5
240
24 1
2.13
243
24
24
24
24
48
48
48
4q
72
72
73
73
96
9b
97
97
120
121
131
122
1 44
i45
145
i46
16b
i6g
169
170
192
193
i(j4
iq4
216
217
318
219
TABLE II. — To find the lime T\ tlie sum of the radii r-\-r", and the chord c being given.
of Ihe Radii r -j- ''
Chor.l
c.
0,00
0,01
0,02
o,o3
o,o4
0,0 5
o,<'(>
o,(i^
o,oti
0,10
0,1 1
0,1 :'
o.ij
0,14
0,1 5
0,16
0,17
0,18
O..If)
0,3(i
0,2 1
0,32
0,23
0,34
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,73
l>ay5 |ilil'.
0,35 8
0,36 8:
0,37
o,38
0,000
0,248
0,497
.0,745
0,993
1,241
1 ,.f()0
1,738
I ,l|S(i
2,234
3.4S1
2,729
2,9
3.224
3,4
3,719
3,965
4,213
4.459
4,7o5
4,951
5,197
5,442
5,688
5,933 A'l
0,74
Days liiif.
6,178
6,423
6,666
6.gio
7, 1 53
7,639
7,881
8,123
8,364
,6o5
,845
l,o85
0,39 9,563
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
0,65
0,70
0,75
9,324
9,801
10,009
10,276
io,5i3
10,748
10,984
I2,i48
13,391
14,407
15,487
i6,5i8
0,000
o,25o
o,5oo
o,75o
1,000
I,25o
i,5oo
i,75o
1. 999
2,349
2,498
2,748
■■'■99'
3,246
3,495
3,-44
3,993
4,241
4,489
4,-37
4,985
5,233
5,480
5,727
5,974
6,221
6,467
6,71 3
6,958
7,2o3
7.448
7,692
7,936
8,180
8,423
8,666
8,908
9,i5o
9,3iii
9,603
9.S72
10,112
io,35i
10,589
10,827
1 1 ,064
12,239
13,393
14,521
5,616
1 6,66 5
18
0,75
Days |iiif.
80
90
101
ii3
127
144
0,000
0,252
O,5o3
0,-55
1 ,007
1,2 58
i,5jo
1,-61
2,Ol3
2,264
2
2 .7(>6
3,0
3,268
3,519
3,769
4,020
4,370
4,520
4,770
5,019
5,269
5,5i8
5,766
6,01 5 4i
0,76
D.iys |iliC.
6,263
6,5 11
6.759
7,006
7,253
7,5oo
7,746
7.995
8,237
8,482
8,727
8,971
9,214
9.458
9,700
9.942
10,184
10,425
io,665
10,905
ii,i44
12,329
1 3,49
i4,634
1 5,743
16,809
17.799
43
M
46
48
5o
5i
53
55
57
59
60
62
64
65
68
70
71
73
76
77
79
90
101
112
125
i4i
169
0.000
0,253
o,5o'
0,760
1, 01 3
1 ,367
1,5
1,7-3
2,026
3,2-9
2,53
2,785
3,o38
3,390
3,542
3,795
4,047
4.399
4.55o
4,802
5,o53
5,3o4
5,555
5,8o5
6,o56
6.3o6
6,5
6,8o5
7,o54
7,3o3
7,55i
7,799
8,047
8,394
8,54i
8,787 60
9,o33 62
0,77
Diiy.i |dif.
0,000
0,255
0,5 10
0,765
1,020
1,375
i,53u
1,785
2,o4o
2,294
2,549
2,8o3
3,o58
3,3 12
3,566
3,820
4,073
4,327
4,58o
4.834
5,087
5,339
5,592
5,844
6,096
6,348
6,599
6,85o
7,101
7,352
7,602
7,852
8,101
8,35o
8,599
9,278
9,52"
9.768
10,012
10,255
1 0,498
10,741
10,982
[1,223
12,419
[3,595
■746
1 5,868
i6,g5o
i-,968
64
66
67
69
71
73
74
77
79
88
99
[1 1
124
139
161
9.095
9,342
9,589
9,835
10,081
10,326
[0,57[
[o,Si5
11,059
[l,3o2
I2,507
13,694
14,857
15,992
17,089
[8,139
i3
i5
16
18
19
21
23
25
27
28
3o
3i
33
35
37
38
4o
42
AÂ
46
47
48
5o
52
54
56
57
59
61
63
65
67
69
71
72
75
76
78
88
98
110
122
i36
1 57
0,78
Duys |dir.
,2665 I ,2738 ,2813 | ,2888 ,2965 ,3042 (?
0,000
0,357
0,5 1 3
0,770
1,027
1,383
1 ,54f
1,796
2,o53
2,309
2,565
2,831
3,077
3,333
3,589
3,845
4,100
4,355
4,610
4,865
5,130
5,374
5,629
5,88:
6, 1 36
6,390
6,643
6,896
7.148
7,4oo
7,652
7.904
8,i55
8,406
8,656
8,906
9,1 56
9,4o5
9,654
9,902
io,i5o
10,397
10,643
10,890
ii,i35
ii,38o
12,595
13,792
14.967
i6,ii4
17,225
18,286
12
i3
i5
17
'9
20
22
33
26'
28
3o
33
33
35
36
39
40
4i
43
45
47
49
5o
52
54
55
57
59
61
63
64
66
68
70
72
73
76
0,0000
0,0001
0,00114
0,0009
O.fKl! fi
0,0025
o,oo36
0,0049
o,oo()4
0,0081
0,0100
0,01 2 I
o,oi44
0,0169
0,0196
0,0225
o,o256
0,0289
o,o324
o,o36i
o,o4oo
0,044 1
0,0484
0,0539
0,0576
0,0635
0,0676
0,0729
0,0784
o,oS4i
0,0900
0,09(11
0,1024
0,1089
0,1 1 56
0,I235
0,1396
0,1369
0,1444
0,l531
0,1600
0,1681
0,1764
0,1849
0,1936
0,2025
O,35oo
o,3o35
o,36oo
0,4225
lO
0,5625
(r -\- r'"^) or r- -f- r"^ nearly.
244 245 246 247 248 249 25o 25i 252 253 254 2551 256| 257
24
49
-3
98
122
i46
171
195
220
25
49
74
98
123
i47
172
196
221
25
49
74
98
123
1 48
172
197
231
25
49
74
99
134
1 48
173
25
5o
74
99
124
i4q
174
198
333
25
5o
75
100
125
i49
174
199
224
25
5o
75
too
125
i5o
175
200
225
25
5o
75
100
126
i5i
176
2or
236
25
5o
76
lOI
[26
i5i
176
202
227
25
5i
76
10
127
l53
177
202
228
25
5i
76
102
127
l53
178
2o3
229
26
5i
77
ro2
128
i53
179
2o4
23o
26
5
77
102
128
i54
179
2o5
2 3o
26
5i
77
io3
129
1 54
180
206
23l
Prop, imrts tor Uio simi ul' tliu Radii,
■ I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9
35
36
37
38
39
4o
4i
42
43
44
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
[OO
0
0
0
1
1
1
0
1
1
2
3
2
2
2
3
3
2
2
3
2
3
4
2
3
4
2
3
4
2
4
5
3
4
5
3
4
6
3
5
6
3
5
6
3
5
7
4
5
7
4
6
8
4
6
8
4
6
8
4
7
9
5
7
9
5
7
10
5
8
10
5
8
10
5
8
I I
6
8
1 1
9 12
10
14
11 i4
i5
[I
8 12
12
i3
i3
i3
i3
i3
i4
i4
i4
16
18
1 o I 30
i3
3
14
i4
i5 17
18
18
19
19
30
20 24
25
25
26
26
21 38 35
3o 36
3i
3i
33
33
3
4
5
5
6
7
8
9
10
13
i3
i4
i4
i5
16
24
35
26
26
27 3i
32
32
33
34
35
37
37
38
38 45
59
59
60
61
55 63
49 56
56 64
63 72
60 1 70 1 80
a5
TABLE H. — To find the time T; the sum of the radii r-\-r"., and the chord c being given.
Sum of the radii r-
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,10
0,1 1
0,1 3
0,1 3
o,i4
o,i5
,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
o,32
0,33
0,34
0,35
0,36
0,37
o,38
0,39
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
o,85
0,79
Day3 jdif.
0,000
0,258
o,5i7
0,775
i,o33
1,292
i,55o
1,808
2,066
2,324
2,582
2,84o
3,097
3.355
3,612
3,869
4,126
4,383
4.640
4,897
5,i53
5,409
5,665
5,92
6,176
6,43]
6,686
6,941
7,195
7.449
7.702
7,956
8,209
8,461
8,7i3
8,965
9'2i7
9,468
9,718
9,968
10,218
10,467
1 0,7 1 5
10,963
11,211
11,457
12,682
1 3,1 _
15,075
16,235
17,360
18,439
0,80
Days \ dif.
0,000
0,260
0,520
0,780
I,o4o
i,3oo
1,559
1,819
2,079
2,339
2,857
3,1 17
3,376
3,635
3,6
4,i53
4,4ii
4.670
4,928
5,186
5,444
5,701
5,959
6,216
6,4
6,729
6,985
7.241
7.497
7,752
8,007
8,262
8,5i6
8,77<
9,024
9.277
9,53o
9,782
io,o34
10,28
io,536
10,786
ii,o36
11,285
11,53
12,769
13,986
i5,i83
16,354
17.493
i8,588
19,609
0,81
Days Idir.
,3121
174
,3-200
0,000
0,262
0,523
0,785
1,046
i,3o8
1 ,569
i,83'i
2,092
2,353
2,61
2,875
3,i36
3,397
3,658
3,918
4,179
4,439
4,699
4,9^9
5,218
5,478
5,737
5,996
6,255
6,5 1 3
6,772
7,o3o
7,287
7,545
7,802
8,059
8,3i5
8,571
8.827
9,082
9,337
9.591
9.845
10,099
10,352
10, 60 5
10,857
11,109
1 1 ,36o
11.610
12,855
14,082
15,289
16,472
17,624
18,736
9,783
0,82
Days |dir.
0,000
0,263
0,526
0,790
I ,o53
i,3i6
1.579
1,842
2,io5
2,368
2,63o
2,893
3,i56
3,4:8
3,680
3,943
4,2o5
4.466
4,728
4,990
5,25i
5,5i2
5,773
6,o34
6.294
6,554
6,814
7.074
7,333
7.592
7,85i
8,109
8,368
8,625
8,883
9,i4o
9,396
9,653
9.908
10,164
10,419
10,673
10,927
11,181
11,434
11,686
12,940
14,177
15,395
16,589
17.754
18,880
19,960
0,83
Days |(lif".
,3281
25
28
29
3o
32
34
35
37
39
4i
42
M
46
47
49
5i
52
54
55
57
59
60
63
64
66
68
70
71
73
75
84
94
io4
116
128
143
162
0,000
0,265
o,53o
0,794
1,059
1,324
1,588
1,853
2,118
2,382
2,646
2,91
3,175
3,439
3,7o3
3,967
4,23o
4.494
4.757
5,020
5,283
5,546
5,808
6,071
6,333
6,595
6,856
7."
7.379
7,639
7,900
8,160
8,420
8,679
8,938
9.197
9.455
9.7 1 3
9.971
10,228
io,485
10,741
10,997
11,252
1 1 ,507
1 1 ,761
i3,024
14,271
1 5,499
i6,7o5
17,882
19,023
20,112
0,84
Days |dir.
5o
52
54
56
57
59
6i
62
64
65
67
69
71
73
75
84
94
104
114
126
i4o
159
0,000
0,266
0,533
0,799
i,o65
1,332
1,598
1,864
2,i3o
2,396
2,662
2.928
3,194
3,460
3,725
3,991
4,256
4,52
4.786
5,o5i
5,3 1 5
5,58o
5,844
6,108
6,3
6,635
6,898
7,161
7,424
7,686
7.948
8,210
8,472
8,733
8,994
9,254
9,5i
9.77
io,o33
10,292
io,55o
0,808
1 1 ,066
11,32
1 1 ,58o
11,836
i3,io8
i4,36
i5,6o3
16,819
1 8,008
19,163
20,271
0,85
Days |dir.
23
o,ooc>
0,268
0,536
0,804
1,072
1 ,340
1 ,608
1,875
2,143
2,4 II
2,678
2,946
3,2i3
3,480
3,747
4,014
4,281
4.548
4,814
5,081
5.347
5,6i3
5,879
6,144
6,4 10
6,675
6,940
7,204
7.469
7,733
7,997
8,260
8,523
8,786
9,049
9,3ii
9,572
9,834
10,095
10,355
0,86
Da
I |dif.
10,616
10,875
ii,i35
71 11 ,394
72 11,652
3362 ,3445 ,3528
11,910
13,191
14,457
1 5,706
i6.q33
18,1 34
19,302
20,427
21,475
0,000
0,270
0,539
0,809
1,078
1,348
1,617
if^^
2,i56
2,425
2,694
2,963
3,232
3,5oi
3,77c
4,o38
4,307
4,575
4,843
5,111
5,379
5,646
5,914
6,181
6,448
6,715
6,98
7.247
7,5i3
7.779
8,044
8,3io
8,574
8,839
g,io3
9,367
9,63o
9,894
io,i56
10,419
10,680
10,942
i,2o3
11,464
11,724
11,983
13,274
14,549
1 5,808
17,045
i8,258
19,439
20,579
21,655
0,87
Days I dif.
,3613
49
5i
53
54
56
58
59
6
62
65
66
68
69
71
73
82
92
101
112
123
i36
i5i
173
0,000
0,271
0,542
0,81 3
1,084
1,355
1,626
1,897
2,168
2,439
2,710
2,980
3,25i
3,521
3,791
4,062
4,332
4,602
4.871
5,i4i
5,4 1
5,679
5,948
6,21
6,486
6,754
7,022
7.290
7,558
7,825
8,092
8,359
8,625
8,892
9,i57
9,423
9,688
9,953
10,21
10,48
10,745
1 1 ,008
11,371
11,533
11,795
i2,o56
i3,356
i4,64i
15,909
17,157
i8,38i
19,575
2o,73o
21,828
0,88
Day 9 [dif.
,3698
0,000
0,273
0,545
0,818
1,091
1 .363
1,636
1,908
2,181
2,453
2,725
2.997
3,269
3,541
3,8 1 3
4,o85
4,357
4,638
4,899
5,170
5,44 1
5,712
5.983
6,253
6,523
6,793
7.o63
7,333
7,602
7,871
8,i39
8,408
8,676
8,944
9,211
9,478
9.745
10,012
10,278
10,543
10,809
1 1 ,0-4
11,338
1 1 .602
11,866
12,120
i3,437
i4,73i
16,009
17,267
i8,5o2
19,709
20,878
21,996
9
II
12
i4
16
17
19
21
22
23
24
26
28
3o
32
33
34
36
38
39
41
42
45
47
49
5o
53
54
56
57
58
60
62
63
65
67
69
70
72
81
90
100
no
121
l32
1 47
164
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
o,<.>o8i
0,0100
0,01 2 1
0,01 44
0,0169
0,0196
0,0225
o,o356
0,0289
o,o324
o,o36i
o,o4oo
0,044 1
0,0484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,084 1
0,0900
0,0961
0,1024
0,1089
0,1 1 56
0,1225
0,1 296
0,1 36g
0,1 444
0,1 52 I
0,1600
0,1681
0,1764
0,1849
0,1936
0,2025
0,2600
o,3o25
0,3600
0,4225
0,4900
0,5635
0,6400
0,7235
,3785 ,3872
. (r -|- ^")" or r^ -J- r"^ nearly.
246
247
248
249
25o
25l
262
253
254
255
256
257
258
359
260
261
262
263
264
265
366
267
268
25
25
25
25
25
25
25
25
25
26
26
26
26
26
26
26
26
26
26
27
27
27
27
49
74
4n
5o
5o
5o
5o
5o
5i
5i
5i
5i
5i
52
52
52
52
52
53
53
53
53
53
64
74
74
75
75
75
76
76
76
77
77
77
77
78
78
78
79
79
79
80
80
80
80
9«
123
99
99
100
100
TOO
lOI
lOI
102
102
102
io3
io3
io4
104
io4
io5
io5
106
106
106
107
107
124
T24
125
125
126
126
137
127
128
128
I2q
1 20
i3o
i3o
i3i
i3i
l32
l32
i33
i33
1 34
134
T-18
r48
t4o
i4q
i5o
i5i
i5i
l52
l52
i53
i54
1 54
i55
i55
1 56
l57
1 57
i58
1 58
i5q
160
160
161
172
173
174
174
175
176
176
177
178
179
179
180
181
181
182
i83
i83
184
i85
186
186
187
188
197
321
198
222
108
199
200
201
202
202
203
204
2o5
206
206
207
208
209
310
210
211
212
3l3
214
2l4
2i3
224
125
226
327
228
229
23o
23o
23 1
232
233
234
235
236
237
238
239
239
240
241
TABLE II
• —
fo find the time T\
the sum
oft
\e radii r-f- r".
ind tlic
chord
c be
ng given
Sum ol tbe Radii r -f- r". j
I'rop. parts for the sum of the Radii. 1
1 |2|3|4|5|6|7|8|9 1
Chord
c.
0,89
0,90
0,91
0,92
0,93
0,94
1 c
2 c
3 <
0
0
I
0
I
1
0 I I
1 I I
12 2
I
I
2
I
2
2
I
2
3
Days |dif.
Da) s \M\
Uays ]dil'.
Ua>s Idil
Uays Idif.
Days |dir.
0,00
o,oou
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,2-4
2
0,276
I
0,277
2
0,279
I 0,280
2
0,282
1
0,0001
4 (
, I
I
2 :
2
3
3
4
0,0-2
0,548
3
0,55l
4
0,555
3
o,558
3 0,56 1
3
o,564
3
0,0004
o,o3
0,823
4
0,827
5
o,832
4
o,836
5 0,84 1
4
0,845
5
o,ooog
5
1
2
2 3| 3|
4
4
5
0,04
i><i9"
6
i,io3
6
1,109
6
i,ii5
6 1,121
6
1,127
6
0,0016
6
1
2
2
i 4
4
5
5
7
I
2
3 4 4
5
6
6
o,o5
1,371
8
1 ,379
7
1 ,386
8
1,394
7 i,4oi
8
1 ,409
7
0,0025
8
2
2
3 4 5
6
6
7
0,06
1,645
9
1 ,654
9
1 ,6(i3
9
1,6-2 I
0 1 ,682
9
1,691
9
o,oo36
9
2
3
4
3 5
6
7
8
0,0-
i>9'9
II
r,93o
10
1,940
1 1
1 ,95 1 1
I 1 ,9(12
10
1 ,972
11
0,0049
3
3 6
0,08
2,193
12
2,205
12
2,217
i3
2,23o I
2 2,242
12
2,254
12
0,0064
10
2
4
7
8
9
0,09
2,46-:
i4
2,481
i3
2,494
14
2,5o8 I
4 2,522
i3
2,535
i4
0,008 1
1 1
1 2
2
2
3
4
4
5
'■> 7
J 7
8
8
9
10
10
1 1
0,10
2,74 1
i5
2,756
i5
2,771
16
2,787 I
5 2,802
i5
2,817
i5
0,01 00
1 3
3
4
5
7 8
9
10
1 2
0,1 1
3,014
17
3,o3i
17
3,o48
17
3,o65 I
7 3,082
16
3,098
17
0,01 2 1
i4
3
4
6
7 8
10
1 1
i3
0,12
3,288
'9
3,3o7
18
3,325
18
3,343 I
8 3.36i
18
3,379
18
0,01 44
i5
16
17
I 3
1 3
I 3
5
5
5
6
6
7
3 9
B 10
d 10
i4
0,1 3
3,562
20
3,582
'9
3,601
20
3,621 2
0 3,64 1
20
3,661
jg
0,0169
1 1
1 2
i3
14
0,1 4
3,835
22
3,857
21
3,878
21
3,899 2
2 3,921
21
3,942
21
0,0196
I 1
12
i5
0,1 5
4,108
23
4,i3i
23
4,1 54
23
4,177 2
3 4,200
23
4,223
22
0,0225
18
i 4
2 4
5
6
7
^ "
i3
i3
14
i5
16
0,16
4,38i
25
4,406
25
4,43 1
24
4,455 2
4 4,479
24
4,5o3
24
0,0256
19
8 io|ii
17
0,17
4,654
27
4,681
26
4,707
26
4,733 2
5 4,758
26
4,784
26
0,0289
20
1 4
2 4
2 4
6
8 I
012
i4
i5
16
18
0,18
4,927
28
4,955
28
4,983
37
5,010 2
8 5,o38
27
5,o65
27
o,o324
21
6
8 1
T ,^
17
18
19
20
0,19
5,200
29
5,229
3o
5,259
29
5,288 2
8 5,3 16
29
5,345
29
o,o36i
22
7
9 1 1| i3
i5
o,ao
5,473
3o
5,5o3
3i
5,534
3i
5,565 3
0 5,595
3o
5,625
3o
o,o4oo
23
24
2 5
2 5
7
7
9 '
10 I
2 l4
2 14
16
I 7
18
'9
21
22
0,21
5,745
32
5,777
33
5,810
32
5,842 3
2 5,874
32
5,906
3i
o,o44i
0,32
6,01-
34
6,o5i
34
6,o85
34
6,119 3
3 6,i52
33
6, 1 85
34
o,o484
25
3 5
8
10 1
3 i5
18
20
23
0,23
6,289
36
6,325
35
6,36o
35
6,395 3
5 6,43o
35
6,465
35
o,o52g
26
3 5
8
lo 1
3 16
18
21
23
0,24
6,56i
37
6,598
37
6,635
37
6,672 i
6 6,708
37
6,745
36
0,0576
27
28
3 5
3 6
8
8
1 1 I
1 1 I
4 16
4 17
19
20
22
22
24
25
0,25
6,832
39
6,871
39
6,gio
38
6,948 I
8 6,986
38
7,024
38
0,0625
29
3 6
9
12 I
5 17
20
23
26
0,26
-,io4
40
7,i44
40
7,lS4
4o
7,224 i
0 7,264
39
7,3o3
40
0,0676
0,27
7-3-5
42
7,417
4i
7,458
42
7,5oo I
I 7,541
4i
7,582
4i
0,0729
3o
3 6
9
9
10
12 I
5 18
21
24
27
0,28
7,646
43
7,689
^^
7,732
M
7,776 ^
2 7,818
43
7,861
42
0,0784
3i
3 6
12 1
6 19
22
25
28
0,29
7>9i6
45
7,961
45
8,006
45
8,o5i I
4 8,095
^A
8,139
A4
0,084 1
32
3 6
i3 1
6 19
22
26
29
33
3 7
10
i3 I
7 20
23
26
3o
o.3o
8,r86
47
8,233
47
8,280
46
8,326 i
'6 8,372
46
8,4i8
45
o,ogoo
34
3 7
10
14 I
7 20
24
27
3i
0,3 1
8,457
48
8,5o5
48
8,553
48
8,601 i
i7 8,648
48
8,696
47
0,0961
0,32
8,726
5o
8,776
5o
8,826
49
8,875 .
i9 8,924
49
8,973
49
0,1024
35
4 7
11
14 I
8 21
25
28
32
0.33
8,996
5i
9,047
52
9,099
5i
9,1 5o ;
)0 9,200
5i
9,25i
5o
0,1089
36
4 7
II
14 I
8 22
25
o9
32
0,34
9,265
53
9,3 1 8
53
9^371
53
9,424
)2 9,476
52
9,528
52
0,1 156
37
38
4 7
4 8
1 1
1 1
i5 I
i5 I
922
9 23
26
27
3o
3o
33
34
0,35
9,534
54
9,588
55
9,643
54
9,697
)4 9,751
54
9,8o5
53
0,1225
39
4 8
12
16 2
0 23
27
3i
35
o,36
9,802
57
9,859
56
9,915
56
9,971
>5 10,026
55
10,081
55
0,1296
0,37
10,070
58
10,128
58
10,186
58
10,244
)7 io,3oi
56
10,357
57
o,i36g
40
4 8
12
16 2
0 24
28
32
36
o,38
io,338
60
10,398
59
10,457
59
io,5i6
ig 10,575
58
io,633
58
0,1 444
4i
4 8
12
16 2
1 25
29
33
37
0,39
io,6o5
62
10,667
6Ï
10,728
61
10,78g (
3o 10,849
60
10,90g
60
0,l521
42
43
4 8
4 9
i3
i3
17 2
17 2
1 25
2 26
'9
3o
34
34
38
39
o,4o
10,872
63
10,935
63
10,998
63
11,061 (
52 11,123
61
11,184
61
0,1600
M
4 9
i3
18 2
2 26
3i
35
4o
0,4 1
ii,i3o
65
II,2o4
64
11,268
64
11,332 f
54 1 1 ,396
63
1 1 ,459
63
0,1681
18 3
32
32
33
34
36
37
38
38
4i
4i
42
43
0,42
1 1 ,4o5
6-
11,4-2
66
11,538
65
ii,6o3 (
56 11,669
64
11,733
65
0,1764
45
I ^
i4
3 27
0,43
11,671
68
11,739
68
1 1 ,807
67
11,874 (
37 11,941
67
12,008
66
0,1849
46
I 9
i4
18 s
3 28
4 28
4 29
0,44
11,936
70
12,006
69
12,075
69
12,144 (
59 I2,2l3
68
12,281
68
0,1936
47
48
I 5
5 10
i4
i4
ig s
19 ''
0,45
12,201
72
12,273
71
12,344
70
i2,4i4
71 12,485
70
12,555
69
0,2025
49
5 10
i5
20 s
529
34
39
U
o,5o
i3,5i8
80
13,598
80
13,678
79
13,757
-9 1 3,836
78
i3,gi4
78
o,25oo
5o
5i
52
5 10
5 10
5 10
i5
i5
16
20 J
5 3o
35
4o
45
0,55
14,821
90
14,911
88
14,999
88
15,087
38 15,175
87
15,262
86
o,3o25
20 ^
6 3i
36
4i
46
0.60
16.109
98
16,207
98
i6,3o5
98
i6,4o3
56 16,499
96
i6,5g5
9!
o,36oo
21 ;
6 3i
36
42
47
0^65
17,377
109
17.486
lOT
17,593
107
17,700 I
37 17,807
io5
17,912
io5
0,4225
53
54
5 11
5 11
16
16
21 ;
732
732
37
38
42
48
0,70
18,623
119
18,742
118
18,860
118
18,978 1
16 19,094
ii5
19,209
ii5
0,4900
22 :
43
49
0,75
19.841
i3i
■9,972
i3o
20,102
128
20,23o I
28 2o,358
126
20,484
125
0,5625
55
6 II
17
17
22 ;
8 33
%
44
5o
0,80
21,025
i44
2i,i6g
i43
2I,3l2
i4i
2M53 I
3g 2i,5g2
1 38
2i,73o
i37
o,64oo
56
6 11
22 -
.8 34
45
5o
o,85
22,l6c
161
22,321
1 59
22,480
1 56
22,636 I
54 22,7go
l52
22,942
i4g
0,7225
57
6 II
17
17
23
g 34
9 35
4c
46
5i
0,90
23,398
i85
23,583
179
23,762 1
73 23,935
170
24,io5
166
0,8100
58
6 12
23
4i
iC
52
59
60
61
62
63
6 12
6 12
6 12
6 12
6 i3
18
18
18
ig
24
24 -
24
25
25
Jo 35
Î0 36
ÎI 37
3i 37
52 38
4i
42
43
43
44
47
48
49
5c
5o
53
54
55
56
57
,3961
,4050
,4141
,42.33
,4325
,4418
â
\ . (r -}- r" ) ^ or r= -f r"^ nearly.
19
269 ■
270
271
272
27
3 27^
275
276
277
278
279
280
281
282
64
6 i3
"j
26
32 38
45
5i
58
I 27
27
2-
7 27
2
7 2-
1 28
28
28
28
28
28
28
28
65
7 i3
2C
26
3339
4f
53
DO
61
62
a 54
54
5^
i 54
5
5 5:
) 55
55
55
56
56
56
56
56
66
7 i3
20
26
33 40
4C
53
54
3 81
81
8
82
8
2 8;
83
83
83
83
84
84
84
85
67
7 i3
20
27
34 40
4-
4 lofi
108
lof
i 109
10
q lie
) no
no
111
III
112
112
112
ii3
68
7 i4
20
27
34 4i
48
48
54
55
5 i35
i35
i3f
; i36
i3
7 .3-
1 i38
i38
139
i3g
i4o
i4o
i4i
i4i
69
7 i4
21
28
35 4i
6 16!
162
16
i i63
16
4 16^
! i65
166
166
167
167
168
169
169
70
7 i4
21
28
35 42
4c,
56
63
7 i8f
189
i9<
) 190
19
I 19:
.93
193
194
195
195
196
197
'97
80
8 16
24
32 .
io 48
5f
64
72
8 31?
216
21
- 218
21
8 2K
) 220
221
222
222
223
224
225
226
go
9 18
27
36 .
i5 54
63
72
81
9 34î
243
24
'i 245
24
6 24-
■ 248
248
249
2 50
25l
253
253
2 54
100 1
0 20
3r
4o
io| 6<"
7f
80
go
TABLE II. — To find the time T\ the sum of the radii r -f r", and tlie chord t being given.
Sum of the Radii r -(- r".
Chord
c,
0,95
Days Idif.
0,00
0,01
0,02
o,o3
o,o4
o,o5
o,o6
0,07
0,08
0,09
0,10
0,1 1
0,12
0,1 3
0,1 4
0,1 5
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
0,3(1
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
0,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
1,00
0,000
0,283
0,567
0,8 5o
i,i33
i,4i6
1,700
1,983
2,266
2,549
2,832
3.1 15
3,397
3,680
3,963
4,245
4,527
4,810
5,092
5,374
5,655
5,937
6,219
6,5oo
6,781
7,062
7.343
7,623
7,903
8.i83
8,463
8,743
9,022
g,3oi
9,58o
9,858
io,i36
io,4i4
io,6gi
10,96g
1 1,245
11,522
11,798
12,074
12,349
2,624
13,992
1 5,348
16,691
18,017
19,324
20,609
21,867
23,091
24,271
25,374
0,96
Days idif.
45
46
48
5o
5i
53
55
56
58
59
61
62
64
65
67
69
77
86
94
io3
!l3
134
1 35
149
i64
191
0,000
0,285
0,570
o,854
1, 139
1,424
1 ,708
1,993
2,278
2,562
2,847
3,i3i
3,4 1 5
3,699
3,984
4,267
4,55i
4,835
5,119
5,4o2
5;685
5,969
6,252
6,534
6,817
7-099
7,382
7,664
7,945
8,227
8,5o8
8,789
9,u70
9,35i
9,63:
9'9ii
10,191
10,470
10,749
11,028
ii,3o6
ir,584
II,!
I2,i39
I2,4i6
1 2 ,693
14,069
1 5,434
16,785
18,120
19,437
,4513
20,733 123
22,002 1 34
23
24
25
26
28
3o
3i
32
35
36
38
38
40
42
43
45
47
48
49
5i
53
54
56
58
59
61
63
64
66
67
68
77
85
94
104
ii3
0,97
Days |iiir.
23,240
24,435
2 5,565
0,000
0,286
0,573
0,859
i,i45
i,43i
1,717
2,oo3
2,290
2,575
3, 1 47
3,433
3,719
4,004
4,290
4,575
4,860
5,i45
5,43o
5,7 1 5
6,000
6,284
6,56g
6,853
7.13?
7,420
7,704
7,987
8,270
8,553
8,836
9,1
9,400
9,682
9>9'54
10,245
10,526
10,807
11,087
1 1 ,367
1 1 ,647
11,926
I2,2o5
12,483
12,761
i4,i46
i5,5i9
16,879
18,224
19,550
20,856
22,1 36
23,386
24,596
25,749
,4608
0,98
Days |dif.
0,000
0,288
0,575
0,86 3
i,i5i
1,439
1,726
2,Ol4
2,3oi
2,589
2,876
3,i63
3,45i
3,738
4,025
4,3 1 2
4,599
4,885
5,172
5,458
5,745
6,o3i
6,3i7
6,6o3
6,888
7,174
7,459
7,744
8,029
8,3i4
8,598
8,882
9,166
9,4 5o
9,733
10,016
0,299
io,58i
10,864
I i,i45
427
1 1 ,708
11,989
12,270
i2,55o
i2,83o
l4,233
1 5,604
16,973
18,326
19,663
20,978
22,269
23,53i
24,756
25,927
,4705
0,99
Days lilif.
0,000
0,28g
0,578
0,868
i,i57
1,446
1,735
2,024
2,3i3
2,603
2,i _
3,180
3,468
3,757
4,045
4,334
4,622
4,910
5,1 ~
5,486
5,774
6,062
6,349
6,637
6,934
7,211
7,497
7,784
8,070
8,356
8,642
8,928
9,2i3
9,499
9,784
10,068
10,353
10,637
10,920
59 11,204
,4802
11,487
1 1 ,770
I2,o53
13,334
12,616
12,897
14,398
1 5,688
17,065
18,428
19,773
2 1 ,099
22,401
23,675
24,913
26,102
1,00
IJays Idif.
0,000
0,291
o,58i
0,872
i,i63
1,453
1,744
2,o34
2,325
2,61 5
2,905
3,196
3,486
3,776
4,066
4,356
4,646
4,935
5,325
5,5i4
5,8o3
6,093
6,382
6,670
6,959
7,347
7,536
7,824
8,112
8,399
8,687
8,974
9,261
9,547
9,834
10,120
10,406
10,691
10,977
11,263
[1,546
11,83
I2,Il5
13,399
12,683
12,965
14,37.'
1 5,772
I7,i58
18,539
19,883
21,219
32,53l
23,817
25,068
26,274
37,404
,4901
3o
3o
32
34
35
37
38
39
4i
43
44
45
47
49
5o
52
53
55
56
57
60
61
62
63
65
67
75
83
91
100
no
119
i3o
i4o
■ 54
169
196
1,01
Days |dif.
,5000
0,292
o,584
0,876
1,168
1,460
1,753
2,044
2,336
2,628
2,920
3,212
3,5o3
3,795
4,086
4,378
4,669
4,960
5,25i
5,542
5.833
6,123
6,4 1
6,704
6,994
7,3i
7,574
7,863
8,i53
8,442
8,73i
9,019
9,3û8
9,596
9,884
10,172
10,459
10,746
ii,o33
ii,3i9
1 1 ,606
11,89:
12,177
12,462
13,747
i3,o32
1 4,449
1 5,855
17,249
18,629
19,99"
21,338
22,661
23,957
25,222
26,443
27,600
1,02
,ys
0,000
0,394
0,587
0,881
1,174
1,468
1,761
3 ,o54
3,348
2,641
2,934
3,228
3,521
3,8 1 4
4,107
4,399
4,692
4,985
5,277
5,569
5,863
6, 1 54
6,446
6,737
7,029
7,320
7,61 1
7,903
8,193
8,484
8,774
9,o()5
9,354
9,644
• 9,934
10,223
IO,5l2
1 0,800
1 1 ,089
11,377
11,66'
11,95'
12,239
13,526
12,813
1 3,099
i4,'^23
1 5,938
17,34
18,729
20,102
21,456
22,789
24,097
25,373
26,610
27,789
,5101 I ,5202
1,03
Days Idif.
49 9,983
0,000
0,295
o,5go
0,885
1,180
1,475
1,770
2,o65
2,359
2,654
2,949
3,243
3,538
3,832
4,127
4,42
4,71 5
5,oog
5,3o3
5,597
5,890
6,1 84
6,477
6,771
7,064
7,356
7,649
7,942
8,234
8,526
8,818
9,1 10
9,401
9,692
1,04
Days |dif.
10,374
io,564
io,E
11,144
11,43
11,723
1 2 ,0 1 2
12,30I
12,589
12,877
i3,i65
14,598
16,020
17,431
18,828
20,210
21,573
22,916
24,235
25,524
26,775
27,972
i3
i4
16
17
'9
20
21
23
24
26
27
29
3o
32
33
34
36
38
39
4o
42
43
44
46
48
49
5i
53
54
56
57
58
60
61
63
65
66
73
82
90
98
107
117
127
■ 37
i48
162
181
,5305
0,000
0,296
0,593
0,889
i,i8b
1,482
1,778
2,075
2,371
2,667
2,963
3,259
3,55:'
3,85i
4,i47
4,442
4,738
5,o33
5,329
5,624
5,91g
6,214
6,5og
6,804
7,098
7,392
7,687
7,981
8,274
8,568
8,861
9,1 54
9,447
9,740
o,o32
10,325
10,617
1 0,908
11,200
11,491
1 1,781
12,073
12,362
12,652
12,942
i3,33i
14,671
16,103
17,521
18,926
20,317
21,690
23,043
24,372
25,672
26,9.37
28,153
74
81
89
98
107
116
125
i36
1 48
160
177
,5408
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
0,0081
0,0100
0,0121
0,01 44
0,0169
0,0196
0,0225
o,o256
0,0389
o,o334
o,o36i
o,o4oo
o,o44
o,o484
o,o52g
0,0576
0,0625
0,0676
0,072g
0,0784
0,084 1
0,0900
0,0961
0,1024
o,io8g
o,n56
0,1225
o, 1 296
0,1369
0,1 444
0,l521
0,1600
0,1681
o, 1 764
0,1849
0,1936
0,3025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
0,6400
0,7225
0,8100
0,9025
1 ,0000
C-
( r + r"
T'' -t- r"" nearly.
275
f
28
1
55
3
83
4
no
«i
,38
6
i65
7
193
K
2 30
9
248
276
277
278
279
280
2S1
282
283
284
385
286
287
288
28g
290
291
292
293
28
28
28
28
28
28
28
28
28
29
29
29
29
29
29
29
29
29
55
55
56
56
56
56
56
57
57
57
57
57
58
58
58
58
58
81
83
83
83
84
84
84
85
85
85
86
86
86
86
87
87
87
88
no
III
III
112
112
112
ii3
ii3
ii4
ii4
114
ii5
ii5
lib
116
116
117
117
T.38
i3g
i3g
i4o
1 40
i4i
i4i
l42
l42
i43
143
144
i44
i45
145
1 46
i46
147
t66
166
167
167
168
i6q
i6q
170
170
171
172
172
173
173
174
175
175
176
ici3
iq4
iq5
iq5
196
Ip7
197
198
iqq
200
300
201
202
202
2o3
204
204
2o5
•>2I
222
222
223
224
225
226
326
227
338
22Q
23o
23o
23l
232
233
234
234
348
249
25o
25l
252
253
254
255
256
257
257
258
25g
260
261
262
263
264
TABLK
n. -
- To fi
1,1 th
e time
T;
the sum of the ra
Hi r
■f
r", nml ll
e chore
c tjciitff given.
tfuill of til»-' Kadil r-\-r".
Prop, parts lur tho sum ol' tlio Kadii. 1
1 l-llfir.r. > >n< 1
Chord
C.
1,05
1,06
1,07
1,08
U.iys |dir.
1,09
Days |dif.
1,10
I 1 2
|0|4|3|t)|7|0
^
1
3
3
0
0
0
0
0
I
I
0
I
I
2
I
I
2
I
1
2
I
2
2
I
2
3
Days |dir.
Days |dif.
Days jdil'.
Days |dir.
OjlKJ
0,000
0,000
0,000
0,000
0.000
0,000
0,OCH)0
o,ui
0,298
I
0,299
2
o,3oi
I
o,303
I
O,3o3
3
o,3o5
I
0,000 1
4
I
2
2
2
3
3
4
0,02
0,596
3
0,599
2
0.60 1
3
o,6o4
3
o,t)07
3
0,610
2
o,ooo4
5
2
2
3
3
4
4
5
4
5
6
5
5
6
o,o3
0,893
5
0,898
4
0.903
4
0.906
4
0.910
5
0.915
4
0,01 ,ci{)
6
2
2
3
4
4
4
0,04
iî'9'
6
',•197
6
I,203
5
1,208
6
I,2l4
5
1,219
6
0,001 ()
2
3
o,o5
1.489
n
1,496
7
i,5o3
7
1,5 10
7
I,5i7
7
1,524
7
0,0025
h
9
2
2
2
3
3
4
4
5
5
5
6
6
7
7
8
u,oO
i.:-«7
8
■.-95
9
1,804
8
1,812
9
1,831
8
1,829
8
0,00 36
0,07
2.084
10
2,094
10
2,1 04
10
2.114
10
2,124
10
3,i34
9
0,0049
10
2
3
4
5
6
-.
8
9
o,<*
2.383
II
2.393
13
2,4o5
11
3.416
11
3,427
II
3,438
II
0,0064
u
2
3
4
6
7
8
9
10
10
0,09
2,680
12
2,692
i3
2,7o5
i3
3,718
12
2,73o
i3
2,743
12
0,008 1
12
2
4
5
6
7
8
II
i3
3
4
5
7
8
9
10
12
0,10
2,977
i4
2,991
i5
3,006
14
3,030
i4
3,o34
i3
3,047
i4
0,0100
i4
3
4
6
7
8
10
11
i3
0,1 1
3.275
i5
3,290
16
3.3o6
i5
3,33 1
16
3.33-7
i5
3,352
i5
0,01 3 1
0,12
3,572
17
3,589
17
3;6o6
17
3.633
17
3.640
16
3,656
17
0,01 44
i5
2
3
5
6
8
9
1 1
12
i4
o,i3
3.869
'9
3,888
18
3.906
18
3.934
'9
3.943
18
3:96.
18
0,0169
16
2
3
5
6
8
10
1 1
i3
i4
o,i4
4,167
20
4,18-
19
4,20(i
30
4,226
20
4,346
'9
4,265
19
0,0196
17
18
2
2
3
4
5
5
7
7
9
9
10
11
12
i3
14
14
i5
16
o,i5
4.464
21
4.485
21
4,5o6
31
4.527
21
4.548
31
4.56c,
31
0,0225
■9
2
4
6
8
10
11
i3
i5
17
0,16
4.761
23
4,784
32
4,806
33
4,839
23
4.85i
33
4,873
33
0,0 2 56
20
4
4
4
5
g
8
10
1 2
14
i5
, r
16
18
0,17
5,o58
24
5.082
24
5,106
34
5,i3o
24
5, 1 54
33
5,177
34
0,0289
21
6
8
i3
i3
14
0,18
5.355
25
5,38o
26
5,4o6
35
5,43 1
25
5.456
25
5,481
35
o,o32.^
3
1 1
17
18
18
'9
0,19
5,65 1
27
5,678
27
5,705
27
5,732
26
5;758
27
5,785
36
o,o36 1
23
3
2
7
7
9
9
1 1
12
10
16
20
21
0,20
5,948
28
5,976
28
6,004
29
6,o33
38
6.061
28
6,089
27
o,o4oo
24
2
5
7
10
12
i4
17
19
22
0,21
6,,244
3o
6,274
3o
6,3o4
29
6,333
3o
6.363
?9
6,392
39
0,044 1
25
3
5
8
10
i3
i5
18
20
23
0,22
6.540
32
6,573
3i
6,6o3
3T
6.634
3i
6.665
3o
6,695
3i
0,0484
26
3
5
8
10
i3
16
18
2 1
23
0,23
6,837
32
6,869
33
6,902
32
6.934
33
6.967
33
6-999
33
0,0529
27
3
5
8
1 1
14
16
■9
20
22
24
35
0,24
7,i32
35
7,167
34
7,201
33
7,234
34
7,268
34
7,3o2
33
0,0576
28
3
6
8
1 1
i4
17
22
0,25
■7,42b
36
7,464
35
7,499
36
7,535
35
7,570
35
7,6o5
34
0,0625
29
3
6
9
12
i5
17
20
23
26
0,26
7,724
37
7. -6 1
37
7,798
37
7.835
36
7.871
36
7,907
37
0,0676
3o
3
6
9
12
i5
18
21
24
27
0,27
8.019
39
8.058
38
8.096
38
8. 1 34
38
8,172
38
8,3IO
38
0,0729
3i
3
6
9
12
16
19
33
25
28
0,2s
8,3i4
4o
8,354
40
8,394
4o
8.434
39
8,473
39
8.5 1 3
39
0,0784
32
3
6
10
■ 3
16
19
33
26
29
0,29
8,610
4i
8,65 1
4i
8,692
4i
8,733
4i
8,774
4i
8,8 1 5
40
0,084 1
33
34
3
3
7
7
10
10
i3
14
17
17
20
20
33
34
26
27
3o
3i
o,3o
8,904
43
8,947
43
8,990
42
9,032
43
9,075
42
9,117
43
0,0900
0,3 1
9: '99
44
9,243
44
9.387
44
9,33i
44
9,375
44
9,419
43
0,0961
35
4
n
1 1
i4
18
21
25
28
32
0,32
9,493
46
9,539
46
9.585
45
9,63o
45
9,675
45
9,720
45
0,1024
36
4
7
11
14
18
22
35
=9
32
0,33
9,787
48
9,835
47
9,882
47
9-929
46
9-975
47
10,023
46
0,1089
37
4
7
11
i5
19
22
36
3o
33
0,34
10,081
49
io,i3o
49
10,179
48
10,227
48
10,375
48
io,323
47
0,1 1 56
38
39
4
4
8
8
II
12
i5
16
'9
20
23
23
27
27
3o
3i
34
35
0,35
10.375
5o
10,425
5o
10,475
5o
10.535
5o
10,575
i^
10,624
P
0,1225
4o
4
4
8
13
16
20
24
25
25
26
26
28
32
36
37
38
39
40
0,36
10.669
5i
10.720
52
10,773
5i
10,823
5i
10,874
5i
10,925
5o
0,1296
8
8
9
9
16
21
33
34
34
35
0,37
10.962
53
r 1 ,0 1 5
53
11.068
53
1 1. 1 20
53
11,173
52
11,225
53
0, 1 369
41
42
43
44
i3
i3
i3
29
o,38
0,39
1 1.255
11,547
54
56
1 1 ,309
1 1 ,6>3
55
56
1 1 ,364
11,659
54
56
ii,4i8
1 1, 71 5
54
55
1 1 ,472
1 1 ,770
53
55
11.525
11,825
54
55
o,i444
0,l53I
4
4
4
17
17
18
3 I
22
22
3^!
3i
o,4o
11,839
58
1 1 .89-
58
11,955
57
12,012
56
12,068
57
12,125
56
0,1600
45
5
9
9
9
10
i4
18
23
27
32
36
4i
4i
42
43
44
0,4 1
I2,l3l
60
12.1QI
59
I3,3 5o
58
i3,3o8
58
13,366
58
12,424
58
0,1681
46
5
i4
18
23
28
32
37
38
042
12,423
61
13,484
6Ô
12,544
60
1 3 ,6o4
60
12,664
60
12,724
59
0,1764
47
48
5
i4
19
'9
20
24
28
33
0,43
12,715
62
12.7-7
62
12.839
61
1 3. goo
63
12.963
60
l3.022
61
o,i84g
5
i4
34
29
29
34
38
0,44
1 3,006
63
■ 3,069
64
i3,i33
63
13,196
63
13,359
63
.3,321
63
0,1936
49
5
10
i5
35
34
39
0,45
13.296
66
1 3.362
65
13.427
64
13,491
64
i3,555
64
i3,6ig
64
0,2035
5o
5
10
i5
20
35
3o
35
40
45
o,5o
14.745
73
i4,8i8
72
14,890
72
14.963
72
i5.o34
72
i5,io6
71
o,25oo
5i
5
10
i5
20
26
3i
36
4i
46
0,55
i6,i83
81
16,264
80
16,344
80
16,424
80
i6,5o4
79
i6,583
79
o,3o35
52
5
10
16
21
26
3i
36
42
4-
0,60
17,610
89
17,699
89
17,788
87
17,875
88
17,963
87
i8,o5o
86
o,36oo
53
5
II
16
21
27
32
37
4:!
48
o,65
19,024
98
19,122
96
19,218
96
ig,3i4
96
19.410
95
i9,5o5
94
0,4225
54
5
II
16
22
27
32
38
43
49
0,70
20,424
io5
20,52g
106
2o,635
io4
20,739
io4
30,843
io4
20,947
io3
0,4900
55
6
II
17
22
38
33
39
44
5o
0,75
21.806
ii5
21.931
ii4
23.o35
ii4
22,149
112
23,361
112
22,373
113
0,5625
56
6
II
17
22
28
34
39
45
5o
0,80
23.16S
125
23,293
124
23,417
133
23,539
123
23,661
131
23,783
121
o,64oo
57
6
1 1
17
23
29
34
4o
46
5i
0,85
24,5o8
i35
24,643
i33
24,776
i33
34.90g
l32
25.o4i
i3i
25,173
129
0,7325
58
6
12
17
23
29
35
4i
46
52
0,90
25,820
1 46
25,966
144
26,110
144
26,254
142
26,396
i4i
36.537
i4o
0,8100
59
6
12
18
24
3o
35
4i
47
53
0,95
27.097
i5g
27.356
1 57
27,4 1 3
1 56
27,569
1 54
27,723
I 52
37,875
i5i
0,9035
18
18
'9
19
19
36
37
37
38
38
1,00
28,330
174
28,5o4
172
28,676
.69
28,845
168
29,013
i65
29,178
1 64
1 ,0000
60
61
63
63
64
6
6
6
6
6
12
12
13
i3
i3
24
24
25
25
26
3o
3i
3i
32
32
42
43
43
44
45
48
49
5o
54
55
56
57
58
.5513
,5618
,5725
,5832
,5941 1
,6050
c"
i . (r + r"«) or r" + r"" nearly.
294
295
296
297 1 29^
299
3oo
3oi
302
3., 3
3o4
3o5
J J
—
—
—
—
65
7
i3
30
26
33
39
46
52
5g
I
=9
3o
3o
3<
) 3c
) 3o
3o
3o
3o
3c
3o
3i
66
7
1 3
30
26
33
4o
46
53
59
2
59
59
59
5<
, 6c
) 60
60
60
60
61
61
61
67
7
i3
30
27
34
4o
47
54
65
3
88
89
89
8(
i 8ç
) 90
! 90
90
91
91
9'
92
68
7
i4
30
27
34
4i
48
54
61
4
118
118
118
lie
) II'
1 120
120
120
121
121
122
122
6g
7
i4
21
28
35
4i
48
55
62
5
l47
1 48
1 48
i4(
) i4(
) i5o
i5o
i5i
i5i
l53
l52
1 53
6
176
177
178
17'
1 17c
) '79
180
181
181
182
182
i83
70
7
i4
21
28
35
42
49
56
63
7
206
207
207
20f
i 30(
) 2og
210
211
211
212
2l3
3l4
80
8
16
24
32
4o
48
56
64
72
8
235
236
237
23
i 23f
; 239
240
241
242
242
343
244
90
9
18
27
36
45
54
63
72
81
9
265
266
266
26
- 26(
5 369
370
271
272
273
274
375
ICX)
10
20
3o
4o
5o
60
70
80
_9o
a6
TABLE II.
— To find the time T
the sum 0
f the radii
■ + .",
and the cb
51-d c
being
given.
Sum of the Itadii r -)- r". i
Chord
C.
1,11
1,12
1,13
1,14
1,15
1,16
1,17
1,18
1,19
1,20
Duys |dif.
Days |dif.
Days |dil'.
Days |dir.
Days |clif.
Kays |dil'.
Days |ilif.
Days Idif.
Days |.lif.
Days |dif.
0,00
0,000
0,000
OjOOC»
0,000
0,000
0,000
0,000
0,000
0,000
0,000
u,uooo
0,01
0,3o6
2
o.3o8
I
0,309
I
o,3io
2
0,3l2
I
o,3l3
I
0,3l4
2
o,3l6
I
o,3i7
I
0,3l8
2
0,0001
0,02
0,612
3
0,61 5
3
0,618
3
0,621
2
0,623
3
0,626
3
0,629
2
o,63l
3
o,634
3
0,637
2
o,ooo4
o,o3
0,919
4
0,923
4
0,927
4
0,931
4
0,935
4
0,939
4
0,943
4
°'947
4
0.951
4
0,955
4
(.,,0009
o,o4
1,225
5
I,23o
6
1,236
5
1,241
6
1,347
5
1,253
6
1,2 58
5
1,263
5
1,268
6
1,274
5
0,0016
o,o5
1, 53 1
7
1,538
7
1,545
7
1.552
6
1,558
7
1,565
7
1,572
7
1,579
6
1,585
7
1 ,592
7
0,0025
0,06
1.837
8
1,845
9
1.854
8
1,862
8
1,870
8
1,878
8
1,886
8
1,894
8
1 ,902
8
1,910
8
o,oo36
0,07
2,143
10
2;i53
10
2.i63
9
2,173
10
3,183
9
2,191
9
3,200
10
2,310
9
2,219
10
2,229
9
0,0049
o,oS
2,449
II
2,460
II
2.471
1 1
2,482
II
2,493
II
2,5o4
II
2,5i5
10
3,525
II
2,536
1 1
2,547
10
(.1,0064
0,09
2,755
i3
2,768
12
2,780
12
2,792
i3
2,8o5
12
3,817
12
2,829
12
2,84 1
13
2,853
12
3,865
12
0,0081
0,10
3.061
i4
3.075
14
3,089
i3
3,102
i4
3,116
i4
3,i3o
i3
3,143
1 3
3,i56
i4
3,170
i3
3,i83
i3
0,0 1 00
0,1 I
3.367
i5
3.382
i5
3,397
i5
3,4i2
i5
3,427
i5
3,443
i5
3,457
i5
3,472
i5
3,487
i4
3,5oi
i5
U,012I
0,12
3,673
17
3,690
16
3,706
16
3.723
17
3,739
16
3,755
16
3,771
16
3,787
16
3,8o3
16
3,819
16
0,0 [44
0,1 3
3.979
18
3.997
17
4.014
18
4,o32
18
4.o5o
18
4,068
17
4,o85
18
4,io3
17
4,120
17
4,i37
17
0,0 [69
0,1 4
4,284
20
4,3o4
'9
4,323
19
4,342
19
4,36i
19
4,38o
■9
4,399
19
4,418
18
4.436
19
4,455
'9
0,0196
0,1 5
4,590
21
4,611
20
4,63 1
21
4,652
20
4,672
21
4,693
20
4,71 3
30
4,733
30
4,753
20
4,773
20
0,0225
0,16
4,095
23
4.918
22
4.940
21
4,961
22
4.983
22
5,oo5
21
5,026
22
5,048
21
5,060
22
5,09]
21
o,o256
0,17
5,201
23
5.224
24
5,248
23
5,271
23
5.394
23
5,3 1 7
23
5,340
23
5,363
23
5,38b
22
5,408
23
0,0389
0,18
5,5o6
25
5,53i
25
5.556
24
5.58o
25
5,6o5
24
5,629
25
5,654
24
5,678
24
5,702
34
5,726
24
0,0834
0,19
5,811
27
5,838
26
5,864
26
5,890
26
5,916
25
5,941
26
5,967
26
5,993
25
6,018
35
6,043
26
o,o36i
0,30
6.116
28
6.144
27
6.1 71
28
6,19g
27
6,236
27
6,253
27
6,280
27
6,3o7
27
6,334
27
6,36i
26
o,o4oo
0,21
6,421
29
6.45o
29
6.479
29
6.5o8
29
6,537
28
6,565
28
6,593
29
6,622
28
6,65o
28
6,678
28
0,044 1
U,22
6,726
3o
6,756
3i
6,787
3<.
6.817
3o
6,847
3o
6,877
3o
6,907
29
6,936
3o
6,966
29
6.995
29
o,o484
0,23
7,o3i
3i
7,062
32
7,094
33
7,126
3i
7,1 57
3i
7,188
3i
7,219
3i
7,25û
3i
7,281
31
7,3 1 2
3i
o,o52g
0,24
7,335
33
7,368
33
7,401
33
7,434
33
7,467
33
7,5oo
32
7,532
33
7,565
32
7,597
32
7,639
32
0,0576
0,25
7,639
35
7,674
35
7,709
34
7,743
34
7,777
M
7,811
34
7,845
34
7,879
33
7,912
34
7,946
33
0,0625
0,26
7,944
36
7.980
36
8.oi(i
35
8.o5i
36
8,087
35
8,122
35
8,i57
35
8,192
35
8,227
55
8,362
35
0,0676
0,27
8,248
37
8.285
37
8,322
37
8.359
37
8,396
37
8,433
37
8,470
36
8,5o6
36
8,542
37
8,579
36
0,0729
0,28
8,55i
39
8:590
39
8,629
38
8,667
39
8,706
38
8,744
38
8,782
38
8,820
37
8,857
38
8,895
37
0,0784
0,29
8,855
40
8,895
40
8,935
40
8,975
40
9,01 5
40
9,o55
39
9,094
39
9,i33
39
9.172
39
9,211
39
o,o84i
o,3o
9>i59
4i
9,200
42
9.242
4i
9,283
4i
9,334
4i
9,365
41
9,406
40
9,446
4i
9,487
40
9,527
4o
0,0900
0,3 1
9,462
43
9:505
43
9,54»'
43
9-591
42
9,633
42
9.675
42
9,717
42
9.759
43
g,8oi
42
9,843
4i
0,0961
0,32
9,765
44
9,809
45
9,854
44
9,898
44
9,94?
43
9,985
44
10,029
43
10,072
43
io,ii5
43
io,i58
43
0,1024
0,33
10,068
46
io,ii4
45
10,1 59
46
IO,3o5
45
I0,250
45
10,395
45
io,34o
45
[o,385
A4
10,429
45
10,474
44
o,io8g
0,34
10,370
48
10,418
47
10,465
4i
IO,5l2
47
10,559
46
io,6o5
46
I o,65 1
46
10,697
46
10,743
46
10,789
45
0,11 56
0,35
10,673
49
10.722
48
10,770
49
10.819
48
10,867
48
10,915
47
10,963
48
n,oio
47
1 1 ,o57
47
11,104
47
0,1225
o,36
10,975
5o
11,025
5o
11,075
5o
n,i25
5o
r 1,175
49
n,224
49
11,273
49
11,322
48
11,370
49
11,419
48
0,1296
0,37
11,277
52
11.329
5i
1 1 ,38o
5i
n.43[
5i
11,482
5i
11,533
50
11,583
5i
11,634
5o
11,684
49
11,733
5o
o,i36g
o,38
[ 1 ,579
53
11.632
53
1 1 ,685
52
.1:737
53
1 1 .790
52
11,842
53
11,894
5i
11,945
52
11,997
5i
12,048
5i
o,i444
0,39
1 1 ,880
55
11.935
54
1 1 ,989
54
I2,o43
54
1 3 ,097
53
1 2 , 1 5o
54
12,204
53
13,357
52
12,809
53
12,362
52
[i,[52I
o,4o
12,181
56
12,237
56
12.293
55
12.348
56
i2,4o4
55
12,459
54
i2,5i3
55
12,568
54
12,622
54
12,676
54
0,1600
0,4 1
12.482
58
12.540
57
12,597
57
12,654
56
[2,710
57
12,767
56
13,833
56
12,879
55
12,934
56
12,990
55
n,i68i
0,42
12:783
59
12.842
58
12.900
59
i2.95[.
58
l3,OI-7
58
13,075
57
1 3, 1 32
57
13,189
57
1 3,246
57
i3,3o3
57
0,1764
0,43
i3.o83
61
i3,i44
60
i3.2o4
"■9
13.263
60
[3,333
59
1 3,382
59
T 3,441
59
i3,5oo
58
i3,558
58
i3,6i6
58
o,i84g
0,44
1 3,383
62
1 3,445
62
i3,507
(11
1 3,568
61
13,629
60
13,689
61
i3,75o
60
i3,8io
60
18,870
59
18,929
59
0,1986
0,45
1 3,683
63
1 3,746
63
13.809
63
13,87?
62
13.934
62
1 3, 096
63
i4,o58
62
l4,130
6,
i4,i8i
61
14,242
61
0,2025
o,5o
1 5, 1 77
70
1 5,247
71
i5.3i8
70
1 5.388
70
1 5,458
69
1 5,527
69
i5,5g6
C9
1 5,665
68
15,733
68
1 5,801
68
o,25oo
0,55
16.662
78
i6,74o
78
i6.8iK
77
16.895
77
16.973
77
17,049
77
17,126
75
17,201
76
17,277
75
17,352
75
o,3o25
0,60
i8,i36
86
18,222
86
i8,3o8
85
18,393
85
18,478
84
i8,563
84
[8,646
83
18,729
83
18,812
87
i8,8g5
83
o,36oo
0,65
19,599
94
19,693
94
19.787
93
19,880
92
19-972
92
20,064
92
20,1 56
91
20,347
90
20,337
91
30,438
89
0,4225
0,70
2i,o5o
102
2I,l52
lOI
21,253
lOI
21,354
lOI
21,455
100
21,555
99
31,654
99
21,753
98
2 1, 85 1
98
21,949
97
0,4900
0,75
22,485
no
22,595
1 10
22,705
no
22,8l5
108
22,933
109
23,o33
107
23,139
107
23,246
!06
23,352
106
23,458
io5
o,5635
0,80
23,903
119
24.022
i'9
24,i4i
118
24.259
117
24,376
117
24,493
116
24,609
(i5
24,724
ii5
24,889
i-iA
34.053
ii3
o,64oo
0,85
25,3oi
I '9
25:430
12S
25.558
127
2 5,685
127
25,812
135
25,937
125
36,062
124
36,186
133
36,309
123
26,433
121
0,7225
0,90
26.677
1 39
26,816
1 38
26.954
■37
37.091
1 36
27,227
i35
27,362
1 34
27,496
1 33
37,639
l32
27,761
l32
27,893
i3o
0,8100
0,95
28.026
i5o
28,176
■49
28,:325
1 47
28,472
146
38,618
i45
28,763
i44
28,907
i43
39,o5o
142
29,192
i4i
29,333
i4o
0,9025
I, ou
:9,342
162
29.504
161
29,665
.59
29.82Z
I 57
29.981
1 56
3o,i37
i55
30,293
[54
30.446
I 52
30.598
r5i
3o,749
i5o
1 ,0000
,6161
,6272
,638.5
,6498
,6613
,6728
,6845
,6962
,7081
,7200
c2
4 . ( r + !■" ,= or )-' -)- r"' nf:irly. |
3oo
3oi
302
3o3
3o4
3o5
3o6
3o7
3o8
309
3[o
3ii
3l2
3i3
3i4
3i5
3i6
3o
3o
3o
3o
3o
3i
3i
3i
3i
3i
3i
3i
3i
3i
3i
32
32
fio
60
60
61
61
61
61
61
62
62
62
62
63
63
63
63
63
QO
9"
91
9'
91
92
92
92
92
93
93
93
94
94
94
95
gb
120
120
121
121
122
133
122
123
123
134
134
124
125
125
126
126
130
i5o
i5i
i5i
l52
l52
1 53
1 53
1 54
1 54
1 55
1 55
i56
1 56
1 57
1 57
1 58
1 58
I So
181
181
182
182
i83
184
184
i85
i85
186
187
187
188
188
189
I go
210
211
21 1
212
2l3
2l4
214
2l5
216
216
217
218
218
219
220
221
221
240
241
242
242
243
244
245
246
246
247
248
249
2 50
2 5o
35l
252
253
270
271
272
273
374
275
275
376
277
278
279
280
381
282
283
284
284
TABLE 11. — To liml llie time T\ the sum of the radii r-|-r", and tlie cliord c being c;iven.
Sum ui' tlie Radii r -\- r'
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
o,oC
0,07
0,08
0,09
0,10
0,1 1
0,13
o,i3
0,1 4
0,1 5
0,16
0,1-
0,1 S
o,ig
0,30
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
0,36
0,37
o,38
o,3g
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
0,65
0,70
0,75
0,80
o,85
0,90
0,95
1,00
\^i
1
!).,)«
idif.
t»,tKH)
t>,3-.>ti
I
i>,()Jy
3
0,919
4
i>J7y
5
■.599
6
1,0 tb
«
2.a3t
q
2,55-
1 1
2,87:
12
3,19(1
i4
3,5 16
14
3,835
16
4,1 54
18
4,4:'4
18
4.-C.3
20
3,1 ij
21
5,43 1
22
5,-50
24
6,069
25
6,38-
27
6,706
28
-.024
2Q
7.343
3o
-,661
32
7,979
33
8,29-
34
8,61 5
35
8.932
37
9,2Do
38
9,567
40
9,884
4i
10,201
Ai
io,5i8
44
io,834
46
1 i,i5i
46
1 1 ,46-
48
11,-83
49
12,099
bi
i3,4i4
D2
i2,73o
53
1 3,045
55
i3,36o
56
13,674
58
13,988
60
i4.3o3
60
15,869
67
'7>427
7^
18,977
82
20.317
Kq
22,o46
97
23,563
io5
25,066
ii3
26,553
122
28,023
i3o
29,473
.3q
30,899
149
,73S
Î1
1,-22
Dhvs
d.r.
o,OOC:
0,321
I
0,642
3
o,9(>3
4
1,284
5
i,6<:.5
-,
1 ,926
8
2,24-
9
2,568
10
2,889
12
3,aio
i3
3,53o
i5
3,S5i
16
4,173
17
4,.-i92
19
4,8 1 3
iq
5,1 33
21
5,453
23
5,774
23
6,094
25
6,4 14
26
6,734
27
7,o53
3q
7,373
3o
7.693
3i
8,012
33
8,33i
35
8,65o
36
8,q6Q
37
9,28»
39
9,607
4o
9,925
41
10,244
43
10,562
44
10,880
45
11,197
47
11,313
48
11,832
5o
I2,l5o
5o
12,466
52
12,783
53
i3,ioo
54
i3,4i6
56
i3,732
57
i4,o48
58
i4,363
60
1 5,936
67
17,503
74
19,059
81
20,606
8q
22,l43
96
23,668
io4
25,179
112
26,675
120
28,153
I2q
29,612
1 38
3i,o48
1 48
,74^
12
1,23
Uuys |djr.
o,ix)o
0,322
0,645
(.1,96-
1,289
1,612
1 ,934
2,256
2,578
2,901
3,333
3,545
3,86-
4,189
4,5ii
4,833
5,i54
5,476
5,797
6,119
6,44o
6,761
7,083
7,4o3
7,724
8,045
8,366
8,686
9,006
g,32
9,647
9,966
10,286
1 0,606
10,925
11,344
11,563
11,882
12,200
i2,5i8
12,836
i3,i54
1 3,472
13,789
i4,io6
14,423
1 6,oo3
17,576
19,140
20,695
22,239
23,772
25,291
26,795
28,282
39,750
31,196
,7565
1,24
Duyâ |dit'.
0,000
0,324
0,647
0,97 1
1,295
1,618
1,942
2,265
2,589
2,9
3,236
3,559
3,882
4,206
4,529
4,853
5,175
5,498
5,821
6,144
6,4C6
6,789
7,1 1 1
7,434
7,756
8,078
8,400
8,723
9,043
9,36-
9,686
10,007
10,328
10,649
10,970
1 1 ,290
11,610
11,930
I2,25o
13,889
13,209
1 3,528
1 3,846
i4,i65
1 4,483
16,070
i7,65o
19,321
20,784
22,335
23,875
25,402
26,915
28,410
9,8
3 1,343
,7688
1,25
UiijS |dil'.
0,000
0,325
o,65o
0,975
1 ,3oo
1,625
1,950
2,274
2,599
2,924
3,349
3,574
3,898
4,323
4,547
4,872
5,196
5,520
5,844
6,168
6,492
6,816
7,i4o
7,464
7,787
8,111
8,434
8,757
9,080
9,4o3
9,725
10,048
10,370
1 0,692
1 1, 014
11,336
11,658
11,979
I2,300
13,621
12,942
1 3,263
1 3,583
i3,qo3
l4,223
14,543
i6,i37
17,724
19,302
30,873
23,43l
23,978
35,5i3
27.034
28,538
3o,o24
3i,488
i3
i4
16
17
18
■9
21
22
24
25
26
28
29
3o
3i
32
34
35
37
38
40
40
42
44
45
46
47
49
5o
52
53
54
55
57
58
59
66
73
80
87
95
io3
no
n8
126
i35
i45
,7813
1,26
Daya |dit'.
0,000
0,326
o,653
0,979
i,3o5
i,63i
1,957
2,384
2,610
3,936
3,262
3,588
3,914
4,240
4,565
4,6
5,3l7
5,542
5,868
6,193
6,5i8
6,844
7,169
7,494
7,818
8,i43
8,468
8,793
9,"7
9,441
9,765
10,08b
10,4 13
10,736
1 1 .059
11,382
1 1 ,7o5
12,028
i2,35o
13,673
12,995
!3,3i7
i3,638
i3,g6o
14,281
1 4.60 3
i6,2o3
17,797
19,382
20,959
22,526
24,081
25,633
37,l52
28,664
3o,i59
3 1,633
33
25
26
27
28
3o
32
33
34
35
36
37
39
4i
43
43
44
46
47
48
5o
5i
52
53
55
5f)
58
59
66
73
80
87
94
102
no
118
127
i35
1 44
,7938
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0025
o,oo36
0,0049
o,t)o()4
0,0081
0,0100
0,<H2I
0,01 44
0,0169
0,0196
0,0235
o,o256
0,0289
o,o324
o,o36i
o,o4oo
o,o44 i
0,0484
0,0^29
0,0570
0,0625
0,0676
0,0729
0,0784
0,084 1
o,oyoo
0,0961
0,1024
0,1089
0,1 1 56
0,I225
0,1296
o, 1 369
0,1444
0,1 52 I
0,l6l)0
o, 1 68 1
o, 1 7(i4
0,1849
0,1936
0,2025
0,2 5oo
o,3o25
o,36oo
0,4225
o,49"o
Ï • (r -}- r'2) or r= -f r" - nearly.
3i7
3i8
3ig
320
331
322
323
354
325
326
327
32
32
33
32
32
32
32
32
33
33
33
63
64
64
64
64
64
65
65
65
65
65
95
q5
q6
q6
q6
97
97
97
98
q8
q8
127
127
128
128
128
l2q
I2q
i3o
i3o
i5o
i3i
l5q
:5q
160
1 60
161
161
162
162
i63
i63
164
190
'9'
19'
192
iq3
iq3
194
194
195
196
196
222
223
223
224
225
225
226
227
228
228
239
254
254
255
2 56
257
258
258
239
260
261
263
285
286
287
288
289
390
391
392
293
393
394
1'
"1'
Piirlslor tlif
sum of
ll.c
Kai
li.
1 |2
3
4 1 5 1 6 1 7 1 8
9
I
0
0
0
0
I
1
I
I
I
3
0
0
I
1
I
I
1
2
2
3
0
1
1
1
2
2
2
2
3
4
0
I
1
2
2
2
3
3
4
5
1
2
2
3
3
4
4
5
6
1
2
2
3
4
4
5
5
7
'
2
3
4
4
5
6
6
8
3
2
3
4
5
6
6
7
9
3
3
4
5
5
6
7
8
10
3
3
4
5
6
7
8
q
1 1
2
3
4
6
7
8
q
10
1 2
2
4
5
6
8
10
1 1
i3
3
4
5
7
8
q
10
12
i4
3
4
6
7
8
10
1 1
i3
i5
2
3
5
6
8
q
1 1
12
i4
16
3
3
5
6
8
ll.»
1 1
i3
i4
17
3
3
5
7
q
10
12
14
i5
18
2
4
5
7
q
1 1
i3
14
lb
19
2
4
6
8
10
I 1
i3
i5
17
20
2
4
6
8
10
1 2
i4
i(
18
21
2
4
6
8
1 1
i3
i5
17
iq
22
2
4
7
q
1 1
i3
i5
18
20
23
2
5
7
q
12
i4
16
18
21
34
2
5
7
10
12
i4
17
'9
22
25
3
5
S
10
i3
i5
18
21)
23
26
3
5
8
10
i3
16
18
21
23
27
3
5
8
1 1
i4
16
iq
22
24
28
3
6
8
1 1
14
17
20
22
25
29
3
6
9
12
i5
17
20
23
26
3o
3
6
9
q
12
i5
18
21
24
27
3i
3
6
12
16
iq
32
25
28
33
3
6
10
i3
16
iq
2 2
26
2q
33
3
10
i3
'7
20
33
36
3o
34
3
7
10
14
17
20
24
27
il
35
4
7
1 1
i4
18
31
35
38
32
36
4
7
1 1
14
18
22
25
29
32
37
4
7
1 1
i5
iq
22
26
30
ii
38
4
8
I 1
i5
iq
33
27
3o
i4
39
4
8
12
16
30
23
27
3i
35
40
4
8
13
16
20
34
28
32
36
4i
4
8
12
16
21
25
2q
33
37
42
4
t
l3
17
31
35
2q
M
38
4J
4
9
i3
17
32
26
3o
M
iq
44
4
9
i3
18
32
26
3i
35
40
45
5
9
i4
18
23
27
33
36
4i
46
3
q
i4
18
33
28
32
37
41
47
5
q
i4
iq
24
28
iJ
38
42
48
5
10
14
iq
24
2q
M
38
43
49
'j
10
13
20
23
29
M
39
44
5o
5
10
i5
20
25
3o
35
4o
45
5i
5
10
i5
20
26
3i
36
4i
46
52
5
10
16
21
26
3i
36
42
47
53
5
1 1
16
21
27
32
37
42
48
54
5
1 1
16
22
27
32
38
4à
49
55
6
I I
17
22
28
33
3q
44
5o
56
6
1 1
>7
22
38
34
3q
45
5o
57
6
1 I
17
23
3q
M
4o
46
5i
58
6
12
17
23
2q
35
4i
46
52
59
6
12
18
24
3o
35
4i
47
53
60
6
1 3
18
24
3o
36
42
48
54
61
6
13
18
24
3i
37
43
4q
55
62
6
13
iq
25
3i
37
43
5o
56
63
6
i3
iq
25
32
38
44
5o
57
64
6
i3
19
26
32
38
45
5i
58
65
-,
i3
20
26
33
3q
46
53
5q
66
7
i3
20
26
33
4o
46
53
5q
67
7
i3
20
27
34
4o
47
54
60
68
i4
30
27
M
4i
48
54
61
69
7
i4
21
28
35
4i
48
55
62
70
7
i4
21
28
35
42
49
56
63
80
8
16
24
33
4o
48
56
64
72
qo
q
18
27
36
45
54
63
72
81
100
Id
30
3o
40
5o
61.
70
80
90
TABLE II. — To find the time T; the sum of the radii r-f-r", and the chord c being given.
Sum of the radii r-f-î-'. 1
Cliord
C.
1,27
1,28
1,29
1,30
1,31
1,32
1,33
1,34
1,35
1,36
_l
Days |ilif.
Day» \A\f.
Days Iclir.
Days |dir.
Days |dil'.
Uaysldif.
Days |dir.
Days \,
if.
Days |dif.
Days Idif. 1
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
),0000
o,oi
0,328
1
0,329
I
o,33o
I
0,33l
2
0,333
I
0,334
I
0,335
I
o,336
2
o,33&
I
0,339
1
1,0001
0,02
o,655
3
o,658
2
0,660
3
o,663
2
o,665
3
0,668
2
0,670
3
0,673
2
0,675
3
0,67b
1
J ,0004
o,o3
0,983
4
0,987
3
0,990
4
0,994
4
0,998
4
1,002
4
1 ,006
3
[,009
4
i,oi3
4
1,017
A
1,0009
o,o4
I,3lO
5
i,3i5
5
1,320
6
1,326
5
i,33i
5
1,336
5
1,341
5
1,346
5
i,35i
5
1 ,356
5
0,0016
o,o5
1,638
6
1,644
7
i,65i
6
1.657
6
1,663
7
1,670
6
1 ,676
6
[,682
7
1,689
6
1,695
6
0,0026
0,06
i,g65
8
1 ,973
8
1,981
7
1,988
8
1.996
8
2,oo4
7
2,011
8
2,oig
7
3,026
8
3,o34
7
i,oo36
0,07
2,293
9
2,302
9
2,3ll
9
2,320
8
2,328
9
2,337
9
2,346
9
2,355
9
2,364
9
2,373
8
3,0049
0,08
2,620
10
2,63o
1 1
2,641
10
2,65 1
10
2,661
10
2,671
10
2,681
10
2,691
10
2,701
10
2,711
ii>
},oo64
0,09
2,947
12
2.959
12
2,971
11
2,982
12
2.994
11
3,oo5
i[
3,016
12
3,028
11
3,o39
11
3,o5o
"
0,0081
0,10
3,275
i3
3,388
12
3,3oo
i3
3,3 1 3
i3
3,326
i3
3,339
[2
3,35i
i3
3,364
12
3,376
i3
3,389
12
),0100
0,11
3,602
M
3,6[6
i4
3,63o
i4
3,644
i4
3,658
i4
3,672
i4
3,686
i4
3,700
i4
3,714
i4
3,72b
i3
1,0121
0,I2
3,929
16
3,945
i5
3,960
i5
3.975
16
3,991
i5
4,006
[5
4,021
i5
4.036
i5
4,o5i
i5
4,066
i5
0,01 44
o,i3
4,256
17
4,273
17
4,290
16
4,3o6
17
4.323
17
4,34o
[6
4,356
16
4,372
17
4,389
16
4,4o5
16
0,0169
o,i4
4,584
18
4,602
18
4,620
17
4.637
18
4,655
18
4,673
18
4.691
17
4,708
18
4,726
17
4,743
18
0,0196
o,i5
4,911
19
4,930
'9
4,949
19
4,968
19
4,987
19
5,006
■9
5,025
'9
5,044
19
5,o63
19
5,082
19
0,0235
0,16
5,237
2[
5,258
21
5.279
20
5.299
31
5,320
20
5,34o
20
5,36o
20
5,38o
20
5,400
20
5,430
20
0,02 56
0,17
5,564
22
5,586
22
5,608
22
5,63o
22
5,652
21
5,673
22
5,695
21
5,716
21
5,737
22
5,759
21
0,0289
0,18
5,891
23
5,914
23
5,937
24
5,961
22
5,983
23
6,006
23
6,029
23
6,o53
22
6,074
23
6,097
22
o,o324
o,ig
6,218
24
6,242
25
6,267
24
6,291
24
6,3i5
24
6,339
25
6,364
23
6,387
24
6,4 1 1
24
6,435
24
o,o36i
0,20
6,544
26
6,570
26
6,596
26
6,622
25
6,647
25
6,672
26
6,698
25
6,723
25
6,748
25
6,773
25
o,o4oo
0,21
6,871
27
6,898
27
6,925
27
6,952
27
6,979
26
7,oo5
27
7,o"32
27
7.059
26
7,o85
26
7,111
26
0,044 1
0,22
7.197
29
7,226
28
7.254
28
7,282
28
7,3io
28
7,338
28
7,366
28
7,394
28
7/i22
27
7.449
38
o,o484
0,23
7,524
29
7.553
3o
7,583
29
7,612
3o
7,642
29
7,671
29
7,700
29
7,729
29
7,758
29
7,787
29
0,0529
0,24
7,85o
3i
7,88 [
3i
7.9' 2
3o
7.942
3i
7,973
3i
8,004
3o
8,o34
3o
8,064
3i
8,095
3o
8,125
3o
0,0576
0,25
8,176
32
8,208
32
8,240
32
8,272
32
8,3o4
32
8,336
33
8,368
3[
8,399
32
8,43 1
3i
8,462
3i
0,0625
0,26
8,5o2
33
8,535
34
8,569
33
8,602
33
8,635
33
8,668
33
8,701
33
8,734
33
8,767
33
8,800
32
0,0676
0,27
8,827
35
8,862
35
8,897
35
8,932
34
8,966
35
Q'™'
34
9,o35
34
9,069
M
9,io3
34
9.137
34
0,072g
0,28
9>i53
36
9.189
36
9,225
36
9.261
36
9.297
36
9,333
35
9,368
36
9,404
35
9.439
35
9.474
35
0,0784
0,29
9.478
38
9,5[6
37
9,553
38
9.591
37
9,628
37
9,665
37
9,702
36
9,738
37
9-775
36
9,811
37
0,084 i
o,3o
9,804
39
9,843
38
9,881
39
9.920
38
9,958
39
9.997
38
io,o35
38
10,073
38
10,1 11
37
io,i48
38
■1,0900
o,3i
10,129
4o
10,169
4o
10,209
40
10,249
40
10,289
39
10,328
4o
io,368
39
10,407
39
10,446
39
10,485
39
o,ii(j6i
0,32
10,454
4i
10,495
42
10,537
4i
10,578
4i
io,6ig
4i
10,660
4o
1 0,70c,
4i
io,74i
40
10,781
41
10,822
40
11,11124
0,33
' 0.779
42
10,82 [
43
10,864
43
10,907
42
10,949
42
10,991
42
ii,o33
42
1 1 ,075
42
11,117
41
ii,i58
42
0,1089
0,34
ii,io3
Aà
ii,i47
44
11,191
Ai
11,235
AA
11,279
43
11,333
44
11,366
43
1 1 ,409
43
11,452
42
11,494
43
0,1 156
0,35
[1,428
45
11,473
46
11,519
45
11,564
45
1 1 ,609
Ai
11,653
45
11,698
44
11,742
45
11,787
AA
ii,83i
AA
0,1225
o,36
11,752
47
11.799
46
11,845
47
1 1 ,892
46
11,938
46
11,984
46
13,o3o
46
12,076
45
12,121
46
12,167
45
0,1 296
0,37
12,076
48
12,124
48
12,172
48
12,220
48
12,268
47
i2,3i5
47
12,362
47
12,409
47
12,456
47
i3,5o3
46
0,1 36g
o,38
1 2 ,400
49
12,449
5o
[2,499
49
12,548
49
12,597
48
12,645
49
i2,(ig4
48
12,743
48
12,790
48
12,838
48
0,1 444
0,39
12,724
5Ï
12,775
5o
12,825
5o
12,875
5i
12,926
5o
12,976
49
1 3,025
5o
13,075
49
i3,i24
5o
i3,i74
49
0,1 521
o,4o
1 3,047
52
13,09g
52
[3,[5[
52
i3,2o3
5i
i3,354
52
i3,3o6
5i
i3,357
5[
i3,4o8
5o
i3,458
5i
i3,5og
5o
0,1600
0,4 1
i3,370
54
1 3,424
53
(3,477
53
i3.53o
53
i3,583
53
1 3,636
52
1 3,688
52
i3,74o
52
13,792
52
1 3,844
52
0,1681
0,42
13,693
55
1 3,748
55
[3,8o3
54
1 3,857
54
13,91 1
54
13,965
54
14,019
54
14,073
53
14,126
53
14,179
53
0,1764
0,43
i4,oiÉ
56
14,072
56
[4, [28
56
14.184
56
i4,24o
55
14,295
55
i4,35c
55
i4,4o5
54
14,459
55
i4,5i4
54
0,1 84g
0,44
i4,33ç
57
[4,396
58
14,454
57
i4,5ii
56
14,567
57
[4.624
56
1 4,680
57
14,737
55
14,792
56
1 4,848
56
0,1936
0,45
14.661
59
14,72c
59
i4,77f
58
1 4,837
58
14.895
58
[4.953
58
i5,oii
57
1 5,068
57
i5,i25
57
i5,i82
57
0,2025
o.5o
[6,26c
66
16,335
65
16,400
65
i6,465
65
i6,5'3c
65
16,595
64
i6,65g
64
16,723
64
16,787
63
1 6,85c
64
o,25oo
0,55
17.871
72
17,942
73
i8,oi5
71
[8,o8e
72
i8,i5E
71
[8,23Ç
72
i8,3oi
70
18,371
71
18,442
70
i8,5i2
70
o,3o25
0,60
19,46-.
80
19,542
79
19,621
79
19,70c
79
'9.77Ç
78
19.85-
78
■9.93Î
77
20,012
78
20,ogc
77
20,16-
76
o,36oo
0,65
2 1 ,o4(
) 87
21, i3;
86
21,2K
86
2[,3o'
86
21,39
85
2 [,47e
85
2i,56[
84
2 1 ,645
85
2 1 ,73c
83
21,81:
84
0,4325
0,70
22,62c
) 94
22,71^
94
22,8ot
93
22,90
93
22,99-
92
23,o8f:
92
23,178
9-
23,26c
92
23,36i
90
23,45i
91
0,4900
0,75
34,18
I lOI
24,28_
lOI
24,38'
lOI
24,48(
i 100
24,58f
) 100
24,68f
' 99
24.78'
98
34,88'
99
24,98:
98
25,o8c
97
0,5625
0,80
25,73
1 [09
25,84;
109
25,95
108
26,o5(
) 108
26,16-
7 107
26,27^
! [06
26,38<
t [06
26,48f
) 106
26,59:
io5
26,69-
[o5
o,64oo
o,85
27,27(
3I[7
27,38-
1 116
27,50,
i 116
27,61
jii5
27,73,
iii5
37,84c
) \iA
27,96
i iiA
28,07-
7 ii3
28,19c
)ii3
28,30;
112
0,7225
0,90
28,79
[25
28,qi(
3 124
29,o4(
) 12^
29, [6
UM
29,28
3 123
29.4 [<
1123
29,53.
i [21
29,65.
121
29,77-
) 120
29,89'
120
0,8100
0,95
30,29
iiM
3o,42f
ii33
3o,56
l32
30,69
3 i32
30,82
5i3i
3o,95(
1 [3o
3[,o8(
3 [3o
3l,2lf
3 138
3 1, 34.
i 129
31,47;
127
0,9025
1,00
3 [,77
7i43
3 1,92;
1 142
32,06
7 [4l
33,20
3 i4
32,34
1i3c
33,48
3 i3c)
32,62
I [38
3 2, 76c
3i37
32,89
7.37
33,o3^
i35
1 ,0000
,8065
,8192
,8321 1 ,8450
,8581
,8712
1 ,8845
,8978
,9113
1 ,9248
\ . (r -\- r")' or r= + r"= nearly. |
322
323
324
325
326
327
328
329
33o
33i
332
333
334
335
336
337
T
32
32
32
33
33
33
33
33
33
33
33
33
33
34
34
34
2
64
65
65
65
65
65
66
66
66
66
66
67
67
67
67
67
3
97
97
97
98
p8
98
98
99
99
99
100
100
100
101
101
101
A
i2g
129
i3o
i3o
i3o
i3i
i3i
I 32
l32
l32
i33
i33
1 34
1 34
1 34
i35
5
i6[
[63
162
1 63
i63
164
164
i65
i65
166
166
167
167
168
168
169
6
193
194
194
Iq5
196
iq6
197
197
ig8
'99
199
200
200
201
202
202
■7
225
226
227
228
228
229
a3o
23o
23l
233
232
233
234
235
235
236
8
258
258
25g
260
261
262
262
263
264
365
266
266
267
268
269
270
S
290
291
292
293
293
294
295
296
=97
298
299
3oo
3oi
302
302
3o3
TABLE II. — To find the time T; the sum of the radii r-f-»"", nnd the cliord c beinj; given.
of Llio K^kIii r+,
Chonl
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,00
0,07
0,0b
0,09
0,10
0,1 1
0,12
o,i3
o,i4
o,i5
0,16
0,17
0,18
0,19
0,20
0,3 I
0,2 J
0,2 J
0,24
0,2')
0,26
0,37
0,38
0,39
o,3o
o,3i
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,43
0,43
0,44
o,4'J
o,5o
0,55
0,60
o,65
0,70
0,73
0,80
o,85
0,90
o,g5
i,;37
0,l)UO
0,340
o.<i8o
1 ,03 1
i,36i
1,701
3,04 1
3,38i
3,721
3,(161
3,4<u
3,741
4,081
4,421
4.7tJi
5,101
5,440
5,780
6,1 19
6,459
6,798
7,47-
7,816
8,1
8,493
8,832
9
9.-J09
9,848
10,186
10,524
10,862
1 1,300
11,537
11,875
12,212
12,549
12,886
l3,223
13,559
13,896
l4,332
i4,568
14,904
15,239
16,914
i8,583
20,243
2 1 ,897
23,543
25,177
26,802
28,4 1 5
3o,o 1 5
3i,f5oo
33,i6g
18
43
45
46
48
49
5o
5i
53
5.
55
5
63
70
76
83
9'J
9'
io4
III
"9
12-7
1 351
1,3d
Day9 I'Jill
0,000
0,341
o,683
1,024
1,366
1.707
3,049
3,3()0
3,73i
3,073
3,4 1 4
3,7'V",
4,1 '96
4,4 i-
4,778
5,119
5,460
5,801
6,143
6,482
6,823
7,164
7,5o4
7.844
8,184
8,525
8,864
9,204
9.544
9
10,223
10,562
U>,902
1 1,341
ii,58o
1 1,918
13,357
I3,5g5
12,934
13,273
13,609
13,947
i4,285
14,622
14,959
1 5,396
16,977
1 8,652
20,319
2 1 ,980
23,632
25,274
26.906
28,526
3o,i34
3l,727
33,3o4
1,39
Days lilil'.
0,000
0,343
o,685
1,028
1,371
1, 71 3
2,o56
2,399
2,74 1
3,084
3,426
3,-69
4,1 1 1
4,453
4,796
5,i38
5,480
5,833
6,164
6,5o6
6,848
7.19'
7.531
7,873
8,31
8,556
8,897
9,338
9.579
9,930
10,360
10,601
10,941
1 1,282
11,622
1 1,962
13,3o3
I 3 ,64 1
12,981
1 3,320
1 3,659
13,998
14,337
14,676
1 5,01 4
i5,352
17,040
18,721
20,395
33,o63
33,731
35,370
27,009
38,637
30.2 53
3 1,853
33,439
18
1,40
Days |>hl'.
9385 I ,9522
,9661
0,000
0,344
0,688
l,o32
1,376
1.719
2,063
3,407
2,75i
3,095
3,438
3,783
4,126
4,469
4,81 3
5,i56
5,5oo
5,843
6,186
6,529
6,872
7.2i5
7,558
7,901
8,244
8,586
8,929
9.271
9,61 3
9.956
10,398
io,63g
1 0,98 1
11,333
11,664
I2,005
1 3,346
13,687
13,028
i3,36g
13,709
1 4,049
14,389
i4,7'9
15,069
i5,4o8
17,102
18,790
20,47 1
22,145
23,810
25,467
27,11,3
28,748
3o,37f)
3 1,979
1,41
U.iya |dit'.
33,572(133
,9800
0,000
0,345
0,690
i,o35
1,38 1
1,736
2,071
2,4i6
2,761
3,106
3,45 1
3,796
4,i4i
4,48'
4,83o
5,175
5,5i9
5,864
6,208
6,553
6,897
7,241
7,585
7.929
8,373
8,617
8,961
9,3o4
9,648
9.991
10,335
10,678
1 I,031
11,363
11,706
12,049
12,391
13,733
13,075
i3,4i7
13,759
l4,!0O
i4,44i
14,783
i5, 123
1 5,464
17,164
18,859
20,547
22,227
23,899
25,562
27,3l5
28,857
30,487
33,io4
33,7o5
49
5i
52
53
55
56
63
68
75
82
89
96
io3
no
I 33
1,42
Uiiys |ilil'.
0,(J00
0,346
0,693
1 ,0.39
1,385
1,732
2,078
2,434
2,771
3,117
3,463
3,809
4,1 55
4,5oi
4,847
5,ig3
5,539
5,885
6,33o
6,576
6,933
7,367
7,613
7.957
8,3. "
8,648
8,993
9,338
9,683
10,027
10,371
10,716
1 1 ,060
1 1 ,4o4
11,748
12,092
12,435
12,779
1 3,1 22
1 3,465
1 3,808
i4,i5i
14,493
1 4,836
15,178
1 5,520
17.227
18,927
20,633
3 2, 309
23,988
',3i8
1,967
>,6o4
1,238
3,837
25,658 95
27," "
28,
3o
33
33
102
109
116
124
l32
,9941 I 1,0082
0,0025
0,00 36
o,0(vi9
o,of.64
0,0081
0,0100
0,01 3 1
0,0144
0,01(19
0,0196
0,0225
0,0 2 56
0,0289
o,o324
o,o36i
o,o4oo
o,o44 1
o,o484
0,0529
0,0576
0,0625
0,0676
0,0739
0,0784
0,084 1
o,ogoo
0,0961
0,1024
0,1089
0,11 56
0,1235
o, 1 396
0,1 3(19
o,i444
0,l531
0,1600
0,1681
o, 1 764
0,1849
o,ig36
0,3035
0,3 5oo
o,3o25
o,36oo
0,4225
igoo
0,5625
o,64oo
0,7225
0,8100
o,go2 5
1 ,0000
(r -t- r'f or r'+ ,
338
33g
340
341
342
343
344
345
346
347
I
34
34
34
34
34
34
34
35
35
35
3
68
68
68
68
68
69
69
69
69
69
J
lOI
103
I03
102
io3
io3
io3
io4
io4
io4
4
i35
1 36
1 36
1 36
I 37
1 37
1 38
1 38
1 38
■39
5
.69
170
170
171
171
172
172
■73
173
174
6
203
203
2o4
2o5
205
206
206
207
208
208
7
237
237
a38
23q
239
240
241
242
242
243
»
270
371
272
273
274
274
275
276
277
278
9
3o4
3o5
3o6
3o7
3o8
3og
3 10
3,1
3ii
3l2
l'nj|t. |iiirt3 fur lliu sum ol' the Hailii.
I I 2 I 3 1 4 1 5 1 <î I 7 1 « I 9
18 23
18 23
24
28 35
32 40
36 45
4o|5o
24
28
25
25
29
29
36
3o
36
3i
27
32
38
32
38
33
29
34
29
34
3o
35
3i
36
3i
3(i
33
37
32
38
33
3q
34
3q
M
4o
35
4i
35
4i
36
42
37
43
37
43
38
44
38
45
39
46
40
46
4o
47
4i
48
4i
48
42
4q
48
56
54
63
(>o
70
12
i3
14
>4
i5
16
17
18
19
20
21
22
23
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
4o
4i
4i
43
43
44
4o 45
41
42
42
43
TABLE
II.
— To find the time T
the sum of (he radii
r + r".
■ nd the chord e
being giren.
Sum of the Uadii r-\~T". 1
Chord
c.
1,43
1,44
1,45
1,46
1,47
1,48
Duya Idif.
1,49
1,50
1,51
1,52 1
Us)l |dir.
Hay» jilil'.
Days |(lif.
Uh;s |dir.
Days Idif.
Days |dil'.
Days Idif.
Days lilif.
Days
(lit.
0,00
0,000
0,(X)0
0.000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,348
I
0,349
I
o,35o
I
0,35l
I
0,352
2
0,354
I
0,355
1
o,356
1
0,357
1
0,358
2
0,0001
0,0 a
0,695
3
0,698
2
0,700
2
0,702
3
o,7o5
2
0,707
3
0,710
1
0,712
2
0,714
3
0,717
2
0,0004
o,o3
1,043
3
1,046
4
i,o5o
4
i,o54
3
i,o57
4
1,061
3
1 ,064
4
1,068
4
1,072
3
1,075
4
0,0009
o,o4
1,390
5
1,395
5
1, 400
5
i,4o5
5
1,410
4
i,4i4
5
1,419
5
1,424
5
1,429
4
1,433
5
0,0016
o,o5
1,738
6
1,744
6
i,75o
6
1,756
6
1,762
6
1,768
6
1,774
6
1 ,780
6
1,786
6
1.792
6
0,0025
0,06
2,o85
8
2,oq3
7
2,100
7
2,107
7
2,114
7
2,121
8
2,129
7
2,i36
7
2,143
7
2,l5o
■J
o,oo36
0,07
2,433
8
2,441
9
2,45o
8
2,458
9
2,467
8
2,475
8
2,483
9
2,492
8
2,5oo
6
2,5()6
8
0,0049
0,08
2,780
10
2,790
10
2,800
9
2,809
10
2,819
10
2,829
9
2,838
10
2,848
9
2,857
9
2,86ti
10
0,0064
0,09
3,128
1 1
3,139
11
3,i5o
10
3,160
11
3,171
II
3,182
11
3,193
10
3,2o3
3,214
II
3,225
10
0,0081
0,10
3,475
12
3,487
12
3,499
12
3,5ii
12
3,523
12
3,535
12
3,547
12
3,559
12
3,571
12
3.583
12
0,0100
0,1 1
3,822
i4
3,836
i3
3,849
i3
3,862
14
3,876
i3
3,88g
i3
3.902
i3
3,915
i3
3,928
i3
3,941
i3
0,0121
0,12
4,170
i4
4,1 84
i5
4,199
i4
4,2i3
i5
4,228
i4
4,242
14
4^256
i5
4,271
i4
4,285
14
4.399
.4
0,0144
0,1 3
4,5i7
16
4,533
16
4,549
i5
4.564
16
4,58o
i5
4,595
16
4,611
i5
4,626
16
4,642
i5
4.657
.5
0,0169
o,i4
4,864
17
4,881
17
4,898
17
4,91 5
17
4,932
17
4,949
16
4,965
17
4,982
17
4,999
16
5,oi5
17
0,0196
0,1 5
5,211
19
5,23o
i&
5,248
18
5,266
iS
5,284
18
5,3o2
18
5,32c
18
5,338
17
5,355
18
5,373
18
0,0225
0,16
5,558
20
5,578
19
5,597
20
5,617
19
5,636
19
5,655
19
5,674
19
5,693
19
5,712
19
5,73 1
■9
0,02 56
0,17
5,905
21
5,926
21
5,947
20
5,967
21
5,988
20
6,ocjS
20
6,028
21
6,049
20
6,069
20
6,o8g
20
0,0289
0,18
6,252
22
6,274
22
6,296
22
6,3i8
21
6,339
22
6,36 1
21
6,382
22
6,4o4
21
6.425
23
6,447
21
o,o324
o,ig
6,599
23
6,622
23
6,645
23
6,668
23
6,691
23
6,714
23
6,737
22
6,759
23
6,782
22
6,804
23
o,o36i
0,20
6,946
24
6,970
24
6,994
25
7,019
24
7,043
24
7,067
24
7,091
23
7,ii4
24
7,1 38
24
7,162
2 3
o,o4oo
0,21
7.393
25
7,3i8
26
7,344
25
7,369
25
7,394
25
7.419
26
7,445
25
7,470
25
7.495
34
7,5 19
2 5
o,o44i
0,22
7,639
27
7,(366
27
7,693
26
7.719
2-
7,746
26
7.773
26
7,79'^
27
7,825
26
7,85i
26
7,877
26
o,o484
0,23
7,986
2&
8,oi4
28
8,042
27
8,069
28
8,09-
28
8,125
27
8,1 5^
28
8,180
37
8,207
27
8,334
37
0,0529
0,24
8,332
39
8,36i
=9
8,390
=9
8,419
29
8,448
29
8,477
29
8,5ot
29
8,535
28
8,563
28
8,591
29
0,0576
0,35
8,678
3i
8,709
3o
8,739
3o
8.769
3i
8,800
3o
8,83o
29
8,859
3o
8,889
3o
8,919
3o
8,949
29
0,0625
0,26
9.025
3i
9,o56
32
9,088
3i
9,119
35
9,i5i
3i
9,182
3i
9.213
3i
9,344
3i
9.275
3i
9,3o6
3"
0,0676
0,27
9,371
33
9,404
32
9,436
33
9,469
33
g,5o2
32
9,534
32
9^566
33
9,599
32
9,63 1
33
9,663
32
0,0729
0,2«
9.717
34
9,75i
34
9,785
34
9,819
33
9,852
34
9,886
34
9.920
33
9,953
33
9,986
34
10,020
33
0,0784
0,29
10,062
36
10,098
35
io,i33
35
10,168
35
I0,203
35
10,238
35
10,273
34
10,307
35
10,342
34
10,376
35
0,084 I
o,3o
io,4o8
37
10,445
36
10,481
37
io,5i8
36
10,554
36
10,590
36
10,626
36
10,662
35
10.697
36
10,733
35
0,0900
0,3 1
10,754
36
10,792
37
10,829
38
10,867
3?
10,904
38
10,942
37
10,979
37
1 1,016
37
ii;o53
37
1 1 ,090
36
o,fj96i
0,32
1 1 ,099
3q
ii,i38
3g
11,177
39
1 1,216
39
11,255
38
11,293
3g
11,335
38
1 1 ,370
38
1 1 ,408
38
1 1 ,4^16
3fc
0,1024
0,33
11.444
4i
1 1 ,485
4a
11,525
4o
11,565
40
ii,6o5
40
11,645
39
1 1 ,684
4o
11,724
39
11,763
3q
1 1 ,802
39
0,1089
0,34
11,790
4i
ii,83i
42
11,873
4i
11,914
4i
11,955
4i
1 1 ,996
4i
i2,o37
40
12,077
4i
12,118
4o
I2,i58
4i
0,1 1 56
0,35
i2,i35
42
12,177
43
12,220
43
12,263
42
i2,3o5
42
12,347
42
1 2 ,389
42
i2,43i
43
12,473
4i
i2,5i4
42
0,.225
o,36
12,479
44
12,523
44
12,567
44
12,61 1
44
12,655
43
12,698
43
12,74"l
43
12,784
43
12,827
43
1 2 ,870
43
0,1296
0,37
12,824
45
12,869
46
12,915
44
12,959
45
1 3,004
4'.
1 3,049
44
13,093
45
i3,i38
44
I3.I82
44
l3,226
44
0,1369
0,38
13,169
46
i3,2i5
47
13,262
46
i3,3o8
46
i3,354
4fc
i3,4oo
45
1 3,445
46
13,491
45
13.536
45
i3,58i
46
0,1 444
0,39
i3,5i3
48
i3,56i
47
1 3,608
48
1 3,656
47
i3,7o3
4-
i3,75o
47
.3,79-
47
1 3,844
46
13,890
47
13,937
46
0,l52I
o,4o
1 3,857
49
13,906
49
13,955
49
i4,oo4
48
i4,o52
49
i4,ioi
48
i4,i49
48
14,197
48
14,245
47
14,293
48
0,1600
0,4 1
l4,20I
5i
l4,252
5o
14,3C2
49
i4,35i
5c.
i4,4oi
5o
i4,45i
49
i4,5oc
49
14,549
49
14,598
49
14,647
49
0,1681
0,42
r4,545
52
14,597
5i
14,648
5i
14,699
5i
i4,75o
5i
i4-8oi
5o
i4,85i
5i
14,902
5o
14,952
5c
1 5.003
5o
0,1764
0,43
14,889
52
14.941
53
14,994
52
1 5,046
53
i5,oqQ
52
i5,i5i
52
i5.2o3
5i
15.254
53
i5,3o6
5i
1 5,357
5i
0,1849
0,44
I 5,232
54
15,286
54
1 5,340
54
15,394
53
1 5,447
53
i5,5ou
53
1 5,553
53
1 5,606
53
1 5,659
53
i5,7i2
53
0,1936
0,45
15,575
56
1-5,63 1
55
1 5,686
55
1 5,741
54
1 5,795
55
i5,85o
54
15.904
54
15.958
54
16,012
54
16,066
54
0,2025
0,5.,
17,288
62
i7,35o
61
17,411
62
17,473
61
17,534
60
17.594
61
17,655
60
17,715
60
17.775
60
17,835
60
0,25oO
0,55
18,996
68
ig,o64
67
19,1 3 1
68
19,199
67
19,266
67
19,333
67
1 9,400
67
19,467
66
19,533
66
19,590
66
o,3o25
0,60
20,696
75
20,771
74
20,845
74
20,919
74
20,993
73
2 1 ,066
73
21,139
73
21,213
73
2 1,285
72
21,357
73
o,36ûo
0,65
2 2,3go
81
22,471
81
22,552
81
22,632
81
22,7l3
7^)
22.792
8c
22,872
79
22,951
79
23,0 3o
79
23,icq
78
0,4225
0,70
24,076
87
24,i63
88
24,25l
87
24338
87
24,425
8t
24,5ii
86
24,597
86
24,683
85
24,768
85
24,853
85
0,4900
0,75
25,753
94
25,847
94
25,941
94
26,035
93
26,128
9"
26,221
93
26,3i4
92
26,406
92
26,498
92
26.590
9'
o,5635
0,80
27,42f
101
27,521
loi
27,622
101
27,723
100
27,823
lOC
27,923
99
28,022
99
28,121
99
28,220
98
28,3.8
98
0,6400
o,85
29,076
106
29,184
loç
29,293
107
29,400
107
29,507
10-
29.61.;
10É
39,720
lot
29,826
106
29.932
.05
3o,o37
io4
0,7335
0,90
3o,72c
lit
3o,83b
lit
3o,q52
ii4
3 1,066
ii5
3i,i8i
IK
3 1^294
114
3 1,408
112
3i,52c
ii3
31.633
1 12
3 1,745
1 1 1
0,8100
0,95
32,352
12C
32,475
12C
32,598
122
32,72(
121
32,841
12
32.962
121
3 3. 08 3
12c
33,2o3
119
33,322
"9
33.44.
no
0.9035
1,00
33,96c
.3
34,10c
i3c
34,23o
i3ol34,36o
12c
34,489
12f
34,617
126
34,745
128
34,873
136
34,999
136
35,1251.36
1 ,0000
1,0225
1 ,0368
1,0513 1 1,0658
1 ,0805
1 ,0952
1,1101 1 1,1250
1,1401
1,1552
(2
i ■
(r + r-f
or j'^ -j- r"'^ nearly. |
343
344
345
346
347
348
349
35o
35.
352
353
354
355
356
357
358
35y
I
34
U
35
35
35
35
35
35
35
35
35
35
36
36
36
36
36
2
6q
69
69
69
69
70
70
70
70
70
71
71
71
7'
71
72
73
3
io3
io3
1 04
104
io4
104
io5
io5
io5
106
106
106
107
107
107
107
108
4
.37
i38
i38
i38
i3q
.3q
i4o
i4o
i4o
i4i
i4.
142
l42
142
143
143
144
5
173
172
173
173
174
174
175
175
176
176
177
177
178
178
179
179
180
6
206
206
207
208
208
309
309
310
211
211
2.2
212
3l3
ai4
214
2.5
2l5
7
340
241
245
242
--4J
■i44
■i44
245
246
246
247
248
249
249
2 5o
35l
35l
8
274
275
276
277
278
278
279
280
281
281
282
283
284
285
386
286
387
9
309
3io
3ii
3ii
3l2
3i3
3i4
3i5
3i6
3l7
3.8
319
320
320
321
323
323
TABLE II. — To find the time T\ the sum of the radii r-j- r", and the chord c being given.
Suit! ol' the Riulu r-|-7-'.
CliorJ
c.
0,00
0,01
0,1) a
o,o3
o,u4
o,()5
0,06
0,07
o,oS
o,og
0,10
0,1 i
0,1 3
0,1 3
0,1 4
0,1 5
0,16
0,17
0,18
0,19
0,20
0,2 1
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,75
0,80
o,85
o,go
0,91
1,00
1,53
l)u>« l.lit.
0,00(1
o,36o
0,719
1,079
1,438
1,798
2,ID7
2,5l()
2,876
3,235
3.595
3.g'j4
4.3i3
4,673
5,o32
5,391
5,75u
6, 1 09
(i,466
6,83
7,i8
7.544
7,903
8,361
8,620
8,978
9,336
io,o53
io,4i I
10,768
1 1,126
11,484
ii,84i
12,199
12,556
12,913
13,370
13,627
13,983
! 4,340
14,696
i5,o53
1 5,408
15,764
16,120
17,895
19,665
21,439
23,187
34,938
36,681
28,416
3o,i4i
3 1, 856
19
1,54
Days |ilif.
33,56c) 118
35,25i|i35
0,000
o,36i
0,73 1
1 ,082
1,443
1 ,8o3
2,164
2,52
2,885
3,246
3,606
3,y6-
4,J3-
4.688
5,048
5,408
5.769
6,129
6,489
6,849
7,309
7,569
7.929
8.388
8,648
9,008
9,367
9,736
10,086
10,445
io,8o4
1 1 , 1 63
11,522
11,1
12,239
12,597
I2.g55
i3,3i4
1 3,671
14,029
14,387
14,745
l5.I03
15,459
i5,8i6
16,173
17.954
19,731
2I,50I
23,365
35,023
36.773
28,5i3
30.345
31,96-
33,678
35,376
1,1705
23
34
25
27
38
il
32
33
34
35
36
3
39
40
4i
43
43
45
46
47
48
49
5i
52
53
60
6:
72
78
0,000
o,363
0,724
1 ,086
1,44
1,809
2,171
2,533
2,895
3,256
3,618
3,980
4,341
4,703
5,064
5,436
5,787
6,149
6.5io
6,871
7.233
7,593
7.954
8,3i5
8,676
9,o37
9,398
9,758
10,1 19
10,479
10,839
11,199
ii,55q
'i.9'9
12,279
12,638
12,998
1 3,357
i3,7i6
14,075
14,434
14.793
i5,i5i
i5,5io
1 5,868 53
1,1858
16,226
i8,oi4
19,796
21,573
33,343
35,107
36,863
38;6io
3o,349
32.078
33,795
35,5oi
1,56
Days |<lir.
0,000
o,3(vi
0,726
1 ,089
1,452
1,81 5
2,178
2,541
3 ,904
3,267
3,63(
3,993
4,355
4.718
5,o8i
5,443
5,806
6,169
6,53 1
6,893
7,2 50
7,618
7,980
8,342
8,704
g,o66
9,428
9.790
io,i5i
io,5i3
10,874
11,23
1 1 .597
ii,g58
I2,3ig 3y
12,679
i3,o4o
i3,4oi
13,761
l4,I2I
14.48 1
i4,84i
1 5,201
i5,56o
15,920
16,27g
18,073
19,861
21,644
33,43
35,191
90 26,953
97 38,707
" 3o,452
32,1
33,gi3
35,635
124
1,2013
1,57
Uiiys IJlf.
0,000
o,364
0,738
1 ,093
1,831
3,i85
2,549
2,913
3.-^77
3,64 1
4,oo5
4.369
4.733
5,097
5,461
5,835
6,188
6,552
6,916
7.'79
7,643
8,006
8,369
8,732
9.095
9,458
9,821
10,184
10,547
10,909
11.273
11,634
1 1 ,996
12,358
12,720
1 3,083
1 3,444
i3,8o5
46 14,167
1,2168
14,538
i4.i .
i5,25o
1 5,611
15,971
16,332
i8,i3i
19,926
2 1 ,7 1 5
23,4g8
25,274
27,043
38,804
3o,555
32,297
34,039
35,748
18
3(J
31
22
23
35
35
3
28
29
3o
33
33
33
35
36
3'
39
4o
4i
43
43
45
45
4-
48
49
5o
53
53
59
65
71
77
83
90
9O
io3
110
116
123
1,58
Daya Idif.
1 ,2325
0,00c
o,365
0,73 1
I ,og6
1,461
1,827
2,193
2,557
3,923
3,288
3,653
4,018
4.383
4,748
5,ii3
5,47
5,843
6,208
6,573
6,938
7,3o2
7.66-
8,o3i
8,396
8,760
9,124
9,488
9,85-
10,317
io,58o
io,g44
ii,3o8
1 1 ,67 1
i3,o35
12,398
13,761
13,124
i3,48
i3,85o
l4,213
14,575
i4,g37
i5,3gg
1 5,661
16,033
i6,385
18,190
19,991
31,786
33,575
35,357
27,i33
38,900
3o,658
33,4o'
34,145
35,871
0,0000
0,0001
0,0004
0,0009
0,0016
0,0035
,oo36
0,0049
0,0064
Hi
ft, 01 00
0,01 2 1
0,01 44
0,01 6g
0,0196
e),0225
0,0 2 56
0,038g
o,o334
o,o3ôi
o,o4oo
0,044 1
0,0484
o,o53g
0,0576
0,0625
0,0676
0,072g
0,0784
0,084 1
o,ogoo
o,og6i
0,1024
0,1 o8g
0,1 156
0,1225
0,1 2g6
0,1 36g
0,1 444
0,1 521
0,1600
0,1681
0,1764
0,1 84g
0,1936
0,3025
o,25oo
0,3o25
o,36oo
0,4235
0,4900
0,5625
0,64*00
0,7335
0,8100
0,9025
10,000
1,2482
( r -(- r " )^ or r' -f- r ' ' neatly.
I'l'ip. (titrtnlur the sum of ihe Kiidii.
1 I 3 I 3 1 4 I 5 1 6 I 7 I 8 I 9
356
357
358
36
36
36
71
71
72
107
107
107
142
143
i43
178
'79
179
2l4
2l4
2l5
249
25o
231
285
286
286
320
331
332
359
36
72
108
1 44
:8o
2l5
25l
287
323
36o
36
73
108
144
180
216
252
388
334
36 1
36
72
108
1 44
181
217
253
289
335
362
36
72
109
145
181
217
253
290
326
363
36
73
log
i45
182
218
254
290
327
364
365
366
36
37
37
73
73
73
109
no
no
1 46
146
i46
183
i83
i83
218
219
220
255
256
256
291
392
293
338
32g
339
i3
i4
i5
16
17
18
19
20
21
22
23
24
25
26
27
28
29
3o
3i
33
33
34
35
36
37
38
39
40
4i
42
43
AA
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
100
1 1
1 1 16
17
17
17
12 17
12 18
i5 30
i5| 20
16
16
23
24
25
26
26
27
28
29
3o
3o
3i
32
33
34
34
35
36
37
38
38
3g
4o
4i
42
42
43
AA
45
46
46
47
48
49
5^
23
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
40
4r
4i
42
43
AA
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
6i
62
63
72
81
90
TAPLE
11.
— Tofi
11(1 the time
T;
the sum 0
f tlie ra
ciii >
4- r", and
the chc
rd c
being given.
Sum ut" llie Radii r-\-r '. |
Chord
C.
1,59
1,60
1,61
1,62
1,63
1,64
1,65
Days Iclir.
1,66
Days [dit'.
1,67
1,68
Days |dir.
bays |ilil'.
Uays |dlt'.
Days lilil'.
Days |.lir.
Duy.-î |.]ir.
Days |4ir.
Day.s |dir.
0,00
0,000
0,000
0,000
0,000
o,00ù
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,367
I
0,368
I
0,369
1
0,370
1
0,371
I
0,372
1
0,373
3
0,375
1
0,376
1
0,377
1
0,0001
0,02
0,733
2
0,735
3
o,738
3
0,740
2
0,742
2
0,744
3
0,747
2
0,749
2
o,75i
2
0,753
3
0,0004
o,o3
1,100
3
i,io3
3
1,106
4
1,110
3
i,ii3
4
1,117
3
1,120
3
1,123
4
1,127
3
1 , 1 3o
4
0,0009
0,04
1,466
5
^At
4
1,475
5
1, 480
4
1,484
5
1,489
4
1,493
5
1,498
4
I,5o2
5
i,5o7
4
0,0016
o,o5
1,832
6
1,838
6
1,844
6
1 ,85o
5
1,855
6
1,861
6
1,867
5
1,873
6
1,878
6
1,884
5
0,0025
0,06
2,199
7
2,206
7
2,213
7
2,220
6
2,226
7
2,233
7
3,240
7
2,247
7
2,254
6
2,260
7
o,oo36
0,07
2,565
8
2,573
8
2,58i
8
2,589
8
2,597
8
2,6o5
8
2,6i3
8
2,621
8
2.629
8
3,637
J.
0,0049
0,08
2.932
9
2.941
9
2.950
9
2,959
9
2.968
10
2,978
9
2,987
9
2-996
9
3,oo5
9
3,014
9
0,0064
0,09
3,298
11
3,309
10
3,319
xo
3,329
10
3;339
II
3,35o
10
3,36o
10
3,370
10
3,38o
10
3,390
10
0,008 1
0,10
3,665
11
3,676
12
3,688
1 1
3,699
11
3,710
12
3,732
II
3,733
1!
3,744
12
3,756
11
3,767
1 1
0,0100
0,11
4,o3i
12
4,043
i3
4,o56
x3
4,069
12
4,081
i3
4,094
12
4,106
i3
4,119
12
4,i3i
12
4,143
i3
0,01 21
0,12
4,397
i4
4,411
i4
4,425
i3
4,438
i4
4.452
i4
4.466
i3
4,479
i4
4,493
13
4,5o6
i4
4,520
i3
0,01 44
0,1 3
4,763
i5
4,778
i5
4,793
i5
4,808
i5
4,823
i5
4,838
14
4,852
i5
4,867
i5
4,882
i4
4,896
i5
0,0169
o,i4
5,129
17
5, 1 46
16
5,162
16
5,178
16
5,194
16
5,210
16
5,226
i5
5,241
16
5,257
16
5,273
16
0,01 96
o,i5
5,496
17
5.5i3
17
5,53o
17
5,547
17
5,564
17
5.58 1
17
5,598
17
5,6i5
17
5,632
17
5,649
17
0,0225
0,16
5,862
18
5,880
19
5,899
18
5,917
18
5,935
18
5;953
18
5,971
19
5,990
18
6,008
18
6,026
ifc
0,02 56
0,17
6.228
'9
6,247
20
6,267
■9
6,286
20
6,3o6
19
6,335
19
6,344
20
6.364
19
6.383
19
6,402
■9
0,0389
0,18
6,594
20
6,614
21
, 6,635
31
6,656
20
6,676
21
6,697
20
6,717
21
6;738
20
6,758
20
6,778
2C
o,o334
0,19
6,960
21
6,981
22
7,oo3
22
7.025
22
7.047
21
7,068
32
7,090
21
7,111
23
7,.33
21
7,1 54
22
o,o36i
0,20
7,325
23
7,348
23
7,37.
23
7,394
33
7.417
33
7,440
23
7.463
22
7,485
23
7.508
22
7,53o
23
o,o4oo
0,21
7.691
24
7,7 > 5
24
7.739
25
7,764
24
7,788
23
7,811
24
7,835
34
7,859
24
7,883
23
7.906
24
0,044 1
0,22
8,057
25
8,082
2 5
8,107
26
8,i33
25
8,1 58
25
8,i83
25
8,208
35
8,233
3 5
8,258
24
8.283
25
0,0484
0,23
8.422
27
8,449
36
8,475
27
8,5o2
26
8,528
26
8,554
36
8,58o
26
8,606
26
8,632
26
8,658
26
0,0529
0,24
8,788
28
8,816
27
8,843
28
8,871
27
8,898
27
8,925
28
8,953
27
8,980
27
9.007
27
9,o34
27
0,0576
0,25
9.153
29
9,182
29
9,2 1 1
29
9.240
28
9,268
29
9.297
28
9,32 5
28
9,353
29
9,382
38
9.410
28
0,0625
0,26
9^519
3o
9-549
3u
9.579
29
9,608
3o
9,638
3o
9,668
29
9.697
3o
9-727
29
9,756
29
9.785
3o
0,0676
0,27
9,884
3i
9-9 '5
3i
9,946
3i
9-977
3i
10,008
3i
io,o3g
3o
10,069
3i
io,ioo
3i
io,i3i
3o
10,161
3o
0,0739
0,28
10,249
32
10,281
33
10,3 [4
33
10,346
32
10,378
32
io,4io
33
10,442
3i
10,473
32
io,5o5
3i
10,536
33
0,0784
0,29
io,6i4
34
10,648
33
10,681
33
10,714
33
10,747
33
1 0,780
33
10,8 1 3
33
10,846
33
10,879
33
10.912
32
0,084 1
o,3o
10,979
35
ii,oi4
34
1 1 ,048
35
1 1 ,08 3
34
11,1 17
34
ii,i5i
34
ii,i85
34
11,219
34
11,253
34
1 1,387
34
0,0900
0,3 1
11,344
36
ii,38o
35
ii,4i5
36
11.451
35
11,486
36
11,522
35
11,557
35
11,592
35
11,627
35
1 1 ,662
35
0,0961
0,32
1 1 ,708
37
11,745
37
11,782
37
1 1 ,8 1 9
37
11,856
36
11,892
37
1 1 ,929
36
11,965
36
12.001
36
i2,o37
36
0,1034
0,33
12,073
38
12,1 1 1
38
12,i4q
38
12.187
38
12,225
38
12,263
37
i2,3oo
38
12,338
37
I2;375
37
12,412
3-
0,1089
0,34
12,437
4o
12,477
39
X2,5i6
39
12,555
39
12,594
39
12,633
39
12,672
38
12,710
39
12,749
38
12,787
38
0,11 56
0,35
12,802
4o
12.842
4i
12,883
4o
[2,923
4o
12,963
40
i3,oo3
4o
1 3,043
40
i3,o83
39
l3,122
40
i3,i62
39
0,1225
o,36
i3,i66
42
1 3.208
4i
13,249
42
13,291
4i
i3,332
4i
13,373
41
i3,4i4
4i
i3,455
4i
13,496
4o
i3,536
4i
0,1296
0,37
i3,53o
43
13.573
43
i3,6i6
42
1 3,658
43
i3,70i
42
13.743
42
15.785
42
13,827
42
13.869
42
13,91 1
43
0,1369
o,38
13.894
M
1 3,938
44
13,982
44
14.026
43
14,069
44
i4,ii3
43
i4,i56
43
14.199
43
14,242
43
i4,285
43
0,1444
0,39
i4,258
45
i4,3o3
45
14,348
45
14.393
45
i4,438
44
14,482
45
14,527
44
14,571
44
i4,6i5
45
1 4,660
44
0,1 52 I
o,4o
14.621
47
1 4,668
46
14.714
46
14,760
46
i4,8o6
46
i4,852
46
14,898
45
14,943
45
14.988
46
1 5.0 34
45
0,1600
0,4 1
14.985
47
i5.o33
46
1 5,080
47
l5,127
47
i5,i74
47
l5,221
47
1 5,368
47
i5,3i5
46
i5,36i
47
1 5.408
46
0,1681
0,42
1 5.34s
49
15,397
48
1 5,445
49
15,494
48
1 5,542
48
15,590
48
15.638
48
1 5,686
48
15,734
47
15,781
48
0,1764
0,43
15.711
5o
15,761
5o
1 5,8 11
5o
1 5,861
49
15,910
49
15,959
5o
16,009
49
i6,o58
48
16,106
49
i6,i55
49
0,1849
0,44
16,074
5i
16,125
5i
16,176
5i
16,227
5i
16,278
5o
16,328
5i
16,379
DO
16,429
5o
16,479
5o
16,529
49
0,1936
0,45
16,437
52
16,489
53
16,542
52
16,594
5i
16,645
52
16,697
5i
16,748
52
16,800
5i
i6,85i
5i
16.902
5]
0,2025
o,5o
18,249
58
i8,3o7
58
18, 365
58
18,423
58
18,481
57
i8,538
57
18.595
58
1 8,653
57
18,710
56
18,766
57
o,2 5oo
0,55
20.o5"5
65
20,120
64
20,184
63
20,247
64
20,3ll
64
20,375
63
30,438
63
2o.5oi
63
20,564
63
20,626
63
o,3o25
0,60
21,856
71
21,937
7C
21.997
70
22.067
69
22,l36
70
22,206
69
22,275
69
22,344
69
22,4l3
68
22,481
69
o,36oo
o,65
23,652
76
23.728
76
23,8o4
76
23,880
76
2 3,956
76
24,o32
75
24,107
75
24,182
74
24,256
75
24,33 1
74
0,4225
0,70
25,44o
83
25^523
82
25,6o5
83
25,688
81
25,769
82
25,85i
81
25,932
81
36,oi3
81
26,094
80
26,174
81
0,4900
0,75
27,222
89
27,3ii
86
27,399
89
27,488
88
27,576
88
27,664
87
27,751
87
27,838
87
37,925
86
28,011
87
o,5625
0,80
28,995
gt
29,091
95
29,186
94
29,280
95
29.375
94
39,469
93
39,562
94
29,656
93
29.749
92
39,841
93
o,64oo
o,85
30,760
103
30.862
101
30.963
101
3 1,064
101
3i,i65
100
3 1,265
100
3 1, 365
100
3 1, 465
99
3 1,564
99
3 1, 663
99
0,7225
0,90
32,5i5
109
32,624
108
32,732
107
33,839
108
32,947
106
33,o53
107
33.160
106
33,266
io5
33,371
106
33,477
io5
0,8100
o,g5
34.260
1x5
34,375
1x5
34,490
114
34.604
114
34,718
114
34,833
112
34,944
ii3
35,o57
112
35,169
113
35,281
III
0,9025
1 ,00
35,994
122
36,1 16
121
36,2 37
121
36,358
121
36,479
120
36,599
120
36.719
119
36,838
'■9
36,957
118
37,075
118
1 ,0000
1,2641 1 1,2800
1,2961
1,3122
1,3285
1,3448
1,3613 1
1,3778 1
1 ,3945 1
1,41 12 1
«2
i . (r + r")- or 7-'^ 4- r " ^ nearly. |
I
I
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
36
36
37
37
37
37
37
37
37
37
37
37
38
38
38
73
73
73
73
73
74
74
74
74
74
75
75
75
75
75
109
109
no
no
1 10
no
1 11
in
III
112
112
112
ii3
ii3
n3
145
1 46
1 46
146
1 47
1 47
1 48
1 48
1 48
1 49
149
i5o
i5o
i5o
i5i
182
182
i83
1 83
1 84
184
i85
i85
186
186
187
1S7
188
188
189
218
218
219
220
230
321
231
222
223
223
224
224
225
226
226
254
255
2 56
256
257
2.58
258
259
260
260
261
262
263
263
264
200
291
392
2q3
294
294
2q5
396
297
298
298
?99
3oo
3oi
3o2
327
328
329
329
33o
33 1
332
333
334
335
336
337
338
338
339
TAIÎI.V", II. — To fiiul the time T\ the sum of the radii r-|->' "> nnil the chonl c lieins given.
.'"uin «r llnJ Kiulil r-f r".
Chord
c
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,10
0,1 1
0,12
o,i3
,i4
o,i5
0,16
0,17
0,1b
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
o,3a
0,33
0,34
0,35
0,36
0,37
o,38
0,39
o,4o
0,4 1
Oy43
0,44
0,45
o,5o
0,55
0,60
0,65
0,70
0,75
o,bo
0,85
0,90
0,95
1,00
1,69
Dii)s |.lil.
O.OOU
0.3-8
o,756
i.i34
i,5ii
2,267
2,645
3,023
3,400
3,7
4,1 56
4,533
4,91 1
5,289
5,666
6,044
6,421
6,-98
7.176
7,553
7,930
8,307
8,684
9,061
9,438
9,81 5
0,191
o,56fc
0,944
1,321
1,697
2,073
2^449
2,825
3,20 !
3,577
3,953
4,328
4,704
5,079
5,454
5,829
6,204
6,5-8
6,953
8,b23
20,689
22,5 5o
24,4o5
26,255
28,098
29,934
31,762
33,582
35,392
37,193
23
1t
1,4281
1,70
lliiys |ilil'.
0,CKX>
0,379
0,75b
!,l3
i,5i6
1,895
2,274
2,653
3,o3
3,4 10
3,789
4,16b
4,54-
4,935
5,3o4
5,683
6,061
6,440
6,818
7,197
7,575
7,953
8,332
8,710
9,088
9.466
9,844
0,222
o,5ç9
o.y77
1.354
1,732
2,10g
2,4!-'6
2,864
3,241
3,617
3,994
4,371
4,747
5,124
5,5oo
5,876
6,252
6,628
7,004
8,879
20,75 1
22,618
34,479
26,335
28,1
30,026
3 1,860
33,686
35,5o3
37,3io
1,4450
1,71
U.iys |dif.
0,000
o,38o
0,-60
1 , 1 4o
i,5';o
1 .900
2,280
2,6()0
3,o4o
3,420
3,800
4,lb0
4,56o
4,940
5,320
5,700
6,079
6,459
6,83b
7,21b
7.597
7.97
8,356
8,735
9.115
9.494
9,873
0,252
o,63i
1,009
1,388
1,76'
2,145
2,523
2 ,909
3,280
3,658
4,o36
441 3
4,79'
5,1 6g
5,546
5,923
6,3oo
6,677
7,o54
8,936
20,81 3
23,686
24,553
26,414
28,269
3o,ii7
3 1,958
33,790
35,6i4
37,427
23
23
25
26
56
28
39
3c
3i
33
33
36
3
3fc
3o
4c
4i
43
AA
44
46
47
46
49
5o
56
63
67
73
80
86
9'
98
io4
110
"7
1,72
Itiiy.: |dir
0.000
0^381
0,-762
i,i44
1,525
1,906
2,28
2,66b
3,049
3,43o
3,811
4.192
4,573
4,954
5,335
5,716
6,097
6,478
6,858
7.339
7.620
8,000
8,38i
8,761
9,i4i
9,522
9«o2
0,282
0,662
1,042
1,421
1,801
2,181
2,56o
2,940
3,3r9
3,698
4,077
4,456
4,835
5,21 3
5,593
5,970
6,348
6,726
7,104
8.993
20,875
22,753
24,626
36,494
28,355
3o,2C9
32,o56
33,894
35,724
37,544
39
3(,
3i
33
34
35
3
3
37
39
4i
41
45
43
45
45
47
4t
5c-
5i
56
62
68
74
76
85
9'
97
104
ic.g
116
1,46211 1,4792
0,000
0,382
0,765
i,i47
1.539
1.911
3,394
3.676
3,o58
3^440
3,833
4,3o5
4,587
4,969
5,35i
5.733
6,ii5
6,497
6,878
7,360
7,642
8,023
8,4o5
8,787
9,16b
9.931
0,3l2
0,693
1 ,074
1,455
1,836
2,216
2,597
3.977
3,358
3,738
4,11
4,498
4.878
5,2
5,637
6.017
6,396
6,776
7,i55
9,048
20,937
22,821
24,700
26,573
38/i4o
3o,3oo
33,i53
33,998
35,833
37,660
46
4
48
49
5o
55
61
67
73
79
85
91
97
io3
o
116
1,74
Duys |.lil.
O,O0('
0.383
0,767
I.lSo
1,534
1,917
2,3oo
3,684
3,067
3,45o
3,834
4,317
4.60U
4x(S3
5,366
5,749
6,i32
6,5 1 5
6,898
7.381
7.664
8,047
8,439
8,812
9.195
9.577
9.959
0,34
0,724
1,106
1,488
1,870
2,25?
2,633
3,oi5
3,397
3,778
4,i59
4,540
4,931
5,3o2
5,683
6,064
6,444
6,83
7,2o5
9,io3
30,998
22,888
34i773
26,653
28,525
30,391
32,25o
34,101
35,943
37.776
0,OOC)0
0,000 1
o,ooo4
0,00c 9
0,0016
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,01 21
0,01 44
0,0169
0,0196
0,0225
0,02 56
0,0289
o,o324
o,o36i
o,o4oo
o,o44 1
0,0481
0,0539
0,0576
3t 0,0635
0,0676
1,49651 1,5138
0.0739
0,0784
0,084 1
o,ogoo
0,0961
0,1024
0,1089
0,1 156
0,1225
0,1296
o,i3fc9
0,1444
0,1 521
0,1600
0,1681
0,1764
0,1849
0,1936
0,2025
0,2 5oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
0,6400
0,7225
lioo
0,9035
c"
i-
r + r'
)^ or T
'-fr '■ =
nrarly
374
375
376
377
378
379
38o
38 1
382
383
384
37
38
38
38
38
38
38
38
38
38
38
75
75
75
75
76
76
76
76
76
77
77
112
ii3
ii3
ii3
Ii3
ii4
114
ii4
ii5
lib
ii5
i5o
i5o
i5o
i5i
i5i
l52
l52
l52
i53
i53
i54
,87
188
188
i8q
189
190
190
iqi
191
192
192
324
225
236
236
227
227
228
339
229
23o
23o
262
363
263
264
265
265
266
367
267
268
269
309
Joo
3oi
3o2
3o2
3o3
3o4
3o5
3o6
3o6
3o7
337
338
338
339
340
34i
342
343
344
345
346
l'rwp. piirls Inr tin- sum ol' tlic Ituilii.
■ I 3 I 3 I 4 I 5 I 6 I 7|8|9
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
100
25 3o
43 4q
56
54 63
60 70
35
28
35
39
26
3o
37
3o
37
3i
38
32
2q
33
39
34
3o
M
3i
35
23
23
34
35
26
37
28
39
3o
3i
32
32
33
34
35
36
37
38
39
40
4i
4i
43
43
44
36
37
38
38
39
40 45
4i
42
43
43
44
45
46
46
47
55 62
90
a8
TABLE II. — To find the time T; the sum of the radii r + t", and the chord c being given.
Sum of the Radii t -\-r".
Chord
c.
0,00
0,01
0,02
o,o3
0,04
o,o5
0,06
0,07
0,08
0,09
0,10
0,1 !
0,1 2
0,1 3
o,i4
0,1 5
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
0,36
0,37
o,38
0,39
o,4o
0,41
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
0,61
0,70
0,75
0,80
o,85
0,90
0,95
1,00
1,75
Days |dif.
0,000
0,385
0,769
1, 1 54
1,538
1,922
2,307
2,691
3,076
3,460
3,845
4,229
4,6i3
4,997
5,383
5,766
6,i5o
6,534
6,918
7,3o2
7,686
8,070
8,454
8,837
9,221
9,6o5
9,988
0,371
0,755
i,i38
1,521
1,904
2,287
2,670
3,o53
3,435
3,81
4,200
4,582
4:965
5,347
5,728
6,110
6,492
6,873
7,255
9' 1 59
21,059
22.95
24,846
26,730
28,609
3o,48?
32,346
34,2o3
36,o53
37,891
1,76
Days |dif.
1,5313
0,000
o,386
0,771
1, 1 57
1,542
1,928
2,3i4
2,699
3,o85
3,470
3,856
4,241
4,626
5,012
5,397
5,782
6,168
6,553
6,938
7,323
7,708
8,og3
8,478
8,863
9>247
9,632
10,017
io,4oi
10,786
11,170
11,554
11,938
12,322
12,706
13,090
i3,47
i3,857
i4,24i
14,624
1 5,008
15,391
i5,774
i6,i57
16,539
16,922
i7,3o4
19,214
21,120
23,022
24,918
26,809
28,69.
30,572
32,443
34,3o6
36, 160
38,oo6
1,77
Days [dit'.
1 ,5488
0,000
0,387
0,773
1,160
1,547
1,933
2,320
2,707
3,093
3,480
3,866
4,253
4,64o
5,026
5,412
5.799
6,i85
6,571
6,958
7,344
7,730
8,116
8,5o2
8,888
9,274
9'659
0,045
0,43 1
0,816
1,202
1,587
1,972
2,357
2,743
3,127
3,5i2
3,897
4,282
4,666
5,o5i
5,435
5,819
6.2o3
6,587
6,970
7,354
9,270
21,181
23,088
24,991
26,887
28,778
30,662
32,539
34,4oS
36,269
38,121
1,78
Days |ilif.
1,5665
0,000
o,388
0,776
i,i63
i,55i
1,939
2,327
2,714
3,102
3,490
3,877
4,265
4,653
5,o4o
5,428
16
5,81 5
18
6,2o3
19
6,590
19
6,977
21
7,365
22
7,752
23
8,i3q
24
8,526
25
8,QI.
26
9,3oo
9,687
0,074
0,460
0,847
1.233
1,620
2,006
2,393
2,779
3,i65
3,55i
3,937
4,322
4,708
5,093
5,479
5,864
6.249
6,634
7,oJ9
7,4o4
9,325
21,242
23,1 55
25,o63
26,965
28,862
3o,75i
32.634
34,5io
36,377
38,235
1,79
Days Idif.
1,5842
0,000
0,389
0,778
1,167
1,555
1,944
2,333
2,722
3,111
3,5oo
3,8
4,277
4,666
5,o54
5,443
5,83 1
6,220
6,608
6,997
7,385
7,774
8,162
8,55o
8,938
9,326
9,7'4
10,102
10,490
10,877
11,265
11,653
12,o4o
12,427
I2,8i5
1 3,202
1 3,589
13,976
i4,363
14,749
i5,i36
l5,522
1 5,909
16,395
16,68
17,067
17,453
19,380
2i,3o3
23,22!
25,i35
27,043
28,945
3o,f
32,73o
34,611
36,484
38,348
49
55
60
66
71
7
83
89
95
101
107
114
1 ,6021
1,80
Days Idif.
0,000
0,390
0,780
1,170
l,56o
i,95o
2,34o
2,73o
3,119
3,509
3,899
4,28g
4,679
5,068
5,458
5,848
6,237
6,627
7,016
7,406
7,795
8,i85
8,574
8,963
9,352
9,74 1
io,i3o
10,519
10,908
11,297
11,685
12,074
12,462
i2,85i
13,239
13,627
i4,oi5
i4,4o3
14,791
i5,i79
1 5,566
15,954
16,341
16,728
I7,ii5
i7,5o2
19,435
21,363
23,287
25,206
27,120
29,028
30,930
32,825
34,712
36.591
38,462
1 ,6200
1,81
Days |dif.
0,000
0,391
0,782
1,173
1,564
1,955
2,346
2,737
3,128
3,519
3,910
4,3oi
4,692
5,082
5,473
5,864
6,255
6,645
7,o36
7^26
7,817
8,207
8,598
8,988
9,378
9,768
0,1 58
0,548
0,938
1,328
1.718
2,108
2,497
2,887
3,276
3,665
4,o5.
4,443
4,832
5,221
5,610
5,998
6,387
6,775
7,i63
7,55i
9,489
21,423
23,3
25,278
27,198
29,111
31,019
33.920
34,81 3
36,698
38,575
1,6381
1,82
Days |dif.
0,000
0,392
0,784
1,176
1,568
1,961
2,353
2,745
3,i37
3,529
3,921
4,3i3
4,7o5
5,097
5,488
5,880
6,272
6,664
7,o55
7,447
7,839
8,23o
8,621
9,01 3
9,4o4
9,795
0,187
0,578
0,969
i,36o
i,75o
2,l4l
2,532
2,922
3,3i3
3, 703
4,093
4.4«3
4,873
5,263
5,653
6,043
6,432
6,822
7,211
7,600
9,544
21,483
23,419
25,349
27,275
29,194
3i,io8
33,01
34,913
36,8o5
38,688
21
23
34
25
26
27
28
29
3o
3i
33
34
34
36
36
38
39
40
42
43
43
44
46
46
48
49
54
60
65
71
76
83
88
94
101
106
1 12
1,6562
1,83
Days Idif.
0,000
0,393
0,786
1,180
1,573
1,966
2,359
2,752
3,145
3,538
3,932
4,325
4,718
5,111
5,5o3
5,896
6,289
6,682
7,075
7,467
7,860
8,253
8,645
9,o38
9,43o
9,822
0,3l5
0,607
0,999
1,391
1,783
3,175
2,566
3,958
3,349
3,74i
4,i33
4,523
4,915
5,3o6
5,696
6,087
6.478
6,868
7,259
7,649
9,598
21,543
23,484
25,420
27,351
29,277
31,196
33,108
35,oi4
36.911
38.800
1,6745
1,84
Days |dif.
0,000
0,394
0,789
I,i83
1,577
1,971
2,366
2,760
3,i54
3,548
3,942
4,336
4,73o
5,124
5,519
5,912
6,3o6
6,700
7,094
7,488
7,5
8,275
8,669
9,062
9,456
9,849
0,243
o,636
1,029
1,422
2,208
2 ,60 1
2,993
3,386
3,779
4,171
4,563
4.956
5,348
5,740
6,i3i
6,523
6,9 1 5
7,3o6
7,698
g,652
2 1 ,6o3
23,549
25,491
27,428
29,359
3i,284
33,202
35,ii3
37,017
38,912
1,6928
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0025
,oo36
0,0049
0,0064
0,0081
0,0100
0,0121
o,oi44
0,0169
0,01 g6
0,0225
,02 56
O,028_
o,o324
o,o36i
o,o4oo
o,o44i
0,0484
0,0529
0,0576
0,0625
0,0676
0,072g
0,0784
0,084 1
o.ogoo
0,0961
0,1024
o,ic
0,1 156
0,1225
0,1296
0,1369
0,1 444
0,l52I
1600
0,1681
1764
0,1849
0,1936
0,2035
o,25oo
o,3o35
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8lCK>
0,9025
1 ,0000
cr
(r -1- r")» or r'' + r"^ nearly.
38 1
382
383
384
385
386
387
388
38g
3go
3gi
392
3g3
3g4
3g5
38
38
38
38
39
3q
39
39
39
39
39
3q
39
3q
40
76
76
77
77
77
77
77
78
78
78
78
78
79
79
79
ii4
ii5
ii5
• ii5
116
116
116
116
117
117
117
118
118
118
"9
162
i53
1 53
1 54
1 54
i54
i55
1 55
i56
1 56
i56
i57
1 57
1 58
1 58
191
191
192
192
193
193
194
194
iq5
iq5
iq6
196
197
197
igS
229
32g
23o
23o
23l
232
232
233
333
334
235
235
236
236
237
267
367
268
260
370
270
271
272
373
273
374
274
275
276
277
3o5
3o6
3o6
307
3o8
309
3io
3io
3ii
3l3
3i3
3i4
3i4
3i5
3i6
343
344
345
346
.347
347
348
349
35o
35i
352
353
354
355
356
TABLE II. — To find the time T; the sum of the radii r-f-r", and the chord c being given.
Sum of the Uailii r-f-r".
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
o,i3
0,1 4
o,r5
0,16
0,17
0,18
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
o,3i
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,75
0.80
o,85
0,90
0,95
1,85
Pays Idii;
0,000
0,395
0,-9 1
J, 186
1, 58 1
1.977
2,3
2,767
3,162
3,558
3,953
4,348
4,743
5, 1 38
5,533
5,959
6,324
6,718
7,Il3
7,5o8
7,903
8,298
8,692
9,08-
9,482
9,876
10,270
io,665
1 1 ,059
11,453
11,847
12.241
12,635
r 3,029
1 3,423
i3,8i6
l4,2IO
i4,6o3
14,996
15,39.:
1 5,783
16,176
i6,568
16.961
17,354
17.746
19,706
21,662
23,6i4
25,562
27,5o4
1,86
Dnya |dir
29 ,44 1
3 1,372
33,296
35,2i3
37,122
39,023
0,000
0,396
0,793
I,l8q
1,586
1,982
2,378
2,775
3,171
3,567
3.964
4,36o
4,756
5,i53
5,548
5,945
6,341
6,737
7,i33
7,529
7,9'4
8,320
8,716
9,112
g,5o7
9,903
10,398
10,694
1 1 ,089
11,484
11,879
12,274
12,669
13,064
13,459
1 3,854
14,248
14,643
i5,o37
i5,43i
15,826
16,230
i6,6i3
17,007
17,401
17,794
19,760
21,722
23,679
35,632
27.58o
1,87
Days lilil'.
1,7113
29,533
31.459
33,389
35,3 12
37,227
i3
i4
i5
16
17
18
19
20
22
23
23
24
26
26
28
28
3c.
3i
33
34
35
36
36
3
39
39
4i
42
42
43
45
46
4-
49
54
59
65
71
76
39, 1 34I 1 1 1
1,7298
0,000
0,39'
0,795
1,192
1,590
1,987
2,385
2,7'
3,1.
3,5
3,974
4,372
4,7(19
5.160
5,563
5,961
6,358
6,755
7,i52
7,549
7,946
8.343
8,739
9,1 36
9,533
9,929
10,326
10,733
11,119
ii,5i5
11,911
1 3 ,3o8
12,704
i3,ioo
13,495
13.89
14,387
14,682
15,078
1 5,473
1 5,868
16,263
i6,658
i7,o53
17,44s
17.843
19,814
21,781
23,744
25,7o3
27,656
39,604
31.547
33.482
35,4ii
37,332
39,245
1,88
Days Itlif.
1 ,7485
0,000
0,399
o,"9'
1 , 1 96
1,594
1,993
2,391
2,790
3,188
3,586
3.9S5
4,383
4.783
5,180
5,578
5,976
6,375
6,773
7,171
7,569
7.967
8,365
8,763
9,161
9,558
9,956
10,354
io,75i
11,149
11,546
11,943
13,341
12,738
i3,i35
i3,532
13.928
i4,325
14,722
i5,ii8
i5,5i5
15,911
i6,3o7
i6,7o3
17,099
17,495
17,891
19,867
21,840
23,809
25,773
37,732
29,686
3i,633
33,575
35,5io
37,437
39,356
1,89
Days |dif.
1,7672
0,000
0,400
o,"99
',"99
1,598
1,998
2,397
2,797
3,197
3,596
3,995
4,395
4,794
5,194
5,593
5,992
6,393
0,791
7,190
7,589
7,988
8,387
8,786
9,i85
9,584
9,983
1 0,38 1
10,780
11,178
11,577
11,975
12,373
12,772
13,170
1 3,568
13,966
1 4,363
14,761
1 5,1 59
1 5,556
1 5,954
i6,35i
16,748
17,145
17,543
17,939
19,931
21,899
23,873
25,843
27,807
_^90^
Daya jdif.
0,000
o,4oi
0,801
1 ,302
i,6o3
2,oo3
2,4o4
2,804
3,2o5
3,606
29,767
3l,720
33,667
35,608
37,541
39,466
4,006
4,407
4,807
5,207 i4
5,608 i5
6,008 16
16
I
19
6,409
6,809
7,209
7,609
8,009
8,409
8,809
9,209
9,609
10,009
10,409
10,808
11,208
1 1 ,608
13,007
I2,4o6
12,806
t3,2o5
i3,6o4
i4,oo3
i4,4o2
1 4,800
15,199
15,598
15,996
16,394
16,793
17,191
17,589
17,987
19,974
21,958
23,937
25,912
27,883
3Q,848
3 1,807
33,76<i
35,706
37,645
39,576
o,ot,oo
0,0001
0,CKK,4
o,o<:)09
0,0016
0,0025
o,oo36
0,0049
0,0064
0,0081
0,01 00
0,01 2 1
0,01 44
0,0169
0,0196
0,0225
0,0256
0,0289
0,0324
o,o36i
o,o4oo
o,o44i
0,0484
0,0529
21
23
24
25
36 0,0576
1,7861 1,8050
0,0625
0,0676
0,0729
0,0784
0,084 1
0,0900
0,0961
0,1094
o.io8f)
o, 1 1 56
0,1225
0,1296
0,1369
o,i444
l52I
0,1600
0,1681
0,1764
0,1849
0,1936
0,3025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5635
o,64oo
0,7335
0,8100
0,9025
1 ,0000
/•a
( r -(- r " J ^ or r^ -f" ^ ' ^ nearly.
392
393
394
395
396
397
398
399
400
4oi
I
3q
3q
3q
4o
40
4o
40
40
40
4o
2
78
"§
79
79
79
79
80
80
80
80
3
118
118
118
"9
119
119
"9
120
120
120
4
1 57
1 57
1 58
1 58
1 58
l5q
l5q
160
160
160
5
196
197
197
iq8
198
199
300
200
201
b
235
236
236
237
238
238
239
239
240
241
7
274
275
276
277
277
278
279
279
280
281
8
3i4
3i4
3i5
3i6
3l7
3i8
3i8
3iq
320
32 1
9
353
354
355
356
356
357
358
359
36o
36 1
Trop, ports for tlie sum of the Itadii.
1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 9
70
80
90
100
7
I 1 17
17
17
18
10
! I l3
1 1 i3
i4
i4
i3 i5
i3 16
14 16
i4
i5
i5
16
16
17
17
18
18
J9
19
20 23
23
25
26
26
3 1 37
2 2 37
3o 4o
35
35
26
36
27
28
28
29
39
3o
3i
3i
32
32
3
34
34
3
35
36
37
37
38
38
39
40
4o
4i
4i
43
48
54
60
13
i3
14
i4
i5
16
17
18
'9
30
21
22
23
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
4o
4i
4i
42
43
39 44
TABLE II. — To find the time T; the sum of the radii r-\-r", and the chord c being given.
Sum of the Radii r-|-'* '•
Cliord
C.
0,00
0,01
0.02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
o,i3
o,i4
o,i5
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
o,4>
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75'
0,80
0,85
0,90
0,95
1 ,00
1,91 I 1,92
1 I i.i- n..,.= i.i;
Days |.lif.
0,000
0.402
o',8o3
I,2o5
1,607
2,008
2,4lO
2,812
3,2 1 3
3,61 5
4,017
4,418
4,820
5,221
5,623
6,024
6,435
6,827
7,228
7,629
8.o3o
8,432
8,833
9,234
9,635
io,o35
10,436
10,837
11,238
11,638
12,039
12,439
12,839
1 3,240
1 3,640
i4,o4o
i4,44o
i4,84o
15,239
1 5,639
16,039 42
i6,438
i6,837
17,236
17,635
Days l.ar.
i8,o34
20,027
22,016
24,001
25.982
27,958
29,928
31,893
33,852
35,804
37,748
39.685
47
0,000
o,4o3
o,8o5
1,208
1,61 1
2,Ol4
3,4i6
2,819
3,222
3,624
4,027
4,43o
4,832
5,235
5,637
6,o4o
6,442
6,845
7,247
7>649
8,o5i
8.454
8,856
9.2 58
9,660
10,062
10,464
io,865
1 1,267
1 1 ,66()
12,070
12,472
12,873
i3,274
13,676
14,077
14,478
14,87g
1 5,379
1 5,680
16,081
16,481
16,882
17,282
17,682
20,080
22,074
24,065
26,05 1
28,032
30,009
31,979
33,943
35,901
37,85!
39,794
23
23
24
25
26
27
29
29
3^
32
32
34
3
35
37
38
39
4o
4i
42
43
44
45
46
47
53
59
64
69
75
80
86
9
97
io3
1,93
Uays |(lir.
0,000
o.4o4
0,808
1,211
1, 61 5
2,019
3,423
2,826
3.23o
3,634
4,o38
4.441
4,845
5,248
5,652
6,o55
6,459
6,862
7,266
7,669
8,072
8,476
8,879
9,282
9,685
10,088
10,491
10,894
1 1,296
1 1 ,699
12,102
i2,5o4
13,907
1 3,309
i3,7i I
i4,ii4
i4,5i6
14,918
1 5,3 19
15,721
16,123
16,524
16,926
17,337
17,728
18,129
20,1 33
22,l33
24,139
26,1 2(
28,107
30,089
3 2, 06 5
34,o35
35,998
37,9^4
109139,903
0,000
o.4o5
0,810
1,21 5
1,619
3,024
2.439
3,834
3.239
3,643
4,048
4.453
4,857
5,262
5,667
6,071
6,476
6,880
7,285
7,689
8,093
8,498
8,902
9,3o6
9,710
10, 1 14
1 0,5 18
10,922
r 1,326
1 1,730
i2,i33
12,537
1 2 .940
1 3, 34.'
1 3,747
i4.i5o
i4,553
14,956
i5,«9
15,762
16,165 42
16,567
16,970
17,372
■7,775
1,8241 I 1,84321 1,8625 | 1,8818
18,177
20,186
22,191
24,192
26.189
28,i8"i
30,169
32,i5o
34,136
36,095
38,o57
4o,oi 1
1,95
Days |dir.
0,000
0,406
0,8 13
1,218
1,634
2.029
2^435
3, 84 1
3,347
3,653
4,o58
4,464
4,870
5,276
5,681
6,087
6,492
6,898
7,3o3
7,709
8,ii4
8,520
8,925
g,33o
9,735
io.i4o
10,545
10.950
11,355
1 1,760
I2,i65
12,569
13,974
1 i,J7-
I 3,783
14,187
14,591
14,995
15,399
i5,8o3
1,96
Days |iiif.
16,207
42
i6.6[o
43
17,014
44
17,417
45
17,821
46
18,224
47
20,2 38
52
23,249
57
24,255
6A
26,258
68
28,256
74
3o,248
80
32,236
85
34,217
90
36,192
96
38,1 59
102
40,120
107
1,90
13
0,000
0,407
0,814
1,231
1,628
2,o35
2,44i
3.848
3.255
3,662
4,069
4,476
4,882
5,289
5,696
6,102
6,509
6,916
7,323
7,729
8,i35
8.541
8,948
9,354
9,760
io,i66
10,573
10,978
11,384
11,790
13,196
I 3, 603
1 3,007
i3,4i3
i3,8i8
l4,333
14,629
i5,o34
15,439
41 1 5,844
16,349
16.653
7,o58
17,463
17,867
18,27
290
22,3o6
24,319
26,326
28,330
3o,328
32,321
34,307
36,2.
38,36i
62
69
73
79
84
9'
96
_ _, 102
40,3371 Ï08
1,97
Unya |ilil7
0,000
0,408
0,816
1,224
1,632
2,o4o
3,448
2,856
3,363
3,671
4.079
4,487
4,89
5,3o3
5,710
6,118
6,526
6,933
7,341
7,748
8,1 56
8,563
8,971
9,378
9,785
10,192
10,599
1 1 ,006
ii,4i3
11,820
12,227
13,634
i3,o4o
1 3,447
1 3,853
14,260
1 4,666
I 5,07 3
15,478
I 5,884
16,290
16,696
17,102
17,507
17,913
18,3 18
20,343
22,364
24,38 1
26,395
28,4o3
1 ,9208
3i
32
34
34
36
36
38
39
4o
4i
42
43
43
45
46
47
53
57
63
68
74
1,98
Days Idtl'.
3o,4o7 79
32,4o5 "
34,398
36,384
38,363
40,335
0,000
0,409
0,818
1,227
1,636
2,o45
2,454
3,863
3,373
3,681
4,090
4,498
4.907
5,3i6
5,735
6, 1 33
6,542
6,951
7,359
7,768
8,176
8,58
8,993
9,403
9,810
10,218
10,626
1 1 ,o34
ii,'!43
ii,85o
12,258
12,666
13,074
i3,48i
13,889
i4,2g6
14,704
i5,ii I
i5,5i8
i5,935
16,332
16,739
17,145
17,552
17,959
i8,365
20,395
22,421
24,444
26,463
28,477
3o,486
32,490
34,488
36,48o
38,464
40,442
1,9405
1,99
Days jdir.
17
0,000
0,4 10
0,820
i,23o
i,64o
3,o5o
3,460
2,870
3,280
3,690
4,100
4,5io
4,930
5,339
5,739
6,149
6,55q
6,968
7,378
7,788
8,197
8,607
g,oi6
9,435
9,835
10,244
10,653
1 1 ,062
11,471 29
1,880 3o
2,00
Day* |dif.
1 ,9602
12,289
1 2 ,698
i3,i07
i3,5i5
13,924
i4,332
i4,74i
i5,i4o
i5,557
15,965
16,373
16,781
17,1
17,597
18,004
l8,4l2
30,447
22,479
34,507
36,53 1
28,550
3o,565
32,574
34,578
36,575
38,566
40,549
0,000
0,4 1 1
0,822
1,233
1,644
2,o55
3,466
2,877
3,2
3,699
4.110
4,531
4,932
5,343
5,754
6,164
6,575
6,986
7,397
7,807
8,218
8,628
9,039
9,449
9,859
10,270
10,680
1 1 ,090
1 1 ,5oo
11,910
12,320
i2,73ci
i3,i4o
i3,549
13,959
37 14,369
" 14,778
15,187
15,597
16,006
i6,4!5
16,824
17,333
17,641
i8,o5o
i8,458
20,499
32,536
24,569
26,59g
28,624
3o,644
33,658
34,667
36,670
38,666
4o,655
37
38
4o
4i
43
43
43
44
46
46
53
57
63
68
74
79
84
89
95
100
106
1,9801
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0035
o,oo36
o,oo4g
0,0064
0,008 1
0,0100
0,01 3 1
0,01 44
0,01 6g
0,0196
0,0225
o,o256
0,0289
o,o324
o,o36 1
o,o4oo
0,044 1
o,o484
o,o52g
0,0576
0,0625
0,0676
0,072g
0,0784
3o 0,08
2,0000
0,0900
0,0961
0,1024
0,1089
0,11 56
0,1225
0,1296
0,1369
o,i444
0,1 52 I
0,1600
0,16
0,1764
0,18 _
0,1936
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
, (r 4- *■ " )" oi" '■* -\- t'"^ nearly.
399
400
4oi
402
4o3
4o4
4o5
406
407
408
409
4io
40
4o
4o
40
4o
40
4i
41
4i
4i
4i
4i
80
80
80
80
81
81
81
81
81
82
82
82
120
120
120
121
121
121
123
122
123
122
123
123
160
160
160
161
161
162
163
162
i63
1 63
1 64
1 64
30O
200
201
201
202
202
2o3
203
204
2o4
2o5
2o5
239
340
241
341
242
243
243
244
344
245
245
246
279
280
281
281
282
283
284
284
285
286
286
287
3iq
330
321
322
322
323
324
325
326
326
327
328
359
36o
36i
362
363
364
365
365
366
367
368
369
411
4i
82
123
164
206
347
288
339
370
TABLE II. — To fiiiil the time T ; the sum of tlie lailii r-f-r", ami the chord c being given.
^um uf the Ru'iii r-\-r".
ChonI
C.
0,t)0
0,0 1
(1,02
o,o3
o,o4
o,u5
o,u6
0,07
0,08
0,09
0,10
0,1 1
0,12
0,1 3
0,1 4
0,1 5
0,16
17
o,i8
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
o/ii
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,75
0,80
o,85
0,90
0,95
1. 00
2,01
2,060
2,47s
2,884
3,296
3,708
4,120
4,532
5,356
5,768
6,180
6,592
7,oo3
-,4i5
7.827
8,238
8,65o
9.061
9.473
9,884
io,2g5
10.707
11,529
1 1 ,940
i2,35i
12,762
13,173
1 3,583
1 3,994
i4,4o5
i4,8i5
l5,225
1 5,636
i6,o46
1 6,456
16,866
17,276
17,686
18,095
i8,5o5
2o,55o
22,593
24,632
26,666
28,697
30,722
32,742
34.757
36,765
38,767
40,762
■2,0-2
Duys Idii;
2,0201
0,000
0,4 1 3
0,826
1,239
1,652
2,o65
2,479
2,892
3,3o5
3,718
4,i3i
4,544
4,957
5,369
5,782
6,195
6,608
7,021
7,433
7,846
8,259
8.671
9,084
9.496
9,909
10,321
10,733
1 1,146
11,558
11,970
12,382
12,794
1 3,206
1 3,61 7
i4,02g
i4,44i
i4,852
1 5,264
15,675
16,086
16,497
16,908
17,319
17,730
i8,i4o
i8,55i
20,602
22,65o
24,694
26,734
28,769
3o,8oo
32,826
34,846
36,86o
38,86-
40,868
2,03
Dats |dif.
2,0402
0,000
0,4 14
0,828
1,242
1 ,656
2,071
2,485
2,899
3,3i3
3,727
4,i4i
4.555
4,969
5,383
5,797
6,211
6,624
7,o38
7,452
7,866
8,270
8,693
9,106
9,520
9,933
0,347
0,760
1,173
1,586
2,000
2,4i3
2,825
3,238
3,65 1
4,064
4,476
4.889
5,3oi
5,7
6,126
6,538
6,95o
7,362
7,774
8,186
■8,597
20,654
22,706
24,756
26,801
28,842
30,878
32,909
34,935
36,954
38,967
40,973
2,04
Oays Id if.
2,0605
0,000
0,4 1 5
o,83o
1,245
1,661
2,076
2,491
2,906
3,321
3,736
4,i5i
4,566
4,q8l
5,39(i
5,811
6,226
6,641
7,o55
7,470
7,885
8,3oo
8,714
9,129
9,543
9,958
10,372
10,787
11,201
II, 61 5
12,029
12,443
12,857
13,271
1 3,685
14,099
i4,5i2
14,926
15,339
i5,753
16,166
16,579
16,992
i7,4ù5
17,818
i8,23i
18,643
20,705
2 2,763
24,81
26,868
28,915
30,956
32,993
35,023
37,048
39,067
41,079
2,05
Dnys |ilir.
2,0808
0,000
o,4i6
o,832
1 ,248
1,665
2,081
2,497
2,913
3,32q
3,74b
4,161
4,577
4,993
5,409
5,825
6,241
6,657
7,073
7,489
7,904
8,320
8,736
9,i5i
9,567
9,983
0,398
0,81 3
l,22i
1,644
2,059
2,47-
2,88(
3,3o4
3,719
4,i33
4,548
4,963
5,377
5,791
6,206
6,620
7,o34
7.448
7,862
8,276
20,756
22,819
24,879
26,935
28,987
3i,o34
33,075
35,112
37,142
39,166
4i,r
2,06
Days )<lit'.
2,1013
0,000
0,4 17
0,834
1,252
1,669
2,086
2,5o3
2,920
3,337
3,754
4,171
4,588
5,oo5
5,422
5,839
6,256
6,673
7,090
7,5o7
7,924
8,34o
8,757
9, '74
9,590
10,007
10,423
io,83g
11,256
11.672
12,088
i2,5o4
12,920
1 3,336
i3,752
14.168
i4,584
14,999
i5,4i5
i5,83o
16,245
16,661
17,076
17,491
17,906
18,32
18,735
20,807
22,876
24,g4i
27,002
29,o5g
3i,iii
33,i58
35,200
37,236
39,265
41,288
0,0000
0,0001
0,0004
o,ooog
0,0016
0,0025
o,oo36
0,0049
0,0064
0,0081
0,0100
0,01 2 I
0,01 44
o,uifk)
0,0196
0,0225
0,0256
0,0289
o,o324
o,o36 1
o,o4oo
0,044 1
0,0484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,084 1
0,0900
0,09!) 1
0,1024
0,1089
0,11 56
0,1225
0,1296
0,1369
0,1 444
0,l521
o, 1 600
0,1681
0,1764
0,1849
o,ig36
0,2025
0,2 5oo
o,3o25
o,36oo
0,4225
o,4goo
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
2.1218
5 ■ (r -f- r " ) ^ or r^ -{- r"^ nearly.
409
410
4ii
4l2
4i3
4i4
4i5
416
417
4i8
4i
4i
4i
4i
4i
4i
42
42
42
42
82
82
82
82
83
83
83
83
83
84
123
123
123
124
124
124
125
125
125
J25
164
164
i64
16!)
i65
166
166
166
167
167
2o5
2o5
206
206
207
207
208
208
209
200
245
246
247
247
248
248
249
250
250
25Ï
286
287
288
288
289
290
291
2QI
292
2g3
327
328
329
33o
33o
33 1
332
333
334
334
368
369
370
371
372
373
374
374
375
376
Prup. purts fur tlin sum uf tho Kudii.
I |2|3|4 |5|6|7|8|g
I
2
3
4
22
23
24
25
26
27
28
29
3o
3i
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
100
I
I
I
I
2
2
2
3
3
4
4
4
4
5
5
6
5
6
6
7
7
8
7
8
8
9
8
10
9
1 1
10
1 1
10
12
1 1
i3
1 1
i3
12
i4
i3
i5
i3
i5
i4
16
i4
17
i5
18
16
18
16
'9
17
20
17
20
18
21
'9
22
iq
22
20
23
20
24
21
25
22
25
22
26
23
27
23
27
24
28
25
29
2b
29
26
3o
26
3i
27
32
28
32
28
33
2q
34
29
M
3o
35
3i
36
3i
36
32
37
32
38
33
3q
34
3q
M
4o
35
4i
35
4i
36
42
37
43
37
43
38
44
38
45
3q
46
4o
46
4o
47
4i
48
4i
48
42
4q
48
56
54
60
63
70
a9
FABLE II
— To find the time T
; the sum of the radii
r + r".
and the chord e
being give
n.
Sum of the Rudii r -|- r ". 1
Chord
C.
2,07
Days |iJil'.
2,08
2,09
2,10
2,11
2,12
2J3
2,14
2,15
2,16
Uavs |i]if.
Days |dil'.
Hays lilif.
Days
dit".
Days Idif.
Days |.lif.
Days Idif.
Days |dif.
Days |dif.
0,00
0,01
0,02
o,o3
o,o4
0,000
0,4 1 8
0,836
1,255
1,673
I
2
3
4
0,000
0,419
o,838
1,258
i>677
I
3
3
4
0,000
0,420
o,84o
1,261
1,681
I
3
3
4
0,000
0,421
0,843
1,264
1,685
1
3
3
4
0,000
0,422
0,844
1,267
1,689
I
2
3
4
0,000
0,423
0,846
1.270
1,693
I
3
3
4
0,000
0,434
o,84&
1,373
1,697
I
2
3
4
0,000
0,425
o,85o
1,276
1,701
1
2
3
4
0,000
0.426
0352
1,279
i,7o5
I
2
3
4
0,000
0,427
o,854
1,282
1 ,709
1
2
2
4
0,0000
0,0001
o,ooo4
o,ooog
0,0016
o,o5
0,06
0,07
0,08
0,09
2,091
2,5og
2,927
3,345
3,763
5
6
7
8
9
2,096
2,5i5
2,934
3,353
3,772
5
6
7
8
10
2,101
2,521
2,941
3,36i
3,782
5
6
7
8
9
2,106
2,537
2,948
3,369
3,791
5
6
7
8
9
2,111
2,533
3,955
3,377
3,800
5
6
7
8
9
2,116
2,539
2,962
3,385
3,809
5
6
7
8
9
2,121
2,545
2,969
3,3g3
3,818
5
6
7
8
9
2,126
2,55i
2,976
3,401
3,827
5
6
7
8
8
2,l3l
2,557
2,983
3,409
3,835
5
6
7
8
9
2,i36
2,563
2,99"
3,417
3,844
5
6
7
8
9
0,0025
o,ûo36
0,0049
0,0064
0,0081
0,10
0,1 1
0,12
0,1 3
o,i4
4,181
4,600
5.018
5,436
5,854
II
II
12
i3
i4
4,192
4,611
5,o3o
5,449
5,868
10
II
12
i3
i4
4,203
4,622
5,042
5,462
5,882
10
I'
12
i3
i4
4,212
4,633
5,o54
5,475
5,896
10
II
13
i3
i4
4,232
4,644
5,066
5,468
5,910
10
II
12
i3
14
4,333
4,655
5,078
5,5oi
5,924
10
11
13
i3
i4
4.243
4,666
5,090
5,5i4
5.938
10
11
12
i3
i4
4,252
4,677
5,102
5,527
5,952
10
1 1
13
i3
i4
4.263
4,688
5,ii4
5.540
5,g66
9
1 1
12
i3
i4
4,271
4.699
5,126
5,553
5,980
10
10
11
12
i3
0,0100
0,0121
0,01 44
0,0169
0,0196
0,1 5
0,16
0,17
0,18
0,19
6,271
6,689
7,107
7,525
7.943
16
17
17
18
'9
6,287
6,706
7.124
7.543
7,962
i5
16
i8
18
19
6,3o2
6,722
7,142
7,56 1
7,981
i5
.6
17
18
19
6,3i7
6.738
7,159
7.579
8,oou
i5
16
17
18
19
6,332
6,754
7,176
7,597
8,019
i5
16
■7
18
19
6,347
6,770
7,193
7,61 5
8,o38
i5
16
.7
18
19
6,362
6,786
7,210
7,633
8,o57
i5
16
17
18
19
6,377
6,802
7,227
7,65 1
8,076
i5
16
16
18
19
6,3g3
6,818
7.243
7,669
8,og5
14
i5
17
if
19
6,/c6
6,833
7.360
7,687
8,114
i5
16
17
18
'9
0,0225
0,0256
0,0289
o,o334
o,o36i
0,20
0,21
0,22
0,23
0,24
8,36i
8,778
9,196
9,61 3
io,o3i
20
21
22
24
24
8,38 1
8.799
9,218
9,637
io,o55
20
22
22
23
24
8,401
8,821
9,240
9,660
10,079
20
21
22
23
25
8.421
8,842
9,262
9,683
1 0, 1 o4
20
21
22
23
24
8,44i
8,863
9,284
9,706
10,128
20
21
23
23
34
8,461
8,884
9,3o6
9.729
IO,l52
20
21
22
23
24
8,481
8,905
9,338
9.753
10,176
20
21
22
23
23
8,5oi
8,936
9,35o
9.775
10,199
20
21
22
23
24
8,521
8,g47
9,372
9.798
10,223
20
20
22
23
24
8,541
8,967
9,394
9,831
10,247
19
2J
33
22
24
o,o4oo
o,o44i
0,0484
0,0529
0,0576
0,25
0,26
0,27
0,28
0,29
10,448
10,866
11,283
1 1 ,700
12,1 18
26
26
27
=9
29
10,474
10,892
I i,3io
11,729
I2,l47
25
26
28
28
29
10,499
10,918
11,338
11,757
12,176
35
26
27
28
29
10,524
10,944
11,365
11,785
I2,2o5
25
27
27
28
29
10,549
10,971
11,392
ii,8i3
12,234
25
26
27
28
29
10,574
10,997
11,419
ii,84i
13,363
25
26
27
28
29
io,5g9
11,023
1 1 .446
1 1 ,869
12,292
25
25
27
28
29
10,634
ii,o48
11,473
11,897
13,321
25
26
27
28
29
10,649
11,074
1 1 ,5oo
11,925
i2,35o
25
26
36
38
29
10,674
11,100
11,536
1 1 ,953
12,379
34
36
27
27
29
0,0625
0,0676
0,0729
0,0784
o,o84i
o,3o
0,3 1
0,32
0,33
0,34
12,535
12,952
13,369
13,786
1 4,202
3o
3i
32
33
35
12,565
12,983
i3,4oi
i3,8i9
i4,237
3o
3i
33
33
34
12,595
i3,oi4
1 3,433
i3,852
14,271
3o
32
33
34
34
12,625
i3,o46
1 3,466
1 3,886
i4,3o5
3i
3i
32
33
35
12,656
1 3,077
1 3,498
13,919
1 4,340
3o
3i
32
33
34
12,686
i3,io8
i3,53o
13,953
14,374
3o
3i
32
33
34
13,716
i3,i39
i3,562
i3,g85
1 4,408
3o
3i
32
33
34
13,746
13,170
i3,5g4
i4,oi8
14,442
29
3i
32
33
33
12,775
l3,201
13,626
i4,o5i
i4,475
3o
3o
3i
32
34
i3,8o5
i3,23i
1 3,657
i4,o83
i4,5o9
3o
3i
32
33
34
o,ogoo
0,0961
0,1034
0,1 o8g
0,1 156
0,35
o,36
0,37
o,38
0,39
14,619
i5,o36
1 5,452
15,869
16,285
36
36
38
38
40
14.655
15,072
15,490
15,907
i6,325
35
37
37
39
39
14,690
15,109
i5,527
1 5,946
i6,364
35
36
37
38
39
14,725
i5,i45
i5,564
15,984
i6,4o3
35
36
38
38
40
14,760
i5,i8i
1 5,602
16,022
16,443
35
36
37
38
39
14,795
l5.3I7
1 5,639
16,060
16,482
35
36
37
38
39
i4.83o
i5,253
15,676
16,098
16,53 1
35
36
37
38
39
14,865
1 5,28g
i5,7i3
16,1 36
i6,56o
35
36
37
38
39
14,900
1 5,325
i5,75o
16,174
16,599
35
36
36
38
38
14,935
i5,36i
15,786
l6,213
1 6,637
35
35
37
38
39
0,1225
o,i2g6
0, 1 369
0,1 444
0,l52I
o,4o
0,41
0,42
0,43
0,44
16,701
17,118
17,534
i7:95o
i8,365
4i
4i
42
43
45
16,742
17,159
17,576
17:993
18,410
40
42
43
AA
45
16,782
17,201
17,619
i8,o37
18,455
4i
4i
42
43
AA
16,823
17,242
17,661
18,080
18,499
40
4i
Ao
43
AA
i6,863
17,283
17,703
18.123
18,543
4o
4i
43
AA
45
16,903
17.324
17,746
18,167
i8,588
40
4i
43
43
AA
i6,g43
17,365
17,788
18,210
i8,632
4o
4i
42
43
AA
i6,g83
17,406
i7,83o
18,253
18,676
40
4i
41
43
AA
17,023
17.447
17,871
18,296
18,720
40
4i
42
42
43
i7,o63
17,488
17,913
i8,338
18,763
39
4i
42
43
AA
0,1600
0,1681
0,1764
0,1849
0,1936
0,45
0,5r)
0,55
0,60
0,65
0,70
18,781
20,858
22,932
25,002
37,06g
29,i3i
46
5i
56
61
66
71
18,827
20,909
22,988
25,o63
27,i35
29,202
45
5i
56
61
66
72
18,872
20,960
2 3,o44
25,124
27,201
29,274
46
5o
55
61
66
71
18,918
21,010
33.09g
25;i85
27,267
39,345
45
5i
56
61
66
71
18,963
21,061
23,i55
25,246
27,333
29,416
45
5o
55
61
66
71
19,008
21,1 11
23,210
25,3o7
27,399
29,487
46
5o
56
60
66
71
19,054
31, 161
23,366
35,367
37,465
29,558
45
5o
55
60
65
71
19.099
21,211
23,321
25,437
27,530
29.639
AA
5o
55
60
65
70
19,143
21,261
23,376
25,487
27,595
29.699
45
5o
55
60
65
70
19,188
2I,3lI
23,43 1
25,547
27,660
29,769
45
5o
55
60
65
70
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,75
0,80
o,85
0,90
0,95
1,00
3i,i88
33,241
35,288
37,329
39,364
41,393
77
82
88
93
99
io4
3 1, 265
33,323
35,376
37,422
39,463
41,497
77
82
8-^
93
98
io4
31,342
33,4o5
35,463
37,5i5
39,561
41,601
77
82
87
93
98
io3
3i,4i9
33,487
35,55o
37.608
39,65g
41,704
76
82
87
92
98
io4
31.495
33,56q
35,637
37,700
39.757
41,808
76
81
87
92
98
io3
31,571
33,65o
35,724
37.792
39,855
41,911
76
83
87
92
97
103
3 1, 647
33,732
35,811
37,884
39,952
42,oi3
76
81
86
92
97
io3
3i,723
33,8i3
35,897
37,976
40,049
42,116
76
80
86
92
97
102
3 1,799
33,8g3
35,983
38,o68
4o.i46
42,218
75
81
86
9'
97
102
31,874
33,g74
36,o6g
38,1 59
40,243
42,320
75
81
86
96
102
o,5635
o,64oo
0,7325
0,8100
0,9025
1 ,0000
2,1425
2,1632
2,1841
2,2050
2,2261 '
2,2472
2,2685 1
2,2898 1
2,31131
2,3328 1 c2 1
. (1- -\- r")'' or ?•= 4. r"» nearly.
4i5
4i6
417
4i8
419
420
421
422
423
424
425
426
437
428
42
42
42
42
42
42
42
43
42
42
43
43
43
43
83
83
83
84
84
84
84
84
85
85
85
85
85
86
125
125
125
125
136
136
126
137
127
127
128
128
128
128
166
166
167
167
168
168
168
169
169
170
170
170
171
171
208
208
209
200
210
210
211
211
212
212
2l3
3l3
2l4
2l4
249
35o
25o
25I
25l
252
253
253
254
2 54
255
256
356
257
291
291
292
2g3
293
294
295
2q5
296
297
298
298
299
3oo
332
333
334
334
335
336
337
338
338
339
340
341
342
342
374
374
375
376
377
378
379
38o
38i
382
383
383
384
385
TABLE II. — To fiiul the time T\ the sum of the radii i-j^ r", nnil tlie choid <■ being given.
Sum ul' thi! Uuilll
r -f- r ".
Chord
C
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,0-7
0,08
0,09
0,10
0,11
0,12
o,i3
o,i4
o,i5
0,16
o,n
0,18
0,1 g
0,20
0,21
0,2?
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
o,3i
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
Oj4i
0.42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
o,85
o,go
0,95
1,00
2,17
Days |dit'
0,000
0,428
o,856
1,284
i,7i3
2,i4i
2,569
2.997
3,425
3,853
4,281
4,7«»
5,i3"
5,565
5,993
6.421
6,849
7,2
7,-'>5
8,i33
8,56o
8,988
9,416
9,843
10,698
11,126
11,553
1 1 ,980
I2,4o8
12,835
13,262
13,689
i4,ii6
14,543
1 4,9
15,396
1 5,823
i6,25o
16,676
17,102
17,529
17,955
i8,38i
18,807
19,233
2i,36i
23,486
25,607
27,725
29,839
3 1,949
34,o55
36,1 55
38,25o
40,339
42,422
21
21
23
24
25
25
27
28
28
3o
3i
32
33
33
34
36
37
37
3g
40
4f
42
43
Ai
49
54
60
65
70
2,18
Udys jdif.
0,000
0,42g
o,858
1,287
1,717
2,146
3,575
3,004
3,433
3,862
4,2gi
4,720
5,149
5,578
6,007
6,436
6,865
7,294
7,723
8,i5i
8,58o 20
9,009
9,437
9,866
lo,2g5
2,19
Unys jdir.
0,000
o,43o
o,86<;
1,291
2,3545
10,723
ii,i5i
ii,58o
1 2 .008
12,436
12,865
13,293
1 3,721
14,149
14,576
1 5,oo4
i5,432
1 5,860
16,2:
16, -"iS
£7, 142
17,569
17,997
18,424
i8,85i
19,277
2I,4lO
23,540
25,667
27,790
29,909
32,024
34,i35
36, 240
38,34o
40,435
2,523
60
65
70
75
80
86
91
96
loi
2,3762
2,l5l
2,58i
3,011
3,44 1
3,871
4,3oi
4,73i
5,161
5,591
6,021
6,45 1
6,881
7,3 1 1
7,74o
8,170
8,600
9,029
9,459
9,889
io,3i8
10,748
11,177
1 1 ,606
i2,o36
12,465
12,89
i3,323
i3,752
i4,i8i
i4,6io
1 5,039
1 5,468
15.896
16.325
16,753
17.182
17,610
i8,o38
1 8,466
18,89,
19,322
21,460
23,595
25,727
27,855
29,979
2,20
Uuys |ilil'.
32,099
34,21 5
36,326
38,43 1
4o,53i
42,6241
24
24
26
27
27
28
3o
3i
32
33
33
34
3
37
3
39
39
4'
4i
42
43
44
49
54
59
64
70
75
80
85
90
95
101
2,3981
0,000
0.431
0,862
1,293
1,724
2 , 1 56
2.587
3,oi«
3,449
3,880
4,3 1 1
4,742
5,173
5,604
6,o35
6,466
6,896
7,327
7,758
8,189
8,6.9
9,050
9,481
9.91 r
10,343
10,773
II,2o3
1 1 ,633
i2,o63
12,493
12,924
i3,354
13,784
i4,2i4
14,643
1 5,073
i5.5o3
15,933
16,362
16,792
17,221
i7,65o
i8,0"9
i8,5o8
18.937
19,366
21,509
23,64g
25,786
27,919
3o,o4g
2,21
lluya |dir.
0,000
0,433
0,864
1,296
1,728
3, 160
2,5g3
3,035
3,457
3,
4,331
4,753
5,i85
5,617
6,048
6,480
6,g 1 2
7,344
7,776
8,207
8,639
9,071
9,5o2
9-0 34
23 10, 365
32,174
34,295
36,4 1 1
38,521
40,626
2,22
Uuys |dir.
10,797
11,228
11,659
i2,ogi
12,522
I2,g53
i3,384
i3,8i5
14,246
14,677
i5,io8
i5,538
15.969
l6.3gg
i6,83o
39 17,260
17,691
18,121
i8,55i
18,981
19,411
3 7,558
33,7o3
25,845
37.984
3o,ii8
33,248
34,37
36,4g5
38,6ii
40,733
3,826
II
13
i4
i5
16
16
17
19
30
20
22
22
24
34
36
27
27
28
29
3o
3i
32
33
34
36
36
38
38
39
4o
4i
43
43
5o
55
59
64
69
75
80
2,735|lOll
2,4200'| 2,4421
0,000
0,433
0,866
1,299
1,733
2, 1 65
=.598
3,o3i
3,464
3,897
4,33o
4,763
5,ig6
5,629
6,062
6,495
6,928
7,36o
7,793
8,226
8,659
9,091
9,53.
9>956
I o,38g
10,82
11,254
11,686
12,1 1
i2,55o
12,982
i3,4i4
1 3,846
14,278
i4,7>o
i5,i42
1 5,574
1 6,00 5
16,437
16,868
17,29g
17,731
18,162
i8,5g3
19,024
19,455
2 1 ,608
23,758
25,904
28.048
30^187
32,323
34,454
36,58o
38,701
40,817
43,926
12
i3
I
i5
i5
17
lb
I
19
31
21
23
23
25
25
36
27
29
3o
3i
33
33
33
34
35
36
37
38
4o
4o
4i
42
43
0,0000
0,000 1
0,0004
0,0009
0,0016
0,0035
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,01 3 1
0,0144
0,0169
0,0196
0,0225
o,o356
0,0289
0,0324
o,o36i
o,o4oo
0,044 1
0,0484
o,o52g
0,0576
0,0625
0,0676
0,0739
0,0784
0,084 1
o,ogoo
0,0961
0,1034
0,1089
o, 1 1 56
0,1225
0,1296
0,1369
o,i444
0,l52I
0,1600
0,1681
0,1764
o,i84g
0,1936
4A 0,2035
4o o,35oo
o,3o3 5
o,36oo
0,4225
0,4900
69
74
79
84
95
101
0,5625
0,6400
0,7335
0,8100
0,9025
1 ,0000
2,46421 ê
Î . (r -f- r" )^ or r^ 4" '
nearly.
426
43
85
128
170
2l3
256
298
341
383
427
43
85
138
171
2l4
256
299
342
384
428
43
86
128
171
2l4
357
3oo
342
385
439
43
86
129
173
2l5
257
3oo
343
386
43o
43 1
432
433
43
43
43
43
86
86
86
87
129
129
i3o
i3o
173
172
173
173
2l5
216
216
217
258
359
25q
260
3oi
3o2
,302
3o3
344
345
346
346
387
388
38q
3qo
43
87
i3o
174
217
260
3o4
347
3qi
Pro|i. parts (or llic sum of llie Radii.
■ I 2 I 3| 41 5| 6| 7|8|9
i3
i4
i5
16
17
18
'9
30
21
22
23
24
25
26
27
28
29
3o
3i
32
33
34
35
36
37
38
39
4o
4i
A->
A3
A4
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
63
6:
8 10
II
9
9
8 10
8 10
8
'9
19
20
17 20
i4 18
18
19
1 5 20
i5
16
16
16
27 33
38
28
29
129
3o
3o
3i
38
32
36
40
23
23
24
25
2 5
36
27
27
38
29
29
3o
3i
32
32
33
34
34
35
36
36
37
38
13
i3
i4
i4
i5
16
17
20
21
22
23
23
24
25
26
27
28
29
3o
3i
32
32
33
3A
35
36
37
38
39
4o
33 3g
39 46
46
47
4i
I
42
A3
39 AA
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
72
81
90
TABLE II. — To find the time T\ the sum of the radii r -\-r'\ and the chord c being given.
Sum of the Radii r-\-r".
Chord
C.
0,00
0,01
0,02
o,o3
0,04
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
0,1 3
o,i4
o,i5
0,16
0,17
,18
.19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
o,3i
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
1,00
2,23
Days |dif.
0,000
0,434
0,868
I,302
1,736
2,170
2,604
3,o38
3,472
3,906
4,340
4,774
5,208
5,642
6,076
6,5io
6,943
7.377
7,811
8,244
8,678
9,112
9,545
9'979
IO,4l2
10,846
11,279
11,712
12,145
13,579
l3,OI2
1 3,445
13,878
i4,3ii
14,743
15,176
15,609
1 6,04 1
16,474
16,906
17,339
17,771
18,2
1 8, 63 5
19,067
19,499
21,637
23,8l2
25,963
28,1 12
3o,256
32,397
34,533
36,664
38,791
40,912
43,027
2,24
Days Idif.
o,Ouo
0,435
0,870
i,3o5
1, 740
3,175
2,610
3,045
3,480
3,9i5
4,35o
4,785
5,220
5,655
6,089
6,524
6,959
7,394
7,828
8,263
8,698
9,i32
9,567
10,001
10,436
10,870
1 1 ,3o4
1 1 ,738
12,173
12,607
i3,o4i
1 3,475
13,909
32 14,343
34 1 4,777
2,25
Days |dir.
0,000
o,436
0,872
i,3o8
1,744
2,1
2,616
3,o52
3,488
3,924
4,36o
4,795
5,23i
5,667
6,ro3
6,539
6,974
7,4 10
7,846
8,281
8,717
9, 1 53
9,588
10,023
10,459
io,f
11,329
11,765
12,200
12,635
29 13,070 29
30 i3,5o5 3o
2,26
Days |(lir.
34
35
36
37
38
39
4<-
4i
42
43
49
53
59
64
69
74
79
85
89
94
99
2,4865
l5,2IO
1 5,644
16,077
i6,5ii
16.944
17,378
17,811
18,244
18,677
19,110
19,543
2 1 ,706
23,865
26,022
28,176
3o,325
32,471
34,612
36,749
38 ,r-
4 1, 006
43,126
1 3,.
14,375
i4,8io
1 5,244
15,679
i6,ii3
16,548
16,982
17,417
17,85 1
42 18,719
43 19,153
2,5088
0,000
0,437
0,874
1,3 II
1,748
2,i85
2 ,62 2
3,059
3,496
3,932
4,369
4,806
5,243
5,680
6,116
6,553
6,990
7,427
7,863
8,3oo
8,736
9,173
9,609
10,046
10,482
10,918
11,355
11,791
12,227
12,663
13,099
i3,535
'3,971
14,407
14,843
15,278
i5,7i4
16,149
1 6,585
17,020
2,27
Days \A\\\
0,000
o,438
0.876
i,3i4
1,752
2,190
2,627
3,o65
3,5o3
3,941
4,379
4,817
5,254
5,692
6,i3o
6,568
7.oo5
7,443
7,88
8,3i
8,756
9,193
9,63 1
10,068
io,5o5
38 17,455
4o 17
19,587
21,754
23,919
26,or
28,239
30,394
32,545
34,691
36,833
38,969
4i,ioi
43,226
10,943
ii,38o
11,817
12,254
28 12,691
18,326
18,761
'9,196
!9,63o
21 ,8o3
23,973
26,140
28,3o3
3o,463
32,618
34,770
36,917
39,o58
41,195
43,325
2,28
Days|dir.
2,5313 I 2,5538
i3,i28
1 3,565
14,002
14,439
14,876
i5,3i2
1 5,749
i6,i85
16,622
i7,o58
17,494
17,930
I 8, 366
18,802
19,238
19,674
21,852
24,026
26,198
28,366
3o,53i
32,692
34,848
37,000
39,147
4i, "
43,425
0,000
0,439
0,878
i,3i7
1,756
2,194
2,633
3,072
3,5ii
3,950
4,389
4,827
5,266
5,7o5
6,i43
6,582
7,021
7,459
7,898
8,337
8,775
9,21 3
9,652
10,090
10,528
10,967
ii,4o5
1 1 ,843
12,281
12,719
i3,i57
13,595
i4,o33
14,471
14.908
1 5,346
15,784
16,221
i6,658
17,096
17,533
17,970
18,407
18,844
19,281
19,71
2 1 ,900
34,080
26,256
28,430
68 30,599
2,5765
2,29
Day..; |dir.
32,765
34,927
37,084
39,236
41,382
0,000
0,440
0,880
1,320
1,759
2,199
2,639
3,079
3,519
3,958
4,398
4,838
5,278
5,717
6,1 57
6,597
7,o36
7,476
7.915
8,355
8,794
9,234
9,673
10,1 12
10,552
10,991
1 1 ,4 3o
! 1 ,869
i2,3oa
12,747
i3,i86
1 3,625
14,064
i4,5o2
14,941
i5.38o
i5,8i8
16,257
16,695
I7,i33
17,572
18,010
18.448
18,886
19,32"
19,761
21,948
24, 1 33
26,3 1 5
28,493
3o,668
2,30
Days |dif.
43,523 99
2,5992
73 32.838
78 35,oo5
37,167
39,324
41,476
43,622
83
94
23
24
25
26
27
28
29
3o
3i
32
33
33
35
35
37
38
38
39
40
4i
43
Aà
49
53
58
63
67
73
78
83
88
93
99
0,000
0,44 1
0,882
1,322
1,763
2,304
2,645
3,086
3,526
3,967
4,408
4 '
5,289
5,73o
6,170
6,611
7,o52
7,492
7.933
8,373
8,81 3
9,254
9,694
1 0,1 34
10,575
11. 01 5
11,455
ii,8q5
12,335
12,775
i3,3i5
i3,655
14,095
i4,53'
14,974
i5,4i3
1 5,853
16,292
16,732
17,17'
17,610
18,049
18,488
18,927
19,366
19,805
21,997
24,186
26,373
28,556
30,735
32.911
35,oB3
37,25o
39,412
41,569
43,721
2,31
Days |dir.
0,000
0,442
0,884
1,325
1,767
2,209
2,65i
3,092
3,534
3,976
4,417
4,859
5,3oi
5,742
6,1
6,625
7,067
7,5o8
7,950
8,391
8,833
9,274
9,715
10, 1 56
io,5g8
1 1 ,039
1 1 ,480
1 1 ,92 1
1 2 ,362
i2,8o3
1 3,244
1 3,684
i4,i25
1 4,566
1 5,007
1 5,447
15,887
16,328
16,768
37 17,208
2,6221
2,32
Days |dTl'.
0,000
0,443
0,885
1,328
1.771
2,2l4
2,656
3,099
3,542
3,984
4,427
4,869
5,3t2
5,755
6,197
6,640
7,083
7.525
7,967
8,409
8,852
9,294
9,736
10,178
10,621
ii,o63 24
ii,5o5
1 1 ,947
12,389
12,83 1
17,648
18,089
18,529
18,968
19,408
19,848
22,045
24,239
26,431
28,619
3o,8o3
32,984
35,161
37,333
39,500
41,662
98143,819
3o
3i
32
32
34
35
35
37
38
39
39
4o
42
42
43
48
53
57
62
68
73
77
93
2,6450 I 2,6681
13,272
1 3,7 1
i4,i56
14,598
1 5,039
1 5,48 1
15,922
i6,363
i6,8o5
17,246
1 7,687
18,12.
18,569
19,010
19,450
19,891
22,093
24,292
26„""
28,681
30,871
33,o57
35,238
37,41 5
39,588
41,755
43,917
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,0121
0,01 44
0,01 69
0,0196
0,0225
0,02 56
0,0289
o,o324
o,o36 1
o,o4oo
0,044 1
o,o484
0,0529
0,0576
0,0625
0,0676
0,0729
26 0,0784
27 o,o84i
0,0900
0,0961
0,1024
0,1089
0,1 1 56
0,1225
0,1 296
0,1 36g
0,1 444
0,1 521
o, 1 600
0,1681
0,1764
0,1849
0,1936
^ 0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
2,6912
0,5625
0,6400
0,7225
0,8100
0,9025
1 ,0000
(r
è .(r + r")^ or r'
nearly.
432
433
434
435
436
437
438
439
440
44 1
■ 442
43
43
43
M
ÂA
AA
AA
AA
AA
AA
AA
86
87
87
87
87
87
88
88
88
88
88
i3o
i3o
i3o
i3i
i3i
i3i
i3i
l32
l32
l32
i33
173
173
174
174
174
175
175
176
176
176
177
216
217
217
218
218
219
219
220
220
221
221
sSg
260
260
261
262
262
263
263
264
265
265
302
3o3
3o4
3o5
3o5
3o6
307
3o7
3o8
309
3°9
346
346
347
348
349
35o
35o
35i
352
353
354
389
390
391
392
392
393
394
395
396
397
398
443
AA
I
80
2
1 33
3
"77
4
222
5
266
6
3io
7
354
8
399
9
TABLE II. — To fin il the time T: the sum of the iDtlii r-|-r '', and the chord c boinp; given.
Srum of iho Kailii t-\-t"
Choiri
C.
2,33
2,34
2,35
Days lilif.
lliijs lilif.
Ilujs |ilil.
0,00
0,000
0,000
0,000
0,0 1
0,444
1
0,445
1
0,446
I
0,02
0,887
2
0.889
3
0,891
2
o»o3
1, 33 1
3
1.334
3
1,337
3
o,o4
1 {ll'i
4
1,779
3
1,783
4
o,o5
2,318
5
2,333
5
2,228
5
0,06
2,662
6
2,668
5
2.673
6
0,07
3,106
6
3.113
-
3,119
-J
0,08
3,54q
8
3.557
-
3,564
8
o,og
3,593
8
4,001
9
4, OK)
8
0,10
4,436
10
4.446
Q
4,455
10
0,1 I
4,880
10
4.8Q0
II
4,901
10
0,12
5.324
1 1
5.335
1 1
5,346
12
0,1 3
5.-6-
13
5,~c)
i3
5,792
13
0,1 4
6,211
i3
6,224
i3
6,33-7
i3
o,.5
6.654
i4
6,668
i5
6,683
i4
0,16
-.09-
lO
7,ii3
i5
7.128
i5
0,17
-,54i
16
7.557
16
7,573
16
0,18
7,984
I-
8,001
17
8,018
17
0,1 g
8,427
•9
8,446
18
8,464
18
0,20
8.871
19
8,890
iq
8,90g
iq
0,2 1
9.314
2(.
9,334
30
9,354
30
0,22
9'757
31
9:778
31
9>7S9
31
0.23
10.200
22
10,222
32
10,344
33
0,34
10,644
22
10,666
23
10,689
23
0,25
11.087
23
1 1,110
34
ii,i34
24
0,26
ii.53o
24
11,554
25
11.570
25
0,27
1 1 ,973
25
1 1 .998
26
12.034
2b
0,28
i2.4i5
27
13.443
27
13,469
26
0,2g
12,858
2b
12,886
37
12,913
38
o,3o
i3,3oi
29
i3,33o
38
i3,358
3q
0,3 1
1 3,744
29
i3,773
3o
i3.8o3
3Q
0,32
i4,i£6
Jl
14-317
3o
14,247
3i
0,33
14,629
32
1 4,661
3 1
14.692
3i
0,34
15,072
33
i5,io4
32
i5,i36
33
0,35
i5,5i4
33
1 5,547
34
z5,58i
33
o,36
15.956
35
15,991
34
16,025
M
0,37
16,399
35
16,434
35
16,469
35
o,38
16.841
36
16,877
36
16,913
36
0,39
17,283
37
17,330
37
17,357
37
o,4o
17,725
38
i-',763
38
17,801
38
0,4 1
18,167
3g
18.306
3q
18,345
3q
0,42
18,609
4o
i8,64q
40
18,689
40
0,43
1 9,o5 1
4i
19,093
4i
19,133
41
oM
19,493
42
19,535
42
19,577
4i
0,45
19,934
43
I9'977
43
20,020
43
o,5o
22.l4l
48
22,189
47
22,236
48
0.55
24,345
53
24,398
53
24,45o
53
0,60
26,546
58
26,604
57
36,661
57
o,65
28,744
63
28.806
63
38,869
63
0,70
3o,938
68
3 1, 006
67
31,073
67
0,75
33,129
72
33,201
73
33,374
72
0,80
35,3i6
77
35.393
77
35,470
77
o,85
37.498
83
37,580
83
37,662
83
0,90
39,675
88
3q,763
«7
39,85o
87
0,95
4 1, 848
92
41,940
93
42,o32
92
1,00
44,014
98
44,112
97
44,309
97
2,71
45
2,73
78
2,76
13
2,36
0,000
0,447
o,8g3
1, 340
1,786
2,233
2 ,67q
3,126
3,572
4,018
4,465
4,911
5.358
5,804
6,2 5o
6,697
7,143
7,58g
8,o35
8,482
8,928
9,374
9.820
0,266
0,712
1, 1 58
1,604
2 ,o5o
2,495
2,941
3,387
3.832
4,278
4,723
5,169
5,614
6,059
6,5o4
6,949
7,394
7,839
8,284
8,729
9,174
9,618
20,063
22,284
24.5o3
26,718
38,931
3i,i4o
33,346
35,547
37,744
39,937
43,124
44,3o6
2,7848
2,37
Days lilif.
0,000
0,447
0,895
1,342
1,790
2,237
2,685
3.i32
3,58o
4,027
4,474
4.933
5.36q
5.816
6,364
6,711
7,i58
7,6o5
8,o53
8,5oo
8,947
9,394
9,84 1
0.288
0^735
1. 183
1^638
3,075
2,523
2,968
3.41:)
3,862
4.3o8
4,754
5,301
5,647
6,093
6.539
6,985
7,43 1
7,877
8,333
8,76g
9,3i5
9,660
30,106
3 2.332
34,555
26,776
38,gg3
3 1, 207
33,417
35.634
37,836
40,02
42.216
44 .4o3
2,8085
2,38
ll«)s lilif.
t,,ooo
0,448
o,8g7
1,345
i>794
2,242
2,690
3, 1 39
3.587
4,o35
4,484
4,932
5,38o
5,839
6,277
6,725
7,173
7,621
8,069
8,5i8
8,966
9,4i4
9,862
o,3c9
0,757
r,3o5
1,653
3,101
2.548
2-996
3,443
3,891
4,338
4,786
5,233
5,680
6,127
6,574
7,021
7,468
7,9i5
8,362
8,809
9,355
9,702
20, 1 48
2 2,379
24,607
36,833
39.055
31,374
33,489
35,700
37,907
40,110
43,307
44 .500
2,8322
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
,oo36
,oo4g
,0064
,0081
,0100
,0121
,0144
,01 6g
,0196
,0235
,02 56
,028g
,o324
,o36i
,o4oo
,044 1
i0484
io53g
,0576
,0625
,0676
,0729
,0784
.084 1
,ogoo
,0961
1034
1089
ii56
1225
1296
1369
1444
l52I
1600
168 1
1764
i84g
ig3Ô
2025
25oo
3o25
,3600
4225
i4goo
625
,64oo
7235
,8100
9035
,0000
K . (,r -\- r" ) ^ or r^ -j- r' ^ n -iirly.
441
443
443
AM
445
446
447
448
449
iA
44
44
44
45
45
45
45
45
88
88
8q
89
8q
89
8q
90
90
l32
1 33
1 33
i33
i34
1 34
1 34
i34
i35
176
177
177
178
178
178
179
179
180
221
331
222
222
223
223
224
234
225
265
265
266
266
367
268
268
269
269
309
309
3io
3ii
3l2
3l2
3i3
3i4
3i4
353
354
354
355
356
357
358
358
359
397
398
399
400
40 1
4oi
4o2
4o3
4o4
Prop.
parts fur ttif, 8UII1 uf tho Radii. 1
■ 1
2l3|4|5|6|7|8|9 1
1
0
0
0
0
1
I
I
1
I
2
0
0
1
I
1
1
1
3
2
3
0
I
1
I
3
2
3
3
3
4
0
I
1
2
3
2
3
3
4
5
I
3
2
3
3
4
4
5
6
1
3
2
3
4
4
5
5
7
I
2
3
4
4
5
6
fi
8
3
3
3
4
5
6
6
7
9
2
3
4
5
5
6
n
8
10
3
3
4
5
6
-T
8
q
1 1
2
3
4
6
7
8
q
10
12
2
4
5
6
7
8
10
1 1
i3
3
4
5
7
8
q
10
12
14
3
4
6
7
8
10
1 1
i3
i5
2
3
5
6
8
q
1 1
12
i4
16
2
3
5
6
8
10
1 1
i3
i4
17
2
3
5
7
q
10
12
14
i5
18
2
4
5
7
9
1 1
i3
i4
16
'9
2
4
6
8
10
1 1
i3
i5
17
20
2
4
6
8
10
12
i4
16
18
21
2
4
6
8
1 1
i3
i5
17
iq
22
2
4
7
q
1 1
i3
i5
18
20
23
2
5
7
q
12
■ 4
16
18
21
24
2
b
7
10
12
i4
17
■9
22
25
3
5
8
10
i3
i5
18
30
23
26
3
5
8
10
i3
16
18
21
23
27
3
5
8
1 1
i4
16
iq
22
24
28
3
6
8
1 1
i4
17
2t
23
35
29
3
6
9
13
i5
17
20
23
26
3o
3
6
q
12
i5
18
21
24
27
3i
3
6
q
12
16
iq
22
35
28
32
3
6
10
l3
16
iq
32
26
29
■Ji
3
7
10
l3
17
20
23
36
3o
M
3
7
10
14
17
20
24
27
3i
35
4
7
I I
i4
18
21
25
28
32
36
4
7
I 1
14
18
22
25
2q
32
37
4
7
1 1
i5
iq
22
26
3o
33
38
4
8
I 1
i5
iq
33
37
3o
34
39
4
8
12
16
20
23
27
3i
35
40
4
8
12
16
20
24
38
32
36
4i
4
8
12
16
21
25
20
a
37
42
4
8
1 3
17
21
25
2q
M
38
4<i
4
9
i3
17
22
26
3c
M
3q
44
4
9
i3
18
22
26
3i
35
40
45
5
q
i4
18
23
27
33
36
4i
46
5
9
14
18
33
28
32
37
4i
47
5
q
i4
iq
24
28
Si
38
43
48
5
10
14
iq
24
2q
M
38
4i
49
5
10
i5
2Ù
25
29
M
39
44
5o
5
10
i5
30
25
3o
35
4o
45
5i
5
10
i5
20
26
3i
36
4i
46
53
5
10
16
21
26
3i
36
42
47
53
5
1 1
16
21
27
32
37
42
48
54
5
1 1
16
22
27
32
38
4i
49
55
6
1 1
17
32
38
33
3q
44
5o
56
6
1 1
17
22
38
34
3q
45
5o
57
6
1 1
17
33
3q
34
4o
46
5i
58
6
12
17
23
2q
35
4i
46
52
59
6
12
18
24
3o
35
4i
47
53
60
6
I 2
18
34
3o
36
43
48
54
61
6
12
18
24
3i
37
43
4q
55
62
6
12
iq
25
3i
37
43
5o
56
63
6
i3
iq
25
32
38
44
5o
57
64
6
i3
'9
26
32
38
45
5i
58
65
7
i3
20
36
33
3q
46
53
5q
fi6
7
i3
20
26
33
40
46
53
5q
67
7
i3
20
37
34
4o
47
54
60
68
7
i4
20
27
34
4i
48
54
61
69
7
i4
21
28
35
4i
48
55
62
70
7
i4
21
38
35
43
4u
56
63
80
8
16
24
33
40
48
5C
64
72
90
0
18
37
36
45
54
63
72
81
ion
1 r
3f
3c
40
5f
6f
70
80
90
AlO
TABLE II
. — To find the time T"; the sum
of the radii
r + r"
and the chord
c being
given.
tjum of tlie Radii r -\- r". 1
Chord
C.
2,39
Uajs lilif.
2,40
Days |(lir.
2,41
2,42
2,43
Days |dif.
2,44
Days |dif.
2,45
2,46
2,47
2,48
Days |dir.
Days |dif.
Days |dif.
Days l.iif.
Days |dif.
Days |dlf.
0,00
0,0 1
O,03
o,o3
o,o4
0,000
0,449
0,899
1,348
1.797
I
2
3
4
0,000
o,45o
0,901
1, 35 1
1,801
I
I
3
4
0,000
o,45i
0,902
1,354
i,8o5
I
2
2
4
0,000
0,452
0,904
1,356
1,809
I
3
3
3
0,000
0,453
0,906
1,359
1,812
I
2
3
4
0,000
0,454
0,908
1,362
1,816
I
3
3
4
0,000
0,455
0,910
1,365
1,820
1
2
3
4
0,000
o,456
0,912
1,368
1.824
I
2
3
0,000
0,457
0,914
1,370
1,827
I
I
3
4
0,000
o,458
0,915
1,373
1, 83 1
I
3
3
4
0,0000
0,0001
0,0004
0,0009
0,0016
o,o5
0,06
0,07
0,08
0,09
2,247
2,696
3,145
3,595
4,044
4
6
7
7
8
2,25l
2,702
3,i52
3,602
4,o52
5
5
6
8
y
2,256
2.707
3,1 58
3,6ir
4,061
5
6
7
7
8
2,361
2.7i3
3,i65
3.617
4,o6g
4
6
7
8
9
2,265
2.719
3,172
3,625
4,078
5
5
6
7
8
2,270
2,724
3,178
3.632
4,086
5
e
7
6
8
2,275
2,730
3,i85
3,640
4,094
4
5
6
7
9
2.279
2.735
3.191
3,647
4,io3
5
6
7
7
8
2,284
2, 74 1
3,198
3,654
4.111
5
5
6
8
8
2,289
2,746
3,204
3,662
4,119
4
6
6
9
0,0025
0,00 36
0,0049
0,0064
0,0081
0,10
0,1 I
0,12
o,i3
o,i4
4,493
4.942
5,393
5,84i
6,290
10
II
ij
12
i3
4,5o3
4,953
5,4o3
5,853
6,3o3
9
10
1 1
12
i3
4.5 1 2
4,963
5,4i4
5,865
6,3i6
9
10
1 1
12
i3
4.521
4,973
5,425
5,877
6,329
10
11
13
i3
i3
4,53 1
4,984
5,437
5,890
6,342
9
10
II
12
14
4,540
4,9g4
5,448
5,903
6,356
9
10
II
12
i3
4,549
5,004
5,45q
5,914
6,369
lO
lO
II
12
i3
4.559
5,014
5,470
5,926
6,382
9
II
II
13
i3
4,568
5,025
5,481
5,938
6,395
9
10
11
12
13
4,577
5,o35
5,492
5,g5o
6,407
9
10
i3
0,0100
0,0121
o,oi44
0,0169
0,0196
0,1 5
0,16
0,17
0,18
0,1 g
6,739
7,188
7.637
8,086
8,535
i4
i5
16
17
18
6,753
7,2o3
7,653
8,io3
8,553
i4
i5
16
17
18
6,767
7,218
7.669
8,120
8,57.
i4
i5
16
17
18
6,781
7,233
7,685
8,i37
8,589
14
i5
16
17
18
6,795
7,248
7.701
8,i54
8,607
i4
i5
16
17
17
6,809
7,263
7.717
8,171
8,624
i4
i5
16
16
18
6,823
7,278
7.733
8,187
8,642
i4
i5
i5
17
i8
6,837
7,393
7,748
8,3o4
8,660
i4
1 5
16
17
17
6,85 1
7,3o8
7.764
8,221
8,677
i4
i4
16
16
18
6,865
7,323
7,780
8,237
8,695
14
i5
%
17
0,0225
o,o256
0,0289
o,o324
o,o36i
0,20
0,21
0,22
0,23
0,24
8,984
9.433
9,882
io,33i
10,780
'9
20
21
22
22
9,oo3
9.453
9,903
10,353
10,802
19
20
21
21
23
9,023
9.473
9.924
10,374
10,825
19
19
20
22
22
9,o4i
9.492
9.944
10,396
10,847
18
20
21
21
23
9.059
9.5i2
9.965
10,417
10,870
19
20
30
23
22
9,078
9,532
9,985
10,439
10,892
19
'9
21
21
23
9,"97
9,55i
10,006
10,460
10,915
i8
20
20
22
22
9,ii5
9.571
10,026
10,482
10,937
'9
'9
21
21
23
9, 1 34
9,59"
10,047
io,5o3
10,959
18
2r
20
21
23
9,i52
9,610
10,067
10,524
10,981
19
19
20
21
22
o,o4oo
o,o44i
o,o484
0,0529
0,0576
0,25
0,26
0,27
0,28
0,29
11,229
1 1 ,677
1 2 , 1 26
12,575
i3,o23
23
25
25
26
27
11,252
11,702
I3,l5l
I 2 ,60 !
i3,o5o
34
24
26
36
28
1 1,276
11.726
12,177
12,627
13,078
33
25
25
27
27
11,299
1 1, 75 1
12.302
12,654
i3,io5
23
24
25
26
27
I 1,323
11.775
12,227
12,680
i3,i32
24
24
25
26
27
11,346
11.799
12,252
12,706
i3,i59
23
24
26
26
27
1 1 ,369
11,823
12,378
13,733
i3,i86
23
25
25
26
27
11,392
11,848
i2,3o3
12.758
i3;2i3
33
24
25
26
27
ii,4i5
11,873
12,338
12,784
1 3,340
24
25
26
27
11,439
1 1 ,896
13,353
12,810
13,267
23
24
25
36
27
0,0625
0,0676
0,0729
0,0784
o,o84i
o,3o
0,3 1
0,32
0,33
0,34
13,472
13,920
i4,368
14,817
1 5,265
28
3i
3i
32
i3,5oo
13,949
14,39g
14,848
15,297
28
=9
3o
3i
32
i3,528
13,978
14.420
14,87g
i5,32g
28
=9
3o
3i
32
i3,556
14,007
14,459
14.910
i5,36i
28
39
3o
3i
32
1 3,584
i4,o36
14,489
14.941
15,393
28
29
=9
3o
3i
i3,6i2
i4,o65
i4,5i8
14,971
i5,424
28
29
3o
3i
33
i3,64o
14,094
i4,548
1 5,002
1 5,456
28
=9
3o
3i
32
1 3,668
i4,i23
14,578
i5,o33
1 5,488
28
29
3o
3i
3i
13,696
i4,i53
1 4,608
1 5,064
i5,5i9
28
29
3^
32
13.724
i4,i8i
1 4,637
15,094
i5,55i
27
38
3o
3i
3i
0,0900
0,0961
0,1024
0,1089
0,1 i56
0,35
o,36
0,37
o,38
0,39
i5,7i3
16,161
16,609
17.057
i7,5o5
33
34
35
36
37
1 5,746
i6,ig5
16,644
17,093
17,542
33
34
35
36
37
1 5,779
16, 22g
16,67g
17,129
17.579
33
34
35
35
36
i5,8i2
16,263
16,714
17,164
I7,6i5
33
34
34
36
37
1 5,845
16,297
16,748
17,200
17,652
32
33
35
36
36
15,877
i6,33o
16,783
17,236
17,688
33
34
34
35
37
l5.QIO
i6;364
16,817
17,271
17,735
32
33
35
35
36
15,942
16,397
i6,852
i7,3o6
17.761
33
34
34
36
36
15,975
1 6.43 1
16,886
17,342
17,797
32
33
34
35
36
1 6,007
1 6,464
16,920
17.377
17,833
33
33
35
35
36
0,1225
0,1296
0,1 369
0,1 444
0,l52I
o,4o
0,4 1
0,42
0,43
0,44
17,953
i8,4oi
1 8,848
19,296
19.743
38
38
4o
4o
43
18^439
18,888
19,336
19,785
37
39
39
4i
4i
18,028
18,478
i8,g27
19.377
19,826
38
38
40
4o
43
18,066
[8,5i6
■8,967
19,417
19,868
37
39
39
4i
41
i8.To3
18,555
19,006
19,458
19,909
38
38
4o
40
4i
i8,i4i
18,593
19,046
19,498
19,950
37
38
39
4o
4i
18,178
i8,63i
19,085
ig,538
19,991
37
39
39
4o
4i
i8.2i5
18,670
19,124
19.578
20,o33
37
38
39
40
4i
18,353
18,708
ig,i63
ig,6i8
20,073
37
38
3q
4o
4i
18.289
18,746
19,202
■ 9,658
20,1 14
37
38
39
40
40
o,r6oo
0,1681
0,1764
0,1849
0,1936
0,45
o,5o
0,55
0,60
o,65
0,70
20,191
22,426
24,659
26,889
29,117
3 1, 340
42
48
52
57
61
67
20,233
22,474
24,71 1
26,946
29,178
3i,4o7
43
47
52
57
62
66
20,276
22,521
24.763
27,oo3
2g,24o
3 1, 473
42
47
52
56
61
66
2o,3i8
22,568
24,81 5
27,059
29,301
31,539
42
47
52
57
61
66
2o,36o
22,61 5
24,867
27,116
39,362
3i,6o5
42
46
5i
56
61
66
20,402
22,661
24,918
27,172
29,423
31,671
42
47
52
56
61
66
20,444
2 2 ,708
24,970
27,228
39.484
31,737
42
47
5i
57
6i
66
30,486
22,755
25,021
27,285
29,545
3i,8o3
43
46
5i
56
61
65
20,528
22,801
25,073
27,341
29,606
3 1, 868
42
47
5i
55
60
65
20.570
2 2,848
25,123
27,396
29,666
31,933
4i
46
5i
56
61
66
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,75
0,80
o,85
0,90
0,95
1,00
33,56o
35,777
37,989
4o, 1 96
42,399
44,596
72
76
81
86
91
96
33,632
35,853
38,070
40,283
42,490
44,692
71
76
8!
86
9'
96
33,7o3
35,929
38,i5i
4o,368
42,58r
44,788
71
76
81
86
96
33,774
36,oo5
38,232
40,454
43,671
44,884
71
76
80
85
95
33,845
36,o8i
38,3i2
4o,53g
42,762
44 ,g79
71
75
81
86
90
95
33,916
36,1 56
38,3g3
40,625
42,853
45,074
70
76
80
85
90
95
33,986
36,232
38,473
40,710
42,q42
45,i69
71
75
8o
85
9?
95
34,057
36.307
38,553
40,795
43.032
45,264
70
75
80
85
95
34,127
36,383
38,633
40,880
43.122
45,359
70
75
80
84
89
94
34,197
36,457
38,7i3
40,964
43,311
45,453
70
75
79
85
89
94
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
2,8561 1
2,8800 1
2,9041 1
2,9282 1
2,9525 1
2,9768 1
3,0013 1 3,0258 1
3,0505 1
3,0752 1
ê
4 . (r + r")= or r" + r"^ nearly. |
447
448
449
45o
45 1
452
453
454
455
456
457
45
45
45
45
45
45
45
45
46
46
46
89
90
90
90
go
90
91
91
91
91
91
1 34
i34
i35
i35
i35
1 36
1 36
1 36
1 37
1 37
1 37
179
179
180
180
180
181
181
182
182
182
i83
224
224
225
225
226
226
237
227
228
228
239
268
269
269
270
271
271
272
272
273
274
274
3i3
3i4
3i4
3i5
3i6
3i6
3,7
3i8
3ig
3.g
320
358
358
35g
36o
36 1
362
362
363
364
365
366
402
4o3
404
4o5
406
407
4d8
4og
4io
4io
All
458
46
I
92
2
1 37
3
i83
4
220
5
275
6
321
7
366
8
4X2
9
TABLE II. — To find the time T\ the sum of the radii r-}-r ", and the chord c being piven.
tiuni of Iho Kudu r-^-r". 1
Pro[
. purl
lor tito
.urn
oltl
e Radii.
■^"
Churd
C.
0,00
2,49
^1 f^/»
2,51
2,52
2,53
llHya |dir.
2,54
l)«y« |dif.
M2|J|4|5|b|7|Hl9
Af^^JKJ
1
2
3
0 0
0 0
0 1
0
1
1
0
I
1
1
2
1
2
I
I
2
3
2
I
2
3
Dnys |dil'.
U«)s |dif.
Days |dir.
Duys |dir.
0,00(
1
0,000
OjOtiC
0,000
0,000
0,000
0,0000
0,01
o,45c
) I
0,460
0
0,460
1
0,461
1
0,462
1
o,463 I
0,000 1
4
0 1
2
2
3
3
3
4
0,02
0,Q|-
3
0,919
2
0,921
3
0,92;
2
0,925
1
0,926 2
0,0004
o,o3
i,37t
) c
1,379
2
i,38i
3
1,384
3
1,387
3
1 ,390 2
O,O0OC)
5
I 1
2
2
3
3
4
4
5
o,o4
1,83;
^
1,838
4
1,842
4
1,84e
3
1,849
4
1,853 4
0,0016
«)
I I
2
2
3
4
4
5
5
7
1 1
2
3
4
4
5
6
6
o,o5
2,2qC
5
2,298
4
2,3o3
5
2,307
5
2,3l2
4
2,3i6 5
0,0025
8
1 2
2
3
4
5
6
6
7
0,06
2,75:;
5
2,757
(
3,763
5
2,76b
6
2,774
5
2,779 6
o,oo36
9
1 2
3
4
5
5
6
7
8
0,07
3,2IC
3,217
f
3,323
7
3,23c
e
3,236
7
3,243 6
0,0049
3
3
4
4
5
r
6
8
0,08
3,66c
3,676
8
3,684
7
3,691
7
3,698
8
3,706 7
0,0064
10
'
5
6
6
7
8
8
S
0,09
4,12e
6
4, 1 36
8
4,i44
8
4,i52
e
4,161
8
4,169 8
0,0081
1 3
1 2
4
7
7
9
10
11
0,10
4,58C
e
4,5q5
i(
4,6o5
9
4,614
9
4,623
\
4,63:
9
0,0100
i3
i4
1 3
1 3
4
4
5
5
7
8
g
9
10
10
12
i3
0,1 1
5,o45
K
5,<.55
K
5,o65
It
5,075
10
5,o85
11
5,oq5| 10
0,0121
7
1 1
0,15
5,5o3
1 I
5,5i4
I I
5,525
1 1
5,536
11
5,547
1 1
5,55t
1 1
0,01 44
i5
2 3
5
6
8
9
11
12
i4
o,i3
5,963
12
5.9-4
12
5,986
12
5,998
12
6,010
1 1
6,02
12
0,0169
16
2 3
5
6
8
10
1 1
i3
i4
0,14
6,420
i3
6,433
i3
6,446
i3
6,459
i3
6,472
i3
6,48d
12
0,0196
17
18
2 3
2 4
5
5
7
7
9
9
10
1 1
13
i3
i4
■ 4
i5
16
o,i5
6,879
i4
6,8g3
i3
6,906
14
6,Q20
i4
6,934
i4
6,94f
i3
0,0225
19
2 4
6
8
10
11
i3
i5
'7
0,16
7,33-7
i5
7,352
i5
7,367
14
7,38 1
i5
7,3c)6
i5
7,4ii
i4
o,o256
0,17
7,796
i5
7,811
16
7,827
16
7.843
i^
7,(-'58
i()
7,87^
i5
0,0389
20
2 4
6
8
10
12
i4
16
18
0,18
8,254
i-j
8,271
16
8,287
17
8,3o4
16
8,320
I",
8,33-
16
o,o324
21
2 4
6
8
1 1
i3
i5
'7
19
o,ig
8,7"
18
8,73o
17
8,747
18
8,765
17
8,782
I-
8,79(
18
o,o36i
22
2 3
2 4
2 5
7
7
9
9
11
12
i3
i4
i5
16
18
18
30
21
0,20
9:'7I
18
9,189
18
9,207
'9
9.336
if
9,244
18
9,363
19
o,o4oo
24
2 5
7
10
12
14
17
19
22
0,21
9,629
19
9,648
30
9,668
19
9,687
19
9,706
11,
9,73?
19
0,044 1
25
3 5
8
i3
i5
18
23
0,22
10,087
20
1 0, 1 07
31
10,128
30
10,148
30
io,ifi8
3C)
10,188
20
0,0484
3 5
3 5
3 6
i3
i4
i4
16
16
17
18
23
24
25
0,23
10,545
22
10,567
21
io,588
21
10,609
21
io,63o
21
io,65i
21
0,0529
26
8
8
8
10
21
0,24
1 1 ,oo3
13
1 1 ,026
22
11,048
22
1 1 ,070
33
1 1 ,092
32
ii,ii4
21
0,0576
27
28
1 1
1 1
19
20
22
22
0,25
1 1 ,463
23
11,485
23
ii,5o8
23
ii,53i
22
11,553
23
11,576
23
0,0625
29
3 6
9
12
i5
17
30
23
26
0,26
1 1,920
24
1 1 ,944
24
1 1 ,968
33
1 1,991
24
12,Ol5
24
i2,o3g
24
0,0676
3o
3 6
9
9
10
13
i5
18
21
24
27
0,27
12,378
25
i2,4o3
24
12,427
25
12,453
25
12,477
25
12,5o2
24
0,0729
3i
3 6
12
16
19
23
25
28
0,28
12,836
25
13,861
36
12,887
26
12.913
36
12,939
25
12,964
26
0,0784
32
3 6
i3
16
19
22
26
'9
3o
0,29
13,394
26
i3,32o
27
1 3,347
27
1 3,374
26
i3,4oo
27
13,427
26
0,084 1
33
3 7
10
i3
17
20
23
36
34
3 7
10
14
17
20
34
27
3i
o,3o
1 3,75 1
28
1 3,779
28
1 3,807
27
1 3,834
38
1 3,862
27
i3,88g
27
0,0900
0,3 1
14.209
29
i4,238
38
14,266
29
14,295
38
14,333
26
i4,35i
29
o,og6i
35
4 7
11
i4
18
21
25
28
32
0,32
14.667
29
i4,6y6
3o
14,726
29
14,755
3o
14,785
29
i4,8i4
29
0,1024
36
4 7
11
i4
18
22
25
29
33
0,33
i5,i25
35
i5,i55
3o
i5,i85
3i
i5,3i6
3o
1 5,246
3o
15,276
3o
o,io8g
37
4 7
11
i5
19
22
26
3o
33
0,34
1 5,582
32
1 5,614
3i
1 5,645
3i
15,676
3i
15,707
3i
1 5,738
3i
o,ii56
38
39
4 8
4 8
11
12
i5
16
19
20
23
23
27
27
3o
3i
34
35
0,35
i6,o4o
32
16,072
32
i6,io4
33
i6,i36
32
16,168
32
16,200
32
0,1225
o,36
16,497
33
i6,53o
34
i6,564
33
16,597
33
i6,63o
33
1 6,663
32
0,1296
4o
4 8
12
16
20
24
28
32
36
0,37
16,955
34
16.989
34
17,023
M
17,057
34
17,091
34
17,135
33
0,1369
4i
4 8
12
16
21
25
29
33
37
0,38
17,412
35
17:447
35
17,482
35
17,517
35
17,552
35
17,587
34
0,1 444
42
4 8
i3
17
21
25
=9
34
38
0,39
17,869
36
i74)o5
36
1 7,94 >
36
17,977
36
i8,oi3
35
i8,o48
36
0,l521
43
44
4 9
4 9
i3
i3
17
18
22
33
26
26
3o
3i
34
35
39
40
o,4o
18,326
37
i8,363
37
18,400
37
18,437
37
18,474
36
i8,5io
37
0,1600
45
46
47
48
49
I t
i4
i4
i4
i4
i5
18
18
23
23
24
34
25
32
32
33
M
34
36
37
38
38
3g
4i
4i
42
43
A4
0,4 1
18,784
37
18,821
38
18,859
38
18,897
38
18.935
37
18,972
38
0,1681
27
28
0,42
19,241
38
19,279
39
19,318
39
19,357
38
19:395
39
ig,434
38
0,1764
: 9
28
0,43
19,698
39
19,737
40
19,777
3q
19,816
4o
19,856
39
19,895
4o
0,1849
. 9
D 10
5 10
19
0,44
20, 1 54
4i
20,195
4i
20,336
40
30,276
4i
20,3i7
A'-
20,357
40
0,1936
19
20
29
29
0,45
20,611
42
2o,653
4i
30,694
42
30,736
4i
20,777
4t
20,818
4i
0,2025
5o
3 10
i5
20
25
3o
35
4o
45
o,5o
22,894
46
22,940
46
33,986
46
23,o33
46
23,078
46
33,124
46
0,2 5oo
5i
J 10
i5
30
26
3i
36
4i
42
42
46
0,55
25,174
5i
25,225
5i
25,276
5i
25,337
5o
25,377
5i
25,438
5o
o,3o25
52
1 10
16
2 1
26
3i
36
4?
0,60
27,453
56
27,508
55
27,563
56
27,619
55
27,674
55
27,729
55
o,36oo
53 -
' 1 1
16
21
27
27
32
37
38
48
o,65
59-727
60
29,787
60
29,847
61
39,908
59
29,967
60
3o,o37
60
0,4225
54 '-
1 1
16
22
32
43
A9
0,70
3 1, 999
65
32,064
65
32,129
64
32,193
65
32,358
65
32,323
64
o,4goo
55 (
) 11
17
32
38
33
39
44
5o
0,75
34,367
70
34,337
69
34.406
70
34,476
69
34,545
70
34,61 5
69
0,5625
56 t
) II
17
22
28
M
39
45
5o
0,80
36,533
74
36.6o6
75
36;68i
74
36,755
74
36,829
74
36,9o3
74
o,64oo
57 f
11
17
23
29
M
4o
46
5i
o,85
38,793
80
38,872
79
38,95i
79
3g,o3o
79
39,109
79
39.188
78
0,7225
58 f
12
17
23
29
35
4i
46
52
0,90
4 1, 049
84
4i,i33
84
41.217
84
4i,3oi
83
4 1,384
84
4 1, 468
83
0,8100
59 t
12
18
24
3o
35
4i
47
53
0,95
43,3oo
90
43,390
88
43,478
89
43,567
89
43,656
88
43,744
88
0,9025
1,00
45,547
94
45,64 1
94
45,735
94
45,829
93
45,922
93 46.01 51
93
1 ,0000
c2
60 6
61 6
63 6
63 6
64 6
12
13
13
i3
i3
18
18
'9
■9
24
24
25
25
26
3o
3i
3i
32
32
36
37
37
38
38
42
43
4i
44
45
48
t
5o
5i
54
55
56
57
58
3,1001 1
3,1250 1
3,1501 1
3,17f
52
3,2005 1 3,2258 |
\ . {r -\- r'' )'^ or r'* -\- r " "^ nearly. |
456
457
458
459
460
461
462
463
464
■9
—
—
—
—
—
—
65 7
i3
20
26
33
39
46
52
59
I
46
46
46
46
46
46
46
46
46
I
66 7
i3
20
36
33
4o
46
53
59
2
Qi
91
92
92
92
92
V
93
93
2
67 7
i3
20
37
M
40
47
54
60
3
■37
1 37
1 37
1 38
1 38
1 38
1 39
139
139
3
68 7
i4
20
27
34
4i
48
54
61
4
182
i83
i83
184
184
184
i85
i85
186
4
69 7
14
21
28
35
41
48
55
62
5
228
229
229
275
23o
23o
23 1
23l
232
232
5
6
274
274
275
276
277
277
278
278
6
70 7
i4
21
28
35
42
49
56
63
7
319
330
321
321
322
333
323
324
325
7
80 8
16
24
32
4o
48
56
64
72
8
365
366
366
367
368
369
370
370
371
8
90 9
18
27
36 45
54
63
72
81
4io
4ii
4l2
4i3
4i4
4i5
416
417
418
9
|ioo 10
20 1
3o
io 5o
So
70
80
22
TABLE II. — To find the time T; the sum of the radii r-j-i"", and the chord c being given.
Sum of the Radii r-f-r". |
Chord
C.
2,55
2,56
2,57
2,58
2,59
Days |dir.
2,60
2,61
2,62
Days |dir.
2,63
Days |dif.
2,64
Days|dif.
Days |dir.
Days |dil'.
Days |dif.
UaysldiC.
Days |dir.
Day» |dif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
o,464
I
0,465
I
0,466
I
0,467
I
0,468
I
0,469
1
0,470
0
0,470
1
0,471
I
0,472
I
0,0001
0,02
0,928
3
0,930
3
0,932
2
0,934
2
0,936
I
0,937
2
0,939
2
0,941
2
0,943
2
0,945
I
0,0004
o,o3
i,3q2
3
1,395
3
1 ,398
3
I,4oi
2
i.4o3
3
1, 406
3
1,409
2
i,4ii
3
I,4l4
3
1,417
2
o,ooog
o,o4
1,857
3
1,860
4
1,864
3
1,867
4
1,871
4
1,875
3
1,878
4
1,882
3
1,885
4
1,889
4
0,0016
o,o5
2,321
4
2,325
5
2,33o
4
2,334
5
2,33g
4
2,343
5
2,348
4
2,352
5
2,357
4
2, 36 1
5
0,0025
0,06
2,785
5
2,790
6
2,796
5
2,801
6
2,807
5
2,812
5
2,817
6
2,823
5
2,828
6
2,834
5
o,oo36
0,07
3,249
6
3,255
7
3,362
6
3.268
6
3,274
7
3,281
6
3,287
6
3,293
7
3,3oo
6
3,3o6
6
0,0049
0,08
3,7i3
-7
3,720
8
3,728
7
3;735
7
3,742
7
3,749
7
3,756
8
3,764
7
3,771
7
3,778
7
0,0064
o,og
4,177
8
4,i85
8
4,193
9
4,203
8
4,210
8
4,218
8
4,226
8
4,234
8
4,242
8
4,25o
8
0,008 1
0,10
4,64i
9
4,65o
9
4,659
9
4,668
9
4,677
10
4,687
9
4,6g6
9
4,7o5
8
4,7i3
9
4,722
9
0,0100
0,11
5,io5
10
5,ii5
10
5,125
10
5,i35
10
5,i45
10
5,i55
10
5,i65
10
5,175
10
5,i85
10
5,195
9
0,0121
0,13
5,5fi9
II
5,58o
1 1
5.591
II
5,602
11
5,6 1 3
1 1
5,624
10
5,634
11
5,645
II
5,656
1 1
5,667
10
0,01 44
o,i3
6,o33
12
6,045
12
6,o57
12
6,069
1 1
6,080
12
6,092
12
6,104
12
6,116
11
6,127
12
6,139
12
0,01 6g
o,i4
6,497
i3
6,5io
i3
6,523
12
6,535
i3
6,548
i3
6,56i
12
6,573
i3
6,586
12
6,5g8
i3
6,611
i3
0,01 g6
0,1 5
6,961
14
6,975
i4
6,989
i3
7,002
i4
7,016
i3
7,02g
i4
7,043
i3
7,o56
i4
7,070
i3
7,o83
i4
0,0225
0,16
7,425
j5
7,440
i4
7,454
i5
7,469
14
7,483
i5
7,498
i4
7,5i2
14
7,526
i5
7,541
i4
7,555
14
o,0256
0,17
7,889
16
7,9o5
i5
7,920
i5
7,935
16
7,951
i5
7,966
i5
7,981
16
7,997
i5
8.012
i5
8,027
i5
0,028g
0,18
8,353
16
8,36q
17
8,386
16
8,402
16
8.418
17
8,435
16
8,45i
16
8,467
16
8,483
16
8,499
16
o,o324
0,19
8,817
17
8,834
17
8,85i
18
8,869
17
8,886
17
8,903
17
8,930
17
8,937
17
8,g54
17
8,971
17
o,o36i
0,20
9,281
18
9,299
18
9,317
18
g,335
18
9,353
18
9,371
18
9,389
18
9,407
18
9,425
18
9,443
18
o,o4oo
0,21
9-744
20
9,764
19
9,783
19
g,8o3
19
9,831
19
9,84o
18
9,858
'9
9,877
19
9,896
19
9,9' 5
19
0,044 1
0,22
10,208
20
10,228
20
10,248
20
10,268
30
10,388
20
io,3o8
30
10,328
19
10,347
20
10,367
20
10,387
20
0,0484
0,23
10,672
21
10,693
21
10,714
30
10,734
21
10,755
21
10,776
21
10,797
21
10,818
20
io,838
21
io,85g
20
o,o52g
0,24
ii,i35
22
ii,i57
22
11,179
22
1 1,201
22
I 1,223
21
11,244
33
1 1,266
21
11,287
22
11,309
22
11,331
21
0,0576
0,25
11,599
23
11,622
33
11,645
22
1 1 ,667
23
I 1 ,690
22
11,712
33
11,735
22
11,757
23
11,780
22
11,802
23
0,0625
0,26
i2,o63
23
1 2 ,oS6
34
12,1 10
23
i3,i33
24
I2,i57
24
12,181
23
13,204
23
13,337
24
12,25l
23
12,274
23
0,0676
0,27
12,526
35
I2,55i
24
13,575
25
1 3 ,600
34
12,624
25
1 3 ,649
24
12,673
24
12,697
25
12,722
24
12,746
24
0,0729
0,28
12,990
25
1 3,01 5
26
1 3,04 1
25
1 3,066
25
13,091
26
i3,ii7
25
i3,i42
35
13,167
25
13,192
25
i3,2i7
25
0,0784
0,2g
1 3,453
26
1 3,479
27
i3,5o6
26
i3,532
26
1 3,558
27
1 3,585
26
1 3,61 1
26
1 3,637
26
1 3,663
26
13,689
26
0,0841
o,3o
13,916
28
1 3,944
27
13,971
27
13,998
27
i4,025
28
i4,o53
27
14.080
27
14,107
27
i4,i34
26
i4,i6o
27
0,0900
o,3i
i4,38o
28
I 4, 408
28
1 4,436
28
i4,464
28
14,492
28
i4,53o
28
i4,548
28
14,576
28
1 4,604
28
i4,632
28
0,0961
0,32
14,843
29
14,872
29
14,901
29
1 4,930
29
14,959
29
14,988
29
15,017
29
1 5,046
29
15,075
28
i5,io3
29
0,1024
0,33
i5,3o6
3o
1 5,336
3o
1 5,366
3o
15,396
3o
15,426
3o
1 5,456
3o
1 5,486
29
i5,5i5
3o
1 5,545
3o
15,575
29
0,1 o8g
0,34
15,769
3i
1 5,800
3i
i5,83i
3i
15,862
3i
15,893
3i
15,924
3o
15,954
3i
1 5,985
3i
16,016
3o
16,046
3o
0,11 56
0,35
l6,232
33
16,264
33
16,396
32
16,328
32
i6,36o
3i
16,391
32
16,423
3i
16,454
32
16,486
3i
16.517
32
0,1225
o,36
16,695
33
16,728
33
16,761
33
16,794
32
16,826
33
16,859
32
16,891
33
16,924
32
16,956
32
16,988
33
0,1296
0,37
I7,i58
34
17,192
34
17,226
33
17,259
34
17,393
33
17,326
34
17,360
33
I7,3g3
33
17,426
34
1 7,460
33
0.1 36g
o,38
17,621
35
17,656
35
17,691
34
17,725
34
■7,759
35
17,794
34
17,828
34
17,862
35
17,897
34
17,931
34
0,1 444
0,39
18,084
36
18,120
35
i8,i55
36
18,191
35
18,226
35
18,261
35
i8,2g6
36
18,333
35
18,367
35
18,402
35
0,l52I
o,4o
18,547
36
i8,583
37
18,620
36
i8,656
36
18,692
37
18,729
36
18,765
36
18,801
36
18,837
36
18,873
35
0,1600
0,4 1
19,010
37
19,047
37
19,084
38
19,122
37
19,159
37
19,196
37
ig,233
37
19,270
37
19,307
36
19,343
37
0,1681
0,42
19,472
38
19,510
39
19,549
38
19,587
38
19,625
38
ig,663
38
19,701
38
'9,739
38
19,777
37
19,814
38
0,1764
0,43
19,935
39
19-974
39
30,01 3
39
20,o52
39
20,ogi
39
3o,i3o
39
20,169
39
20,208
38
20,346
39
20,285
39
0,1849
0,44
20,397
40
20,437
40
20,477
4o
20,517
40
20,557
40
20,597
4o
20,637
40
20,677
39
20,716
4o
20,756
39
0,1 g36
0,45
20,859
42
20,901
4i
20,942
40
20,982
4i
21,023
4i
2 1 ,064
4i
2i,io5
40
21,145
4i
21,186
4o
21,226
4i
0,2025
o,5o
23,170
46
23,3l6
45
23,261
46
23,307
45
23,352
46
23,398
45
23,443
45
23,488
45
33,533
45
23,578
45
o,25oo
0,55
25,478
5i
25,52g
5o
25,579
5o
25,639
5o
25,679
5o
25,72g
5o
25,779
49
25,838
5o
25,878
49
25,927
5o
o,3o25
0,60
27,784
55
27,839
55
27,894
55
27,949
54
38,oo3
55
28,o58
54
28,1 12
54
28,166
55
28,221
54
28,275
54
o,36oo
o,65
30,087
60
3o,i47
59
3o,2o6
59
3o,265
60
3o,325
59
3o,384
59
3o,443
59
3o,5o2
58
3o,56o
59
30,619
59
0,4225
0,70
33,387
64
32,45i
64
32,5i5
64
32,579
64
32,643
64
32,707
64
32,771
63
32,834
63
32,897
64
32,g6i
63
0,4900
0,75
34,684
69
34,753
68
34,821
69
34,890
69
34,959
68
35,027
68
35,095
68
35,i63
68
35,23i
68
35,299
68
0,5625
0,80
36,977
73
37,o5o
74
37,124
73
37,197
74
37,271
73
37,344
73
37,4.7
72
37,489
73
37,562
72
37,634
73
0,6400
o,85
39,266
79
39,345
78
3g,43 3
78
39,501
78
39,579
78
3g,657
77
39,734
78
39,812
77
39,889
77
39,966
77
0,7225
0,90
4i,55i
84
4 1, 635
83
41,718
83
4 1 ,800
83
4i,883
82
4i,g65
83
42,048
83
42,i3o
82
42,212
83
42,294
81
0,8100
0,95
43,832
88
43,920
88
44,008
88
44,096
87
44,i83
87
44,270
87
44,357
87
44ÀM
87
44,53 1
86
44,617
87
o,go25
1,00
46,108
93
46,201
93
46,294
92
46,386
92
46,478
92
46,570
92
46,663
92
46,754
91
46,845
92
46,937
91
1 ,0000
3,2513
3,2768
3,3025
3,3282
3,3541
3,3800 1
3,4061
3,4322
3,4585 1
3,4848 1
C^
è . (r H- r " )•' or
r^ -{- r"^ nearly.
1
462
463
464
465
466
467
468
469
470
471
472
46
46
46
47
47
47
47
47
47
47
47
92
93
93
93
93
93
94
94
94
94
94
i3g
i3g
i3g
1 40
i4o
i4o
1 40
i4i
i4i
i4i
142
i85
1 85
186
186
186
187
187
188
188
188
189
23l
232
232
233
233
234
234
235
235
236
236
277
278
278
279
280
280
281
281
282
283
283
323
324
325
326
326
327
328
328
329
33o
33o
370
370
371
372
373
374
374
375
376
377
378
4i6
417
418
419
419
420
421
422
423
424
425
473
47
I
P5
3
142
3
189
4
237
5
284
6
33i
7
378
8
426
9
TABLE II. — To linil llie tinu- T : (lie sum of (he lailii r-{-r", and the chord c beino; given.
Sum ol' Iho Ra'lii r-f-r".
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,u5
0,06
0,07
0,08
0,09
0,10
0,1 I
0,I2
0,1 3
0,1 4
o,,5
0,lt)
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0^2
0.43
0,44
0,45
o,5o
0,55
0,60
0,65
0,73
0,80
0,85
o,go
0,95
1,00
2,65
Davs {dif.
0,000
0,4-3
0,946
i,4i(
1,89
2,366
2.839
3.3i2
3,785
4,258
4,73i
5,204
5,6^7
6,i5i
6,624
7,09-
7,569
8,042
8,5i5
8,988
9,461
9,934
10,407
10,879
11,352
11,825
12,297
12,770
13,242
i3,7i5
14,187
14,660
I 5,1 32
i5,6o4
16,076
16,549
17,021
17,493
17,965
18,437
18.908
19.380
19,852
20,324
20,795
2 1 ,267
23,623
25.977
28;'329
30,678
33,024
35,367
37,707
40,043
42,375
44,704
47,028
2,66
Days |dif.
3.5113
0,000
0,474
1
0,948
2
1.422
;
1,896
4
2,3-0
5
2,844
6
3,3i8
-
3,-Q2
-
4,2(10
8
4.740
Q
■3.214
\<
5,(jSS
1 1
6.102
1 )
6,636
12
7,110
t3
7,584
i4
8,o58
t5
8,53i
16
9,oo5
17
9'4-9
18
9,953
18
10,426
20
10,900
20
11,373
22
11,847
22
12,321
23
12,794
24
13,267
25
i3,74i
26
i4,2i4
27
14.687
28
i5,i6i
28
1 5,634
2Q
16,107
3o
i6,58o
3l
i7,o53
32
17,526
33
17,999
34
18,472
34
18,944
36
19.417
37
19,890
3-
20,362
38
2o,835
39
2i,3o7
40
23,668
AÂ
26,026
4q
28,382
54
3o,736
58
33,087
63
35,435
67
37,779
72
4o,I20
7-^
42.457
81
44,790
86
47,119
90
2,67
Unys liUr.
3,5378
0,000
0,475
o.gSo
1,425
1 ,900
2.375
2 ,85o
3,325
3.-99
4.2-4
4.749
5.224
5.699
(J.1-4
6,(548
7.123
7,598
8,073
8,547
9,02
9.497
9.971
10,446
10,920
11,395
12,344
12,81
13,292
13,767
14,241
i4,7i5
15,189
1 5,663
16,1 37
16,611
17,085
17,559
i8,o33
i8,5o6
18,980
19,454
19-927
2O,400
20,874
21,347
23,712
26,075
28,436
30,794
33,i5o
35,5o2
37,85 1
40,197
42,538
44,876
47.209
2,68
Days |dir.
3,5645
0,000
0,476
0,952
1,427
1,903
2,379
2.855
3,33i
3, So
4,282
4,758
5,234
5,"
(■),i85
6,661
7,i3-
7.612
8,088
8,563
9,039
9.5i4
9,990
10,465
10,941
ii,4i6
11,892
1 2 ,367
12,842
i3,3i7
13,792
14.268
1 4,743
i5,2i8
15,693
16,167
16,642
17,117
17,592
18,066
i8,54i
19,016
19,490
19,964
20,439
20,913
21,387
23,757
26,125
28,490
3o,853
33,212
35,56g
37,923
40,273
42.619
44,962
47,3oo
19
3,5912
2,69
Days (dif.
0,000
0,477
0,953
i,43o
'.■907
2.384
2,860
3,337
3,814
4,290
4,767
5,244
5,720
6.19
6,673
7,1 5o
7,626
8,io3
8,579
9,o56
9,532
10,009
io,485
10,961
11,438
11.914
12,390
12,866
13.342
i3,8i8
14,294
14,770
1 5,246
l5,722
16,198
16,673
17.149
17,625
18,100
18,576
i9,o5i
19,527
20,002
20,477
20.952
21,427
23,802
26,174
28,543
30,911
33,275
35,637
37,995
40,349
42,700
45,047
47,390
3,6181
2,70
Days jdif.
0,000
0,47
0,955
1,433
1,910
2,388
2,866
3,343
3,821
4,298
4,776
5,253
5,73i
6,208
6,686
7,i63
7,64i
8,11
8,595
9.073
9,55o
10,027
io,5o4
10,982
11,459
1 1 ,g36
12,4 "
12,890
1 3,367
1 3,844
i4,32i
14.798
15,274
1 5.751
16,228
16,704
17,181
17,658
i8,i34
18,610
19,087
ig,563
20,o3g
20,5i5
20,991
2 1 ,467
23,846
26,223
28,597
30,96g
33,338
35,704
38, 066
40.426
42,781
45,i33
47,480
3,6450
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
o,oo36
0,0049
o,oû(i4
0,0081
0,0100
0,01 2 1
0,01 44
l^Olfxj
0,0196
0,0225
o,o256
0,028g
o,o324
o,o36i
o,o4oo
0,044 1
0,0484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,0841
0,0900
0,0961
0,1024
0,1089
0,1 156
0,1225
0,1296
0,1 36g
0,1444
0,l52I
0,1600
0,1681
0,1764
0,1849
0,1936
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
~72"
(r +
nearly.
471
472
473
474
475
476
477
47
47
47
47
48
48
48
94
94
95
95
95
95
95
i4i
142
142
142
143
143
143
188
189
igo
190
190
191
2 36
236
237
237
238
238
239
283
283
284
284
285
286
286
33o
33o
33 1
332
333
333
334
3-7
378
3-8
379
38o
38 1
382
424
425
426
427
428
428
429
478
48
143
191
23g
287
335
382
43f.
Trop, parts for ilto sum ut' tho Kadii.
I I 2 I 3 I 4 |5|6|7|8|9
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
100
3ol4o
56
54 63
6o| 70
23
23
24
a5
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
4o
4i
4i
42
43
4Â
45
46
47
5,
54
55
49 56
52 59
53 5g
60
61
62
63
72
81
90
aU
TABLE
II.-
-To find the time
T;
the sum of the radii r
+ '•
', and the chord c
being given.
Sum of the Kadii r-\-r". |
Chord
C.
2,71
2,72
2,73
2,74
2,75
2,76
Days |dir.
2,77
Days Idif.
2,78
Days |dif.
2,79
2,80
Daysldi
Days Idif.
Days |dif.
Days |dif.
Days |dif.
Days |dif.
Days|dit'.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,478
I 0,479
I
0,480
I
o,48i
1
0,482
1
o,483
1
0,484
1
o,485
I
0,486
0
0,486
I
0,000 1
0,02
0,957
2 0,959
2
0.96 1
I
0,962
2
0,964
2
0,966
2
0.968
1
0,969
2
0,971
3
0,973
1
0,0004
o,o3
1,435
3 1,438
3
1, 44 1
2
1,443
3
1,446
3
1,449
2
1, 45 1
3
1,454
2
1,456
3
1,459
3
0,0009
o,o4
1,914
3 1,917
4
1,921
4
1,925
3
1,928
4
1,932
3
1,935
3
1,938
4
1,942
3
1,945
4
0,0016
o,o5
2,392
5 2,397
4
2, 40 1
5
2,406
4
2,4l0
4
2,4i4
5
2,419
4
2,423
4
3,427
5
2,432
4
0,0025
0,06
2,871
5 2,876
5
3,881
6
2,887
5
2,892
5
2,897
5
2,902
6
2,908
5
2,913
5
2,918
5
o,oo36
0,07
3,349
7 3,356
6
3,362
6
3,368
6
3,374
6
3,38o
6
3,386
6
3,392
6
3,398
7
3,4o5
6
0,0049
0,08
3,828
7 3,835
7
3,842
7
3,849
7
3.856
7
3,863
7
3,870
7
3,877
7
3,884
7
3.8qi
7
0,0064
0,09
4,3o6
8 4,3 i4
8
4,322
8
4,33o
8
4,338
8
4,346
8
4,354
7
4,36i
8
4,369
8
4,377
8
0,008 1
0,10
4,785
8 4,793
9
4,802
9
4,811
9
4.820
9
4,829
8
4,837
9
4,846
9
4,855
8
4,863
9
0,0100
o,u
5.263 I
0 5,273
9
5,383
10
5,293
10
5,3o2
9
5,3ii
10
5,321
10
5,33i
9
5,340
10
5,35o
9
0,0121
0,12
5,74 1 I
I 5,752
II
5,763
10
5,773
II
5,784
10
5,794
11
5,8o5
10
5,81 5
11
5,826
10
5,836
10
0,01 44
o,i3
6,220 r
I 6,23i
12
6,243
II
6,254
12
6,266
11
6,277
11
6,288
12
6,3oo
11
6,3 11
II
6,322
12
0,01 69
0,1 4
6,698 I
2 6,710
i3
6,733
13
6,735
12
6,747
i3
6,760
13
6,772
12
6,784
12
6,796
12
6,808
i3
0,0196
0,1 5
7,176 I
4 7,190
i3
7,2o3
i3
7,216
i3
7,229
i3
7,242
i3
7,255
i4
7,269
i3
7,282
i3
7,295
i3
0,0225
0,16
7,655 I
4 7,669
i4
7,683
i4
7,697
i4
7,711
i4
7,725
i4
7,739
14
7,753
i4
7,767
i4
7,781
i4
0,0256
0,17
8,i33 I
5 8,i48
i5
8,i63
i5
8,178
i5
8,193
i5
8,208
i5
8.223
i4
8,237
i5
8,252
i5
8.367
i5
0,0289
0,18
8,611 I
6 8,627
16
8,643
16
8,659
16
8,675
i5
8,690
16
8,706
16
8,722
16
8,738
i5
8,753
16
o,o324
0,19
9,o8g I
7 9>'o6
17
9,123
17
9,i4o
16
9,1 56
17
9,173
17
9,190
16
9,206
17
9,223
16
9,239
17
o,o36i
0,20
9,568 I
7 9,585
18
9,6o3
17
9,620
18
9,638
18
9,656
17
9,673
18
9.691
17
9,708
17
9-725
18
o,o4oo
0,21
io,o46 I
8 10,064
19
io,o83
18
10,101
19
10,120
18
io,i38
18
10,1 56
19
10,175
18
10,193
18
10,31 1
'9
o,o44i
0,22
10,524 I
9 10,543
20
io,563
'9
io,583
19
10,601
20
10,621
19
io,64o
19
1 1 ,65q
'9
10,678
19
10,697
'9
o,o484
0,23
1 1 ,002 2
0 11,022
21
1 1,043
20
ii,o63
20
ii,o83
20
1 1 . 1 o3
20
11,123
20
11,143
20
ii,i63
30
ii,iS3
20
0,0529
0,24
1 1 ,480 2
1 ii,5oi
31
11,522
21
11,543
22
11,565
21
11,586
21
1 1 ,607
21
11,628
20
1 1 ,648
21
1 1 ,669
21
0,0576
0,25
11,958 2
2 1 1 ,980
22
I 2 ,002
22
12,024
22
12,046
22
13,068
22
12,090
23
12,112
31
i2,i33
22
I3,i55
22
0,0625
0,26
12,436 2
3 12,459
23
12,482
23
i2,5o5
23
12,528
22
i2,55o
23
12,573
23
12,596
22
12,618
23
13,641
23
0,0676
0,27
12,914 2
4 13,938
34
12,962
23
12,985
24
13,009
24
i3,o33
23
i3,o56
24
1 3,080
23
i3.io3
24
13,127
23
0,0729
0,28
13,392 2
4 i3,4i6
25
1 3, 44 1
25
1 3,466
24
13,490
25
i3,5i5
34
i3,53g
25
1 3,564
24
i3,588
25
i3,6i3
24
0,0784
0,29
13,870 2
5 13,895
26
13,921
25
1 3,946
36
13,972
25
1 3,997
26
i4,o23
25
i4,o48
25
14,073
35
14,098
26
o,o84i
o,3o
14,347 3
7 14,374
26
1 4,400
27
14,427
26
i4,453
26
i4,479
27
i4,5o6
26
i4,532
26
i4,558
26
14,584
26
o,ogoo
0,3 1
14,825 2
7 i4,853
38
14,880
27
14,907
27
14,934
28
14,962
27
14,989
27
i5.oi6
27
1 5,043
27
15.070
27
0,0961
0,32
i5,3o3 ?
8 i5,33i
28
15,359
28
1 5,387
29
i5,4i6
28
1 5,444
28
1 5,472
28
i5,5oo
27
1 5, 5 37
28
1 5,555
28
0,1024
0,33
15,780 3
0 i5.8io
29
i5,83q
29
1 5,868
29
15,897
29
15,936
29
1 5,955
28
15.983
29
16,012
29
1 6,04 1
29
0,1089
0,34
i6,258 3
0 16,288
3o
i6,3i8
3o
16,348
3o
16,378
3o
16,408
29
16,437
3o
16,467
3o
16,497
29
16,526
3o
0,1 1 56
0,35
16,735 3
I 16,766
3i
16,797
3i
16,838
3i
16,859
3i
16,890
3o
16,920
3i
16,951
3o
16,981
3i
17,012
3o
0,1235
o,36
I7,2i3 3
3 17,345
32
17,277
3i
17,308
32
17,340
3i
17.371
33
i7,4o3
3i
17,434
32
17,466
3i
17,497
32
0,1296
0,37
17,690 3
3 17,723
33
17,756
32
17,788
33
17,821
32
17,853
33
17,886
32
17.918
33
17,950
33
17,983
32
0,1369
o,38
18,168 3
3 18,201
34
18,235
33
18,268
34
i8,3o2
33
18,335
33
i8,368
34
i8,4o2
33
18,435
33
18,468
33
0,1 444
0,39
18,645 3
4 18,679
35
18,714
34
18,748
34
18,782
35
18,817
34
i8,85i
34
i8,885
34
18,919
34
18,953
34
0,1 52 I
o,4o
19,122 C
6 19,1 58
35
19,193
35
19,238
35
19,263
35
19,398
35
19,333
35
19,368
35
19,403
35
19,438
35
0,1600
0,4 1
19.599 :
7 19,636
36
19,672
36
19,708
36
19,744
36
19,780
36
19,816
36
19,853
36
19,888
35
19,923
36
0,1681
0,42
20,076 .
8 2o,ii4
37
30,1 5i
37
20,188
37
20,225
36
30,261
37
20,398
37
20,335
37
20,372
36
20,408
37
0,1764
0,43
20,553 ,
8 20,591
38
20,629
38
30,667
38
20,705
38
20,743
38
30,781
37
20,818
38
20,856
37
2o,8q3
38
0,1849
0,44
2 1 ,o3o ,
g 21,069
39
21,108
39
2I,l47
39
21,186
38
21,224
39
21,263
38
2i,3oi
39
21,340
38
2 1 ,378
38
0,1936
0,45
21,507 i
!o 21,547
40
21,587
39
21,626
40
21,666
4o
21,706
39
21,745
39
21,784
40
21,824
39
21,863
39
0,2025
o,5o
23,890 ^
i5 23,935
44
23,979
44
24,023
44
24.067
44
24,111
44
24,i55
44
24,199
43
24,242
44
24,286
44
o,25oo
0,55
26,2-ri i
:9 26,320
49
26,369
48
26,417
49
26,466
48
26,5 1 4
49
26,563
48
26,61 1
48
26,659
48
26,707
48
o,3o25
0,60
28,65o ;
)3 28,703
54
28,757
53
28,810
53
28,863
52
28,915
53
28,968
53
29,021
53
29,073
53
29,126
52
o,36oo
0,65
31,026
J8 3 1, 084
58
3l,!42
57
31,199
58
31.257
57
3i,3i4
57
3i,37i
57
31,428
57
3 1, 485
57
3 1, 542
57
0,4225
0,70
33,400 (
33 33,462
62
33,524
62
33,586
62
33,648
62
33,710
62
33,772
61
33,833
62
33,895
61
33,956
61
0,4900
0,75
35,771 (
56 35,837
67
35,904
67
35,971
66
36,o37
66
36,io3
66
36,169
66
36,235
66
36,3oi
66
36,367
66
0,5625
0,80
38,i38
71 38,209
71
38,280
72
38,352
70
38,422
71
38,493
71
38,564
70
38,634
71
38,7o5
70
38,775
70
o,64oo
o,85
4o,5o2
76 40,578
75
4o,653
76
40,729
76
4o,8o5
75
40,880
75
40,955
75
4i,o3o
75
4i,io5
75
41,180
75
0,7225
0,90
42,862
3o 42,942
81
43,023
80
43,io3
80
43,i83
80
43,263
80
43,343
79
43,422
80
43.5o2
79
43,58i
79
0,8100
o,g5
45,218
35 45,3o3
85
45,388
85
45,473
85
45,558
84
45,642
85
45,727
84
45,811
84
45,895
84
45,979
83
0,9025
1,00
47,570
30 47,660
90
47,750
89
47,839
89
47,928
89
48,017
89
48,106
89
48,195
89
48,284
88
48,372
88
1 ,0000
3,672
1 3,6992
3,7265
3,7538
3,7813
3,8088
3,8365
3,8642
3,8921 1
3,9200
c"
J . (r + r")' 01
7-* + r"^ nearly. 1
477
478
479
480
481
482
483
484
485
486
487
I
48
48
48
48
48
"48
"48
48
49
49
49
I
3
f.
96
96
96
96
96
97
97
97
97
97
3
3
i43
143
144
1 44
144
i45
145
145
1 46
1 46
1 46
3
4
19'
■9'
192
192
192
193
193
194
194
194
iq5
4
5
23q
239
240
240
241
34 1
342
242
243
243
244
5
6
386
3S7
287
288
289
289
290
290
291
292
292
6
7
334
335
335
336
337
337
338
339
340
340
341
7
8
382
382
383
384
385
386
386
387
388
389
3qo
8
9
429
43o
43i
432
433 1
434
4
35
4:
6
43
7
437
438
9
TABLE II. — To find the time 7"; tlie sum of the radii r-\-r ", and the chord c being given.
Siun 01' the Radii r-f-r
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
0,0")
,u6
0,1)7
o.oy
(),
0,10
0,11
0,12
û,i3
o,i4
0,1'j
0,16
0,77
0,I(J
0,30
0,21
0,22
0,23
0,24
0,23
0,26
0,27
0,28
0,29
o,3t)
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
1,00
2,81
Days (liir.
o,oou
0,487
0,974
1,462
1,949
2,436
2,923
3.41 1
3,898
4,385
4.8-2
5,359
5,84'"i
6.334
6,821
7,3o8
-,7o5
8,282
8,-69
9.256
9.743
io,2 3o
10,716
II,2o3
1 1 ,690
12,177
12,664
i3,i5o
1 3,637
14,124
i4,6io
1 5,09'
1 5,583
16.070
i6;556
17,042
17,529
i8,oi5
i8,5oi
18,987
19,473
i9!959
20,445
20,93 1
2i,4i6
21,90:
24,33o
26,755
29, 1 78
31,599
34,017
36,433
38,845
4i,255
43,660
46,062
48,460
iS
2,82
Uays |<lif^
0,000
0,488
0,976
1,464
1,952
2,440
3.929
3,417
3,9o5
4,393
3,9481
4,881
5,3('>9
5.85-
6,345
6,833
-,32I
7,809
8.29"
8,-84
9,272
9,-60
[0,248
10,736
11,223
11,711
12,199
12.686
l3,i74
1 3,66 1
i4,i49
i4,636
i5,i24
1 5,6
16,098
i6,585
17,073
i7,56o
18,047
18,534
19,021
19,508
I9'995
20,481
20,968
21,455
21,941
24,3-3
26,803
2g.23i
3 1 ',656
34,079
36,4'
38,915
41,329
43,739
46,i46
48,548
21
23
23
25
20
36
26
28
29
3o
3o
3
32
33
34
34
35
37
37
38
3q
43
48
52
56
61
66
70
74
79
83
2,83
Days \dit'.
3,9762
0,000
0,489
0,9-8
1 ,467
1,956
2,445
2,934
3,423
3.t)12
4,4oi
4,889
5,378
5,867
6,356
6,845
7,334
7,822
8,3 1 1
8,800
9,289
9
10,26*)
10,755
11,243
11,732
12,220
13,709
13,197
1 3,686
14,174
14,663
i5,i5o
1 5,639
16,137
i6,6i5
i7,io3
17,591
18,079
18,567
19,055
19,543
20,o3o
20,5i8
2 1 ,oo5
21,493
21.980
24,4'6
26,85i
29,283
3i,7i2
34,i4o
36,564
38,q85
4i,4o3
43,818
46,229
48,636
2,84
Days jdil'.
0,000
o,4go
0,980
1,470
1,959
2,449
2,939
3,429
3,9Iq
4,408
4,898
5,388
5,878
6,36-
6,857
7,347
7,836
8,326
8,81
9,3o5
9,795
10,284
10,774
11,263
11,752
12,243
I2,73i
l3,220
1 3,710
14,199
1 4,688
r5.i7
1 5,666
16, 1 55
16,64.
I7,i33
17,633
18,111
18,600
19,088 34
4,0045
19,57'
20,066
20,554
21,043
2 1,53 1
22,019
34,460
26,898
29,335
3 1 ,769
34,200
36,629
39.055
41,478
43,897
46,3
48,724
2,85
Days I dit'.
4,0328
0,000
0,491
0,981
1,472
1 ,963
2,453
2.944
3,435
3,925
4,416
4,907
5,397
5,888
6,3-8
6,869
7,36o
7,85o
8,341
8,83 1
9,321
9,812
lO,302
10,793
11,283
11,773
12,263
12,754
i3,244
i3,-34
l4,234
14.-^14
1 5,204
1 5,6(),
16,184
16,674
17,163
n,653
i8,i43
1 8,632
19,1
ig,6i3
20,101
20,590
3 1 .080
21,569
22.o58
24,5o3
36,946
29,387
3i,825
34,261
36,694
3g,i25
4i,552
43,975
46,395
48,812
2,86
Diiy3 |dir.
4,0613
0,000
0,492
0,983
1,475
1 ,966
2,458
2,949
3,441
3,932
4:424
4,915
5,407
5,898
6,39c
6,881
7,372
7,864
8,355
8,847
9,338
9,829
10,320
10,812
ii,3o3
11,794
12,285
12,776
13,267
i3,758
14,249
1 4,740
i5,23i
i5,72i
16, 312
16,703
17,194
17,684
18,175
i8,665
19,156
ig,646
20,1 36
20,627
21,117
2 1 ,607
22,097
24,546
26,994
29,439
31,882
34,322
36,759
3g,ig4
41,626
44,o54
46,478
48,899
'9
0,0000
0,000 1
o,of)o4
o,<ioot)
0,0016
0,0035
o,oo36
o,oo4g
0,0064
0,008 1
0,0100
0,01 3 1
0,01 44
0,01 6g
0,0196
0,03 3 5
o,o356
0,028g
o,o324
o,o36i
o,o4oo
o,o44 1
o,o484
o,o52g
0,0576
0,0625
0,0676
0,0729
0,0784
0,084 1
0,0900
0,0961
0,1024
0,1089
0,11 56
0,1225
0,1 396
0,1 36g
0,1444
0,l521
0,1600
0,1681
0,1764
o,i84g
o,ig36
0,3025
0,2 5oo
o,3o25
0,3600
0,4225
o,4goo
0,5625
0,6400
0,7225
0,8100
0,9025
,0000
Prop, parts for tho fioiii of tho Kadii.
1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9
4,0898
(r -i- r")^ or r^ 4" '" " ^ nearly.
485
486
487
488
48g
490
491
49
49
49
49
49
49
49
97
97
97
98
98
98
98
i46
i46
i46
i46
i47
1 47
l47
194
194
195
195
196
196
196
243
243
244
244
245
245
246
291
292
292
2g3
293
294
2g5
340
340
341
342
342
343
344
388
389
390
3qo
391
392
393
437
437
438
439
44o
44 1
443
492
148
197
246
295
344
394
443
10
8
9
9
10 II
22 26
20 23
5
5
6
7
8
9
10
11
12
i3
i4
i4
i5
16
17
'9
20
21
22
33
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
4o
4i
4i
42
43
44
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
72
81
90
TABLE ir
. — Tofinil the time T
; the sura of the radii
r + r",
ant
the chord
c being
given.
Sum of tlie Raiiii r -j- r". 1
Chord
c.
2,87
2,88
2,89
2,90
2,91
2,92
2,93
2,94
2,95
2,96
Days Idif.
Diiys Idif.
D.iy8 Irlif.
Days |dif.
Days IdiC.
Days |dif.
Days |dir.
Days Idit".
Days |dif.
Days {dir.
0,00
0,01
0,02
o,o3
o,o4
0,000
0,493
0,985
1,477
1,970
I
2
3
3
0,000
0,493
o,9«7
1,480
1 ,973
I
I
2
3
0,000
0,494
0,988
1,482
1,976
I
2
3
4
0,000
0,495
0,990
1,485
1,980
I
3
2
3
0,000
0,496
0,993
1,487
1,983
1
3
4
O,ùù0
0,497
0,993
1,490
1,987
I
2
3
3
0,000
0,498
0,995
1,493
1,990
0
2
2
4
0,000
0,498
0,997
1,495
1,994
I
I
3
3
0,000
o,4gg
0,990
i,4g8
■,997
1
3
2
3
0,000
o,5oo
1,000
i,5oo
2,000
I
2
3
4
0,0000
0,0001
0,0004
0,0009
0,0016
o,o5
0,06
0,07
0,08
0,09
2,462
2,954
3,447
3,939
4,432
4
6
6
7
7
2,466
2,960
3,453
3,946
4,439
5
5
6
7
8
2,471
2.965
3:459
3.953
4,447
4
5
()
7
8
2,475
3,970
3,465
3, 960
4,455
4
5
6
7
7
2,479
2.975
3,471
3,967
4,462
4
5
6
6
8
2,483
3, 980
3,477
3,973
4,470
5
5
6
7
8
2,488
2,985
3,483
3,980
4,478
4
5
6
7
7
2,492
2,990
3,489
3,987
4,485
4
5
5
7
8
2,496
2,995
3,494
3,994
4,4g3
4
5
6
6
7
2,5oo
3,000
3,5oo
4,000
4,5oo
5
5
6
7
8
0,0025
ù,oo36
0,0049
0,0064
0,0081
0,10
0,1 1
0,12
0,1 3
0,1 4
4,924
5,4i6
5,909
6,4oi
6,893
8
10
10
II
12
4.933
5,426
5,919
6,4 1 2
6,905
9
9
10
II
12
4.941
5,435
5,929
6,423
6,917
9
9
10
II
12
4,95o
5,444
5,939
6,434
6,929
8
10
II
11
12
4,958
5,454
5,950
6,445
6,941
9
9
10
11
12
4,967
5,463
5,g6o
6,456
6,953
8
10
10
II
12
4,975
5,473
5,970
6,467
6,965
9
9
10
11
13
4,984
5,482
5,980
6,478
6,977
8
9
10
11
13
4,gg2
5,4gi
5,ggo
6,489
6,989
8
9
10
1 1
1 1
5,000
5,5oo
6,000
6,5oo
7,000
9
10
1 1
11
12
0,0100
0,0121
0,0 1 44
0,0169
0,0196
0,1 5
0,16
0,17
0,18
0,19
7,385
7,878
8,370
8,863
9,354
i3
i3
i4
i5
16
7,398
7,891
8,384
8,877
9,370
i3
i4
i5
16
17
7,4 n
7,905
8,399
8,893
9,387
1 3
i4
i4
i5
16
7.434
7.919
8,4i3
8.908
9,4o3
i3
i3
i5
16
16
7,437
7,932
8,428
8,934
9,419
12
i4
i4
i5
16
7,449
7,946
8,443
8,939
9,435
1 3
i4
i5
i5
16
7,462
7,960
8,457
8,954
9,45i
i3
i3
i4
i5
17
7,475
7,973
8,471
8,96g
9,468
i3
i4
i5
16
16
7,488
7,987
8,486
8,985
9,484
12
i3
14
i5
16
7,5oo
8,000
8,5oo
9,000
9,5oo
i3
i4
i4
i5
16
0,0225
0,02 56
0,0289
0,0824
o,o36i
0,20
0,21
0,22
0,23
0,24
9,846
io,338
10,8 3o
11,322
11,814
17
18
19
20
21
9,863
io,356
10,849
11,34?
11,835
18
18
'9
20
31
9,88.
10,374
10,868
11,362
1 1 ,856
17
18
19
20
30
9,898
10,393
10,887
1 1 ,383
1 1 ,876
17
18
19
19
21
9.915
io,4io
10,906
1 1, 401
11,897
17
18
18
20
20
9,932
10,428
10,924
1 1 ,43 1
11,917
17
18
19
19
20
9,949
(0,446
10,943
1 1 ,44o
11,937
17
18
19
20
31
9,966
10,46^
10,962
1 1 ,460
ii,g58
17
18
18
19
20
9.983
10,482
10,980
11,479
11,978
17
17
19
20
2f
10,000
10,499
10,999
11,499
1 1 ,998
16
i&
19
19
21
o,o4oo
0,044 1
o,o484
o,o52g
0,0576
0,25
0,26
0,27
0,28
0,29
i2,3o6
12,798
13,290
13,782
14,274
22
23
23
24
35
12,338
13,821
1 3,3 1 3
1 3,806
14,399
21
22
24
34
25
12,349
i3,84'3
1 3,337
i3,83o
1 4,324
33
22
33
24
34
13,371
13,865
i3,36o
13.854
14,348
31
33
33
24
25
13,393
13,887
13.383
13,878
14,373
21
22
23
24
25
I3,4i3
12,909
I 3, 406
13,903
i4,3g8
22
23
23
24
25
12,435
12,933
13,429
13,926
i4,423
31
32
3 3
23
24
12,456
12,954
1 3,452
13,949
1 4,447
21
22
22
24
25
'2,477
12,976
1 3,474
13,973
14,472
21
22
23
24
24
12,498
12,998
1 3,497
1 3,997
14,496
21
22
23
24
25
0,0625
0,0676
0,0729
0,0784
0,084 1
o,3o
0,3 1
0,32
0,33
0,34
14,766
1 5,257
1 5,749
16,241
16,732
25
27
27
28
29
14,791
1 5,284
15,776
16,369
16,761
26
27
28
28
3u
14,817
i5,3ii
1 5,804
16,297
16,791
26
36
27
38
29
14,843
1 5,337
1 5,83 1
16,335
16,830
25
27
28
29
29
14,868
1 5,364
1 5,859
16,354
16,849
26
26
27
28
29
14,894
15,390
1 5,886
i6,383
16,878
25
26
27
28
29
14.919
i5,4i6
i5,9i3
i6,4io
16,907
36
27
27
28
28
14,945
1 5,443
1 5,g4o
i6,438
16,935
25
26
27
28
29
14.Q70
15,469
15,967
16,466
1(5,964
26
26
28
28
29
14,996
15,495
15,995
16,494
16,993
25
26
27
28
29
0,0900
0,0961
0,1034
0,1089
0,1 156
0,35
o,36
0,37
0,38
o.Sg
17,224
17,715
18,207
18,698
19,189
3o
3i
3 1
33
34
17,^54
17.746
i8,238
i8,73i
19,223
3o
3i
33
32
33
17,284
17,777
18,270
18,763
19,256
3o
3 1
32
33
34
i7,3i4
17,808
i8,3o3
18,796
19,290
3o
3i
3 1
32
33
17,344
17,839
18,333
18,828
19,323
3o
32
33
33
17,373
i7,86q
i8,365
1 8,861
ig,356
3o
3i
3i
32
33
i7,4o3
17,900
18,396
18,893
19,389
3o
3o
32
32
34
17,433
17,930
18.428
18,925
19,423
3o
3i
3i
33
33
17,463
17,961
18,459
18,958
19,456
=9
3i
32
32
33
17,492
17,992
18,491
18.990
19,489
3o
3o
3i
32
33
0,1225
0, 1 296
0, 1 36g
o,i444
0,l52I
o,4o
0,4 1
0,42
0,43
0,44
19,680
20,172
20.663
2i,i54
21,645
35
35
36
37
38
19,715
20,207
20,699
21. 191
31,683
34
35
36
37
37
19:749
20,343
20,735
21,238
31,720
34
35
36
37
38
19,783
20,377
20,771
21,265
21,758
35
35
36
36
38
19,818
20,3 1 2
20,807
2I,30I
21,796
34
35
36
37
37
19,853
20.347
20,843
2 1,338
21,833
34
35
35
37
38
19,886
20,382
20.878
21,375
21,871
34
35
36
36
37
19,920
20,417
2o,gi4
2I,4lI
2 1 ,908
34
35
36
37
38
19,954
20,453
20.950
31,448
2 1 ,946
34
35
35
36
37
19,988
20,487
20,985
21,484
21,983
34
34
36
37
37
0,1600
0,1681
0,1764
0,1849
0,1936
0,45
o,5o
0,55
0,60
0,65
0,70
22,l36
24,589
27,041
29.490
3 1, 938
34,382
38
43
47
52
56
61
32,174
24,632
27,088
29,542
3 1, 994
34,443
39
43
48
52
56
60
23,2l3
24,675
27,1 36
29,594
33,o5o
34,5o3
39
An
5i
56
60
22,252
24,718
27,183
29,645
32,106
34,563
38
43
47
53
55
61
22,290
34.761
37,33o
39,697
32,161
34,624
47
5i
56
60
23,329
24,804
27,277
29,748
32,217
34,684
38
42
47
5i
56
60
32.367
24,846
27,324
29,799
32,273
34,744
38
43
47
53
55
59
22,4o5
24,889
27,371
2g,85i
32,328
34,8o3
38
42
47
5i
56
60
33,443
34,981
27,418
29,902
32,384
34,863
46
5i
55
60
22,482
24,974
27,464
29,953
32,439
34,923
38
42
47
5i
55
59
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,75
0,80
o,85
0,90
0,95
1,00
36,82 4
39,264
41,699
44,1 32
46,56 1
48,986
65
69
74
78
83
87
36,889
39,333
41,773
44,210
46,644
49,073
65
69
74
78
82
87
36,954
39,403
41^847
44,288
46,726
49,160
65
%
78
82
87
37,019
39,471
41.920
44,366
46,808
49.247
64
78
83
87
37,083
39,540
41,993
44,444
46,890
49,334
65
69
74
77
83
86
37,148
89,609
43,067
44,521
46,973
49,420
64
68
73
78
81
86
37,212
39,677
43,140
44,599
47,o54
49,506
64
69
72
77
82
86
37,276
39,746
42,212
44,676
47,1 36
49,592
64
68
73
77
82
86
37,340
39,814
43,385
44,753
47,218
49,678
64
68
73
77
81
86
37,4o4
39,882
42,358
44,83o
47,299
49,764
64
69
72
77
81
86
0.5625
0,6400
0,7235
0,8100
o,go35
1 ,0000
4,1185
4,1472
4,1761
4,2050
4,2341
4,2632
4,2925
4,3218
4,3513 1
4,3808 1
â
^ . (r -f- r")^ or t^ -\- r""^ nearly.
491
492
493
494
4g5
496
497
498
499
5oo
I
49
49
49
49
5o
5o
5o
5o
5o
5o
I
3
98
98
99
99
99
99
99
100
100
100
3
3
i47
i48
l48
i48
149
i4g
149
i5o
i5o
3
4
196
197
197
198
198
199
199
200
200
4
5
246
346
247
247
248
248
249
249
25o
25o
5
6
295
395
3q5
296
297
298
398
299
299
3oo
6
7
344
344
345
346
347
347
348
349
349
35o
7
8
393
394
3g4
395
396
397
398
398
399
400
8
9
442
443
iA'i
445
446
446
447
448
44g
45o
9
TABLE II. — To finil the time T; the sum of the radii r-\-r", and llie chord c beiiiK given.
Sum of the Railii r-f-r". j
Clioiil
C.
2,97
2,98
nays iiiir.
2,99
3,00
3,01
3,02
Days |dif.
Days Idil".
Days |ciif.
Days lilif.
Days |.iif.
0,00
o,ouo
0,000
0,000
0,000
0,000
0,000
o,ouoo
0,01
o,5oi
1
0,502
I
o,5o3
0
o,5o3
1
o,5o4
1
o,5o5
I
0,0001
0,02
1,002
u
i,oo4
1
i,oo5
2
1,007
2
1,009
1
1,010
2
0,0004
o,o3
i,5o3
2
i,5o5
3
i,5o8
2
i,5io
3
i,5i3
2
i,5i5
3
o,ooog
o,o4
2,oo4
3
2,007
3
2,010
4
2,014
3
2,017
3
2,020
4
0,0016
o,o5
2,5o5
4
3,509
4
2,5i3
4
2,5i7
4
2,521
5
2,526
4
0,0025
o,o(5
3,oo5
6
3,011
5
3,016
5
3.021
5
3,026
5
3,o3i
5
o,oo36
0,07
3,5oC
6
3,5i2
6
3,5i8
6
3,524
6
3,53o
6
3,536
6
0,0049
0,08
4,00-'
7
4,014
7
4.021
6
4,027
7
4,o34
7
4,04 1
7
0,0064
0,09
4,5o8
8
4,5i6
7
4,523
8
4,53i
7
4,538
8
4.546
7
0,0081
0,10
5,oog
8
5,017
9
5,026
8
5.o34
9
5,043
8
5,o5i
8
0,0100
0,1 1
5,5io
9
5,519
9
5,528
ii>
5,538
9
5,547
9
5,556
9
0,0121
o,r2
6,011
10
6,021
10
6,o3i
10
6,04 1
10
6,o5i
10
6,061
10
0,01 44
0,1 3
6,5ii
II
6,522
11
6,533
1 1
6,544
11
6,555
1 1
6,566
1 1
0,0169
o,i4
7,012
13
7,024
12
7,o36
13
7.048
II
7,059
12
7.071
12
0,0196
0,1 5
7,5i3
i3
7,526
12
7,538
i3
7,55i
12
7,563
i3
7.576
12
0,0225
0,16
8,014
i3
8,027
i4
8,04 1
i3
8,o54
i4
8,068
i3
8,081
i3
o,o256
0,17
8,5i4
i5
8,529
i4
8,543
i4
8,557
i5
8,572
i4
8,586
14
0,0289
0,18
g.oi5
i5
9,o3o
i5
9,045
16
g,o6i
i5
9,076
i5
9,09 '
i5
o,o324
o,ig
9,5i6
16
9,532
16
9.548
16
9,564
16
g,58o
16
9,596
16
o,o36i
o,ao
10,016
17
io,o33
17
io,o5o
17
10,067
17
10,084
16
10,100
17
o,o4oo
0,21
10,517
18
10,535
17
10,552
i&
10,570
18
io,588
17
io,6o5
18
0,044 1
0,22
1 1,018
18
ii,o36
'9
11.055
18
1 1,073
19
1 1 ,092
18
11,110
19
o,o484
0,23
ii,5i8
30
11,538
19
11,557
19
11,576
20
1 1 ^96
19
1 1, 61 5
19
o,o52g
0,24
12,019
30
12,039
20
i2,o5g
20
12,079
21
12,100
20
I2,I30
20
0,0576
0,25
I2,5ig
21
i3,54o
21
i3,56i
21
13,583
21
iD,6o3
21
12,624
21
0,0625
0,26
1 3,020
22
13,042
22
1 3,064
21
i3,o85
32
i3,i07
22
i3,iP9
22
0,0676
0,27
i3,520
23
1 3,543
23
i3,566
22
1 3,588
23
i3,6ii
23
i3,634
22
0,0729
0,28
14,021
23
i4,o44
24
i4,o68
23
14,091
24
i4,ii5
23
i4,i38
24
0,0784
0,29
i4,52i
24
14.545
25
14,570
24
14,594
24
i4,6i8
25
14,643
24
o,o84i
o,3o
1 5,021
25
1 5.046
26
15,072
25
15,097
25
l5,123
25
i5,i47
25
0,0900
o,3i
i5,52i
27
1 5,548
26
1 5,574
26
1 5,600
26
15,626
26
1 5,652
26
0,0961
0,32
16,022
37
16,049
27
16,076
26
16,102
27
16.I2C)
27
i6,i56
27
0,1024
0,33
16,522
28
i6,55o
2-!
16,577
38
i6,6o5
28
1 6,633
28
16,661
27
0,1089
0,34
17,022
=9
i7,o5i
28
17,079
29
17,108
28
I7,i36
29
I7,i65
28
0,11 56
0,35
17,522
29
i7,55i
3o
i7,58i
29
17,610
3o
17,640
29
17,669
29
0,1225
o,36
18,022
3o
i8,o52
3i
i8,o83
3o
i8,ii3
3o
18,143
3o
18,173
3i
0,1296
0,37
18,522
3i
18,553
3i
i8,584
3i
i8,6i5
32
18,647
3i
18,678
3i
0,1369
o,38
19,022
32
19,054
32
19,086
32
19,118
32
ig,i5o
32
19,182
32
0,1 444
0,39
ig,522
33
19,555
33
19,588
32
1 9,620
33
ig,653
33
19,686
32
0,l521
0,40
20,022
33
2o,o55
34
20,089
34
20,123
33
30,1 56
34
20,190
33
0,1600
0,4 1
20,521
35
2o,556
34
2o,5go
35
20,625
34
20,659
35
2o,6g4
34
0,1681
0,42
2 1 ,02 1
35
2 1 ,o56
36
21,092
35
21,127
35
21,162
36
21,198
35
0,1764
0,43
21,521
36
21,557
36
21,593
36
21,629
37
21.666
36
21,702
36
0,1849
0,44
22,020
37
22,057
37
22,094
37
22,l3l
37
22;i68
37
22,2o5
37
0,1936
0,45
22,520
38
22,558
38
22,596
38
22,634
37
23,671
38
22,70g
38
0,2025
o,5o
25,016
42
25,o58
43
25,101
43
25,143
42
25,i85
43
25,227
42
o,25oo
0,55
27,5 11
46
27,557
47
27,604
46
27,65o
47
27,697
46
37,743
46
o,3o25
0,60
3o,oo4
5o
3o,o54
5i
3o,io5
5i
3o,i56
5o
3o,2o6
5i
3o,357
5o
o,36oo
o,65
32,494
55
32,549
55
32,6o4
55
32,659
55
32,7i4
55
32,76g
54
0,4225
0,70
34,982
60
35,042
59
35,101
59
35,160
59
35,2ig
59
35,278
59
0,4900
0,75
37,468
63
37,53i
64
37,595
64
37,659
63
37,722
63
37,785
63
0,5625
0,80
39.951
68
40,019
68
40,087
67
40, 1 54
68
40,222
68
40,290
67
o,64oo
o,85
43,430
73
42,5o3
72
42,575
72
42,647
72
42,71g
72
42,791
72
0,7225
0,90
44,907
77
44,984
76
45.060
77
45,i37
76
45,2i3
76
45,289
76
0,8100
0,95
47,380
81
47,461
81
47.542
81
47,623
81
47,704
80
47,784
81
0,9025
1,00
49,85o
85
49,935
86
50,02 1
85
5o,io6
85
50,191
85
50,276
85
1 ,0000
1 4,4105
4,4402
4,4701
4,5000
4,5301
4,5602
i.(r+r"J= or r'-f-r'" nearly. |
499
5oo
5oi
502
5o3
5o4
5o5
5(,6
5o
5o
5o
5o
5o
5o
5i
5i
100
100
100
100
101
lOI
101
101
i5o
i5o
i5o
i5i
i5i
i5i
l52
l52
200
200
200
201
201
202
202
202
25o
25o
25l
25l
252
252
253
253
209
3oo
3oi
3oi
302
303
3o3
3o4
34q
35o
35i
35i
352
353
354
354
3qq
400
4oi
402
402
4o3
4o4
4o5
449
45o
45i
452
453
454
455
455
Prup. purts I'or tlio sum of tho Radii.
I |3|3|4 |5|6|7|8|9
60
61
6?
63
64
65
66
67
68
69
70
80
90
100
23
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
40
4i
4i
42
43
44
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
72
81
90
a12
TABLE II. — To find the time T; the sum of the radii r-f»'", and the chord c heing p;iven.
Sum of tlie Iladil r-\~r'
Chord
c.
0,00
0,01
0,02
o,o3
o,(
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
o,i3
o,i4
o,i5
o,i6
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,38
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
0,36
0,37
o,38
0,39
o,4o
o,4i
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
0,65
0,70
0,75
0,80
0,85
o,go
o,()5
3,03
Days |dif.
0,000
O,5o6
1,012
i,5i8
2,024
2,53o
3,o36
3,542
4,o48
4,553
5,059
5,565
6,071
6,577
7,o83
7,588
8,094
8,600
9,106
9,612
10,117
10,623
11,129
11,634
12,140
12,645
i3,i5i
1 3,656
14,162
14,667
15,172
15,678
i6,i83
16,688
I7>i93
1 7,698
18,204
18,709
19,214
19,718
20,223
20.728
21,233
31,738
23,342
22,747
25,269
27,789
3o,3o7
32,823
35,337
37,848
40,357
42,863
45,365
47,865
5o,36i
4,5905
3,04
Days |dit'.
0,0(
0,507
i,oi4
1,520
2,037
2,534
3,o4i
3,547
4,o54
4,56i
5,068
5,574
6,081
6,588
7.094
7,601
8,108
8,6i4
9,121
9,637
10,1 34
io,64i'
11,147
11;653
12,160
12,666
13,172
13,679
i4,i85
14,691
15,197
1 5,704
16,210
16,716
17,322
17,798
18,234
18,739
19,245
19,751
30,257
30,762
21,268
21,774
22,279
22,784
25, 3i 1
37,835
30,35?
32.878
35,3g6
37,911
0,424
42,934
45,441
47,94'i
5o,445
3,05
Daysldil'.
4,6208
0,000
o,5o8
1, 01 5
1,533
2,o3o
2,538
3,046
3,553
4,061
4,568
5,076
5,584
6,091
6,599
7,106
7,614
8,131
8,628
9,1 36
9,643
io,i5i
10,658
ii,i65
11,6'
12,l8o
12,687
13,194
1 3,701
i4,2o8
i4,7i5
l5,223
15,739
16, 2 36
16,743
17,250
I7>757
18,264
18,770
I9>277
19,784
20,290
20,797
2i,3o3
2 1 ,809
22,3l6
22,833
25,353
27,881
3o,4o8
33,933
35,455
37,974
40,492
43,006
45,517
48,025
5o,53o
3,06
Days Idif.
4,6513
0,000
o,5o8
1,017
1,525
2,o34
2,542
3,o5i
3,559
4,067
4,576
5,084
5,593
6,101
6,609
7,118
7,626
8,i34
8,643
9,i5i
9.659
10,167
10,675
1 1 , 1 84
1 1 ,692
12,708
i3,2i6
1 3,724
l4,232
1 4,740
1 5,247
1 5,755
i6,263
16,771
17,278
17,786
18,294
18,801
19,309
19,816
30,324
20,83i
31,338
21,845
22,352
3,07
Days[dif.
3,08
0,000
0,509
1,019
1,528
2,o37
3,546
3,o56
3,565
4,074
4,583
5,093
5,602
6,111
6,620
7,129
7,638
8, 1 48
8,657
9,166
9,675
10,184
10,693
11,202
,86o| 37
25,394
37,937
3o,458
32,987
35,5i3
38,o37
0,559
43,077
45,593
48,io5
5o,6i4
4,6818
19,729
13,237
1 3,746
i4,255
14,764
15,372
15,781
16,290
16,798
i7,3o7
i7,8i5
18,324
i8,S32
19,340
19,849
30,357
30,865
21,373
21,881
23,389
23,897
25,436
27,973
3o,5o8
33,o4i
35,573
38,100
40,626
43,149
45,668
48,i85
50,6981
Days |dir.
4,7125
0,000
0,5 10
1,020
i,53o
2,o4o
2,55i
3,061
3,571
4,081
4,591
5,101
5,611
6,121
6,63 1
7>i4i
7,65 1
8,161
8,671
9,181
9,691
10,300
10,710
11,220
1 1 ,73o
12,240
",749
i3,359
13,769
14,278
14,788
1 5,297
1 5,807
i6,3i6
16,826
17,335
17,844
18,353
i8,863
19,372
19,881
20,3go
20,899
2 1 ,4oS
21,917
22,426
22,934
25,477
28,019
3o,558
33,095
35,63o
38,i63
40,693
43,220
45,744
48,265
50,782!
3,09
Days Idif.
0,000
o,5i 1
1,022
1,533
2,o44
2,555
3,066
3,577
4,087
4,598
5,109
5,620
6,i3i
6,642
7,i53
7,663
8,174
8,685
9,196
9,706
4,7432
10,728
11,238
11,749
13,359
12,770
1 3,280
13,791
i4,3oi
i4,8i2
l5,322
i5,832
16,343
i6,853
17,363
17,873
i8,383
18,893
19,403
19,913
20,423
20,933
21,443
31,953
22,462
22,972
25,519
28,064
3o,6o8
33,i49
35,688
38,335
40,759
43.291
45^819
■",344
5o,866
23
3,10
Days Idir.
4,7741
0,000
0,5l2
1,034
1,535
2,047
2,559
3,071
3;582
4,094
4,606
5,117
5,639
6,i4i
6,652
7,164
7,676
8,187
8,699
9,210
9,733
10,233
10.745
11,256
11,768
12,27g
12,791
j3,3o2
i3,8i3
14,324
i4,836
1 5,347
1 5,858
16,369
16,880
17,391 28
17.903
i8,4i3
18,924
19,435
19,946
20,4 56
20,967
21,478
21,988
22,499
23,009
25,56o
38,110
3o,657
33,3o3
35,747
38,288
40.826
43,362
45,894
48,424
5o,95o 83
3,11
Days |dif.
0,000
0,5 1 3
1,025
1,538
2,o5o
2,563
3,075
3,588
4,101
4,61 3
5,126
5,638
6,1 5 1
6,663
7,176
7,688
8,201
8,71 3
9,225
9,738
10,25o
10,762
11,275
1 1 ,787
'2,299
12,8
i3,323
1 3,836
14,348
14,860
15,373 24
1 5,884
16,396
16,907
3,12
Dav!
4,8050
'7,419
17,931
18,443
18,955
19,466
19:978
31,001
2I,5l2
22,024
22,535
23,046
25,602
28,155
30,707
33,257
35,8o5
38,35o
40,893
43,432
45,969
48,5o3
5i.o33
0,000
o,5i3
1.027
1, 540
2,o54
2,567
3,080
3,59.
4,107
4,621
5,i34
5,647
6,161
6,674
7,187
7,700
8,2
8,727
9,240
9,753
10,266
10,780
1 1,393
11,806
12,319
12,832
1 3,345
1 3,858
14,371
15,396
1 5;909
16,422
16,935
17,447
17.960
18^473
18,985
19,498
20,010
20,522
3i,o35
21,547
33,o5g
22,571
33,o83
35,643
38,201
3o,757
33,3ii
35,863
38,4i2
4o,g5g
43,5o3
46,044
48,582
5i,ii7
37 0,1936
4,8361 I 4,8672
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,013 1
0,01 44
0,0169
0,0196
0,0225
0,0256
0,028g
0,0824
o,o36 1
o,o4oo
0,044 1
o,o484
0,0529
0,0576
0,0625
0,0676
0,0739
0,0784
0,08
0,0900
0,0961
0,1024
0,1c
0,1 1 56
0,1225
0,1296
0,1369
0,1444
0,1 52 I
1600
0,1681
0,1764
0,1849
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
IF
. (r -{- r")' or r'^ -^ r " ^ nearly.
5o4
5o5
5o6
5o7
5o8
5o
5i
5i
5i
5i
101
101
101
101
102
i5i
l52
l52
l52
l52
202
202
3f)2
303
2o3
252
253
253
2,54
254
3o2
3o3
3o4
3o4
3o5
353
354
354
355
356
40 3
4o4
4o5
4o6
4o6
454
455
455
456
457
Sog
5io
5ii
5i
5i
5i
102
102
102
1 53
i53
i53
204
204
204
255
355
2 56
3o5
3o6
3o7
356
357
358
407
408
409
458
459
460
5l3
5 1
102
1 54
3o5
356
307
358
4io
46 1
5i3
5i
io3
1 54
205
257
3o8
359
4io
462
5i4
5i
io3
1 54
206
257
3o8
36o
4ii
463
TABI.K II. — To find llie time T\ tlie sum of the radii r-\-r", and the chord c being given.
TABLE
II.
— To find the time
T;
the sum of the radii t
+ r".
ind the chord c
being given.
Suni of the Radii r -\-r". I
Chord
c.
3,19
3,20
3,21
3,22 1
3,23
3,24
3,25
3,26
3,27
3,28
Days |dif.
Days |ilir.
Days |dir.
Daysl
dif.
Days |dif.
Days |dif.
Days |dif.
Days Idif.
Days |d.f.
Days |dif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
o,5ig
I
0,520
1
0,5: 1
I
0,522
0
0,522
I
0,523
I
0,524
1
o,525
I
0,526
0
0,526
I
0,0001
0,02
I ,o38
2
i,o4o
2
1,042
I
1,043
2
i,o45
I
i,o46
2
1,048
2
i,o5o
I
l,o5l
3
I,o53
1
0,0004
o,o3
1,557
3
I,56o
2
1,562
3
1,565
2
1,567
3
1,570
2
1,572
2
1,574
3
1,577
2
1,579
3
0,0009
0,04
2,077
3
2,080
3
2,o83
3
2,086
4
2, ego
3
2,og3
3
2,og6
3
2,099
3
2,102
4
3,106
3
0,0016
o,o5
2,596
4
2,600
4
2,6o4
4
2,608
4
2,612
4
2,616
4
2,620
4
2,624
4
2,628
4
3,632
4
0,0025
0,06
3,ii5
5
3,120
5
3,125
4
3,I2Q
5
3,i34
5
3, 139
5
3,144
5
3,i4g
5
3,i54
4
3.i58
5
o,oo36
0,07
3,634
6
3,64o
5
3,645
6
3,65î
6
3,657
5
3,662
6
3,668
6
3,674
5
3,67g
6
3,685
5
0,0049
0,08
4,i53
7
4,160
6
4,166
6
4,172
7
4,179
6
4,i85
7
4,192
6
4,1 g8
7
4,2o5
6
4,21 1
7
0,0064
0,09
4,672
7
4,679
8
4,687
7
4,694
7
4,701
8
4,709
7
4,716
7
4,723
7
4,73o
8
4,738
7
0,0081
0,10
5,191
8
5,199
8
5,207
9
5,216
8
5,224
8
5.232
8
5.240
8
5,248
8
5,256
8
5,264
8
0,0100
0,1 I
5,710
9
5,719
9
5,728
9
5,737
9
5,746
9
5,755
g
5,764
9
5,773
8
5,781
9
5.790
9
0,0121
0,12
6,22g
10
6,23g
10
6,249
ID
6,25g
9
6,268
10
6,278
10
6,288
9
6,2g7
10
6,3o7
10
6,3.7
9
0,01 44
0,1 3
6,748
II
6,759
10
6,769
I 1
6,780
II
6,791
10
6,801
II
6,812
10
6,822
10
6,832
II
6,843
10
0,0169
o,i4
7.267
12
7,279
II
7,290
11
7,3oi
12
7,3i3
II
7,324
II
7,335
12
7,347
11
7,358
II
7,369
II
0,0196
0,1 5
7,786
i3
7,799
12
7,811
12
7,823
12
7,835
12
7,847
12
7,859
12
7,871
12
7,883
12
7,895
i3
0,0225
0,16
8,3o5
i3
8,3i8
1 3
8,33i
l3
8.344
i3
8.357
i3
8,370
i3
8,383
i3
8,396
i3
8,4og
i3
8,422
i3
0,02 56
0,17
8,824
i4
8,838
1 4
8,852
i4
8,866
1 3
8,879
i4
8,893
14
8.907
14
8.921
i3
8,g34
14
8.948
14
0,0289
0,18
9,343
i5
9,358
i5
9,373
i4
9,387
i5
9,402
14
9,416
i5
9,43i
i4
9,445
i5
g,46o
i4
9.474
i5
o,o324
0)i9
9,862
16
9,878
i5
9,893
i5
9,908
16
9.924
1 5
9,939
16
9,955
i5
9,970
i5
9.985
i5
10,000
16
o,o36i
0,20
io,38i
16
10,397
17
1 0,4 1 4
16
io,43o
16
10,446
16
10,462
16
10,478
16
10,494
17
io,5ii
16
10,527
16
o,o4oo
0,21
10,900
17
10,917
17
io,g34
17
10,951
17
10,968
17
10,985
17
1 1,002
'7
11,0 1 g
17
1 1 ,o36
17
ii.o53
17
0,044 1
0,22
11,419
18
11,437
18
11,455
17
11,472
18
1 1 ,490
18
ii,5o8
18
11,526
17
11,543
18
ii,56i
18
11,570
17
o,o484
0,23
11.938
18
1 1 .g56
19
11,975
19
1 1 ,gg4
18
12.012
19
I2.o3l
18
1 2 ,o4q
19
12,068
18
12, 086
19
12,lo5
iS
0,0629
0,24
12,456
20
12,476
19
12,495
20
I2,5i5
19
i2;534
20
12,554
19
12,573
19
12,592
20
!2,6l2
19
12,63 1
19
0,0676
0,25
12,975
20
12,995
21
i3,oi6
20
i3,o36
20
i3,o56
21
1 3,077
20
13,097
20
i3,ii7
20
i3,i37
20
i3,i57
20
0,0626
0,26
13,494
21
i3,5i5
21
1 3,536
21
1 3,557
21
13,578
21
1 3,599
21
13,620
21
i3.64i
21
13,662
21
1 3,683
21
0,0676
0,27
i4,oi3
22
i4,o35
21
i4,o5(i
22
14,078
22
i4,ioo
22
l4,I22
22
i4,i44
22
i4,i66
21
14,187
32
14.209
22
0,0729
0,28
i4,53i
23
14,554
23
14,577
22
14,59g
23
14,622
23
i4,645
22
14,667
23
14,690
23
i4,7i3
22
14,735
23
0,0784
0,29
i5,o5o
23
15,073
24
15,097
24
l5,I2I
23
i5,i44
23
15,167
24
16,191
23
i5,2i4
24
1 5,238
23
1 5,261
23
0,084 1
o,3o
1 5,568
25
15,593
24
15.617
25
1 5.642
24
1 5,666
24
15,690
24
i5,7i4
25
15,739
24
1 5,763
24
15,787
24
0,0900
0,3 1
16,087
25
16,112
25
16, 1 37
26
i6,i63
25
16,188
25
i6,2i3
25
i6,238
25
16,263
25
16,288
25
i6.3i3
25
0,0961
0,32
i6,6o5
27
1 6,63 2
26
1 6,658
25
i6,683
26
16,709
26
16,735
26
16,761
26
16,787
26
i6,8i3
25
i6,838
26
0,1024
0,33
17,124
27
I7,i5i
27
17,178
26
17,204
27
I7,23i
27
17,258
26
17,284
27
17,311
27
17^338
26
17,364
27
0,1089
0,34
17,642
28
17,670
28
17,698
27
17,725
28
.7,753
27
17,780
28
17,808
27
17,835
28
17,863
27
17,890
27
0,1 1 56
0,35
18,161
28
18,189
29
18,218
28
18,346
28
18,274
29
i8.3o3
28
i8,33i
28
18,359
28
18,387
29
18,416
28
0,1226
0,36
18,679
29
18,708
3o
18,738
29
18,767
29
18,796
29
18,825
29
i8,854
29
i8,883
29
i8,gi2
29
18,941
29
0,1296
0,37
19.197
3o
19,227
3i
19,258
3o
19,288
3o
ig,3i8
3o
19,348
29
19,377
3o
19,407
3o
19,437
3o
19.467
3o
0,1369
o,38
19,716
3i
19,747
5o
19,777
3i
19,808
3i
19,839
3i
19,870
3i
19,901
3o
19,931
3i
19,962
3o
19,992
3i
0,1 444
0,39
20,234
32
20,266
3i
20,297
32
3o,32g
32
20,36i
3i
20,392
32
20,424
3i
20,455
32
20,487
3i
20,5 1 8
3i
0,1621
o,4o
20,752
32
20,784
33
20,817
33
20,85o
32
20,882
3a
20,gi4
33
20,947
32
20,979
32
21,011
33
2 1 ,043
33
0,1600
o,4i
2I,27<.
33
2i,3o3
34
21,337
33
21,370
33
2i,4o2
34
21,437
33
2 1 ,470
33
2i,5o3
33
21,536
33
21,569
33
0,1681
0,42
21,788
34
21,822
34
21,856
35
2 1 ,89 1
3/
21,925
34
21,959
34
2 1 ,993
33
22,026
34
22,060
34
22,094
34
0, 1 764
0,43
22,3o6
3^
22,341
35
22,376
35
22,4ll
35
2 2,446
35
22,481
34
22,5l5
35
22,550
35
23,585
34
22,619
35
0,1 84g
0,44
22,824
3t
22,860
36
22,896
35
22,931
36
22,967
36
23,oo3
35
23,o38
36
23,074
35
23,109
36
33,145
35
o,ig36
0,45
23,342
3t
23,378
37
23,4i5
37
23,452
36
23,488
37
23,525
36
23,56i
36
23,597
37
23.634
36
23,670
36
0,3025
o,5o
2 5,93c
4
25,971
4i
26,012
4o
26,052
4i
26,09;
4c
26,133
4i
26,174
40
26.214
4i
26,255
4o
26,395
4c
0,2600
0,55
28,517
e
28,562
45
28,607
44
28,65i
45
28,696
45
28,741
44
28,785
45
28,83o
44
28,874
45
28.91g
44
o,3o25
0,60
3 1,1 02
4<
3i,i5i
4ç
3 1, 200
49
31,249
4s
3 1, 298
48
3 1, 346
49
31.395
48
3 1, 443
49
31,492
48
3 1,540
49
o,36oo
o,65
33,685
5:
33,738
5:
33,7qi
53
33,844
53
33,897
53
33.g5c
53
34,oo3
52
34.055
53
34,108
52
34,160
53
0,4226
0,70
36,26e
5-
36,323
se
36,38 1
57
36,438
57
36,495
5-
36,552
56
36,6o8
57
36,665
57
36,72 2
56
36,778
57
0,4900
0,75
38,845
61
38 ,906
62
38.q68
61
39,02g
61
39,090
61
3g,i5i
61
3g.2i2
61
39,273
60
39,333
61
39.394
61
0,6626
0,80
41,421
6f
41,487
65
41.552
65
4i,6i-
6f
41.683
65
41,748
65
4i;8i3
65
41,878
65
4i,q43
64
42,007
65
o,64oo
o,85
43,994
7c:
44,064
!<:
44,i34
6g
44,2o3
7r
44,273
6(;
44,342
6Ç
44,4ii
69
44,48c
69
44,54a
69
44,618
69
0,7226
0,90
46,565
lA
46,639
T\
46,71 3
74
46,787
73
46,860
74
46,934
73
47,007
73
47,080
73
47,i53
73
47.226
73
0,8100
0,95
4q,T33
78
49,21!
It
49,28c
78
49.367
78
49,445
7-
4g,522
78
49,600
77
49,677
77
49,754
77
49,83 1
77
0,9026
1,00
51,69-
83
51,780
83
51,862
82
5 1 ,944
83
52,026
82
52,108
81
52,189
82
52,271
81
52,352
82
52,434
81
1 ,0000
5,0881
5,1200
5,1521
5,18
42
5,21651 5,2488
5,2813
5,3138 1 5,3465
5,3792
c2
(,■ -)- r")' or r' + r'"' nporly.
5l8
52
io4
i65
207
269
3ii
363
4i4
466
619
52
io4
1 56
208
260
3ii
363
4i5
467
620
52
104
1 56
208
260
3l2
364
416
468
621
62
io4
i56
208
261
3i3
366
417
469
623
52
io4
167
209
261
3i3
365
4i8
470
533
52
io5
167
209
262
3i4
366
418
471
624
52
io5
167
3IO
262
3i4
367
4ig
472
626
53
106
1 58
210
263
3i5
368
420
473
626
53
106
1 58
210
263
3i6
368
421
473
627
53
io5
1 58
211
264
3i6
369
422
474
TABLE II. — TofinJ the time T; the sum of the raJii r-\-r'', ami the chord r. heing given.
8uni ol
the
Uu.lii r+r".
Clu.i.l
c.
0,00
3,^29
3,30
3,31
3,32
3,33 3,34 1
D.iys l.lir.
Uins |dir.
Du>s
.lit.
" I)uys]iï7r.
ll.iys iiiir.
Uuys |dif 1
0,000
0,000
0,000
0,000
0,000
0,000
t>,00<.10
0,01
0,527
1
0,528
1
0,52lj
I
o,53o
0
o,53o
1
0,53 1
1
0.0001
0,0a
i.o54
2
i,o56
2
1 ,o5»
I
1 ,059
2
1,061
1
1,062
2
o,ooo4
0,0 3
1,582
2
1,584
2
1 ,58(1
3
1,589
2
1,591
3
..594
a
0,0009
o,o4
2,log
3
2,112
3
2,1 15
3
2,11b
4
2,122
3
2,125
3
0,0016
0,0 5
2,636
4
2,64ci
4
2,644
4
2,648
4
2,652
4
2,656
4
0,0025
0,06
3,i63
5
3,168
5
3,173
5
3,178
4
3,182
5
3,187
5
o,oo36
0,07
3,690
6
3,696
6
3,702
5
3,707
6
3,7.3
5
3,7.8
6
0,0049
o,oS
4^2] 8
6
4,224
6
4,23o
7
4,237
6
4,243
7
4,25o
6
0,0064
0,09
4,745
7
4,752
7
4,759
7
4,766
b
4,774
7
4,78.
7
0,0081
0,10
5,272
8
5,280
8
5,288
8
5,296
8
5,3o4
8
5,3i2
8
0,0100
0,1 1
5.799
9
5,808
9
5,817
8
5,825
9
5,834
9
5,843
9
0,01 2 1
0,1 a
6,326
10
6.336
9
6,345
10
6,355
10
6,365
9
'6,374
10
0,01 44
0,1 3
6,853
1 1
6,864
10
6,874
lu
6,884
1 1
6,895
10
6,go5
1 1
0,0169
o,i4
7.3SO
12
7,392
1 1
7,4o3
11
7,4i4
1 1
7,425
1 1
7,436
1 1
0,0196
0,1 5
-T.Q08
1 2
7,920
12
7,93'
11
7,943
12
7.955
12
7,967
12
0,0225
0,16
8,435
13
8,447
i3
8,460
i3
8,473
i3
8,486
13
8,498
i3
o,0256
0,17
8,962
i3
8.9-5
i4
8,989
i3
9,002
14
9,016
i3
9,"29
i4
0,0289
0,1 «
9,489
14
9,5o3
i4
9,517
i5
9,532
14
9,546
i5
9,56i
i4
o,o324
0,19
10,016
i5
io,o3i
i5
10,046
i5
10,061
i5
10,076
16
10,092
i5
o,o36i
0,20
10,543
iC
10,559
16
10,575
16
10,591
16
10,607
16
10,623
i5
o,o4oo
0,21
1 1 ,070
16
1 1 ,086
17
ii,io3
17
11,120
17
1 1,1 37
16
ii,i53
17
0,044 1
0,22
1 1,596
lb
1 1,614
16
1 1 ,632
17
1 1 ,649
18
1 1 ,667
17
11,684
18
0,0484
0,23
12,123
'9
12,142
18
13,160
19
12,179
ifc
12,197
18
12,21 5
19
o,o529
0,24
1 2 .65o
2<i
1 2 ,670
19
12,689
19
12,708
19
12,727
19
.2,746
'9
0,0576
0,25
1 3,177
20
13,197
20
i3,2i7
20
i3,237
20
i3,257
20
.3,277
20
0,0625
0,26
1 3,704
2 I
i3,7?5
21
1 3,746
20
13,766
21
13,787
21
1 3,808
30
0,0676
0,27
i4,23i
21
l4,252
22
14,274
22
14,296
21
i4,3i7
22
14,339
21
0,0729
0,28
1 4,758
22
14,780
22
14,802
23
14,825
23
14,847
22
14,869
33
0,0784
0,29
0,284
23
1 5,3o7
24
i5,33i
23
i5,354
23
.5,377
23
1 5,400
33
0,084 1
o,3o
1 5,81 1
24
1 5,835
24
1 5,859
24
1 5,883
24
.5,907
24
15.931
24
0,0900
0,3 1
1 6,338
24
16,362
2 5
16,387
25
i6,4i2
25
.6,437
34
16,461
a5
0,0961
0,32
16,864
26
16,890
25
16,915
26
i6,g4i
26
16,967
35
16,992
25
0,1024
0,33
17,391
26
17,417
27
17,444
26
17,470
26
.7,496
37
17,523
26
0,1089
0,34
17,9'7
28
17,945
27
17,97'
37
17,999
37
18,026
27
i8,o53
27
0,1 156
0,35
18,444
28
18,472
28
i8,5oo
28
18,528
28
i8,556
28
i8,584
37
0,1225
o,36
18,970
'9
18,999
29
19,028
39
19,057
28
19,085
39
19,114
39
0,1296
0,37
19.497
'9
19.526
3o
19,556
'9
19,585
3o
19,615
29
.9,644
3o
0,1 369
o,38
20,023
3o
20 o53
3i
20,084
3o
20,Il4
3i
20,145
3o
20,175
3o
0,1444
0,39
20,549
32
2o,58i
3i
20,612
3i
20,643
3!
20,674
3i
20,705
3i
0,l52I
o,4o
2 1 ,076
32
21,108
32
2I,l4o
33
21,172
32
2I,304
3i
21,235
32
0,1600
0,4 1
2 1 ,602
33
2 1,635
32
21,667
33
2 1 ,700
33
21,733
33
3 1 ,766
32
0,1681
0,42
22,128
34
22,162
33
21,195
34
22,229
33
23,262
34
22,296
33
0,1764
0,43
22,654
34
22,688
35
33,733
34
22,757
35
22,792
34
22,826
34
0,1849
0,44
23,180
35
23,2l5
36
23,25l
35
23,286
35
23,321
35
23,356
35
0,1936
0,45
23,706
36
23,742
36
33,778
36
23,8i4
36
33,85o
36
33,886
36
0,2025
o,5o
26.335
40
26,375
4o
26,41 5
40
26,455
4o
26,495
40
26,535
4o
o,25oo
0,55
28.963
ii
39,007
AA
29,05 1
Ai
29,095
AA
29,139
AA
29,183
AA
o,3o35
0,60
31,589
48
3 1,637
48
3i,685
48
31.733
48
31,781
48
31,829
48
o,36oo
o,65
34,21 3
52
34,265
52
34,3i7
53
34,369
52
34,421
52
34,473
52
o,42a5
o,7t<
36,835
56
36,891
56
36,947
57
37,004
56
37,060
56
37,116
56
0,4900
0,75
39,455
60
39,51 5
60
39,575
61
39,636
60
39,696
60
39,756
60
o,56a5
0,80
42,072
64
42,i36
65
42,201
64
42,265
64
42,329
65
4?, 394
64
o,64oo
o,85
44,687
68
44,755
69
44,834
68
44,893
69
44,961
68
45,029
68
o,7aa5
0,90
47,399
73
47,373
73
47,445
73
47,517
73
47,589
73
47,662
73
0,8100
0,95
49.908
77
49,985
77
50,062
77
5o,i39
76
5o,2i5
77
50,292
76
0,9035
1,00
52,5i5
81
52,596
81
52,677
80
53,757
81
53,838
81
52,919
80
1 ,0000
5,4121
5,4450
5,4781
5,5112
5,5445
5,5778 1 c^ 1
4 . ( r + r " ) ^ I
r r' + r'--
nearly. |
l'ri>|>. purts liir tli(> sum uf tlio Itiiilii.
1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9
536
537
528
539
53o
53i
53
53
53
53
53
53
io5
io5
106
106
106
106
1 58
1 58
1 58
1 59
1 59
1 59
210
211
211
312
212
212
263
264
264
365
265
266
3.6
3i6
3<7
3i7
3i8
3.9
368
369
370
370
37.
373
421
433
422
433
434
425
473
474
475
476
477
478
532
53
106
160
2l3
266
3,9
373
426
479
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
100
10
16
1 1
16
1 1
16
1 1
17
1 1
17
1 1
.7
12
17
12
.8
I 2
18
12
.8
12
19
l3
.9
i3
19
i3
20
i3
20
i3
20
i4
20
.4
21
.4
2 I
16
24
18
37
2<
3.^
23
23
24
25
26
27
28
29
3o
3i
32
32
33
34
35
36
37
38
39
4o
4i
4i
42
43
AA
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
73
81
90
Al3
TABLE
II.
— To find the time
T
the sum of the rad
■>»•+/■",
ind
the chord
c being given.
Sum oflhe lUtiii T-}-r". |
Chord
C.
0,00
3,35
3,36
3,37
3,38
3,39
Days |dir.
3,40
3,41
3,42
3,43
Days |dir.
3,44
Day^ |(lir.
Days |dil.
Days |dil'.
Days \i\L
Days |d
it'. Days lilif.
Days|dil'.
llaysldir.
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,532
1
0,533
1
0,534
0
0,534
1
0,535
I
o,536
I 0,537
1
o,538
0
o,538
I
0,539
I
0,000 1
0.02
1,064
2
1,066
1
1,067
2
i,o6g
I
1,070
3
1,072
I 1 ,073
2
1,075
2
1,077
1
1,078
3
o,ooo4
0,n3
1,596
2
1,598
3
1,601
2
i,6c;3
3
i,6o5
3
1,608
3 1,610
3
i,6i3
2
1, 61 5
2
1,617
3
0,0009
o,o4
2,128
3
3,l3l
3
2,i34
3
3,1 37
4
3,l4l
3
2,144
3 2,147
3
2,i5o
3
3,i53
3
3,1 56
4
0,001 6
o,o5
3,660
4
2,664
4
3,668
4
3,672
4
2,676
4
3,680
4 3,684
4
2,688
4
2,692
3
3,6g5
4
0,0025
0,06
3,192
5
3,197
4
3,301
5
3.206
5
3,2 1 I
5
3,316
4 3,220
5
3,325
5
3,23o
5
3,335
4
o,oo36
0,07
3,724
5
3,73g
6
3,735
6
3;74i
5
3,746
e
3,752
5 3,757
6
3,763
5
3,768
6
3,774
5
0,0049
0,08
4,356
6
4,262
7
4,269
6
4,275
6
4,281
7
4,388
6 4,294
6
4,3oo
6
4,3o6
7
4,3i3
6
0,0064
0,09
4,788
7
4,795
7
4,802
7
4,809
7
4,816
7
4,833
8 4,83 1
7
4,838
7
4,845
7
4,853
7
0,008 1
0,10
5,320
8
5,328
8
5,336
8
5,344
7
5,35i
8
5,359
8 5,367
8
5,375
8
5,383
8
5,391
8
0,0100
0,11
5,852
8
5,860
9
5,869
9
5,878
9
5,887
8
5,895
9 5,go4
9
5,9 1 3
8
5,921
9
5,930
8
0,01 2 1
0,12
6,384
9
6,393
10
6,4o3
9
6,4 1 3
10
6,423
g
6,43 1
10 6,44 1
9
6,45o
9
6,459
10
6,469
9
0,01 44
0,1 3
6,916
10
6,926
10
6,936
10
6,g46
11
6,957
IC
6,967
10 6,977
10
6,987
1 1
6,998
10
7,008
10
0,0169
o,i4
7>447
12
7,459
II
7.470
II
7,481
11
7.493
II
7,5o3
II 7,5i4
II
7,535
II
7,536
1 1
7.547
1 1
0,0196
o,i5
7.979
12
7-99'
13
8,oo3
13
8,oi5
13
8,027
12
8,039
II 8,o5o
12
8,063
13
8,074
12
8,086
12
0,0335
0,16
8,5ii
i3
8,524
i3
8,537
12
8,54g
i3
8,562
13
8,574
1 3 8,587
i3
8,600
13
8,612
i3
8,625
12
o,0256
0,17
9,043
i3
9,o56
i4
9,070
i3
g,o83
IÂ
9,097
i3
g,i 10
i4 9,124
i3
9. '37
l3
9,i5o
i4
9.164
i3
0,0389
0,18
9,575
i4
9,58g
i4
9,6o3
i5
g,6i8
i4
9,632
i4
9,646
i4 9,660
i4
9. M
i5
9,689
i4
9.703
i4
o,o324
0,19
10,107
i5
10,122
i5
io,i37
i5
10,l53
i5
10,167
i5
10,183
i5 10,197
i5
10,212
i5
10,227
i5
10,242
i4
o,o36i
0,20
1 0,6 38
16
10,654
16
1 0,670
16
10,686
16
10,703
16
10,718
i5 10,733
16
10,749
.6
10,765
i5
10,780
16
o,o4oo
0,21
1 1 , 1 70
17
11,187
16
Il,203
17
I 1,330
17
1 1,337
16
11,253
17 11.370
16
11,286
17
ii,3o3
16
11,3,9
17
0,044 1
0,22
1 1 ,703
17
11.719
18
11,737
■7
11,754
18
■1,772
17
11,78g
17 11 ,806
18
11,824
17
ii,84i
17
1 1 ,858
17
0,0484
0,23
12,234
18
12,352
18
12,270
18
12,288
18
i2,3o6
■9
12,325
18 13,343
18
I2,36i
18
13,379
19
13,397
18
0,0529
0,24
12,765
19
12,784
19
i2,8o3
19
12,822
19
I2,84l
■9
13,860
19 13,879
19
12,898
'9
12,917
■9
13,936
19
0,0576
0,25
13,397
20
1 3,3 1 7
30
i3,337
19
1 3,356
30
13,376
20
i3,3g6
30 i3,4i6
19
1 3,435
30
1 3,455
'9
1 3,474
20
0,0625
0,26
13,828
21
1 3,849
21
13,870
20
1 3,890
31
13,911
30
1 3,931
31 13,953
30
i3,g73
31
■3,993
30
i4.oi3
21
0,0676
0,27
i4,36o
21
i4,38i
33
i4,4o3
21
14,424
33
i4,446
31
14,467
21 14,488
21
i4,5o9
3 2
i4,53i
31
i4,553
31
0,0729
0,28
14,892
22
14,914
22
14.936
23
14.958
22
i4,g8o
32
1 5,002
33 i5 025
22
i5,o47
33
15,069
22
15,091
31
0,0784
0,2g
1 5,433
33
1 5,446
33
15,469
33
15,492
33
i5,5i5
33
1 5,538
23 i5,56i
33
1 5,584
33
1 5,606
23
1 5,63g
23
0,0841
o,3o
15,955
33
15,978
34
16,002
34
16,026
24
i6,o5o
23
16,073
24 16,097
24
16,121
23
i6,i44
24
16,168
23
0,0900
0,3 1
16,486
25
i6,5ii
34
16,535
25
i6,56u
24
1 6,584
25
16,609
34 i6,633
35
i6,658
24
16,683
24
16,706
25
0,096 1
0,32
17,017
36
17,043
25
17,068
36
17,094
35
17,119
25
17,144
35 17,169
26
17,195
25
17,330
25
17,345
25
0,1034
0,33
17,549
26
17,575
26
17,601
26
17,627
36
17,653
27
17,680
26 17,706
36
17,733
25
17.757
36
17,783
26
0,1089
0,34
18,080
27
18,107
37
i8,i34
27
18,161
27
18,188
27
i8,2i5
27 18,242
26
18,268
27
18,395
37
18,333
26
0,1 1 56
0,35
18,611
38
18,639
28
18,667
38
18,695
27
18,733
28
18,750
28 18,778
27
i8,8o5
28
i8,833
27
18,860
28
0,1 325
o,36
19,143
38
19,171
29
ig.200
28
19,338
29
19,257
38
19,285
29 i9,3i4
28
19,342
28
19,370
29
19.399
38
0,1396
0,37
19,674
59
19,703
3o
19,733
29
19,762
29
•9.79'
39
19,830
3o 19,850
29
19,87g
29
19,908
o9
19.937
29
0,1 36g
0,38
2O,305
3o
30,235
3o
20,265
3i
20,296
3o
20,326
3o
20,356
3o 2o,386
3o
20,416
29
20,445
3o
30,475
3o
0,1 444
0,39
30,736
3i
30,767
3i
20,798
3i
20,82g
3i
30,860
3i
3o,8gi
3i 30,922
3o
2o,g52
3i
20,983
3i
31,Ol4
3o
0,1 52 I
o,4o
21,267
32
21,299
32
2i,33i
32
21.363
3i
21,394
32
31,436
3i 31,457
32
2i,48g
3i
21,520
33
31,552
3i
0,1600
0,4 1
21,798
33
2 1,83.
32
3 1,863
33
21,896
33
21,928
33
2i,g6i
32 21,993
32
22,025
33
33,058
33
33,090
33
0,1681
0,42
22,329
34
23,363
33
33,396
33
22,42g
34
22,463
33
23,496
33 33,539
33
32,503
33
22,5g5
33
33,638
33
0,1764
0,43
33,860
34
22,894
35
32,939
34
22,963
34
22,997
34
23,o3i
34 23,o65
33
33,og8
34
23,l32
34
23,166
34
0,1849
0,44
23,391
35
33,426
35
33,461
35
23,496
35
33,53i
34
23,565
35 23,600
35
33,635
34
23,669
35
33,704
35
0,1936
0,45
23,g32
36
23,958
35
23,993
36
34,039
36
34,o65
35
24,100
36 24, I 36
35
34,171
36
24,207
35
34,342
35
0,2025
o,5o
26,575
40
26,615
40
36,655
39
36,694
40
26,734
39
26,773
4o 26,81 3
39
26,852
40
26,893
39
26,g3i
39
o,25oo
0,55
39,337
•iA
29,371
43
29.3i4
Aà
2g,358
44
29,402
43
39,445
44 29,48g
43
39,533
43
29,575
44
39,619
43
o,3o25
0,60
31,877
48
31,935
48
3'i,973
47
33,030
48
33,068
47
33,ii5
48 33,i63
47
33,210
47
33.357
48
32,3o5
47
o,36oo
0,65
34,525
53
34,577
52
34,62g
5i
34,680
52
34,732
52
34,784
5i 34.835
5i
34,886
53
34,938
5i
34.989
5i
0,4225
0,70
37,173
55
37,327
56
37,283
56
37,339
55
37,394
56
37,450
55 37.5o5
56
37,56i
55
37,616
55
37,671
55
0,4900
0,75
39,816
60
39,876
59
39.935
60
39.995
60
4o,o55
59
4o,ii4
60 40,174
59
4o,233
59
40,393
59
40,35 1
60
o,5635
0,80
43,458
64
42,522
63
42,585
64
42,649
64
43,713
63
42,776
54 43,840
63
42,C)o3
63
43.966
64
43,o3o
63
o,64oo
0,85
45,097
68
45,i65
68
45,333
68
45,3oi
67
45,368
68
45,436
67 45.5o3
68
45,571
67
45,638
67
45,7o5
67
0,7235
o,go
47.734
73
47,806
72
47.878
72
47.950
72
48,022
71
48,093
72 48,i65
71
48,236
71
48,3o7
72
48,379
71
0,8100
0,95
5o,368
76
5o,444
76
5o,53o
76
50,596
76
50,672
76
5o,748
75 50,823
76
50,89g
75
50,974
75
5i,o4g
75
0,9035
1 ,( )i )
52.999
80
53,07g
80
53,i5g
80
53,239
80
53,319
80
53,399
5,780
80 53,479
79
53,558
80
53,638
79
53,717
79
1 ,0000
5,6113
5,6448
5,6785
5,7122
5,7461
0 5,8141 1
5,8482 1
5,8825 1 5,91681
c"
^ . (r -j- r")'* or r^ -y r'"^ nearly. |
53i
533
533
534
535
536
537
538
539
540
I
53
53
53
53
54
54
54
54
54
54
I
2
106
106
107
107
107
107
107
108
108
108
2
3
1 59
160
160
160
161
161
161
161
163
162
3
4 ■
212
2l3
3l3
214
214
3l4
2l5
2l5
316
216
4
5
266
266
267
367
368
268
269
269
370
270
5
6
3i9
319
320
320
331
333
32?
323
333
334
6
7
373
37?
373
374
375
375
376
377
377
378
7
8
435
436
4a6
427
428
439
43o
43o
43 1
432
8
9
478
A
i79
4
80
48
I 1
48
2 1
482
483
484
485
486
9
TABLE II. — To find the time T; the sum of the radii r-j-r", and the chord c being given.
tfuni v)t' the Kudil r -^ r '
CJlurd
C
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,01)
0,10
0,11
0,12
0,1 3
o,i4
0,1 5
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,39
o,3o
o,3i
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
0,40
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
o,85
0,90
0,95
3,45
Duys |ilil
0,796
1,336
1,875
3,4i5
2,955
3,4q4
4,0.34
4,573
5,1 12
5,652
6,191
6,-3 1
7,370
7,809
8,348
8,888
9,427
9,966
2o,5o5
21,044
21,583
22.122
22,661
23,200
23,739
24,377
26,970
29,663
33,353
35,040
37,726
4o,4t I
43,093
45,773
48,45o
5 1, 1 24
53,796
0,000
o,54o
1,080
1,620
2,160
2,699
3,239
3,779
4,319
4,859
5,3qQ
5,()38
6,478
7,018
-.558
8, 098
8,637
9,1--
9r'"i '4
10,2 56
i3
5,9513
0,000
0,541
1,081
1,622
2,i63
2,703
3,244
3,785
4,325
4,866
5,4o6
5,947
6,488
-,028
-,56(i
8.iotj
8,650
9- '9"
9,-3.
10,271
10,812
11,352
1 1 ,893
12,433
12,973
i3,5i4
i4,o54
14,594
i5,i34
15,675
i6,2i5
16,755
17,295
17,835
18,375
18,915
19.455
'9,995
30,535
21,075
2 1, 61 4
22,l54
32,694
23,333
23,773
24,3 1 3
27,010
29,705
32,3g9
35,ogi
37,781
40,470
43,i56
45,839
48,531
51,199
53,875
3,47
Udys |dir.
0,000
0,54 1
1 ,08 3
1 ,634
2,l6t)
2,707
3,249
3,790
4,332
4,873
5,4i4
5,956
6,497
7,0 38
7,58o
8,662
9,204
9,745
10,286
10,827
1 1 ,369
11,910
i2,45i
12,993
i3,533
14,074
i4,6i5
i5,i56
15,697
i6,238
16,779
17,320
17,861
18,402
i8,g42
19,483
20,024
20,565
2 1 , 1 o5
2 1 ,646
22,186
23.737
33,267
23,807
24,348
27,049
29,748
32,446
35,i43
37,836
40,528
43,219
45.906
48,592
5 1, 274
53,954
3,48
Days Idii;
5,9858 1 6,0205
0,000
0,543
1,084
1,627
2,169
2,711
3,253
3,796
4,338
4,880
5,422
5,964
6,5o6
7,048
7,5yl
8,i33
8,675
9,217
9,759
io,3oi
10,843
11,38
11,927
1 2 ,469
i3,oi 1
1 3,553
14,095
i4,636
15,178
15,720
16,262
i6,8o3
17,345
17,887
18,428
18,970
i9,5ii
2o,o53
30,594
3i,i36
21,6-
23,318
22,759
23,3oi
23,842
24,383
27,088
29,791
32.493
35,193
37,891
40,587
43,281
45,973
48,662
5 1, 349
54,o33i
3,49
Dnys jdif.
0,000
0,543
1,086
1,629
2,172
2,7l5
3,258
3,801
4,344
4,887
5,43o
5,973
6,5 1 6
-,o5y
7,601
8,144
8,687
9,23'
9,^7
io,3i6
10,859
1 1 ,40 1
1 1 ,944
12,487
' 3,039
13,572
i4,i i5
i4,657
1 5,200
1 5,742
i6,385
16,838
17,370
17,912
18,455
18,997
19,539
20,082
20,624
21,166
21,708
23,35o
22,792
23,334
23,876
24:4>8
27,127
39,834
32,540
35,244
37,946
40,646
43,344
46,o4o
48,733
5r,424
54,11!
i3
6,0552
6,0901
3,50
I)ay4 |dir.
o,coo
0,544
1
1 ,088
I
i,63i
:
2,175
3
2,719
4
3,363
4
3,8c6
6
4,35o
6
4,894
7
5,438
7
5,981
9
6,53 5
9
7.C 69
10
7,6 1 3
11
8,1 56
12
8,700
12
9.243
i4
9,787
i4
1 0,33 1
14
10,874
16
ii,4i8
16
ii,g6i
17
i3,5o5
18
1 3,048
'9
13,593
19
i4,i35
20
14.678
21
l5,333
31
1 5,765
23
i6,3o8
24
16,85?
34
17,395
35
17.938
26
18,481
27
19,024
27
ig,567
38
30,110
3Q
30,653
3o
2i,ig6
3i
21,739
3t
22,383
33
22,825
33
23,368
33
23,910
35
24,453
35
27,166
3q
39,877
43
32,587
46
35,394
5i
38 ,001
54
4o,7o5
58
43,407
62
46, r 06
67
48,804
70
51,498
75
54,190
78
6,1250
0,001 >o
0,0001
0,0004
0,0(Ki9
0,0016
0,0025
o,oo36
0,0049
0,0064
0,0081
0,0I 00
0,01 3 1
o,oi44
0,0169
0,01 g6
*,0225
0,02 56
0,0289
o,o324
o,o36i
o,o4oo
o,o44 1
0,0484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,0841
0,0900
0,0961
0,1024
o,io8g
0,1 1 56
0,1225
0,1396
o, 1 369
0,1 444
0,1 521
0,1600
0,1681
0,1764
0,1 84g
0,1936
0,3035
0,2 5oo
o,3o25
0.3600
0,4225
0,4900
(r -t- r' )^ or r'^ 4" '' " ^ nearly.
538
539
54
54
108
108
161
162
2l5
216
269
270
323
323
377
377
43o
43 1
484
485
540
541
542
543
54
54
54
54
108
108
108
log
162
162
i63
i63
216
216
217
217
270
371
271
272
324
325
335
326
378
379
379
38o
432
433
434
434
486
487
488
48q
544
54
log
1 63
218
272
336
38 1
435
490
Prni>. imrts (i>r tlie Kuiii of (lie Uudii.
, I 2 I 3 I 4 I 5 I 6 I 7 1 8 I 9
25
3
26
3
27
3
28
3
29
3
3o
3
3i
3
32
3
33
3
34
3
35
4
36
4
37
4
38
4
39
4
4o
4
4i
4
42
4
43
4
44
4
45
5
46
5
47
5
48
b
49
5
5o
5
5f
5
52
5
53
5
54
5
55
6
56
6
57
6
58
b
59
6
60
6
61
6
63
6
63
6
64
6
65
7
66
7
67
7
68
7
69
7
70
7
80
8
90
9
00
10
28 35
32 40
36 1 45
40
I
1
2
1
2
2
2
3
3
3
4
4
4
5
4
5
5
5
6
6
6
6
7
6
7
8
7
8
9
8
9
10
8
10
1 1
9
10
12
10
1 1
i3
1 1
13
i4
II
i3
i4
13
i4
i5
i3
14
16
i3
i5
17
14
16
18
i5
17
'9
i5
18
20
16
18
21
17
19
22
18
20
23
18
21
23
19
22
24
30
23
25
20
23
26
21
24
27
22
2 5
28
23
26
29
23
26
3o
24
27
3i
25
28
33
25
29
32
26
3o
33
37
3o
34
27
3i
35
28
32
36
29
33
37
29
34
38
3o
34
3q
3i
35
40
32
36
4i
32
37
4i
33
38
42
34
38
43
34
39
44
35
4o
45
36
4i
46
36
43
4"
37
43
48
38
43
49
3q
44
5o
3q
45
5o
4o
46
5i
4i
46
52
4i
47
53
42
48
54
43
49
55
43
5o
56
44
5o
57
45
5i
58
46
52
59
46
53
59
47
54
60
48
54
61
48
55
62
49
56
63
5f)
64
72
63
72
81
70
80
90
TABLE II.
— To find the time T
the sum 0
f the ra
(lii
' + r".
am! (he chord c being given.
t^uin .il' the Kiiclii r -|- r". 1
Chord
C.
3,51
3,52
3,53
3,54 1
3,55
3,56
3,57
3,58
3,59
Days l.lil'.
3,60
Day» |(iil'.
I)ilj3 IcIlT.
Days |ilil'.
Dny» 1
III'.
Days |.lif.
Day* \m:
Days |clil'.
Days |.lir.
Days |ilif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0U0
0,000
0,0000
0,01
0,545
0
0,545
I
0,546
1
0,547
I
0,548
0
0,548
I
0,549
I
o,55o
1
o,55i
I
0,552
0
0,0001
0,02
1,089
2
1,091
1
1,092
2
1,094
1
1,095
2
1,097
I
1,098
2
1,100
1
1,101
2
i,io3
2
0,0004
o,o3
1,634
2
1.636
2
1,638
3
i,64i
2
1,643
2
1,645
3
1,648
2
i,65o
2
1,653
2
1,654
3
0,0009
o,o4
2,178
3
2,181
3
2,184
3
2,187
4
2,191
3
2,194
3
2,197
3
2,200
3
2,2o3
3
2,206
3
0,0016
o,o5
2,723
4
2,727
3
2,730
4
2,734
4
2,738
4
2,742
4
2,746
4
2,75o
4
2,754
3
2,757
4
0,0025
0,06
3,267
<;
3,272
5
3,277
4
3,281
5
3,286
5
3,291
4
3,295
5
3,3oo
4
3,3o4
5
3,309
4
o,oo36
0,07
3,812
5
3,817
6
3,823
5
3,828
6
3,834
5
3,839
5
3,844
6
3,85o
5
3,855
5
3,860
6
0,0049
0,08
4,356
7
4,363
6
4,36y
6
4.375
6
4,38i
6
4,387
6
4,393
7
4,400
6
4,406
6
4,4 1 2
6
0,0064
o,og
4,901
7
4,908
7
4,915
7
4,922
7
4,929
7
4,936
7
4,943
6
4,949
7
4,956
7
4,963
7
0,008 1
0,10
5,445
8
5,453
8
5,461
8
5,469
7
5,476
8
5,484
8
5,492
7
5,499
8
5,5o7
8
5,5i5
7
0,0100
0,1 1
5,990
8
5,908
9
6,007
8
6,01 5
9
6,024
8
6,o32
9
6,o4i
8
6,049
9
6,o58
8
6,066
9
0,0121
0,12
6,5'34
10
6,544
9
6,553
9
6,562
9
6,571
10
6,58 1
9
6,590
9
6,599
9
6,608
10
6,618
9
0,0 1 44
0,1 3
7,079
10
7,089
10
7,099
10
7,109
10
7,119
10
7,129
10
7,139
10
7,149
10
7,1 5g
10
7,169
10
0,0169
o,i4
7,623
II
7,634
II
7,645
II
7,656
1 1
7,667
10
ifin
1 1
7,688
1 1
7,69g
11
7,710
10
7,720
II
0,0196
0,1 5
8,168
1,
8,179
12
8,191
12
8,2o3
1 1
8,2i4
12
8,226
1 1
8,237
12
8,249
1 1
8,260
12
8,272
II
0,0225
0,16
8,712
1 3
8,725
12
8,737
12
8,749
i3
8,762
12
8,774
12
8,786
i3
8,799
12
8,811
12
8,823
12
0,0256
0,17
9,257
i3
9,270
i3
9,283
i3
9'296
i3
9,309
i3
9,3i2
i3
9,335
i3
9,348
i3
g,36i
14
9.375
i3
0,028g
0,18
9,801
i4
9,8 1 5
i4
9,829
i4
9,843
14
9,857
14
9,871
i3
9,884
i4
9,898
i4
9.912
14
9,926
i4
0,0324
0,19
10,345
i5
io,36o
i5
10,375
i4
10,389
i5
I o,4o4
i5
10,419
i4
10,433
i5
10,448
i5
io,463
14
10,477
i5
o,o36i
0,20
10,890
i5
10,905
16
10,921
i5
1 o.g36
16
10,952
i5
10,967
i5
10,982
16
1 0,998
.5
1 1, 01 3
1 5
11,028
16
o,o4oo
0,21
11,434
16
ii,45o
16
11,466
17
11,483
16
1 1 ,499
16
ii,5i5
16
ii,53i
16
11,547
'7
11,564
16
1 1 ,58o
16
o,o44i
0,22
1 1 ,978
17
I 1 :995
■7
12,012
17
1 2 ,029
17
i2,o46
17
1 2, 06 3
17
1 2 ,080
17
i2,<->97
17
12,1 14
17
12,l3l
17
o,o484
0,23
12,52.3
17
12,540
18
12,558
18
13,576
18
12,594
17
12,61 1
18
12,62g
18
12,647
18
12,665
17
12,682
18
o,o52g
0,24
13,067
18
i3,o85
19
i3,io4
19
i3,i23
18
i3,i4i
19
i3,i6o
18
13,178
19
13,197
18
i3,2i5
18
i3,233
19
0,0576
0,25
1 3,61 1
19
i3,63o
30
i3,65o
'9
13,669
19
1 3,688
20
1 3,708
■9
13,727
19
1 3,746
'9
13,765
20
1 3,785
19
0,0625
0,26
i4,i55
20
14,175
20
14,195
21
14,216
20
i4,236
20
14,356
20
14,276
20
14,396
20
i4,3i6
20
i4,336
20
0,0676
0,27
14,699
21
14,720
21
1 4,74 1
21
14,762
21
14,783
21
1 4,804
21
14,825
20
14,845
21
14,866
21
14,887
20
0,0729
0,28
1 5,243
22
i5,265
22
15,287
22
1 5,309
21
i5,33o
22
i5,352
21
15,373
22
15,395
21
i5,4i6
22
1 5,438
21
0,0784
o>29
15,788
22
i5,8io
23
1 5,833
22
1 5,855
22
15,877
23
15,900
22
15,922
22
i5,g44
23
15,967
22
1 5,989
22
0,084 1
o,3o
16,332
23
16,355
23
16,378
23
16,401
24
16,425
23
16,448
23
16,471
23
i6,4g4
23
i6,5i7
23
16,540
23
o,ogoo
0,3 1
16,876
24
16,900
24
16,924
24
16,948
24
16,972
24
16,996
24
17,020
23
17,043
H
17,067
24
17,091
24
0,0961
0,32
17,420
25
17,445
24
1 7,469
25
17,494
25
17,519
25
17,544
24
17,568
35
i7,5g3
24
17,617
25
1 7,642
24
0,1024
0,33
I ■',964
25
17.989
26
i8,oi5
25
i8,o4o
26
18,066
25
18,091
26
18,117
25
18,143
26
18,168
25
18,193
25
0,1089
0,34
i8,5o8
26
18,534
26
i8,56o
27
18,587
26
i8,6i3
26
18,639
26
1 8,665
27
i8,6g2
26
18,718
26
18,744
26
0,1 156
0,35
19,05 1
28
19,079
27
19,106
27
i9,i33
27
19,160
27
19,187
27
ig,2i4
27
ig,24i
27
19,268
27
19.295
26
0,1225
o,36
19,595
28
19,623
28
I9,65i
28
19,679
28
19-707
28
19.735
27
19,762
28
19,790
28
19,818
27
1 9,845
28
0,1296
0,37
20,139
29
20,168
29
20,197
28
20,225
29
20,254
38
20,282
29
30,3 1 1
28
30,339
29
20,368
28
20,396
29
0,1369
o,38
2o,683
29
20,712
3û
20,742
59
20,771
3o
20,801
29
2o,83o
'9
20,8 5g
3o
20,88g
29
20,918
29
20,947
29
0,1 444
0,39
21,227
3o
21,257
3o
21,287
3o
2 1,3 1 7
3i
2 1,348
3o
21,378
3o
2 1 ,4û8
3o
21,438
3o
21,468
3o
21,498
3o
0,l52I
o,4o
21,770
3i
21,801
3i
21,832
3i
21,863
3i
21,894
3i
21,925
3i
2 1 ,956
3i
3 1 ,g87
3i
22,Olfi
3o
2 3,048
3i
0,1600
o,4i
22,3l4
32
22,346
32
22,378
3i
22,409
32
22.441
32
22,473
3i
22,5o4
33
22,536
3 1
22,567
32
22,599
3i
0,1681
0,42
22,858
32
22,890
33
22,923
32
22,955
33
22,988
32
23,020
33
23,o53
32
23,o85
32
23,1 17
33
33.i5o
32
0, 1 764
0,43
23,4oi
34
23,435
33
23,468
33
23,5oi
33
23,534
34
23,568
33
23,601
33
23,634
33
23,667
33
23,700
33
0,1849
0,44
23,945
34
23,979
34
24,01 3
34
24,047
34
24,081
34
24,ii5
34
24,149
34
24,i83
34
24,217
34
24,25l
33
0,1936
0,45
24,488
35
24,523
35
24,558
35
24,593
35
24,628
34
24,662
35
24,697
35
24,732
34
24,766
35
24,801
34
0,2025
o,5o
27,2o5
39
27,244
38
27,282
39
27,321
39
27,36o
38
27,398
39
27,437
38
27,475
39
37,5i4
38
27,552
39
o,25oo
0,55
29,920
42
29,962
43
3o,oo5
43
3o,o48
42
30,090
43
3o,i33
42
30,175
43
3o,2i8
42
3o,2fio
42
3o,3o2
43
o,3o25
0,60
32,633
47
33,680
47
32,727
46
32,773
47
32,820
46
32,866
46
32,912
47
32,959
46
33,oo5
46
33,o5i
46
o,36oo
o,65
35,345
5i
35,396
5o
35,446
5i
35,497
5o
35,547
5o
35,597
5i
35,648
5<
35,6()8
5o
35,748
5o
35,798
5o
0,4225
0,70
38,o55
55
38, no
54
38, 164
55
38,219
54
38,273
54
38,327
54
38,38i
54
38,435
54
38,48y
54
38,543
54
o,4goo
0,75
40,763
5g
40,822
58
40,880
58
40,938
59
40,997
58
4i,o55
58
4i,ii3
58
41,171
58
41,229
57
41,286
58
o,5625
0,80
43,469
63
43,532
62
43,594
62
43,656
62
43,718
62
43,780
62
43,842
63
43,904
62
43,966
62
44,028
61
o,64oo
o,85
46,173
66
46,239
66
46,3o5
67
46,372
66
46,438
66
46,5o4
66
46,570
65
46,635
66
46,701
66
46,767
65
0,7225
0,90
48,874
7C
48,944
71
49,0 1 5
70
49,085
70
49,1 55
70
49,225
70
49,295
69
49,364
70
49,434
69
49,5o3
70
0,8100
0,95
51,573
74 5; 647
74
51,721
74
51,795
74
5i.86g
74
51,943
74
52,017
74
52,ogi
73
52,164
74
52,238
73
o,go3 5
1,00
54,268
79 54,347
78
54,425
78
54,5o3
78
58
54,58 1
78
54,659
78
54,737
1 ,,
54,814
78
54.892I 77
54,969
78
1 ,0000
6,16
01
1 6,19
52
6,23
05
6,26
6,30
13
6,33
68
6,37
25
6,4082
6,44
41
6,48
00
C^
(r -|- r")^ or T^ -\- r"' nearly.
543
54
109
1 63
317
272
326
38o
434
489
54
log
i63
218
273
326
38 1
435
4go
545
55
log
164
218
273
327
382
436
4gi
546
547
548
54g
55o
55i
552
55
55
55
55
55
55
55
109
log
no
no
no
no
no
1 64
164
1 64
i65
i65
i65
166
218
219
219
230
220
220
221
273
274
374
275
275
276
276
328
328
329
339
33o
33 1
33 1
382
383
384
384
385
386
386
437
438
438
439
440
44 1
442
4gi
492
4g3
494
4g5
496
497
TABLE II. — To fiiul tlic time T; the sum of the radii r-fr', and the chord <■ beini; given.
Sum of tJio
RQ.lii r+r ■
Choiil
c.
3,61
3,62
Days Idif.
3,63
3,64
3,65
3,66
Days Idif.
Days |dir.
Days Idif
Days Idif.
Days |dif.
0,00
0,000
0,000
0.000
0,000
0,000
0,000
0,0000
0,01
0,552
1
0,553
I
0,554
I
0,555
c
0,555
1
o,556
I
O.OOOI
0,03
i,io5
I
1,106
2
I,IO&
I
1,109
2
I, III
I
1,1 12
2
0,0004
o,o3
1,657
2
1,659
2
1,661
3
1,664
2
1,666
2
1,668
2
0,0009
o,o4
2,209
3
2,212
3
2,2l5
3
2,21b
3
2,221
3
2,224
3
0,0016
o,o5
2,761
4
2,765
4
2,760
4
2,773
4
2,777
3
2,780
4
0,0025
0,06
3,3i3
5
3,3i8
5
3323
4
3,327
5
3,333
4
3,336
5
o,oo36
0,07
3,866
5
3,871
5
3,876
6
3,882
5
3,887
5
3,892
6
0,0049
0,08
4,4 1 S
6
4,424
6
4.430
6
4,436
6
4,442
6
4,448
n
0,0064
0,09
4,970
7
4,977
"
4,984
7
4,99'
"
4,998
7
5,oo5
6
0,0081
0,10
5.522
8
5,53o
8
5,538
-
5,545
8
5,553
8
5,56i
7
0,0100
0,1 1
6:075
8
6,o83
8
6,091
9
6,100
8
6,108
9
6,117
s
0,0121
0,1 3
6,627
9
6,636
g
6,645
9
6,654
9
6,663
10
6,673
9
0,01 44
0,1 3
7.i7q
10
7,189
i^,
7,199
10
7,209
10
7,219
IC
7,229
9
0,0169
0,1 4
7,73i
II
7,742
II
7,753
10
7,763
11
7,774
II
7,785
10
0,01 96
0,1 5
8.283
12
8,295
11
8.3o6
12
8,3i8
II
8,329
11
8,340
12
0,0225
0,16
8.835
i3
8,848
12
8;86o
12
8,872
12
8,884
12
8,896
i3
0,02 56
0,17
9,388
i3
9,401
i3
9,4i4
12
9,426
i3
9,439
i3
9,452
i3
0,0289
0,1 S
9,940
i3
9,953
i4
9,967
14
9,981
i4
9,995
i3
10,008
i4
o,o324
0,19
10,492
i4
io,5o6
i5
10,521
i4
10,535
i5
io,55o
i4
10,564
i5
o,o36i
0,20
1 1 ,044
i5
11,059
i5
11,074
16
1 1 ,090
i5
ii,io5
i5
11,120
i5
o,o4oo
0,21
11,596
16
1 1, 61 2
16
11,628
16
11,644
16
1 1 ,660
16
11,676
16
0,044 1
0,22
I2,i48
17
I2,i65
16
12,181
17
12,198
17
I2,2l5
17
12,232
16
o,o484
0,23
12,700
17
13,717
18
12,735
17
12,752
18
12,770
17
12,787
18
0,0529
0,24
13,252
18
13,270
18
13,288
19
i3,307
18
i3,325
18
1 3,343
19
0,0576
0,25
1 3,804
'9
i3,823
'9
1 3,842
19
i3,S6i
19
1 3,880
19
13,899
19
0,0625
0,26
i4,356
19
i4,*75
2(j
14,395
20
i4,4i5
20
14,435
20
14,455
20
0,0676
0,27
i4,qo7
21
14.928
21
14,949
20
14,969
21
14,990
20
i5,oio
21
0,0729
0,28
15,459
22
1 5,48]
21
i5,502
22
i5,524
21
i5,545
21
1 5,566
21
0,0784
0,29
16,011
22
i6,o33
23
i6,o56
22
16,078
22
16,100
22
16,122
22
0,084 1
o,3o
i6,563
23
i6,586
23
16,609
23
i6,632
23
1 6,655
22
16,677
23
0,0900
0,3 1
17.115
23
I7,i38
24
17,162
24
17,186
23
17,209
24
17,233
24
0,0961
0,32
17,666
25
17,691
24
I7,7i5
25
17,740
24
17,764
25
17,789
24
0,1024
0,33
18,218
25
18,243
26
18,269
25
18,294
25
18,319
25
18,344
25
o,io8g
0,34
18,770
26
18,796
26
18,822
26
18,848
26
18,874
26
18,900
25
o,ii56
0,35
19,321
27
19,348
27
19,375
27
19,402
26
19,428
27
19,455
27
0,1225
o,36
19,873
28
19,901
27
19.928
28
19,956
27
19,983
27
20,010
28
0, 1 296
0,37
20,425
28
20,453
28
20,481
28
20,509
29
20,538
28
20,566
28
0,1369
o,38
20,976
29
2I,oo5
29
2i,o34
29
2 1 ,063
29
21,092
29
21,121
29
0,1 444
0,39
21,528
29
21,^57
3o
21,587
So
21,617
3o
21,647
29
21,676
3o
0,1 52 I
o,4o
22,079
3i
32,110
3o
22,l4o
3i
22,171
3o
22,201
3i
22,232
3o
0,1600
o,4i
22,63o
32
22,662
3i
22,6g3
3i
22,724
32
22,756
3i
22,787
3i
0,1681
0,42
23,182
32
23,2l4
32
23,246
32
23,378
32
23,3io
32
33,342
32
0,1764
0,43
23,733
33
23,766
33
23,799
33
23,832
32
23,864
33
23,897
33
0,1849
0,44
24,284
34
24,3i8
34
24,352
33
24,385
34
24,419
33
24,452
M
0,1936
0,45
24,835
35
24,870
34
24,904
35
24,939
34
24,973
34
25,007
35
0,2025
o,5o
27.591
38
27,629
38
27,667
38
27,705
39
27,744
38
27,782
38
o,25oo
0,55
3o,345
42
3o,387
42
30,429
42
3o,47i
42
3o.5i3
42
3o,555
42
o,3o25
0,60
33,097
46
33,143
46
33,189
46
33,235
46
33:281
46
33,327
45
o,36oo
0,65
35.848
5o
35,898
5o
35,948
49
35,997
5o
36,o47
5o
36,097
^9
0,4225
0,70
38,597
54
38,65 1
53
38,7o4
54
38,758
54
38,8i2
53
38,865
53
0,4900
0,75
41,344
58
4i,4o2
57
41,459
58
4i,5i7
57
41,574
58
4i,632
57
0,5625
0,80
44,089
62
44,i5i
61
44,212
62
44,274
61
44,335
61
44,3y6
61
o,64oo
o,85
46,832
66
46,898
65
46,963
65
47,028
65
47,093
66
47,159
65
0,7225
0,90
49,573
69
49,642
70
49,712
6q
49,781
69
49,8'5o
69
49,919
68
0,8100
0,95
52,3ii
73
52,384
73
52,457
74
52,53i
7a
52,6o3
73
52,676
73
0,9025
1. 00
55,047
77
55,124
77
55,201
77
55,278
77
55,355
76
55,43i
77
1 ,0000
6,5161 1
6,552-2 1
6,5885 1
6,6248 1
6,66131
6,6978 1
Î . C J- + r " ) = or r' -f- r « nearly. |
55i
552
553
55
"55
55
no
no
III
1 65
166
166
220
221
221
376
276
277
33i
33i
332
386
386
387
44i
442
442
4q6
497
498
554
55
111
166
222
277
332
388
443
499
555
56
III
167
222
278
333
389
5oo
556
56
III
167
222
278
334
389
445
^■-■7
56
III
167
223
279
334
390
446
5oi
Trop, purts fur tbo sun) uf tlio Radii.
i|2|3|4|5|6|7|8|9
I
2
3
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
80
90
9|i4
9
i3
i3
i4
i4
i4
i5
i5
16
16
16
17
17
18
18
i4 18
i4 19
i4 19
i5 20
25 3o
22
23
24
25
26
26
27
25 28
56
56
64
63|72 81
70I 80I00
23
23
24
25
26
27
28
29
3û
3i
32
32
33
34
35
36
37
38
39
40
4i
4i
43
43
A4
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
72
a14
FABLE II
— To find the time T
, the sum 0
f the radii
r + ,-",
and the chord c
being give
n.
Sum of the Radii r + r ". 1
Chord
C.
3,67
3,68
3,69
3,70
3,71
Days |dif.
3,72
3,73
3,74
3,75
3,76
Days \M:
Days
die.
Days |dif.
Days
dif.
Days |dif.
Days Idif.
Days |dir.
Days jdif.
Days |dif.
0,00
0,01
0,02
o,o3
o,o4
0,000
0,557
i,ii4
1,670
2,227
I
I
3
3
0.000
o;558
i,n5
1,673
2,23o
0
2
2
3
0.000
o;558
1,117
1,675
2,233
I
I
2
3
0,000
0,559
1,118
1,677
2,236
I
2
3
3
0,000
o,56o
1.120
1,680
2,239
I
I
2
3
0,000
o,56i
1.121
i;682
3,242
0
2
2
3
0,000
0,56 1
1,123
1,684
2,245
1
I
3
3
0,000
0,562
1,124
1,686
2,248
1
2
3
3
0,000
o,563
1,136
1,689
2,25l
I
I
3
3
0,000
o,564
1,127
1,691
2,254
0
2
2
3
0,0000
0,0001
0,0004
0,0009
0,0016
o,o5
0,06
0,07
0,08
0,09
2,784
3,341
3,898
4,455
5,011
4
4
5
6
3,788
3,345
3,903
4,46i
5,018
4
5
5
6
7
2,792
3,35o
3,908
4,467
5,035
3
5
6
6
7
2,795
3,355
3,914
4,473
5,o32
4
4
5
6
2,799
3,359
3,919
4,479
5,039
4
5
5
6
6
2,8o3
3,364
3,934
4,485
5,045
4
4
5
6
7
2,807
3,368
3,929
4,491
5,o53
4
5
6
6
2,811
3,373
3,935
4,497
5,059
3
4
5
6
7
2,8l4
3,377
3,940
4,5o3
5,066
4
5
5
6
6
2,818
3,382
3,945
4,509
5,072
4
4
5
6
7
0,0025
o,oo36
0,0049
0,0064
0,0081
0,IO
0,11
0,12
0,1 3
o,i4
5,568
6,125
6,682
7,238
7..795
8
8
9
10
11
5,576
6,i33
6,691
7,248
7,806
7
9
9
10
10
5,583
6,143
6,700
7,358
7,816
8
8
9
10
II
5,591
6,i5o
6,709
7,268
7,827
7
8
9
ro
10
5,598
6, 1 58
6,718
7,278
7,837
8
8
9
10
11
5,606
6,166
6,727
7,288
7,848
7
9
9
9
II
5,6i3
6,175
6,736
7,297
7,859
8
8
i^
10
5,621
6,i83
6,745
7,3o7
7,869
7
8
9
10
u
5,628
6,191
6,754
7,3l7
7,880
8
9
9
10
10
5,636
6,200
6,763
7,337
7,890
8
9
9
II
0,0100
0,013I
0,0144
0,0169
0,0196
0,1 5
o,t6
0,17
0,18
0,19
8,352
8,9"9
9,465
10,022
10,579
n
12
i3
i4
i4
8,363
8,921
9.478
io,o36
10,593
12
12
i3
i3
i4
8,375
8,933
9,491
10,049
10,607
II
12
i3
i4
i5
8,386
8,945
9,5o4
io,o63
10,622
II
12
i3
i3
1 4
8:5?
9,5i7
10.076
io,636
12
12
i3
i4
i4
8,409
8,969
g,53o
10,090
io,65o
II
12
12
i4
i5
8,420
8,981
9,542
io,io4
io,665
11
13
i3
i4
8,43i
8.993
9,555
10,117
1 0,679
II
12
i3
i4
i4
8,442
9,oo5
9,568
io,i3i
10,693
13
13
l3
i3
i5
8,454
9,017
9,58i
io,i44
10,708
II
12
12
i4
i4
0,0225
o,o256
0,0289
0,0824
o,o36i
0,20
0,21
0,22
0,23
0,24
ii,i35
1 1 ,693
12,248
t2,8o5
i3,362
i5
16
17
17
18
ii.i5o
ii;7o8
12,265
13,822
i3,38o
16
16
17
18
18
11,166
11,724
12,282
i3,84o
13,398
i5
16
16
17
18
11,181
1 1 ,74o
12,298
13,857
i3,4i6
i5
i5
17
18
18
11,196
11,755
I2,3i5
12,875
1 3,434
i5
16
i7
17
18
11,211
11,771
12,332
12,893
1 3,453
i5
16
16
17
18
11.236
11,787
12,348
1 2 ,909
1 3,470
i5
16
17
18
18
11.241
ii,8o3
12,365
12,027
1 3,488
i5
16
16
17
18
1 1,256
1 1 ,8 1 9
I2,38i
12,944
i3,5o6
1 5
i5
17
17
18
11,271
11.834
i2,3q8
12,961
1 3,524
i5
16
16
17
18
o,o4oo
0,044 1
o,o484
0,0529
0,0576
0,25
0,26
0,27
0,28
0,29
13,918
14,475
i5,o3i
1 5,587
16,144
19
19
30
22
22
13,937
14,494
i5,o5i
1 5,609
16,166
19
20
21
21
23
i3,o56
i4,5i4
15,072
i5,63o
16,188
19
20
20
21
22
i3,975
14,534
15,092
i5,6'5i
16,210
19
19
21
21
22
1 3,994
14,553
i5,ii3
15,672
16,232
19
20
20
21
22
i4,oi3
14,573
i5,i33
15.693
16,254
18
19
20
21
21
i4,o3i
14,593
i5,i53
i5,7i4
16,275
19
20
31
22
23
i4,o5o
i4,6i2
i5,i74
1 5,736
16,297
19
30
20
21
22
14,069
i4,632
15,194
15,757
16,319
19
19
20
21
23
i4,o88
1 4,65 1
i5,2i4
1 5,778
i6,34i
19
20
21
21
21
0,0625
0,0676
0,0739
0,0784
0,084 I
o,3o
0,3 1
0,32
0,33
0,34
16,700
17,257
I7,8i3
18,369
18,925
33
23
24
25
26
16,723
17,280
17,837
18,394
18,951
23
24
24
25
26
16,746
i7,3o4
17,861
18,419
18,977
22
23
25
25
26
16,768
17,327
17,886
18,444
19,003
33
33
24
25
25
16,791
17,350
17,910
18,469
19,028
23
24
24
25
26
i6,8i4
17,374
17,934
18,494
19,054
23
23
34
25
26
i6,836
17,397
17,958
18,519
19,080
23
24
24
35
35
16,859
17,421
17,982
18,544
ig,io5
22
23
24
35
26
16,881
17,444
18.006
18,569
ig,i3i
33
33
24
24
35
16,904
17,467
i8,o3o
18,593
I9,i56
22
23
24
25
26
o,ogoo
0,0961
0,1024
0,1089
o,ii56
0,35
0,36
0,37
0,38
0,39
19,482
2o,o38
20,594
2i,i5o
21,706
26
27
28
3o
19,508
20,065
20,622
31,179
31,736
27
27
28
29
29
19,535
20,092
2o,65o
21,208
2 1 ,765
26
38
28
28
3o
19,561
20,120
20,678
21,236
2 1 ,795
27
27
28
29
29
19,588
20,147
20,706
21,365
21,824
26
27
28
=9
3o
19,614
20,174
20,734
31,294
21,854
26
27
28
28
29
19,640
20,201
30,762
31,333
21,883
27
27
28
29
29
19,667
20,228
20,790
2i,35i
21,912
26
27
28
^9
3o
19,693
20.355
20,818
2i,38o
21,942
26
27
27
28
29
19,719
20,282
20,845
21,408
21,971
27
27
28
29
29
0,1225
0,1296
o,i36g
0,1444
0,l52I
o,4o
0,4 1
0,42
0,43
0,44
22,262
22,818
23,374
23,930
24,486
3o
3.
32
33
33
22,292
33,849
23,4o6
23,963
24,519
3i
3i
32
32
34
22,323
23,88o
23,438
23,995
24,553
3o
3i
32
33
33
33,353
22,911
23,470
24,028
24,586
3o
3.
3i
32
33
22,383
22,942
23,5oi
24,060
24,619
3i
3i
32
33
33
32,4l4
23,973
23,533
24,093
24,652
3o
3i
32
^4
22,444
23,004
23,565
24,125
24,686
3o
3i
3i
33
33
22,474
23,o35
23,596
24,i58
24,719
3o
3i
32
32
33
33,5o4
33.066
23,628
24,190
24,752
3o
3i
3i
32
33
22,534
23.097
23,659
24,232
24,785
30
3o
32
32
33
0,1600
0,1681
0, 1 764
0,1849
0,1986
0,45
o,5o
0,55
0,60
0,65
0,70
25,042
37,820
30,597
33.372
36, 1 46
38,918
34
38
43
46
5o
54
25,076
27,858
3o,63ci
33,418
36,196
38,972
34
38
4i
45
49
53
25,110
27,896
3o,68o
33,463
36,345
39,025
34
38
42
46
25,i44
37,934
3o,722
33,5o9
36,294
39,078
34
37
42
45
5o
53
25,178
27,971
30,764
33,554
36,344
39,i3i
34
38
4i
46
49
53
25,212
28,009
3o,8o5
33,600
36,393
39,184
34
38
42
45
49
53
25,246
28,047
3o,847
33,645
36,442
39,237
34
38
4i
45
49
53
25,280
28,085
3o,888
33.690
36,491
39,290
34
37
42
46
49
53
25,3i4
28,122
3o,93o
33,736
36,540
39,343
34
38
4i
45
È
25,347
28,160
3o,97i
33,781
36,589
39,396
34
37
4i
45
52
0,2025
0,3 5oo
o,3o35
o,36oo
0,4225
0,4900
0,75
0,80
0,85
o,go
0,95
1 1,00
41,689
44,457
47,334
49,987
53,749
55,5o8
57
61
64
69
73
77
4 1, 746
44.5i8
47,288
5o.o56
52,833
55,585
57
61
65
69
72
76
4i,8o3
44,579
47,353
5o,i25
52,894
55,661
57
61
65
68
73
76
4 1, 860
44,64o
47,418
50,193
52,967
55,737
57
61
64
69
73
77
41,917
44,701
47,482
50,262
53,039
55,8x4
57
60
65
68
72
76
41,974
44,761
47,547
5o,33o
53,111
55,890
57
61
64
69
72
76
42,o3i
44,822
47,611
50,399
53,i83
55,966
56
61
65
68
72
76
42,087
44,883
47,676
50.467
53;255
56,o42
57
60
64
68
72
75
42,144
44,943
47,740
5o,535
53,337
56,117
56
60
64
68
72
76
43,300
45,oo3
47,804
5o,6o3
53,399
56,193
57
61
64
68
72
76
0,5635
o,64oo
0,7225
0.8100
0,9025
1 ,0000
6,7345
6,7712
6,8081
6,84
50
6,8821
6,9192
6,9565
6,99
38
7,0313
7,0688
i
(r -\~ r")^ or r"^ -\~ r"^ nearly.
556
56
III
167
222
278
334
389
445
5oo
557
56
II I
167
233
279
334
390
446
5oi
558
56
112
167
223
279
335
391
446
502
559
56
112
168
224
280
335
391
447
5o3
56o
56
112
168
224
280
336
392
448
5o4
56i
56
112
168
324
281
337
393
449
5o5
562
56
112
169
225
281
337
393
45o
5o6
563
56
ii3
169
225
282
338
3g4
45o
507
564
56
ii3
169
226
282
338
395
45 1
5o8
TABLE II. =^ To find the time T; the sum of the radii r -j-r ", and the chord c being given.
Sum of the Rmlii r-l-r". |
Prop, parts for the sum of the Radii. 1
1 ITI /ICI/'I— 1C>1„I
Cliord
C.
3,77
3,78
3,79
3,80
3,81 '
3,82
I 1 2 1 3 1 4 1 5 1 6 1
7 1 0 1 9 1
1
2
3
0
0
0
0
0
0
I
1
0
I
1
I
2
1
I
2
I
1
2
I
2
2
I
2
3
Days |ilir.
Days |dif.
Days Idir.
Days |dir.
Days |dif.
Days |dil'.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
o,564
1
o,565
I
o,566
1
0,567
0
0,567
I
0,568
I
0,000 1
4
0
I
2
2
3
3
3
4
0,02
1,129
I
i,i3o
2
l,l32
1
i,i33
2
I,l35
I
i,i36
3
0,0004
o,o3
1,693
2
1,695
3
1,698
2
1,700
2
1,702
2
1,704
3
0,0009
5
2
2
3
3
4
4
5
o,o4
3,257
3
2,260
3
2,263
3
2,266
3
2,369
3
2,272
3
0,0016
6
7
2
2
2
3
3
4
4
4
4
5
5
6
5
6
o,o5
2,823
4
2,826
3
2,829
4
2,833
4
2,837
3
2,84o
4
0,0025
8
2
2
3
4
5
6
6
7
0,06
3,386
5
3.391
4
3.395
5
3,4oo
4
3,4o4
5
3,409
4
o,oo36
9
2
3
4
5
5
6
7
8
0,07
3,950
6
3i956
5
3,961
5
3,966
4,533
5
3,971
6
3.977
5
0,0049
10
2
3
4
5
G
7
8
8
9
10
0,08
4,5 1 5
6
4,521
6
4,527
6
6
4,539
6
4,545
6
0,0064
1 1
2
3
4
6
7
7
9
10
0,09
5,079
7
5,086
5,093
6
5,099
7
5,106
7
5,u3
6
0,008 1
12
2
4
5
6
8
11
0,10
5.643
8
5,65 1
~
5,658
8
5,666
7
5,673
8
5,681
7
0,0100
i3
i4
3
3
4
4
5
6
7
7
8
8
9
10
10
1 1
12
i3
o,u
6,208
S
6,216
8
6,224
8
6,232
9
6,241
8
6,249
8
0,01 2 1
0,12
6,772
9
6,781
9
6,790
9
C,799
9
6,808
9
6,817
9
0,01 44
i5
2
3
5
6
8
9
II
12
i4
o,i3
-,336
10
7,346
10
7.356
10
7,366
9
7,375
10
7,385
10
0,01 6g
16
2
3
5
6
8
10
II
i3
i4
o,i4
7, go I
10
7.9"
11
7,922
10
7,935
10
7.942
11
7,953
10
0,01 96
17
18
3
2
3
4
5
5
7
7
9
9
10
u
12
i3
i4
i4
i5
16
0,1 5
8,465
1 1
8,476
11
8,487
12
8,499
11
8,5io
11
8,521
II
0,0225
■9
2
4
6
8
10
1 1
i3
i5
17
0,16
9>"20
12
9.041
12
9,o53
12
9,065
12
9.077
12
9.089
13
o,o256
16
18
0,17
9.593
i3
9,606
i3
9.619
12
9,63 1
i3
9.644
i3
9,657
12
0,028g
20
2
4
6
8
10
13
i3
i4
0,18
io,i58
i3
10,171
i3
10,184
i4
10,198
i3
10,211
i4
10,225
l3
o,o324
21
2
4
6
8
II
i5
17
19
0,19
10,722
i4
10,736
i4
io,75o
i4
10,764
i5
10,779
i4
10,793
i4
o,o36i
22
23
2
2
4
5
7
n
9
9
II
12
i3
i4
i5
16
18
18
20
21
0,20
11,286
i5
ii,3oi
i5
ii,3i6
i5
II, 33 1
i5
11,346
i5
ii,36i
i4
o,o4oo
24
2
5
7
10
12
i4
17
19
22
0,21
ii,85o
16
11,866
16
11,882
i5
11,897
16
11,913
i5
11,928
16
o,o44 1
35
3
5
5
5
6
8
8
8
8
10
i3
i5
18
20
23
0,22
12,414
17
i2,43i
16
12,447
17
12,464
16
1 3,480
16
12,496
17
o,o484
26
3
3
3
10
i3
16
18
2 1
23
0,23
".978
18
12,996
17
i3,oi3
17
i3,o3o
17
1 3,047
17
1 3,064
17
0,0529
i4
i4
16
19
30
22
24
25
0,24
i3,542
18
i3,56o
18
13,578
18
13,596
18
i3,6i4
18
1 3,632
i8
0,0576
27
28
1 1
II
17
22
0,25
14,107
18
i4,i25
19
i4,i44
'9
i4,i63
18
i4,i8i
19
l4>200
18
0,0635
29
3
6
9
12
i5
17
20
23
26
0,26
14,671
19
14,690
19
14,709
20
14,729
'9
i4,748
20
14,768
19
0,0676
3o
3
6
9
9
10
12
i5
18
21
24
27
0,27
i5,235
20
i5,255
20
15,275
20
15,295
30
1 5,3 1 5
30
1 5,335
20
0,0729
3i
3
6
12
16
'9
23
35
38
0,28
1 5,799
20
15,819
21
i5,84o
21
1 5,861
21
1 5,882
31
1 5,903
21
0,0784
32
3
6
i3
16
'9
33
36
29
0.29
16,362
22
1 6,384
22
16,406
32
16,428
21
16,449
32
16,471
21
0,084 1
33
34
3
3
7
7
10
10
i3
i4
17
17
30
30
33
24
36
27
3o
3i
o,3o
16.926
23
16,949
22
16,971
23
16,994
22
17,016
22
i7,o38
23
o,ogoo
0,3 1
17,490
24
i7,5i4
23
17,537
23
17,560
23
17,583
23
17,606
23
0,0961
35
4
7
II
i4
18
21
25
28
33
0,32
i8,o54
24
18,078
24
18,102
24
18,126
24
i8,i5o
24
18,174
23
0,1024
36
4
7
II
i4
18
23
25
29
33
0,33
18,618
25
18,643
24
18,667
25
18,692
25
18,717
24
i8,74i
25
0,1089
37
4
7
II
i5
■9
33
26
3o
33
0,34
19,182
25
19,207
26
19,233
25
19,258
25
19,283
26
19,309
35
o,n56
38
39
4
4
8
8
II
12
i5
16
'9
20
23
23
27
27
3o
3i
34
35
0,35
19,746
26
19.772
26
19,798
26
19,824
26
i9,85o
26
19,876
26
0,1225
32
33
34
36
37
38
o,36
20,309
27
2o,336
27
2o,363
27
20,390
37
20.417
27
20,444
27
0,1296
4o
4
8
12
16
20
24
25
28
0,37
3o'.873
28
20,901
27
20,928
28
20,956
28
2o',984
27
21,011
38
0,1389
4i
4
8
12
16
21
29
0,38
21,437
28
21,465
29
2 ' ,494
28
21,522
28
2i,55o
29
21,57g
38
0,1 444
42
4
8
i3
17
21
25
19
0,39
22,000
3o
22,o3o
29
22,059
29
22,088
29
33,117
29
22,l46
29
0,l52I
43
44
4
4
9
9
i3
i3
17
18
22
22
26
26
3o
3i
34
35
39
40
o,4o
0,4 1
22,564
23,127
3o
3i
22,594
23,1 58
3o
3i
22,624
23,189
3o
3o
23,654
23,219
3o
3i
22,684
23,25o
19
3i
22,7l3
23,281
3o
3o
0,1600
0,1681
45
5
5
5
5
5
9
i4
18
18
23
23
„ /
27
28
28
32
32
33
34
34
36
37
38
38
39
4i
4i
42
43
44
0,42
23,691
3i
23,722
32
23,754
3i
23,785
32
23,817
3i
23,848
3i
0,1764
46
9
i4
i4
0,43
24.254
33
24,287
32
24,319
32
24,35i
32
24,383
32
24,41 5
32
o,i84g
47
9
19
24
24
25
0,44
24,818
33
24,85i
33
24.884
33
24,917
32
24,949
33
24,982
33
0,1936
48
49
10
10
i4
i5
19
20
29
29
0,45
25.38i
34
25.4i5
34
25,44s
33
25,482
34
25,5i6
33
25.549
34
0,2025
5o
5i
52
53
54
5
10
i5
20
25
3o
35
4o
45
o,5o
28,197
38
28.235
37
28,272
38
28,310
37
28,347
37
28;384
38
o,35oo
5
10
i5
20
26
3i
36
4i
46
0,55
3l,OI2
42
3i;o54
4i
3 1 ,095
4i
3i,i36
4i
3i,i77
4i
3i,2i8
4r
o,3o25
5
10
16
21
26
3i
36
42
4-'
0,60
33,826
45
33,871
45
33,91c
45
33,961
45
34,006
44
34,o5o
45
o,36oo
5
16
21
27
27
32
37
38
42
48
0,65
36.638
4g
36,687
48
36,735
49
36,784
49
36,833
48
36,88 1
49
0,4335
5
I J
16
33
33
43
49
0,70
39,448
53
39,501
52
39,553
53
39,606
52
39,658
53
39,711
52
0,4900
55
6
II
17
32
28
33
39
44
5o
0,75
42,257
56
42,3i3
57
42,37r
56
42,426
56
42,482
56
42,538
56
0,5625
56
6
1 1
17
22
38
34
39
45
5o
0,80
45,064
6c
45,124
60
45,18^
60
45,244
60
45,3o4
60
45,364
60
o,64oo
57
58
6
1 1
17
23
29
34
40
46
5i
0,85
47,868
64
47,932
64
47.99f
64
48,060
64
48,124
63
48,187
64
0,7335
6
12
17
23
20
35
4i
46
52
0,90
50,671
68
50,739
67
5o,8oe
68
50,874
67
5o,94i
68
51,009
67
0,8100
5g
6
12
18
24
3o
35
4i
47
53
0,95
53,471
-2
53,543
71
53,6m
72
53,686
71
53,757
71
53,828
71
o,go25
1,00
56,26gl 70
56,344
75
56,4191 76
56,495
75
56,570
75
56,645
75
I
,0000
60
61
62
63
6
6
6
6
12
12
12
i3
18
18
19
19
24
24
25
25
3o
3i
3i
32
36
37
37
38
38
41
43
43
44
45
48
5o
5o
5i
54
55
56
57
58
7,1065
7,1442
7,1821
7,2200
7,2581 1
7,2962
c2
,^ . (r -f- r" ]^ or r^ -\- r"^ nearly.
563
564
565
566
567
568
56g
64
6
i3
19
26
32
—
—
65
7
i3
20
26
33
39
46
52
59
I
56
56
57
57
57
57
57
I
66
7
i3
20
26
33
4o
46
53
59
2
ii3
ii3
ii3
1x3
ii3
ii4
114
2
67
7
i3
20
27
34
40
47
54
60
3
160
,6g
170
170
170
170
171
3
68
7
i4
30
27
34
4i
48
54
61
4
225
226
226
226
227
227
228
4
69
7
i4
31
28
35
4i
48
55
62
5
282
282
283
283
284
284
285
5
6
338
338
339
340
340
341
34i
6
70
7
i4
31
28
35
42
49
56
63
7
394
395
396
396
397
398
3o8
7
80
8
16
24
32
40
48
56
64
72
8
45o
45i
452
453
454
454
455
8
90
9
18
27
36
45
54
63
72
81
Q
507
5o8
509
509
5io
5ii
5l2
9
100
10
20
3o
40
5o
60
70
80
iS.
TABLE II. — To find the time T\ the sum of the radii r -)-»■", and the chord c being given.
Sum of the Radii r'\-r".
Ciiord
C.
3,83
3,84
3,85
3,86
Days Idif.
3,87
3,88
3,89
Days Idif.
3,90
3,91
3,92
Days |dir.
Days |dif.
Days idif.
Days Idif.
Days Idif.
Daysjdif.
Days |dif.
Days |dif.
0,00
0,01
0,02
o,o3
0,04
0,000
0,569
i,i38
1,706
2,275
I
I
3
3
0,000
0,570
1, 139
1,709
2,278
0
2
2
0
0,000
0,570
I,l4l
1,711
2,281
I
I
2
3
0,000
0,571
I,l42
i,7i3
2,284
I
2
2
3
0,000
0,572
1. 144
i,7i5
2,287
I
I
3
3
0,000
0,573
i,i45
1,718
2,290
0
2
2
3
0,000
0,573
i,i47
1,720
2,293
1
I
2
3
0,000
0,574
I,l48
1,722
2,296
1
2
3
0,000
0,575
1,1 5o
1.724
2,299
0
I
3
0,000
0,575
i,i5i
1,726
2,302
I
I
3
3
0,0000
0,000 1
0,0004
0,0009
0,0016
o,o5
0,06
0,07
0,08
0,09
2,844
3,4:3
3,982
4,55i
5,119
4
4
5
6
7
2,848
3,417
3,987
4,557
5,126
4
c
r
7
2,852
3,422
3,992
4,562
5,i33
3
4
5
6
6
2,855
3,426
3,997
4,568
5,139
4
5
6
6
7
2,859
3,43 1
4,oo3
4,574
5,i46
4
4
5
6
7
2,863
3,435
4,008
4,58o
5,i53
3
5
5
6
6
2,866
3,440
4,oi3
4,586
5,159
4
4
5
6
7
2,870
3,444
4,018
4,592
5,166
4
4
5
6
7
2.874
3.448
4,023
4,598
5,173
3
5
5
6
6
2,877
3,453
4,028
4,604
5,179
4
4
5
6
7
0,0025
0,00 36
0,0049
0,0064
0,0081
0,10
0,11
0,12
o,i3
0,1 4
5,688
6,237
6,826
7,395
7,963
8
8
9
9
II
5,696
6,265
6,835
7>4o4
7.974
8
9
10
.10
5,7o3
6,273
6,844
7,4 1 4
7,984
7
8
8
9
10
5,710
6.281
6;852
7.423
7.994
8
I
10
II
5,718
6,290
6,861
7,433
8,oo5
8
9
10
10
5,725
6,298
6,870
7.443
8,oi5
8
8
9
9
10
5,733
6,3o6
6,879
7.452
8,025
7
8
9
10
II
5,74f
6,3i4
6,888
7.462
8,o36
•-j
8
9
9
10
5,747
6,322
6,897
7,471
8,046
8
8
8
10
10
5,755
6,33o
6,905
7.481
8,o56
7
8
9
9
11
0,0100
0,0121
0,0144
0,0169
0,0196
o,i5
0,16
0,17
0,18
0,19
8,532
9,101
io,238
10,807
II
12
1 3
i3
i4
8,543
9,ii3
9,682
IO,25l
10,821
II
II
i3
i4
i4
8,554
g,i24
9.695
10,265
io,835
II
12
12
i3
i4
8,565
9, 1 36
9.707
10,278
10,849
II
12
i3
i3
i4
8,576
9,148
9.720
10,291
10,863
12
12
12
14
i4
8,588
9,160
9,732
io,3o5
10,877
II
12
i3
i3
i4
8,599
9.172
9.745
io,3i8
10,891
II
12
12
i3
i4
8,610
9,184
9.757
io,33i
10,905
II
11
i3
i4
i4
8,621
9.195
9.770
10,345
10,919
II
12
12
i3
i4
8,632
9.207
9,782
io,358
10,933
n
12
i3
i3
i4
0,02 25
o,o256
0,0289
o,o324
o,o36i
0,20
0,21
0,22
0,23
0,24
11,375
11,944
I2,5i3
1 3,081
i3,65o
i5
16
16
17
18
1 1 ,390
1 1 ,960
12,529
i3,oq8
i3,668
i5
i5
16
17
17
ii,4o5
11,975
12,545
i3,ii5
1 3,685
i5
16
17
17
18
11,420
11,991
12,562
i3,i32
1 3,703
i5
i5
16
17
18
11,435
12,006
12,578
1 3,149
i3,72i
i4
16
16
17
18
11,449
12,022
12,594
i3,i66
13,739
i5
i5
16
17
17
11,464
i2,o37
12,610
i3,iS3
i3,755
i5
16
17
17
iS
11.479
i2,o53
12,627
l3,200
1 3,774
i5
i5
16
17
18
11.494
12,068
12,643
i3,2i7
13,792
i4
16
16
17
17
ii,5o8
12,084
12.659
i3;234
13,809
i5
i5
16
17
18
o,o4oo
0,044 1
o,o484
0,0529
0,0576
0,25
0,26
0,27
0,28
0,29
14,218
14,787
i5,355
15,924
16,492
19
19
20
21
22
14,237
14,806
15,375
1 5,945
i6,5i4
18
19
20
20
21
i4,255
14,825
15,395
15,965
16,535
'9
20
20
21
22
14,274
i4,845
i5,4i5
15,986
16,557
18
19
20
21
21
14,292
14.864
i5,435
16,007
16,578
19
19
20
21
22
i4,3ii
i4,883
14.455
16,028
16,600
18
19
20
20
21
i4,329
14.902
14,475
16,048
16,621
19
20
20
21
21
14,348
14,922
15,495
16,069
16,642
18
19
20
20
22
1 4,366
i4,94i
i5,5i5
16,089
i6,664
19
19
20
21
21
i4,385
14,960
i5,535
16,110
1 6,685
i8
19
20
21
21
0,0625
0,0676
0,0729
0,0784
o,o84i
o,3o
o,3i
0,32
0,33
0,34
17,061
17,629
18,197
18,766
19,334
22
23
24
24
25
i7,o83
17,652
l8,22I
18,790
19,359
22
23
24
25
26
i7,io5
17,675
18,245
i8,8i5
19,385
22
23
24
24
25
17.127
17,698
18,269
18,839
19,410
23
23
23
25
25
i7,i5o
17,721
18,292
18,864
19,435
22
23
24
24
25
17.172
17.744
i8,3i6
18,888
19,460
22
23
24
24
25
17.194
17,767
18,340
18,912
19,485
22
23
23
25
25
17,216
17.790
1 8,363
18,937
19,510
22
22
24
24
25
17,238
17.812
18,387
18,961
19,535
22
23
23
24
25
17,260
17,835
18,410
18,985
19,560
22
23
24
25
25
0,0900
0,0961
0,1024
0,1089
0,1 1 56
0.35
o;36
0,37
o,38
0,39
19,902
20,471
2 1 ,039
2 1 ,607
22,175
26
26
27
28
29
19,928
20,497
2 1 ,066
21,635
22,204
26
27
28
28
29
19,954
20,524
2 1 ,094
21.663
22,233
26
27
27
29
29
19,980
20,55 1
21,121
21,692
22,262
26
26
28
28
29
20,006
20,577
21,149
21,720
22,291
26
27
27
28
29
20,o32
20,604
21,176
21,748
22,320
26
27
27
28
28
2o,o58
20,63i
2I,203
21,776
22,348
26
26
27
28
29
20,084
20,637
2I,23o
21,804
22,377
25
27
28
28
29
20,109
20,684
21,258
21,832
22,406
26
26
27
28
29
20,i35
20,710
21,285
21,860
22,435
26
26
27
28
28
0,1225
0,1296
0,1369
0,1444
0,l52I
o,4o
0,4 1
0,42
0,43
0,44
22,743
23,3ii
23,879
24,447
25,01 5
3o
3i
3i
32
33
22,773
23,342
23,910
24,479
25,o48
3o
3o
32
32
32
22,8o3
23,372
23,942
24,5ii
25,080
^9
3o
3i
32
33
22,832
23,402
23,973
24,543
25,ii3
3o
3i
3i
32
33
22,862
23,433
24,004
24,575
25,i46
'9
3o
3i
3i
32
22,891
23,463
24,o35
24,606
25,178
3o
3o
3i
32
33
22,921
23,493
24,066
24,638
25,211
^9
3i
3i
32
32
22.950
23,524
24,097
24,670
25,243
3o
3o
3i
32
32
22,980
23,554
24,128
24,702
25,275
29
3o
3i
3i
33
23,009
23,584
24,159
24.733
25,3o8
3o
3o
3o
32
'32
0,1600
0,1681
0,1764
0,1849
0,1936
0,45
o,5o
0,55
0,60
o,65
0,70
25,583
28,422
3i,259
34,095
36,93o
39,763
33
37
4i
45
48
52
25,616
28,459
3i,3oo
34, 1 40
36,978
39,815
34
37
4i
44
48
52
25,65o
28,496
3i,34i
34,184
37,026
39,867
33
37
4i
45
t
25.683
28;533
3 1, 382
34,229
37.075
39.919
33
37
40
44
48
52
25,716
28,570
3l,422
34,273
37,123
39.971
34
37
4i
45
48
52
25,75o
28,607
3 1,463
34,3i8
37,171
40,023
33
37
4i
48
52
25,783
28,644
3i.5o4
34,362
37.219
40,075
33
37
4o
45
48
5i
25,816
28,681
3 1, 544
34,407
37,267
40,126
33
37
4i
AA
48
52
25,849
28,718
31.585
34,45i
37,3 1 5
40,178
33
37
4o
AA
48
52
25,882
28,755
3i,625
34,495
37,363
4o,23o
33
36
4i
AA
48
5i
0,2025
0,2 5oo
o,3o2 5
o,36oo
0,4225
0,4900
0,75
0,80
0,85
0,90
o,g5
1,00
42,594
45,424
48,25i
5 1 ,076
53,899
56,720
56
59
63
67
71
75
42,65o
45,483
48,3 1 4
5i,i43
53,970
56,795
56
60
64
68
71
75
42,706
45,543
48,378
5l,2II
54,o4i
56,870
56
59
63
67
71
74
42,762
45,602
48 ,44 1
51,278
54,112
56,944
55
60
63
67
71
75
42,817
45,662
48,5o4
5 1,345
54,1 83
57,019
56
5q
63
67
71
74
42,873
45,721
48,567
5i,4i2
54,254
57,093
56
66
70
75
42,929
45,780
4S,63o
51,478
54,324
57,168
55
60
64
67
71
74
42,984
45,840
48,694
5i,545
54,395
57,242
55
59
62
67
70
74
43,039
45,899
48,756
5i,6i2
54,465
57,3i6
56
59
63
66
70
74
43,095
45,958
48,819
51,678
54,535
57,390
55
1%
67
71
74
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
7,3345 1
7,3728
7,41131
7,44981
7,4885 1
7,527:21
7,5661 1
7,6050 1
7,6441 1
7,68321
i""
^ . (r + '*")* or Î-" -\- r"~ nearly. 1
568
~1
ii4
170
227
284
341
398
454
5ii
569
57
ii4
171
228
285
341
398
455
5l2
070
57
ii4
171
228
285
342
399
456
5i3
571
572
573
574
575
576
57
57
57
57
58
58
ii4
ii4
ii5
ii5
ii5
ii5
171
172
172
172
173
173
228
229
220
23o
23o
23o
286
286
287
287
288
288
343
343
344
344
345
346
400
400
4oi
402
4o3
4o3
457
458
458
459
460
461
5i4
5i5
5i6
5i7
5i8
5i8
TABLE II. — TofinJ the time T; the sum of the radii r-)-r", and tlie chord c bcinp given.
Sum of tlie Ra'iii r-f-r".
Choril
c.
0,00
0,01
0,02
o,o3
o,o4
o,c5
0,0(1
0,07
0,08
o,og
0,10
0,1 I
0,12
0,1 3
0,1 4
0,1 5
o,j6
0,17
0,18
0,19
0,30
0,31
0,23
0,33
0,24
0,25
0,36
0,37
0,28
0.29
o,3o
0,3 1
0,32
0,33
0,34
0,35
0,36
0,37
o,38
o,3g
o,4o
0,4 1
0,43
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
1. 00
3,93
Days lilif
0,000
0,576
l,l52
1.7=9
2,3o5
2,88 1
3,457
4,o33
4,610
5,186
5,762
6.338
6.9,4
7,4il"
8,067
8,ti43
9-" y
9,79:1
10,371
ro,947
11,523
12,099
12,675
l3,2 )I
13,837
r4,4o
14,979
15,355
16,1 J 1
16,706
17,383
17,858
18,434
19,010
10,585
20,l6l
20,736
3I,3l3
21,888
22,463
23,039
23,6i4
24,189
24,765
35,340
2J,Ql5
28,791
3 1, 666
34,539
37,411
40,381
43,130
46,017
48,882
5 1, 74 5
54,606 70
57,4641 74
3,94
Days |(iif.
7,72:
3.Î
0,000
0,577
i,i54
i,73i
2,3o8
2,885
3,462
4.039
4,616
5,193
5,769
6,346
6.93
7,5oo
8,077
8,654
0,23 I
5.807
10, 384
■ 0.96
11,538
12,1 14
12,691
13.26S
i3,84]
i4>42i
14,998
1 5,5-5
16, i5i
16,728
i7,3o4
17,881
18,457
'9,o34
ig,6io
20,187
20.761
21339
21,913
22.493
23,068
3 3,644
24,220
24,796
25,372
25.948
28,838
3r,7o6
34,j83
37,'i59
4o,33r
43,3oj
46,076
48,945
5i,8ii
54,676
57,538
3,95
Days |dir.
7.7618
1 1,332
1 2 , 1 3o
I2,70'
1 3,285
1 3,862
i4,44o
1 5,0 1
15,594
16,172
if>,749
17,326
1 7,903
18.481
i9,o58
19,635
20,212
20,789
21.36b
2 ' ,943
22,320
23,09'
23,674
24,331
24,828
25,4o5
35.9S1
28,865
31,747
34,627
37,507
4o,384
43.260
46; 1.35
49.007
5^878
54,746
57,612
0,000
0,578
i,i55
1,73.
2,3l
2.888
3,466
4,o44
4,621
5,199
5,777
6,354
6,()32
7,309
8,087
8,665
9,242
9,820
10,3971 i3
io,975| i4
3,96
Days liiif
7,8013
0,000
0,378
l,l57
1,735
2,3i4
3,892
3,470
4,049
4,637
5,206
5,784
6,363
6.941
7,5i9
8,097
8,676
9,254
9,833
io,4io
■ 0,989
1 1 ,567
12,145
12,723
i3.3o2
1 3,S8o
(4,458
1 5,0 36
1 5,61 4
16,192
16,7
17,'
17,926
i8,5o4
19,082
19,660
20,2.38
20.816
' 1^393
31,971
22,549
23,704
24,382
34:839
23,437
26,014
28,901
31,787
34,671
37,554
40,436
43.3i5
46,193
49,070
5 1,944
54,816
57,68')
0,371^
1,138
1,737
2,3i7
2.896
3,-175
4,o54
4,633
5,212
3,791
6,370,
6,949 9
7,328
S,io8
7,8408
8,687
9,366
9,84
10,434
I 1 ,003
11,582
12,161
12,739
1 3,3 18
1 3,897
14,476
13,03
i5,63-
i6,2i3
16.79
17,370
17,949
18,527
19,106
19,683
30,263
20,843
21,420
21,999
22,57
23.1 56
23;734
3.'<,3l3
34,89
35,469
26,047
28.938
3 1, 83'
34,71 5
37,603
40,487
43,370
46,252
49,1 32
32,010
54,886
57,759
3,98
0,0(K)
o,58o
I
1,160
I
1 ,740
2,319
2
3
■-',899
4
3,479
4,0 5(1
5
5
4,639
5,219
6
6
5,799
(i,378
7
8
6,958
7,538
9
9
8,118
10
8,698
10
9,277
12
9,857
10,437
12
i3
i>,oi6
i4
11,596
i5
12.176
i5
12.733
17
1 3,335
17
13.9) 5
17
) 4,494
) 5,074
19
19
15.653
16,233
1 6,8 1 2
17,392
'7,971
i8,55i
i9,i3o
19,710
20,289
30,868
21,447
22,027
33,606
33,i85
33,764
24,343
24,923
25,5oi
36,080
28,974
31,867
34,759
37,649
4o,538
43,425
46,3x1
49,194
53,076
54,955
57,833
7,8805 I 7,9202
0,0000
0,tK)01
0,0004
0,0009
0,0016
0,0035
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,01 3 1
0,01 44
0,0169
0,0196
0,0335
0,0256
0,0389
o,o324
o,o36i
o,o4oo
0,044 1
0,0484
0,0529
t),o576
0,0625
0,0676
0,0729
0,0784
o,o84i
0,0900
0,0961
0,1024
0,1089
0,11 56
0,1225
o, 1 296
o, 1 369
0,1 444
0,l521
0,1600
0,1 68 1
0,1764
0,1849
0,1936
0,2025
o,25oo
o,3o25
o,36oo
0,4335
0,4900
0,5635
o,64oo
0,7335
0,8 1 00
0,9025
1 ,0000
^2^
. ( r -f- r " ) ^ or r'^ -{- r' '^ nearly.
373
576
577
578
579
58o
I
58
58
58
58
58
58
I
2
113
ii5
ii5
116
116
116
2
3
173
173
173
173
174
174
3
4
23o
23o
33l
23l
232
232
4
5
388
288
389
289
390
290
5
6
345
346
346
347
347
348
6
7
4o3
4o3
4o4
4o5
4o5
406
7
8
460
461
462
463
463
464
8
9
5i8
5(8
5,9
520
521
523
0
Proi». \n\t\^ Inr tln! sum uf tliu Uadii.
. I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9
6
6
6
7
7 10
1 1
i& 21
'9
21
16 21
16 22 27
22
33
23
24
35
33 26
24
25
26
26
27
28
3^
3o
3i
32
33
34
34
35
36
37
38
38
39
40
41
43
43
43
44
45
46
46
47
48
t
5o
5i
53
53
54
54
27
28
29
3o
3i
33
33
33
34
35
36
37
38
39
4o
4i
4i
43
43
45
46
47
48
49
5o
5o
5i
52
53
54
55
56
57
58
59
59
60
61
62
63
64 721
73 81
80I90I
a15
TABLE II
. — To find the time T
; the sum
3f the radii
r-\-r", and the chord c being
given.
Sum of the Radii r-\-r".
Chord
C.
3,99
4,00
4,01
4,02
4,03
4,04
4,05
1 4,06
4,07
4,08
Days |dif.
Days |dif.
Days |dif.
Days|dif.
Days |dif.
Days|dif.
Days |dir.
Days |dif. 1 Days |d
if. Days |dif.
0,00
0,00c
0,00c
0,00c
)
o,ooc
0,00c
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
o,58i
r
o,58i
o,58:
I
o,583
c
o,58:
o,584
1
o,585
I 0,586
0 0,586
I
0,587
I
0,000 1
0,02
1,161
3
i,i63
1
1,164
:;
I,i6(
1
1,16-
1
1,168
1,170
I 1,171
2 1,173
I
1,17^
2
0,0004
o,o3
1,742
2
1,744
i,74f
s
1,746
i,75c
1,753
1,755
2 1,757
3 i,75g
3
1,761
2
0,0009
0,04
2,323
3
2,325
-
2,32fi
-
2,33i
3,334
C
3,337
'-
2,340
3 2,343
3 2,346
2
3,348
3
0,0016
o,o5
2,90;
4
¥^ïl
z
2,910
A
3,qi4
3,917
A
3,931
A
2,935
3 3,g38
4 2,932
4
2,936
3
0,0025
0,06
3,484
4
3,488
A
3,493
5
3,497
4
3,5oi
A
3,5o5
c
3,5io
4 3;5i4
4 3,5i8
5
3,523
4
o,oo36
0,07
4,064
5
4,069
5
4,074
5
4,079
5
4,084
f
4,090
(;
4,og5
5 4,100
5 4,io5
5
4,1 10
5
0,0049
0,0064
0,08
4,645
6
4.65 1
5
4,656
e
4,662
€
4,668
6
4,674
ë
4,680
5 4,685
6 4,691
6
4,697
6
0,09
5,225
7
5,232
e
5,338
7
5,245
6
5,25i
7
5,358
e
5,364
7 5,371
6 5,277
7
5,284
6
0,0081
0,10
5,806
7
5,8 1 3
7
5,830
8
5,828
7
5,835
7
5,842
7
5,849
8 5,857
7 5,864
7
5,871
7
0,0100
0,11
6,386
8
6,394
&
6,4o2
8
6,4 10
8
6,4i8
8
6,426
e
0,434
8 6,443
8 6,45o
8
6,458
8
0,01 3 1
0,12
6,967
9
6,976
8
6,984
9
6,993
9
7,002
8
7,0 1 c
6
7,019
9 7,038
8 7,o36
9
7,045
9
0,01 44
0,1.3
7.547
10
7,557
9
7,566
10
7,576
9
7,585
10
7,595
s
7,604
9 7>6i3
lo 7,623
9
7,632
9
0,0169
0,14
8,128
10
8,i38
10
8,i48
10
8,i58
II
8,i6g
10
8,179
10
8,189
10 8,199
10 8,309
IC
8,219
10
0,01 96
o,i5
8,708
II
8,719
II
8,73o
II
8,74 1
11
8,752
II
8,763
II
8,774
II 8,785
10 8,795
11
8,806
II
0,0325
0,16
9,289
12
9,3oi
1 1
9,3 12
12
9,324
1 1
9,335
13
9,347
13
9,359
II 9.370
12 9,382
II
9,393
12
0,0256
0,17
9,8f>9
1 3
9,882
13
9,894
12
9,906
i3
9,919
13
9,931
13
9,943
i3 9,956
1 3 9,968
13
9,980
12
0,0289
0,18
io,45o
i3
io,463
i3
10,476
i3
10,489
i3
lO,502
l3
io,5i5
l3
10,538
i3 io,54i
1 3 i 0,554
i3
10,567
i3
o,o324
0,19
ii,o3o
i4
1 1 ,044
14
1 1 ,ù58
i4
1 1 ,072
i3
1 1 ,o85
i4
11,099
14
1 1,1 T 3
i4 11,127
i3 ii,i4o
i4
ii,i54
14
o,o36i
0,20
1 1 ,6 1 1
i4
11,625
i5
1 1 ,640
i4
1 1 ,654
i5
1 1 ,669
i4
11,683
i5
11,698
i4 11,712
i5 11,727
i4
II,74l
i4
o,o4oo
0,21
I3,igi
i5
12,206
16
12,322
i5
12,237
i5
13,353
i5
13,367
16
13,283
1 5 13,298
i5 i3,3i3
i5
13,338
i5
0,044 1
0,22
13,772
16
12,788
16
i3,8o4
i5
i2,8ig
16
12,835
16
1 3,85 1
16
13,867
16 12,883
16 13,899
16
I3,gi5
16
o,o484
0,23
i3,353
17
1 3,369
16
i3,385
17
l3,403
17
i3,4ig
16
i3,435
17
i3,452
17 13,469
16 i3,485
17
l3,S03
16
o,o53g
0,24
13,932
18
i3,g5o
17
13,967
18
13,985
17
l4,003
17
14,019
18
i4,o37
17 i4,o54
17 14,071
18
i4,o8g
17
0,0576
0,25
1 4,5 1 3
18
i4,53i
18
14.549
18
14,567
18
i4,585
18
i4,6o3
18
14,621
18 i4,639
18 i4,657
18
14,675
18
0,0625
0,36
i5,og3
19
l5,I 13
19
i5,i3i
19
i5,i5o
i8
1 5,168
'9
15,187
19
1 5,206
ig i5,325
18 r5,343
19
15,363
19
0,0676
0,27
15,673
20
15,693
'9
i5,7i3
30
1 5,733
20
1 5,752
19
i5,77>
30
1 5,79 1
19 i5,8io
30 i5,83o
19
i5,84g
19
0,072g
0,28
16,253
21
16,274
30
i6,3q4
30
16,3 1 4
2!
16,335
20
i6,355
30
16,375
30 i6,3g5
21 i6,4i6
30
1 6,436
20
0,0784
0,2g
1 6,834
31
i6,855
31
16,876
21
16,897
21
i6,gi8
21
16,939
21
1 6,g6o
21 16,981
21 17,003
31
17,033
20
o,o84i
o,3o
I7,4i4
22
17.436
21
17,457
33
17,479
33
i7,5oi
32
17,533
21
17,544
23 17,566
33 17,588
3!
17,609
22
0,0900
0,3 1
17,994
33
18,017
22
i8,o39
23
18,063
33
18,084
22
iS,io6
33
i8,i2g
33 i8,i5i
23 18,174
23
i8,ig6
32
0,096 1
0,32
18,574
33
18,597
24
i8,62'i
23
18,644
33
18,667
23
i8,6go
23
18,7.3
24 18,737
23 18,760
33
1 8,783
23
0,1024
0,33
19,154
34
19,178
24
19,303
34
19,336
34
ig,3 5o
24
19,274
24
19,298
24 19,322
24 19,346
33
19,369
34
0,1089
0,34
19,734
25
19,759
25
19,784
24
19,808
25
19,833
25
19,858
24
19,882
2 5 19,907
24 19,931
25
ig,g56
34
0,1 1 56
0,35
20.3 1 4
26
20,340
35
20,365
26
3o,3gi
25
3o,4i6
25
3o,44i
26
20,467
25 30,492
35 20,5l7
25
20,543
36
0,1225
o,36
20,894
27
20,921
26
20,947
36
20,973
26
2o,qqq
26
21,025
26
2I,o5l
26 21,077
26 2i,io3
26
2i,i3g
36
0,1296
0,37
21,474
37
2 1, loi
27
31,538
27
31,555
27
21,582
27
21,609
26
21,635
27 21,663
27 21,689
27
21,716
26
0,1 369
o,38
2 2,o54
38
23,083
28
22,110
27
22,l37
28
22,l65
27
22,193
28
22,220
27 32,247
28 22,275
27
22.302
37
0,1 444
0,39
22,634
29
22,663
28
22,691
28
22,719
29
22,748
28
33,776
28
22,804
28 22,832
28 22,860
28
33,888
39
0,l52I
0,40
23,2l4
?9
33,243
29
33,273
29
33,3oi
29
23,33o
29
23,359
20
33,388
29 23,417
29 23,446
29
33,475
39
0, 1 600
0,4 1
33,794
3o
23,824
3o
23,854
39
23,883
3u
33,913
3o
23,943
3Ô
23,973
29 24,002
3o 24,o33
39
34,061
3o
0,1681
0,42
34,374
3o
24,404
3i
24,435
3o
24,465
3i
34,496
3o
34,536
3i
24,557 .
ÎO 24,587
3o 34,617
3,
24,648
3o
0,1764
0,43
24,954
3 1
24,985
3i
25,016
3i
25,o47
33
25,079
3i
25,110
3i
35,i4i
jl 25,172
3i 2 5,2o3
3i
35,334
3i
0,1849
0,44
25,533
32
25,565
33
3 5,5g7
33
35,62g
32
25,661
33
25,693
32
25,735
32 25,757
3 1 25,788
32
25,820
33
0,1936
0,45
26,113
33
26,146
32
26,178
33
36,311
33
36,344
33
26,276
33
36,309
Î2 26,341
33 26,374
32
36,406
33
0,3025
o,5o
29,011
36
29,047
37
3g,o84
36
29,120
36
39, 1 56
36
39,193
37
29,22g
Î6 29,265
36 29,301
36
29,337
36
0,2 5oo
0,55
31,907
4i
3 1, 948
40
31.988
4o
33,038
39
33,c)67
4o
33,107
4"
32,i47 '
io 32,187
^0 32,237
39
32,266
4o
o,3o25
0,60
34,8o3
ÂA
34,847
43
34,890
AA
34,934
AA
34,978
43
35,021
44
35,o65 /
13 35,108
43 35,i5i
44
35,195
43
o,36oo
o,65
37,697
47
37,744
48
37,792
47
37,839
47
37,886
47
37,933
47
37,980 t
iS 38,028
il 38,075
46
38,121
47
0,4335
0,70
40,589
5i
4o,64o
5i
40,691
5i
40,742
5i
40,793
5i
40,844
5i
40,895
ji 40,946
5o 40,996
5i
4i,o47
5o
o,4goo
0,75
43,480
55
43,535
55
43,590
54
43,644
55
43,699
54
43,753
55
43,808
54 43,862
54 43,916
55
43,971
54
o,5635
0,80
46,369
59
46,438
58
46,486
58
46,544
59
46,6o3
58
46,661
58
46,719
58 46,777
58 46,835
58
46,Sg3
58
c,64oo
o,85
49,357
62
49,319
63
49,38i
62
49,443
62
4q,5o5
62
49,567
61
49,628 (
32 49,690 (
33 49,753
61
4g,8i3
62
0,7225
0,90
52, 142
66
52,308
65
53,273
66
53,339
66
5i,4o5
65
52,470
66
53,536 (
55 52,601 (
55 52,666
66
52.732
65
0,8100
0,95
55,035
70
55,095
69
55,164
69
55,333
70
55,3o3
69
55,372
69
55,441 (
39 55,5io
59 55,579
69
55,648
69
0,9025
1,00
57,906
73
57,979
73
58,o52
73
58,125
73
58,198
73
58,271
73
58,344 •
73 58,4 17
73 58,490
72
58,562
73
1 ,0000
7,9601 1
8,0000 1
8,0401 1
8,0802 1
8,12051
8,16081
8,201.
3 8,241
3 8,28251
8,3232 1
c2
\ . {r •\- r")'^ or r^ + r"' nearly. |
579
58o
58 1
582
583
584
585
586
587
588
I
58
58
58
58
58
58
~9
~59
"59
"5^
I
2
116
116
116
116
117
117
117
117
117
118
a
3
174
174
174
175
175
175
176
- 176
176
176
3
4
233
332
333
233
233
334
234
234
235
235
4
5
390
290
391
2qi
392
392
293
2q3
294
294
5
6
347
348
349
349
35o
35o
35i
352
352
353
6
7
4o5
406
407
407
408
409
4io
4io
4ii
4l2
7
8
463
464
465
466
466
467
468
469
470
470
8
9
r
21
5
32
c
23
5
241
5
35 1
5:
6 1
527
537
528
529
9
TABLE II. -
To find the time T;
the sum
of the rad
ir + ,
", and th
e chord
c being
given.
Sum ot" the Kaihi r-\-r'\ |
Prop, parts lur the sum of the Kadii. i
. 1 ^ 1 J 1 /I r, 1 /; 1 -, I i; 1 ^ 1
ChorJ
c.
4,09
4,10
4,11
4,1-2
4,13
4,14
I 1 2 1 J 1 4 1 J 1 6 1 7 1 8 1
y.
1
2
3
0
0
0
0
I
I
0
1
I
I
2
1
I
2
1
I
3
I
2
2
I
2
3
Days |dif.
Days |dif.
Days Idif.
Duya |dir.
Uays |dir.
Days |dif.
0,00
0,000
0,000
o,o< )0
0,000
0,000
OjOOt.)
<.),ouoo
0,01
o,58«
I
0,589
0
0,589
I
0,590
I
0,591
0
0,591
I
<.),0l.i01
4
1
3
2
2
3
3
4
0,02
1,17(1
1
1. 177
2
1. 1 -9
I
1,180
I
1,181
2
I,i83
1
0,0004
o,o3
1,763
3
1,766
2
1,768
2
1.770
2
..773
2
1,774
2
0,0009
5
2
2
3
3
4
4
5
o,o4
2,35i
3
2.354
3
2.357
3
2,36o
0
2,363
3
2,366
2
0,0016
6
7
2
2
2
3
3
4
4
4
4
5
5
6
5
6
o,o5
2,939
4
2,943
3
2,946
4
2,950
3
2,953
4
2,957
4
0,0025
8
2
2
3
4
5
6
6
7
0,06
3.527
4
3,53 1
5
3,536
4
3,540
4
3,544
4
3,548
5
o,oo36
9
2
3
4
5
5
6
7
8
0,07
4,ii5
5
4,120
5
4.125
5
4,i3o
5
4,i35
5
4,i4o
5
0,0049
10
2
3
4
4
5
5
f,
7
8
8
9
10
0,08
4,7o3
5
4,708
6
4,714
6
4,720
5
4,725
6
4,73 1
6
0,0064
2
3
fi
7
7
9
10
0,09
5,290
7
5,297
6
5,3o3
7
5,3io
6
5,3i6
7
5,323
6
0,008 1
12
2
4
()
8
1 1
0,10
5,8-8
-
5,885
7
5,892
8
5,900
7
5,907
7
5,914
7
0,0100
i3
i4
3
3
4
5
6
7
7
8
8
9
10
10
1 1
12
i3
0,11
6,466
S
6,474
8
6,482
8
6,490
7
6,497
8
6,5o5
8
0,01 3 I
0,12
7,o54
8
7,062
9
7,071
8
7.079
9
7,088
9
7,097
8
0,0.44
i5
2
3
5
6
8
9
11
12
.4
o,i3
7,64 1
10
7,65 1
9
7,660
9
7,669
10
7,679
9
7,688
9
0,0169
16
2
3
5
6
8
10
II
i3
.4
o,i4
8,229
10
8,239
10
8,249
10
8,259
10
8,269
10
8,279
10
0,0196
17
18
2
2
3
4
5
5
7
7
9
g
10
II
12
i3
i4
■ 4
i5
16
0,1^1
8,8 1-
II
8,828
10
8,838
11
8,849
II
8,860
II
8,871
10
0,033 5
■9
2
4
6
8
10
II
i3
i5
.7
0,16
9,4o5
II
9,416
12
9.428
II
9,439
12
9,45 1
I.
9,462
11
o,o256
16
18
0,17
9.993
i3
io,oo5
12
10,017
12
10,029
12
io,o4i
12
io,o53
12
0,0289
20
2
4
6
8
10
12
.4
0,18
io,58o
i3
10,593
i3
10,606
i3
io,6ig
i3
io,632
12
10,644
i3
o,o334
21
2
4
6
8
II
i3
i5
'7
•9
0,19
11,168
.3
11,181
i4
11,195
.4
11,209
i3
11,222
i4
1 1 ,236
i3
o,o36i
22
23
2
2
4
5
7
7
9
9
II
12
i3
i4
i5
16
18
18
20
21
0,50
.1,735
i5
11,770
i4
11,784
i4
11,798
i5
ii,8i3
i4
11,827
.4
o,o4oo
24
3
5
7
10
12
i4
'7
19
22
0,21
0,22
13,343
12,931
.5
i5
12,358
12,946
i5
16
12,373
12^62
i5
16
12,388
12,978
i5
16
1 2,4o3
12,994
i5
i5
12,418
13,009
i5
16
0,044 1
0,0484
25
26
3
3
5
5
5
6
8
8
8
8
10
i3
i3
i5
16
18
18
20
23
23
0,23
i3,5i8
■7
i3,535
16
1 3,55 1
17
1 3,568
16
1 3,584
17
i3,6oi
16
0,0529
3
3
10
i4
i4
16
24
25
0,24
i4,io6
17
i4,i23
17
i4,i4o
18
i4,.58
17
14,175
17
14,192
17
0,0576
27
28
1 1
1 1
17
19
30
22
0,25
14,693
18
14,711
18
i4,72<)
iG
.4,747
18
14,765
18
14,783
18
0,0625
29
3
6
9
12
i5
.7
30
23
26
0,26
1 5,281
'9
i5,3oo
18
i5,3i8
'9
1 5,337
'9
1 5,356
18
.5,374
19
0,0676
3o
3
6
9
9
10
12
i5
18
21
24
27
0,27
1 5,868
20
1 5,888
19
15,907
20
.5,927
■9
1 5,946
19
15,965
20
0,0739
3i
3
6
12
16
'9
22
25
28
0,28
16,4 56
20
16,476
20
16,496
20
i6,5i6
20
1 6,536
20
1 6,556
20
0,0784
32
3
6
i3
16
'9
22
26
29
0,29
17,043
21
17,064
21
i7,o85
21
17,106
21
17,127
20
17,147
21
0,084 1
33
34
3
3
7
7
10
10
i3
i4
.7
17
30
20
23
24
26
27
3o
3i
o,3o
i7,63i
21
17,652
22
17.674
21
17,695
22
17,7.7
21
17,738
22
0,0900
0,3 1
18,218
23
18,241
22
1 8.363
23
i8,285
22
18,307
22
18,329
23
0,096 1
35
4
7
II
i4
18
21
25
28
32
0,32
i8,So6
23
18,829
23
i8,852
23
18,875
23
18,897
23
18,920
23
0,1034
36
4
7
II
.4
18
22
25
29
32
0,33
19,393
24
.9>4i7
23
19,440
24
19.464
24
1 9,488
23
19,51 1
24
o,io8g
37
4
7
11
i5
■9
33
26
3o
33
0,34
19,980
25
20,005
24
20,029
25
2o,o54
24
20,078
24
20,102
25
0,11 56
38
39
4
4
8
8
II
12
i5
16
19
20
23
23
27
27
3o
3i
34
35
0,35
2o,568
25
20,593
25
20,618
25
20,643
25
20,668
25
20,693
25
0,I335
32
36
37
38
39
40
0,36
2i,i55
26
21,181
26
21,207
25
21,232
26
21,358
26
21,284
26
o,i3g6
40
4
8
12
16
20
24
28
0,37
21,742
27
21,769
26
21,795
27
21,822
26
2 1 ,848
27
21,875
26
0,1369
4i
4
8
12
16
21
25
25
29
33
34
34
35
0,38
22,329
28
22,357
27
2 2,384
27
92,4ll
27
22,438
28
2 2,466
27
o,t444
43
4
8
i3
17
31
=9
0,39
22,917
28
22,945
98
22,973
28
23,001
28
23,039
27
23,o56
28
0,l521
43
44
4
4
9
9
i3
i3
17
18
33
22
26
26
3o
3i
o,4o
23,5o4
28
23,532
29
23,56i
29
23,590
29
23,619
28
23,647
=9
0,1600
45
46
5
T /,
18
18
23
33
24
24
25
32
32
33
34
34
36
37
38
38
39
4i
4.
42
43
44
0,4 r
24,091
29
24,120
3o
24,1 5o
29
24,179
3o
24,209
'9
24,238
29
0,1681
5
5
5
5
9
14
i4
i4
i4
i5
27
28
28
0,42
24,678
3o
24,708
3o
24,738
So
24.768
3o
24,798
3i
24,829
3o
0,1764
9
0,43
25,265
3i
25,296
3i
25,327
3i
25.358
3<
25,388
3i
25,419
3i
0,1849
47
9
'9
0,44
25,852
32
25,884
3i
25,9.5
32
25,947
3i
25,978
32
26,010
3i
o,ig36
48
49
10
10
19
20
29
29
0,45
26,439
32
26,471
33
26,504
32
26,536
32
26,568
32
26,600
32
0,3025
5o
5
If»
1 5
20
25
3o
35
40
45
o,5o
29,373
36
29,409
36
29,445
36
29,481
36
39,5.7
35
3g,552
36
o,35oo
5i
5
10
i5
30
26
3i
36
4i
46
0,55
32,3o6
4o
32,346
39
32,385
40
32,425
3g
32,464
4(
32,5o4
39
o,3o35
52
5
10
16
31
26
3i
36
42
47
0,60
35,238
43
35,281
43
35,324
43
35,367
43
35,410
43
35,453
43
o,36oo
53
5
1 1
16
31
27
27
32
37
38
42
48
o,65
38, 168
47
38,2i5
47
38,262
47
38,3o9
4C
38,355
47
38,4o2
46
0,4225
54
5
1 1
16
22
32
43
49
0,70
41,097
5i
4.,i48
5o
41,198
5i
41,249
5o
41,29g
5o
41,349
5o
0,4900
55
6
II
17
22
28
33
39
44
5o
0,75
44,025
54
44,079
54
44,i33
54
44,187
54
44,241
54
44.295
53
0,5625
56
6
II
17
32
28
34
39
45
5o
0,80
46,95 1
57
47,008
58
47,066
58
47,124
5-
47,. 8.
58
47,239
57
0,64*10
57
6
II
17
23
29
34
4o
46
5i
0,85
49,875
61
49.936
61
49,997
62
5o,o59
61
5o,i2o
61
5o,i8i
61
0,7225
58
6
12
.7
23
29
35
4i
46
52
0,90
52,797
65
52,862
65
52,927
65
52,992
64
53,o56
65
53,121
65
0,8100
59
6
12
18
24
3o
35
4.
47
53
0,95
55,7.7
68
55,785
69
55,854
69
55,923
6t
55,991
68
56,o59
69
0,9025
1,00
58,635
72
58,707
72
58,779
72
58,85i
73
58,924
72
58,996
7.
I
,0000
60
61
62
63
64
6
6
6
6
6
12
12
12
i3
i3
18
18
19
19
24
24
25
35
26
3o
3 1
3i
32
32
36
37
37
38
38
42
43
43
44
45
48
49
5o
5o
5i
54
55
56
57
58
8,3641
8,4050
8,4461
8,4872
8,5285
8,5698
" C^
^ . (r + r")=^ or r^-^ r"'^ nearly.
587
588
589
590
5g<
592
.9
—
—
65
7
i3
20
26
33
3g
46
52
59
I
59
59
59
59
^§
59
I
66
7
i3
20
26
33
40
46
53
59
1
117
118
118
118
118
118
2
67
7
1 3
20
37
34
4o
47
54
60
3
176
176
177
177
177
178
3
68
7
i4
20
27
34
4i
48
54
61
4
235
235
236
236
236
237
4
69
7
i4
21
28
35
4i
48
55
62
5
294
294
2g5
295
296
296
5
6
352
353
353
354
355
355
6
70
7
i4
21
28
35
42
49
56
63
7
4ii
4l2
4l2
4i3
4i4
4i4
7
80
8
16
24
32
4o
48
56
64
72
8
470
470
471
472
473
474
8
90
9
18
27
36
45
54
63
72
81
9
528
529
53o
53i
532
533
9
100
10
20
_3o_
4o
5o
60
70
80
9^
TABLE II. — To find the time T; the sum of the radii r-\-r", and the chord c being given.
ISutn of tlie Radii r -\- t".
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,1 3
o,i4
0,1 5
o,t6
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
4,1.5
Days |<lir.
0,000
0,592
1,184
I,77f'
2,368
2, 961
3,553
4,i45
4,737
5,32g
5,921
6,5 1 3
7,io5
7,697
8,289
8,881
9:473
io,o65
10,657
11,249
ii,84i
12,433
1 3,025
13,617
14,209
i4,8oi
15,393
1J.985
16,576
17,168
17,760
18,352
18,943
19,535
20,127
20,718
a 1 ,3 1 o
2i,goi
22.^93
23,084
23,676
24,267
24,859
25,i5o
26,041
26,633
29,588
32,543
35,496
38,448
41,399
44,348
47,296
50,242
53,r86
56,128
59,067
4,1(5
Days |dir,
8,611:3
0,000
0,593
1,186
1,778
2,371
2,964
3,557
4,i5o
4,743
5,335
5,928
6,521
7,1'^
7,707
8,299
8,892
9,485
1 0,078
10,670
11,263
11.856
12,448
1 3,633
14,226
14,819
i5,4ii
16,004
16,596
17,189
17,781
18,37-1
18,966
19,558
20,l5l
20,743
21,335
21,928
22,520
a3,ii2
23,704
24,296
24.88g
25,48 1
26,073
26,665
29,624
32,583
35,539
38,495
41,449
44,402
:Î7,353
5u,3o3
53,25o
56,196
59. '39
4,17
Days III if.
8,6528
0,000
o,5g4
1,187
1,781
2,374
2,968
3,56i
4,i55
4,748
5,342
5,935
6,539
7,122
7,716
8,3og
8,903
9>49'5
10,090
1 0,68 i
11,276
17,870
1 2,46-;
1 3,007
i3,65o
14,243
1 4,836
r5,4;'o
1 6.02 ■;
^6:616
17,209
17.803
18,396
18,989
19,583
20,175
20,768
3i,36i
21,954
22,547
23,i4o
23,733
24,326
24,918
25, 5i I
26,ioi
26,697
29,660
32,621
35,582
38,541
41=499
,456
47,4ii
5o,364
53,3i5
56,264
59,21 1
4,18
Days |dif.
8,6945
0,000
0.59.
li'Sg
1,783
2,377
2,971
3,566
4,160
4,7'i4
5,348
5,943
6,537
7,i3i
7,725
8,3)9
8,913
9,5oS
10,102
10,696
1 1 ,290
11,884
12,478
1 3,072
1 3,666
14,260
i4,85
1 5,44s
i(i,o42
1 6.636
(7, 230
17,82
i8,4i8
19,012
19,605
20,199
20,793
2i,3S7
2 1 ,980
22,374
23,168
23,761
24,355
24,948
25,542
26, 1 35
26,72g
29,695
32,661
35.625
38,588
41,549
44,5oq
17,468
5o, 124
53,379
56.332
59,283
4,19
Days |.lif.
8,7362
0,000
0,595
i,igo
1.785
2,3So
2,975
3,570
4,i65
4,760
5,355
5,g5o
6,j44
7,1 39
7,734
8,329
8,924
9,519
io,ii4
10,709
ii,3o3
11,898
12,493
i3,oS8
i3,68J
'4,277
14,872
15,467
i6.o(i
1 6,656
'7,25i
17,845
i8,4':o
19,034
19.629
20;223
20,818
2l,4l2
23,007
2 2,(j<Ji
ai, 195
23,790
3 4,38 <
2'î,978
25,572
36,167
26,76 1
29,731
3 a. 700
35/568
38,634
41,59g
44,563
47,525
50.485
53,fi3
56, -lOO
59.354
4,20
Days ]dif.
26
38
8,7781
0,000
o,5g6
1,191
1,787
2,383
2,978
3,574
4,170
4,765
5,36i
5,957
6,552
7,148
7.744
8,339
8,935
9,53o
10,126
10,721
ii,3i7
ii,gi2
i2,5o8
i3,io3
13,69g
14,294
1 4.8go
1 5,485
16.0S1
1 6,676
17,271
1 7,867
18,462
19,057
19,65
ao,248
20,843
31,438
22,033
32,628
33,233
23.818
24.<l3
25,008
3 5,6o3
26,198
26,793
29,766
3 ',739
35.710
38.680
4 1,649
44,616
47.532
5o,546
53,5oS
56,468
5g,l26
4,21
Days |dif.
0,000
o,5g6
1,193
',789
3,386
2,982
3,578
4,175
4,77'
5,367
5,96
6,56o
7,1 56
7,753
8,349
8,g45
9,542
io,i38
10,734
ii,33o
1 5 11,927
i5 13,533
8,8200
i3,i 19
13.71J
i4|3ii
1 4,908
1 5,5o4
1 6, 1 00
16,696
17,292
17,888
18,484
19,080
19,676
20,272
20,868
2i,,|63
22,059
22,655
23.r!5l
23,847
24,442
2 5,0)8
a 5,634
26.22g
26,825
20.802
32,778
35,753
38,727
41,699
44,670
47,639
5o,6o6
53,5-2
56,536
30,497
4,22
Days Idil'.
8,8621
0,000
0,597
1,194
1,791
2,388
a.985
3,583
4,180
4,777
5,374
5,971
6,568
7,1 65
7,762
8,359
8,956
9,553
io,i5o
10,747
",344
11,941
13,538
1 3,i35
1 3,732
14,328
4,23
Days |dir.
0,000
0,598
r , I g6
1,793
2,391
2,989
3,587
4,i85
4,782
5,38o
5,978
6,576
7,173
7,77'
8,36g
8,967
9.564
10,162
10,760
11,357
11,955
12,553
i3,i5o
1 3,74s
14,345
14,925
18
'4,943
.8
1 3.323
lb
13,540
IQ
io,i 19
19
i6,i38
IQ
16,710
10
16,735
20
17,3l2
21
17,33;
20
'7.909
21
17,930
aa
i8,3ob
22
18,528
22
19.103
22
19,125
23
19,699
23
19,722
24
20,396
24
20,320
21
20,893
2 5
20,917
23
2i,4Sa
2J
2i,5i4
26
32.083
27
22,113
26
3 a. 68 a
27
22,709
27
23,278
38
23,3o6
28
23,875
28
2 3-ooj
aq
2 (.471
2Q
34.300
aq
25.068
29
23,097
3o
3 5,664
3o
35.694
3/
26,2(X)
3i
36,291
3?
26,85-
3i
26.8S8
32
29,837
36
29;S73
33
32,8.7
39
32 856
3q
33,795
a
35,838
Â2
38,773
46
38,819
46
41,749
49
41,798
30
44:733
53
44.776
53
47,6{j6
56
47.7"
37
50,6(17
60
30,727
60
53,636
64
53.700
64
56,6o3
68
36,671
67
5g,i68
72
39,640 71
8,9465
8,90
12
4,24
Days Idif.
0,000
0,598
','97
1,796
2,394
2,993
3,591
4,190
4,788
5,386
5,c
6,583
7,183
7,780
8,379
8«77
9,576
10,174
10,773
11,371
"«69
12.567
1 3, 1 66
1 3,764
i4,362
14,961
1 5,559
16,137
16.755
17,353
r7,o52
i8,53o
'9:'
'9.746
20,344
20,942
21,540
22,l38
2 2,736
23,334
23,932
24,52g
25,127
25.725
26,323
36,930
29,908
32,893
35,880
38,865
4 1,848
'14,829
47.809
30,787
53,764
56,738
59,7"
0,0000
0,0001
o,ooo4
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,0100
0,0121
0,0 1 44
o,oi()g
0,0196
0,0235
0,02 56
0,0389
o,o324
o,o36i
r4 o,o4oo
1 5 o,o44'
0,0484
0,0529
0,0576
8,9888
0,062 5
0,0676
0,072g
0,0784
0,084 1
0,0900
0,0961
0,1024
0,1089
0,1 156
0,1335
o, 1 396
o, 1 369
0,1 444
0,l52I
0,1600
o, 1 68 1
0,1764
o, 1 84g
o,ig36
0,2025
0,2 5oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
0,6400
0,7225
0,8100
o,go25
T ,0000
■ Ir -\- r-')-
+
nearly.
591
592
593
594
5g
59
59
59
ti8
118
119
"9
177
178
178
178
2 36
237
237
238
2q6
296
297
297
355
355
356
356
4i4
4i4
4i5
4i6
473
474
474
475
533
533
534
535
595
60
"9
179
238
298
357
417
476
536
596
60
119
179
238
298
358
4'7
477
536
597
60
"9
179
239
299
358
418
478
537
598
60
120
179
239
299
359
419
478
538
399
60
120
180
240
3oo
35g
4'9
479
539
TABLE II.-
- To tmJ the time T;
lie suir
of the latl
i r-f-r", am! the
chord e heiiif; given
Sum of Iho Ka'lil r-j-r".
Prop, parts Ibl' tlio aolii uf tlif Itudii. 1
ll2iSI/il5l6i-,l8lol
Chord
C.
0,00
4,25
4,26
4,27
4,28
4,29
4,30
2
3
0 0
0 0
0 1
0 0
1 I
I I
I
I
2
I
2
I I
?
I
2
3
Days Idif.
Dii>3 |Jif.
Days Idir.
Days |dir
Dnys Idif.
l)uy3 Idif.
I
2
2
3
0,000
0,000
0,000
0,000
0,000
0,000
0,OUtiO
0,01
0,599
I
0,600
I
0,601
0
0,601
I
0,602
1
o,(io3
0
O.CtOOI
4
0 1
I 2
2
2
3
3
4
0,02
1,198
2
1,200
1
1,201
2
1,20J
1
1,204
1
l,205
2
0,0004
o,o3
iryS
2
1,800
2
1 ,802
2
1,804
2
I,8o()
2
1,808
2
0,0009
5
I 1
2 2
3
3
4
4
5
o,o4
2,397
3
2,4oo
2
2,402
3
2,4o5
3
2,408
3
2,4lJ
3
0,0016
6
7
I 1
2 2
2 3
3
4
4
4
4
5
5
6
5
6
o,o5
2.qq6
4
3,000
3
3.003
4
3,007
3
3,010
4
3,oi4
3
0,0025
8
I 2
2 3
4
5
6
C
7
0,06
3.5^
4
3,599
5
3,604
4
3,608
4
3,612
4
3,616
5
o,oo36
9
I 2
3 4
5
5
6
7
8
0,07
4,>94
5
4,199
5
4,204
5
4,209
5
4,3 14
5
4,219
5
0,0049
3 4
3 4
4 5
5
6
8
0,08
4.794
5
4,-99
6
4,8o5
6
4.811
t
4,816
6
4,822
5
0,0064
10
I 2
7
9
0,09
5,393
6
5,399
7
5,406
6
5,412
b
5,418
6
5,424
7
0,0081
1 1
1 3
I 3
I 2
6
6
7
7
8
8
i(-
10
1 1
0,10
5.Q92
_
5.999
-.
6,006
^
6,01 3
7
6,020
7
6,027
7
0,0100
i3
1 3
I 3
4 5
7
8
&
9
u
1 2
i3
0,1 I
6,591
8
6,599
8
6,607
7
6,6 1 4
b
6,622
8
6,63o
8
0,01 2 1
.4
4 6
7
10
1 1
0,12
7,190
9
7.199
8
7.207
9
7,216
b
7,224
9
7,233
8
0,01 44
i5
2 3
5 6
8
9
10
1 1
12
.4
0,1 3
7,-8q
10
7.-99
9
7,808
9
7,817
9
7,826
9
7,835
9
0,0169
16
3 3
5 6
h
1 1
iC
.4
o,i4
8,389
9
8,398
10
8,4o8
10
8,418
i<
8,428
10
8,438
10
0,0196
17
2 3
5 7
g
to
12
U
i5
0,1 5
8,988
10
8,998
II
9,009
10
9,019
II
g,o3o
IC
9.040
II
0,0225
18
19
2 4
2 4
5 7
6 8
9
10
1 1
1 1
i3
i3
14
it
16
17
0,16
9,58-
11
9.598
1 1
9,609
12
9,621
1 1
9,632
11
9,643
11
0,02 56
0,17
10,186
12
10,198
12
10,210
12
10,222
12
10,234
12
10,246
12
o,<;)28g
20
2 4
6 8
10
1 3
14
it
18
0,18
io,-85
i3
10,798
12
10,810
i3
10,823
i3
io,836
12
10,848
i3
o,o324
21
2 4
6 8
1 1
i3
i5
f
.9
0,19
11,384
i4
11,398
i3
ii,4ii
i3
11,424
14
11,438
i3
ii,45i
i3
o,o36i
22
23
2 4
2 5
7 9
7 g
1 1
12
.3
i4
i5
iG
I^
I^
20
21
0,20
11,983
i4
".997
.4
12,011
14
12,025
.4
12,039
i4
i2,o53
i5
o,o4oo
24
2 5
7 10
I 2
i4
'7
IÇ
22
0,21
12,582
i5
12,597
i5
12,612
i5
12,637
14
12,641
i5
12,656
i5
0,044 1
0,22
i3,i8i
16
13,197
i5
l3,212
16
l3,228
i5
13.243
16
i3,25g
i5
0,0484
25
3 5
8 10
i3
.5
18
21
23
0,23
13,780
16
13,796
'7
i3.8i3
16
13,829
16
i3,845
16
1 3,861
16
o,o52g
26
3 5
8 10
i3
16
lb
2
23
0,24
14,379
17
14,396
17
i4,4i3
17
.4,430
17
14,447
17
14,464
16
0,0576
27
28
3 5
3 6
8 II
8 II
i4
i4
16
17
19
2Û
2:
25
24
25
0,25
14,978
18
14,996
17
i5,oi3
18
i5,o3i
ife
.5,049
17
1 5,066
18
0,0625
29
3 6
9 '2
i5
17
20
2;
26
0,26
.5,577
18
15,595
19
i5,6i4
18
1 5,632
ifc
1 5,650
19
1 5,66g
18
0,0676
0,27
16,176
'9
16,195
'9
16,214
'9
16,233
19
16,252
19
16,271
19
0,0729
3o
3 6
9 12
i5
18
21
24
27
0,28
.6,775
20
16.-95
19
16,814
20
i6,834
2(
i6,854
'9
.6,873
20
0,0784
3i
3 6
9 "
16
19
22
25
28
0,2g
17.374
20
17,394
21
17.4.5
20
.7,435
2(
17,455
21
17,476
20
0,084 1
32
33
3 6 ,
3 7 I
0 i3
0 i3
16
17
'9
20
23
23
2f,
2t
29
3o
o,3o
17>973
21
17,994
21
i8,oi5
21
i8,o36
21
18,057
21
18,078
21
o,ogoo
34
3 7 I
0 .4
17
20
34
27
3i
0,3 1
18,572
21
.8,593
32
i8,6i5
22
1 8,637
22
18,659
22
i8,6Si
21
o,og6i
35
28
32
32
33
34
0,32
19.170
23
19,193
22
ig,2i5
33
19,238
22
19,360
23
ig,283
22
0,1024
35
4 7 I
1 i4
18
21
0,33
0,34
20,368
23
34
19.79'
2o,3g2
24
24
19,816
20,4.6
23
24
ig,83g
20,440
23
34
19^862
20,464
23
33
ig,885
20,487
?3
24
0,1 o8g
0,1 1 56
36
37
38
4 7 1
4 7 I
4 8 I
1 .4
1 .5
1 .5
18
'9
.9
22
22
23
25
26
27
3^
3c
0,35
20,967
24
20,991
25
21,016
25
2i,o4i
24
3 1 ,o65
25
2 1 ,090
24
0,1225
39
4 8 I
2 16
20
23
27
3i
35
0,36
2 1,565
26
2i,5gi
25
21,616
25
2 1 ,64 1
26
3 1 ,667
25
21,69?
25
0,1 2g6
4.,
4.
42
43
4 8 I
4 8 I
4 8 I
4 9 I
2 16
2 16
3 .7
3 17
24
25
25
26
28
29
3p
36
37
38
0,37
o,38
22,164
22,763
26
26
22,igo
22,789
26
27
22,216
22,816
26
27
22,242
22,843
26
27
22,268
22,870
26
26
22,2g4
22,8g6
26
27
0,1 3(39
0,1 444
30
21
33
34
34
0,39
23,36i
28
23,389
27
23,4i6
28
23,444
27
23,47.
27
23,4g8
28
0,1 52 I
21
23
=9
3o
39
o,4o
23,960
28
23,988
28
24,016
28
24,044
28
24,072
28
24,. 00
28
0, 1 600
44
4 g I
3 .8
22
26
3i
35
40
0,4 1
24,558
=9
24,587
29
24,616
29
24,645
29
24,674
29
24.703
28
0,1681
45
46
47
48
5 r, T
418
4 i&
4 '9
4 '9
23
27
33
36
4i
0,42
25,i57
29
25,186
3o
25,216
3o
25,246
29
25,275
3o
25,3o5
29
0,1764
5 a
23
28
32
37
4i
0,43
25,755
3.
25,786
3o
25,816
3o
25,846
3o
25,876
3i
25,go7
3o
0,1849
5 g I
24
24
28
33
38
42
0,44
26,354
3i
26,385
3i
26,416
3.
26,447
3i
26,478
3o
26,508
3i
o,ig36
J g I
5 10 I
29
34
38
43
0,45
26,952
32
26,984
3i
27,01 5
32
27,047
33
27,07g
3i
27,110
32
0,2025
49
5 10 I
5 20
25
29
34
39
44
o,5o
29,943
36
59.979
35
3o,oi4
35
3o,o49
35
3o,o84
35
3o, 1 1 9
36
o,35oo
5o
5 10 I
5 20
25
3o
35
40
45
0,55
32,934
39
32,g73
38
33,011
39
33,o5o
39
33,089
38
33,127
39
o,3o2 5
5i
5 10 I
5 20
26
3i
36
4.
46
0,60
35,923
42
35,g65
43
36,oo8
42
36,o5o
42
36,og2
42
36, 1 34
42
o,36oo
52
5 10 I
6 21
26
3i
36
42
47
0,65
38.911
46
38,g57
46
39,003
45
39,048
46
3g,og4
46
3g,i4o
46
0,4225
53
5 II 1
6 21
27
32
37
42
48
o,7<j
41,897
5o
4., 947
49
4i,gg6
5o
42,046
49
42,og5
49
42,144
49
o,4goo
54
5 II I
6 22
27
32
38
43
49
0,75
44,882
54
44,936
52
44,988
53
45,o4i
53
45,og4
53
45,147
53
o,5625
55
5 II I
7 22
28
33
39
44
5o
0,80
47.866
57
47,923
56
47.979
57
48,o36
56
48,092
56
48,i48
57
0,6400
56
5 II I
7 22
28
34
39
45
5o
o,85
5o,848
60
50,908
60
50,968
60
5 1, 028
60
5 1, 088
60
5i,i48
60
0,7225
57
5 I I I
7 23
29
34
4o
46
5i
0,90
53,8p8
63
53,891
64
53,955
64
54,019
63
54,082
64
54,146
63
0,8 1 00
58
5 12 I
723
29
35
4.
46
52
0,95
56,8o6
67
56,873
67
56,940
68
57,008
67
57,075
67
57,142
67
o,go2 5
59
i 12 I
8 24
3o
35
4.
47
53
1,00
59,782
7.
59,853
71
59,924
70
59,994
7.
6o,o65
71
60, 1 36
70
1 ,0000
60
61
62
63
64
5 12 I
5 12 I
5 12 I
5 i3 1
3 i3 I
8 24
S 24
9 25
?, 25
3o
3i
3 1
32
36
42
43
43
44
48
t
5o
54
55
56
57
58
9,0313 1
9,0738
9,1165"
9,1592
9,2021
9,2450 1
"?~
37
37
38
^ . (r -|- I- ") '^ or r"^ -{- r'"^ nejirty. |
598
599
600
601
602
6o3
9 23
9 26
32
38
45
5.
I
60
60
60
60
60
60
I
65
66
7 i3 2
7 i3 2
0 26
a 26
33
33
39
40
46
46
52
53
59
2
3
120
■79
120
180
120
180
120
180
120
181
121
181
2
3
67
68
7 i3 2
^ i4 2
T 27
D 27
34
34
4o
4i
47
48
54
54
60
61
4
5
239
240
3oo
240
3oo
340
3oi
241
3oi
241
302
4
5
69
7 i4 2
I 28
35
4i
48
55
62
6
359
36o
36 1
36i
362
6
70
7 i4 2
I 28
35
42
49
56
63
7
419
4.9
420
421
421
422
7
80 f
i 16 2
^ 32
40
48
56
64
72
8
4-'8
479
480
48 1
482
482
8
90 (
) 18 2
- 36
45
54
63
2'
81
9
538
539
540
54 1
542
543
9
00 11
20 3
4o
5<
6o|
70 eoi9o|
il6
rABLE
II
— To find the time T
, the sum of the
radii
r + r",
and
the chord
e being given.
SumoftheKadii r+i-".
Chord
C.
4,31
Days |dir.
4,32
4,33
4,34
4,35
4,36
4,37
4,38
Days|ilif.
4^39
4,40
Days Idif.
Days Idif.
Days Idif.
Days Idif.
Days|dif.
Days Idif.
Days |dif.
Days |dif.
0,00
0,000
0,00c
0,000
0,000
0,000
0,00c
o,ooo|
0,000
0,00c
0,000
0,0000
0,01
o,6o3
1
0,60^
I
0,60 5
I
0,606
0
0,606
I
0,607
I
o,6of
0
0,608 1
o,6oc
I
0,610
0
0,000 1
0.02
1 ,207
1
1,208
3
1,310
I
1,311
I
1,212
3
1,31^
I
I,2l5| 2
1,217 1
I,3lt
I
l,2lg
2
0,0004
o,o3
1,810
2
1,812
3
i,8i4
3
1,817
2
1.81&
2
1,821
3
1,82-
2
1,82
) 2
1,83-
2
1,829
2,439
2
0,000c
0,0016
o,o4
2,4i4
2
2,416
3
3,4ig
3
2,422
3
3,435
3
2,428
3
2,43o 3
2,433 3
2,43e
3
3
o,o5
3,017
4
3,021
3
3,024
4
3,028
3
3,o3i
4
3,o35
3
3,o3f
4
3,04
3
3,045
3
3,o48
4
0,0025
0,06
3,621
4
3,625
4
3,629
4
3,633
4
3,637
4
3,64i
5
3,64e
4
3,65o 4
3,654
4
3,658
4
o,oo36
0,07
4,224
5
4,229
5
4,334
5
4,339
5
4,M4
4
4,248
5
4,25:
5
4,258 5
4,263
5
4,268
5
0,0049
0,0064
0,08
4,827
6
4,833
6
4,83q
5
4.844
6
4.85o
5
4,855
6
4,861
5
4,866 6
4,872
6
4,878
5
o,og
5,43 1
6
5,437
6
5,443
7
5,45o
6
5,456
6
5,463
6
5,46£
7
5,47;
6
5,481
6
5,487
6
0,008 1
0,10
6,o34
7
6,o4i
7
6,o48
7
6,o55
7
6,062
7
6,069
7
6,07e
7
6,08;
7
6,ogo
7
6,097
7
0,0100
0,11
6,638
7
6,645
8
6,653
8
6,661
7
6,668
8
6,676
8
6,684
7
6,691
8
6,6gg
7
6,706
8
0,01 2 1
0,12
7.241
8
7,24q
9
7,358
8
7,366
8
7,274
9
7.283
8
7.291
8
7,29c
9
7,3o8
8
7,3i6
8
0,01 44
o,i3
7,844
9
7,853
9
7,862
10
7.873
9
7.881
9
7.890
9
7.89c
9
7,90s
9
7,gi7
9
7,g26
g
0,01 6g
0,14
8,448
9
8,457
10
8,467
10
8,477
10
8,487
g
8,496
10
8,5oè
10
8,5 r(
10
8,526
9
8,535
!0
0,0196
o,i5
9,o5i
10
9,061
II
9.072
10
g,o82
II
9,093
10
g,io3
II
9,114
10
Q,I2^
II
g,i35
10
9,145
. 10
0,0225
0,16
9.654
13
9,666
11
9.677
II
9,688
II
9.699
II
9,710
II
9.721
II
9.73.
13
9,744
II
9.755
II
0,02 56
0,17
10, 258
12
10,270
"
10,281
12
10,293
12
!o,3o5
12
io,3i7
12
10,33g
12
10,341
II
10,352
12
io,364
13
0,028g
0,18
10,861
i3
10,874
13
10,886
i3
10,89g
12
io,gii
i3
io,g34
13
10,936
i3
io,94c
12
io,g6i
i3
10,974
13
o,o324
0,19
1 1 ,464
i4
11,478
1 3
11,491
i3
ii,5o4
i3
ii,5i7
i4
ii,53i
i3
11,544
i3
11,55:
i3
11,570
i3
11,583
i3
o,o36i
0,20
12,068
i4
12,083
i3
13,095
i4
1 3 , 1 og
i4
12,123
14
I3,i37
i4
I3,l5l
i4
I3,i65
i4
12,17g
i4
13,193
i4
o,o4oo
0,21
12,671
i4
13,685
i5
13,700
i5
I2,7i5
i4
12,73g
i5
13,744
i5
13,759
i4
12,773
i5
12,788
i4
13,802
i5
o,o44i
0,22
•3,274
i5
13,289
16
i3,3o5
i5
1 3,320
i5
1 3,335
16
i3,35i
i5
1 3,366
i5
i3,38i
16
13,397
i5
i3,4i2
i5
o,o484
0,23
13,877
16
13,893
16
13,909
16
1 3,935
17
i3,g43
16
i3,g58
16
1 3,974
16
i3.gqr
16
14,006
i5
I4.03I
r6
0,0529
0,24
i4,48o
17
14,497
17
i4,5i4
17
i4,53i
17
14,548
16
i4,564
17
i4,58i
17
14,598
16
i4,6i4
17
i4,63i
17
0,0576
0,25
1 5,084
17
i5,ioi
18
i5,iig
17
i5,i36
17
i5,i53
18
15,171
17
i5,i88
18
i5,3o6
'7
I 5,223
17
1 5,240
18
0,0625
0,26
15,687
18
i5,7o5
18
15,733
18
1 5,74 1
18
■5,759
'9
15,778
18
15,796
18
i5,8i4
18
1 5,832
18
i5,85o
18
0,0676
0,27
16,290
19
16,309
19
16,338
19
16,347
18
i6,365
'9
1 6,384
19
1 6,4^3
19
16,422
'9
16,44 1
18
16,459
19
0,0729
0,28
16,893
20
16,913
19
i6,q32
30
i6,q53
■9
16,971
20
16,991
19
17,010
20
i7,o3o
ly
17,049
20
17,069
19
0,0784
0,29
17,496
20
I7,5i6
21
17,537
30
17.557
20
17,577
20
17,597
31
17,618
20
17,638
20
17,658
20
17,678
20
o,o84i
o,3o
18,09g
21
l8,130
21
i8,i4i
31
18,163
21
i8,i83
21
18,204
21
l8,325
21
18,246
31
18,267
20
18,287
31
0,0900
0,3 1
18,702
22
18,734
22
18,746
31
18,767
22
18,789
22
18,811
21
i8,833
22
18,8 54
21
18,875
22
18,897
21
0,0961
0,32
i9,3o5
23
19,338
22
19,350
32
19.372
23
19,395
33
19,417
22
19,439
33
19,463
22
19,484
22
ig,5o6
22
0,1024
0,33
19,908
23
ig,93i
23
19,954
24
19,978
23
20,001
23
20,024
23
20,047
22
2o,o6g
23
30,og3
23
20,Il5
23
0,1089
0,34
20,5 11
24
20,535
24
2o,55g
24
3o,583
23
20,606
24
2o,63o
24
20,654
23
20,677
24
30,701
24
20,735
23
0,1 1 56
0,35
2I,Il4
25
3i,i3g
24
2i,i63
25
21,188
24
21,312
34
31,236
25
21,261
24
21,285
34
3 1 ,309
25
3 1,334
24
0,1335
o,36
21,717
25
31,742
26
21,768
25
21.793
25
21,818
35
21,843
25
21,868
25
2i,8g3
25
3 1 ,9 1 8
25
31,943
25
0,1 296
0,37
22,320
26
3 2,346
36
23,373
36
22,398
25
22,433
26
2 3,44q
26
33,475
26
22,5oi
25
22,526
26
22,552
2t3
0,1 369
o,38
22,923
27
32,950
26
22,976
2T
33,oo3
26
33,029
37
3 3,o56
36
23,083
27
23, log
26
23,i35
26
23,161
27
0,1 444
0,39
23,526
27
23,553
27
23,58o
28
33,608
27
23,635
27
33,663
27
33,689
27
23,716
27
23,743
27
23,770
27
0,1 53 I
o,4o
24.128
=9
24,157
27
24,184
28
34,212
28
24,240
38
24,368
38
34,396
28
24,324
38
34,352
27
24,379
28
0,1600
o,4i
24,73i
29
24,760
29
24,789
28
34.817
29
24,846
?9
24.875
28
34,go3
29
24,932
28
24,960
29
24.989
2«
0,1681
0,42
25,334
29
25,363
3o
25,393
29
35,422
29
35,45i
3o
35,481
29
35,5io
29
25,539
29
2 5,568
3o
25,5g8
29
0,1764
0,43
25,937
3o
25,967
3û
25,997
3o
26,027
3o
36,057
3o
26,087
3o
36,117
3o
26,147
3o
26,177
3o
36,207
29
0,1849
0,44
26,539
3i
26,570
3i
26,601
3i
26,633
3o
36,663
3i
26,693
3i
26,724
3o
26,754
3i
26,785
3o
26,815
3\
0,1936
0,45
27,142
3i
37,173
32
27,205
3i
27,236
32
37,368
3i
27,299
32
37,331
3i
27,362
3i
27,393
3i
27,424
32
0,2025
o,5o
3o,i55
35
30,190
35
3o,225
35
30,260
34
30,294
35
3o,339
35
3o,364
35
3o,3q9
35
30,434
34
3o,468
35
o,25oo
0,55
33,166
39
33,3o5
38
33,243
39
33,282
38
33,3io
38
33,358
39
33,3g7
38
33,435
38
33,473
38
33,5ii
39
o,3o35
0,60
36,176
43
36,319
43
36,261
42
36,3o3
4i
36,344
42
36,386
42
36,438
42
36,470
42
36,5 12
4i
36,553
42
o,36oo
o,65
39,186
45
39,231
46
39.277
45
39,322
46
3g,368
45
39,41 3
45
39,458
46
39,504
45
39,54g
45
39,594
45
0,4225
0,70
42,193
5o
42,343
49
42,292
49
42,341
49
42,390
49
42,439
48
43,487
49
42,536
49
42,585
49
42,634
48
o,4goo
0,75
45,200
52
45,353
53
45,3o5
53
45,358
52
45,4io
53
45,463
53
45,5i5
53
45,567
52
45,619
53
45,672
53
o,5625
0,80
48,2o5
56
48,261
56
48,317
56
48,373
56
48,42Q
56
48,485
56
48,541
56
48,597
56
48.653
55
48,708
56
o,64oo
o,85
5 1,208
60
51,268
5g
5i,337
60
51,387
59
5 1, 446
60
5i,5o6
59
5 1, 565
60
51,625
59
5 1, 684
59
5 1, 743
59
0,7335
0,90
54,209
63
54,273
64
54,336
63
54,3gg
63
54,462
63
54,525
63
54,588
63
54,65 1
63
54.714
60.
54,776
63
0,8100
0,95
57,209
67
57,276
66
57.342
67
57,4og
67
57,476
66
57,543
67
57,609
66
57,675
67
57,742
66
57,808
66 o,go25
1,00
60,206 71
60,277
70
60,347
70
60,4 1 7
70
60,
9;
487I
71
6o,558
70
60,638 70
6o,6g8
69
60,767
70
60,837
70 1,0000
9,2881
9,33121
9,3745 1
9,41781
4613 1
9,5048 1
9,5485
9,5922 1
9,6361 1
9,6800 1 c2 1
k-tr^T")" or r^ -|- r"= nearly. |
602
6o3
604
6o5
606
607
608
609
610
I
60
60
60
61
61
61
61
61
61
I
2
120
121
lai
121
121
121
132
122
122
3
3
181
181
181
182
182
182
183
i83
i83
3
4
241
241
242
342
342
243
243
244
244
4
5
3oi
3o2
3o2
3o3
3o3
3o4
3o4
3o5
3o5
5
6
36 1
362
362
363
364
364
365
365
366
6
7
421
422
433
434
434
435
436
426
427
7
8
482
482
483
484
485
486
486
487
488
8
9
5
42
543
544
5
45
545
546 1
547 1
548 1
549
1
9
TV BLE II. — To fiml the time T\ the sum of the r;i(lii ?-(-)• ", anil the chord <• hehig t^ivcn.
tfum ol t ho Rati II r -f- r ". |
Chord
C.
4,41
4,42
4,43
4,44
4,45
4,46
Days |dif.
Days |dir.
Days Idif.
Days |dil'.
Days |dil".
Days |dif.
0,00
o,ui
o,o:i
o,u3
o,u4
0,000
0,610
1,23 1
1,83 1
2,442
1
1
2
2
0,000
0,611
1,222
1,833
2 AAA
1
2
3
3
0,000
0,612
1,234
1,835
2,447
0
1
3
3
0,000
0,612
1,235
1,837
2,45o
I
I
3
3
0,000
0,61 3
1,226
1,83()
2,453
I
3
3
3
0,000
0,6l4
1,328
1,843
2,455
1
I
2
3
0,0000
0,0001
0,0004
0,0009
0,0016
o,o5
0,06
0,07
o,ufci
0,uy
3,o52
3,(562
4,273
4.883
5,493
3
4
5
6
3,o55
3,666
4,378
4,889
5,5oo
4
5
4
5
6
3,o5q
3,671
4,282
4,894
5,5o6
3
4
5
6
6
3,062
3,675
4.387
4,900
5,5i2
4
4
5
5
6
3,066
3,679
4.392
4,905
5,5i8
3
4
5
6
6
3,069
3,683
4,397
4,911
5,524
4
4
5
7
0,0025
o,oo36
0,0049
0,0064
0,008 1
0,10
0,1 I
0,1 :'
o,i3
0,14
6,104
6,714
7,334
8
9
9
10
6,111
6.722
7,333
7,944
8,555
8
9
9
6,118
6,729
7.341
7,953
8,564
7
8
8
9
lO
6,125
6.737
7,349
7.963
8,574
6
7
9
9
10
6,i3i
6,744
7,358
7.97'
8,584
7
8
8
9
9
6,1 38
6,753
7.366
7.980
8,593
7
8
8
9
10
0,0100
0,0121
0,01 44
0,0169
0,0196
o,i5
0,16
0,17
0,18
o,Iy
9.- 50
9,-66
10,3-6
10,986
11,596
1 1
11
13
i3
14
9,i6()
9>777
io,388
■0,999
1 i,6io
10
11
II
12
i3
9,176
9,788
10,399
11,01 1
11,623
1 1
1 1
13
i3
i3
9,187
9.799
10,41 1
1 1,034
1 1 ,636
10
1 1
12
12
i3
9. '97
9,810
10,423
1 1 ,o36
1 1 ,649
10
11
13
13
l3
9,207
9.821
10,435
1 1 ,048
1 1 ,662
.1
11
11
i3
i3
0,0225
o,o256
0,0289
o,o324
o,o36i
0,30
0,21
0,11
0,23
0,24
12,207
12,817
.3,427
i4,o37
1 4,648
i4
i5
i5
16
16
12,221
12,832
1 3,443
i4.o53
14,664
i3
14
16
16
17
12,234
12,846
1 3,458
14,069
14,681
i4
i5
i5
16
16
13,348
12,861
1 3,473
i4,o85
14,697
14
14
1 5
16
17
12,262
12,875
1 3,488
i4,ioi
14,714
i4
i4
i5
16
16
12.276
12,889
i3,5o3
'4,1 '7
14.730
i4
i5
i5
16
17
o,o4oo
o,o44 1
0,0484
0,0539
0,0576
0,35
0,36
0,27
0,38
0,29
1 5,258
1 5,868
16,4-8
17,088
17.698
17
iS
'y
'9
20
1 5,275
1 5,886
16,497
17,107
17,718
17
18
18
20
20
15,393
15,904
i6,5i5
17,127
17,738
18
18
19
19
20
i5,3io
15,922
16,534
17.146
17,758
17
18
19
'9
20
1 5,327
1 5,940
16,553
I7,i65
17,778
17
18
18
20
20
1 5,344
1 5,9 58
16,571
I7,i85
17,798
17
'7
'9
'9
20
0,0635
0,0676
0,0729
0,0784
0,084 1
o,3o
0,3 1
0,32
0,33
0,34
i8,3o8
18,918
19,528
20,i38
20.748
21
22
22
23
24
18,329
18,940
19,550
20,161
20,772
21
21
22
23
23
i8,35o
18,961
19,572
30,184
20^795
30
21
23
23
24
18,370
18,983
19.595
20,307
20,819
21
22
22
22
23
18,391
19,004
19,617
20,339
30,843
21
21
23
23
24
i8,4i2
19,035
19,639
30,352
30,866
20
22
22
23
3 3
0,0900
0,0961
0,1024
0,1089
0,11 56
0,35
o,36
0,37
o,38
0,39
21,358
21,968
22,578
33,188
= 3,797
24
35
35
26
2-
21,382
21,993
2 2,6o3
23 2l4
23.824
24
35
26
26
27
2i,4o6
22,018
22,639
33,240
23,85i
25
25
26
26
27
2!,43l
22,043
2 2,655
23,266
23,878
24
24
25
27
27
21,455
22,067
32,680
23.293
33,905
24
25
26
36
27
21,479
22,092
22.706
2,3,319
33,932
24
25
25
26
27
0,1225
0,1296
0,1 36g
0,1444
0,l52I
0,40
0,4 1
0,42
0,43
0,44
24,407
35,017
25,627
26,236
26,846
28
28
3o
3o
24,435
25,045
25,656
26.266
26,876
28
29
3o
3i
24,463
25,074
25,685
26,296
26,907
27
28
3o
3o
24,490
25.I02
25,714
26,336
26,937
28
28
29
29
3i
34.518
35,i3o
25,743
36.355
26,968
27
29
^9
3o
3o
24,545
25,159
25,7-;2
26,385
26,998
28
28
3o
3o
0,1600
0,1681
0,1764
0,1849
0,1936
0,4'i
o,5o
0,55
0,60
o,65
0,70
37,456
3o,5o3
33,55o
36,595
39,639
43,682
3i
35
38
43
45
49
27.487
3o,538
33,588
36,637
39,684
42,731
3i
34
38
4i
45
48
27,518
30,572
33,626
36,678
39,729
42,779
3i
35
38
42
45
49
27,549
30,607
33,664
36,720
39.774
42,828
3i
34
38
4i
45
48
27,580
3o,64i
33,702
36,761
39,819
42,876
3i
35
38
42
45
49
27,611
30,676
33.740
36,8o3
39,864
42,925
3i
34
38
4i
45
48
0,2025
o,25oo
o,3o2 5
o,36oo
0,4225
0,4900
0,75
0,80
o,85
0,90
0,95
1,00
45,734
48,764
5 1.802
54,839
57,874
60,907
53
55
59
63
66
70
45,776
48,819
5i,86i
54,903
57.940
60,977
53
56
59
63
69
45,828
48,875
51,920
54,964
58,oo6
61,046
53
55
'^
66
70
45,880
48.930
5,,979
55,037
58,072
61,116
53
56
63
66
69
45,932
48,986
52,o38
55,089
58,1 38
6i,i85
5i
55
63
66
69
45,983
49,041
52,097
55,i5i
58,2o4
61,254
52
55
59
62
65
69
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
9,7241
9,76821
9,8125
9,8568
9,9013
9,9458
C^
1 . (r + r' )' or ?•= -f r " ' nearly. |
609
610
61
61
122
122
i83
i83
244
244
3o5
3o5
365
366
426
427
487
488
548
549
61!
61
122
i83
244
3o6
367
55o
612
61
122
J 84
245
3o6
367
438
490
55 1
6i3
61
133
184
245
3o7
363
429
490
552
6i4
61
123
1 84
246
307
368
43o
491
553
Prup. purls tur the ëuid ut' the- KuUii.
i|2|3|4|5|6|7|8|9
1
1
1
2
3 1
2
2
3
3
3
4
4
4
5
4
5
5
5
6
6
6
6
7
6
7
8
7
8
9
8
9
10
8
10
1 1
9
10
12
10
11
i3
11
12
'4
11
i3
'4
12
14
i5
i3
i4
16
i3
.5
'7
i4
iG
18
i5
'7
'9
lb
18
20
16
18
21
17
19
22
18
20
33
18
21
33
iq
22
24
20
23
25
30
23
26
31
24
27
22
25
28
33
36
29
33
26
3o
24
27
3i
25
28
33
2 5
29
32
26
3o
33
27
3o
34
27
3i
35
28
32
36
29
33
37
29
34
38
3o
M
39
3i
35
40
32
36
4i
32
37
4i
33
38
42
34
38
43
M
39
44
35
40
45
36
4i
46
36
42
4-
37
42
48
38
Ai
49
3q
4A
5o
3q
45
5o
40
46
5i
4i
46
32
4i
47
53
42
48
54
43
49
55
43
5o
56
M
5o
57
45
5i
58
46
52
59
46
53
59
47
54
60
48
54
hi
48
55
63
49
56
63
56
64
72
63
72
81
70
80
90
FABLt
II
— To find the Unie T
, tl
le sum 0
f th
3 radii
r + r".
and the chord <
being
çiven.
Sura of llie lladii r -\- r". |
Chord
C.
4,47
Days |ilil".
4,48
4,4
Days
0,000
9
ÏÏTfT
4,50
Days |dir.
4,51
Days |dir.
4,52
4,53
Days {dif.
4,54
4,55
Days |dir.
4,56
Days |dil'.
Days Idif.
Days |dir.
Days |dil'.
0,00
0,000
0,000
0,000
0,000
0,00(_)
0,000
0,000
0,000
0,000
0,0000
0,01
0,6 1 5
0
0,6 1 5
1
0,616
1
0,617
0
0,617
1
0,618
I
0,619
0
0,619
1
0,620
I
0,621
0
0,0001
0,02
1,229
I
i,23o
2
1,232
1
1,233
2
1,335
1
1,236
I
1,237
2
1,239
I
1 ,240
I
l,24l
3
o,ooo4
o,o3
1,844
2
1 ,846
2
1,848
2
1 ,85o
2
1,852
2
1,854
3
1,856
2
1,858
2
1,860
2
1,862
2
0,0009
0,04
2,458
3
2,461
3
2,464
3
2,466
3
2,469
3
2,472
3
2,475
2
2,477
3
2,480
3
2,483
2
0,0016
o,o5
3,073
3
3,076
4
3,080
3
3,o83
3
3,086
4
3,090
3
3,093
4
3,097
3
3,100
3
3,io3
4
0,0025
0,06
3,687
4
3,691
4
3,695
4
3,699
5
3,704
4
3,708
4
3,712
4
3,716
4
3,720
4
3,724
4
o,oo36
0,07
4,3o2
4
4,3o6
5
4,3ii
5
4,3 16
5
4,321
5
4,326
4
4,33o
5
4,335
5
4,340
5
4,345
5
(_i,oo49
0,08
4,916
6
4,922
5
4,937
6
4,933
5
4,938
6
4,944
5
4,949
6
4,955
5
4,960
5
4,965
6
0,0064
0,09
5,53 1
6
5,537
6
5,543
6
5,549
6
5,555
7
5,563
6
5,568
6
5,574
6
5,58o
6
5,586
6
0,0081
0,10
6,i45
7
6,i52
7
6,1 59
7
6,166
7
6,173
6
6,179
7
6,186
7
6,193
7
6,200
7
6,207
7
0,0100
0,11
6,760
7
6,767
8
6,775
7
6,782
8
6,790
7
6,797
8
6,8o5
7
6,812
8
6,830
7
6,837
8
0,0121
0,12
7,374
8
7,382
9
7,391
8
7,399
8
7,407
8
7,4 1 5
8
7,433
9
7,432
8
7,44o
8
7,448
8
o,ui44
0,1 3
7,9*^9
9
7,998
8
8,006
9
8,c,i5
9
8.024
9
8,o33
9
8,043
9
8,o5i
9
8,060
9
8,069
8
0,0169
o,i4
8,6o3
10
8,61 3
9
8,622
10
8,632
9
8,64 1
10
8,65i
10
8,661
9
8,670
10
8,680
9
8,689
10
0,0196
0,1 5
9,218
10
9,228
10
9,238
10
9,248
11
9,259
10
9,269
10
9,279
10
9,289
II
9,3oo
10
9,3 10
10
0,0225
0,16
9,832
1 1
9.843
1 1
9,854
II
9,865
II
9,876
11
9,887
1 1
9,898
11
9-909
11
9-920
10
9,930
II
0,0256
0,17
10,446
12
io,458
13
10.470
1 1
10,481
12
10,493
12
io,5o5
1 1
10,5 16
12
10,528
1 1
10,539
12
io,55i
12
0,0289
0,18
ii,û6i
12
1 1 ,073
12
ii,o85
i3
1 1 ,098
12
11,110
12
11,133
i3
ii,i35
12
ii,i47
12
1 1 , 1 59
i3
11,172
12
o,o324
0,19
11,675
i3
11,688
i3
11,701
i3
11,714
i3
I ',727
i3
11,740
i3
11,753
i3
1 1 ,766
i3
1 1 ,779
i3
11,792
i3
o,o36i
0,20
12,290
i3
i2,3o3
i4
I 3,3 1 7
i4
i2,33i
i3
12,344
14
12,358
i4
12,372
i3
12,385
i4
12,399
i4
i2,4i3
i3
o,o4oo
0,21
12,904
i4
12,918
i5
I2,q33
i4
I2,q47
i5
12,962
14
12,976
i4
1 2 ,990
i5
!3.oo5
14
13,019
14
i3,o33
i5
o,o44i
0,22
i3,5i8
i5
i3,533
i5
1 3,548
16
1 3,564
i5
1 3,579
i5
i3,5g4
i5
i3,6cJ9
i5
13,624
i5
1 3,639
i5
1 3,654
i5
o,o484
0,23
i4,i33
i5
i4,i48
16
i4,i64
ifi
1 4, 1 80
16
14,196
i5
14.21 1
16
14,227
16
14,243
16
14,259
i5
14.274
16
o,o52g
0,24
14,747
16
14,763
17
14,780
16
14,796
17
l4:8l3
16
14,829
17
1 4,846
16
14,862
16
14,878
17
14,895
16
0,0576
0,25
i5,36i
17
15,378
18
15,396
17
i5,4i3
17
i5,43o
17
1 5,447
17
1 5,464
17
1 5,481
17
'5,498
17
i5,5i5
17
0,0625
0,26
15,975
18
15,993
18
16,01 1
18
i6,o3q
18
16,047
i&
i6,o65
17
16,082
18
1 6, 1 00
18
16,118
18
i6,i36
17
0,0676
0,27
16,590
18
i6,(x)8
'9
16,627
18
16,645
19
16.664
16
16,682
19
16,701
18
16,719
'9
16,738
18
1 6,756
18
0,0729
0,28
17,204
19
17,223
'9
17,343
20
17,262
'9
17.281
19
i7,3oo
19
17,519
19
17,338
19
17,357
19
17,376
19
0,0784
0,29
17,818
20
17,838
20
17,858
20
17,878
20
17,898
20
i7,gi8
19
17,937
20
'7,9^7
20
17,977
20
17,997
20
0,084 1
o,3o
18,432
21
18,453
21
18,474
3C»
18,494
21
i8,5i5
20
18,535
21
i8,556
20
18,576
21
18,597
20
18,617
21
0,0900
0,3 1
19,047
21
19,068
21
19.089
21
19,1 10
22
19,132
21
i9;i53
21
19,174
21
19,195
21
19.216
22
19,338
31
0,0961
0,32
19,661
22
19,683
23
19,705
22
'9,727
22
'9,74g
21
19,770
22
'9,793
22
19,814
22
19,836
23
19,858
22
0,1034
0,33
20,275
23
20,298
22
20,330
23
20,343
22
2o,365
23
20,388
23
20,4 1 1
22
20,433
23
2o,456
23
20,478
23
0,1089
0,34
20,889
23
2 1 ,9 1 2
24
20,936
23
20,959
23
20,982
24
2 1 ,006
25
21,029
23
21,o52
23
21,075
33
2 1 ,098
23
0,1 156
0,35
2i,5o3
24
21,527
34
2i,55i
34
21,575
24
31,599
24
31,623
24
21,647
24
2 1 ,67 1
24
2 1 ,695
24
21,719
23
0,1235
o,36
22,117
25
22,142
24
22,166
25
22,191
25
22,216
25
22,241
24
22,265
25
22,290
24
32,3l4
25
22,339
34
0,1296
0,37
22,73l
25
22,756
26
22,782
25
32,807
26
22,833
35
22,858
25
22,883
26
22.909
25
22,9 34
25
22,959
35
0,1369
o,38
23,345
26
23,371
26
23,397
26
23,433
26
23,449
26
23,475
26
23,5oi
36
23,527
36
23,553
26
23,579
26
0,1444
0,39
23,959
27
23,986
27
24,oi3
26
34,039
27
24,066
27
34,093
26
24,119
27
24,i46
27
24,173
26
24,199
27
0,l52I
o,4o
24,573
27
24,600
28
24,628
27
34,655
28
24,683
27
34,710
27
24,737
28
24,765
27
24,792
37
34,819
28
0,1600
o,4i
25,187
28
25,2l5
28
25,343
28
25,271
28
25,209
28
25,327
29
35,356
28
25,384
28
25,412
27
25,439
28
0,1681
0,42
25,801
29
25.830
28
25,858
29
25,887
29
35,916
29
25.945
29
25,974
28
26,002
29
26,03 1
39
26,060
28
0,1764
0,43
26,4 1 5
29
26,444
3o
26,474
39
26,5o3
3(,
26,533
29
26,562
29
26,591
3o
36,631
29
36,65o
39
26,679
3o
0,1849
0,44
27,028
3Î
27,o5g
3o
27,089
3o
27,119
3o
27,149
3o
27,179
3o
27,209
3o
27,239
3o
27,269
3o
27,399
3o
o,ig36
0,45
27,642
3i
27,673
3i
37,704
3i
27,735
3i
27,766
3i
27,797
3o
37,837
3i
27,858
3i
37,88q
3o
37,919
3i
0,2025
o,5o
3o,7io
35
3o,745
34
3o,7-9
34
3o,8i3
35
3o,848
34
30,882
34
3o,gi6
34
3o,95o
35
30,985
34
31,019
34
o,25oo
0,55
33,778
38
33.816
37
33,853
38
33.891
38
33,929
38
33,967
37
34,004
38
34,042
37
34,079
38
34,117
37
o,3o25
0,60
36,844
4r
36;885
4i
36,026
42
36,968
4i
37,000
4i
37,o5o
41
37,091
4i
37,i32
4i
37,173
4'
37,214
4i
o,36oo
o,65
39,909
45
39,954
45
39,999
AA
4o,o43
45
40,088
AA
4o,i33
45
40,177
AA
40,22I
45
40,266
AA
4o,3io
AA
0,4225
0,70
42,973
48
43,021
48
48
43,117
48
43,i65
48
43,21 3
48
43,361
48
43,309
48
43,357
48
43,4o5
48
0,4900
0,75
46,o35
52
46,087
52
46,139
5i
46,190
52
46,242
5i
46,293
52
46,345
5i
46,396
5i
46,447
52
46,499
5i
0,5625
0,80
49,096
55
49.i5i
56
49,207
55
49,362
55
49,317
55
4g,372
54
49,426
55
49,481
55
49,536
55
49,591
54
o,64oo
o,85
52,i56
58
52,214
59
52,273
58
52,33i
59
52,390
58
52,448
59
53,507
58
52,565
58
52,623
58
53,681
58
0,7225
0,90
55,2 13
63
55,276
6?
55,338
63
55.400
62
55,462
61
55,523
62
55,585
62
55,647
62
55,7oq
61
55,770
62
0,8100
0,95
58,269
66
58,335
66
58,4oi
65
58,466
65
58,53 1
66
58,597
65
58,662
65
58,727
65
58,793
65
58,857
65
0,9025
1,00
61,323
70
6i,3g3
69
61,462
69
6i,53i
69
61,600
68
61,668
69
61,737
69
61,806
68
61,874
J9
61,943
68
1 ,0000
9,9905
10,0352
10,0801
10,1250
10,1701
10,2152
10,2605
10,3058
10,3513
10,3968]
^ . (r -(- r")^ or î^ -f r"^ nearly. |
6i3
6r4
6i5
616
617
618
619
620
621
I
61
61
62
62
62
62
62
62
62
I
2
123
123
123
123
123
124
124
124
124
2
3
1 84
184
i85
i85
i85
i85
186
186
186
3
4
245
246
246
246
247
247
248
248
248
4
5
307
3o7
3ù8
3o8
3og
309
3io
3io
3ii
5
6
368
368
36g
370
370
37.
371
372
373
6
7
429
43o
43i
43 1
432
433
433
434
435
7
8
4qo
491
4q3
493
494
494
495
496
497
8
9
5
5^ 1
553
554
5
54
,55
556 1
557 1
558
559
9
T.VRLE II. — To tinJ the time T; the sum of (lie lailii r-f-c'', and the chord <• beinf; given.
Sum ol
the
Kn.lii r+r".
Clionl
4,57
4,58
4,59
Days |Hir.
4,60
4,61
4,62
Days l.lil".
c.
Days |dil".
Days |.lcl'.
Days IJil".
Days lilil".
o,t)o
0,000
0,000
o,ooc
0,000
0,000
0,000
0,00t)0
(),()!
0.(32 1
1
0,632
I
0,623
0
0,623
1
0,624
I
0,635
0
0,0001
0,(I2
ii243
1
1,344
1
1,345
2
1,247
I
1,248
2
l,25o
I
o,o<»o4
o,<>3
t,8fi4
2
1 ,866
2
1 ,868
2
1,870
2
1,872
3
1 ,874
3
0,0(109
o,o4
2,485
3
2,488
3
2,491
3
2,494
'2
2,496
3
2,499
3
0,0016
o,o5
3,107
3
3,110
4
3,..4
3
3,117
3
3,120
4
3,134
3
0,0025
0,06
3,728
4
3,732
4
3,736
4
3,740
4
3,744
5
3,749
4
o,oo36
0,07
4,35o
4
4,354
5
4,359
5
4,364
4
4,368
5
4,373
5
0,0049
o,oi)
4,971
5
4,976
()
4,982
5
4,987
6
4,993
5
4,998
5
0,0064
0,09
5,592
6
5,598
fa
5,604
7
5,611
6
5,617
6
5,623
6
0,008 1
0,10
6.214
6
6,230
-
6,227
7
6,234
-
6,241
6
6,247
7
0,0100
0,1 1
6.835
7
6,843
8
6;85o
7
6,857
8
6,865
7
6,872
8
0,0121
0,I3
-,456
8
7,464
8
7,47'
9
7,48.
8
-,48q
8
7,497
8
0,01 44
0,1 3
8,077
9
8,086
9
8.095
9
8.104
9
8,ii3
9
8,122
8
0,0169
o,i4
8,699
9
8,708
10
8,718
9
8:727
10
8,737
9
8,746
10
o,oi<j6
0,1 5
9,330
10
9,33o
10
9,340
1 1
9,35i
10
9,361
10
9,371
10
0,0335
0,16
9'94i
II
9,953
1 1
9,963
11
9'974
11
9«85
11
9,996
10
o,o256
0>i7
io,563
1 1
10,574
12
io,586
11
10,597
13
10,609
11
10,620
12
0,0289
0,18
11,184
12
1 1,196
12
1 1,208
12
1 1.220
1 3
1 1,333
13
11,245
12
o,o324
0,19
1 1 ,8o5
i3
11,818
1 3
I! ,83!
i3
1 1 ,844
i3
1 1 ,857
i3
11,870
12
o,o36i
0,30
12,426
i4
1 3 ,440
i3
12,453
14
12,467
14
13,481
i3
12,494
i4
o,o4oo
0,21
1 3.048
14
13.062
14
13,076
14
13,090
1 4
i3,io4
i5
13.119
i4
o,o44 1
0,22
i3;66o
i5
1 3,684
i5
1 3,699
14
i3.-i3
i5
13,738
i5
i3,743
i5
o,o484
0,23
14,290
i5
i4,3o5
16
1 4,32 1
16
i4,33-
i5
14,353
16
1 4,368
i5
0,0529
0,24
14,911
16
14,927
I-
i4:944
ifa
14,960
16
14,9-6
16
14,992
17
0,0576
0,25
i5,532
I-
1 5,549
i~
1 5,566
17
1 5,583
17
1 5,600
17
i5,6i7
17
0,0625
0,26
i6,i53
18
16,171
18
16,189
17
16.306
18
16,224
17
16,241
18
0,0676
0,27
16,774
19
16,793
18
16,811
lb
16,820
19
1 6.848
18
16,866
18
0,0739
0,28
i-,395
20
17.41 5
19
17,434
19
17,453
18
17,471
19
17,490
'9
0,0784
0,29
18,017
'9
i8,o36
20
1 8,0 56
20
18,076
19
18,095
30
i8,ii5
20
0,084 1
o,3o
1 8,638
20
1 8,658
20
18,678
21
18,699
30
18,719
20
18,739
21
0,0900
0,3 1
19,259
21
19,280
21
19,301
21
19,333
31
19,343
31
19,364
21
0,0961
0,32
19,880
21
19.901
22
19,923
33
19.945
21
19.966
23
19,988
22
0,1034
0,33
2o,5oi
23
30,523
22
20,545
23
2o,568
32
2o,5go
22
20,612
23
0,1089
0,34
21,121
24
2i,i45
33
21,168
23
21,191
23
2(,2l4
23
21,237
23
0,1 156
0,35
2 1,742
34
21,766
24
2 1 ,790
24
3i,8i4
23
21,837
24
31,861
24
0,1225
o,36
22.363
35
22,388
34
22,412
25
33,437
24
22,461
34
3 3,485
25
0,1296
0,37
22.984
35
23,009
35
23.034
26
23:060
25
23,o85
25
33,1 10
25
0, 1 36g
o,38
2 3,6o5
36
33,63'i
36
23,657
25
23,682
26
23,708
36
33,734
26
0,1 444
0,39
24,226
36
24,252
37
24,27g
26
24,3o5
27
24,332
36
34,358
27
0,l521
0,40
24,847
37
34,874
27
34,901
27
24,938
27
24.955
27
24,983
27
0,1600
0,4 1
25,467
38
35,495
38
35,323
38
25,55i
28
25,579
27
35,606
38
0,1681
0,42
26,088
39
36.1 17
3fc
26,145
59
26,174
28
26,302
29
36,33l
38
0,1764
0,43
36,709
39
36.738
39
26,767
29
36,7g6
3o
26,836
29
26,855
29
0,1849
0,44
27,329
3o
27,3 5g
3o
27,389
3o
27,419
3o
27,449
3o
'7,479
3o
o,ig36
0,45
27.950
3i
37,981
3o
28,011
3i
28,042
3o
38,073
3i
28,103
3o
0,2025
o,5o
3i,o53
34
31,087
34
3l,I21
34
3i.i55
34
3i,i8o
33
3l,222
34
o,25oo
0,55
34, 1 54
38
34,193
3-
34,339
38
34,267
37
34,3o4
37
34,341
37
o,3o25
0,60
37,255
4i
37,396
4i
57.33-
4o
37,377
4 1
37.41&
4i
37,459
40
o,36oo
0,65
40,354
45
4o,3g9
U
40,443
Aâ
40.48-
Ai
40,53 1
AA
40,575
AA
0,4225
0,70
43,453
47
43,5oo
48
43,548
48
43,5g6
Al
43,643
48
43,691
Al
0,4900
0,75
46,55o
5i
46,601
5i
46,652
5i
46,703
5i
46,754
5i
46,8o5
5i
o,5625
0,80
49,645
55
49,700
54
49.754
55
49,809
54
49,863
55
49,918
54
o,64oo
o,85
52,739
58
53,-97
58
52,855
58
52,gi3
58
53,971
58
53,029
57
0,7325
0,90
55.832
61
55,893
62
55,955
61
56,oi6
61
56,077
61
56,1 38
61
0,8100
0,95
58,923
65
58,987
65
59,052
65
59,117
65
59,183
64
59,246
65
0,9035
1,00
62,01 1
fig
62,080
68
62,148
6f-
63,316
68
63,384
68
63,352
68
1 ,0000
10,4425
l(),4ç
82
10,53
"4Ï
10.5>=
00
10,62
61
10.67
22
"?^
r' + .
630
62
124
186
348
3io
372
434
496
558
621
63
134
186
348
3ii
373
435
497
55n
622
62
124
187
'49
3ii
373
435
498
56o
a17
623
62
125
187
249
3l2
374
436
498
56 1
624
625
62
63
125
125
187
188
25o
25o
3l2
3i3
374
375
437
438
4qq
5oo
503
563
1 I ' 3 I 4 I 5 I (i I 7 I 8 I 9 1
I
0
2
0
3
0
4
0
5
6
-T
8
9
10
1 1
12
i3
i4
i5
2
16
2
17
2
18
2
19
2
30
2
31
2
32
3
23
2
24
2
35
3
36
3
27
3
28
3
29
3
3o
3
i5 I
16 19 22
19
49
I
3
3
4
5
5
6
7
8
9
10
1 1
12
i3
14
i4
i5
16
17
23
33
34
3 5
26
38
29
3o
3i
32
32
33
34
35
36
37
38
39
40
4i
4i
42
43
AA
45
46
47
56
63
5o
5o
5i
53
53
54
55
56
57
58
59
59
60
61
62
63
3
60
64
7281
80I90
TABLE II. — To fi ml the time T; the sum of the radii )■ -f-r", and the chord c being given.
Sum (
r Uie Rai
il r
+ '■'
Chord
C.
4,63
Days |(]ir.
4,64
4,65 4,66
4,67
4,68
Daysldif.
4,69
4,70
Daysldif.
4,71
4,72
Days |(lir.
Days |dif.
Days |dir.
Days |dir.
Days |dif.
Days |dir.
Days Idif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,625
I
0,626
I
0,637
0
0,627
1
0,628
1
0,629
0
0,639
I
o,63o
I
o,63i
0
0,63 1
I
0,0001
0,02
T,25l
I
1,252
2
1,2 54
1
1,255
1
1.256
2
1,358
I
1,259
1
1 ,260
2
1 ,362
1
1,363
1
o,ooo4
o,n3
1,876
■1
1,878
2
i,S8o
2
1,882
2
1,884
2
1,886
2
1,888
2
i,8go
2
1,892
2
1 ,894
2
0,0009
0,04
2,502
2
2,5o4
3
2,5o7
3
2,5lO
2
2,5l2
3
2,5i5
3
a,5i8
3
2,521
2
2,523
3
3,536
3
0,0016
o,o5
3,127
3
3,i3o
4
3, 1 34
3
3,i37
4
3,i4i
3
3,144
3
3,147
4
3,i5i
3
3,1 54
3
3,i57
4
0,0025
0,06
3,753
4
3,757
4
3,761
4
3.765
4
3,769
4
3,773
4
3,777
4
3,781
4
3,785
4
3,789
4
o,oo36
0,07
4,378
5
4,383
4
4,387
5
4,392
5
4,397
5
4,402
4
4,406
5
4,4 11
5
4,416
4
4,430
5
0,0049
0,08
5,oo3
6
5,009
5
5,oi4
6
5,020
5
5,025
5
5,o3o
6
5,o36
5
5,04 1
5
5.046
6
5,o53
5
0,0064
0,09
5,629
6
5,635
6
5,64i
6
5,647
6
5,653
6
5,659
6
5,665
6
5,671
6
5,677
6
5,683
6
0,0081
0,10
6,254
7
6,261
7
6,268
6
6,274
7
6,281
7
6,288
7
6,295
6
6,3oi
7
6,3o8
7
6,3 1 5
6
0,0100
0,11
6,880
7
6.887
7
6,894
8
6,902
7
6,909
8
6,917
7
6,924
7
6.931
8
6,939
7
6,946
7
0,01 3 1
0,12
7,5o5
8
7,5i3
8
7,52,
8
7,529
8
7,537
8
7,545
8
7,553
8
7,56i
9
7,570
8
7.578
8
0,01 44
o,i3
8,i3o
9
8,139
9
8,i48
9
8,i57
8
8,i65
9
8,174
9
8,i83
9
8,192
8
8,200
9
8,209
9
0,0169
o,i4
8,756
9
8,765
10
8,775
9
8,784
9
8,793
10
8,8o3
9
8,812
10
8,822
9
8,83 1
9
8,840
10
0,0196
o,i5
9,38i
10
9>39!
10
9,401
10
9,4 11
10
9,431
II
9,432
10
9,442
10
9,453
10
9,462
10
9,472
10
0,0225
0,16
10,006
11
10,017
i 1
10,028
1 1
10,039
II
io,o5o
10
10,060
II
10,071
1 1
10,082
10
10,092
II
io,io3
II
0,0256
0,17
io,632
II
10,643
12
10,655
1 1
10,666
12
10,678
II
10,689
11
10,700
13
10,713
1 1
10,733
12
10,735
11
0,038g
0,18
11,257
12
11,269
12
11,281
12
11,293
i3
ii,3o6
13
ii,3i8
13
ii,33o
13
11,342
12
11,354
12
1 1 ,366
12
o,o324
0,19
11,882
i3
11,895
l3
1 1 ,go8
i3
11,921
i3
11,934
12
11,946
i3
11,959
i3
":972
i3
11,985
12
",997
i3
o,o36i
0,20
i2,5o8
i3
12,531
14
12,535
i3
13,548
i4
12,562
l3
12,575
i3
13,588
14
I 2 ,602
i3
i2,6i5
i4
12,629
i3
o,o4oo
0,21
1 3,1 33
i4
i3,i47
i4
1 3,161
i4
13,175
i5
13,190
i4
1 3,204
i4
i3,3i8
14
l3,233
i4
i3,246
14
13,260
i4
0,044 1
0,32
1 3,75s
i5
1 3,773
i5
13,788
i5
i3,8o3
i4
i3,8.7
i5
i3,833
i5
1 3,847
i5
1 3,863
i5
13,877
14
13,891
i5
o,o4S4
0,23
i4,383
16
14,399
i5
i4,4i4
16
i4,43o
i5
i4,445
16
i4,46i
i5
14,476
16
14,493
i5
i4,5o7
16
14,523
i5
0,0529
0,24
15,009
16
i5,025
16
i5,o4i
16
1 5,057
16
1 5,073
16
l5,oSg
17
i5,io6
16
1 5,1 22
16
i5,i38
16
i5,i54
16
0,0576
0,25
1 5,634
17
1 5,65 1
17
1 5,668
16
1 5,684
17
1 5,701
17
15.718
17
i5,735
17
1 5,752
16
15,768
17
1 5,785
17
0,0625
0,26
16,259
18
16,277
T7
16,294
18
1 6,3 1 3
17
16,329
18
16,347
17
1 6,364
18
16,382
17
16,399
'7
i6,4i6
18
0,0676
0,27
16,884
18
16,902
'9
16,921
18
16.939
18
16,957
18
16,975
18
16.993
18
17,01 1
19
i7,o3r
18
17,048
18
0,0739
0,28
17,509
19
.7,528
19
17,547
19
17,566
19
17,585
19
17,604
19
17,633
18
17,641
19
17,660
19
17,679
19
0,0784
0,2g
i8,i35
19
i8,i54
20
18,174
'9
18,193
20
i8,2i3
19
18,232
20
18,353
19
18,271
20
18,291
'9
i8,3io
19
o,o84i
o,3o
18,760
20
18,780
20
18,800
20
18,820
21
18,841
20
18,861
20
18,881
20
i8,goi
30
i8,g2i
30
18,941
20
0,0900
o,3i
19,385
21
19,406
21
19,427
20
19,447
21
19,468
21
19,48g
21
■9,5io
2!
ig,53i
31
19,552
20
19,572
21
0,0961
0,32
20,010
21
2O,03l
22
20,o53
22
30,075
21
20,096
22
30,! 18
21
20,139
22
20,161
21
20,182
21
20,203
33
0,1024
0,33
20,635
22
20,657
22
30,679
23
30,703
23
20,724
22
30,746
22
20,768
22
20,7gc
33
20,812
23
20,835
33
o,io8g
0,34
21,260
23
21,283
23
3 1 ,3o6
23
31,339
23
21,352
22
21,374
23
21,397
23
2 1 ,420
33
21,443
23
21,466
33
0,1 156
0,35
21,885
23
3 1 ,908
24
21,932
24
21,956
23
21,979
24
22,003
23
32,026
24
22,o5o
23
22,073
24
33,097
23
0,1335
0,36
22,5lO
24
33,534
24
22,558
25
22,583
24
2 2 ,607
24
2 2 ,63 I
24
22,655
24
22,67g
25
22,704
24
33,738
24
0,1396
0,37
23,i35
25
33, 160
25
23,i85
25
23,210
25
33,335
24
23,25g
25
23,284
25
23,309
25
23,334
25
33,35g
24
0,1369
o,38
23,760
25
23,785
26
23,8ii
26
23,837
25
33,863
26
23,888
25
23,913
26
23,939
25
23,964
26
23,ggo
25
0,1444
0,39
24,385
36
24,411
26
34,437
27
34,464
26
34,490
26
24,5i6
26
24,542
26
24,568
27
24,595
36
24,621
26
0,l531
o,4o
25,009
27
25,o36
27
3 5,o63
27
25,090
27
35,117
27
25,i44
27
25,171
27
25,ig8
27
35,335
27
25,353
26
0,1600
0,4 1
25,634
28
25,663
28
25,690
27
35,717
38
25,745
27
25,772
28
25,800
28
25,828
27
35,855
27
35,883
28
0,1681
0,42
26,25g
28
36,387
29
26,3 16
28
36,344
38
26,372
29
26,401
28
36,43g
28
26,457
38
26,485
28
36,5i3
38
0,1764
0,43
26,884
=9
26,913
29
26,942
29
26,971
29
27,000
29
37,02g
29
37,o58
29
27,087
38
27,ii5
29
37,144
29
0,1849
0,44
27,509
29
27,538
3o
27,568
3o
27,598
29
27,627
3o
27,657
29
27,686
3o
27,716
3o
27,746
29
27,775
29
0,1 g36
0,45
28,i33
3i
28,164
3o
28,194
3o
28,224
3i
28,255
3o
28,385
3o
28,3i5
3o
28,345
3i
28,376
3o
28,406
3o
0,2025
o,5o
3i,256
34
31,290
34
3i,324
34
3 1, 358
33
3 1 ,39 1
34
3i,435
34
31,459
33
31,492
34
3 1, 526
33
31,559
34
0,2 5oo
0,55
34,378
38
34,416
37
34,453
37
34,490
37
34,527
37
34,564
37
34,601
37
34,638
37
34,675
37
34,712
36
o,3o25
0,60
37,499
4i
37,540
4i
37,581
4o
37,621
4i
37,662
4o
37,703
40
37,743
4i
37,783
4o
37,823
4o
37,863
40
o,36oo
o,65
4o,6ig
Ai
4o,663
AA
40,707
AA
4o,75i
AA
40,795
AA
4o,83g
AA
4o,883
43
40,936
AA
40,970
AA
4i,oi4
43
0,4225
0,70
43,738
48
43,786
47
43,833
Ai
43,880
Ai
43,927
48
43,975
Ai
44,022
47
44,069
Ai
44,116
Al
44,i63
47
o,4goo
0,75
46,856
5o
46,906
5i
46,957
5i
47,no8
5i
47,059
5o
47,109
5i
47,160
5o
47,210
5o
47,260
5i
47,3ii
5o
0,5625
0,80
49.972
54
50,026
54
5o,o8o
54
5o,i34
54
5o,i88
54
5o,343
54
50,396
54
5o,35o
54
5o,4o4
54
50.458
53
o,64oo
0,85
53,086
58
53,144
58
53,202
57
53,359
57
53,3i6
58
53,374
57
53,43i
57
53,488
58
53,546
57
53,6o3
57
0,7225
0,90
56,199
61
56,36o
61
56,321
61
56,383
6r
56,443
61
56,5o4
61
56,565
60
56,625
61
56,686
60
56,746
61
0,8100
0,95
59,3ii
64
59,375
65
59,440
64
59.504
64
59,568
64
59,633
65
59,697
64
59,761
64
5g,825
63
5g,888
64
0,9025
1,00
62,420
68
63,488
68
62,556
68
6
2,624
67
62,691
68
63,759
68
63,837
67
62,894
67
62,961
68
63,02g
67
1 ,0000
10,7185
10,76481
10,8113
10,8578
10,9045
10,9512
10,9981
1 1 ,0450
11,0921
11,1392
(?
è . (r + r " )' or r= + J
"^ nearly. |
624
635
626
637
628
629
63o
63 1
632
I
62
63
63
63
63
63
63
63
63
I
2
125
135
125
135
126
126
126
126
126
3
3
187
188
788
188
188
i8g
189
189
190
3
4
25o
25o
2 5o
25l
25l
252
252
253
253
4
5
3l2
3i3
3i3
3i4
3i4
3i5
3i5
3i6
3i6
5
6
374
375
376
376
377
377
378
379
379
6
7
437
438
438
439
440
440
44 1
442
442
7
8
499
5oo
5oi
502
502
5o3
5o4
5o5
5o6
8
9
562
563
56
3
5(
54
5
65
566
567 1
568 1
569
9
TABLE II. — To find the time T\ tlie sum of the radii r-}-»' ", and tlie cliord c beinf; f^iven.
Sum of llio Kadii r~\-r".
r
..p.
imrl
Itir
llio
bum
ul ll
,,■ lit
Jii.
Cliuri
4,7;j
4,7-^
[ 4,75
4,76 1 4,77
1 4,78
OjOOtIO
1 1 3 1 3 1 4 1 5 1 (i 1 7 1 8 1 g
1
2
3
0
0
0
0
0
0
I
1
0
I
I
2
I
I
3
I
I
3
I
3
3
I
2
3
C.
Days 1
lif. Days |i
if. Days |d
1'. Days 1
lir Days |(lif. | Days |dif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
o,ui
o,632
I o,633
I o,634
0 0,634
1 o,635
0 0,635
I
0,000 1
4
1
2
2
2
3
3
4
0,02
1,264
3 1,366
I 1,367
1 1,368
3 1,370
I 1,271
I
o,ooo4
o,o3
1,896
3 1,898
2 I ,QOO
I ■■9"'
3 I ,go4
2 1 ,9o(
2
0,0009
5
2
2
3
3
4
4
5
0,04
2,529
3 2,53l
3 2,534
3 2,537
2 2,53g
3 2,542
3
0,0016
6
2
3
3
3
3
4
4
4
4
r
5
6
5
6
0,uf>
3,161
3 3,164
3 3,167
4 3,171
3 3,174
3 3,177
4
0,0035
8
3
2
3
4
5
()
6
7
t),u6
3,793
4 3,-97
4 3,801
4 3,8o5
4 3,8og
4 3,8i3
4
o,oo36
9
3
3
4
5
5
6
7
8
0,07
4,425
5 4,43o
4 4,434
5 4,439
5 4,444
4 4,448
5
0,0049
3
0,0b
5, 057
5 5.063
6 5,068
5 5.073
5 5,078
6 5,084
5
0,0064
10
3
4
5
6
7
8
9
o,oy
5,689
C 5,(^95
6 5,701
6 5,707
6 5,7.3
6 5,71g
6
0,008 1
1 1
13
3
3
3
4
4
5
6
6
7
7
8
8
9
10
10
1 1
0,10
6.321
7 6,338
7 6,335
6 6,341
7 6,348
7 6,355
6
0,0100
i3
i4
3
3
4
5
6
7
8
9
10
12
i3
U,I I
6,o53
8 6,961
7 6,968
7 6,975
8 6,g83
7 6,990
7
0,01 3 1
4
7
8
10
II
0,12
7,586
8 7.594
8 7,603
8 7,610
8 7,618
8 7,626
8
0,0144
i5
2
3
5
6
8
9
10
1 1
13
'4
û,i3
8,218
8 8,336
9 8,335
9 8,244
8 8,252
9 8,261
9
0,01 6g
16
2
3
5
6
8
1 1
i3
'4
0,14
8,85o
9 8,85g
9 8,868
10 8,878
g 8,887
9 8,896
10
0,01 96
17
3
3
5
7
g
10
12
i4
i5
o,i5
9,482
10 9,492
10 g,5o2
10 9,5i3
10 g,522
10 9,532
10
0,0225
18
'9
3
2
4
4
5
6
7
8
9
10
11
1 1
i3
i3
i4
i5
16
17
0,16
io,ii4
11 10,125
10 io,i35
II io,i46
II io,i57
10 10,167
II
o,o256
0,17
10.746
11 10,757
13 10,769
II 10,780
11 10,791
12 io,8o3
II
0,0289
30
2
4
6
8
10
13
i4
i6
18
O.ltl
11,378
12 11,390
13 II,403
12 ii,4i4
12 II ,426
12 11,438
12
o,o324
31
2
4
6
8
11
i3
i5
'7
'9
o,iy
12,010
l3 12,023
13 I3,035
i3 12,048
1 3 13,061
12 12,073
i3
o,o36i
33
2
4
7
9
11
i3
i5
18
20
33
2
5
7
9
13
i4
16
18
21
0,30
12,642
i3 13,655
i4 i2,66g
i3 13.683
i3 12,695
i4 12,709
i3
o,f>4oo
24
2
5
7
10
12
i4
17
19
22
0,3 1
i3,274
14 i3,388
i4 i3,3o2
4 i3,3i6
1 4 i3,33o
i4 1 3,344
i4
0,044 1
0,22
13,906
1 5 13.931
14 i3,g35
5 i3,g5ù
1 5 13,965
i4 13.979
i5
o,o484
25
36
3
5
8
10
l3
i5
18
30
23
0,23
i4,538
1 5 i4,553
16 14,569
5 i4,584
i5 14,599
16 f4,6i5
i5
o,o52g
3
5
8
10
i3
16
18
31
23
0,24
15,170
16 i5,i86
16 l5,203
6 i5,3i8
16 i5,234
16 i5,35o
16
0,0576
27
3
5
8
u
i4
16
'9
33
24
38
3
6
8
1 1
i4
17
20
3 2
25
0,25
1 5,802
17 i5,8ig
16 1 5,835
7 1 5,852
17 15,869
16 i5,885
17
0,0625
29
3
6
9
13
i5
17
20
33
26
0,26
16,434
17 i6,45i
18 16,469
7 16,486
17 i6,5o3
17 16,530
18
0,0676
3o
3
6
9
9
10
I 3
i5
18
3 1
34
25
36
27
28
59
3o
0,27
0,28
17,066
17.698
18 17,084
18 17,716
18 17,102
ig 17,735
8 17,120
9 '7,754
18 I7,i38
18 17,772
18 I7,i56
19 17,79'
18
'9
0,073g
0,0784
3i
33
3
3
6
6
12
i3
16
16
19
19
20
2 3
32
0,2y
i8,32g
3o i8,34g
19 1 8,368 :
0 i8,388
19 18,407
19 18,436
19
0,084 1
33
3
7
10
i3
17
33
36
o,3o
18,961
20 18,981
20 19,001 ;
0 19,021
20 19,041
30 19,061
30
o,ogoo
34
3
7
10
i4
17
20
34
27
3i
o,3i
19,593
31 ig,6i4
20 19,634 2
1 19,655
21 19,676
30 19,696
31
0,0961
35
4
7
II
14
18
21
35
28
32
0,32
20,225
31 30,346
33 20,268 3
1 30,289
21 20,3 10
33 30,332
31
0,1024
36
4
7
1 1
i4
18
23
35
29
32
0,33
20.857
33 20,879
33 20,901 3
2 20,933
22 20,945
2 2 20,967
23
0,1 o8g
37
4
7
II
i5
'9
22
36
3o
33
0,34
21,488
23 2I,5ll
33 21,534 3
3 31,556
33 31,57g
33 3 1, 603
23
0,1 1 56
38
4
8
II
i5
'9
23
27
3o
34
39
4
8
12
16
30
23
27
3i
35
0,35
22,120
24 33,l44
23 22,167 a
3 32,190
34 22,214
23 23,237
23
0,1335
0,36
23,752
24 32,776
24 22,800 2
4 32,824
24 22,848
24 32,872
34
0,1296
40
4
8
12
16
30
34
38
32
36
0,37
23,383
25 23,408
25 23,433 2
5 23,458
24 33,483
25 23,507
34
o,i3(k)
4i
4
8
12
i(j
21
35
29
33
37
o,38
24,01 5
36 24,041
3 5 34,066 3
5 24.091
36 34,117
25 34,142
25
0,1 444
43
4
8
i3
17
31
25
29
34
38
o,3g
24,647
26 34,673
26 24,699 2
6 34,735
36 34,75i
36 24,777
26
0,l52I
43
4
9
i3
17
33
26
3o
34
39
44
4
9
i3
18
33
36
3i
35
40
o,4o
25,278
27 35,3o5
J7 25,332 2
7 25,35g
26 25,385
27 25,4l2
26
0,1600
o,4i
25,910
i-j 35,937
28 35,g65 2
7 25,gg2
i-j 36,01g
!8 26,047
27
0,1681
45
5
9
i4
18
2 3
27
32
36
4i
0,42
26,541
îg 26,570
28 26,598 2
8 26,626
!8 26,654
!8 26,683
27
0,1764
46
5
9
i4
18
23
38
33
37
4i
0,43
27.173
!g 37,303
18 27,230 3
9 27,25g
jg 27,288
!8 37,3 1 6
=9
0,1849
47
5
9
i4
19
24
38
33
38
43
0,44
37,804
3o 27,834
!g 27,863 3
0 27,893
iQ 37,923
!g 37,g5i
3o
0,1936
48
5
10
14
'9
34
29
34
38
43
49
5
10
i5
20
35
29
34
39
44
0,45
28,436 ,
ÎO 28,466 .
ÎO 38,496 3
0 28,526 .
3o 28,556 .
3o 38,586
3o
0,2025
r
35
26
36
3o
3i
3i
33
33
35
36
36
37
38
4o
4i
42
45
46
47
48
49
o,5o
3i,5g3 .
Î3 31,626
33 3 1, 659 3
4 31,693 .
33 31,726 .
33 31,759
34
o,25oo
5o
5i
53
53
54
J
r
10
i5
i5
16
16
16
2C)
0,55
34,748
37 34,785
37 34,822 3
7 34,859
36 34,895 .
37 34,933
37
o,3o2 5
J
5
10
20
0,60
37,903 ;
U 37,944 <
io 37,984 4
0 38,024 t
jo 38,o64 ^
fo 38,io4
4o
o,36oo
r.
10
31
0,65
4i,o57 .
i4 41.101 I
<3 41, 144 4
4 4i,i88 i
i3 4i,23i ,
i3 4 1, 274
44
0,4225
f,
1 1
3 1
27
42
43
0,70
44.210 1
ij 44,257 I
Î6 44, 3o3 4
7 44,35o i
[7 44,397 i
i7 44,444
46
0,4900
J
1 1
22
27
0,75
47,36i
5o 47,4 1 1
5i 47,462 5
0 47,5i2 '
)o 47,562 t
)0 47,612
5o
0,5625
55
56
6
6
II
17
17
17
17
18
22
22
38
38
33
34
39
39
4o
4i
44
45
5o
5o
0,80
5o,5ii
54 5o,565
53 5o,6i8 5
4 50,673 '
)3 50,725 ;
)4 5o,77g
53
0,6400
57
58
6
Ï I
23
29
3^
M
46
46
5i
o,85
53,660
37 53,717
J7 53,774 5
7 53,83 1
)6 53,887 !
>7 53,g44
57
0,7235
6
13
33
35
53
0,90
56,8o7
5o 56,867 (
3i 56,938 6
0 56,988 (
io 57,048 (
X) 57,108
60
0,8100
59
6
1 2
34
35
4i
47
53
o.gS
59.953 (
Î4 60,016 (
54 60,080 6
3 60,143 (
)4 60,207 t
)3 60,270
64
o,go25
1,00
63,096 (
57 63,i63 f
J7 63,33o 6
7 63,397 (
>7 63,364 t
)7 63,43 1
67
1 ,0000
60
61
62
63
64
6
6
6
6
6
12
12
12
l3
l3
18
18
'9
19
19
24
34
35
3 5
36
3o
3i
3i
33
33
36
37
37
38
38
42
43
43
44
45
48
49
5o
5o
5i
54
55
56
57
58
11,186
5 11,233
8 ll,28i;
3 11,328
8 11,376
5 11,42421
\ . {r -\- t")'* or r^-\- r"^ nearly. |
63 1
632
633
634
635
636
— ■
65
7
l3
20
36
33
39
46
53
59
I
63
63
63
63
64
64
I
66
7
i3
20
26
33
4o
46
53
59
2
136
126
137
127
127
137
2
67
7
i3
20
'7
34
40
47
54
60
3
189
I go
190
190
191
191
3
68
7
14
20
27
34
4i
48
54
61
4
353
253
353
2 54
254
254
4
69
7
i4
21
38
35
4i
48
55
62
5
3i6
3i6
3.7
3t7
3i8
3i8
5
6
379
379
38o
38o
38i
382
6
70
7
i4
21
38
35
43
49
56
63
7
442
442
443
444
445
445
7
80
8
16
24
33
4o
48
56
M
72
8
5o5
5o6
5o6
5o7
5o8
5og
8
90
9
18
27
36
45
54
63
72
81
9
568
569
670
571
572
572
9
00
0
20
3o
4o
5o
60
7"
80
9°!
TABLE II. — To find the time T; the sum of the radii r + >■". and the chord e being given.
t>uin of llie Railii r -\- r". 1
Chord
C.
4,79
4,80
4,81
4,82 1
4,83
4,84
4,85
4,86
4,87 1
4,88
Days |.l(l'.
Days Idir.
Days 1.1 if.
Days 1
dif.
Days Idif.
Uaya |dir.
Tj^sfdrr.
Days Irtif.
Day
3 |dif.
Daya |dif.
0,00
0,000
0,000
0,0(Ki
0,000
0,000
0,000
0,000
0,00
0
0,0c
0
0,000
0,0000
0,01
o,636
I
o,637
0
",637
I
0,638
I
0,639
0
0,639
I
0,640
I
0,64
I 0
0,64
1 I
0.642
I
0,0001
0,0 a
1,279
2
1,274
I
1,275
I
1,276
3
1,278
I
1,279
1
1,280
2
I,2&
2 I
1,28
3 t
1,284
2
0,0004
o,o3
i,go8
2
1,910
2
1,912
2
1.914
2
1.916
2
i,gi8
2
1,920
2
1,92
2 2
1,93
4 2
1,926
2
o,ooog
0,04
2,545
3
2,547
3
2,55o
3
2,553
3
2;555
3
2,558
2
2,56o
3
2,56
3 3
2,56
6 2
2,568
3
0,0016
o,o5
3,181
3
3,184
3
3,187
4
3,191
3
3,194
3
3,197
4
3,201
3
3,2c
4 3
3,20
7 3
3,310
4
0,0025
0,06
3,817
4
3,821
4
3,825
4
3,82g
4
3,833
4
3,837
4
3,84i
4
3,84
5 4
3,84
9 4
3,853
3
o,oo36
0,07
4,453
5
4,458
4
4,462
5
4,467
5
4,472
4
4,476
5
4,481
4
4,48
5 5
4,4g
0 5
4,495
4
o,oo4g
0,08
5,089
5
5,094
6
5,100
5
5,io5
5
5,110
6
5,116
5
5.121
5
5.12
6 5
5,i3
1 6
5,i37
5
0,0064
0,09
5,725
6
5,73.
6
5,737
6
5,743
6
5,74g
6
5,755
6
5^761
6
5;7e
7 6
5,77
3 6
5,779
6
0,0081
0,10
6,36i
T
6,368
7
6,375
6
6,38 1
7
6,388
6
6,3g4
-1
6,4oi
7
6,4c
8 6
6,4 1
4 7
6,421
6
0,0100
o,i I
6,997
8
7, 00 5
7
7 .0 1 3
7
7,oig
8
7,027
7
7,"34
7
7,o4i
7
7,0-j
8 8
7,o5
6 7
7, 06 3
7
0,0121
0,12
7,634
7
7,64 1
8
7,649
8
7,657
8
7,665
8
7,673
8
7,681
8
7,6fc
9 8
7,6cj
7 8
7,705
8
0,0144
0,1 3
8,270
8
8,278
9
8,28-j
9
8.996
8
8,3o4
9
8,3 1 3
8
8,321
9
8,3:
0 8
8,33
8 9
8,347
9
0,0 1 6g
0,1 4
8,906
9
8,915
9
8,924
10
8,934
9
8,g43
9
8,953
9
8,y6i
10
8,9-
■ 9
8,98
" 9
8.98g
9
0,0196
0,1 5
9.'^42
10
9,559
10
9,563
10
9,572
10
g,582
9
9,59'
10
9,601
10
9.61
I 10
g.69
1 IO
g,63i
10
0,0225
0,16
10,178
10
10,188
II
10,199
11
10,210
10
10,330
11
IO,23l
10
10,241
1 1
10,3^
2 II
IO,2f
3 10
10,273
1 1
o,o256
0,17
10,814
II
10,825
1 1
io,836
13
io,848
II
10,8 5g
1 1
10,870
1 1
10,881
19
10,8c
3 II
io,gr
4 11
io,gi5
II
0,0289
o,i3
I i,45o
13
11,462
13
11,474
13
11,486
12
1 1 ,498
12
1 1, 5 10
II
11,521
19
11,5;
3 19
■ ■,5^
5 12
11,557
12
o,o324
0,19
12,086
i3
12,099
13
12,11 1
i3
12,124
12
I2,i36
i3
12,149
13
l2,lGl
i3
r2,i-
4 19
I2,lt
6 i3
12,199
12
o,o36i
o,ao
19,723
i3
12,735
i4
■2,749
i3
12,762
1 3
12,775
i3
12,788
i3
I 2 .80 1
■ 4
12,8
5 i3
12,83
8 i3
i2,84i
i3
o,o4oo
0,21
i3,358
i4
13,372
i4
i3,386
i4
1 3,400
14
i3,4i4
1 4
13,428
i3
■ 3,44i
i4
i3,4'
5 14
1 3,4c
9 i4
13483
i4
0,044 1
0,22
1 3,994
i5
14,009
i4
l4,023
i5
i4,o38
■ 4
i4,o52
15
14,067
i4
14,081
i5
1 4,0c
)6 14
i4,i I
0 i5
i4,i25
i4
0,0484
0,23
i4,63o
i5
i4,645
i5
1 4,660
16
14,676
i5
■ 4,691
i5
14.706
i5
14,72 ■
i5
■ 4,7^
Î6 16
■ 4,7'
2 i5
14.767
i5
0,0529
0,24
1 5,266
16
15,283
16
15,298
16
i5,3i4
16
i5,33o
i5
■ 5,345
16
i5,36i
16
i53-
7 16
1 5,3c
3 16
1 5,409
i5
o,o5-6
0,25
i5,go9
16
1 5,91 8
17
■5,935
■7
1 5,953
16
15,968
17
1 5,985
16
i6/)oi
17
16,0
8 16
i6,o'
4 17
i6,o5i
16
0,0625
0,26
i6,538
■7
16,555
17
16,573
17
16,589
18
16,607
17
16,624
■ 7
16,64 1
17
16,6:
8 17
16,6-
5 17
16.692
18
0,0676
0,27
17.174
18
17,192
17
17,209
18
■ 7.237
18
17,245
18
17,263
18
17,281
18
17,2c
)9 18
■7,3 1
7 17
17,334
18
0,0739
0,28
17,810
18
17,838
'9
17-847
18
17,865
■9
17,884
18
■ 7,912
■9
17.921
18
17,9'
9 19
i7,g-
8 18
i7-g76
■9
0,0784
0,2g
18,445
30
i8,465
19
■ 8,484
'9
i8,5o3
ig
l8,523
■9
i8,54i
20
i8,56i
■9
i8,5t
io I g
iS,5ç
9 '9
18,618
■9
o,o84i
o,3o
19,081
20
19,101
30
ig,i2i
20
ig,i4i
90
19,161
20
ig.tSi
19
19.200
20
19,2:
0 90
■9-2-
0 20
ig,26o
■9
0,0900
0,3 1
I9'7I7
21
19,738
30
19,758
21
■ 9.779
20
■9.799
21
19,820
20
19,840
21
i9,8(
)I 20
ig,8f
1 21
19,903
20
0,0961
0,32
30,353
21
90,374
21
90,395
23
30,4l7
91
2o,438
21
20,459
21
30,480
21
90, 5c
)I 91
20, 5;
3 21
20,543
21
0,1034
0,33
30,989
23
2 1 ,0 1 1
21
9 1,o32
9 9
2 r ,o54
22
3 1 ,076
29
2 1 ,098
99
21,190
22
i\,\i
9 91
21, if
3 22
2i,i85
22
0,1089
0,34
21,624
23
21,647
23
2 1 ,670
23
3 1 ,693
23
21,715
22
21,737
93
2 1 ,760
22
2i,7f
i9 23
2 1, 8c
4 23
21,827
22
0,1 1 56
0,35
22,360
23
32,283
24
22,307
2 3
2 2,33o
23
23,353
23
29,376
23
29,3gg
23
224
2 23
22^:!^
'5 23
22,468
23
0,1335
o,36
22,896
24
32,920
24
22,g44
23
2 2 ,967
24
3 3,ggi
24
33,oi5
24
2 3,o3g
24
23,of
>3 23
2 3, of
G 24
23,1 10
24
0,1296
0,37
23,53 1
25
23,556
25
33,58i
24
33,6o5
2 5
2 3,63o
24
2 3,654
3 5
23,679
24
23.7c
j3 34
23,7:
7 25
23,752
24
0,1369
o,38
24,167
25
24,199
36
34,218
25
24.243
25
34,268
25
24,2g3
35
24.3 1 S
25
24,3.
f3 2 5
24,3t
)8 25
24,393
25
0,1 444
0,39
24,8o3
26
24,829
36
24,855
25
34y88o
26
24,906
26
24,932
26
24,958
26
24,9*
i4 25
2 5,0c
9 26
25,o35
26
0,l52I
o,4o
25,438
27
3 5,465
26
25,491
27
35,5i8
27
25,545
26
25,571
26
25,597
27
25,6
4 26
25,6:
0 27
25,677
26
0,1600
0,4 1
36,074
27
26,101
27
36.128
28
36,1 56
27
26,183
27
26,210
27
26.237
27
26,9(
54 27
26,9c
I ii
26,318
27
0,1681
0,42
26,709
28
26,737
28
26,765
28
26,793
28
96,821
28
26,849
27
26,876
28
26,91
.4 28
26,g"
2 28
26,960
27
0,1764
0,43
27,345
29
27,374
28
27,402
29
27,431
28
27,45g
29
27,488
28
27.516
28
27,5.
i4 29
27.5-
3 28
27,601
28
0,1 84g
0,44
27,981
=9
28,010
29
28,039
29
28,068
29
28,og7
29
28,126
29
28,155
29
28,1!
M 3o
28,2
4 29
28,243
28
0,1 g36
0,45
98,616
3o
28,646
3o
38,676
3c
28,706
29
28,735
3o
38,765
3o
28,795
3o
38,8
25 2g
38,8
)4 3o
28,884
3o
O,3035
o,5o
31,793
33
3 1 ,826
33
31.859
33
3 1, 893
33
31.995
34
31,959
35,1 5 1
33
31,993
33
32,0
!5 33
32,0
)8 33
32.ogi
33
o,25oo
0,55
34,969
36
35,oo5
37
35,049
36
35,078
37
35;ii5
36
36
35,187
37
35,3
!4 36
35,2f
)o 36
35,996
37
o,3o25
0,60
38,i44
4o
38,i84
39
38,333
4'
38.263
4o
38,3o3
4o
38,343
39
38,389
40
38,4
22 40
38, 4(
>2 39
38,5oi
40
o,36oo
o,65
4i,3i8
43
4i,36i
43
4 1 ,4o4
43
4 1, 447
43
41,490
43
41,533
43
41,576
43
4i,6
g A'i
4i,6f
)2 43
4 1, 70 5
43
0,4225
0,70
44,49"
47
44,537
46
44383
47
44,63o
46
44,676
47
44.723
46
44,76g
46
44,8
5 47
44,8f
J2 46
44,go8
46
0,4900
0,75
47.662
5o
47.7 1 2
5o
47,762
5o
47,812
4g
47,861
5o
47,gii
5o
47 ,9'' ■
49
48 ,0
0 5o
48, of
5o 49
48,iog
5o
o,5625
0,80
5o,832
53
5o,885
54
5o,g39
53
5o,gg3
53
5 1, 045
53
5 1 .09S
53
5i.i5i
53
5 1,2
d4 53
5l,2
J7 53
5i,3io
52
0,6400
o,85
54,00 T
57
54,o58
56
54, ■M
57
54,i7i
56
54,227
57
54,284
56
54,340
56
54,3<
)6 56
54,4
52 57
54,509
56
0,7225
0,90
57,168
60
57,228
60
57,288
60
57,348
60
57,4o8
60
57,468
59
57,527
60
57,5
il 59
57,6
\& 60
57,706
r?
0,8100
0,95
60,334
63
60,397
64
60,461
63
60,52^
63
60,587
63
6o,65o
63
60,71 3
63
60,7
76 63
60,8
39 63
60,902
63
o,go35
1,00
63,498
67
63,565
66
63,63 1
67
63,698
66
62
63,76/
67
63,83 1
66
63,897
67
63,9
54 66
64, 0
3o 66
64 ,096
66
1 ,0000
1 1 ,4721
11,5200
11,5681
1 1 ,'6"
11,6645
11,7128
11,7613
11,8098
11,8585111,9072
<?
r . ( r -f- r" j^ or r^ -j- r" '^ nearly.
1
635
63(
) 637
638
639
(i4o
64 1
642
643
I
64
%i
i 64
64
64
^64
~A
64
&A
I
Î
127
12-
127
138
128
128
128
128
12g
3
3
'9'
191
191
igi
192
192
192
193
,93
3
4
254
25^
! 255
255
2 55
256
256
257
257
4
5
3i8
3iS
319
3i9
320
320
321
321
322
5
6
38 1
38:
382
383
383
384
385
385
386
6
7
445
44'
446
447
447
448
449
449
45o
7
8
5o8
5oc
) 5io
5io
5ii
5l2
5i3
5i4
5i4
8
9
572
57:
5-
?3
t
74
S-
.75
576
577
578
579
9
TABLE II. — Toliiul tlic time T; the sum of the vailii r-
-)■'', and the chord e being given.
Sum ut' iho [iQ'lii r-f-r".
Clioiil
4,8i)
4,90
4,91
4,92
4,93
4,94
c.
l);iys Iclif.
Days lUif.
Buys |(lil'.
Days |dif.
Days lilir.
Duys l.lir.
0,00
0,000
0,000
0,000
0,000
0,000
o.oot»
0,0000
0,01
0,643
0
0,643
I
0,644
I
o,645
0
0,645
1
0,646
I
0,0001
0,05
1 ,286
I
1,287
I
1,28s
1
1,289
2
1,291
1
1,392
I
0,0004
o,o3
1,928
2
1,930
2
1.933
3
1,934
2
1 ,936
2
i.g38
3
o,ooog
o,o4
2,571
3
2,574
2
2,576
3
2,579
2
2,58i
3
2,584
3
0,0016
o,o5
3,2i4
3
3,217
3
3,230
4
3,224
3
3,227
3
3,23o
3
0,0025
0,06
3,85()
4
3,860
4
3,864
4
3,868
4
3,872
4
3,876
4
o,oo36
0,07
4,490
5
4,5o4
4
4,5o8
5
4,513
5
4,5i8
4
4,522
5
0,0049
0,08
5,142
5
3,147
5
5,i52
6
5,1 58
5
5,i63
5
5,168
5
0,0064
0,09
5,785
6
5,791
5
5,79(5
6
5,802
6
5,808
6
5,8i4
6
0,008 1
o,io
0,427
7
6,434
7
6,441
6
6,447
7
6,454
6
6,460
7
0,0100
0,1 1
7,070
7
7>077
8
7,o85
7
7,092
7
7,099
7
7,106
7
0,0121
0,12
7,7" 3
8
7,721
8
7,729
7
7,736
8
7,744
8
7,752
8
o,oi44
o,i3
8,356
8
8,364
9
8,37!
8
8,38 1
9
8,390
8
8,398
9
0,0169
o,i4
8:99»
9
9.-O07
10
9,<"7
9
9,026
9
9,o35
9
9,044
9
0,01 96
0,1 5
9,64 1
10
9,65 1
10
9,66 1
9
9,670
10
9,680
10
9690
10
0,0225
0,16
10,284
10
10,294
II
io,3o5
10
io,3i5
II
10,326
10
io;336
10
o,o256
0,17
10,926
II
10.937
12
10,949
II
10,960
II
10,971
II
10,982
11
0,038g
0,18
11,569
12
ii,58i
12
11,593
1 1
1 1 ,604
12
11,616
12
11,628
12
o,o324
0,19
12,211
i3
12,224
12
12,236
i3
12,249
12
12,261
i3
12,274
12
o,o36 1
0,30
12.854
1 3
12,867
i3
12,880
i4
12,894
i3
12,907
i3
12,920
i3
o,o4oo
0,21
1 3,497
i4
i3,5ii
i3
i3,524
14
i3,538
i4
i3,552
14
1 3,566
i3
0,044 1
0,22
i4,i39
i5
i4,i54
i4
i4,i68
i5
i4,i83
14
14,197
14
l4,2II
i5
o,o484
0,23
14,782
i5
14,797
i5
i4,8i2
i5
14,827
i5
i4,842
i5
i4,85;
i5
0,0529
0,24
15,424
16
i5,44o
16
1 5,456
16
15,472
16
1 5,488
i5
i5,5o3
16
0,0576
0,25
1 6,067
16
i6,o83
17
16,100
16
16,116
17
i6,i33
16
16,149
16
0,0625
0,26
16,710
17
16,727
17
16,744
17
16,761
17
16,778
17
16,795
17
0,0676
0,27
17,352
18
17,370
18
17,388
17
i7,4o5
18
17,423
18
17,441
17
0,072g
0,28
17,995
18
i8,oi3
18
i8,o3i
■9
i8,o5o
18
18,068
18
18,086
19
0,0784
0,29
18,637
'9
1 8,656
'9
18,675
'9
i8,6g4
19
18,713
19
18,732
19
0,084 1
o,3o
'9^'79
20
I9-299
20
19,319
30
19,339
19
19,358
20
19,378
19
0,0900
0,3 1
19,922
20
19,942
21
19,963
30
19,983
30
30,Oo3
21
20,024
20
0,0961
0.32
20, 564
21
20,585
21
30,606
31
20,627
31
30/348
21
20,669
21
0,1024
0,33
21,207
21
21,228
22
3I,25o
22
21,272
31
21,293
22
2i,3i5
22
o,io8g
0,34
21,849
22
21,871
23
31,894
22
2i,gi6
22
21,938
33
2i,g6i
22
0,11 56
0,35
22,491
23
22,5l4
23
22,537
23
22,56o
23
22,583
33
22,606
23
0,1225
o,36
23,i34
23
33,157
34
23,181
34
23,2o5
23
23,228
24
23,253
23
0,1296
0,37
23,776
24
33,800
25
33,835
24
23.849
34
33,873
24
23,897
25
0,1369
o,38
24,418
25
54,443
25
34,468
25
24,493
25
24,5i8
25
24,543
25
0,1 444
o,3g
25,061
25
25,086
26
25,113
25
25,i37
26
25,i63
26
25,189
25
0,l52I
o,4o
2 5,7o3
26
25,729
26
25,755
27
25,782
36
25,808
26
25,834
26
0,1600
0,4 1
26.345
27
26,372
27
26,399
27
26,436
27
26,453
27
26,480
36
0,1681
0,42
26,987
28
27,01 5
27
27,042
28
27,070
28
27,098
27
27,135
27
0,1764
0,43
27,629
59
27,658
28
27,686
28
27,714
28
27,742
28
27,770
29
0,1849
0,44
28,271
29
28,300
29
28,329
29
38,358
29
28,387
29
28,416
29
o,ig36
0,45
28.914
29
28,943
3o
28,973
29
39,003
3o
29,033
29
29,061
3o
0,2025
o,5o
32.124
32
32.156
33
32,i8q
33
33.332
33
32,255
33
33,388
32
o,25oo
0,55
35.333
36
35;369
36
35,4o5
36
35:44.
36
35,477
36
35,5i3
36
o,3o35
0,60
38,54 1
39
38,58o
4o
38,620
39
38,659
39
38,698
40
38,738
39
o,36oo
o,65
4 1, 748
43
41,791
42
4 1, 833
43
41,876
43
41,919
43
41,961
43
0,4225
0,70
44:954
46
45,000
46
45,046
46
45,093
46
45,i38
46
45,184
46
0,4900
0,75
48,1 59
49
48.208
5o
48,258
49
48,3o7
49
48,356
49
48,4o5
5o
0,5635
0,80
5 1, 362
53
5i;4i5
53
5 1, 468
52
5i,52o
53
51,573
53
51,626
52
o,64oo
0,85
54,565
56
54,621
56
54,677
56
54,733
55
54,788
56
54,844
56
0,7325
0,90
57,765
60
57,835
59
57,884
59
57,943
59
58,002
60
58,o63
59
0,8100
0,95
60.965
62
61,027
63
61,090
62
6i,i52
63
6i,2i5
63
61,277
63
0,9035
1,00
64.162
66
64,228
66
64,294
66
64,36o
66
64.426
66
64,4g3
65
1 ,0000
1 1 .9.:
.61
12,0c
)50
12,0E
>41
12,H
);32
V2,\l
Î25
i2;2f
)18
~?~
642
64
128
193
257
331
385
449
5i4
5-S
643
64
129
193
257
322
386
45o
5i4
644
64
129
193
258
322
386
45 1
5i5
58o
nenrly.
645
65
129
194
258
323
387
452
5i6
58 1
646
65
I2g
194
258
323
388
452
5i7
58 1
Prop.
turts fur tlin sum ui' tliu Itiulii. |
■ 1
2 3 1 4 1 5 1 6 1 7 1 8
_9
I
0
0
0
0
I
I
I
I
I
2
0
0
I
I
I
I
I
2
2
3
0
I
I
I
2
2
2
2
3
4
0
I
I
2
2
2
3
3
4
5
1
3
2
3
3
4
4
5
6
I
2
2
3
4
4
5
5
7
I
2
3
4
4
5
6
6
8
2
2
3
4
5
6
6
7
9
3
3
4
5
5
6
7
8
10
2
3
4
fj
6
7
8
9
1 1
2
3
4
(>
7
8
9
10
1 3
2
4
5
6
7
8
10
1 1
i3
3
4
5
7
8
9
10
12
i4
3
4
0
7
8
10
1 1
i3
i5
3
3
5
6
8
9
I I
12
i4
16
2
3
5
6
8
10
I I
i3
i4
'7
3
3
5
7
g
10
13
i4
i5
18
2
4
5
7
9
1 1
i3
i4
16
19
2
4
6
8
!0
1 1
i3
i5
17
20
2
4
6
8
10
12
i4
16
18
21
2
4
6
8
1 1
i3
i5
17
'9
22
2
4
7
9
I 1
i3
i5
18
30
23
3
5
7
9
12
i4
16
18
31
24
2
5
7
10
I 2
i4
17
■9
33
25
3
5
8
10
l3
i5
18
20
33
26
3
5
8
10
i3
16
18
21
23
27
3
5
8
1 1
i4
16
>9
22
24
28
3
6
8
1 1
i4
17
20
23
25
29
3
6
9
12
i5
17
20
23
26
3o
3
6
g
12
i5
18
21
24
27
3i
3
6
9
1 2
16
'9
22
25
28
32
3
6
10
i3
16
19
23
26
29
33
3
7
10
i3
17
20
33
26
3o
34
3
7
10
i4
17
20
24
27
3i
35
4
7
1 1
14
18
21
25
28
32
36
4
7
1 1
i4
18
22
25
29
32
37
4
7
1 1
i5
19
22
26
3o
33
38
4
8
II
i5
19
23
27
3o
34
39
4
8
13
16
20
23
27
3i
35
40
4
8
12
16
20
24
28
32
36
4i
4
8
12
16
21
25
29
33
37
42
4
8
l3
17
21
25
29
34
38
43
4
9
l3
17
23
26
3o
34
39
44
4
9
i3
18
22
26
3i
35
40
45
5
9
i4
18
33
27
32
36
4i
46
5
9
i4
18
23
28
32
37
4i
47
5
g
i4
19
24
28
33
38
42
48
5
10
i4
19
24
29
34
38
43
49
5
10
i5
20
35
29
34
39
44
5o
5
10
i5
20
35
3o
35
4o
45
5i
5
10
i5
20
26
3i
36
4i
46
52
5
10
16
21
26
3i
36
42
47
53
5
1 1
16
21
27
32
37
42
48
54
5
1 1
16
22
27
32
38
43
49
55
6
1 1
17
32
28
33
39
44
5o
56
6
1 1
17
22
28
34
39
45
5o
57
6
1 1
17
23
29
34
4o
46
5i
58
6
12
17
23
29
35
4i
46
52
59
6
12
18
24
3o
35
4i
47
53
60
6
12
18
24
3o
36
42
48
54
61
6
13
18
24
3i
37
43
49
55
62
6
12
'9
35
3i
37
43
5o
56
63
6
l3
19
25
33
38
44
5o
57
64
6
i3
19
26
32
38
45
5i
58
65
7
i3
20
26
33
39
46
5:
5g
66
7
i3
20
26
33
40
46
53
59
67
7
i3
20
27
34
4o
47
54
60
68
7
i4
20
27
34
4i
48
54
61
69
7
i4
21
28
35
4i
48
55
62
70
7
i4
31
28
35
42
49
56
63
80
8
16
24
32
40
48
56
64
72
90
9
18
27
36
45
54
63
72
81
1 (10
10
30
3o
4o
5o
60
70
80
go
lis
TABLE
II.
— To find the time
T;
the sum of the rac
iir
4-,-", and
the chord (
being given.
Chord
C.
4,95
4,96
4,97
Days IJil'.
4,98
4,99
5,00
5,01
5,02
5,03
5,04
Days |(lil".
Days Idir.
Days |dif.
Days |ilir.
Daysldif.
Days |dif.
Daysldif.
Days Idif.
Days |dif.
0,00
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
0,647
0
0,647
I
o,648
1
0,649
0
0,649
I
o,65o
I
o,65i
0
o,65i
I
o,652
I
o,653
0
0,0001
0,02
1,293
2
1,295
I
1 ,296
I
1,297
2
1,299
I
i,3oo
1
i,3oi
1
I,302
2
1 ,3o4
I
i,3o5
1
0,0004
o,o3
1 .940
2
1.942
2
1,944
2
1,946
2
1,948
3
i,95o
3
1,952
2
1.954
3
1,956
3
1,958
2
0,0009
o,o4
2,587
2
2,589
3
2,592
3
2,595
2
2,597
3
2,600
2
2,602
3
3,6o5
3
2,608
3
2,610
3
0,0016
o,o5
3,233
4
3,237
3
3,240
3
3,243
3
3,246
4
3,25o
3
3,253
3
3,356
3
3.25g
4
3,363
3
0,0025
0,06
3,880
4
3,884
4
3,888
4
3,892
4
3,896
4
3,900
4
3,go4
3
3,907
4
3,911
4
3,qi5
4
o,oo36
0,07
4,527
4
4,53i
5
4.5.36
4
4,540
5
4,545
5
4,55o
4
4,554
5
4,559
4
4,563
5
4,568
4
0,0049
0,08
5,173
6
5,179
5
5,i84
5
5,189
5
5,194
5
5,199
6
5,2o5
5
5,310
5
5,2i5
5
5,230
5
0,0064
0,09
5,820
6
5,826
6
5,832
6
5,838
6
5,844
5
5,849
6
5,855
6
5,861
6
5,867
6
5,873
6
0,008 1
0,10
6,467
6
6,473
7
6,480
6
6,486
7
6,493
6
6,499
7
6,5o6
6
6,5 1 2
7
6,5ig
6
6,525
7
0,0100
0,11
7,ii3
8
7,121
7
7,128
7
7,i35
7
7.142
7
7.149
7
7,1 56
7
7,i63
8
7. 171
7
7,178
7
0,013 1
0,12
7,760
8
7,768
8
7,776
1
7,783
8
7.791
8
7.799
8
7,807
8
7,8i5
7
7,822
8
7,83o
8
0,01 44
o,i3
8,407
8
8,4 1 5
9
8,424
8
8,432
9
8,441
8
8,449
8
8,457
9
8,466
8
8,474
9
8,483
8
0,0169
o,i4
9,o53
9
9,062
ID
9,072
9
9,081
9
9,090
9
9.099
9
g,io8
9
9.117
9
9.126
9
g,i35
9
0,01 96
o,i5
9,700
10
9,710
9
9>7i9
10
9.729
10
9.739
10
9.749
9
9,758
10
9.768
10
9,778
10
9,788
9
0,0225
0,16
10,346
II
10,357
10
10,367
11
10,378
10
io,388
11
10,399
10
io,4og
10
10,419
11
io,43o
10
10,440
10
o,o256
0,17
10,993
II
1 1 ,oo4
II
11, 01 5
11
11,026
II
ii,o37
II
11,048
11
1 1 ,o5g
12
1 1 ,07 1
11
11,082
II
11,093
11
0,0389
0,18
1 1 ,64o
II
ii,65i
12
11,663
12
11,675
12
11,687
11
1 1 ,698
12
11,710
12
11,723
II
11,733
13
11,745
12
o,o334
0,19
12,286
i3
12,299
12
I2,3l I
12
12,323
i3
12,336
12
12,348
12
i2,36o
i3
13,373
13
12,385
13
I3,3g7
i3
o,o36i
0,30
12,933
i3
12,946
i3
12,959
i3
12,972
i3
12,985
i3
12,998
i3
i3,oii
i3
i3,o34
i3
i3,o37
l3
i3,o5o
i3
o,o4oo
0,21
13,579
i4
13,593
i4
1 3,607
i3
1 3,620
i4
1 3,634
i4
I 3, 648
i3
i3,66i
i4
13,675
i4
i3,68g
i3
1 3,703
i4
0,044 1
0,22
14,226
i4
i4,24o
i5
i4,255
i4
14,269
.4
14,283
i5
14,298
i4
i4,3i2
i4
14,326
i4
1 4,340
i5
14,355
i4
o,o484
0,23
14,872
i5
14,887
i5
14,902
i5
'4,917
i5
14,933
i5
14,947
i5
14,962
i5
14,977
i5
i4,gg3
i5
1 5,007
i5
0,0539
0,24
i5,5i9
16
i5,535
i5
i5,55o
16
1 5,566
15
i5,58i
16
15,597
16
i5,6i3
i5
1 5,638
16
1 5,644
i5
1 5,659
16
0,0576
0,25
i6,i65
17
16,182
16
16,198
16
16,214
17
i6,23i
16
16,247
16
16,263
16
16,379
16
i6,3g5
17
1 6,3 1 3
16
0,0625
0,26
16,812
17
16,829
17
16,846
17
1 6,863
17
1 6,S8o
17
16,897
16
16,913
17
16,930
17
i6,g47
17
1 6,964
17
0,0676
0,27
■ 7,458
18
17,476
17
17,493
18
17,511
18
17,539
17
17,546
18
17,564
17
I7,58i
18
i7.5gg
17
17,616
18
0,072g
0,28
i8,io5
18
18,123
18
i8,i4i
18
i8,i5q
19
18,178
18
18,196
18
18,214
18
l8,232
18
i8,25o
19
18,369
18
0,0784
0,29
i8,75i
19
18,770
'9
18,789
19
18,808
'9
18,837
19
i8,846
18
18,864
'9
1 8,883
19
18,902
19
i8,g2i
19
o,o84i
o,3o
19,397
20
i9'4i7
20
19,437
19
19,456
20
19,476
19
19,495
30
I9,5i5
'9
T9,534
20
19,554
19
19,573
20
o,ogoo
0,3 1
20,044
20
20,064
20
20,084
21
20,I05
20
30,125
20
20,145
3r
2o,i65
20
2o,i85
20
20,305
20
20,225
20
0,0961
0,32
20,690
21
20,711
21
20,732
21
20,753
21
20,774
21
30,795
30
30,81 5
21
20,836
21
30,857
21
20,878
20
0,1034
0,33
21,337
21
21,358
22
2i,38o
21
21,401
22
21,423
21
31,444
33
21,466
21
21,487
31
2i,5o8
33
2i,53o
21
0,1089
0,34
21,983
22
2 2,005
22
22,027
22
32,049
23
22,072
22
33,094
32
22,116
22
22,l38
33
22,160
33
32,182
22
o,ii56
0,35
22,629
23
22,652
23
22,675
23
22,698
22
23,730
23
22,743
23
22,766
23
22,789
32
23,8 I I
23
22,834
23
0,1 335
o,36
23,275
24
23,299
23
23,322
24
23,346
23
23,369
24
23,393
23
23,4i6
24
23,440
33
33,463
23
23,486
24
0,1296
0,37
23,922
24
23,946
24
23,970
24
23,994
24
24,018
24
24,042
24
24,066
24
24,090
24
34.114
24
34,i38
34
0,1 36g
o,38
24,568
25
24,593
25
24,618
24
24,642
25
24,667
25
24,692
24
24,716
35
24,741
25
24,766
24
24,790
35
0,1 444
o,3g
25,2l4
26
25,240
25
25,265
25
25,290
26
25,3i6
25
2 5,34 1
26
25,367
25
35,3g3
25
25,417
35
25,443
36
0,l521
0,40
25.860
26
25,886
36
25,912
27
25.939
26
25,965
26
25,991
26
26,017
26
26,043
26
26,069
26
26,095
25
0,1600
0,4 1
26,506
27
26,533
27
26,560
27
26,587
26
36,61 3
27
26,640
27
26,667
36
26,6g3
27
26,720
27
26,747
36
0,1681
0,42
27,l52
28
27,180
27
27,207
28
27,235
37
27,262
27
27,289
28
27,317
37
27,344
27
37,371
27
27,3g8
38
0,1764
0,43
27,799
28
27,827
28
27,855
28
27,883
28
27,911
28
27,939
28
27,967
38
27,995
28
38,033
27
28,o5o
28
0,1849
0,44
28,445
28
28,473
29
28,502
29
28,531
28
28,559
29
38,588
29
28,617
38
28,645
29
38,674
38
28,703
29
0,1936
0,45
29,091
29
29,120
29
29)149
3o
29'i79
29
29,208
29
29,237
3o
29,267
29
29.296
29
39,325
29
2g,354
29
0,2025
o,5o
3i,32o
33
32,353
33
32,386
32
32,4i8
33
32,45i
32
32,483
33
32,5i6
33
32,548
33
32,58i
32
32,6i3
33
0,2 5oo
0,55
35,549
36
35,585
36
35,62 1
36
35,657
36
35,693
36
35,729
35
35,764
36
35,800
36
35,836
36
35,872
35
o,3o25
0,60
38,777
39
38,8i6
40
38,856
39
38,895
39
38,934
39
38,973
39
39,012
39
3g,o5i
39
39,090
3g
3g,i3g
39
o,36oo
o,65
42,004
43
42,047
42
42,089
42
42,i3i
43
42,174
42
42,216
43
42,25g
43
42,3oi
42
42,343
42
42,385
42
0,4225
0,70
45,230
46
45,276
45
45,321
46
45,367
46
45,4i3
45
45,458
46
45,5o4
46
45,55o
45
45,5g5
46
45,641
45
0,4900
0,75
48,455
49
48,5o4
49
48,553
49
48,602
49
48,65 1
49
48,700
48
48,748
49
48,797
49
48,846
49
48,8g5
48
0,5625
0,80
51,678
52
5i,73o
53
5 1, 783
52
5 1, 835
52
51,887
52
51,939
53
5 1 ,gg2
53
52,o44
53
52,096
53
52,i48
53
o,64oo
0,85
54,900
56
54,956
55
55,011
56
55,067
56
55,123
55
55,178
55
55,233
56
55,280
55
55344
55
55,399
56
0,7225
0,90
58,121
59
58,i8o
59
58,239
59
58,298
58
58,356
59
58,4 1 5
59
58,474
58
58,532
5g
58,591
59
58,65o
58
0,8100
o,g5
61,340
62
6 1 ,402
62
61, 464
63
61,527
62
61,589
62
6i,65i
62
6i,7i3
63
61,775
63
61,83-
61
61,898
62 0,9020 1
1,00
64,557
66
64,623
66
64,689
65
64,754
65
64,819
66
64,885
65
64,g5o
65
65,oi5
65
65,o8o
66
65,i46 65|i,oooo|
12,2513
12,3008
12,3505
12,4002
12,45
01
12,5000
12,5501
12,60021
12,65051 12,70081 c" I
■(r + r" y
•j- r"^ nejirly.
646
647
648
65
65
65
129
129
i3o
194
194
ig4
258
359
359
333
324
334
388
388
38g
453
453
454
5l7
5i8
5i8
58r
582
583
64g
65
i3o
195
2Ô0
335
38g
454
5ig
584
65o
65i
652
653
65
65
65
65
T,3o
i3o
i3o
i3i
iq5
ig5
196
196
260
260
261
261
325
326
326
327
3go
39.
3gi
392
455
456
456
457
520
521
522
522
585
586
587
588
TABLE
I. -
-To find the time 7"; the sum
of the rail
ii )• -|- '■ ", ami the choi-d c bein
; given
Sum ot tlie lladil r-\-r '.
Prtip. parts lor the sum ol" the Kodti.
Choid
c.
5,05
5,06
UajS |dir.
5,07
5,08 5,09 I 5,10
I 1 3 1 3| 41 5| 6| 7|8 9
1
2
3
0 0
0 0
0 I
0
I
I
0
1
T < -
I
I
2
1
3
2
2
3
Days |dir.
Days |.lir.
Days |dil'. Days |d
if. Days |dif.
I
I
2
1
2
0,00
0,000
0,000
0,00c
0.000
0,000
0,000
0,0000
0,01
0,653
I
o,654
c
0,6 54
I
o;655
I o,656
0 o,656
I
0,000 1
4
0 I
1
3
2
2
3
3
4
0,02
i,3oO
2
i,3ob
1
I,3tH|
I
1 .3 1 0
3 I,3l3
I I,3i3
I
0,0004
o,o3
1,960
I
1. 96 1
2
1 ,96c
1:965
2 1,967
2 1 ,g6c
2
0,0009
5
2
2
3
3
4
4
5
o,o4
2,6i3
2
2,6i5
3
3,618
2
2 ,620
3 2,623
3 2,626
2
0,0016
6
2
2
2
3
3
4
4
4
4
5
5
6
5
6
o,o5
3,266
3
3,261)
3
3,273
4
3,276
3 3,279
3 3,282
3
0,0025
8
I 2
2
3
4
5
6
6
7
OjOti
3,9 '9
4
3.923
4
3,927
4
3.931
4 3,935
3 3,938
4
o,oo36
9
I 2
3
4
5
5
6
7
8
0,07
4,572
5
4,5-
4
4,58i
5
4^586
4 4,590
5 4,5g5
4
o,oo4g
10
I 2
3
3
4
4
4
5
5
6
6
6
8
o,o«
5,225
6
5,33i
5
5,236
5
5.241
5 5,246
5 5,25i
5
0,0064
7
8
8
9
OjOy
5,879
5
5,884
6
5,890
C
5,896
6 5,go2
6 5,908
5
0,0081
1 1
12
1 2
I 3
7
■7
9
10
10
II
0,10
0,1 1
6,532
7>i85
6
6,538
7,192
7
6,545
7,199
6
7
6,55i
7,206
7 6,558
7 7,2i3
6 6,564
7 7,220
6
n
0,01 00
0,01 2 1
1 3
i4
I 3
I 3
4
4
5
6
7
7
8
8
9
10
10
1 1
12
i3
0,12
7,83b
6
7,846
b
7,854
7
7.861
8 7,869
8 7,877
7
0,01 44
i5
3 3
5
6
8
9
10
II
12
i4
o,i3
8,491
9
8,5oo
b
8,5o8
8
8;5i6
9 8,525
8 8,533
8
0,01 6g
16
2 3
5
6
8
II
i3
i4
P,i4
9ii44
9
9,1 53
9
9,162
9
9,171
9 9,180
9 9,189
9
0,0196
17
18
2 3
2 4
5
5
7
7
9
9
10
10
11
12
i3
i4
i4
i5
16
o,i5
9.797
10
9,807
10
9,817
9
9,826
10 9,836
10 9,846
9
0,0225
19
2 4
6
8
II
i3
i5
17
0,16
io,45o
11
10,461
10
10,471
10
10,481
II 10,492
10 IO,503
10
o,o256
0,17
11,104
II
ii,ii5
II
11,126
10
ii,i36
II Il,l47
II ii,i58
II
0,0289
20
2 4
6
8
10
12
i4
16
18
0,18
11,757
II
11,768
12
11,780
12
11,-92
11 11 ,8o3
12 1 1 ,8 1 5
II
o,o324
21
2 4
6
8
II
i3
i5
17
19
0,19
l2,4lO
12
12,422
12
12,434
i3
12,447
13 13,459
12 13,471
12
o,o36i
22
23
2 4
2 5
7
7
9
9
II
12
i3
14
i5
16
18
18
20
21
0,30
i3,o63
i3
13,076
i3
13,089
i3
l3,102
12 i3,ii4
i3 i3,i27
i3
o,o4oo
24
2 5
7
10
12
i4
17
19
22
0,21
1 3,716
i3
13,729
i4
1 3,743
i4
13,757
1 3 13,770
i4 13,784
i3
0,044 1
25
26
3 5
3 5
3 5
3 6
8
8
8
8
i3
i3
i4
i4
i5
i()
16
17
18
18
23
23
24
25
0,22
0,23
14,369
1 5,022
14
i5
1 4,383
1 5,037
14
i5
14,397
i5,o52
i4
i4
i4,4i 1
1 5,066
i5 14,426
i5 i5,o8i
i4 i4,44o
1 5 i5,oq6
i4
i5
o,o484
0,0539
10
10
20
21
0,24
15,675
i5
15,690
16
1 5,706
i5
i5,72i
16 15,737
1 5 1 5,752
i6
0,0576
27
28
1 1
II
19
20
22
32
0,25
16,328
16
16,344
16
i6,36o
16
16,376
16 16,392
17 i6,4og
16
0,0625
29
3 6
9
12
i5
17
20
23
26
0,26
i6.q8i
17
1 6,998
16
17,014
17
I7,o3l
17 17,048
17 17,065
16
0,0676
3o
3 6
9
9
10
12
i5
18
21
24
27
0,2-
1^,634
I-
i7,65i
18
17,669
IT
17,686
18 17,704
17 '7,721
17
0,0729
3i
3 6
1 2
16
19
19
20
22
25
28
0,28
18,287
18
t8,3o5
18
18,323
18
18,341
18 18,359
18 18,377
iS
0,0784
32
3 6
i3
16
22
26
=9
3o
0,29
18.940
18
18,958
19
18,977
19
18,996
ig 19,015
18 ig,o33
19
0,084 1
33
3 7
10
i3
17
23
26
34
3 7
10
i4
17
20
24
27
3i
o,3o
19,593
19
19,612
19
19,63 1
20
1 9,65 1
19 19,670
19 19,68g
20
0,0900
o,3i
20.245
20
20,265
21
20,286
20
2o,3o6
20 20,336
ig 20,345
20
0,0961
35
4 7
II
i4
18
21
25
28
32
0,32
20,898
21
20,919
21
20,940
20
20,960
21 20,981
21 21,002
20
0,1024
36
4 7
II
i4
18
22
25
29
32
0,33
2i,55i
22
21,573
21
21,594
21
2i,6i5
21 21,636
22 21,658
21
o,io8g
37
4 7
II
i5
19
33
26
3o
33
0,34
22,204
22
22,226
23
22,248
22
22,270
22 22,292
22 22,3l4
22
0,1 156
38
39
4 8
4 8
II
12
i5
16
19
20
33
23
27
27
3o
3i
34
35
0,35
22,857
22
22,87g
23
22,902
23
22,925
22 22,947
23 22,970
22
0,1225
o,36
23,5io
23
23,533
23
23,556
23
23,579
24 23,6o3
23 23,626
23
0,1296
4o
4 8
12
16
20
24
28
32
36
0,37
24,162
24
24,186
24
24,210
24
24,334
24 24,258
24 24,282
34
o,i36g
4i
4 8
12
16
21
35
29
33
37
o,38
24.815
25
-:4,84o
24
24,864
25
24,889
24 24,Ql3
25 24,938
34
0,1444
42
4 8
i3
17
21
25
=9
34
38
0,39
25;468
25
25,493
25
25,5i8
25
25,543
25 25,568
26 25,594
25
0,l52I
43
44
j 9
4 9
i3
i3
17
18
22
22
26
26
3o
3i
34
35
39
40
o,4o
26,120
26
26,146
36
26,172
26
26,198
26 26,234
26 26,250
25
0,1600
5 9
5 9
5 10
5 10
18
18
19
23
23
24
24
25
32
32
33
34
34
36
37
38
38
39
4i
4i
42
43
44
0,4 1
26,773
27
26,800
26
26,826
27
26;853
26 26.87g
26 26,905
27
0,1681
45
46
47
i4
i4
i4
27
28
28
0,42
0,43
27,426
28.078
27
28
27,453
28,106
27
28
27,480
28,134
27
28
27,507
28,162
27 27,534
27 28,189
27 27,561
28 28,217
27
38
0,1764
0,1849
0,44
28,731
28
28,759
29
28,788
28
28,816
sg 28,845
28 28,873
38
o,ig36
48
49
i4
i5
19
20
29
29
0,45
29,383
3o
29,4i3
29
39,442
29
29,471
29 29,500
2g 29,529
29
0,2025
5o
5 10
i5
20
25
3o
35
40
45
o,5o
3?,646
32
32,678
32
32,710
33
32,743
32 32,775
32 32.807
33
o,25oo
5i
52
5 10
i5
20
26
3i
36
4i
46
0,55
35,907
36
35,943
35
35,978
36
36.0I4
35 36,o49
36 36,o85
35
o,3o25
J 10
16
21
26
3i
36
42
47
0,60
39,168
39
39,207
38
39,345
39
39,284
39 39,323
39 3g,362
38
o,36oo
53
5 1 1
16
21
27
32
37
42
48
0,65
42,427
42
42,469
43
43,5l3
42
42,554 ^
12 42,596 i
il 42,637
42
0,4225
54
5 II
16
22
27
32
38
43
49
0,70
45,686
45
45,731
46
45,777
45
45,822 i
i5 45,867 .
i5 45,912
45
o,4goo
55 f
5 II
17
22
28
33
39
44
5o
0,75
48,943
49
48,992
49
49,041
48
49,089 i
ig 4g,i38 .
i8 49,186
48
0,5625
56 (
3 II
17
23
28
34
39
45
5o
0,80
52,200
5i
52,25l
52
52,3o3
52
52,355
)2 52,407 ,
ji 52,458
52
o,64oo
57 (
) 11
17
23
29
34
40
46
5i
o,85
55,455
55
55,5io
55
55,565
55
55,620 .
)5 55,675
J5 55,730
55
0,7225
58 t
12
17
23
29
35
4i
46
52
0,90
58,708
58
58,766
59
58,835
58
58,883 '
)8 58 ,94 1
39 59,000
58
0,8100
59 c
12
18
34
3o
35
4i
47
53
0,95
61,960 62
65,2 11 65
62,022
61
63,o83
62
62,145 t
5i 62,306 t
)2 62,268
61
0,9025
1,00
65,276
64
65,34o
65
65,4o5 (
12,903
55 65,470 (
55 65,535
64 I, 0000 1
60 (
61 t
62 r
63 (J
64 6
12
12
13
1 3
i3
18
18
19
19
24
24
25
25
26
3o
3i
3i
32
36
37
37
38
38
42
43
43
44
45
48
5o
5o
5i
54
55
56
57
58
12,7513
12,8018'
12,85251
2 12,954
1 13,00501 c2 1
h . {r + r"y or r'^-\- r"^ nearly. |
652
653
654
655
656
657
19
32
—
—
—
—
65 7
i3
20
26
33
39
46
52
59
I
65
65
65
66
66
66
I
66 7
1 3
20
26
33
40
46
53
59
2
i3o
i3i
i3i
i3i
i3i
i3i
2
67 7
i3
20
27
34
4o
47
54
60
3
196
196
,96
197
197
197
3
68 7
i4
20
27
34
4i
48
54
61
4
261
261
262
262
263
263
4
69 7
i4
21
28
35
4i
48
55
62
5
326
327
327
328
338
329
5
6
3qi
392
3q2
393
394
394
6
70 7
i4
21
28
35
42
49
56
63
7
456
457
458
459
459
460
7
80 8
16
24
32
40
48
56
64
72
8
522
522
523
524
525
526
8
90 9
18
27
36
45
54
63
72
81
90
9
587
588
58o
590
Sgo
591
9
100 10
20
3o
40
5o
60
70
80
TABLE II.
— To find the time T
the sum of the radii
r + r",
and the chord c
be
ng given.
Sum of the Radii r-)-r".
Cliord
c.
5,11
5,12
5,13
5,14
5,15
5,16
5,17
5,18
5,19
5,20
Days |dif.
Days |dir.
Days |dif.
Days |dif.
Days |dif.
Days Idif.
Days Idif.
Days Idif.
Days Idif.
Days.
0,00
0,000
0,000
0,000
0,000
0,000
0;000
0,000
0,000
0,000
0,000
0,0000
0,01
o,657
I
0,658
0
0,658
I
0,659
I
0,660
0
0,660
I
0,661
I
0,662
0
0,662
I
o;663
0,0001
0,02
1,314
I
i,3i5
2
i,3i7
I
I,3l8
I
1,319
2
1,321
I
1,322
I
1,323
1
1,334
2
1,326
0,0004
o,o3
1. 97 1
2
1,973
2
i>975
2
''977
2
'.979
2
1,981
2
1,983
2
1,985
2
1,987
I
1,988
0,0009
o,o4
2,628
3
2,63 1
2
2,633
3
2,636
2
2,638
3
2,64 1
3
2,644
2
2,646
3
2,649
2
2,65 r
0,0016
o,o5
3,285
3
3,288
4
3,292
3
3,295
3
3,298
3
3,3oi
3
3,3o4
4
3,3o8
3
3,3ii
3
3,3i4
0,0025
0,06
3,942
4
3,946
4
3,950
4
3,954
4
3,958
4
3,962
3
3,965
4
3,969
4
3,973
4
3,977
o,oo36
0,07
4,599
5
4,604
4
4,608
5
4,6i3
4
4,617
5
4,622
4
4,626
5
4,63 1
4
4,635
5
4.640
0,0049
0,08
5,256
5
5,261
6
5,267
5
5,272
5
5,277
5
5,282
5
5,287
5
5,393
5
5,397
5
5.302
0,0064
0,09
5,913
6
5,919
6
5,925
6
5,931
5
5,936
6
5,942
6
5,948
6
5,954
5
5,959
6
5^965
0,0081
0,10
6,570
7
6,577
6
6,583
7
6,590
6
6,596
6
6,602
7
6,609
6
6,6 1 5
7
6,622
e
6,638
0,0100
0,11
7,227
7
7,234
8
7,242
7
7,249
7
7,256
7
7,263
7
7,270
7
7,277
7
7,284
7
7,291
0,0121
0,12
7,884
8
7,892
8
7,900
8
7,908
7
7,915
8
7,923
8
7,931
7
7,938
8
7,q46
8
7,954
0,0 1 44
0,1 3
8,541
9
8,55o
8
8,558
8
5,566
9
8,575
8
8,583
8
8.591
9
8,600
8
8,608
8
8,616
0,0169
0,1 4
9)198
9
9.207
9
9,216
9
9,225
9
9,234
9
9,243
9
9,252
9
9,261
9
9,270
9
9,279
0,0196
0,1 5
9,855
10
9,865
10
9,875
9
9,884
10
9,894
10
9,904
9
9,9'3
10
9,923
9
9,932
10
9,942
0,0225
o,t6
IO,5l2
II
io,523
10
10,533
10
10,543
10
10,553
II
io,564
10
10,574
10
io,584
10
10,594
II
io,6o5
0,03 56
0,17
11,169
II
11,180
11
11,191
II
11,202
II
1 1,2 1 3
II
11,224
II
11,235
II
11,346
10
11,256
11
11,267
0,0289
0,18
11,826
12
11,838
II
11,849
12
11,861
II
1 1 ,872
12
11,884
12
1 1 ,896
11
11,907
12
11,919
II
1 1 ,930
o,o324
0,19
12,483
12
12,495
i3
i2,5o8
12
12,520
12
12,532
12
12,544
12
13,556
12
13,568
1 3
12,58,
12
12,593
o,o36i
0,20
i3,i4o
i3
i3,i53
i3
i3,i66
i3
i3,i79
i3
13,192
12
i3,2o4
i3
i3,2i7
i3
i3,33o
i3
1 3,243
12
1 3,255
o,o4oo
0,21
1 3,797
i4
i3,8ii
i3
13,824
i4
1 3,838
i3
i3,85i
i3
1 3,864
14
13,878
i3
13,891
i4
1 3,905
i3
13,918
0,044 I
0,22
14,454
i4
i4,468
i4
14,482
i4
14,496
i4
i4,5io
i5
i4,535
14
14,539
i4
14,553
i4
14,567
14
i4,58i
0,0484
0,23
i5,iii
i5
15,126
i4
i5,i4o
i5
i5,i55
i5
15,170
i5
i5,i85
i4
15,199
i5
i5,2i4
i5
l5,22g
14
1 5,343
o,o529
0,24
15,768
i5
1 5,783
16
1 5,799
i5
i5,8i4
i5
15,829
16
1 5,845
i5
1 5,860
i5
15,875
16
15,891
i5
15,906
0,0576
0,25
16,425
16
16,441
16
16,457
16
16,473
16
i6,48q
16
i6,5o5
16
16,521
16
16,537
16
16,553
16
16,569
0,0635
0,26
17,081
17
17,098
17
I7,:i5
17
I7,i32
16
17,148
17
i7,i65
16
17,181
'7
17,198
17
I7,2i5
16
I7,23r
0,0676
0,27
17,738
18
17,756
17
17.773
17
17.790
18
17,808
17
17,825
17
17.842
17
17,859
18
17,877
17
I7,8q4
0,0729
0,28
18,395
18
i8,4i3
18
i8,43i
18
i8,44y
18
18,467
18
1 8,485
18
i8,5o3
18
18,521
18
18,539
17
1 8, 556
0,0784
0,29
19,052
19
19,071
18
19,089
'9
19,108
18
19,126
19
19,145
18
19,163
19
19,182
'9
19,201
18
'9,219
0,084 1
o,3o
19,709
19
19,728
19
'9.747
'9
19,766
20
19,786
'9
19,805
'9
19.824
'9
19.843
'9
19,862
20
19,882
0,0900
0,3 1
20,365
20
20,385
20
20,4o5
20
20,425
20
20,445
20
2o,465
20
20,485
20
2o,5o5
'9
20,524
20
20,544
0,0961
0,32
21,022
21
3 1 ,043
20
2 1 ,o63
21
21,084
20
2i,io4
21
21,125
30
21,145
21
2 1 , 1 66
20
21,186
31
21,307
0,1024
0,33
21,679
21
21,700
21
21,721
21
21,742
22
21,764
21
21,785
21
21,806
21
31,837
21
21,848
31
21,869
0,1089
0,34
22,336
21
22,357
22
22,379
22
22,401
22
22,423
22
22,445
21
22,466
23
3 3,488
22
22,510
33
22,533
0,1 156
0,35
22,992
23
2 3,01 5
22
23,o37
23
23,060
22
23,082
23
23,io5
22
23,127
23
33,149
23
23,173
23
23,194
0,1225
o,36
2 3,649
23
23,672
23
23,695
23
23,718
23
23,74i
23
23,764
23
23,787
23
33,810
23
23,833
23
23,856
0,1296
0,37
24,3o6
23
24,329
24
24,353
24
24,377
24
24,401
23
24,424
24
24,448
24
34,47'
23
24,495
24
24,519
0, 1 369
0,38
24,962
25
24,987
24
25,011
24
25,o35
25
25,060
24
25,084
24
25,108
25
35,i33
24
35,i57
24
25,181
û,i444
0,39
25,619
25
25,644
25
25,669
25
25,694
25
25,719
25
2 5,744
25
25,769
25
25,794
25
35,819
25
25,844
0,l52I
o,4o
26,275
26
26,301
26
26,327
25
26,352
26
26,378
26
26,404
25
36,429
36
26,455
25
36,480
36
36,506
0,1600
0,4 1
26,932
26
26,958
27
26,985
26
27,011
26
27,037
26
27,063
27
37,090
26
27,116
36
27,142
26
27,168
0,1681
0,42
27;588
27
27,61 5
27
27,642
27
27,66g
27
27,696
27
27,723
27
27,750
27
27,777
27
27,804
26
27,83o
0,1764
0,43
28,245
27
28,272
28
28,300
28
28,328
27
28,355
28
28,383
27
38,410
28
28,438
27
28,465
28
28,493
0,1849
0,44
28,901
29
28,930
28
28,958
28
28,986
28
29,014
28
29,042
29
29,071
38
29,099
28
39,137
28
29,155
0,1936
0,45
29,558
29
29,587
29
29,616
28
29,644
29
29,673
29
29,702
29
29,731
29
29.760
28
29,788
29
29,817
0,2025
o,5o
32,839
33
32,872
32
32,904
32
32,936
32
32,968
32
33,000
32
33,o32
33
33,n64
32
33,096
32
33,128
o,25oo
0,55
36,120
36
36,i56
35
36,191
35
36,226
36
36,262
35
36,297
35
36,332
35
36,367
35
36,4o3
36
36,438
o,3o25
0,60
39,400
39
39,439
39
39,478
38
39,516
39
39,555
38
39,593
38
39,63 1
39
39,670
38
39,708
39
39,747
o,36oo
o,65
42,679
42
42,721
42
42,763
42
42,8o5
42
42,847
4i
42,888
42
42,930
41
43,971
42
43.013
42
43,055
0,4225
0,70
45,957
45
46,002
46
46,048
45
46,093
45
46,i38
44
46,182
45
46,227
45
46,272
45
46,3 1 7
45
46,362
0,4900
0,75
49,234
49
49,283
48
49,33i
48
49,379
48
49,427
48
49,475
49
49,524
48
49,572
48
49,620
48
49,668
0,5625
0,80
52,5io
52
52,562
5i
52,6i3
52
52,665
5i
52,716
5i
53,767
52
52,819
5i
52,87c
5i
52,921
5i
52,972
0,6400
o,85
55,785
54
55,839
55
55,894
55
55,949
54
56,oo3
55
56,û58
55
56,1 13
54
56,167
55
56,222
54
56.376
0,7225
0,90
59,o58
58
59,116
58
59,174
58
59,232
58
59,290
57
59,347
58
59,405
58
59,463
58
59,521
57
59',578
0,8100
0,95
62,329
62
62,391
61
62,452
61
62,5i3
61
62,574
61
62,635
61
62.696
61
62,757
61
62,818
61
62,879
0,9025
1,00
65,599
65
65,664
64
65,728
65
65,7931 64
65,857
65165,92^
64
65,9861 64
66,o5o
6/
66,1 14
64
66,178
1 ,0000
13,0561
13,1072
13,1585
1 13,2098
13,2613113,3128
13,3645
13,4162
13,4681
13,5200
^2
h ■('■ + '■")' or r' + r"= nearly. |
656
657
658
659
660
661
662
663
I
6G
66
66
66
66
66
66
66
I
2
i3,
i3i
l32
l32
l32
l32
l32
i33
2
3
'97
'97
197
198
198
198
199
199
3
4
262
263
263
264
264
264
265
265
4
5
328
Sag
329
33o
33o
33i
33 1
332
5
6
394
394
395
395
396
397
397
398
6
7
459
460
461
461
462
463
463
464
7
8
525
526
526
527
528
52q
53o
53o
8
9
59
0
'9'
592 1
593
5Ç
)4 1
595
5(
36
597
9
TA RLE
II
— To find the time T
, the sum of tlic v;
dii
^■Vr",
and
the chord
c being given.
Sum of the Radii r-f-r". 1
Chord
c.
5,->0
5,30
5,40
5,50
5,60
5,70
5,80
5,90
6,00
6,10
Days |dir.
Huy*. |dir.
Days |dir.
Days |dir.
Dnysldil'.
Days lilif.
Days|,lir.
Day» Idif.
Days |dil.
Days Idir.
0,00
0,0(K)
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,0000
0,01
o,fi63
6
0,669
6
0,675
7
0,682
6
o,68S
6
0,694
6
0,700
6
0,706
6
0,713
6
0,718
6
0,0001
0.02
1.326
13
1,338
i3
I,35l
12
1,363
i3
1,376
12
1,388
13
1 ,4ot.
12
I .4 1 2
12
1,434
13
1,436
II
o,ooo4
o,o3
i.il88
'9
2,007
HI
3,03.6
19
2,045
.8
2, 06 3
'9
2,082
18
2 , 1 00
18
2.118
18
2,1 36
18
2,i54
17
o,ooog
0,04
3,()5l
26
2,677
25
3,702
35
2,727
24
2,75i
25
2,776
24
2,800
24
2,834
24
2,848
24
2,872
23
0,0016
o,o5
3,3i4
32
3,346
3i
3,377
3i
3,4o8
3i
3,43g
3i
3,470
3o
3,5oo
3o
3,53o
3o
3,56o
29
3,58g
3o
0,0026
0,1)6
3,977
38
4,01 5
38
4,o53
37
4,ogo
37
4,^27
37
4,164
36
4,300
36
4,236
36
4,272
35
4,3o7
35
o,oo36
0,07
4.64"
44
4,684
44
4,738
44
4.772
43
4,8 ■ 5
43
4,858
42
4,goo
42
4.943
43
4,984
4i
5,025
4i
o,oo4g
0,08
5;3o2
5i
5,353
5o
5,4o3
5o
5,453
DO
5,5o3
49
5,552
48
5,600
48
5,648
48
5,696
47
5,743
47
0,0064
0,09
5,965
57
6,022
57
6,079
56
6,1 35
55
6,190
55
6,245
55
6,3oo
.54
6,354
54
6,408
53
6,461
53
0,0081
0,10
6,628
63
6,691
63
6,754
63
6.817
61
6,878
61
6,939
61
7,000
60
7,060
60
7,130
59
7,179
58
0,0i00
0,11
7,'9'
70
7,36i
69
7,43o
68
7,498
68
7,566
67
7,633
67
7,700
66
7,766
66
7,833
65
7,897
64
0,0121
0,I5
7.ii54
76
8,o3o
75
8,io5
75
8,180
74
8,254
73
8,327
73
8,400
72
8,472
T-
8,544
70
8,6i4
71
0,01 44
o,i3
8,616
83
8,699
81
8,780
81
8,861
81
8,g42
79
9,021
79
g,ioo
78
9.178
7-
9,255
77
9,332
76
o.oifig
o,i4
9>279
89
9,368
88
9.456
87
9,543
86
9,629
86
9,7 15
85
g,8oo
84
9,884
83
9,967
83
io,o5o
82
0,0196
o,i5
9.945
95
TO,o37
94
io,i3i
94
10,225
92
10,317
92
io,4og
91
io,5oo
90
10,590
89
10,67g
89
10,768
88
0,0325
0,16
io,6o5
101
10,706
101
10,807
99
10,906
99
1 1 ,oo5
98
ii,io3
97
I l,300
96
11,396
95
■ i,3gi
95
11,486
94
o,0356
0,17
11,267
108
11,375
107
11,482
106
1 1 ,588
io5
1 1 ,693
104
■1,797
io3
1 1 ,goo
103
12,002
101
12,lo3
101
I2,204
0,028g
0,18
1 1 ,930
114
12,044
ii3
i2,i57
112
12,269
1 1 1
i3,38o
II I
■ 2,491
loq
12,600
108
12,708
107
i2,8i5
106
12,921
10Ê
o,o324
o,ig
12,593
120
I2,7i3
120
12,833
118
■ 2,951
117
1 3,068
116
i3,i84
116
i3,3oo
114
■ 3,4i4
ii3
■ 3,527
113
■ 3,639
112
o,o36i
0,20
i3,355
127
i3,382
126
i3,5o8
125
i3,633
123
1 3,756
122
13,878
131
■3,999
121
l4,^30
■'9
■4,23g
116
■ 4,357
117
o,o4oo
0,3 1
i3,c|i8
i33
i4,o5i
I 32
i4,i83
i3i
■4,3i4
i3o
■ 4,444
128
■4,572
127
■4,69g
■27
14.836
135
i4,g5i
124
■ 5,075
123
0,044 1
0,23
i4,58i
139
14,720
■39
14,859
■37
■4,996
i35
i5,i3i
i35
■ 5,266
i33
■5,399
l33
i5;53i
l33
1 5,663
i3o
15,793
■29
0,0484
0,23
i5,243
1 46
1 5,389
145
1 5,534
i43
■ 5,677
■ 42
15,819
■ 4^
1 5, 960
i3g
16,099
i38
16,337
■ 37
16,374
i36
i6,5io
i35
0,0529
0,24
15,906
l52
i6,o58
i5i
16,209
i5o
16,359
i48
16,507
146
1 6,653
1 46
■6,799
144
16,943
143
17,086
142
17,228
lAi
0,0576
0,25
16,569
1 58
16,727
■57
16,884
1 56
1 7,o4o
■ 54
17,194
1 53
'7,347
l53
■7,49g
i5o
17,649
l4g
■7,798
148
17,946
1 46
0,0625
0,26
I7,23i
i65
17,396
164
17,560
162
■ 7,722
160
17,883
■ 59
i8,o4^
■ 58
■8,199
i56
18,355
1 55
i8,5io
i54
18,664
I 52
0,0676
0,27
17,894
171
i8,o65
170
18,335
168
i8,4o3
167
18,570
.65
■8,735
i63
i8,8gb
i63
19,061
161
iq,222
■ 59
ig,38i
■ 58
0,072g
0,28
18, 556
178
18,734
176
i8,gio
174
19,084
173
19,257
■72
19,429
169
19,598
■69
19,767
166
19,933
166
20,099
■ 64
0,0784
0,29
■9.2'9
i84
19,403
183
ig,5S5
iSi
19,766
■ 79
19,945
177
20,122
176
20,298
■74
20,472
173
20,645
■72
30,817
170
0,0841
o,3o
19,882
190
20,072
189
20,261
186
20,447
i85
20,632
184
30,816
182
20,998
180
21,178
179
21,357
■77
21,534
176
o,ogoo
0,3 1
20,544
197
20,74l
■95
2o,g36
193
21,12g
■9'
21,330
190
2 1,5 10
188
2i,6g8
186
31,884
i85
2 2 ,069
i83
22,252
182
o,og6i
0,32
21,20^
2o3
21,410
201
2 1 ,6 1 1
■99
2I,8lO
198
22,008
195
22,2o3
194
22,397
'93
33,590
.190
22,780
190
2 2,g70
187
0,1024
0,33
21,869
210
22,079
207
22,386
2o5
22,491
2o4
22,695
202
22,897
200
23,097
198
23,2g5
197
23,493
195
23,687
ig4
0,1 o8g
0,34
22,532
2l5
22,747
214
22,961
212
23,173
210
23,383
208
23,591
206
23,797
204
24,001
203
24,204
301
24,4o5
'99
0,1 1 56
0,35
23,194
223
23,4i6
220
23,636
218
23,854
216
24,070
314
24,284
3l3
24,497
310
24,707
209
24,916
206
35,122
206
0,1225
o,36
23,856
339
24,o85
226
24,3 1 1
224
24,535
223
24,758
220
24,978
3l8
35,196
217
35,4i3
214
25,627
3l3
2 5,84o
211
0,1 296
0,37
24,519
2 35
24,754
232
24,986
33 1
25,217
223
25,445
226
25,671
335
25,896
222
26,118
221
26,339
319
26,558
216
0,1369
o,38
25,r8i
241
25,422
239
25,661
237
25,898
235
26,133
232
26,365
33o
36,5g5
229
36,834
226
37,o5o
225
27,275
323
0,1 444
0,39
25,844
247
2'5,og 1
245
26,336
243
26,57g
241
26,820
239
27,o5g
236
27,295
235
27,530
232
27,762
33l
27,993
228
0,1 521
o,4u
26,506
254
26,760
25l
27,011
249
37,360
247
27,507
245
27,752
243
27,995
34o
28,235
2 3g
28,474
336
28,710
235
0,1600
0,4 1
27,168
260
27,428
258
27,686
256
27,942
353
28,195
25l
28,446
248
28,694
247
28,941
244
29,185
343
29,428
240
0,1681
0,42
37,83o
267
28,097
264
38,36i
262
28,623
25g
28,883
257
2g,i3g
355
39,394
353
29,646
25l
'9.897
248
3o,i45
246
0,1764
0,43
28,493
273
28,^66
270
29,o36
368
2g,3o4
265
29,569
264
3g,833
360
3o,og3
2 5g
3o,352
356
3o,6o8
355
3o,863
252
0,1849
0,44
29,155
279
29,434
277
29,711
274
29,985
372
3o,257
269
30,526
367
30,793
264
3 1, 057
363
3 1, 320
260
3i,58o
258
o,ig36
0,4a
39,817
286
3o,io3
283
3o,386
280
3o,666
378
3o,g44
275
3i,2ig
273
31,492
371
3 1, 763
268
32,o3i
266
32,2g7
264
0,2025
o,5o
33,138
3i7
33,445
3i5
33,760
3ii
34,071
3og
34,38o
3o6
34,686
3o3
34,98g
3oi
35,390
298
35,588
296
35,884
293
o,25oo
0,55
36,438
349
36,787
346
37,i33
343
37,476
33g
37,81 5
337
38,i52
334
38,486
33 1
38,8i7
3Ï8
39.145
325
3g,470
323
o,3o25
0,60
39.747
38 1
40,128
377
4o,5o5
374
40,879
371
4i,25o
367
4i,6i7
365
4i,g82
36 1
43,343
358
42.701
355
43,o56
352
o,36oo
o,65
43.n55
4i3
43.468
409
43,877
4o5
44,282
4o2
44,684
398
45,082
3g5
45,477
3gi
45,868
388
46,256
384
46,640
383
0,4225
0,70
46,363
445
46,807
440
47,247
437
47,684
433
48,117
42g
48,546
425
48,g7i
43.
49,393
4i8
4g,8io
4i4
5o,224
4ii
o,4goo
0,75
40,668
476
5o,[44
473
5o,6i7
468
5r,o85
464
5i,54g
459
52,008
456
52,464
45 ■
52.9,5
448
53,363
AU
53,807
440
0,5625
0,80
52,972
5ou
53,481
5o4
53,g85
5oo
54,485
495
54,g8o
4go
55,470
486
55,g56
482
56,438
478
56,gi6
Ali
57,38g
470
o,64oo
o,85
56,276
54.
56,817
536
57,353
53o
57,883
526
58,4o9
521
58,93o
5^7
59,447
5l2
59,959
5o8
60,467
5o3
6o,g70
5oo
0,7225
0,90
59.578
573
6o,i5i
568
60,71g
563
61.281
557
6 1,838
552
62.390
547
62,937
542
63,479
538
64,017
534
64,55 1
538
0,8100
0,95
62,879
6o5
63,484
599
64,o83
594
64,677
588
65,365
583
65;848
578
66,436
573
66,999
567
67,566
564
68,i3o
558
o,go25
1,00
66,178
637
66,8 1 5
632
67,447
625
68,073
619
68,6gi
6i4
69,305
608
69,913
6o4
70,517
597
71,114
593
71,707
588
1 ,0000
13,5200
14,0450
14,5â00
15,1250
15,6800
16,2450
16,8200
17,4050
18,0000
18,6050
â
1 5 . (r -)- r")- or r^ -|- r "^ nearly. |
662
665
668
671
674
677
680
683
686
689
692
6g5
698
701
704
707
710
7i3
7^6
71g
66
67
67
67
67
68
68
68
69
69
6q
70
70
70
70
?■
71
1'
72
72
l32
i33
1 34
1 34
i35
1 35
1 36
1 37
■37
1 38
1 38
i3g
i4o
i4o
i4i
i4^
143
143
143
1 44
199
200
200
201
202
203
204
205
206
207
208
2oq
2og
210
211
312
3l3
214
2l5
216
265
266
267
268
270
271
272
273
374
276
277
278
279
280
282
283
284
285
386
288
33i
333
334
336
337
33q
340
342
343
345
346
348
34q
35 1
352
354
355
357
358
36o
397
399
4oi
4o3
4o4
406
408
4io
4l2
4i3
4i5
417
419
421
422
424
426
428
43o
43 1
463
466
468
470
472
474
476
478
480
482
484
487
48q
4qi
4q3
4q5
497
499
5oi
5o3
53o
532
534
537
53q
542
544
546
549
55i
554
556
558
56i
563
566
568
570
573
575
596
599
601
604
607
609
612
6i5
6,7
620
623
626
628
63 1
634
636
639
642
644
647
il9
TABLE II. — To find the time T"; the sum of the radii r-\-r", and the chord c being given.
Sum of the Radii r-\-r'
Chord
c.
OjOO
0,0 1
0,02
o,o3
0,04
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
0,1 3
o,i4
0,1 5
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
0,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
o,65
0,70
0,75
0,80
0,85
0,90
0,95
r,oo
6,20
Days |ilif.
O,0CO
0,724
3,619
2q
4,343
35
5,066
4>
5,790
46
b,5i4
52
7.237
58
7,961
64
8,685
70
Q,4o8
76
IO,l32
82
io,856
ii,58o
i2,3o3
1 3,037
i3,75i
14,474
15,198
15,932
16,645
17,369
18,093
18,816
19,539
20,263
20,987
21,710
22,434
23,i57
23,881
24,604
25,328
26,05 1
26,774
37,498
28,221
28,945
29,668
30,391
3i,ii5
3 1,838
33,56i
36,177
39,793
43,408
47,022
5o,635
54,247
57,859
61,470
65,079
68,688
73,295
116
132
127
1 34
139
146
i5i
157
i63
168
175
180
186
193
198
2o3
209
316
221
227
233
238
245
25o
256
262
29
320
349
378
408
437
466
495
525
554
584
19,2200
6,30
Days lilif.
0,000
0,730
1,459
2,i8y
3,918
3,648
4,377
5,107
5,836
6,566
7,295
8,035
8,755
9,484
JO,2l4
10,943
I 1,673
12,402
i3,i33
i3,86i
14,590
1 5,320
16,049
16,779
i7,5oS
i8,238
18,967
19,696
20,426
2i,i55
21,885
23,6t4
23,343
24,073
24,803
25,53i
26,260
26,990
37,719
28,448
29,177
39.906
3o,636
3 1, 365
32,094
33,823
36,468
4o.ii3
43,757
47,400
5 1,043
54,684
58,335
61,965
65, 604
69,343
72,879
5
13
1
33
29
35
4(
4-
53
58
63
eg
75
80
87
93
98
io3
no
116
121
127
i33
139
i44
i5o
1 56
161
168
173
179
i85
1 go
196
302
208
2l3
219
225
23l
237
243
2
2 54
260
289
3i7
346
375
4o4
434
462
491
520
549
578
19,8450
6,40
Days |dif.
0,000
0,735
1,471
2,206
3,941
3,677
4,4 1 3
5, 1 47
5,883
6,618
7,353
8,0!
8,824
9,559
io,3g4
1 i,o3(
1 1 ,765
i3,5oc.
i3,235
i3,97i
14,706
1 5,441
16,176
16,912
17,647
18,382
19,117
1 9,852
20,587
21,323
22,o58
22,793
33,528
24,263
34,gg8
25,733
36,468
27,203
27,938
28,673
29, 408
3o,i43
30,878
3i,6i3
32,348
33,o83
36,757
4o,43o
44, io3
47,775
5 1, 447
55,118
58,787
63,456
66,124
69,791
73,437
23
38
34
40
45
5i
57
63
68
74
80
8
9
97
io3
108
114
120
126
1 3
1 3
i43
149
1 55
161
166
171
177
i83
189
195
200
206
212
218
223
229
235
240
346
252
257
286
3i5
344
373
2q
459
488
5i7
545
574
20,4800
6,50
Days 111 if.
0,000
0,741
T,482
2,223
2,964
3,7o5
4.446
5,187
5,928
6,66g
7,4io
8,i5i
8,892
9,633
10,37.
11,11 5
11,856
12,597
1 3,338
14,079
i4,83o
i5,56i
i6,3o3
17,043
17,784
18,535
19,266
20,007
3o,748
21,489
22,229
22,970
23,711
24,453
25,193
25,933
26,674
37,4 1 5
28,1 56
39,637
30,378
3i,ii8
3i,85g
33,600
33,340
37,043
40.745
44,44
4S,i48
5 1, 848
55,547
59,246
62.944
66,64 1
70,336
74,o3i
86
g
g7
io3
108
114
120
125
i3i
1 36
143
1 48
1 53
i5q
164
171
176
182
187
193
199
205
210
216
22i
333
23g
344
2 5o
2 56
284
3i3
341
369
398
43
455
484
5l2
541
570
21,1250
6,60
Days |dif.
0,000
0,747
1,493
2,240
2,987
3,734
4,480
5,337
5,974
6,720
7,467
8,2i4
8,961
9,707
10,454
11,201
11-947
12,694
i3,44i
14,187
14,934
1 5,681
16,427 124
17,174 129
17,920 i36
17
22
28
S4
40
45
5i
57
63
67
74
79
84
90
96
101
107
18,667
19,414
20,160
20,90-
31,653
22,400
23,i46
23,893
24,639
25,386
26,132
26,879
27,625
28,372
29,118
29,864
3o,6i I
3 1, 357
32,10.3
32,85û
33,596
37,327
4i,o58
44,7
48,5i7
52,246
55,974
59,701
63,428
67,153
70,877
74,601
1 46
l52
i58
164
169
175
180
186
191
1 98
2o3
209
21
220
226
23l
237
243
254
282
3io
338
36
395
423
452
48o
5o8
537
565
21,7800
6,70
Days Idif.
0,000
0,752
I,5o5
3,357
3,009
3,762
4,5 1 4
5,267
6,019
6,771
7,534
8,376
9,028
9,781
10,533
11,385
12,037
1 2 ,790
1 3,542
14,294
i5,o47
1 5,799
i6,55i
i7,3o3
i8,o56
18,808
19,560
20,3t2
2i,o65
22,569
23,321
24,073
34,835
25,577
26,330
27,082
27.834
28,586
29,338
30,090
3o,842
31,594
32,346
33,098
33,85o
37,609
4 1, 368
45,126
48,884
52,64i
56,397
6o,î53
63, 008
67,661
7i>4i4
75,166
1 46
i5i
[56
162
168
174
179
i85
191
195
201
207
212
218
224
329
235
240
246
252
280
3o8
336
364
393
431
448
476
5o5
533
56i
22,4450
6,80
Days Id if.
0,000
o,758
i,5i6
2,274
3,o32
3,790
4,548
5,3o6
6,064
6,822
7,579
8,33
9,095
9,853
10,61 1
11,369
12.137
i2;885
1 3,643
i4,4oi
i5,i59
15,916
16,67
17,433
18,190
18,948
19,706
20,463
21,221
21,979
22,737
23,495
24-,253
25,010
25,768
26,535
27,283
28,041
28,798
29,556
3o,3i4
3 1 ,07 1
3i,82q
32,586
33,344
34,102
37 „
41,676
45,463
49,248
53,o33
56,8 1 8
60,601
64,384
68,166
71,947
75,737
28
33
3g
44
5o
56
61
67
72
78
83
89
94
100
io5
III
117
122
12(
1 33
1 39
1 44
i5o
i56
161
166
173
178
i83
[89
195
200
2o5
212
21-
333
238
333
339
245
25o
278
3o6
334
362
389
417
445
473
5oi
529
557
23,1200
6,90
Days [dif.
0,000
0,764
1,527
2,291
3,o54
3,818
4,58 1
5,345
6
6,872
7,635
8,3y8
9,163
9,925
10,1'"
11,453
I3,3l6
12,979
i3,743
i4,5o6
15,270
i6,o33
16,796
17,560
18,323
19,087
i9,85o
20,61 3
21,377
22,l4o
22,903
23,667
24,43o
25,ig3
25,95
26,721
27,483
28,246
29,0 n
39,773
3o,536
31,399
33,062
32.825
33,589
34,352
38,167
41,982
45,796
49,610
53,422
57,235
61,046
64.857
68,667
72,476
76,284
27
33
38
44
4g
55
61
66
72
77
83
88
94
99
io5
no
116
122
137
1 33
l37
1 43
i4g
i54
160
166
171
176
183
187
193
199
3o4
209
2l5
221
226
232
2 38
242
248
■6
3o3
33i
358
387
4i4
442
469
497
"25
552
23,8050
7,00
Days |dif.
0,76c
i,53b
2,3o'
3,076
3,845
4,6 1 4
5,383
6,i52
6,921
7,690
8,459
9,338
9,997
10,766
11,535
1 2 ,3o4
13,073
1 3,842
1 4,61 1
i5,38o
16,149
1 6.9 1 8
17^687
1 8,456
19,224
19-993
20,762
2 1, 53 1
22,3oo
23,069
23,838
24,606
25,375
26,144
26,913
27,682
28,450
29,219
39,988
30,757
3i,535
33,2g4
33,o63
33,83 1
34,600
38,443
42,285
46,127
49,968
53,809
93
98
io4
109
ii5
120
126
i3i
1 37
143
1 48
1 53
i5g
164
169
176
181
186
191
ig7
2o3
208
2l3
219
225
23o
235
241
246
274
3oi
329
356
383
57,649411
61,488439
65,326466
69,164494
73,001 521
76,8361549
24,5000
7,10
Days [dif.
0,000
0,774
,.549
2.323
3,098
3.872
4,647
5,421
6,196
6,970
7,745
8,519
9,294
10,068
10,843
11,617
12,393
1 3, 166
1 3,940
14,71 5
i5,4 _
16,364
i7,o38
I7,8i3
18,587
ig,36i
20,1 36
30,gT0
21,684
22,45g
23,233
24,007
24, ""82
25,556
26,330
27,104
27,879
28,653
29,427
3o,20I
30,976
3i,75o
32,524
33,398
34,072
34,846
38,717
42,586
46,456
5o,334
54,192
58,o6o
6i,g37
65,7g3
6g,658
73,533
77,385
6
11
■7
22
28
33
38
43
49
54
60
65
7'
76
83
87
92
98
io3
109
1 1
120
125
i3o
i36
i4i
1 47
i53
i57
1 63
169
174
179
i85
191
195
201
20-
217
223
234
239
245
273
3oo
326
354
38 1
4o8
435
463
490
5l7
545
25.2050
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
o,oo36
0,0049
0,0064
0,0081
0,0100
0,0121
o,oi44
0,0169
0,0196
0,0225
o,o256
0,0289
o,o334
o,o36i
o,o4oo
o,o44i
o,o484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,084 1
0,0900
o,og6i
0,1024
o,io8g
0,1 156
0,1225
0,1396
0,1369
0,1 444
0,l53I
0,1600
o, 1 68 1
o, 1 764
0,1 84g
o,ig36
O,3035
o,25oo
o,3o35
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
( Î- 4- r")- or
+
nearly.
723
726
729
732
735
733
741
744
747
75o
753
756
759
762
765
768
771
774
777
72
73
73
73
74
74
74
74
75
75
75
76
76
76
77
77
77
77
78
i45
i45
i46
i46
l47
1 48
i48
1 49
1 49
i5o
i5i
i5i
l52
1 52
I53
1 54
1 54
1 55
i55
217
218
219
220
221
221
222
223
224
225
226
227
228
229
230
23o
23l
232
233
289
290
292
293
294
295
296
298
299
3oo
3oi
3o2
3o4
3o5
3o6
307
3o8
3io
3ii
362
363
365
356
368
35o
.371
372
374
375
377
378
3So
38 1
383
384
386
387
38q
434
436
437
439
441
443
445
446
448
45o
452
454
455
457
459
461
463
464
466
5o6
5o8
5io
5l3
5i5
517
5 10
521
523
525
527
52q
5Ji
533
536
538
540
542
544
578
58t
583
586
588
590
5q3
5q5
5qS
600
603
6o5
607
610
bl2
614
6.7
619
622
65 1
653
656
659
652
664
667
670
672
675
678
b8o
68J
686
689
691
694
697
6g9
TABLE
II.
— To fi
ml tlie tiiiifc
T
the sum of (lie railii r + )■'',
and the cliord i
being given.
Sum of the Ita.lii r-f-r". |
Cliord
c.
7,-20
7,30
Days |dif.
7,40
7,50
7,60
7,7
0
7,80
Days Idif.
7,90
8,00
8,10
Days |(lil".
Days |(lif.
Days Idif
Days Idif.
Days
dir:
Days Idif.
Days Idif.
Days |dif.
0,00
0,0(I0
0,000
0,000
0,000
0,000
0.f)00
0,000
0,0O(j
0,000
0,000
o,cooo
0,01
0,780
5
0,785
6
0,701
5
0,-96
5
0,801
6
oi8o-
5
0,812
5
0,817
5
0,822
5
0,827
5
0,0001
0,02
i,56o
1 1
.,571
10
i,58i
11
1,592
11
i,6o3
i(ï
i,6i3
1 1
1,()34
10
1.634
10
1,644
10
1,654
11
o,ooo4
o,o3
2,34o
16
2,356
16
2,372
16
2,388
16
2,4o4
16
2,420
i5
2,435
i(i
2,45i
i5
2,466
16
2,482
i5
0,0009
o,o4
3,120
21
3,i4i
22
3,i63
21
3,184
21
3,2o5
21
3,226
21
3,247
21
3,268
20
3,288
21
3,309
20
0,0016
o,o5
3,900
27
3,927
26
3,953
27
3,980
27
4,007
26
4,o33
26
4,059
26
4,o85
26
4,111
25
4,1 36
26
0,0025
0,06
4,680
32
4,712
32
4,744
32
4,776
32
4,808
3i
4,839
32
4,871
3i
4,902
3i
4,933
3o
4,963
3i
o,oo36
0,07
5,459
38
5,497
38
5,535
37
5,572
37
5,609
37
5,646
36
5,682
37
5.719
36
5,755
36
5,791
35
0,0049
0,08
6,239
44
6,283
42
6,325
43
6,368
42
6,410
42
6,452
42
6,494
42
6,536
41
6,577
4i
6,618
4i
0,0064
o,og
7,019
49
7,068
48
7,116
48
7,164
48
7,212
47
7.259
47
7,3o6
47
7,353
46
7,399
46
7,445
46
0,0081
0,10
".799
54
7,853
54
7.907
53
7,960
53
8,0 1 3
52
8.065
53
8,118
52
8,170
5i
8,221
5i
8,272
5i
0,0100
0,1 1
8,579
5o
8,638
59
8,(197
59
8,756
58
8,81 4
58
8,872
57
8,929
58
8,987
56
9,043
57
g,ioo
56
0,0121
0,12
9,359
65
9.424
64
9,488
64
9,552
63
9,61 5
64
9, •'79
62
9.741
62
9,8o3
63
9,865
62
9.927
61
0,01 44
o,i3
10,139
70
10,209
70
10,279
69
10,348
69
io,4i7
68
io,485
68
10,553
67
10,620
67
10,687
67
10,754
66
0,0169
o,i4
10,919
75
10,994
75
11,069
75
ii,i44
74
11,218
74
11,292
73
11,365
72
11.437
72
ii,5og
72
ii,58i
71
0,01 96
0,1 5
11,699
81
11.780
80
11,860
80
1 1 ,940
79
12,019
79
1 2 ,098
78
12,176
78
12,254
78
12,332
76
12,408
77
0,0225
0,16
12,4-9
86
12,565
86
1 2,65 1
85
12,736
85
12.821
84
1 2 ,905
83
1 2, 988
83
13,071
83
i3;i54
82
i3,236
81
0,02 56
0,17
1 3,208
92
i3,35o
9'
i3,44i
9'
i3,532
90
13,622
89
l3,7Il
89
1 3,800
88
1 3,888
88
■3,976
87
i4,o63
86
0,0289
0,18
i4.o38
q8
i4,i36
96
l4.232
96
14,328
95
i4,423
95
i4i5i8
94
i4,6i2
93
i4,7o5
93
14,798
92
i4,8go
92
o,o324
0,19
14,818
io3
14,921
102
1 5,023
101
i5,i24
100
1 5,224
100
1 5,324
99
i5,423
99
l5,522
98
15,620
97
15,717
97
o,o36i
0,20
15,598
108
15,706
107
i5,8i3
107
15.920
106
16,026
io5
i6,i3i
io4
16,235
io4
16,339
io3
16.443
102
16,544
102
o,o4oo
0,21
16,3-8
ii3
16.491
ii3
16,604
112
16,716
111
16,827
no
16,937
£10
17,047
log
17,1 56
108
17,264
iû8
17,372
106
o,o44 1
0,22
1 7, 1 58
118
17,276
118
17,394
118
I7,5i3
116
17,628
116
17,744
ii4
17,858
ii5
17-973
ii3
18,086
ii3
18,199
112
o,o484
0,23
I -.938
124
18,062
123
i8,i85
123
i8,3o8
121
18,429
121
i8,55o
120
18.670
'■9
18,789
119
l8,go8
118
1 9,026
117
0,0529
0,24
18.71:
i3o
18,847
129
18,976
127
ig,io3
127
1 9,23o
127
19,357
125
19,482
124
ig,6o6
124
19,730
123
19,853
122
0,0576
0,25
19.497
i35
19,632
i34
19,766
i33
19,899
i33
20,o32
i3i
20,i63
i3o
20,393
i3o
20,423
12g
20,552
128
20,680
127
0,0625
0,26
20,277
i4o
20,417
t4o
20,557
1 38
20,695
1 38
2o,833
1 36
20,96g
1 36
2I,105
i35
21,240
i34
21,374
1 33
2I,507
i33
0,0676
0,27
2l,o57
i46
2!,203
i44
21,347
144
2M91
i43
21,634
142
21,776
i4i
21,917
i4o
2 2,o57
1 39
22,196
i38
22,334
i38
0,072g
0,28
21,837
i5i
21,988
i5o
22,l38
i4g
22,287
148
22.435
147
22,582
i46
22,728
i46
22,874
1 44
23,018
i44
23,162
142
0,0784
0,29
22,616
1 57
22,773
1 55
22,928
1 55
23,o83
1 53
23;236
i53
23,389
IDl
23,540
i5i
23,691
149
23,840
i4g
23,989
i47
0,084 1
o,3o
23,396
162
23,558
161
23,719
i6o
23,879
1 58
24,037
i58
24,195
1 57
24,352
i55
24,507
i55
24,662
i54
34,816
l52
0,0900
0,3 1
24,176
167
24,343
166
24,509
166
24,675
164
24,889
162
25,001
162
25,i63
161
25,324
16c
25,484
159
25,643
i58
0,0961
0,32
24,956
172
25,128
172
25,3oo
170
25,470
170
25.640
168
25.808
167
25.975
166
26,141
i65
26,306
164
26,470
i63
0,1024
0,33
25-35
179
25.914
176
26,090
176
26,266
175
26,441
173
26,614
173
26:787
171
26,958
170
27,128
169
27,297
168
0,1089
0,34
26,5i5
184
26,699
182
26,881
181
27,062
180
27,242
179
27,421
177
27,598
177
27,775
175
27,950
174
28,124
173
0,11 56
0,35
27,295
189
27,484
187
27,671
187
27,858
i85
28,043
184
28,227
i83
28,4i<;'
181
28,5gi
181
28,772
179
28,95]
.78
0,1225
o,36
28,074
195
28,269
ig3
28,462
192
28,654
190
28,844
189
2g,o33
186
29,221
187
29,408
186
29.594
184
29,778
i83
0,1296
0,37
28,854
200
29.054
198
29,252
197
2gMg
196
29,645
195
29.840
ig3
3o,o33
192
3o,225
'9'
3o,4i6
189
3o,6o5
i8g
0,1369
o,38
29,634
205
29,839
204
3o.o43
202
30,245
201
3o,446
200
3b;646
198
3o,844
198
3 1, 042
195
3 1,237
ig5
3 1,432
ig4
0,1444
0,39
3o,4t3
211
30,624
209
3o;833
208
3i,o4i
206
3 1, 247
205
3i,452
204
3 1, 656
202
3 1, 858
201
32,o5g
200
32,259
199
0,l52I
0,40
31,193
216
3 1 ,409
2l5
31,624
2l3
3 1, 837
211
32,o48
211
32,259
208
33,467
208
32,675
2g6
33,881
2o5
33,086
204
0,1600
o,4i
31.973
221
32,ig4
220
32,4i4
218
32,632
217
32.840
216
33,o65
214
33,279
2l3
33,492
211
33,7o3
210
33,913
3t)g
0,1681
0,42
32,752
227
32,979
225
33,204
'>o4
33,428
222
33,650
221
33,871
219
34,090
218
34,3o8
217
34,525
2l5
34,740
2l4
0, 1 764
0,43
33,532
232
33,764
23l
33,995
229
34,224
227
34,451
226
34,677
225
34,902
223
35,125
222
35,347
220
35,567
2iq
0,1849
0,44
34,3 11
238
34,549
236
34,785
234
35,019
233
35,252
232
35,484
22g
35,7i3
229
35,942
227
36,i6g
225
36,394
224
0,1936
0,45
35,091
243
35,334
241
35,575
240
35,8i5
238
36,o53
237
36,390
235
36,525
233
36,758
232
36,990
23l
37,221
229
0,3035
o,5o
38,989
270
39,259
268
39,527
266
39,793
265
4o,o58
263
4o,32i
261
40,582
259
4o,84i
258
41,099
256
41,355
255
0,2 5oo
0,55
42,886
297
43,183
295
43,478
293
43,77.
291
44,062
289
44,35 1
287
44,638
28(3
44,924
283
45,207
282
45,489
380
o,3o25
0,60
46,782
324
47,106
332
47,428
320
47,748
3i7
48,o65
3i6
48,38 1
3i3
48,6g4
3l2
49,006
309
4g,3i5
3o8
4g,623
3o6
o,36oo
o,65
50,678
35i
51,029
349
51,378
346
51,724
345
52,069
34!
52.410
340
52,750
337
53,087
336
53,423
333
53.756
33i
0,4225
0,70
54,573
379
54,952
375
55,327
373
55,700
371
56,071
368
56;439
366
56,8o5
364
57,169
36i
57,53o
359
57,88g
356
0,4900
0,75
58,468
4o5
58,873
4o3
59,276
400
59,676
397
60,073
395
60,468
392
60,860
389
61,249
387
6 1,636
385
63,021
382
o,5625
0,80
62,362
432
62,794
43o
63,224
427
63,65 1
423
64,074
421
64,495
418
64 91 3
4i6
65,32y
4i3
65,742
4io
66,1 52
408
o,64oo
0,85
66,255
460
66,7 1 5
456
67,171
454
67,625
45o
68,075
447
68,5"22
445
68.967
441
69,408
439
69,847
436
70,283
433
0,7325
0,90
70,148
486
70,634
484
71,118
480
71,598
477
72,075
473
72,548
471
73,019
468
73,487
464
73,951
462
74,4 1 3
45g
0,8100
0,95
74,039
5i4
74,553
5io
75,o63
5o7
75,570
5o4
76,074
5oo
76,574
497
77,071
4g3
77,564
491
78,055
487
78,542
485
0,9025
1,00
77,930
54 1
78,471
537
79,008
534
79,542
53o
80,072
527
80,599
523
81,122
519
8i,64i
5i7
83,i58
5i3
82,671
5io
1 ,0000
25,9200l
26,64501
27,38001
28,12501
28,88001
29,64501
30,42
00
31,20501
32,0000|
32,80501
i . ( r -f- r " J ^ or f^ -f* ** '^ nearly. |
7-9
-82
7S5
7S8
I
78
78
79
79
2
i56
1 56
1D7
1 58
3
234
235
236
236
4
3l2
3i3
3i4
3i5
5
390
391
393
3q4
6
467
469
471
473
7
545
547
55o
552
8
623
626
628
63o
9
701
704
707
709
79'
794
797
800
£o3
806
8cg
812
8i5
818
821
834
79
79
80
80
80
81
81
81
82
82
82
82
1 58
i5q
i59
160
161
161
162
162
1 63
164
164
1 65
237
23ii
339
340
241
242
s43
244
245
245
246
247
3i6
3i8
3ip
320
321
322
324
325
326
327
328
33o
396
397
3qc,
400
4o2
4o3
4o5
4c6
4o8
409
4ii
4l2
475
4-6
478
480
482
484
485
487
489
4qi
4q3
494
554
556
558
56o
562
564
566
568
571
573
575
577
633
635
638
640
642
645
647
65o
652
654
657
65q
712
7i5
717
720
723
725
728
73i
734
736
739
742
827
83o
83
83
i65
166
248
249
33i
332
4i4
4i5
4q6
498
579
58 1
662
664
744
747
TABLE II. — To find the time T; the sum of the radii r-\-r'\ and the chord c being given.
Sum of, the Radii r-\-r".
Chord
C.
0,00
0,01
0,02
o,o3
o,o4
o,o5
0,06
0,07
0,08
0,09
0,10
0,11
0,12
0,1 3
o,i4
o,i5
0,16
0,17
o,iS
'•9
0,30
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
o,3o
0,3 1
0,32
0,33
0,34
0,35
o,36
0,37
o,38
0,39
o,4o
0,4 1
0,42
0,43
0,44
0,45
o,5o
0,55
0,60
0,65
0,70
0,75
0,80
o,85
0,90
0,95
8,20
Days |dir.
0,000
0,832
1,665
2>497
3,329
4,162
4,994
5,826
6,659
7
8,323
9, 1 56
9,988
10,820
11,652
12,485
1 3,3 1 7
i4,i49
14,983
t5,8i4
1 6,646
17,478
i8,3ii
i9''43
'9.975
20,807
2 1 ,640
22,472
23,3o4
24,i36
24,968
25,801
26,633
27,465
28,297
29,129
29,961
30,794
31,626
32,458
33,290
34,122
34,954
35,786
36,6 1 8
37,450
4 1 ,6 [ o
45,769
49,929
54,087
58,245
62,4o3
66,56o
70,716
74,873
79,037
83,i8i
i5
35
3o
36
40
45
5i
55
61
66
71
76
81
86
91
96
lOI
107
I II
116
122
127
i3i
1 36
143
147
I 53
1 57
162
167
172
177
i83
.87
192
197
202
307
2l3
218
233
328
253
279
3o3
339
355
38o
4o5
43
456
4i
507
8,30
Diiya Idir
0,000
0,837
1,675
2,5l2
3,35o
4,187
5,024
5,862
6,699
7,536
8,374
9,211
10,049
10,886
11,723
i2,56i
13,398
i4,235
15,073
15,910
16,747
17,585
18,432
19,259
20,097
30.934
21,771
33,608
33,446
24,283
25,120
2 5,958
26,795
27,633
28,469
29,306
3o, 1 44
30,981
3i,8i8
32.655
33,492
34,339
35,167
36,oo4
36,84 1
37,678
4 1, 863
46,o48
5o,333
54,4i6
58,6oo
62,783
66,965
71,147
75,328
7q.5o8
83,688
i5
3o
35
4o
46
5o
56
6<
65
71
75
80
86
90
9G
101
io5
11 1
116
130
136
i3i
i36
i4i
1 46
i5i
■ 55
161
166
171
176
33,6200
19'
196
202
207
211
316
221
326
252
277
3o2
327
352
377
40 3
43
453
479
5o4
8,40
Duys Idif.
0,000
0,842
1,685
2,527
3,370
4,212
5,o54
5,897
6,739
7,583
8,434
9,367
10,109
10,951
1 1 ,794
13,636
13,478
i4,33i
i5,i63
16,006
16,848
17,690
18,533
19,375
20,217
2 1 ,060
21,902
3 2,744
23,587
24,429
25,271
26,1 13
26,956
27,798
28,640
39,483
3o,325
31,167
33,009
32,85i
33,694
34,536
35,378
36, 220
37,063
37,904
43,ii5
46,335
5o,534
54,743
5o
55
60
65
70
75
80
85
90
95
100
io5
110
ii5
120
125
i3o
i35
i4o
145
i5o
i55
160
i65
170
176
180
i85
19(
195
200
205
210
2l5
220
225
25o
375
3oo
326
34,4450
58,952 35o 59,302
63, 160
67,368
71,575
75,781
79-987
84,192
8,50
"Days |dil'.
376
4oo
435
45i
476
5oi
35,2800
0,000
0,847
1,695
2,542
3,390
4,237
5,û84
5,932
6,779
7,627
8,474
9,322
10,169
11,016
11,864
13,711
1 3,558
i4,4o6
1 5,253
16,101
16,948
17,795
18,643
19,490
20,337
2i,i85
3 2,o33
22,879
23,727
24,574
25,421
26,268
27,1 16
27,963
28,810
29,658
3o,5o5
3i,352
32,199
33,046
33,894
34,741
35,588
36,435
37,282
38,129
42,365
46,600
5o,834
55,069
63,536
67,768
73,000
76,232
8o,463
84,693
i5
20
25
3o
35
40
5o
54
60
65
69
75
80
84
9"
94
99
io5
109
114
120
134
129
35
139
i44
i49
i55
159
164
169
174
179
184
189
194
198
3o4
209
2l4
219
224
2
373
299
323
349
373
398
423
448
473
4q8
8,60
Days |dif.
0,000
0,853
i,7o5
3,557
3,4io
4,363
5,114
5,967
6,819
7,671
8,524
9,376
10,329
1 1 ,08 1
1 1,933
12,786
1 3,638
14,490
15,343
16,195
17,047
17,900
18,752
19,604
20,457
21,309
22,161
23,014
2 3,866
34,71
35,570
26,433
27,275
28,127
28,979
39,833
3o,684
3i,536
32,388
33,240
34,092
34,945
35,79
36,649
37,501
38,353
42,61 3
46,873
5i,i33
55,393
59,65i
36,1250
94
99
io3
109
114
118
134
'29
33
i3S
143
149
i53
1 58
i63
173
178
i83
188
193
1Ç
202
207
313
217
223
248
272
297
331
346
63,90g 371 64,280
68,166
72,423
76,680
80,936
85,191
8,70
Days Irtif.
0,000
0,857
1,71 5
2,572
3,429
4,287
5,144
6,001
6,859
7,716
8,573
9,43 1
10,288
11,145
13,002
T 2,860
13,717
1 4,574
i5,432
16,289
17,146
i8,oo3
18,861
19,718
20,575
21,433
22,290
23,147
24.004
24,861
25,719
26,576
27,433
28,290
29,147
3o,oo4
30,862
31,719
32,576
33,433
34,290
35,147
36 ,004
36,86i
37,718
38,576
42,861
47,145
5i,43o
55,7i3
59,997
396
421
445
470
495
36,9800
562
72,844
77,125
81,406
85,686
127
1 33
i38
i43
1 47
I 52
'57
162
167
173
177
1S2
it
192
'97
202
207
212
217
231
245
371
295
320
369
394
418
443
467
492
8,80
Days Idif.
0,000
0,862
1,724
2,587
3.449
37,8450
4,3ii
5,173
6,o36
6,898
7,760
8,622
9,485
10,347
1 1,209
12,071
12,933
1 3,796
1 4,658
1 5,520
1 6,38 2
17,244
18,107
18,969
i9,83i
2o,6g3
21,555
22,417
33,280
24,143
25,oo4
25,866
26,728
27,590
28,452
29,314
30,176
3 1, 039
3 1 ,90 1
32,763
33,625
34.487
35,349
36,2 1 1
37,073
37,935
38,797
43,106
47,416
5i,725
56,o33
60,341
64,649
68,956
73,262
77,568
81,873
86,178
38,7200
25
3ù
34
39
AA
49
53
58
64
69
74
78
83
88
93
98
102
107
112
117
122
128
l32
'37
142
■47
I 52
'57
162
167
172
175
180
i85
190
195
200
2o5
310
2l5
220
245
268
293
3'r
342
366
391
4i6
440
465
,90
Days
0,000
0,867
1,734
2,601
3,469
4,336
5,2o3
6,070
6,937
7,804
8,671
9,538
io,4o5
1 1,273
12,l4o
13,007
13,874
'4,74'
1 5,608
16,475
17,342
18,209
19,076
19,943
20,810
21,677
22,545
23,4 12
24,279
25,146
26,013
26,880
27,747
28,614
29,481
3o,348
3 1,2 1 4
32,081
32,948
33,81 5
34,683
35,549
36,4' 6
37,283
38,1 5o
39,017
43,35i
47,684
52,018
56,35 1
6o,683
65,oi5
69,347
73,678
78,000
82,338
86,667
5
10
i5
'9
24
29
34
39
AÂ
49
54
59
63
68
73
78
83
87
92
97
102
107
112
117
122
126
i3i
1 36
i4o
145
i5o
i55
160
i65
170
175
180
i85
190
'95
199
204
209
2l4
21
343
368
391
3i6
341
36:
389
4i3
438
462
487
39,6050
9,00 9,10
Days |dir.
Days
0,000
0,000
0,873
5
0,877
1,744
10
1,754
3,616
14
3,63o
3,488
19
3,5o7
4,36o
24
4,384
5,333
29
5,361
6,104
M
6,1 38
6,976
3q
7,01 5
7,848
Ai
7,891
8,730
48
8,768
9,592
53
9,645
10,464
58
10,522
11,336
63
1 1 ,399
12,208
67
12,275
1 3,080
72
i3,i52
13,952
77
14,039
14,824
82
14,906
15,695
87
15,782
16,567
92
16,659
17,439
97
17,536
i8.3i[
102
i8,4i3
19,183
lOb
19,289
2o,o55
1 11
20, 1 66
20,927
116
2 1 ,043
21,799
121
2 1 ,920
22,671
125
22,796
23,543
i3o
23,673
24,41 5
i35
24,55o
25,286
i4i
25,427
36,1 58
145
26,3o3
27,o3o
i5o
27,180
27,902
i55
28,057
28,774
159
28,933
29,646
104
29,810
3o,5i8
i6q
30,687
3 1,389
174
3 1,563
32,261
'79
32,44o
33,i33
1 84
33,3 17
34,oo5
188
34,193
34,877
iq3
35,070
35,748
,p8
35,946
36,620
20 J
36,823
37,492
308
37,700
38,364
212
38,576
39.235
218
39.453
43,594
341
•43.835
47,952
2fab
48,218
52,309
390
52,5g9
56,667
3i4
56,981
61,024
338
61,362
65,38o
363
65,743
69,736
^87
70,123
74,091
4.1
74,5o2
78,446
435
78,881
82,800
460
83,260
87,154
483
87,637
40,5(
)00
41,40.50
0,0000
0,0001
0,0004
0,0009
0,0016
0,0025
o,oo36
0,0049
o,où64
0,008 1
0,0100
0,0121
0,01 44
0,0169
0,0196
0,0225
0,0256
0,0289
o,o324
o,o36i
o,o4oo
o,o44 1
o,o484
0,0529
0,0576
0,0625
0,0676
0,0729
0,0784
0,0841
0,0900
0,0961
0,1024
0,10
0,11 56
0,1225
0,1296
0,1369
0,1444
0,l52I
0,1600
0,1681
0,1764
0,1849
0,1936
0,2025
o,25oo
o,3o25
o,36oo
0,4225
0,4900
0,5625
o,64oo
0,7225
0,8100
0,9025
1 ,0000
\
■ (»■ +
r" )^ 0
r= +
" - no;t
rlv.
832
835
838
84'
844
847
85o
853
856
859
S62
865
868
871
874
877
83
84
84
84
84
85
85
85
86
86
86
87
87
87
87
88
166
167
168
168
i6q
i6q
170
171
171
172
172
173
174
174
175
175
25o
25l
25l
252
253
254
255
256
257
258
2 5q
260
260
261
262
263
333
334
335
336
333
339
34o
34 1
342
344
345
346
347
348
35o
35i
416
4i8
4'9
421
422
424
425
427
428
43o
43i
Aii
434
436
437
fî
4q9
5oi
5o3
5o5
5o6
5o8
5io
5l2
5i4
5i5
5l7
519
521
523
524
526
582
585
587
589
5qi
5q3
5q5
5q7
5qq
601
60 3
606
608
610
612
6i4
666
668
670
673
675
678
6S0
63 2
685
687
6go
6q2
694
697
699
702
749
752
754
757
760
762
765
768
770
773
776
779
781
784
787
789
TABLE III.
This fable gives llie true anomaly U of a comet, moving in a parabolic orbit, whose perihelion distance is equal to
the ine.in ili<taiice of the sun from tlie earth or uniti/ ; tlic time from the perihelion being /' days. It was computed
by Uurckhardt, by means of the formula in book ii. § 23, Mécanique Celeste, [693 &.C.], namely,
<' = 27''''-",4038 . . . X^Stang. àt/+tang.3 )^u\.
If the perihelion distance be -D, and the time from the perihelion t days, we must put< = i»3't' [693a]. If [7 be
given, we must iind, in this table, the corresponding value of log. t', and then the value of t from the formula,
log. / = log. r+f log. I>.
But if t he given, we must first find
log. r = log. t — f log. D ;
and then from this table the value of U, corresponding to this value of log. ('.
When t' is less than 5 days, the différences of log. (' vary so rapidly, that it is found convenient to vary the form of this
part of the table. This is done by two ditierent methods ; the one proposed by Burckhardt, the other by Carlini ; by
means of the fust six columns of the first page of Table III. Burckhardt's method consists in finding (', from log. (' ;
and then, with the argument t', we obtain the corresponding value of U, as in the first three columns of the table. In
the next three columns, which contain the table of Carlini, the argument is log. (', as in all the rest of the table, and the
corresponding number is log. - , or log. U — log. t' ; U being expressed in sexagesimal minutes, and t', in days. This
method of Carlini is very convenient, in the case which most frequently occurs; namely, where t is given to find U ;
for we have,
log. V = log. I — I- log. D ;
log. U'ln minutes = log. f -(-tab. number corresponding to log. t.
In the determination of the cimslant factor 97 ^'',4038, in the above value of (', we have neglected, as in [G92'], the
mass of the earth in comparison with that of the sun ; as is usually done in computing tables of this kind. This omission
may be rectified, by adding 0,0000006 to the argument log. (' in the table ; or by subtracting 0,0000006 from the logarithm
oit, in finding the log. t'.
Tables of this kind have been given by several authors, as Halley, La Caille, Zach, Pingre, &c. ; but above all, by
Delambre, who improved and extended this table very much, giving the values of U, corresponding to the argument I',
taken at convenient intervals from ^'^0 to t'^ 200,000 days. Burckhardt made an important improvement in Table
III. ; by taking for the argument log. «', ivhich is given by a previous calculation, and by this means he saves the labor
of finding t' from log. t'.
Barker published a general table of the paraboUc motion of a comet, in which the argument is the true anomaly U,
taken at intervals of 5"' ; the corresponding numbers are what he calls the logarithms of the mean motion represented by
log. mean motion = log. t' — 0,039871(j,
and the numbers in Barker's table may be deduced from those of Burckhardt's in Table III., by putting
Barker's log. = Burckhardt's log. (Table III.) — 0,0398710 ;
so that Table III., may be considered as an improvement on Barker's table, and may be used for the same purposes ; the
arguments, however, are in an inverted order. The argument in Barker's table being the true anomaly U ; and in
Burckhardt's table, the argument is the logarithm of the time t'.
EXAMPLES OF THE USE OF TABLE III.
EXAMPLE I.
Given the log. of perihelion distance, or log. D. =9,76565oo
Time from perihelion ' =49 ^^^,26281
To find the true anomaly U.
f X log. D, 9,6484750
log. t, 1, 69243 10
log. f:=\og. t— f log. D, 2,0439560
In Table III. 90'' 16'" 29^,3 corresponds to 2,o43
5 01 ,9 = 3i 5',8 X 0,9561
f=9o 21 3 1 ,2
EXA3IPI.E III.
Given the log. of perihelion distance, or log. D = o,t 35oooo
Time from perihelion, i = 2 days.
To find the true anomaly U, by Burckhardt's method.
I- X log. -D,
log. t,
log. (' = log. t — J log. D,
t' = i'^''\2gS-633
In Table III. x "^,2 corresponds to
Tab. diif. 5oI^6x 0,987633 = 493^4
Sum is
0,1875000
o,3oio3oo
= o,ii353oo
1 4o 20 ,6
= 8 i5,4
U=i 48 36 ,0
EXAMPLE II.
Given the log. of perihelion distance, or log. i) = 9,76565oo
True anomaly, C7'==9o 2i"'3i,2
To find (.
Log. 2,043 Table III., corresponds to 90 16 29 ,3
Difference, 3oi',9^ 5 01 ,9
Tabular difference, 3i5 ,8 : 3oi ,9 :: 0,001 : 0,0009560
Hence log. t', =2,0439560
Add I log. D, = 9,6484750
Sum is log. ( = log. 49 ^^',2528
; 1,6924310
EXAMPLE IV.
Given D, t' as in example iii., to find U, by Carlini's
method.
Table III., Carlini, log. If— log. t',
Sura is log. [/■= 108 ,600 = 1 48 36 ,0
o,ii353o
1,922298
2,o35828
a20
To find the true anomaly U, corresponding
perihelion distance is the same as the mean di
TABLE III.
to the time t' from the perihelion in days, in
stance of the sun from the earth.
a parabolic orbit, whose
Days
t'.
days
0,0
0,1
0,2
0,3
0,5
0,6
0.7
0,8
0,9
r,o
2,2
2,3
2,4
2,5
2,6
2.7
2,8
2,9
3,0
3,1
3,2
3,3
3,4
3,5
3,6
3,7
3,8
3,9
4,0
4,1
4,2
AÀ
4,5
4,6
4,7
4,8
4,9
5,0
5,r
5,2
5,3
5,4
5,5
5,6
5,7
5,8
5,9
6,0
True
Anom. U.
0,00,00,0
0,08,21,8
0,16,43,6
o,25,o5,4
0,33,27,2
0,41,48,9
o,5o,io,7
0,58,32,4
I ,o6,54;
I,l5,l5;
1,23,37,4
1,31,59,0
1,40,20,6
1,48,42,2
1,57,03,7
2,o5,25,2
2, 1 3,46,6
2,22,08,0
2,30,29,4
2,38,5o,7
2,47,11.9
2,55,33,1
3,o3,54,2
3,!2,l5,2
3,20,36,2
3,28,57,1
3,37,17,9
3,45,38,6
3,53,39,3
4,02,19,9
4,10,40,4
4,19,00,8
4,27,21,1
4,35,4i,3
4,44,01,4
4,52,21,5
5,oo,4i,4
5,09,01,2
5,17,20,9
5,25,40,6
5,34,00,1
5,42,19,5
5,5o,38,7
5,58,57,8
6,07,16,8
6,15,35,7
6,23,54,4
6,32,i3,o
6,4o,3i,5
6,48,49.8
6,57,08,0
7,05,26,0
7,13,43,9
7,22,01,7
7,30,19,3
7.38,36,7
7,46,53,9
7,55,11,0
8,03,37,9
8,11,44,7
8,20,01,3
Diff.
5oi,8
5oi,8
5oi,8
5oi,8
5oi,7
5oi,8
5oi,7
5oi,7
5ox,7
5oi,6
5oi,6
5oi,6
5oi,6
5oi,5
5oi,5
5oi,4
5oi,4
5oi,4
5oi,3
5oi,2
5oi,2
5oi,i
5oi,o
5oi,o
5oo,9
5oo,8
5oo,7
5oo,7
5oo,6
5oo,5
5oo,4
5oo,3
5oo,2
5oo,i
5oo,i
499'9
499,8
499 >7
499.7
499.5
499.4
499.2
499.1
499,0
498,9
498,7
498,6
498,5
498,3
498,0
497.9
497.8
497,6
497.4
497.2
497.1
496.9
496,8
496,6
(lay?
9,00
9,10
9.20
9,3o
9.40
9,5o
9,60
9.70
9,80
9.90
0,00
0,10
0,20
o,3o
0,40
o,4o
0,4 1
0,42
0,43
0,44
0,45
0,46
0,47
0,48
0^49
o,5o
o,5i
0,52
0,53
0,54
0,55
o,56
0,57
o,58
0,59
0,60
0,61
0,62
o,63
o,64
0.65
0,66
0,67
0,68
0,69
0,70
0,71
l.o^. u
nùnulns, jjjg.^
imnu/t log
t' in days
1,922370
1,922370
1,922370
1,92236g
1,92236b
1,922366
1,922364
1,932360
1,922354
1,9223 '
1,922338
i,9223o3
1,922264
1,922201
1,933102
,922102
,922089
,923076
,923063
,922048
,922032
,922016
.921999
,921981
,921963
,931944
,921924
,921903
,921881
,921858
,921834
,92 1 808
,921782
,921754
,921725
,921695
,93 1 663
,921630
,921595
,921558
,921520
.921480
,921437
,921393
,921347
,92 1 3oo
,g2i25i
Log. ol
t days.
0,700
0,701
0,703
0,703
0,704
0,705
0,706
0,707
0,708
0,70g
0,710
0,71 1
0,712
0,713
0,714
0,71 5
0,716
0,717
0,718
0,719
0,720
0,721
0,733
0,723
0,724
0,735
0,726
0,727
0,728
0,729
0,730
0,731
0,732
0,733
0,734
0,735
0,736
0,73
0,738
0,739
o,74o
0,741
0,742
0,743
0,744
0,745
0,746
0,747
0,748
0,749
0,750
0,75 1
0,752
0,753
0,754
0,755
0,756
0,757
o,758
0,759
0,760
True
Anom. U.
6,58,07,1
6,59,04,6
7,00,02,3
7,01,00,1
7,01,58,0
7,02,56,1
7,o3,54,3
7,04,53,7
7,o5,5i,i
7,06,49,7
7,07,48,5
7,08,47,3
7,09,46,3
7,10,45,5
7,11,44,8
7.^2,44,2
7.i3,43,7
7,14,43,4
7,i5,43,2
7,16,43,2
7,17,43,2
7,18,43,4
7.19.43,7
7,20,44,2
7,21,44,8
7,22,45,6
7.23,46,5
7,24.47.5
7.25.48,7
7,26,50,0
7,27,51,5
7.28,53,1
7,29,54,8
7,3o,56,7
7.31.58,7
7,33,00,9
7,34.o3,3
7,35;o5,6
7,36,o8,2
7,37,10,9
7,38,i3,8
7,39,16,8
7,40,19.9
7,41,23,2
7,42,26,6
7,43,3o,2
7,44,33,9
7,45,37.8
7,46,41,8
7,47,45,9
7,48,50,2
7,49,54,7
7.50,59,3
7,52,o4,o
7,53,08,9
7,54,13.9
7,55,19,1
7,56,24,4
7>57.29,9
7,58,35,5
7,59.41,3
Diff.
57.5
57,7
57,8
57.9
58,1
58,2
58,4
58,4
58,6
58,8
58,8
59,0
5q,2
59,3
59,4
59,5
59,7
59,8
60,0
60,0
60,2
60,3
60,5
60,6
60,8
60,9
61,0
61,2
61,3
61,5
61,6
61,7
61,9
62,0
62,2
62,3
62,4
62,6
62,7
62,9
63,0
63,1
63,3
63,4
63,6
63,7
63,9
64,o
64.3
64.5
64,6
64.7
64.9
65,0
65,2
65,3
65,5
65,6
65,8
65,9
Log. oC
(' day!
0,760
0,761
0,762
0,763
0,764
0,765
0,766
0,767
0,768
0,769
0,770
0,771
0,772
0,773
0,774
0,775
0,776
0.777
0,778
0,779
0,780
0.781
0,782
0,783
0,784
0,785
0,786
0,787
0,7'
0,789
0.790
o,79'
0.792
0,793
0,794
0.795
0.796
0.797
0.7
0,799
0,800
0,801
0,802
o,8o3
0,804
o,8o5
0,8
0,807
0,808
0,809
0,810
0,811
0,812
0,8 1 3
0,81 4
0,81 5
0,816
0,817
0818
0,819
0,820
True
Anom. U.
d m i
7.59.41
8,00,47.2
8,01,53,3
8,03,59,5
8,o4,o5,9
8,o5,i2,4
8,06,19,1
8.07,25,9
8^08,32,8
8,09,39,9
8,10,47,3
8,11,54,6
8,l3,02,2
8,14,10,0
8,15,17,8
8,16,25,9
8,17.34,0
8,18,43,4
8,19,50,9
8,20,59,5
8,22,08,3
8,33,17,3
8,34,36,4
8,35,35,7
8,26,45,1
8,27,54,7
8.29,04.4
8,3o,i4i3
8,3i,24,4
8,32,34,6
8,33,44,9
8.34,55,5
8,36,06,2
8,37,17,0
8,38,28,0
8,39,39,2
8,4o,5o,5
8,43,02,0
8.43,i3,6
8,44,25,5
8,45,37.4
8,46,49.6
8,48,01,9
8,49,14.3
8,50,36,9
8,51,39,7
8,52,52,7
8.54,o5,8
8,55,19,1
8,56,33,5
8,57,46,1
8,58,59,9
9,00,1 3,8
9,01,27,9
9,02,42,2
9,o3,56,7
9,05,11,3
9,06,26,0
9,07,41,0
9,08,56,1
9,10,11,4
65,9
66,1
66,2
66,4
66,5
66,7
66,8
66,9
67,1
67,3
67,.
67,6
67,8
67,1
68,1
68,1
68,4
68,5
68,6
68,8
69,0
69,1
69,3
69,4
69,6
69.7
69.9
70,1
70,2
70,3
70,6
70,7
70,8
71,0
71,2
71,3
71,5
71,6
71,9
7i>9
72,2
72,3
72.4
72,6
72,8
73,0
73.1
73,3
73,4
73,6
73,8
73.9
74.1
74,3
74,5
74,6
74,7
75,0
75.1
75,3
75,4
Log. of True
t' days. Anom. U.
0,820
0,821
0,833
0,823
0,824
0,825
0,826
0,837
0,828
0,829
o,83o
o,83i
o,832
o,833
0,834
o,83
o,836
0,837
0,838
0,839
0.84
0,842
0,843
0,844
0,845
o,846
o,f
o,84&
0,84
o.85o
o,85i
o,852
0,853
0,854
o,855
o,856
0,857
o,858
0,859
0,860
0.861
0,863
o,663
0,865
0,866
0,867
0,8
0,869
0.870
0^871
0,872
0,873
0,874
0,875
0,876
0,877
0,878
0,879
0,880
Diff.
9,10,11,4
9,11,26,8
9,12,42,4
9.13,58,2
9,i5,i4,2
9,i6,3o,3
9,17,46,6
9,19,03,1
9,20,19,7
9,21,36,6
9,22,53,5
9,24,10,7
9,25,28,c
9,26,45,5
9,28,03,3
9,29,31,1
9,30,39,1
9,3i,57,3
9,33,15,7
9,34,34,3
9,35,53,0
9,37,11.9
9, 38,3 1,0
9,39,50,3
9,41,09,7
9,42,39,3
9,43,49,1
9,45,09,1
9.46,29,3
9,47.49.6
9,49,10,1
9,5o,3o,8
9,5 1,5 1, 7
9,53,12,7
9,54,34,0
9,55,55,4
9,57,17,0
9.58,38,7
10,00,00,
10,01,22,9
10,02,45,2
10,04,07
io,o5,3o,4
10.06,53,3
10,08,16,4
10,09,39,6
io,ii,o3,)
10,12,26,7
io,i3,5o,5
io,i5,i4.5
io,i6,.38,7
io,i8,o3,i
10,19,27,6
10,20,53,4
10,22,17,3
10,23,43,5
10,25,07,8
10,26,33,3
10,27,59,0
10,29,34,9
io,3o,5i,o
75.4
75,6
75,8
76,0
76,1
76.3
76,5
76,6
76,9
76,9
77,2
77.3
77.5
77.7
77,9
78,0
78,2
78,4
78,6
78,7
78,9
79,'
79,3
79,4
79,6
79,8
80,0
80,2
80,3
80,5
80,7
80,9
81,0
81,3
81,4
81,6
81,7
82,0
82,2
82,3
82,5
83,7
82,9
83,1
83,2
83,5
83,6
83,8
84,0
84,2
84,9
85,2
85,3
85,5
85,7
85,9
86,1
86,3
TABLE III.
To finil the truc anomaly f, con-esponilinn to the time /'from the peiihelion in days, in a parabohc orbit, whose perihelion
ilijt.incc is tliesanie as the mean ilistance of tlie sun from the eartli.
t' days.
0,880
0,881
0,882
0,88
0,884
0,885
0,886
0,887
0,888
0,889
0,890
0,891
0,895
0,893
0,894
0,895
o,8g6
0,897
0,898
0.899
0,900
0,901
0,902
0,903
0,904
0,905
0,906
0,907
0,908
0,909
0,910
o.gii
0,912
0,913
0,914
0,915
0,916
0,91
0,918
0,919
0,920
0,921
0,923
0,923
0,924
0,925
0,926
0.927
0,928
0,929
0,930
0,931
o,g33
0,933
0,934
0,935
0,936
o,g3-
0,938
0,939
0,940
Truo
Anoiii. U.
o,3o,5i ,0
0,32,17,3
0,33,43,7
0,35,10,4
0,36,37,2
0,38,04,3
0,39,31,5
0,40,59,0
0,43,26,6
0,43,54,4
0,45,22,4
o,46,5o,7
0,48,19,1
0,49,47,7
o,5i,i6,5
0,52,45,5
0,54,14,7
0,55,44,1
0.57,13,7
o;58,43,5
,oo,i3,5
,01,43,7
,o3.i4, 1
,04,44,7
,06,1 5,5
,07,46,5
,09,17,7
,10,49,1
,12,20,7
,i3,52,5
, 1 5,24,6
,i6,56,8
,18,29,2
,20,01 ,9
,21,34,7
,23,07,7
,24,41,0
,26,14,4
,27,48,1
,29,22,0
,3o,56,o
,32,3o,3
,34,04,8
,35,39,5
,37,14,4
,38,49,5
,40,24.8
,43,00,4
,43,36,1
,45,12,1
.46,48,3
,48,24,6
,5o.oi,3
,5i,38,i
,53,i5,i
,54,53,3
,56,29,8
,58,07,5
-,59,45.3
3,01,23,5
2,o3,oi,S
86.3
86,4
86,7
86,8
87,1
87,2
87,5
87,6
87,8
88,3
88,4
88,6
88,8
89,0
89,3
89,4
89,6
90,0
90,2
90,4
90,6
90,8
91,0
91,2
91 '4
91.6
91,8
92,1
92,2
93,4
93,7
93,8
93,0
93,3
93,4
93,7
93=9
94,0
94,3
94.5
94,7
94.9
95,1
95,3
95,6
95,7
96,0
96,3
96,3
96,6
96>9
97.0
97,3
97,5
97.7
97,8
98,3
98,3
q8,5
iMg of
t' (Jays.
0,940
0,941
0,942
0,943
0,944
0,945
0,946]
o,94'
0,948
0,949
0,950
0,951
0,952
0.953
0,954
0,955
0,956
0,957
0,958
0,959
0,960
0,961
0.96
0,963
0,964
0,965
0,966
0,967
0,968
0,969
0,970
0,971
0,973
0,973
0,974
0,975
0,976
0,97
0,978
0,979
0,980
0,981
0,982
0,983
0,984
0,985
0,986
0,987
0,988
0,989
0,990
0,991
0,992
0,993
0,994
0.995
0,996
0,99
0,998
0,999
1,000
'I'rue
A nom. U<
I2,o3,Ol,&
I2,o4,4o,3
I2,c6,i9,<
i2,o7,5b,o
13,09,37,2
I2.ii,>6,6
12,12,56,3
I3,i4,36,i
12,16,16,3
12,17,56,4
12,19,37,0
12,31,17
12,22,58,6
12,34,39,8
12,26,21,3
12,28,02,8
12,29,44,7
12,31,26,7
12,33,09,0
I2,34,5i,5
13,36,34,3
12,38,17,3
12,40,00,4
12,41,43,8
12,43,27,5
12,45,11,3
12,46,55,4
1 2^8,39,6
i3,5o,34,3
13,53,09,1
12,53,54,1
12,55,39,3
12,57,24,8
12,59,10,5
i3,oo,56,4
l3,02,42,6
i3,o4,29,o
1 3,06, 1 5,6
i3,o8,o2,5
13,09,49,6
i3, 1 1,36,9
i3,r3,24,5
t3,i5,i2,3
i3, 17,00,3
i3,i8,48,6
i3,20,37,i
i3, 22, 25,9
i3,24,i4,8
i3, 26,04, 1
13,27,53,5
13,39,43,2
i3,3i,33,3
i3,33,23,3
i3,35,i3,7
1 3,37,04,4
i3,38,55,3
i3,4o,46,4
1 3,42,37,8
13,44,29,4
13.46.3 1,3
i3,48,i3,4
98,5
98,7
99,0
99.2
99.4
99.7
99.8
100,1
100,2
100,6
100,7
100,9
101,2
101,4
101,6
101,9
103,0
102,3
102,5
102,8
102,9
Io3,2
io3,4
io3,7
io3,8
io4,i
io4,4
io4,5
io5,o
io5,2
io5,5
io5,7
io5,9
106,3
106,4
106,6
io6,g
107,1
107,3
107,6
107,8
108,0
108,3
108,5
108,8
108,9
109,3
109,4
109,7
110,0
110,1
110,4
110,7
110,9
111,1
111,4
111,6
111,9
112,1
1 12,3
Log. >,1
t' duys.
1 ,000
1,001
1,002
i,oo3
i,oo4
i,oo5
1,006
I,00'
1,008
1,009
1,010
1,011
1,013
1,01 3
1,014
1,01 5
1,016
1,01
1,018
1,019
1,020
1,021
1,022
1,023
1,024
1,025
1,026
1,027
1,038
1,039
i,o3o
i,o3i
l,o32
i,o33
i,o34
i,o35
i,o36
1.037
i,o38
1,039
i,o4o
1. 04 1
1. 042
i,o43
i,o44
1,045
1,046
I,o47
i,o48
1,049
i,o5o
i,o5i
I,o52
i,o53
i,o54
i,o55
1,0 56
i,o57
i,o58
1,059
1,060
True
Anom. U.
tt m s
3,48,i3,4
3,5o,o5,-
3,5i,58,3
3,53,5i,2
3,55,44,3
3,57,37,6
3,5g,3i,2
4,01,35,0
4,03,19,1
4,o5,i3,4
4,07,07,9
4,09,02,7
4,10,57,8
4,13,53.1
4,14,48,6
4,i6,44,4
4,18,40,5
4,30,36,8
4,32,33,4
4,24,3o,2
4,26,27,3
4,28,24,6
4,3o,23,2
4,32,20,0
4,34,18,1
4,36,16,5
4,38,i5,i
4,4o,i3,g
4,43, i3,o
4,44,13,4
4,46,12,0
4,48,1 1, g
4,5o,i2,o
4,52,12,4
4,54,1 3
4,56,14,0
4,58,i5,3
5,00,16,7
5,03,18,4
5,04,20,4
5,06,22,6
5,08,25,1
5,10,27,8
5,i2,3o,7
5,14,34,0
5,16,37,5
5,18,41,3
5,20,45,4
5,22,4g,7
5,24,54,3
5,26,59,1
5,29,04,2
5,31,09,6
5,33,15,3
5,35,21,2
5,37,27,4
5,39,33,9
5, 4 1, 40,7
5,43,47,7
5,45,55,0
5,48,02,6
Diff.
3,3
2,6
2.9
3,1
3,3
3,6
3,8
4,1
4,3
4,5
4,8
5,1
5,3
5,5
5,8
6.1
6;3
6,6
6,8
7>i
7.3
7,6
7,8
8,1
8,4
8,6
8,8
9.1
9.4
9.6
9.9
20,1
30,4
30,7
21,2
21,5
31,7
22,0
22,3
22,5
23,7
23,9
33,3
33,5
23,8
24,1
34,3
24,6
24,8
25,1
25,4
25,7
25,9
26,2
26,5
26,8
27,0
27,3
27,6
37,8
Log. of
V days.
1,060
1,061
1,062
1 ,o63
1,064
i,o65
1,066
i,o&
1,068
1,069
1,070
1,071
1,073
1,073
1,074
1,075
1,076
1.077
1,078
1,079
1,080
1,081
1,083
i,o83
1,084
i,o85
1,086
1,087
1,088
1,089
i,ogo
1,091
1,093
i,og3
1,094
I,og5
1,096
1,097
1,098
1.099
1,100
1,101
1,102
i,io3
i,io4
i,io5
1,106
1,107
i,io8
I, log
1,110
1,111
1,113
i,ii3
i,ii4
i,ii5
1,116
1,117
1,118
1,119
1,120
True
Anum. U.
5,48,02,6
5,5o,io,4
5,52,18,5
5,54,26.9
5,56,35,5
5,58,44,4
6,00,53,6
6,o3,o3,o
6,o5,i2,8
6,07,22,8
6,09,33,1
6,11,43,7
6,i3,54,6
6,i6,o5,7
6,18,17,1
6,30,38,8
6,22,40,8
6,24,53,1
6,27,05,6
6,29,18,5
6.3 1.3 1.6
6,33,44,9
6,35,58.6
6,38,12,5
6,40,26,7
6,43.41.3
6,44;55;9
6,47,11,1
6,49,26,5
6,5i,42,i
6.53,58.0
6,56,14.3
6,58,30,8
7,00,47,6
7,o3,o4,6
7,05,22,0
7.07,39,7
7,09,57,7
7,12, i5,c)
7.14.34,4
7,16,53,2
7,19,13,3
7.3 1.3 1. 7
7,33,5i,4
7,36,11,4
7,28,31,7
7,3o,52,3
7,33,i3,i
7,35.34,3
7,37,55,8
7,40,17,6
7,42,39,6
7,45,01,9
7,47,24,6
7,49,47.5
7,53,10,8
7,54,34,3
7,56,58,1
7,59,22,3
8,01,46,7
8,04,11,5
Diff.
37,8
28,1
28,4
28,6
28,9
39.2
29,4
29.8
3o,o
3o,3
3o,6
3o,g
3i,i
3i,4
3i,7
32,0
33,3
32,5
32,9
33,1
33,3
33,7
33,9
34,2
34,5
34,7
35,2
35,4
35,6
35,9
36,3
36,5
36,8
37,0
37.4
37.7
38.0
38,3
38,5
38,8
39,1
3g,4
39:7
40,0
40,3
40,6
40,8
4l,2
4i,5
4i,8
42,0
43,3
43,7
42,9
43,3
43,5
43,8
44,3
44.4
44,8
45,0
Log. ol
(' day.s
1,130
J, I
1,122
1,123
1,124
1,125
1,126
1,127
1,128
1.129
i,i3o
i,i3i
l,l33
i,i33
i,i34
i,i35
I, I
i,i37
1,1 38
i,i39
i,i4i
1,1
1,143
i,i44
1, 145
i,i46
i,i47
i,i48
1,149
i,i5o
i,i5i
I,l52
i,i53
1,1 54
i,i55
1,1 56
i,i57
1,1 58
1,159
1,160
1,161
1,163
i,i63
1,164
i,i65
1,166
1,167
1,168
1,169
1,170
1,171
1,173
1,173
1,174
1,175
1,176
1,177
1,178
1,170
I,ll
True
Anom. U,
8,04,11,5
8,o6,36,5
8, eg ,01 ,9
8,1 1,37,5
8,13,53,5
8,16,19,7
8,18,46,3
8,31, i3. 1
8,33,40,3
8,26,07,8
8,38,35,5
8,3i,o3,6
8,33,32,0
8,36,00,7
8,38,29,7
8,40,59,0
8,43,28,6
8,45,58,5
8,48,38,8
8,50,59,3
8.53,3o,2
8,56,01,4
8,58,32,8
g,oi,o4,6
g,o3,36,7
9.06,09,1
9,08,41,8
9,11,14,8
g,i3,48,2
9, 16,3 1, g
9,18,55,9
9,21,30,2
9,24,04,8
9,36,3g,8
9.29.15.0
9,3i,5o,6
9,34,36,5
9,37,02,7
9,39,39,3
9,43,16,0
9,44,53,3
9,47,3o,8
g,5o,o8,7
g,52,46,g
g,55,25,3
ig,58,o4,i
20,00,43,2
2O,03,22,6
30,06,02,3
20,08,42,4
20,1 1,23,1
2o,i4,o3,5
20,16,44,5
20,19,35,9
20,33,07,6
20,34,49,6
30,27,32,0
2o,3o,i4,7
20,32,57,7
2o,35,4i,o
20,38,24,7
Diff.
45,0
45,4
45,6
46,0
46,3
46,6
46,8
47,3
47.5
47.7
48,1
48,4
48,7
49,0
49,3
49,6
49.9
5o,3
5o,5
5o,g
5l,2
5i,4
5i,8
52
52,4
52,7
53,0
53,4
53,7
54,0
54,3
54,6
55,0
55,2
55,6
55,9
56,2
56,5
56,8
57,3
57,6
57,9
58,3
58,4
58,8
5g,i
59,4
59.7
60,1
60,4
60,7
61,0
61,4
61,7
63,0
63,4
63,7
63 ,0
63,3
63,7
64,0
TABLE IJI.
To find the true anomaly U, corresponding to the time f from the perihelion in days, in a paraholic orbit, whose perihelion
distance is the same as tlie mean distance of the sun from the earth.
Log. of
t' days.
i,i8o
i,i8i
1,182
i,i83
1, 184
i,i85
1,186
1,187
1,188
i,ie
1,190
1,191
1,192
1,193
1. 194
1,195
1,196
1. 197
1,198
1. 199
1,200
1,201
1,202
I,203
1,204
I,2o5
1,206
1,207
1,20
1,209
1,210
1,211
1,212
1,2 1 3
I,2l4
1,2 1 5
1,216
1,217
1,218
I.2I9
1,220
1,221
1,222
1,223
1,224
1,225
1,226
1,227
1,22
Truo
Anom. U.
1,22g
1,280
1,23 1
1,232
1,233
1,234
1,235
1,236
1,237
1,238
1,239
1,240
20,38,24,7
20,4i,o8,7
20,43,53,0
20,46,37,6
20,49,22,6
20,52,07,9
20,54,53,6
20,57,39,6
21,00,25,9
2I,03,I2,6
21,05,59,6
21,08,46,9
21,11,34,6
21, r4, 22,6
21,17,10,9
21,19,59,6
21,22,48,6
21,25,37,9
21,28,27,6
2i,3i,i7,6
21,34,08,0
21,36,58,7
21,39,49,7
21,42,41,1
21,45,32,8
21,48,24,8
2I,5l,17,2
21,54,10,0
2i,57,o3,i
21,59,56,6
22,03,5o,4
22,o5,44,5
22,08,39,0
22,11,33,8
22,14,29,0
32,17,24,5
22,20,20,3
22,23,16,5
22,2D,l3,0
22,29,09,9
22,32,07,1
22,35,04,7
22,38,02,6
22,4l,00,9
22,43,59,5
22,46,58,5
22,4g,57,8
22,52,57,4
22,55,57,4
22,58,57,8
23,01,58,5
23,04,59,6
23,08,01,0
23,11,02,7
23,i4,o4,8
23,17,07,3
23,20,10,1
23,23, i3,3
23,26,16,8
23,29,20,7
33,32,24,9
i64,o
i64,3
i64,6
i65,o
i65,3
i65,7
166,0
166,3
166,7
167,0
167,3
167,7
168,0
168,3
168,7
169,0
169,3
169,7
170,0
170,4
170.7
171,0
171.4
171.7
172,0
172.4
173,8
173..
173,5
173,8
174.1
174,5
174
175,3
175,5
175,8
176,2
176,5
176,9
177,2
177,6
177,9
178,3
178,6
179,0
179,3
179,6
180,0
180,4
180,7
181
181,4
181.7
182,1
182,5
183,8
i83.2
i83,5
i83,9
1 34.2
184,6
Log. ol
V days.
True
Anora. U.
1,240
1,24 1
1,242
1,243
1,244
1,245
1,246
1,247
1,248
1,249
I,25o
I,25l
1,252
1,253
1,254
1,255
1,256
1,257
1,258
1,359
1,260
1,261
1,262
1,263
1,354
1,265
1,366
1,267
1,368
1,269
1,2'
1,371
1,373
1,373
1,274
1,275
1,276
1,277
1,278
1,279
1.280]
1,281
1.282
1^283
1,384
1,285
1,286
1,287
1,2
1,289
1,290
1,291
1,292
1,293
1,294
1,395
1,296
1,29-
1,299
1 ,3oo
d m s
23,33,24,9
33,35,29,5
23,38,34,5
23,41,39,8
23,47,5i,4
23,50,57,8
23,54,04,5
23,57,11,6
24.00,19,0
24,03,26,8
24,06,35,0
24,09,43,5
24,13,52,4
34,16,01,6
24,19,11,2
24,22,31,2
34,25,3l,5
24,28,42,2
24,3i,53,3
24,35,04,7
24,38,16,5
24.41,28,6
34,44.41,1
24,47,54,0
34,51,07,3
24,54,20,9
24,57,34,9
25,00,49,2
25,o4,o3,9
25,07,19,0
25,10,34,4
25,i3,5o,3
25,17,06,4
25,20,23,0
25,23,39,9
25,26,57,2
25,3o,i4,g
25,33,32,9
25,36,5i,3
25,4o,io,i
25,43,39,3
25,46,48,8
25,5o,o8,7
25,53,29,0
25,56,49,6
26,00,10,6
26,03,32,0
36,06,53,8
2D,io,i5,9
36,i3,38,5
26,17,01,4
26,30,34,7
26,23,48,4
26,27,12,4
36,3o,36,8
26,34,01,6
26,37,26,7
26,40,52,3
36,44,18,3
36,47,44,5
Diff.
i84,6
i85,o
i85,3
1 85,6
186,0
186.4
186,7
187,1
187,4
187,8
188,3
188,5
188,9
189,2
189,6
190,0
190,3
190,7
191,1
191,4
191,8
192,1
192,5
'92,9
193,3
193,6
194,0
194,3
194,7
195,1
195,4
195,8
196,2
196,6
196,9
197,3
197.7
198,0
198,4
198,8
199,2
199,5
199,9
200,3
200,6
201,0
201,4
201,8
202,1
202,6
203,9
2o3,3
2o3,7
204,0
2o4,4
2o4,8
2o5,i
2o5,6
2o5,9
206,3
206,6
Log. of
t' days.
I,3oo
i,3oi
I,303
i,3o3
i,3o4
i,3o5
i,3o6
1 ,307
i,3o8
1,809
i,3io
i,3ii
I,3l2
i,3i3
i,3i4
i,3i5
i,3i6
1,817
1,818
1,819
1,820
1,331
1,833
1,333
1,324
1,325
1,336
1,337
1,328
1,339
i,33o
1,33 1
1,333
1,333
1,334
1,335
1,336
1,337
1,838
1,339
1,340
1,341
1,343
1,343
1,344
1,345
1,846
1,347
1,348
I,-
i,35o
i,35i
1,353
1,353
1,354
1,355
1,356
1,35
1,358
1,359
i,36o
True
Anom. U.
26,47,44,5
26,51,11,1
26,54,38,1
26,58,o5,5
27,01,33,3
27,05,01,4
27,08,29,9
27,11,58,8
27,15,38,1
27,18,57,8
27,23,37,8
27,35,58,3
27,29,29,0
27,88,00,2
27,86,81,9
27,4o,o3,S
27,43,86,1
27,47,08,8
27,50,4 1, g
27,54,15,4
27,57,49,3
38,01,28,5
38,04,58,1
28,08,33,1
28,12,08,5
28,15,44,2
28,19,20,3
28,23,56,9
28,36,88,8
28,80,11,1
28,33,48,7
28,87,26,8
28,41,05,3
28,44,44,1
28,48,28,3
28,52,02,9
28,55,42,9
28,59,28,3
2g,o3,o4,o
29,06,45,2
39,10,26,7
29,14,08,6
39,17,50,9
29,31,33,6
29,25,16,7
29,29,00,2
29,82,44,0
29,86,38,8
29,40,13,9
29,48,57,9
39,47,43,3
29,51,29,1
39,55,15,3
29,59,01,8
3o,o2,48,8
3o,o6,36,i
30,10,38,9
3o,i4,i3,o
80,18,00,5
80,31,49,4
3o,35,38,7
Log, of
t' diiys.
3o6,6
207,0
207,4
307,8
208,1
208,5
208,9
209,3
209,7
210,0
210,4
210,8
311,2
211,7
211,9
212,3
212,7
2l3,l
21 3,5
218,9
2l4,2
2i4,6
21 5,0
2 1 5,4
21 5,7
216,1
216,6
216,9
217,8
217,6
218,1
218,5
218,8
219,2
21 9,6
230,0
220,4
220,7
331,2
221,5
221,9
222,3
222,7
228,1
228,5
233,8
234,3
334,6
235,0
225,4
335,8
226,2
336,5
227,0
227,3
227,8
228,1
228,5
338,g
22g,3
32Q,
True
Anom. U.
1 ,860
1,36 1
1,363
1,863
1,364
1,365
1,366
1,867
1,368
i,36g
1,870
1,371
1,873
1,878
1,874
1,375
1,876
1,377
1,378
1,379
i,38o
i,38i
1,382
1.388
1,384
1,385
1,386
1,887
1,888
1,389
1 ,890
1,891
1,892
1,393
1,394
1,895
1,396
i,3g7
1,898
1 ,399
i,4oo
1,401
1,402
i,4o3
i,4o4
i,4o5
i,4o6
1,407
i,4o8
i;4og
i,4io
i,4i
I,4l2
i,4i3
i,4i4
i,4i5
i,4i6
1,417
i,4i
1.419
1,420
Diff.
3o,35,38,7
80,39,28,4
80,33,18,5
80,87,08,9
80,40,59,8
3o,44,5i,o
80,48,42,7
80,52,34,7
80,56,27,1
31,00,19,9
3i,o4,i3,i
81,08,06,7
81,12,00,7
3i,i5,55,i
31,19,49,8
81,28,45,0
81,37,40,5
8i,3i,36,5
81,35,82,8
81,39,29,5
3 [,43,36,6
81,47,34,1
3l,5l,32,0
81,55,20,3
81,59,19,0
32,o3,i8,o
32,07,17,5
82,1 1,17,3
83,15,17,5
32,19,18,2
82,28,19,2
82,27,20,6
82,81,22,4
82,35,24,6
83,89,37,3
33,43,80,1
82,47,33,5
82,51,87,3
83,55,41,4
33,59,46,0
38,o3,5o,9
33,07,56,3
88,13,02,0
33,16,08,1
33,20,14,6
33,24,21,5
33,38,38,7
33,33,36,4
33,36,44,5
33,40,52,9
88,45,01,8
88,49,11,0
83,53,20,7
88,57,80,7
34,01,41,1
84,o5,5i,9
84,10,08,1
34.14,14,7
84,18,26,6
34,22,89,0
84,26,51,8
Log. of
(' du
339,7
280,1
280,4
23o,9
23l,3
281,7
282,0
282,4
233,8
283,3
333,6
334,0
334,4
234,7
235,2
235,5
286,0
286,8
236,7
287,1
287,5
287,9
238,3
288,7
289,0
289,5
289,8
240,2
240,7
241,0
241,4
241,8
242,2
243,6
242,9
243,4
243,8
244,1
244,6
244,9
245,3
245,8
246,1
246,5
246,9
247,2
347,7
348,1
248,4
248,9
249,3
249,7
25o,o
25o,4
25o,8
25l,3
3 5 1, 6
i5i,9
253,4
2 53,8
253,1
1,420
1,421
1,422
1^438
1,424
1,425
1,426
1,427
1,428
1,439
1,480
i,43i
1,482
1,433
1,434
1,435
1,436
1,437
1,438
1,439
True
Anom, U.
34,36,5 1,8
34,81,04,9
34,35,18 "
34,89,82,4
34,43,46,7
34,48,01,4
84,52,16,5
84,56,83,0
85,00,47,8
35,o5,o4,i
35,09,20,7
35,18,37,7
35,17,55,3
35,33,18,0
35,26,81,2
35,80,49,7
35,85,08,7
35,3g,2i
85,43,47,8
35,48,08,0
i,44o 35,52,28,5
i,44i 35,56,49,4
1,442
1,443
1,444
1,445
1,446
1,447
1,448
i,45o
1,45:
1,452
1,453
1,454
1,455
1,456
1,457
1,458
1,459
1,460
1,461
1,462
1,468
1,464
1,465
1,466
1,467
1,468
1,469
1,470
1,471
1,472
1,473
1,474
1,475
1,476
1,477
1,478
1,479
1,480
36,01,10,7
86,05,82,4
86,09,54,4
36, 1 4, 1 6,9
86,18,89,8
86,33,08,0
86,27,36,6
36,3i,5o,6
36,36,15,0
36,40,89,8
86,45,04,9
86,4o,8o,5
86,53,56,4
36,58,22,7
87,02,49,4
87,07,16,5
87,11,43,9
87,16,11,8
87,20,40,0
87,25,08,6
87,39,37,6
87,84,07,0
87,88,86,7
87,43,06,9
87,47,37.4
87,52,08,3
87,56,39,6
38,01,11,3
88,o5,43,3
38,io,i5,7
38, 14,48,5
88,19,21,7
38,23,55,3
88,38,29,3
38,38,o3,5
88,87,88,3
88,43,18,2
38,46,48,7
88,5i,24,5
253,1
353,6
253,9
354,3
254,7
255,1
255,5
355,8
3 56,3
356,6
357,0
357,5
257,8
258,2
258,5
259,0
259,4
359,7
360,2
260,5
260,9
261,8
261
262,0
262,5
262,9
268,2
363,6
264,0
364,4
264,8
265,1
265,6
265,9
266,3
266,7
267,1
367,4
267,9
368,2
268,6
369,0
369,4
269,7
270,2
270,5
270,9
371,3
271,7
273,0
272,4
272,8
278,2
378,6
273,9
274,3
274,7
375,0
375,5
275,8
276,1
TABLE III.
To fin
distance
1 tlie true anonuily
is the same as tlie
V, corresponiliiig to llie time (' from the
iiiCiHi ilislrtiice ol tlie sim from the earth.
perihelion in days, in a parabolic orbit, whose perihelion
Log. I.I
I' duvs
True
.-\iiom. U.
l,4So
i,48i
1,485
1,483
1,484
1,485
1,466
1,48'
1,481
1,489
1,490
1.491
1,492
1,493
1.494
1,495
1,496
1,497
1,498
1.499
i,5oo
i.5oi
i,5o3
i,5o3
i,5o4
i,5o5
i,5o6
I,507
i,5o8
1,509
i,5io
i,5i I
I,5l3
i,5i3
i,5i4
i,5i5
i,5i6
I,5i7
I,5i8
1,519
1,530
1,52 1
1,522
1,523
1,524
1,525
1,526
1,527
1,528
1,529
i,53o
1,53 1
1,532
1,533
1,534
1,535
1,536
1,537
1,538
i,53q
i,54n
38,5 1, 24,5
38,56,00,6
39,00,37,3
39,o5,i4,i
39,09,51,4
39,14,29,1
39,19,07,1
3g,23,45,5
39,28,24,3
39,33,03,5
39,37,43,0
39-42,33,9
39,47,03,3
39,51,43,8
39,56,24,8
40,01,06,2
4o,o5,48,o
4o,io,3o,i
4o,l5,!3,6
40,19,55,4
40,34,38,6
4o,2g,22,2
40,34,06,2
4o,38,5o,5
40,43,35,2
40,48,20,2
4o,53,o5,6
40,57,5 1,4
41,02,37,6
41,07,24,1
41,12,10,9
4i,i6,58,
41,21,45,7
41,26,33,6
4i,3i,2i,9
4i, 36, 10,6
41,40,59,6
41,45,49,0
4i,5o,38,7
4i,55;:S,8
42,00,19,2
42,o5,io,o
42,10,01,1
43,14,52,6
42,19,44,4
43,24,36,6
43,39,39,2
42,34,22,1
43,39,15,3
42,44,08,9
43,49.02,9
42,53,57,2
42, 58,5 1, 9
43,03,46,9
43,08,42,3
43,i3,38,o
43,18,34,0
43,33,304
43.38,27,1
43,33,24,3
43,38,2
Uiff.
276,1
376,6
276,9
277,3
277,7
278,0
278,4
278,8
279,2
279,5
279-9
280,3
380,6
281,0
281,4
381,8
282,1
283,5
382,8
383,2
283,6
284,0
284,3
284,7
285,0
285,4
285,8
286,3
286,5
286,8
287,2
287,6
287,9
288,3
288,7
289,0
289,4
289,7
290,1
290,4
290,8
291,1
391,5
291.8
292,3
293,6
292,9
293,3
293,6
294,0
294,3
294,7
395,0
295,4
295,7
296,0
296,4
296,7
297-1
297,5
U,g. of
(' tlujs.
,540
,541
,543
,543
,544
,545
,546
,547
,548
,549
,55o
,55i
,552
,553
,554
,555
,556
,557
,558
.559
,56o
,56i
,562
,563
,564
,565
,566
,567
.568
,569
,570
,571
,572
.573
,574
,575
,576
,577
,578
,570
,5So
,58 1
,582
,583
,584
,585
,586
,587
,588
,589
,590
,591
,592
,593
,594
,595
,596
,597
,598
,599
,600
True
Aiiom. U.
43,38,31,7
43,43,19,5
43,48,17,5
43,53,1 5,8
43,58,i4,5
44,o3,i3,6
44,08,1 3,0
44, i3,i2,7
44,18.13,8
44. 23,i3,2
44,28,14,1
44,33, i5, 3
44,38, 16,6
44,43,18,4
44,48,20,5
44,53,22,9
44,58,35,7
45,03,38,8
45,08,33,2
45,1 3,36,0
45,i8,4o,i
45,33,44,6
45,38,49,4
45,33,54,5
45,38,59,9
45,44,05,6
45,49,11,7
45,54,18,0
45,59,34,7
46,o4,3i,7
46,09,39,0
46,14,46,6
46,rg,54,6
46,35,03,9
46,3o,ii,5
46,35,20,3
46,40,39,5
46,45,39,0
46,5o,48,8
46,55,59,0
47,01,09,4
47,06,20,1
47,1 1,3l,2
47,16.42,6
47,21,54,3
47,27,06,3
47,32,18,6
47,37,31,2
47,42,44,1
47.47,57,2
47,53,10,7
47,58,24.5
48,o3.38,6
48,08,52,9
48.14,07,6
48,19,32.6
48,24,37.9
48,39,53,^
48,35.09.3
48 4o,35..<
48,4 5,4 1, t
a21
297,8
298,0
298,3
298,7
299,'
299-4
299-7
3oo,i
3oo,4
3oo,g
3oi,i
3o 1 ,4
3oi,8
3o2,i
3o2,4
3o3,8
3o3,i
3o3,4
3o3,8
3o4,2
3o4,4
3o4,8
3o5,i
3o5,4
3o5,7
3o6,i
3o6,3
3o6,7
307,0
3o7,3
3o7,6
3o8.o
3o8.3
3o8,6
3o8,8
309,3
309,5
309,8
3lO,3
3io,4
3 1 0,7
3ii,i
3u,4
3ii,7
3l3,0
3i3,3
3 1 3,6
3i2,9
3i3,i
3i3,5
3i3,8
3i4.i
3i4,3
3i4,7
3t5,o
3i5,3
3i5,5
3i5.g
3i6.i
3 1 6,5
3i6.7
Log. 01"
(' days.
1,600
1,601
1,602
1 ,6o3
i,6o4
i,6o5
1,606
1,607
1,606
1,609
1,610
1,611
1,612
i,6i3
i,6i4
i,6i5
1,616
1,617
i,6iS
1.619
1,620
1,621
1,622
1,623
1.624
1,635
7,636
1,637
1,638
1,639
i,63o
1,63 1
1,632
1,633
1,634
1,635
1,636
1,637
1,638
1,639
1,640
i,64i
1 ,642
1,643
1,644
1,645
1,646
1,647
1,648
1,649
i,65o
J, 65 1
1,652
1,653
1,654
1,655
1 ,656
1,657
1,658
1 -659
1,660
'i'rue
Anoni. U,
48,45,41,9
48,5o,58,6
48,56,1 5,6
49,0 1, 32, g
49,06,50,4
49,12,08,3
49,17,36,5
49,33,44,g
49,28,03,6
49,33,33,6
49,38,41,8
4g,44,oi ,4
4g.4g,3i,3
49,54,41,3
5o,oo,oi,7
5o,o5,23,3
5o, 10,43, 3
5o,i6,o4,4
5o, 31, 35,9
50,36,47,6
50,32,09,6
5o,37,3i,8
50,43,54,3
5o,48,i7,i
5o,53,4o,2
5o, 59,03, 5
51,04,27, 1
5i,og,5o,9
5i,i5,i5,o
51,20,39,4
5i ,26,04,0
5i, 31,28,9
5 1, 36,54,0
51,42,19,4
5 1. 47.45.0
5i,53,io,Q
51,58,37,0
52,o4,o3,4
52,09,30,0
52,i4,56,9
52,20,34,0
53,35, 5i,3
53.3i,r8,9
52,36,46,7
52,42,14,8
52,47,43,1
52,53,11,6
53,58,4o,4
53,o4,og,4
53,og,38,7
53,i5,o8,2
53,20,37,9
53,26,07,9
53.31.38. 1
53,37,08,5
53,42,39.
53,48,09.9
53,53.41,0
53.59 13,3
54,04,43,9
54,io,i5,6
DifT,
3 1 6,7
3i7,o
3i7,3
3i7,5
3i7,9
3i8,2
3 18.4
3i8;7
319,0
319,3
319,6
3,g,8
320,1
320,4
320,6
32o,g
321,3
331,5
321,7
333,0
323,3
323,5
323,8
323,1
323,3
323,6
323,8
324,1
324,4
324,6
324,g
325,1
325,4
325,6
325,g
326,1
326,4
336,6
326,9
327,1
327.3
337,6
337,8
328,1
338,3
338,5
338,8
339,0
329,3
32g,5
33g,7
33o,o
33o,3
33o,4
33o,6
33o.8
33i.i
33 1, 3
331.6
33 1, 7
33t,q
f days
,660
,661
.663
,663
,664
.665
,666
,667
,668
,66g
,670
.671
;673
,673
,674
.675
,676
,677
,678
,679
,680
,681
.682
,683
,684
,685
,686
,68-j
,688
,689
,690
,6gi
,6g2
.693
,694
,695
,696
,697
,6g8
.699
,700
,7UI
,702
,703
,704
,705
,706
,707
,708
,709
,710
f7ii
,712
,71 3
,7 '4
,71 5
,716
^717
,718
-719
Tiuo
Anom. U.
d m s
54,io,i5,6
54,15,47,5
54,21,19,7
54,26,53,1
54,32,24,7
54.37,57,5
54,43,3o,5
54,4g,o3,7
54,54,37,1
55,00,10,8
55,o5,44,7
55,11,18,8
55,16,53,0
55,23,27,5
55,28,02,2
55^3,37,7
55, 3g, J 3,3
55,44,47^4
55,5o,23,g
55,55,58,6
56,01,34,4
56,07,10,4
56,13.46.7
56,18,33,1
56,33,5g,7
56,39,36.5
56;35,i3,5
56,4o,5o,6
56,46,37,9
56,53,o5,4
56,57,43,3
57,o3,3r,i
57,08,59,1
57,14,37,3
57,20,15,7
57,35,54,3
57, 3 1, 33,0
57,37,11,9
57,42,50.9
57,48,3o,i
57,54,09,5
57,59,49,2
58,05,29,0
58, 1 1, 08 ,9
58,i6,4g,o
58,23,39,2
58,38.09,6
58.33;5o.i
58,39,3o,8
58,45,11,6
58,5o,52,5
58,56,33,6
5g,02,i4,9
5g,07,56,3
59,-i 3,37,9
59,19,19,7
5g. 25,01, 6
5g,3o,43,6
59,36,35.7
5g,42,o8,o
59.47,50,4'
33 1,9
332,2
332,4
333,6
333,8
333,0
333,3
333,4
333,7
333,g
334.1
334.3
334,5
334,7
334,g
335,1
335,3
335,5
335,7
335,8
336.0
336,3
336.4
336,6
336,8
337,0
337.1
3373
337,5
337,8
337,g
338.0
338,3
338,4
338.6
338,7
338,9
339,0
339,3
339,4
33g,7
33g,8
339,9
340,1
340,3
34o,4
340.5
340,7
340,8
340,9
341,1
341,3
341,4
341,6
34 1. 8
34 1. 9
343,0
343,1
343,3
343.4
343.5
Log. of
1/ «lays
,720
,73 1
,723
,723
,724
.735
;736
,727
.72-
,72g
.730
,73,
,733
,733
,734
,735
,736
,737
,738
,73g
,740
,74 1
.743
,743
,744
,745
,746
•74'
,748
,749
,750
;75i
.753
;753
,754
,755
,756
,757
,758
,759
,760
,761
,763
,763
,764
,765
.766
.767
,768
,769
,770
,771
:772
,773
,774
,775
,776
,777
,778
,779
,780
Truo
.^nuiii. U.
59,47,50,4
59,53,33,9
59,5g,i5,6
6o,o4,58,4
60,10,4
60,16,24,4
60,33,07,6
6o,37,5o,g
60,33,34,3
60,39,17,8
60,45,01,5
6o,5o,45,3
60,56,29,3
6l,03,l3,3
61,07,57,4
6i,i3,4i,7
61,19,26,1
61, 35, 10,6
6i,3o,55,3
6i,36,3g,9
61,42, 34,i
61 ,48,09,8
61,53,54,9
6i,5g,4o,i
63,o5,35,3
63,11,10,6
62,16,56,0
63,33,41,5
63,38,37,1
63,34,12,8
62,3g,58,6
62,45,44,5
62,5i,3o,5
63,57,16,6
63,o3,03,8
63,o8,4g.o
63,14,35,3
63,30,31,7
63.36,08.3
63,3i,54,8
63,37 ,4i, 4
63,43,38,1
63,4q,i4:9
63,55,01,8
64,00,48,7
64,06,35,7
64,13,33,8
64,i8,og,g
64,33,57,1
64,3g,44,4
64.35.3 1. 8
64,4i,ig,3
64,47,06,7
64.53,54,3
64,58,41,7
65,o4,3g,3
65. 10.16.9
65,i6,o4,6
65,31,53,4
65,37.40.3
65,33,28'
1
343,5
342,7
342,8
342,9
343,1
343,2
343,3
343,4
343,5
343,7
343,8
343. g
344io
344,2
344,3
344,4
344,5
344,6
344,7
344,9
345,0
345,1
345,2
345,2
345,3
345,4
345.5
345.6
345,7
345,8
345.9
346,0
346,1
346,2
346,2
346,3
346,4
346,5
346,6
346,6
346,7
346,8
346.9
346^9
347,0
347,1
347,1
347,2
347,3
347,4
347,4
347,5
347,5
347,5
347,6
347,6
347,7
347,8
347,8
347,9
347,9
To find the true
distance is the same
TABLE III.
anomaly U, coiTCsponding to the time (' from the perihelion in
as the mean dislance of tlie sun from the eajth.
days, in a parabolic orbit, whose perihelion
Log. of
(' days.
1.780
1,781
1,782
1,783
1,784
1,785
1,786
1,787
1,768
1,789
1,7911
1,791
1,792
1,793
1,794
1,795
1,796
',797
1,798
1.799
1,800
1,801
1,802
i,8o3
i,8o4
i,8o5
1,806
1,807
1,808
i.E
i,8io
1,812
i,8i3
i,8r4
i,8i5
i,8i6
1,817
1,818
1,819
1,820
1,821
1,822
1,823
1.824
1,825
1,826
1,827
1,828
1,829
i,83o
1, 83 1
1,832
1,833
1,834
True
Anom. U.
65,33,28,1
65,39,16,0
65,45,o3,9
65,5o,5i,9
65,56,39,9
66,02,28,0
66,08.16.1
66,i4,o4,3
66,19,52,5
66,25,40,7
66,31,29,0
66,37,17,3
66,43,o5,6
66,48,53,9
66,54,42,3
67,00,30,7
67,06,19,2
67,12,07,7
67,17,56,1
67,23,44,5
67,29,32,9
67,35,21,4
67,41,09,9
67,46,58,4
67,52,47,0
67,58,35,6
68,04,24,1
68,10,12,7
68,16,01,3
68,21,49,9
68,27,38,5
68,33,27,1
68,3g,i5,7
68,45,04,2
68,50,52
68,56,4:
69-02,29,9
69,08,18,4
69,14,07,0
69,19,55,5
69,25,44,0
69,31,32,5
69,37,21,0
69,43,09,5
69,48,58,0
69,54,46,5
70,00,34,9
70,06,23,3
70,12,11,7
70,18,00,0
70,23,48,3
70,29,36,6
70,35,24,9
70,41, i3,i
70,47,01,3
1.835 70,52,49,5
1.836 70,58,37,6
1.837 71,04,25,7
1.838 7i,io,i3,l
1.839 71,16,01,1
i,84o 71,21,49,8
347,9
347,9
348,0
348,0
348,1
348,1
348,2
348,2
348,2
343,3
348,3
348,3
348,3
348,4
348,4
348,5
348,5
348,4
348,4
348,4
348,5
348,5
348,5
348,6
348,6
348,5
348,6
348.6
348,6
348,6
348,6
348,6
348,5
348,6
348,6
348,5
348,5
348,6
348,5
348,5
348,5
348,5
348,5
348,5
348,5
348.4
348,4
348,4
348,3
348,3
348,3
348,3
348,2
348,2
348,2
348,1
348,1
348,1
348,0
348,0
348,0
Loj. of
t' days
True
Anom. U.
I,84o
1, 84 1
1,842
1,843
1,844
1,846
1,847
1,848
1,849
i,85o
1. 85 1
1,852
1,853
1,854
1,855
1,856
1,857
1,858
1,859
1,860
1,861
1,862
1,863
1,864
1,865
1,866
1,867
1,868
1,869
1,870
1,871
1,872
1,873
1,87
1,875
1,876
1,877
1,878
1,879
1,880
1,881
1,882
1,883
1,884
1,885
1,886
1,887
1.888
1,890
1,891
1,892
1,893
1,895
1,896
1,897
i,8g8
1,899
1,900
71,21,49,8
71,27,37,8
71,33,25,7
7i,3g,i3,6
71,45,01,4
71,50,49,1
71,56,36,8
72,02,24,5
72,08,12,1
72,13,59,7
72,19,47,2
72,25,34,7
72,31,22,1
72,37,09,4
72,42,56,7
72,48><i3,9
72,54,31,0
73,00,18,0
73,06,05,0
73,1 1,52,0
73,17,38,9
73,23,25,7
73,29,12,5
73,34,59,2
73,40,45,8
73,46,32,3
73,52,18
73,58,o5,i
74,o3,5i,4
74,09,37,6
74,1 5,23,7
74,21,09,7
74,26,55,6
74,32,4i
74,38,27,3
74,44, 1 3,0
74,49,58,6
74,55,44,1
75,01,29,6
75,07,15,0
75,i3,oo,3
75,18,45,5
75,24,30,6
75,3o,i5,5
75,36,00,4
75,41,45,2
75,47,29,9
75,53,14,4
75,58,58,9
76,04,43,3
76,10,27,5
76,16,1 1,6
76,21,55,6
76,27,39,5
76,33,23,3
76,39,07,0
76,44,50,6
76,50,34,1
76,56,17,5
77,02,00
77,07,43,8
348,0
347,9
347,9
347,8
347,7
347,7
347,7
347,6
347,6
347,5
347,5
347,4
347,3
347.3
347,2
347,1
347,0
347,0
347,0
346,9
346,8
346,8
346,7
346,6
346,5
346,4
346,4
346,3
346,2
346,1
346,0
345,9
345,9
345,8
345,7
345,6
345,5
345,5
345,4
345,3
345,2
345,1
344,9
344,9
344,8
344,7
344,5
344,5
344,4
344,2
344,1
344,0
343,9
343,8
343,7
343,6
343,5
343,4
343,2
343,1
343.0
Log. of
t' diiys.
I, goo
IfJOl
1,902
1,903
1,904
i,go5
1,906
1,907
i,go8
1,909
1,910
i,9H
1,912
1,913
1,914
1,915
1,916
i,9'7
1,918
i,9'9
1,920
1,921
1,922
1,923
1,924
1,925
1,926
1,927
1,928
1,92g
1,930
1,931
1,932
1,933
True
Anom. U.
1,935
1,936
1,937
1,938
1,939
1,940
1,941
1,942
1,943
1,944
1,945
1,946
1,947
1,948
1,949
1,950
1,951
1,952
1.953
1,954
1,955
1,956
1,957
1.958
1,959
1,960
77,07,43,8
77,13,26,8
77,19,09,7
77,24,52,5
77,3o,35,2
77,36,17,8
77,42,00,2
77,47,42,5
77,53,24,6
77,59,06,6
78,04,48,5
78,10,30,2
78,16.11,7
78,21,53,1
78,27,34,4
78,33,15,5
78,38,56,5
78,44,37,4
78,50,18,2
78,55,58,9
79,01,39,4
79,07,19,8
79,1 3,00,0
79,18,40,0
79,24,19,9
79,29,59,6
79,35,3g,2
7g,4i,i8,7
7g,46,58,o
7g,52,37,2
7g, 58, 16,2
8o,o3,55,i
8o,og,33,8
8o,i5,i2,3
8o,20,5o,7
8o,26,28,g
8o,32,o6,g
80,37,44^8
80,43,22,5
80,49,00,1
80,54,37,6
81,00,14,9
8 1, o5, 52,0
81,11,28,9
81,17.05,7
81,22,42,3
81,28,18,8
81,33,55,1
81,39,31,2
81,45,07,1
8i,5o,42,8
8i,56,i8,4
82,01,53,8
82,07,29,0
82,l3,04;I
82,i8,3g,o
82,24,13,7
82,29,48,2
82,35,22,5
82,40,56,6
82,46,30,5
Diff.
Log. of
L' days.
343,0
342,9
342,8
342,7
342,6
342,4
342,3
342,1
342,0
341,9
341,7
341,5
341,4
341,3
341,1
341,0
340,9
340,8
340,7
340,5
340,4
340,2
340,0
339,9
339,7
339,6
339,5
339,3
339,2
339,0
338,9
338,7
338,5
338,4
338,2
338,0
337,9
337,7
337,6
337,5
337,3
337,1
336,9
336,8
336,6
336,5
336,3
336,1
335,g
335,7
335,6
335,4
335,2
335,1
334,9
334,7
334,5
334,3
334,1
333,g
333,8
True
Anom. U.
i,g6o
1,961
1,962
1,963
1,964
1,965
1,966
1,967
1, 968
1,969
1,970
1,971
1,972
1,973
1,974
1,975
1,976
1.97'
1,978
1 ,979
1,980
1,981
1,982
1,983
1,985
1,986
1,987
1,990
1.99'
1-99
1,993
1,994
1,995
1,996
1,997
1 ,999
2,000
2,001
2,002
2,oo3
2,004
2,oo5
2,006
2,007
2,008
2,oog
2,010
2,01 1
2,012
2,Ol3
2,Ol4
2,Ol5
2)01
2,017
2,018
2,oig
2,020
Uiff.
82,46,3o,5
82,52,04,3
82,57,37,9
83,o3,ii,3
83,08,44,5
83,i4,i7,5
83,i9,5o,3
83,25.23,^
83,3o,55,
83,36,27,8
83,41,59.9
83,47,3i,7
83,53,o3,4
83,58,34,9
84,04,06,2
84,09,37,3
84.1 5.08.2
84,20,38,9
84, 26,0g ,4
84,3i,3g,8
84,37,09,9
84,42,39,8
84,48,09,5
84,53,39,0
84,59,08,3
85,04,37,4
85. 10.06.3
85,i5,35,o
85,2i,o3,5
85,26,3
85,31,59,9
85,37,27,7
85,42,55,4
85,48,22,8
85,53,5o,o
85,59,17,0
86,04,43,8
86,10,10,4
86,1 5,36,8
86,21,02,9
86,26,28,8
86,3 1, 54,5
86,37,20,0
86,42,45,3
86,48,10,4
86,53,35,2
86,58,59,8
87,04,24,2
87,09,48,4
87,15,12,3
87,20,36,0
87,25,59,5
87,31,22,8
87,36,45,9
87,42,08,7
87,47,31 ,3
87,52,53,7
87,58,1 5,8
88,o3,3
88,08,59.4
88,i4,2o,g
333,8
333,6
333,4
333,2
333,0
332,8
332,7
332,5
332,3
332,1
33 1, 8
33 1,7
33i,5
33 1,3
33i,i
33o,9
33o,7
33o,5
33o,4
33o,i
329,9
329,7
329,5
329,3
329,1
328,9
328,7
328,5
328,3
328,1
327,8
327,7
327,4
327,2
327,0
326,8
326,6
326,4
326,1
325,9
325,7
325,5
325,3
325,1
324,8
324,6
324,4
324,2
323,9
323,7
323,5
323,3
323,1
322,8
322,6
322,4
322,1
321, g
331,7
32 1,5
321,2
Log. of
i' days.
2,020
2,021
2,022
2,023
2,024
2,025
2,026
2,027
2,028
2,029
2,o3o
2,o3l
2,032
2,o33
2,o34
2,o3
2,o36
2,o37
2,o38
2,039
2,o4o
2,04 1
2
2,043
2,044
2,045
2,o46
2,047
2,o48
2,o4g
2,o5o
2,05l
2,o52
2,o53
2,o54
2,o55
2,o56
2,067
2,o58
2,059
2,060
2,061
2,062
2 ,o63
2,064
2,o65
2,066
2,067
2,068
2,o6g
2,070
2,071
2,072
2,073
2,074
2,075
2,076
2,077
2,078
2,070
2,080
True
Anom. U.
Diff.
88,14,20.9
88,ig,42;i
88,25,03,1
88,30,23.9
88,35,44,4
88,4i,o4,7
88.46,24,8
88,51,44,6
88,57,04,2
8g,o2,23,6
89,07,42,7
89,13,01,6
8g, 18, 20,2
89,23,38,6
89,28,56,7
89,34,14,6
89,39,32,3
89,44,49,8
89,50,07,0
89,55,23,9
90,00,4^
90,05,57,1
90,11, i3,3
90,16,29,3
90,21,45,
90,27,00,6
90,32,15,8
90,37 ,3o,8
gg,42,45,6
90,48,00,1
90,53,14,4
90,58,28,4
91,03,42,1
91,08,55,6
91,14,08,9
gi,ig,22,o
gi, 24,34,8
91,29,47,3
gi,34,5g,6
gi,4o,ii,6
91,45,23,4
91,50,34,9
91,55,46,1
92,00,57,1
92,06,07,8
92,11,18,3
92,16,2'
92,21,38,5
92,26,48,2
92,31,57,7
92,37,06,9
92,42,15,8
92,47,24,5
92,52,32,9
92,57,41,1
g3,02,49,o
93,07,56,'
93,13,04,1
93,18,11
g3,23,i8,o
93,28,24,5
321,2
321,0
320,8
320,5
320,3
320,1
7,9
7,7
7,5
7
6,9
6,7
6,5
6,2
6,0
5,8
5,5
5,2
5,0
4,8
4,5
4,3
4,0
3,7
3,5
3,3
3,1
2,8
2,5
2,3
2,0
1,8
1,5
1,2
1,0
0,7
0,5
0,2
0,0
'9.7
309,5
309,2
3o8,9
3o8,7
3o8,4
3o8,2
307,9
3o7,7
3o7,4
3o7
3o6,8
3o6,5
3o6,3
TABLE m.
To firiil ttio truc aiiuiiialv f^ coriespuiulinfi to the time t' from the peiihclioii in days, in a parabolic orbit, whose perihelion
distance is tlic same as tlie mean distance oi' the sun Iroiii tbe cartb.
Log. of
t' days
2,oSo
2,081
2,082
2,o83
2,084
a,o85
a, 086
2,087
2,o8S
2,089
2,090
2,091
2,092
2,og3
2.094
2,og5
2,096
2,097
2,098
2,099
2,100
2,101
2,102
2,io3
2,io4
2,I05
2,106
2,107
2,108
2,109
2,110
2,11 I
2,112
2,1 l3
2,Il4
2,ir5
2,116
2,117
2,118
2,119
2,130
3,121
2,122
2,123
2,124
2,125
2,126
2,127
2,128
2,129
3,i3o
2,l3l
2.l33
2;i33
2,i34
2,i35
2,i36
2,l37
2,i38
2,139
2,l4o
Truo
Anoin. U.
93,28,24,5
93,33,30,8
93,38,36,9
93,43,42,7
93,48,48,3
93,53,53,6
93,58,58,6
94,o4,o3,4
94,09,07,9
94,14,12,1
94,19,16,1
94,24,19,8
94,29,23,2
94,34,26,4
94,39,29,3
94,44,32,0
94,49,34,4
94,54,36,5
94,59,38,3
95,04,39,8
95,09,41,1
95,14,42,1
95,19,42,9
93,24,43,4
95,29,43,6
95,34,43,5
95,39,43,2
95,44,42,6
95,49,41,7
95,54,40,6
95,39,39,2
96,04,37,5
96,09,35,6
96,14,33,4
96,19,30,9
96,24,28,1
96,29,25,1
96,34,21,8
96,39,18,2
96,44,14,3
96,49,10,2
96,54,05,8
96,59,01,1
97,o3,56,i
97,08,50,9
97,i3,45,4
97,18,39,6
97,23,33,5
97,28,27,2
97,33,20,6
97,38, i3,7
97,43,06,5
97,47,59,1
97 ,52,5 1,3
97,57,43,3
98,02,35,0
98,07,26,5
98,12,17,6
98,17,08,5
98,21,59,1
98,26,49,4
3o6,3
3o6,t
3o5,8
3o5,6
3o5,3
3o5,o
3o4,S
3o4,5
3o4,2
3o4,o
3o3,7
3o3,4
3o3,2
3o2,9
3o2,7
3o2,4
3o2,I
3oi,8
3oi,5
3oi,3
3oi,o
3oo,8
3oo,5
3oo,2
299,9
299,7
299.4
299,1
298,9
298,6
398,3
298,1
297,8
297,5
297,2
297,0
296,7
296,4
256,1
295,9
295.6
295,3
295,0
294,8
294,5
294,2
293,9
293,7
293,4
293,1
292,8
292,6
292,2
292,0
291,7
291,5
291,1
290,9
290,6
290,3
290,1
Loi'. or
(' days,
2,l4o
2,l4l
2,142
2,143
2,i44
2,145
2,i46
2,l47
2,14s
2,149
2,i5o
2,l5l
2,l52
2,i53
2,1 54
2,i55
2,1 56
2,i57
2. 1 58
2. 1 59
2,160
2,161
2,163
2,i63
2,164
2,i65
2,166
2,167
2,168
2,169
2,170
2,171
2,172
2,173
2,174
2,175
2,176
2,177
2,178
2,179
2,180
2,181
2,183
2,i83
2,1
2,i85
2,186
2,187
2,188
2,189
2,igo
2,191
2,192
2,193
2,194
2,195
a,ig6
2,197
2,198
2,199
2,200
'I'iUO
Anom. U.
98,26,49,4
98,31,39,5
98,36,29,3
98,41, 1«,8
98,46,08,0
98,50,56,9
98,55,45,5
99,00,33,9
99,o5,22,o
9g,I0,09,8
99,14,57,3
99,19,44,6
99,24,31,5
gg, 29,18,3
99,34,04,7
99,38,5o,8
99,43,36,6
gg,48,22,3
gg,53,07,5
99,57,52,5
00,02,37,2
00,07,2
oo,i2,o5,9
00,16,49,8
00,21,33,4
00,26,16,7
00,30,59,7
00,35,42,5
oo,4o,23',o
00,45,07,2
00,49,49,1
oo,54,3o,7
00,59,12,0
oi,o3,53,i
01,08,33,9
oi,i3,i4,4
01,17,54,6
01,22,34,6
01,27,14,2
oi,3i,53,6
01,36,32,7
01,41,11,5
01,45, 5o,o
01,50,28,2
01,55,06,2
01,59,43,9
02,04.21,3
02,08^8,4
o2,i3,35,2
02,18,11,7
02,22,48,0
02,27,24,0
02,31,59,6
02,36,35,0
02,41,10,1
02,45,45,0
o2,5o,ig,6
02,54,53,8
03,59,27,8
o3,o4,oi,5
o3,o8,34,9
Diff.
290,1
289,8
289,5
289,2
288,9
288,6
288,4
288,1
287,8
287,5
287,3
286,9
286,7
286,5
286,1
285,8
285,6
285,3
285,0
284,7
284,5
284,2
283,9
283,6
283,3
283,0
282,8
282,5
282,2
281,9
281,6
281,3
281,1
280,8
280,5
280,2
280,0
279,6
279,4
279'!
278,8
278,5
278,2
278,0
277,7
277,4
277, t
276,8
276,5
276,3
276,0
275,6
275,4
275,1
274,9
274,6
274,2
274,0
273,7
373,4
273,1
Log
f days.
2,200
2,201
2,202
2,2o3
2,204
2,2o5
2,206
2,207
2,208
2,209
2,210
2,21
2,212
2,2l3
2,2l4
2,2l5
2,216
2,217
2,2lS
2,219
2,220
2,221
3,222
2,223
2,224
2,235
3,226
2,227
2,228
2,229
2,23o
3,23l
2,232
2,233
3,234
2,235
2,236
2,237
3,238
2,239
2,240
2,241
2,242
2,243
2,244
2,345
2,246
2,247
2,248
2,249
2,25o
2,25l
2,252
2,253
2,254
2,255
2,256
2,257
2,2 58
2,259
2,260
'I'rue
Aiioni. C/,
o3,o8,34,9
o3,i 3,08,0
o3, 1 7,40,9
o3,22,i3,5
03,26,45,8
o3,3i,i7,8
03,35,49,5
o3,4o,20,g
03,44,52,
03,49,23,0
o3,53,53,6
o3, 58, 23,9
04,02,53,9
04,07,23,6
04,11,53,1
04,16,22,3
04,20,5 1,3
04,25,19,8
04,29,48,1
04,34,16,1
o4,38,43,9
04,43,11,3
04,47,38,5
o4,52,o5,4
o4,56,32,
o5,oo,58,4
o5,o5,24,5
05,09,50,3
o5,i4;i5,7
o5, 1 8,40,9
o5,23,o5,g
o5,27,3o,5
o5,3 1,54,9
05,36,19,0
o5,4o,43,8
o5,45,o6,3
05,49,29,5
o5,53,52,5
o5,58,i5,i
06,02,37,5
06,06,59,6
06,11,21,5
06,1 5,43,0
06,20,04,3
06,24,25,3
06,28,46,0
06,33,06,4
06,37,36,5
06,41,46,3
o6,46,o5,g
),5o,25,2
),54,44,3
1,59,02,9
/,o3,2i,4
[07,07,39,6
,11,57,4
,i6,i5,o
,20,32,4
,24,49,4
,29,06,2
,33,33,7
273,1
272,9
272,6
272,3
271,7
271,4
271,2
270,9
270,6
270,3
270,0
269,7
269,5
269,2
368,9
268,6
268,3
268,0
267,8
267,4
267,2
266,9
266,7
266,3
266,1
265,7
365,5
365,2
265,0
264,6
264,4
264,1
263,8
263,5
263,2
363,0
262,6
262,4
363,1
261,9
261,5
261,3
261,0
260,7
260,4
260,1
259,8
25g,6
25g,3
259,0
258,7
258,5
258,3
257,8
257,6
257,4
357,0
256,8
256,5
256,2
jOg. ot'
' days.
3,360
3,26
2,262
2,363
2,264
2,365
2,266
2,267
2,268
2,26g
2,270
2,371
2,272
2,273
2,274
2,375
2,276
2,277
2,278
2,279
2,280
2,281
2,282
2,283
2,284
True
Anom, U-
107,33,33,7
107,37,38,9
107,41,54,8
107,46,10,5
107,50,25,9
107,54,41,0
107,58,55,8
io8,o3,io,3
108,07,24,6
108,1 1,38,5
108,1 5,52,2
io8,2o,o5,6
108,24,18
108,28,31,6
108,32,44.2
io8,36,56,5
108,41,08,5
108,45,20,3
108,49,31,7
108,53,42,9
108,57,53,8
109,02,0-
109,06,14,8
109,10,24,9
109,14,34,7
2,283 109,18,44,2
,286 log, 32, 53, 5
2,287
2,288
2,28g
2,2go
2,391
2,392
2,293
2,294
2,295
2,2g6
2,297
2,398
2,29g
2,3oo
2,3oi
2,302
2,3o3
2,3o4
2,3o5
2,3o6
2,3o7
2,3o8
2,3og
2,3lO
2,3ll
2,3l2
3,3i3
3,3i4
2,3 1 5
2,3i6
2,317
2,3i8
2,3i9
2,320
109,27,03,4
log, 3i, 11, 1
iog,35,ig,6
109,39,27,7
109,43,35,6
109,47,43,2
io9,5i,5o,6
109,55,57,6
110,00;
110,04,11,0
110,08,17,2
110,12,23,3
110,16,28,9
110,20,34,4
110,24,39,5
110,28,44,4
1 10,32,49,0
110,36,53,3
1 10,40,57,4
110,45,01,1
110,49,04,6
110,53,07,9
110,57,10,8
111,01, i3, 5
iii,o5,i5,9
111,09,18,1
iii,i3,ig,9
111,17,21
111,21,22,9
111,25,33,9
111,29,24,71
111,33,25,2
111,37,25,5
iii,4i,25,5
256,2
255,9
255,7
255,4
255,1
254,8
254,5
354,3
253,9
253,7
253,4
253,1
253,9
252,6
252,3
252,0
25i,8
25i,4
25l,2
25o,g
25o,6
25o,4
25o,i
24g,8
249,5
249,3
248,g
248,7
348,5
248,1
247,9
247,6
247,4
247,0
246,8
246,6
246,2
246,0
245,7
245,5
245,1
244.9
244,6
244,3
244,1
243,7
243,5
243,3
242,9
242,7
343,4
343,3
241,8
341,6
241,4
24 1,0
240,8
240,5
240,3
24o,o
339,7
Loy. of
V (lays.
2,330
2,321
2,323
2,323
2,324
2,325
2,326
2,327
2,328
2,329
2,33o
2,33i
3,333
3,333
2,334
2,335
2,336
2,337
2,338
2,339
2,34o
2,341
2,342
2,343
2,344
2.345
2,346
2,347
2,348
2,349
2,35o
2,35i
2,352
2,353
2,354
2,355
2,356
2,357
2,358
2,35g
2,36o
2,36i
2,362
2,363
2,364
2,365
2,366
2,367
2,368
2,369
2,370
2,371
3,373
2,373
2,374
2,375
2,376
2,377
2,378
2,379
3,38o
True
Anom. U.
111,41,25,5
111,45,35,3
111,49,34,6
111,53,23.8
111,57,22,7
112,01,21,3
112,05,19,7
112,09,17,8
Ii2,i3,i5,6
H2,i7,i3,i
112,21,10,4
112,25,07,4
112,29,04,3
113,33,00,7
112,36,56,9
112,40,52,8
112,44,/' "
112,48,43,9
ii2,52,3g,i
112,56,34,0
1x3,00,28,6
ii3,o4,33,o
1 1 3,08, 17,0
ii3,i2,io,8
1 13, 16,04,4
113,19,57,7
ii3,23,5o,7
113,27,43,4
Ii3,3i,35,g
113,35,28,1
ii3,3g,2o,i
ii3,43,ii,8
ii3,47,o3,3
ii3,5o,54,5
1 1 3,54,45,4
ii3,58,36,i
114,02,36,5
114,06,16,6
114,10,06,5
ii4,i3,56,i
114,17,45,5
114,31,34,6
114,25,33,4
114,29,11,9
ii4,33,oo,2
114,36,48,3
ii4,4o,36,i
114,44,23,6
ii4,48,io,g
114,51,57,9
114,55,44,6
ii4,59,3i,i
ii5,o3,i7,3
ii5,07,o3,3
115,10,49,0
ii5, 14,34,4
115,18,19,6
ii5,33,o4,6
11 5,25,49,3
115,29,33,7
1 1 5,33,17,9
Diff.
239,7
239,4
239,2
238,9
238,6
238,4
238,1
237,8
287,5
237,3
237,0
236,8
236,5
236,2
235,9
235,7
235,4
235,3
234,9
234,6
234,4
234,0
333,8
233,6
233,3
233,0
232,7
232,5
232,2
232,0
23 1,7
23i,5
23l,2
23o,9
23o,7
23o,4
23o,i
229,9
229,6
229,4
229,1
228,8
228,5
228,3
238,1
227,8
227,5
227,3
227,0
226,7
226,5
226,2
226,0
225,7
225,4
225,2
225,0
334,7
234,4
324,2
333,9
TABLE III.
To find the true anomaly U, corresponding to the time (' from the perihelion in days, in a parabolic orbit, whose perihelion
distance is the same as the mean distance of the sun from the earth.
Log. ol
(' days
2,38o
2.38i
2,382
2,383
2,384
2,385
2,386
2,387
2,388
2,389
2,390
2,3yl
2,392
2,393
2,394
2,395
2,396
2,397
2,398
2,399
2,400
2,4oi
2,4o2
2,4o3
2,4o4
2,4o5
2,4o6
2,407
2,4o8
2,409
2,4lO
2,4ll
2,4l2
2,4l3
2,4i4
2,4i5
2,4r6
2,417
2,4i8
2,419
2,420
2,421
2,422
2,423
2,424
2,425
2,426
2,427
2,428
2,429
2,43o
2, 43 1
2,43?
2,433
2,434
2,435
2,436
2,437
2,438
2,43g
2,44o
True
Anoin. U.
115,33,17,9
1 1 5,37,01,8
II 5,40,45,5
115,44,28,9
ii5,48,i2,o
ii5,5i,54jg
115,55,37,6
115,59,20,0
ii6,o3,o2,
116,06,43,9
116,10,25,5
116,14,06,9
116,17,48,0
116,21,28,8
116,25,09,4
116,28,49,7
116,32,29,8
116,36,09,6
116,39,49,2
116,43,28,5
116,47,07,7
ii6,5o,46,5
116,54,25,2
ii6,58,o3,5
117,01,41,6
117,05,19,5
117,08,57,2
117,12,34,5
117,16,11,6
117,19,48,5
117,23,25,2
117,27,01,5
117,30,37,7
117,34,13,6
117,37,49,1
117,41,24,4
117,44,59,5
1 17,48,34,3
117,52,08,9
117,55,43,3
117,59,17,3
ii8,o2,5i
118,06,24,8
118,09,58,1
Ii8,i3,3i,2
118,17,04,0
118,20,36,7
118,24,09,1
118,27,41,2
ii8,3i,i3,
118,34,44,8
ii8',38,i6,2
118,41,47,4
118,45,18,3
118,48,49,0
118,52,19,5
118,55,49,7
118,59,19,7
119,02,49,4
119,06,18,9
119,09,48,1
223,9
223,7
223,4
223,1
222,9
222,7
222,4
222,1
221,8
221,6
221,4
221,1
220,8
220,6
220,3
220,1
219,8
219,6
219,3
219,2
218,8
218,7
218,3
218,1
217.9
217,7
217,3
217,1
216.9
216,7
216,3
216,2
21 5,9
2i5,5
2 1 5.3
2l5,I
2i4,8
2i4,6
2 1 4.4
2l4,0
213.9
2i3^6
2i3,3
2l3,l
212,8
212,7
212,4
212,1
211,9
211,7
211,4
211,2
210,9
210,7
210,5
210,2
210,0
2og,7
209,5
209,2
209.0
Log. or
£ days.
2,44o
2,44 1
2,442
2,443
2,444
2,445
2,446
2,447
2,448
2,449
2,45o
2,45i
2,452
2,453
2,454
2,455
2,456
2,457
2,458
2,459
2,46u
2,461
2,462
2,463
2,464
2.465
2,466
2,467
2,468
2,469
2,470
2,471
2,472
2,473
2,474
2,475
2,476
2,477
2,478
2,479
2,480
2,481
2,482
2.483
2,485
2,486
2,487
2,4
2,4
2,490
2,491
2,49a
2,493
2,494
2,495
2,496
2,497
2,498
2,499
2,5oc
True
Anoiii. U.
19,09,48,1
19,13,17,1
19,16,45,9
19,20,14,4
19,23,42,7
19,27,10,7
19,30,38,5
19,34,06,1
19,37,33,4
ig,4i,oo,5
19,44,27,3
19,47,53,9
19,51,20,3
19,54,46,4
ig,58,i2,3
20,01,38,0
2o,o5,o3,4
20,08,28,6
20,11,53,5
20,l5,l8,2
,18,42,7
,22,07,0
>,25,3i,o
1,28,54,8
,32,18,3
20,35,41,6
20,39,04,7
20,42,27,5
20,45,50,1
20,49,12,5
20,52,34,7
20,55,56,6
20,59,18,3
2i,02,3g,7
21,06,00,9
2 1 ,09,2 1 ,9
21,12,42,7
21,l6,o3,2
21.19.23. 5
21,22,43,6
21,26,03,4
21,29,23,0
21,32,42,4
2 1,36,01,5
21,39,20,4
2i,42,3g,i
21,45,57,6
21,49,15,.
21,52,33,8
2 1. 55.5 1. 6
21,59.09,1
22,02,26,4
22,05,43,5
22,09,00,4
22,12,17,0
22,i5,33,4
22,18,49,6
22,22,o5,5
22,25,21,3
22,28,36,8
22, 3l, 52,0
Dlff.
209,0
208,8
208,5
208,3
208,0
207,6
207,3
207,1
206,8
206,6
206,4
206,1
205.9
2o5,7
2o5,4
205,2
204r9
204,7
2o4,5
204,3
2o4,0
2o3,8
2o3,5
2o3,3
2o3,I
202,8
202,6
202,4
202,2
201,9
201,7
201,4
201,2
201,0
200,8
200,5
200,3
200,1
99,8
99,6
99,4
99,1
98.9
98,7
98,5
98,2
g8,o
97,8
g7,5
97,3
97,.
96,9
96,6
96,4
96,2
95,9
95,8
95,5
95,2
95.1
Log. ol
2,5oo
2,5oi
2,502
2,5o3
2,5o4
2,5o5
2,5o6
2,507
2,5o8
2,509
2,5lO
2,5ll
2,5l2
2,5i3
2,5i4
2,5i5
2,5i6
2,5i7
2,5i8
2,519
2,520
2,521
2,522
2,523
2,524
2,525
2,526
2,527
2,528
2,52g
2,53o
2,53i
2,532
2,533
2,534
2,535
2,536
2,537
2,538
2,539
2,540
2, 54 1
2,542
2,543
2,544
2,545
2,546
2,547
2,548
2,549
2,55o
2,55i
2,552
2,553
2,554
2,555
2,556
2,557
2,558
2,559
2,56o
True
Anoni . U.
d m s
22,3l,52,0
22,35,07,1
22,38,22,0
2 2, 4 1,36,6
22,44, 5i,o
22,48,05,2
22,5l,ig,2
22,54,33,0
22,57,46,6
23,oo,5g,9
23,o4,i3,o
23,07,25,8
23,10,38,4
23,i3,5o,7
23,17,02,9
23,20,14,8
23,23,26,5
23,26,38,0
23,2g,4g,2
23,33,00,3
23,36,11,1
23, 3g, 2 1,7
23,42,32,1
23,45,42,3
23,48,52,3
23,52,02,0
23,55,11.5
23,58,20,8
24,01,29,9
24,04,38,8
24,07,47,5
24,10,55,9
24,14,04,2
24,17,12,2
24,20,20,0
24,23,27,6
24,26,35,0
24,29,42,2
24,32,4g,i
24,35,55,9
24,39,02,4
24,42,08,7
24,45,14,8
24,48,20,7
24,51,26,4
24,54,31,8
24,57,37,1
25,00,42,1
25,o3,47,o
25,o6,5i,6
25,09,56,0
25,l3,00,2
25,16,04,2
25,19,08,0
25,22,11,5
25,25, i4-Q
25,28,18,0
25, 3 1,2 1,0
25,34,23,7
25,37,26,2
25,40,28,5
Diff.
95,1
94:9
94,6
94,4
94,2
g4,o
g3,8
93,6
g3,3
g3,i
92,8
92,6
92,3
92,2
9'>9
gi,7
91,5
91,2
91,1
go,8
90,6
90,4
90,2
90,0
89,7
8g,5
8g,3
69,1
88,9
88,7
88,4
88,3
88,0
87,8
87,6
87,4
87,2
86,9
86,8
86,5
86,3
86,1
85,9
85,7
85,4
85,3
85. o
84.9
84,6
84,4
84,2
84,0
83,8
83,5
83,4
83,1
83,o
82,7
82,5
82,3
82,1
Log. of
( (luys
2,56o
2,56l
2,562
2,563
2,564
2,565
2,566
3,567
2,568
2,56g
2,570
2,571
2,572
2,573
2,574
2,575
2,576
2,577
2,578
2,579
2,58o
2,58i
2,582
2,583
2,584
2,585
2,586
2,587
2,588
2,589
2,590
2,591
2,5g2
2,593
2,594
2,595
2 ,596
2,597
2,5g8
2,5g9
2,600
2,601
2,602
2,6o3
2,604
2,6o5
2,606
2,607
2,608
2,6og
2,610
2,611
2,612
2,6i3
2,614
2,6i5
2,616
2,617
2,618
2,6ig
2,620
Ti uc
Au 0111. U.
25,40,28,5
25,43,30,6
25,46,32,5
25,49,34,2
25,52,35,7
25,55,37,0
25,58,38,1
26,01,38,9
26,04,39,6
26,07^^0,1
26,10,40,3
26,13,40,4
26,16,40,2
26,19,39,8
26,22,39,3
26,25,38,5
26,28,37,5
26, 3 1,36,3
26,34,35,0
26,37,33,4
26,4o,3i,6
26,43,29,6
26,46,27,4
26,49,25,0
26,52,22,4
26,55,19,6
26,58,16,6
27,01, i3,4
27,04,10,0
27,07,06,4
27,10,02,6
27,12,58,6
27,15,54,4
27,18,50,0
27,21,45,4
27,24,40,5
27,27,35,5
27,3o,3o,3
27,33,24,9
27,36,19,3
27,3g,i3,5
27,42,07,5
27,45,01,3
27,47,54,9
27,50,48,3
27,53,41,6
27,56,34,6
27,59,27,4
28,02,20,1
28,05,12,5
28,08.04.7
28,10,56,8
28,13,48,7
28,16,40,3
28,19,31,8
28,22,23,1
28,25,14,2
28,28,05,1
28,30,55,8
28,33,46,3
28,36.36,7
82,1
81,9
81,7
81,5
81,3
81,1
80,8
80,7
80,5
80,2
80,1
79,8
79,6
79,5
79,2
79,0
-8.8
78,7
78,4
78,2
78.0
77,8
77,6
77,4
77.2
77,0
76.8
76.6
76^
76,2
76,0
75,8
75,6
75,4
75,1
75,0
74,8
74,6
74,4
74,2
74,0
73,8
73.6
73,4
73,3
73,0
72,8
72,7
72,4
72,2
72,1
71,9
71.6
71,5
71,3
71,1
70,9
70,7
70,5
70,4
70,2
Log. oi
I' days.
2,620
2,621
2,622
2,623
2,624
2,625
2,626
2.627
2,628
2,629
2,63o
2,63 1
2.632
2,633
2,634
2.635
21636
2,637
2,638
2,63g
2,640
2,64i
2 ,642
2.643
2,644
2,645
2,646
2,647
2,648
2,64g
2,65o
2,65 1
2,652
2,653
2,654
2,655
2,656
2,657
2,658
2,659
2,660
2,661
2,662
2,663
2,664
2,665
2,666
2.667
2;668
2,669
2,670
2,671
2,672
2,673
2,674
2,675
2,676
2,677
2,678
2,679
2,680
True
Anoni. U.
28,36,36
28,3g,26,g
28.42,16.8
28,45.06,5
28,47,56,0
28,50,45.4
28,53,34.5
28,56,23,5
28,59,12
29,02,00,9
29,04,49,3
29,07,37.5
29,10,25,5
29,13, i3,3
29,16,01,0
29,18,48,4
29,21,35,7
29,24,22,8
29,27,0g
29,29,56,4
29,32,43,0
2g,35,2g.3
29,38,15,5
29,41,01,4
29,43,47,2
29,46,32,8
29,49,18,3
29,52,03.5
29,54,48,6
29,57,33,4
3o,oo,]8,i
3o,o3,02.6
30,05,46^8
3o,o8,3o,9
3o,i 1,14,8
3o,i3,58,6
3o, 16,42, 1
3o,ig,25,5
3o,22,or
30,24,5
30,27,34,5
3o, 30,17, 3
30,32,59,6
3o,35,4i,9
3o,38,24,o
3o,4i,o6,o
3o,43,47,7
30,46,29,3
30,49,10,7
3o,5i,52,o
3o, 54,33,0
3o,57,i3,S
30,59,54,5
3i,o2,35,o
3i,o5,i5,3
31,07,55,5
3i, 10,35, 5
3i,i3,i5,3
3i,i5,54,9
3i,i8,34,3
3i,2i,i3.6
Dim
70,2
6g,9
6g,7
6g,5
6g,4
69,1
69,0
68,8
68,6
68,4
68,2
68,0
67,8
67,7
67,4
67,3
67,1
66.9
66,7
66,6
66,3
66,2
65.9
65,8
65,6
65,5
65,2
65,1
64,8
64,7
64,5
64,2
64,1
63,9
63,8
63,5
63,4
63,2
63 ,0
62,8
62,7
62,4
62,3
62,1
62,0
61,7
61,6
61,4
61,3
61,0
60,8
60,7
60,5
60,3
60,2
60,0
59,8
59,6
59,4
59,3
59,0
TABLE 111.
To find llie true anomaly U, corresponding to the time t' from llie perilielion in days, in a parabolic orbit,
distance is llie same as ihe mean distance of the sun Irom the earth.
whose perilielion
I (lay-
2,6So
2,681
2,683
2,684
2,685
2,686
2,687
2,688
2,689
2,6go
2,691
2,693
2,693
2,(59-1
2,695
2,696
2 ,697
2,(k)6
21699
2,700
2,701
2,703
2,703
2,704
2,705
2,706
2,707
2,708
2,709
2,710
2,71 1
2,712
2,7 1 3
2,714
2,71 5
2,716
2,71
2,718
2,719
2,720
2,721
2,722
2,723
2,724
2,725
2,736
2,727
2,728
2,739
2,73o
2,73i
2,733
2,733
2,73i
2,735
2,736
2,737
2,738
2,739
2,740
'i'rue
Aiioiii. U.
1 3 1.2 1. 1 3.6
i3i,23,52,6
i3i,26,3i,5
131,29,10,2
1 3 1, 3 1, 48,8
1 3 1, 34,27, 1
1 3 1.37.05. 3
1 3 1.39.43.4
1 3 1,42, 2 1, 3
i3i,44,58,8
1 3 1, 47,36,3
i3i,5o,i3,6
i3i,52,5o,8
1 3 1.55.27.7
1 3 1. 58 .04.5
i32,oo,4r,i
1 32.03.17.6
i32,o5,53,8
132,08,29,9
i32,ii,o5,8
i32,i3,4i,6
132,16,17,3
i32, 18,52,6
l32,2t,27,
132,24,02,9
133,26,37,8
i32,2g,i2,5
i32,3i,47,i
i32,34,3i,5
132,36,55,7
132,39,29,7
i32,42,o3,6
132,44,37,3
i32,47,io,8
132,49,44,2
133,52,17,4
i32,54,5o,5
i32,57,23,3
i32, 59,56,0
133,02,28,5
i33,o5,oo,g
133,07,33,1
i33,io,o5,i
i33,i2,36,9
i33,i5,o8,6
133,17,40,1
i33,2o,i [,4
133,22,43,6
i33,25,i3,6
133,27,44,4
i33,3o,i5,i
133,32,45,6
1 33,35, 16,0
133,37,46,2
i33,4o,i6,2
i33,42,46,i
i33.45,i5.8
1 33,47,45,3
i33,5o,i4,7
133,53,43,9
i33,55,i2,9
59,0
58,9
58,7
58,6
58,3
58,3
58,1
57,8
57,6
57,5
57,3
57,2
56,9
56,8
56,6
56,5
56,2
56,1
55,9
55,8
55,6
55,4
55,2
55,1
54,9
54,7
54,6
54,4
54,3
54,0
53,9
53,7
53,5
53,4
53,2
53,1
52,8
52,7
52,5
52,4
53,2
52,0
5i,8
5. ,7
5i,5
5i,3
5i ,2
5i,o
5o,8
5o,7
5o,5
5o,4
5o,2
5o,o
49,9
49,7
49,5
49,4
49.2
49,0
48. q
L d&yi.
2,740
2,741
2,743
2,743
2,744
2,745
2,746
2,747
2,748
2,749
2,75o
2,75l
2,753
2,753
2,754
2,755
2,756
2,757
2,758
2,759
2,760
2,761
3,763
3,763
2,764
2,765
2,766
3,767
3,768
3,769
2,770
2,771
2,772
2,773
2,774
2,775
3,776
2,777
2,778
2,779
3,780
2,-81
2,783
2,783
2,784
2,785
2,786
2,78-
2,788
2,789
2,790
2,791
2,793
3,793
2,794
2,795
3,796
2,797
3,798
2,799
2,800
T,u.j
Ailun). £/.
l33,55,I3,9
i33,57,4i,8
1 34 ,00, 1 0,6
134,03,39,3
1 34,05,07,6
134,07,35,8
1 34,10,03,9
i34,i2,3i,8
1 34, 14,59,6
134,17.27,2
134,19,54,7
l34,22,22,0
134,34,49,1
134,27,16,0
134,39,42,8
i34,32,og,4
134,34,35,9
134,37,02,3
134,39,38,4
1 34,4 1, 54,4
1 34,44,30,3
134,46,45,9
i34,4g,ii,4
1 34.5 1.36.8
134,54,02,0
1 34.56.27.0
i34,58,5i.9
i35,oi,i6,6
i35,o3,4i,3
i35,o6,o5,6
135,08,39,8
i35,io,53,9
i35, 13,17,9
i35,i5,4i,7
i35,i8,o5,3
135,30,28,8
135. 33.53.1
i35,35,i5,3
i35,27,38,2
i35,3o,oi,
i35,32,23,8
1 35,34,46,3
135,37,08,7
1 35.39.30.9
i35,4i,53,o
i35,44,i4,9
135,46,36,7
135,48,58,3
i35, 51,19,8
1 35,53,4 1,3
135,56,03,3
135,58,33,3
1 36,00,44,3
i36,o3,o4,9
i36,o5,25,5
1 36,07,45,9
1 36, 10,06, 3
1 36, 13, 36,3
1 36, 1 4,46,3
i36,i7,o6,i
i36,i9,35,8
a22
148,9
148,»
i48,6
i48,4
1 48,2
i48,i
147,9
i47,a
147,6
147,5
147.3
147.1
i46,9
1 46,8
1 46,6
i46,5
i46,3
i46,3
1 46,0
145,8
■ 45,7
i45,5
i45,4
145,3
145,0
144,9
144,7
144,6
i44,4
i44,3
i44,i
144.0
143,8
143,6
i43,5
143,3
143,1
143,0
142,9
143,7
i43,5
142,4
143,3
143,1
i4i,9
i4i,8
i4i,6
i4i,5
i4i,4
i4i,i
i4r,o
140.9
1 4" ,7
1 40,6
i4o,4
i4o,3
1 40,1
i4o,o
139,8
139,7
i3q.5
3,800
3,801
3,8o2
3,8o3
2,804
2,8o5
2 ,806
2,807
2,808
3,809
2,810
2,811
2.812
2,8i3
2,814
2,8i5
2,816
3,817
2,818
3,819
3,820
2,821
2.822
2;823
2,824
2,835
3,826
2,827
2,828
2,829
2,83o
2,83 1
2,833
3,833
2,834
2,835
2,836
2.837
3,838
3,839
3,84o
2, 84 1
2,843
2,843
2,844
2,845
2,846
2,847
3,6
2,849
2,85o
3,85 1
2,853
2,853
2,854
2,855
2,856
2,85
3,858
3,859
3,860
'I'rue
A null). {/,
36,19,35,8
36,31,45,3
36,24,04,7
36,36,23,9
36,28,42,9
36,3 1,0 1, 8
36,33,30,6
36,35,39,3
36,37,57,7
36,4o,i6,o
36,43,34,3
36,44,52,2
36,47,10,1
36,49,27,8
36,5 1,45,4
36,54,02,8
36,56,30,1
36,58,37,2
37,00,54,2
37,03,1 1,1
37,05,27,8
37,07,44,4
37,10,00,8
37,12,17,0
37,14,33,1
37,16,49,1
37,19,04,9
37,31,20,6
37,23,36,2
37,25,5i,6
37,28,06,8
37,3o,2i,9
37,32,36,9
37,34,51,7
37,37,06,4
37,39,20,9
37,41,35,3
37,43,49,6
37,46,03,7
37,48,17,6
37,50,3 1, 4
37,52,45,1
37,54,58,7
37,57,12,1
37,59,25,3
38,01,38,4
38,o3,5i,4
38,o6,o4,3
38,08,16,9
38,10,39,4
38,i2,4i,8
38,14,54,1
38,17,06,2
38,19,18,3
38,3i,3o,o
38,33,4i,7
38,35,53,3
38, 38,04
38,3o,i6,o
38,33,37,3
38,34,38,3
39,5
39,4
39,2
39,0
38,9
38,8
38,6
38,5
38,3
38,3
38,0
37,9
37,7
37,6
37,4
37,3
37,1
37,0
36,9
36,7
36.6
36,4
36,3
36,1
36,0
35,8
35,7
35,6
35,4
35,2
35,1
35,0
34,8
34,7
34,5
34,4
34,3
34,1
33,9
33,8
33,7
33,6
33,4
33,2
33,1
33,0
32,8
33,7
33,5
33,4
33,3
33,1
33,0
3i,8
3t,7
3i,6
3 1. 4
3i,3
3l,3
3 1,0
3o,q
Log. al
I .luy-~.
3,660
3, 861
2,863
3,863
2,864
3,865
2,866
3,867
2,868
2,869
3,870
2,871
2,873
3,873
2,874
2,875
2,876
2,877
2,878
2,879
2,880
3,881
2,882
3,883
2,884
2,885
2,886
3,887
3,8
2,890
3,8gi
3,892
2,8g3
2,895
2,896
2,897
2,898
2,899
2, goo
2,901
2,902
2,903
2,904
3,9o5
2 ,906
3,90
2,908
2,909
2,910
2.9"
3,913
2913
2,914
2.915
2,916
2.917
2,918
2.919
3,930
Ti uo
Anoni, U.
38,34,38,3
38,36,49,1
38,38,59,8
38,4 1, 10,4
38,43,30,9
38,45,31,3
38,47:41,4
38,4g,5i,4
38,53,01,3
38,54,11,1
38,56,20,8
38,58,30,3
3g,oo,3g,6
39,03,48,9
39,04,58,0
39,07,06,9
3g,og,i5,7
39,1 1,34,4
39,13,33,0
3g,i5,4i,4
39,17,49,7
39,19,57,8
39,23,05,8
3g,34,i3,7
3g,36,2i,5
39,38,39,1
3g,3o,36,6
39,33,43,9
39,34,51
39,36,58,3
39,39,05,1
3g,4i,i i.g
3g,43,i8,6
3g,45,35,3
39,47,31,6
39,49,37,9
39,51,44,0
39,53,50,0
3g,55,55,9
39,58,01,7
40,00,07,3
40,03,I3,8
4o,o4,i8,3
40,06,33,4
40,08,28,5
40,10,33,5
4o,i 2,38,3
4o, 14,43,0
40,16,47,6
40,18,53,0
40,20, 56, 4
4o, 23,00,6
40,35,04.6
40,37,08,6
40,39,13,4
4o,3i,t6,i
40,33.19,6
4o,35,33,i
40,37,36.4
40.39,39,6
40.41.32.6
3o,9
3o,7
3o.6
3o,5
3o,3
3o,2
3o,o
29.9
39,8
29,7
29,5
39,3
39,3
29,'
28,g
28,8
38,7
28,6
28,4
28,3
27.9
27,8
27,6
37.5
37,3
37,2
27,1
26,9
36,8
26,7
36,6
36,4
36,3
36,1
36,0
25,9
35,8
25,6
25,5
35,4
35,3
25,1
25,0
34,8
34,7
34,6
24:
24,4
24,2
34,0
34,0
33,8
33,7
33,5
33,5
33,3
33,3
33,0
33.Q
1..1J;. t
I' (l.iys
2,920
2,931
2,923
2,933
2,924
2,925
2 ,926
2,927
2,938
2,929
2,930
3,931
2,933
2,933
2,934
2,935
2,936
2,937
2,g38
2,93g
3,g4o
2,941
3.943
3,943
2,944
2,945
2.946
2,947
3,948
2,949
3,950
3,951
3,953
2,953
2,954
2,956
2-957
3,958
2,959
3,960
2,961
3,963
3,963
3,964
2,965
2,966
2,967
2,g68
2,969
2,970
2,971
2,9'
2,973
2, .974
2,975
2,976
2,977
2,978
2-979
2.<
'J'rue
Anom. U.
40,4 1,32, 6
40,43,35,5
4o,45,38,3
40,47,41,0
40,49,43,5
40,51,45.9
4o,53,48,3
4o,55,5o,3
40,57,52,4
40,59,54,3
4 1.01.56.0
41,03,57,7
4 1, 05,59, 3
41,08,00,6
4i,io,oi,g
4i,i2,o3,o
4i,i4,o4,i
4i,i6,o5,o
4i,i8,o5,7
41,20,06,4
4i,22,o6,g
41,24.07,3
4i .26,07,6
4iJ38,o7,8
4 1,30,07,8
41,32,07,7
41,34,07,5
41,36,07,1
4i,38,o6,7
4 1.40.06.1
4i,43,o5,4
4 1, 44 .04 ,6
4i,46,o3,6
4 1,48,03,5
4i,5o,oi,3
41,52,00,0
4r,53,58,6
41,55,57,0
41,57,55,3
41,59,53,5
43,01, 5i,6
43,o3,4g,6
43,o5,47,4
43,07,45,1
43,09,42,7
43,1 i,4o,3
43,13,37,5
43,i5,34,8
43,17,31,9
42,19,28,9
42,21,35,7
43,33,33,5
43,35,19,1
42,27,1 5,f
42,29,I2,C
42,3i,o8,3
42,33,04,4
42,35,00,4
43,36,56,4
43,38,53,3
43,40,47.'
22,9
33,8
3 2,7
33,5
3 2,4
33,3
33,1
33,1
31,9
21,7
21,7
31,5
21,4
21,3
21,1
21,1
20,9
20,7
20,7
20,5
20,4
20,3
20,2
20,0
19,9
19,8
19,6
19,6
19,4
19,3
19,2
19,0
18,9
18,8
.8,7
18,6
18,4
18,3
18,2
18,0
17,8
17,7
17,6
17,5
17,3
.7,3
17,1
17,0
16,8
16,8
16,6
16,5
16,4
16,3
16,1
16,0
16,0
1 5,8
1 5,6
1 5.6
TABLE III.
To finil the true anomaly U, correspondin» to the time V liora the peiihelion in days, in a p.iiab6lic orbit, whose perihelioB
distance is tlie same as tlie mean distance of the sun from the earth.
2,982
2,9"-
2,9
2,985
2,986
2,987
2,988
2,989
2,990
2-99'
2,992
2.993
2,994
2,995
2,996
2.997
2,998
2>999
3,000
True
Ailuin. U.
142,40,47,8
142,42.43,4
142,44.38,8
142,46,34,3
142,48,29,4
142, 5o, 24, 5
142,52,19,4
142,54,14,3
142,56,09,1
i42,58,o3,7
142,59,58,2
143,01,52,6
i43,o3,46,9
i43,o5,4i,i
143,07,35,2
143,09,29,1
143, 1 1,23,0
143, i3, 16, 7
i43,i5,io,3
i43,i7,o3.8
143,18,57,3
ii5,6
1 1 5,4
1 1 5,4
Il5,2
1 1 5, 1
"i4,9
1 14,9
ii4,8
1 14.6
114,5
ii4,4
1 1 4,3
Il4,2
ii4,i
1 1 3,9
1 1 3,9
1 1 3.7
1 1 3,6
ii3,5
ii3,5
£' liay
3,00
3,0
3,02
3,o3
3,04
3,o5
3,06
3,07
3,oS
3,oy
3,10
3,11
3,12
3,i3
3,i4
3.
3,16
3,17
3,18
3,19
3,20
3,21
3,22
3,23
3,24
3,25
True
Anoin. U.
143,18,57,3
143,37,44,7
i43,56,2o,8
i44,i4,45,6
144,32,59,4
i44,5i,o2,5
145,08,54,8
i45,a6,36,5
145,44,07,7
146,01,28,(5
146,18,39,3
146,35.39,9
i46,52,3o,5
147,09,11,3
147.25,42,5
147,43,04,1
i47,58,i6,2
148,14.19,0
i48,3o,i2,5
148,45,56,9
149,01,32,3
149,16,58,9
149,33,16,6
149.47.25,7
1 50,02,26,3
i5o,i7,i8,2
1127,4
1 1 16,1
1 104,8
1093,8
[o83,i
1072,3
1 06 1 ,7
105l,2
1 o4o,9
io3o,7
1020,6
I o ! 0,6
1000,8
991.2
981,6
973,1
962,8
953,5
944,4
935,4
926,6
9 '7.7
909.'
900,5
892,0
883,7
Luj;. ol
/' days.
3,25
3,26
3,2-
3,2b
3,29
3,3o
3,3i
3,33
3,33
3,34
3,35
3,36
3,37
3,38
3,39
3,40
3,41
3,42
3,43
3,44
3,45
3,46
3.4-
3,48
3,4o
3,5o
'I'rue
Anom. V.
l5o,I7,l8,2
l5o,32,ol,c,
1 50,46,37, j
1 5 1, 01, 04,5
i5i,i5,23,5
151,29,34,7
i5i,43,38,o
1 5 1. 57.33.5
l52,II,21,l
i52, 25,01,3
i52,38,34,o
152,51,59,1
1 53,05,16.9
i53,i8,27;5
i53,3i,3o,8
153,44.27,1
153,57,16,3
154,09,58,5
i54,22,33,g
1 54.35.02.6
154,47.24,4
154,59,39,6
1 55, 1 1,48,
i55,23,5o,5
155,35,46.2
1 55,47,35,6
I) I ft'.
883,7
875,4
867,2
859,0
85i,2
843,3
835,5
827,6
820,2
812,7
8o5,i
797.8
790,6
783,3
776,3
769.2
762,2
755,4
748,7
74 1, 8
735,2
728,8
722,1
7 1 5,7
709,4
7o3,i
l.„g. ..!■
1 (J.iy3.
3,5o
3,5i
3,53
3,53
3,54
3,55
3,56
3,57
3,58
3,59
3,60
3,61
3,63
3,63
3,64
3,65
3,66
3.67
3,68
3,69
3',70
3,71
3,73
3,73
3.7
3i75
Anoin. U.
55,47,35,f
55,59,18,7
56,10,55,5
56,32,26,3
56,33,5o,8
56,45,09,4
56,56,22,1
57,o7,28,fc
57,18,29,7
57,29,24,8
57,40,14,3
57,5o,58,o
58,01,36,3
58,i2,o8,t
58,22,36,0
58.32,57,7
58,43,14,1
58,53,25,3
59,o3,3i,f
59,1 3,3 1,6
59,23,27,1
59,33,17,5
59,43,02,^
59,53,43,3
60,02,18,6
60,11,49,1
703,1
696,8
690,7
684,6
678,6
672,7
666,7
660, g
655,1
649.5
643,7
638,2
633,6
637,3
621,7
616,4
611,1
6o5,8
600,6
595,5
590,4
585,3
58o,4
575,4
570,5
565,6
Lug. ul
t days.
3,75
3,76
3,77
3,7b
3,79
3,80
3,81
3,82
3,83
3,84
3,85
3.86
3,87
3,8t
3,89
3,90
3,91
3.93
3,93
3,94
3,95
3,96
3,07
3,cjb
3,9t
4,00
'J'rue
Anorn. t/.
60,11,49,1
60,21,14,7
60. 30. 35.6
60.39.51 .7
6o,4g,o3,(.
6o,58,og,7
61,07,11,9
61,16,09.5
61 ,25,02,5
6i,33,5i,i
61,42,35,3
6i,5i,i5,o
61,59,50,4
63,08,31
62,16,48,3
62,25,11,0
62,33,39,4
63,41,43.6
63,49,53,b
62,58,00,1
63,o6,o2,o
63,i4,oo,i
63,21,54,2
63,39,44,5
63,37,3o,t
63 ,45,1 3,4
5,00 172,32,09,2
565,6
56o,9
556,1
55i,3
546,7
543,2
537,6
533,0
528,6
524,1
519,8
5i5,4
5ii,i
5o6,8
5o2,7
498,4
494,3
490,3
486,3
478,1
474,1
470,3
466,3
462,6
TABLE IV.
This tahle is given for the purpose of computing the true anomaly k, from the time t from the perihelion ; in a very excentrical
orbit, whether it be an ellipsis or hyperbola. The excentricity is represented by c, and the periheliim distance by D. In using
this table we must first compute, by means of Table III., the anomaly U, corresponding to the time t from the peiihelion, in a parabola,
whose perihelion distance is D. To this value of U, we must apply a correction, of the first order, S. (1 — f) ; first proposed by
Simpson, and which corresponds to the function [697]. When 1 — e is somewhat large, and great accuracy is required, we must apply
a correction of the second order B. (1 — e)3 ; first computed by Bessel. The logariihms of the values of S, B, in se.'iagesimal seconds,
are given in Table IV., for every degree of the anomaly U, with their differences; and when any one of these values is negative, the
letter n is annexed to its logarithm. For intermediate values of U, we must use the common rules o( interpolation. The logarithms
of S are given to seven places of decimals, and those of B to five places; but in most cases it will be sufficiently accurate, if we reject
the two last of these ligures. The logarithm of S, added to the log. (1 — e), gives the logarithm of Simpson's correction; and the
logarithm of B, added to 2 log. (1 — e), gives the logarithm of Bessel's correction. In symbols we have,
v= V'-{-S.{l—e) + B . (1— e)2; [In an ellipsis].
v=U—S.{e — l)-i-^ ■ (<:— 1)-; ['n a hyperbola].
EXAMPLE.
We shall suppose that with the lime t from passing the perihelion, and the perihelion distance D, the anomaly in a parabola is found,
by means of Table III., to be i!7= SO"*. Then it is required to find the true anomaly v ; in an ellipsis, whose excentricity is e = 0,99 ;
and in a hyperbola, whose excentricity is e= 1, 01.
In an ellipsis.
Given e:=o,gg and J7=5o'' to find i).
S log. 4,383 i7„
I — e log. 8,00000
• 24i ,6 log. 2,383i7h
B log. 3,8341 7«
(i — e) log. 8,00000
same 8,00000
-o*,7 log. g,824i7„
Simpson's correction,
Bessel's correction.
1/= 5o 00 00 ,0
— 4 01 ,6
— o ,7
Tiue anoTialy 0 = 49 55 57,7
III a hyperbola.
Given e = i, 01 and U^ 5o to find v.
The calculation of the corrections of Simpson and Bessel, is the
same as in the ellipsis; the only difference is in the sign of Simpson's
correction.
C7'=: 5o' 00'" 00'' ,0
Simpson's correction, -f" 4 01 ,6
Bessel's correction, — o ,7
True anomaly » = 5o o4 00 ,9
TABLE IV.
To fiiui tlio line nnomaiy v, in a very eccentric ellipsis or hyperbola, fiom the corresponding anomaly ?7in a parabola ; according
to Simpsoir.s inelhoil, improved by Hessel. ^^
lO
II
12
i3
i4
i5
i6
17
18
19
20
21
22
23
24
25
26
27
28
29
3o
3i
3a
33
34
35
36
37
38
39
40
4i
42
43
44
45
46
47
48
49
5o
5i
52
53
54
55
56
57
58
59
60
i.ug. or s.
sex. seconds.
Infîn. neg.
2,g5425,35n
3,35491,94.1
3,43o56,98„
3,5548g,o5„
3,65ioo,44n
3,72941,16,1
3,79500,5611
3,85i66,48n
3^0 1 30,3971
3,94536,62n
3,98488,07,.
4,02060, 54ii
4,o53i 1, 8 In
4,08285,95,.
4,iioi9,93n
4, 1 354o,67n
4,15871,98»
4,i8o3j,95n
4,20039,68,1
4,21905,61»
4,23642,13,.
4,2535g,o6n
4,36764,78„
4,28166,46,.
4,29470,54»
4,30683,33»
4, 3 1806,61,.
4,33847.5o„
4,33808,67,,
4,34693. 32„
4.355o3,8i„
4,36343,83,,
4,36912,33,1
4,37513,93,,
4, 38049, 1 4n
4,385iç,o3„
4,38924,52,,
4.39266,33,,
4,39544,73,,
4,39760,06,,
4,3991 3, 36„
4,4oooi,i3„
4,4ooj6,oi„
4,39986,33„
4,39881,03,,
4,39708,93,,
4.39468,40»
4,3yi57,gon
4,38774,94»
4,383i7,o6»
4,37781,33,,
4,37164,40,1
4,36462,19»
4,35670,34»
4,34783,17»
4,33795,o3„
4,33698,74„
4,3i486,o8„
4,30147.45,,
4,3867i,76„
I'irsl Dili'.
Infinite.
3oo(i6,59
i7565,o4
13432,07
9611,39
7840,73
6559,40
5665,9a
4963,91
44o6,23
3951,45
3572,47
325i,27
2974,14
2733,98
25ao,74
233i,3i
2160,97
2006,7^
1865,93
i736,5i
1616,94
i5o5,72
i4oi,68
i3o4,o8
1 2 1 1 ,79
I I34,3B
io4o,8g
961,17
884,55
810,59
739,03
669,49
601,61
535,21
469,89
4o5,49
341,80
278,41
2i5,33
I 52,20
88,93
34,83
— 39,69
105,39
172,10
— 240,53
— 3 1 o,5o
— 382,96
— 457,88
— 535,74
— 616,93
— 703,31
— 79 '^95
— 887,07
— 988,14
— 1096,39
1313,66
— 1 338,63
— 1475,69
— 1 636,, 6
Second Difl;
Infin. neg.
— ia5oi,55
— 5i32,g7
— 2820,68
— 1770,67
— I28i,3a
— 893,48
— 702,01
— 557,68
— 454,78
— 378,98
— 321, ao
— 277,13
— 240,16
— 2l3,34
— 189,43
— 170,34
— 1 54,34
— i4o,8o
— 139,42
— 119.57
Ill ,32
— io4,o4
— 97,60
— 92,29
— 87,51
— 83,39
— 79'72
— 76,63
— 73,96
— 7>,57
— 69,53
— 67,88
— 66,4o
— 65,33
— 64,4o
— 63,69
— 63,39
— 63 ,08
— 63.i3
— 63,28
— 64,09
— 64,52
— 65, 60
— 66,81
— 68,43
— 69,97
— 72,46
— 74.92
— 77.86
— 81,,
— 85,29
— 89,74
— 95,12
— 101,07
— 108, 1 5
— 116,37
— I35,g
— 1 37,06
— i5o,37
l^ug, of B.
Infin. neg.
2,o5ig4»
3,35333,1
2,53oog„
2,656 10,1
2, 75450,,
a, 8356 1,1
2,90473»
2,9651 3»
3,01903»
3,o6777„
3,1 i24o„
3,1 5364,,
3,19205,,
3,228o5„
3,36198,,
3,3940g,,
3,33458,,
3,35365„
3,38i42„
3,4o8oo„
3,43348^
3,45794»
3,48 i46„
3,5o4o7«
3,5258i„
3,54673,,
3,56684„
3,586ig„
3,6o479„
3,62264,,
3,63o76„
3,656 17,,
3,67187,,
3,68685,,
3,701 13„
3,71473,,
3,72758,,
3,73g7i,,
3,75 no»
3,76174»
3,77 165„
3,78080»
3, 78g 1 9»
3,79679"
3,8o356„
3,80949»
3,8 1 454»
3,81870»
3,82194»
3,82417»
3,82538»
3,82548,,
3,82445/1
3,83220»
3,8i863»
3,81 365,,
3,807 1 3«
3,7g8g7»
3,78897»
3,77695»
Infin.
3oi39
1 7676
12601
9840
8111
6912
6o4o
53go
4874
4463
4ia4
384 1
3600
3393
331 I
3o4g
3907
2777
2658
2548
3446
3353
2361
2174
2091
3012
1935
i860
1785
I7I2
i64i
1570
i4g8
1428
i35g
1286
I2l3
II 39
1064
991
9i5
83g
760
677
593
5o5
4i6
324
223
— io3
— 235
-357
— 653
— 816
— 1000
— 1303
— 1438
60
61
63
63
64
65
66
67
68
69
70
71
72
73
74
75
76
7
78
79
80
81
83
83
84
85
86
87
90
9'
9'
93
94
95
96
9-
9i
99
100
101
103
io3
io4
io5
106
109
ITO
1 I 1
113
ii3
ii4
ii5
116
117
118
"9
i.i.g. or «.
sex. seconds
4,28671,76.
4,37045,70.
4,35a53,4on
4,33375,93.
4,21090,35»
4,1 8668,38.
4,15974,94»
4,1 2965,58.
4,og582,g6„
4,o575i,66,
4,01 368 ,88.
3,g639i,i 1.
3,90307,53.
3,83ogi,43n
3,74094,99»
3,62292,660
3,45382,45.
3,16757,57»
0,87017,73.
3,16902,43
3.47716,74
3,65978,86
3,79098,67
3,89400,55
3,979 '7.62
4,o53oi,79
4,1 i582,oi
4,17270,58
4,23412,29
4,371 1 1,80
4.3i443,5i
4,35465,59
4.3g325,36
4,43757,53
4,46ogo,g5
4,4925o,5i
4,53354,88
4,55120,92
4,57862,82
4,60492,71
4,63020,78
4,65456,52
4,67807,55
4,70080,97
4,72283,19
4,74419,74
4,76495,65
4,785 1 5,45
4,8o483,28
4,82402,87
4,84277,69
4,86110,88
4,87905,38
4,89663,98
4,9 1389, 1 3
4,93083,36
4,94748,58
4.96387,30
4,98001,1 1
4,99593,23
5,01 162,34
1626,06
1792,30
- 1977.47
- 2i85,58
242 1 ,97
2693,44
3009,36
- 3383,63
- 383i,3o
- 4382,78
- 5077,77
- 5983,59
7216,09
- 8996,44
- 11802,33
- i6gio,2
- 28634,88
-2,39739,84
3o8i4,3i
18262,12
i3i ig,8i
io3oi,88
8517,07
7284,17
638o,23
5688,57
5i4i,7i
46gg,5i
4330,71
4o33,o8
375g,67
3532,37
3333,43
3i5g,56
3oo4,37
2866,04
2741,90
2629,8g
2528,07
2435,74
235i,o3
2273,43
2202,33
2i36,55
2075,91
2019.80
1967,83
igig,5ci
1874,82
1833,19
i794,5o
1758,60
1725,15
i6g4,i3
i665,32
I 638 ,63
1613,91
1 59 1 , 1 3
1 570, 1 1
i55o.gi
Second Dili'.
- i5o,37
- 166,24
- 185,17
- 208,11
- 236,3g
- 271,47
- 3i5.g2
- 373,26
- 448,68
- 55i,48
- 694,99
- 905,82
-1232. 5o
-i78o;35
-2805,89
— 5i42,3i
— 2817,93
— 1784,81
— i233,go
— 9"3,95
— 6gi,65
— 546,86
— 443,30
— 368,8o
— 307 ,63
— 263,4 1
— 237,40
— 198,85
— 173,86
— 155,19
— 1 38,33
— 124,14
— 112,01
— 101,82
— 92,33
— 84,71
— 77,61
— 71,20
— 65,67
— 60,64
— 56,11
— 51,97
— 48,24
— 44,77
— 4i,63
— 38. 6g
— 35,90
— 33,45
— 3 1, 02
— 28,81
— 26,70
— 24,71
— 22,79
— 2 1 ,0 1
— 19.20
..r B.
fconds.
3,776g5„
3,76267,,
3,7458i„
3,72602,,
3,70282,,
3,6756i„
3,64354»
3,6o555n
3,56oo7n
3,5o47i,„
3,4357g„
3,347o5„
3,22612,,
3, 043 5 1
2,68g54n
2,2 1 £60
2,g34i7
3,3f)257
3,37493
3,5o38o
3,60770
3,69533
3,771 54
3,83938
3,90043
3,9563o
4,00782
4,o558o
4,10078
4,i43i4
4,i8325
4,23 I 39
4,35765
4,2g244
4,32585
4,35799
4,3889a
4,41887
4,44786
4,47599
4,5o333
4,52993
4,55585
4,58 116
4,6o5gi
4,63oi3
4,65383
4,67711
4,6999g
4,72248
4,74463
4,76647
4,78803
4,8og3o
4,83o36
4,85i3i
4,87189
4,89242
4,91281
4,g33o8
4,g532g
- 142B
- 1686
- '979
- 2320
- 2721
- 3207
- 3799
-4548
- 5536
- 68g2
- 8874
-i2og3
-18261
-35397
+71557
26840
17235
12888
10390
8763
7621
6774
6114
5588
5i52
4798
4498
4236
401
38o4
3636
3479
334i
32i4
3093
2995
2899
2813
2734
2659
25g3
253i
2475
2421
237.
2328
2288
2249
22l5
2184
2i55
2138
2106
2o85
2068
2o53
2039
2027
ao2i
201 5
TABLE IV.
To find the true anomaly «, in a very eccentric ellipsis or liyperbola, from the corresponding anomaly 17 in a parabola ;
accordini; to Simpson's method, improved by Bessel.
U.
d
1 20
121
122
123
124
125
126
127
128
129
i3o
i3i
l32
i33
1 34
i35
1 36
1 37
i38
139
i4o
i4r
l42
143
i44
i45
1 46
l47
148
149
i5o
i5i
l52
1 53
1 54
i55
1 56
1 57
1 58
159
160
161
162
1 63
1 64
i65
166
167
168
169
170
171
172
173
174
sex. seconds
5,01162,34
5,02713,25
5,04246.44
5,05763,87
5,07266,81
5,08757,06
5,10235,87
5,11705,01
5,i3i65,35
5,14618,91
5,16066,94
5,17510,76
5,18952,15
5,20392,32
5,2i832,5o
5,23274,66
5,24719,89
5,26170,05
5,27626,64
5,29091,30
5,3o565,78
5,32o5i,87
5,3355i 43
5,35o66,42
5,36598,87
5,38i5i,i7
5,39725,43
5,4 1 324,24
5,42950,27
5,44606,4 1
5 46295,75
5,48021,66
5,49787,97
5,5 1 598,58
5,53458.03
5,55371,25
5,57343,74
5,59381 ,69
5,61492,02
5,63682,66
5,65962,52
5,68341,99
5,70833,00
5,73449,49
5,76207,92
5,79127,92
5,82233,12
5,85552,33
5,89121,30
5,92985,02
5,97201,42
6,01 846,63
6,07023,80
6,12877,42
6,19619,12
175 6,27576,94
176 6,37300.24
[77 6,49819,30
178 6,67546,42
179 6,97560. [8
180 Infinite
1 550,91
i533,iQ
■ 5i7,43
1 502,94
1490,25
1478,81
1469,14
1460.34
1453^56
i448,o3
1443,82
1441,39
1440,17
i44o,i8
1442,16
i445,23
i45o,!6
1456,59
1464,66
i474,48
1486,09
i4g9,56
i5i4,99
i532,45
i552,3o
1574,26
1598,81
1626,03
1 656, 1 4
1689,34
1725,9
1766,3.
1810,61
1859,45
1913,22
1972,49
2037, g5
2110,33
2 1 go,64
2279,86
2379,47
2491,01
2616,49
2758,43
2920,00
3io5,2o
33i9,2i
3568,97
3863,72
4216,40
4645,21
5i77,i7
5853,62
674 1 ,70
7957,82
9723,30
12519,06
17727,12
3oo 13,76
Intinite
Seeond
Ditr.
— 17,72
— 15,76
— 14,49
— 12,69
— 11,44
— 9-67
— 8,80
— 6,78
— 5,53
— 4,21
— 2,43
+
1,22
0,01
1,98
3,07
4,93
6,43
8,07
9,82
11,61
i3,47
i5,43
17,46
19,85
21,96
24,55
27,22
3o,ii
33,20
36,57
4o,4o
44, 3o
53,77
59,27
65,46
72,38
80,3 1
89,22
99,61
1 1 1 ,54
125,48
i4i,94
161,57
i85,2o
2l4,01
249,76
294,75
352,68
428,81
531,96
676.45
888,08
1216,12
1765,48
2795,76
5208,06
12286.64
Infinite.
Lo;. of «.
sex.seLOnds.
4,9532g
4,97344
4,gg355
5,01 36
5,03379
5,05395
5,07419
5,09451
5,1 1496
5,i3556
5,1 5634
5,17734
5,ig855
5,22004
5,24182
5,26395
5,28649
5.3o()4i
5,33278
5,35664
5,38io6
5, 4061 1
5,43 1 80
5,45821
5,48539
5,5i34i
5,54236
5,57230
5.6o33i
5,63547
5,66893
5,70375
5,74010
5,77806
5,81781
5,85948
5.90332
5,94946
5,gg8i6
6,04965
6,10426
6,16232
6,224i4
6,29016
6,36 1 11
6,43737
6,51976
6,60914
6^70666
6,81 364
6,93190
7,06371
7,21220
7,38173
7,57867
7.81288
8,ioo85
8,47350
9,00022
201 5
201 1
2012
2012
2016
2024
2o32
2045
2060
2078
2100
2121
2149
2178
22l3
2254
2292
2337
2386
2442
25o5
2569
2641
2718
2802
2895
2gg4
3ioi
3216
3346
3482
3635
37g6
3975
4167
4384
46i4
4870
5i4g
5461
58o6
6182
6602
7og5
7626
8239
8938
9752
10698
11826
i3i8i
i484q
16953
19694
23421
28797
37265
52672
In the extreme and middle parts of the table,
the first differences vary rapidly, in which case
we may use the valuer, of .S', i>, instead of tiieir
logarithms, as in the following auxiliary table.
AUXILIARY TABLE IV.
Î7.
s.
sex. seconds.
Diff.
B.
sex. seconds.
Diff.
d
0
1
2
3
4
5
70
7'
72
73
74
75
76
77
78
79
80
81
82
s
— 0,0
— 900.0
— 1798:5
— 26g5,o
— 3588,3
— 4477,2
— 55o7,4
— 4iq6,9
— 2843,3
— 1470,9
— 7,4
1475,8
3ooo,3
4568,7
6i8o,c
— goo,o
— 8g8,5
— 896.5
— 893,3
— 888,9
1 3 10.5
1 353.6
1372,4
i463,5
i483,2
i524,5
1 568,4
1611,3
s
— 0,0
— II2.7
— 225,6
— 338,9
— 453,0
— 568,2
— 2727,7
2223,6
— i683,i
— iio5,4
— 489,3
1 65, 4
85q,3
1594,3
2370,9
— 112,7
— 112,9
— ii3 3
— ii4,i
— Il5,2
5o4,i
540,5
577,7
616,1
654,7
693,9
735,0
776,6
TABLE V. — For an Ellipsfs.
Tliis table is to be used in finding the true anomaly v, currespomling to the time t from the perihelion, in a very
excentrical ellipsis ; the cxcentricity e and the perihelion distance D being given. In paint of accuracy, it is not restricted to
the tirst and second powers of 1 — e, lilic Tal)le IV., but includes all llie power.-* of tliat quantity. Tliis table is nearly in the
same form as it was first given by Professor Gauss.
Rl-le. From e find a=l — e, a'- = 0,1 + 0,9 . f, and then find the approximate value of log. (', by the following formula;
Appiox. log. i'=log. (-j-log. a.' — 2^. log. £).
With this value of (' find the corresponding value of t'iu Table III. ; also,
log. /3 = log. a + arith. log. co. a'^ -j- 9,6939700 — 10,0000000 ;
Approx. log. .1= log. /2-(-2 log. tang. ^ C.
Entfir Table V., with the natural number corresponding to this value of log. j9, and find the corresponding log. S, which is to
be subtracted from the approximate log. (' to obtain the corrected value of log. t'. With this corrected value find, in Table
III., the corrected value of U, and for the sake of distinction, we shall represent it by w ; then the corrected value of log. Ji
is found by the following formula, which is similar to the preceding one, changing {/into w ;
Correct, log. Jl = log. /3 -)- 2 log. lang.^ w.
It will very rarely be necessary to repeat again this operation to get a more accurate value oi A ; we may therefore, with this
value of ^, find the correct value of C, in Table V., and then,
tang.a 4 ti =
.4
C — 0,i
.Î ■ 1 — e '
■0,8.^
JO . sec.a è V.
[Anomaly v].
[Radius vector r].
C + 0,2.^
We may observe that in computing a large number of observations, it will frequently happen that the value of Bis very nearly
known, at the commencement uf the operation ; in this case the correction B, may be applied to the first process, in finding the
approximate value oJ t'.
EXAMPLE.
Given the excentricity « = 0,96764567 ; log. perihelion distance i3 = 9,7656500; f = 63''''^^544 ; to find d and ;
From the value of e we get, a=i — e=:o,o3235433; a'2 = 0,1 4-0,9 6^0,970881103.
t = 63"'*y',544
Approximate Operation.
a'2^o,i +0,9 . e log.
a' log.
D log. CO.
its half
log.
Approx. log. t'
Hence U = gQ6'', in Table III.
a= 1 — e log.
a'2 log. CO.
Constant log.
Sum gives /3 log.
.i Î7=49''33" tang.
same
Approx. .4 = 0,022923 log.
Corresponding log. B=o,ooooo4o, Table V.
9,9871661
9,9935830
o, 2343500
0,1171750
i,8o3o745
2,1481825
8,5o993a5
0,0128339
9,6989700
8,2217364
0,06927
0,06927
8,36027
a23
Corrected Operation.
Subtract log. £ := o,ooooo4o gives correct log. ('
Hence U or u; = 99'' 6°* 1 3', 4 in Table III.
i 10 = 49'' 33" 6', 7
Corrected Jl = 0,0229361
same
tang,
same
log.
log.
2,1481785
8,2217364
0,0692972
0,0692972
8,36o33o8
0,8 JÎ = 0,01 83409
.
C =; 1 ,0000242
log. CO.
•
C — 0,8.^ = 0,9816833
0,0080286
I +«=1,96764547
log-
0,293946g
I— e = a
log. CO.
1,4900675
Sum is
2 log.
tang. 4 V
0,1523738
4 0 := 5o 0 0,1
tang.
0,0761869
d m t
»= too 0 0,2
C + 0,2 JÎ = 1 ,0046094
log. CO.
9,9980028
C — 0,8 jî = 0,981 6833
log.
9,9919714
D
log.
9,76565oo
io = 5o''o°'o',i
sec.
0,1919327
aame
0,1919327
0,1394896
TABLE V. — For an Ellipsis.
In the inverse problem, we have given, the true anomaly v, the perihelion distance O, and the excentriciiy e, to find the time
t from the perihelion in days. This is obtained by the following rule.
Rule. With e and » find T =
1 — e
1+e
. tang.3 J 11, and then by Table V., the corresponding value of C. Also,
Log. A= log. T-f log. C -Y arith. comp. log. (1 -f- 0,8 . T) ;
from which we find log. B, by means of Table V. Then we find,
log. ij = 2,0654486 + J log. 2) + i log. ./Î + log. 5 — è log. (1 — e) ;
log. <2 = log. <i + 8,8239087 + log. A -\- log. (1 -f 9 <■) — log. (1 — c) ;
'=', + '2-
EXAMPLE.
Given as before « = 0,96764567; log. perihelion distance X) = 9,7656500 ; and the true anomaly »^ 100'' o^o', 2 ; to find the
time ( from the perihelion in days.
I — e
4 0 ^ 5o o"" o , I
T = 0,0233539
Hence C= 1,0000242 Table V
1-1-0,8T= i,oi8683r
.4= 0,0229261
Corresponding log. B in Table V.
log-
8,5099325
log. CO.
9,706053 1
tang.
0,0761869
same tang.
0,0761869
log.
8,3683594
log-
0,00001 o5
log. CO.
9,9919609
log.
8,36o33o8
o,ooooo4o
: 43"'^J'%564
-^dftyg
<2=i9""'%98o
,,4-,2 = 63''»>-S544 = (
Constant log.
t log. D
è log. Jl
log. B
J log. (i — e) arith. co.
log.
Constant
A log.
+ 9 « = 9,7088110 log.
(i — e) log. CO.
log.
2,0654486
9,6484750
9,1 801 654
0,0000040
o,745o337
1,6391267
8,8339087
8,36o33o8
0,9871661
1 ,4900675
i,3oo5gg8
Lo<j. B
0,000
001
002
00 3
004
o,oo5
006
007
oo3
009
0,010
oil
012
oi3
oi4
0,0 1 5
016
017
018
019
0,020
021
022
023
024
0,025
026
027
028
029
o,o3o
o3i
032
o33
o34
o,o35
o36
o37
o38
039
o,o4'
0,0000000
000
000
001
001
0,0000002
oo3
oo4
oo5
006
0,0000007
009
oil
oi3
oi5
0,000001 7
019
022
024
027
o,ooooo3o
o33
o36
o4o
043
0,0000047
o5i
o55
oSg
o63
0,0000067
072
077
082
087
0,0000092
097
io3
108
ii4
0.0000120
1 ,0000000
1 ,0000000
1,0000002
1 ,0000004
1 ,0000007
1 ,00000 1 1
1,0000016
1,0000022
1,000002g
1,0000037
1 ,0000046
I ,ooooo56
1 ,0000066
1 ,0000078
1 ,0000090
0,00000
0,00100
0,00200
o,oo3oi
o, 00401
o,oo5o2
o,oo6o3
0,00704
o,oo8o5
0,00907
0,01008
0,01110
0,01212
o,oi3i4
o,oi4i6
i,ooooio3 o,oi5i8
i,oooon8 0,01621
1,0000 1 33
1,0000149
1,0000166
1,0000184
1,0000203
1,0000223
1 ,0000244
1 ,0000265
1 ,0000288
1 ,oooo3 1 2
i,oooo336
1, 0000362
I ,oooo388
1, 00004 16
1 ,0000444
1 ,0000473
i,oooo5o3
i,oooo535
1 ,0000567
1 ,0000600
1 ,0000634
1 ,0000669
1 ,0000704
1 .000074 1
0,01723
0,01826
0,01929
O,02o32
0,021 36
0,02239
0,02343
0,02447
o,o255i
0,02655
0,02760
0,02864
0,02969
o,o3o74
o,o3i79
0,03284
0,03389
0,03495
o,o36oi
0.03707
o,o38i3
0,03919
O,o402 5
o,o4i 32
Lo£. B
o,o4o
o4i
042
o43
044
0,045
o46
o47
048
049
o,o5o
o5i
o52
o53
o54
o,o55
o56
o57
o58
059
0,060
061
062
o63
064
o,o65
066
067
068
069
0,070
071
072
073
074
0,075
076
077
078
079
0,080
0,0000120
126
i33
139
i46
0,00001 52
i59
166
173
18
0,0000181
196
204
212
220
0,0000228
236
245
254
263
0,0000272
281
290
3oo
309
o,oooo3i9
339
339
35o
36o
0,0000371
38 1
392
4o3
4i5
0,0000426
437
449
461
473|
0,000048 5 1
1 ,000074 1
1 ,0000779
1 ,00008 1 8
i,oooo858
1 ,0000898
1 ,0000940
1,0000982
1,0001026
1,0001070
1,0001116
1,0001163
1,0001210
1,0001258
1 ,000 1 307
1, 0001 358
1,0001409
1, 000 1 461
i,oooi5i4
1,0001 568
1,0001623
1,000167g
1,0001736
1 ,000 1 794
1,0001853
1,0001913
1,0001974
i,ooo2o36
1,0002099
1,00021 63
1,0002228
1,0002394
1, 0002360
1,0002428
1 ,0002497
1,0002567
1 ,0002638
1,0002709
1,0002782
1 ,0002856
1 ,0002930
1 ,ooo3oo6
o,o4i3ig
0,042387
0,043457
0,044528
o,o456oi
0,046676
0,047753
o,o4883i
0,0499 1 1
0,050993
0,052077
o,o53i63
o,o5425o
0,055339
o,o5643o
0,057523
o,o586i8
0,059714
0,060812
0,061912
o,o63oi4
0,064 1 18
o,o65223
o,o6633i
0,067440
o,o6855i
0,069664
0,070779
0,071896
0,073014
0,0741 35
0,075257
0,076381
0,077507
0,078635
0,079765
0,080897
o,o82o3o
o,o83i66
o,o843o3
o,o85443
0,080
081
082
o83
084
o,o85
086
087
088
089
0,090
091
092
093
094
0,095
096
097
098
099
0,100
101
102
io3
1 04
o,io5
106
107
108
109
0,1 10
11 1
113
II
114
0,1 15
116
117
ii8
i'9
0,120
Los. B
0,00004
4_
5io
523
535
o,oooo548
56 1
575
588
602
0,000061 5
629
643
658
672
0,0000687
701
716
73i
746
0,0000762
777
793
809
825
0,0000841
857
873
890
907
0,0000924
94
958
975
993
0,000101
1029
io47
io65
io83
0,0001102
,ooo3oo6
,ooo3o83
,ooo3i6o
,0003239
,ooo33i9
,0003399
,ooo348i
,ûoo3564
,ooo3647
,0003732
,ooo38i8
,0003904
,0003993
,ooo4o8 1
,ooo4 1 70
,0004261
,0004353
,0004446
,0004539
,ooo4634
,0004730
,0004826
,0004924
,ooo5o23
,ooo5 123
,ooo5224
,ooo5325
,0005428
,ooo5532
,ooo5637
,0005743
,ooo585o
,ooo5g58
,0006067
,0006177
,0006288
,0006400
,00065 1 3
,0006627
,0006743
.0006858
o,o85443
o,o86584
0,087737
0,088872
0,090019
0,091168
0,092319
0,093472
0,094627
0,095784
0,096943
0,098104
0,099266
0,1 0043 1
0,101598
0,102766
0,103937
o,io5i 10
0,106284
o, 1 0746 1
0,108640
0,109820
0,1 1 ioo3
0,112188
0,1 13375
0,1 1 4563
0,115754
0,116947
0,1 18142
0,119339
O,i2o538
0,131739
0,122942
o, 124148
0,125355
0,126564
0,127776
0,138989
O,l3o3o5
o,i3i423
0,132643
TABLE v. — For an Ellipsis.
To find the true anomaly
n a very excentric elli
psis, by th
J method of Gauss
A
Log. B
c
T
A
Log. B
c
T
A
Log. B
c
T
0,120
0,0001102
i,ooo6858
o,i32643
0,180
0,00025 1 5
1,0016764
0,209894
0,240
0,0004537
1 ,0028644
0,296980
121
1121
1 ,0006976
0,1 33865
181
2543
1,0016945
0,211253
241
4576
i,oo288g4
0,297498
122
1.39
1 ,0007094
0,135089
182
2672
1,0016128
0,212614
24j
461 5
i,oo2gi46
0,299018
123
ii58
1,0007213
o,i363i5
i83
2601
i,ooi63ii
o,2i3g77
243
4664
1 ,0039397
o,3oo542
124
1 178
1 ,0007334
0,137543
184
263o
1,0016496
0,2 15343
244
4694
1,0039651
o,3o2o68
0,125
0,0001197
1,0007455
0,i38774
o,i85
0,0002660
1,0016682
0,216712
0,245
0,0004734
I ,oo3g9o5
o,3o35g7
126
1217
1,0007577
0,140007
186
2689
1,0016868
o,2i8o83
246
4774
i,oo3oi6i
o,3o5i39
127
1236
1,0007701
o,i4i34i
18--
2719
1,0017067
0,219466
247
48i4
i,oo3o4i8
o,3o6664
128
1256
1,0007825
0,142478
188
2749
1,0017246
0,220832
248
4864
1 ,0030676
0,308202
129
1276
1,0007951
0,143717
189
2779
1,0017436
0,222211
249
4894
1 ,0030936
0,309743
0,1 3o
0,0001206
1 ,0008077
0,144959
0,190
0,0002809
1,0017637
0,223692
0,260
0,0004936
1,0031196
0,3 1 1286
i3i
.3.7
1,0008205
0,146202
19'
283q
1,0017830
o,2 24g75
261
4976
i,oo3i458
o,3i2833
l32
1337
i,ooo8334
0,1 47448
192
2870
i,ooi8oi3
o,33636i
263
5017
1, 0031721
o,3i4382
1 33
i358
i,ooo8463
0,148695
193
2900
1,0018208
0,337760
263
6o58
1,0031986
0,316935
1 34
1378
1,0008594
0,149945
194
3931
1, 0018404
0,229141
264
5099
1, 0032260
0,3 17490
0,1 35
0,000139g
1 ,0008736
o,i5ii97
0,195
0,0002963
1,0018601
o,23o535
0,255
o,ooo5i4i
1,0032617
0,3 1 9048
1 36
1421
1 ,0008859
o,i53452
196
2993
1,001879g
o,33i93i
256
5182
1,0032784
0,320610
1 37
i442
1 ,0008993
0,1 53708
197
3o25
i,ooi8gg8
0,233329
267
6224
1 ,oo33o53
0,322174
1 38
1463
1,0009128
0,154967
198
3o56
1, 001 g 198
o,23473i
258
6266
1, 0033323
0,323741
■ 39
1 485
1 ,0009264
0,156228
199
3o88
i,ooig4oo
o,336i35
269
53og
1,0033596
o,3253i2
o,i4o
0,0001 5o7
i,ooog4oi
0,167491
0,200
O,0003l2O
i,ooig6o2
0,237641
0,260
o,ooo535i
1,0033867
0,326885
i4i
l52Q
1 ,0009539
0,1 58756
201
3i53
1,0019806
0,338960
261
6394
i,oo34i4i
0,328461
142
i55i
1 ,0009678
0,160024
203
3i84
1,0020011
o,24o36i
262
5436
i,oo344iG
o,33oo4i
i43
1573
1,0009819
0,161294
2o3
3216
l,OC'>02I7
0,24177c
263
5479
i,oo346g2
o,33i623
144
1596
1 ,0009960
0,162566
204
3249
1,0020424
0,243193
264
5533
I ,oo34g7o
0,333208
0,145
0,0001618
1,0010103
o,i6384o
o,2o5
0,0003282
1, 0020632
0,344612
0,266
o,ooo6566
1,0036248
0,334797
i46
i64i
1,0010246
o,i65i 16
206
33i5
1 ,0020842
0,246034
266
66og
1,0036628
o,336388
1 47
1664
1,0010390
0,166395
207
3348
1,0021062
0,347468
267
5653
1 ,oo358og
0,337983
1 48
1687
i,ooio536
0,167676
208
338i
1,0021264
0,248885
268
56g7
1,0036091
0,339680
149
1710
1, 00 10683
0,168969
209
3414
l,oo2i477
o,25o3i6
26g
5741
1 ,0036376
0,341181
0,1 5o
0,0001734
I,ooio83o
0,170345
0,210
o,ooo3448
1,0021690
0,261748
0,270
0,0006786
1 ,0036669
0,342786
i5i
1757
1,0010979
0,171533
211
3482
i,oo2igo5
o,253i83
271
682g
i,oo36g45
0,344392
l52
1781
1,0011 129
0,172823
212
35i6
1,0022122
0,354620
272
5874
1,0037232
0,346002
1 53
i8o5
1,0011280
0,1741 15
2l3
355o
1,0022339
0,266061
273
^919
1,0037621
0,347615
i54
1829
1,0011432
0,175410
214
3584
1 ,0022567
0,267604
274
5964
1,0037810
0,349231
o,i55
0,0001854
1,0011 585
0,176707
0,2 I 5
o,ooo36i8
1,0022777
o,268g5o
0,276
0,000600g
i,oo38ioi
o,35o85o
1 56
1878
1,0011739
0,178006
216
3653
i,oo229g8
0,360398
276
6064
i,oo383g3
0,362473
1 57
1903
1,0011894
0,179308
217
3688
1,0023220
0,361849
277
6100
i,oo38686
0,354098
1 58
'927
I,001205l
0,180612
218
3733
1,0023443
o,2633o3
278
6145
1,0038981
0,355727
.59
1952
1, 00 1 2 208
0,181918
219
3758
1 ,0023667
0,264769
279
6191
1 ,0039277
o,35736g
0,160
0,0001977
1, 001 2366
0,183336
0,220
0,0003793
1,0023892
0,366218
0,380
0,0006337
1,0039673
o,368g94
161
2003
1,0012526
0,184537
231
383g
1,0024119
0,267680
281
6283
1,0039873
o,36o632
162
2028
1 ,00 1 2686
o,i8585o
222
3865
1,0024347
0,269145
282
633o
1,0040171
0,363374
1 63
2o54
1,0012848
0,187166
223
3900
1,0024576
0,270613
283
6376
1,0040472
0,363918
164
2080
i,ooi3oi I
o,i88484
224
3g36
1 ,0024806
0,273083
284
6423
1,0040774
o,365566
0,1 65
0,0002106
i,ooi3i75
0,189804
0,225
0,0003973
1,0026037
0,273555
0,286
0,0006470
1,0041077
0,367217
166
2l32
1,00 1 3340
0,191137
226
4009
1,0026269
0,276031
286
6617
i,oo4i38i
0,368871
167
2i58
i,ooi35o6
0,193453
227
4o46
I,0025502
0,276609
287
6664
1,0041687
0,370629
168
2184
1,0013673
0,193779
228
4082
1,0026737
0.277990
288
6612
i,oo4i9g4
0,373189
169
2211
1,00 1 384 1
0,196109
229
4119
1,0026973
0,279474
28g
6660
1, 0042302
0,373863
0,170
0,0002238
i,ooi4oio
0,196441
o,23o
o,ooo4i56
1,0036210
0,280960
0,390
0,0006708
1,0042611
0,375521
171
2265
i,ooi4i8i
0,197775
33l
4iq4
1 ,0026448
0,282450
291
6766
1,0043933
0,377191
172
2292
1,0014353
0,1991 13
233
423 1
1,0026687
0,283943
392
6804
1,0043333
0,378865
173
2319
i,ooi4535
o,3oo45i
2 33
4269
1,0026928
0,285437
3g3
6862
1 ,0043547
o,38o642
174
2347
1,0014699
0,201793
234
43o6
1,0027169
0,286935
394
6901
1, 0043861
0,382222
0,175
0,0002374
1,0014873
o,2o3i37
0,235
0,0004344
1,0027412
0,288435
0,296
0,0006960
1,0044177
o,383go6
176
2402
I ,001 5o49
o,2o4484
2 36
4382
1 ,0027666
0,289939
296
6999
1 ,0044493
o,3855g3
177
243o
1,001 5226
o,2o5833
237
4421
1,0027901
0,391446
397
7048
1 ,00448 1 3
0,387283
178
2458
1. 001 5404
0,207184
238
445g
1,0028148
0,292964
298
7097
i,oo45i3i
0,388977
7 79
2486
1, 001 5583
o,3o8538
239
4498
i,oo283q5
0,294466
'99
7147
1,0045462
0,390673
0,180
0,00025 1 5
1,001 5-64
0,209894
0,240
0,000453-
1,0038644
0,395980
o,3oo
0,0007196
1 .0046774
o.3q9374
TABLE VI. —For an Hyperbola.
This table is used in finding the true anomaly v of a comet, moving in a hyperbolic orbit, which approaches very nearly to
the form nf a parabola; the excentricity e, the perihelion distance D, and the lime t from passing the -perihelion being
given. Like the preceding table, it is not restricted to the first and second powers of e — 1, but includes nil the powers of that
quantity.
Rule. From e find 0.-'2 = 0,1 -j-0,9 . e; and then the approximate value of log. /' from the formula,
Approx. log. (' = log. t-\-\cg. a' — f^'og- D.
With this value of (', find the corresponding value of U, in Table IIL, also,
log. /3 = log. (e — l)+arith. co. log. a'2 + 9,6989700 — 10,0000000;
Approx. log. .4 = log. ^-\-2 log. tang.J U.
Enter Table VL, with the natural number, corresponding to this value of log. A, and find in it the corresponding log. B ; which
is to be subtracted from the approximate log. (', to obtain the corrected value of log. ('. With this corrected value, find in Table
IIL, the corrected value of log. U; and for distinction, we shall call it w; then the corrected value of log. A is found by the
following formula, which is similar to the preceding;
Correct, log. A = log. /2 -|- 2 log. tang.^ w.
It will very rarely be necessary to repeat the operation, to get a more accurate value of A ; we shall therefore use it, in
finding the correct value of C, in Table VI., and then,
A «4-1 _
1 '
tang.2 i t!
C-|-0,8.^ ■ e-
C-\-a,S.A
[Anomaly v\.
. D.sec.Si V.
[Radius vector r].
C — 0,2.A
In computing a large number of observations, it will frequently happen that the value of B is very nearly known, at the
commencement of the operation; in this case, the correction B, may be applied in the first process, for finding the approximate
value of t'.
Given the excentricity e= 1,261882; log. perihelion distance
Approximate Operation.
a'2 = 0,1 +0,9. e^ 1,2356938 log.
a' log.
D log. CO.
its half
log.
Approx. log. f
EXAMPLE.
0,0201667; t = 65''''^',4i236 ; to find e and r.
t = 65^'')",4i236
Hence ^7= 70" 32"", nearly, in Table III.
e — 1=0,261882 log.
a'2 log. CO.
Constant log.
Sum gives y3 log.
à £7=35'' 1 6"" tang.
same
Approx. .4 = 0,05299 log.
Corresponding log. B = 0,0000207, Table VI.
0,0919108
0,0459554
9,9798343
9,9899171
1,8 1 56598
1, 83 1 3666
9,4i8io56
9,9080892
9,6989700
9,0251648
9,84952
9,84962
8,72420
Corrected Operation.
Subtract log. 5 = 0,0000207 gives correct log. t'= i, 83 1 3469
Hence î/ort/)^70 3i 87^,0 in Table III.
è«) = 35S5'"48',5
Corrected A = 0,0629792
0,8 A = 0,0423834
C = 1,0001261
same
tang,
same
log.
C -|- 0.8 -^ = 1,0426095
e -j- I = 2,261882
e — I ^ 0,261882
Sum is 2 log. tang. J v
u = 33 3i 3o
f. d .,771 S
V = 67 o3 00
C — 0,2 ./Î = 0,9895303
C4-0)8-^= 1.0426096
D
4 0 = 33 3 1 3o
tang.
9,0261648
9,8494702
9,8494702
8,7241062
log. CO.
9,9819200
log.
0,3544699
log. CO.
0,6818944
9,6423896
9,8211947
g. CO.
0,0046709
log-
0,0180800
log.
0,0201667
sec.
0,0790189
same
0,0790189
log.
0,2008544
TABLE VI. —For an Hyperbola.
/» the inverse problem, we have given, the true anomali/ v, the perihelion distance D, and the excentricily e, to find the lime
tfrom the perihelion in days. This is obtained by the following rule, which is similar to that for an ellipsis, in the last table.
Ç \
Rule. With e and » find T = — — . tang.-' 4 p, and then by Table VI., the corresponding value of C. Also,
Log. A = log. T + log. C -f arilh. co. log. (1 — 0,8 . T) ;
and the corresponding log. B, in Table VL Then find,
log. t^ = 2,0654486 -j- 1 log. D + i\og.A + log. B — A log. (e — 1 ) ;
log. «2 = log. <i + 8,8239087 + log. A + log. (1 _(- 9 e) — log. (e — 1) ;
EXAMPLE.
Given a-s before £=1,261882; log. perihelion distance i)= 0.0201657 ; and the true anomaly t! = 67'' o3"o' ; to find th
time from the perihelion (.
e — i=a=o,26i882 log. 9,4i8io56
e-|-i =2,261
è»=33"'3i"'3o'
82 log. CO. 9,6455301
tang. 9,8aiig46
same tang. 9,8211946
T = 0,0508189 log. 8,7060249
Hence C^ 1,0001261 Table VL log. o,oooo548
1 — 0,8 T =: 0,9593449 log. CO. 0,0180232
.4=: 0,0529791 log. 8,7241049
Corresponding log. B in Table VI. 0,0000207
f J = 56'^°ySo683o
f„= g*"'^', 34407
Itj-f (,=65''^y% 41237 = f.
TABLE VI.
Constant log. 2,o654486
§ log. D o,o3o2485
è log. A 9,362o5s4
log. B 0,0000207
• àIog-(« — l)arith.co. 0,3909472
log. 1,7487174
Constant 8,8239087
A log. 8,7241049
I -|-9 e= I2,356g38 log. 1,0919108
(e — i) log. CO. 0,5818944
log. 0,9705362
A
Log. B
c
T
0,00000
A
Log. B
c
T
A
Log. B
c
T
0,000
0,0000000
1 ,0000000
o,o4o
0,000011 8
1,0000722
0,038757
0,080
0,0000468
1, 0002850
0,075168
001
000
1 ,0000000
0,00100
o4i
124
1,0000758
o,o3g6g5
081
480
1,0002921
0,076050
002
000
1,0000002
0,00200
042
i3o
1, 000075g
o,o4o632
082
492
I ,ooo3gg2
0,076930
oo3
001
1 ,0000004
0,00299
043
1 36
1 ,0000833
o,o4i567
o83
5o4
1 ,ooo3o65
0,077810
004
001
1 ,0000007
0,00399
044
143
1,0000872
o,o425oo
084
5 16
I ,ooo3 1 38
0,078688
o,oo5
0,0000002
1,0000011
o,oo4g8
0,045
0,0000149
1,0000912
0,043432
o,o85
0,0000528
I,00032I2
0,079564
006
oo3
1,0000016
0,00597
o46
i56
1,0000953
o,o44363
086
540
1 ,0003287
0,080439
007
oo4
1,0000022
0,00696
047
i63
1 ,0000994
o,o452g2
087
553
1, 0003363
o,o8i3i3
008
oo5
1 ,0000029
o,oo7g5
048
170
1, 0001037
0,046220
088
566
1 ,ooo344o
0,082186
009
006
1 ,0000037
o,oo8g4
049
177
1,0001080
0,047147
089
578
i,ooo35i7
o,o83o57
0,010
0,0000007
1 ,0000046
0,00992
o,o5o
0,0000184
1,0001124
0,048072
0,090
o,oooo5gi
1,0003595
0,083927
on
009
1, 0000055
0,01090
o5i
19'
1,0001 i6g
o,o489g5
ogi
6o4
1 ,0003674
0,084796
012
on
1 ,0000066
0,01 i8g
o52
199
l,00012l5
O,o4ggi7
092
618
1,0003754
o,o85663
oi3
oi3
1 ,0000077
0,01287
o53
207
1,0001262
o,o5o838
og3
63 1
i,ooo3835
o,o8652g
oi4
oi5
1 ,000008g
0,01 384
o54
2l5
i,oooi3io
o,o5i757
094
645
1,0003917
o,o873g4
0,0 1 5
0,00000:7
1,0000102
0,01482
o,o55
0,0000223
1, 0001 338
0,052675
0,095
o,oooo658
1,0003999
0,088257
016
0T9
1,0000116
o,oi58o
o56
23 I
1,0001407
o,o535g2
096
673
i,ooo4o83
0,089119
017
021
1,00001 3 1
0,01677
o57
23g
1,0001 4 58
o,o545o7
097
686
1,0004167
0,089980
018
024
1,0000147
0,01774
o58
247
i,oooi5og
o,o5542o
098
700
1,0004252
O,ogo84o
019
027
1,0000164
0,01872
o5g
256
i,oooi56i
o,o56332
099
714
1,0004338
0,091698
0,020
o,ooooo3o
1,0000182
0,01 g68
0,060
0,0000265
1,0001614
0,057243
0,100
0,0000728
1 ,0004424
0,092555
021
o33
1 ,0000200
0,02065
061
273
1,0001667
o,o58i52
101
743
1,00045 12
0,093410
022
o36
1,0000220
0,02162
062
282
1,0001722
o,o5go6o
102
758
1 ,0004600
0,094265
023
039
1 ,0000240
0,02258
o63
2gi
1,0001777
0,059967
io3
772
1 ,0004689
o,og5ii8
024
043
1 ,0000261
0,02355
064
3oi
i,oooi833
0,060872
104
787
1,0004779
0,o95g6g
0,025
o,ooooo46
1,0000283
0,0245 1
o,o65
o,oooo3io
I, 000 I 8g I
0,061776
o,io5
0,0000802
1,0004820
0,096820
026
o5o
1 ,oooo3o6
0,02547
066
320
i,ooorg4g
0,062678
106
8.7
1 ,0004962
0,09766g
027
o54
1 ,oooo33o
0,02643
067
329
1,0002007
0,063579
107
833
i,ooo5o54
o,og85i7
028
o58
i,oooo355
0,02739
068
33g
1,0002067
0,064479
108
848
i,ooo5i48
0,099364
029
062
i,oooo38i
0,02834
069
349
1,0002128
0,065377
109
864
1,0005242
0,100209
o,o3o
0,0000067
1 ,0000407
0,02930
0,070
0,0000359
1,0002189
0,066274
0,110
0,0000880
1 ,0005337
o,ioio53
o3i
071
1,0000435
o,o3o25
071
370
1, 000225]
0,067170
11 1
895
i,ooo5432
o,ioi8g6
o32
076
1 ,oooo463
o,o3 1 20
072
382
1,00023 1 4
0,068064
112
911
1,0005539
0,102738
o33
080
1 ,0000492
o,o32i5
073
390
1,0003378
o,o68g57
ii3
928
1,000 5626
0,103578
o34
o85
i,oooo523
o,o33io
074
401
1,0003443
o,o6g848
114
944
1,0005724
o,io44i7
o,o35
0,0000091
1, 0000554
o,o34o4
0,075
0,00004l2
i,ooo25og
0,070738
o,ii5
0,0000960
i,ooo5823
o,io5255
o36
096
i,oooo585
0, 03499
076
423
1,0002575
0,071627
116
0977
i,ooo5g23
0,106092
o37
lOI
1,0000618
o,o35g3
077
434
1 ,0003643
0,072514
117
0994
1 ,0006034
0,106927
o38
107
1, 0000652
o,o3688
078
445
1,0002711
0.073400
118
1010
1,0006135
0,107761
039
112
1 ,0000686
0,03782
079
457
1,0002780
0,074285
119
1027
1,0006328
o,io85g4
o,o4o
0,0000118
1.0000722
0,03876
0,080
0,0000468
1,0002850
0.075168
0,120
0,0001045
1. 00063 3 1
o,iog426
a24
TABLE VI. — For an Hyperbola.
To find the true anomaly in a hyperbolic orbit, which is nearly of a parabolic form, by the method of Gauss.
Log. B
0,120
121
123
123
124
0,125
126
127
128
129
0,1 3o
i3i
l32
i33
1 34
o,i35
1 36
1 37
i38
1 39
0,1 4o
i4i
i4a
i43
144
0,145
i46
147
I
1 49
o,i5o
i5i
iSa
i53
1 54
o,i55
1 56
l57
1 58
159
0,160
161
163
1 63
164
0,1 65
166
1 6
168
169
0,170
171
172
173
174
0,175
176
177
178
179
0,180
0,0001045
1062
1079
1097
iii4
0,0001 133
ii5o
1 168
1 186
I2o5
0,0001223
1242
I26I
1280
1299
0,0001 3 18
i337
i357
1376
1396
o,oooi4i6
1 436
i456
1476
1497
o,oooi5i7
1 538
1 559
i58o
1601
0,0001623
1643
i665
1686
1708
0,0001730
1752
1774
1797
1819
0,0001843
1864
1887
1910
1933
0,0001956
1980
2oo3
2027
205l
0,0002075
2099
2133
2l47
2172
0,0002196
2221
2246
2271
2296
0,0003321
1, 000633 I
1,0006435
1 ,0006539
1,0006645
1,0006751
I ,ooo6858
1 ,0006966
1,0007075
1,0007185
1,0007295
1 ,0007406
1 ,00075 1 8
1 ,000763 1
1 ,0007745
1,0007859
1 ,0007974
1 ,0008090
1,0008207
1,0008325
1 ,0008443
0,109426
o,iio356
o,iiio85
0,111913
0,112740
o,ii3566
0,114390
o,ii52i3
o,ii6o35
o,ii6855
0,117675
0,118493
0,119310
0,120126
0,130940
0,121754
o, 122566
0,123377
o,i24r86
0,124995
,0008562 0,1 25802
1,0008682 0,126609
I ,ooo88o3
1,0008935
1 ,0009047
1,0009170
1 ,0009294
1 ,00094 1 9
1 ,0009545
1,0009671
1,0009798
1,0009926
i,ooioo55
i,ooioi85
i,ooio3i5
1,0010578
1,001071 1
1,0010844
1,0010978
1,001 1 1 13
1,0011349
1,001 1 386
1,001 1 523
1,0011661
1,0011800
1,0011940
1,0012081
1,0012222
1,0012364
1,0012507
1, 00 1 265 1
1,0012795
1,0013940
i,ooi3o86
i,ooi3333
i,ooi338o
1,0013539
1,0013678
1, 001 3827
1,0013978
o,i274i4
0,128217
0,139020
0,129822
o,r3o622
o,i3i42i
0,132219
o,i33oi6
0,i338t2
0,1 34606
0,1 3 5399
0,136191
0,13698a
i,ooio446 0,137772
o,i3856i
o,i3g349
o,i4oi35
0,140920
o,r4i7o4
0,142487
0,143269
o,i44o5ù
0,144829
o,i456o8
o,i46385
0,147161
0,147937
0,148710
o,i49483
o,i5o255
o,i5io26
o,i5i795
o,i52564
o,i5333i
0,154097
0,154862
0,155626
0,1 56389
0,i57i5i
Log. B
0,180
181
182
1 83
184
o,i85
186
187
189
0,190
191
192
193
194
0,195
1 96
197
198
199
0,200
201
202
2o3
204
o,2o5
206
207
208
209
0,210
211
212
2l3
2l4
0,2 I 5
216
217
3l8
219
0,220
331
222
223
224
0,225
226
227
228
229
o,23o
23l
232
233
234
0,235
236
287
238
239
0,340
0,0002321
2346
2872
2398
2423
0,0002449
2475
25o2
2528
2554
o,ooo258i
2608
2634
2661
2688
0,0002716
2743
2771
2798
2826
0,0002854
2882
2910
2938
2967
0,0003995
3o24
3o53
3082
3iii
o,ooo3i4o
3169
3199
3228
3258
0,0003288
33i8
3348
3378
3409
0,0003439
3470
35oo
353
3562
0,000359.
362 5
3656
3688
3719
0,000875 1
3783
38i5
3847
388o
0,0008912
8945
3977
4oio
4o43
0,0004076
1,0018978
1,0014129
1,0014281
1,0014434
i,ooi4588
1,0014742
1,00
i,ooi5o54
1, 001 52 10
1, 001 5368
1,0015526
1,001 5685
I, 001 5845
i,ooi6oo5
1,0016167
1,0016829
1,0016491
1,0016655
1,0016819
1,0016984
i,ooi7i5o
1,0017817
1,0017484
1,0017652
1,0017821
1,0017991
1,0018161
i,ooiS333
i,ooi85o4
1,0018677
i,ooi885o
1,0019024
1,0019199
1,00x9875
1,1019551
1,0019728
1,0019906
1 ,0020084
1 ,0020264
1 ,0020444
1,0020625
1,0020806
1 ,0020988
1,00211
1, 002 1 355
1 ,002 1 540
1,0021725
1,0021911
1,0022098
1,0022285
1,0022478
1,0022662
1, 002 285a
1,0028043
1,0028284
1,0023425
1,0028618
1,0028811
I ,oo24oo5
1,0024200
1,0034396
o,i57i5i
0,157911
0,158671
0,159439
0,160187
0,160948
0,161698
0,162453
0,168206
0,163958
0,164709
o, 165458
0,166207
0,166955
0,167702
0,168447
0,169192
o,i6gg35
0,170678
0,171419
0,172159
0,173899
0,178637
0,174374
0,175110
0,175845
0,176579
0,177813
0,178044
0,178775
o, 179505
0,180334
0,180962
0,181688
0,182414
0,188189
o, 188863
o,i84585
o,t858o7
0,186028
0,186747
0,187466
0,188184
0,188900
0,189616
o, 190881
o,igio44
0,191757
0,192468
0,198179
0,198889
0,194597
o, ig53o5
o,jg6oi2
0,196717
0,197433
0,198136
0,198839
o,igg53o
0,200281
0,200981
Loff. B
0,240
241
242
248
244
0,245
246
247
248
249
o,25o
25l
252
253
254
0,255
2 56
257
258
259
0,260
261
262
268
264
0,265
266
267
268
269
0,270
271
272
278
274
0,275
276
277
278
279
0,280
281
282
288
284
0,285
286
387
288
289
0,390
291
292
298
294
0,0004076
4iio
4i43
4176
42
0,0004244
4277
43ii
4846
438o
o,ooo44i4
4483
45i8
4553
0,0004588
4628
4658
4644
472g
0.0004765
4801
4838
4878
o,ooo4g45
4g8i
5oi8
5o55
5ogi
o,ooo5 128
5i65
5302
5340
5277
o,ooo53i5
5352
5390
5428
5466
o,ooo55o4
5542
558 1
56iQ
5658
0,0005697
5786
5775
58i4
5853
0,0005898
5982
5972
6012
6o52
1,0034896
1, 0024593
1,002478g
1,0034987
i,oo25i85
1,0025334
1,0025584
1,0025785
1,0025986
1,00261"
1,0026891
1,0036594
1 ,0036799
1,0027004
1,0027209
100,27416
1,0027628
1,0027880
1,0028089
1,0028248
1,0028458
1,0028669
1,00288"
1,0029093
1,0029305
1,0029519
1 ,0029733
1,0029948
1, 0080164
1 ,0080880
I ,oo3o8 1 5
I,oo3io38
1,0081253
1,0081478
0,200981
0,201680
0,202828
0,308025
0,208721
o,2o44i6
0,205lI0
o,2o58o3
0,206495
0,207186
0,207876
o, 208565
0,209254
0,209941
0,210627
0,21 i3i8
0,211997
0,213681
0,2 1 3864
n,2i4o45
0,214736
0,21 5406
o,3i6o85
0,216768
0,217440
0,218116
0,218791
0,219465
0,220l38
0,220811
1,0080597 0,221482
0,295 0,0006093
396
297
298
=99
o,3oo
6182
6172
6218
63 53
0.0006394
i,oo3igi5
1,0082187
1,0082859
1,0082583
1 ,0082807
1,0088082
1, 0088257
1 ,0083484
1,0088711
I ,oo33g38
1,0084167
I ;oo84396
1,0084626
1, 00848 56
1, 0035087
1, 0085819
i,oo35552
1,0035785
1,0086019
I, 0086253
I ,0086489
1,0086725
1,008696!
1,0087199
1,003^437
0,223l58
0,222822
o,2234gi
o, 224159
1,0031693 0,224826
0,335492
0,226157
0,226821
0,227484
9,228147
0,228808
0,239469
0,280128
0,280787
0,281445
0,282102
0,282758
0,283418
0,234068
0,234721
0,285874
0,286025
0,286676
0,287826
0,287975
0,288628
0,289371
0,389917
o,24o563
0,241207
TABLE VII. —For a Parabola.
This table is for co7nputing Ihe time t in days, for a cornel to describe, in a parabolic orbit, an arc of
the true anomaly, re2irese7ited by v' — v = 2f. This arc 2 f being given, together with the extreme
radii r, r'.
Rule. Put tang, z = i/- ; cos. y = cos. /. sin. 2 z.
With this value of y, find in Table VII. the corresponding log. C; then we have,
log. t = Iog. C-(-log. sin.èy + S.log. (^).
EXAMPLE.
Given log. >• = 9,9ii5i4o. log. 7^= 9,7902520 ; 2/=i 11 44'" 22' ; to find ( in days.
4 log. r" 9,8951260
è log- r 9,9557670
= 4i°
22 = 82"
1 o s
37
f= 5 52 n
y= 9 53 22
8' ,5 tang. 9,9393690
■ ''" sin. 9,9957814
COS. 9,9977170
cos. 9,9934984
è log. r 9,955757c
z COS. 9,8776911
^^^ log. 0,0780659
Multiplied by 3 . log. 0,2341977
Table VII. log. C. 1,7622613
è 2/ = 4"* 56"" 4i' sin. 8,9354800
ï=8 ",54g4 log. 0,9319390
With the two radii r,
to describe that arc, in
TABLE VII. — For a Parabola.
r, )■', and the included arc »' — v= if, to find the time t in days, for a comet
parabolic orbit.
d m
0,00
0,10
0,20
o,3o
o,4o
o,5o
1,00
1,10
1,20
i,3o
Mo
T,5o
2,00
■2,10
2,20
2,3o
2,40
2,5o
3,00
3,
3,20
3,3o
3,40
3,5o
4,00
4,10
4,20
4,3o
4,4o
4,5o
5,00
1,7644177
1,7644171
1,7644153
1,7644122
1 ,7644079
1,7644024
1,7643957
1,7643877
1,7643785
1,7643681
1,7643565
1,7643436
1,7643395
1,7643142
1,7642977
1,7642799
1,7642610
1,7642408
1,7642193
1,764196'
1,7641728
1,7641477
1,7641213
1 ,7640938
1, 7640650
1, 76403 5o
1,7640037
1,7639713
1 ,7639376
1 ,7639027
1,7638665
6
18
3i
43
55
67
80
92
io4
116
129
i4i
i53
i65
178
189
202
2X5
226
239
25l
264
275
288
3oo
3i3
324
337
349
362
373
Log. C
d 111
5,00
5,10
5,20
5,3o
5,40
5,5o
6,00
6,10
6,20
6,3o
6,4o
6,5o
7,00
7,10
7,20
7,3o
7^0
7,5o
8,00
8,10
8,20
8,3o
8,4o
8,5o
9,00
9;I0
9.20
9,3o
9,40
9,5o
10,00
1,7638665
1,7638292
1 ,7637906
1 ,7637508
1,7637097
1 ,7636675
1 ,7636240
1,7635793
1,7635334
1,7634862
1 ,7634378
1,7633882
1 ,7633374
1,7632853
1,7632320
1,7631775
1,7631217
i,763o648
1 ,7630066
1,7629471
1,7628865
1,7628247
1,7627616
1,7626973
1,7626318
1,7625550
1,7624970
1,7624278
1,7623574
1,7622858
1,7622129
373
386
398
411
422
435
447
459
472
484
496
5o8
521
533
545
558
569
582
595
606
618
63 1
643
655
668
680
692
704
716
729
-4 1
d m
10,00
10,10
10,20
IO,3o
10,;
10, 5ù
11,00
11,10
11,20
II, 3o
II, 40
11, 5o
12,00
12,10
12,20
12, 3o
i2,4o
12,5o
1 3,00
i3,io
l3,20
i3,3o
i3,4o
i3,5o
1 4,00
i4,io
i4,2o
i4,3o
1 4,40
i4,5o
1 5,00
liOg. C
1,7622129
1,762138s
1 ,7620634
1,7619869
1,7619091
1,7618301
1,7617498
1,7616684
1,7616857
1,7616017
1,7614166
i,76i33o3
1,7612427
1,7611539
1,7610638
1 ,7609726
1 ,760880 1
1 ,7607864
1,7606916
1,7606963
1 ,7604980
1 ,7603994
1,7602996
1,7601986
1 ,7600962
1.7599927
1,7598880
1,7697820
1,76967 '
1 ,7696664
1,7694568
Diff.
neg.
74 1
754
766
778
790
So3
814
827
84o
861
863
876
901
gi2
926
937
062
973
986
999
lOIO
1023
io35
io47
1060
1073
1084
1096
1109I
16,00
i5,io
1 5,20
i5,3o
16,40
i5,5o
16,00
i6,io
16,20
16, 3o
i6,4o
16,60
1,7594568
1,7693469
1,7692338
1,7691206
1,7690060
1,7688903
1,7687733
1,758655 1
1,7685357
i,7584i5o
1,7682931
1,7681700
17,00 1,7680457
17,10 1,7679201
17,20
i7,3o
17,40
17,60
18,00
18,10
18,20
i8,3o
18,40
i8,5o
19,00
19,10
19,20
19,30
19,40
19,60
20,00
1,7577933
1,7676663
1,7676361
1,7674067
1,7672740
1,7671411
1 ,7670070
1,7668716
1,7567361
1,7665973
1,7564583
1,7663 1 80
1,7661766
1, 7660338
1,7668899
1,7557448
1,7555984
neg.
1 1 09
II2I
Ii33
1 145
1167
1 170
1182
1 194
1207
1219
123l
1243
1266
1268
1280
1292
i3o4
I3I7
1329
i34i
i364
i366
1378
1 390
i4o3
i4i5
1427
1439
i45i
i464|
USES OF TABLES VIII. IX. AND X.
Table VIII. combined with Table IX., for an elliptical orbit, anJ witli Table X., for a hyperbolic orbit, are used in findin»; the
elements of the orbit; when we have given, the two radii r, )-', the included heliocentric arc «' — ti=2/, and the time t of describing
that arc, expressed in days. These tables are limited to the most useful values of h, H, which do not exceed 0,6 ; and to values of x, z,
which do not exceed 0,3. These limits include the most common cases ; and in observations which do not fall within them, we can use
the indirect solutions explained in this appendix. VVkeii h or H exceeds 0,040, and log. yy, or log. YV, is required to be correct in
the seventh decimal place, ice must use the second differences.
PRECEPTS FOR TABLES VIII., IX., IN AN ELLIPTICAL ORBIT.
The particular object of these tables is to facilitate the computation of the value of O §: = £' — E, representing the diflference between
the two excentric anomalies E', E; corresponding respectively to the true anomalies v',v; which is an important part of the
preliminary process, in computing the elements of the orbit. After g has been found, the elements may be computed by the methods,
given in this appendix ; we shall not however enter here upon this subject, but shall restrict our remarks to the mere explanation of the
method of computing the value of g, by means of the tables.
Iq the calculation of g, there are two separate cases ; the one when / is acute, or t)' — ti between 0 and 180 ; the other when J is
obtuse or v' v between ISO'' and 360''. We shall give the precepts, in both the.^e cases, at full length, for convenience of reference ;
remarking, however, that the case of/ being acute, is that which occurs most frequently in practice, and is that for which these tables
are particularly designed.
When f is acute.
We must find w, I, mm, h, by the following formulas ;
tang. (45 +ii>)=\/ -;
1 =
sin. 2 hf tang.a 2 u)
cos. / ' COS. /
log. mm = 5,5680729 + 2 log. t — 3 log. cos./— f log. {rr') ;
Approx. log. 7» = log. mm — log. (7 -f f ) •
With this approximate value of h, find, in Table VIII., the
coresponding approximate value of loi
Approx. value oi x^
yy, also,
mm
men f is obtuse.
We must find w, L, MM, H, by the following formulas ;
tang. (45'' -|- w)=y/-;
r
sin.2 4/ tang.2 2 w
L =
yy
— I.
With this approximate value of x, find, in Table, IX., the
corresponding approximate value of *, and then the corrected
value of ft, from the formula,
corrected log. 7t = log. mm — log. (' + | + H •
With this corrected value of h, find a new value of log. yy, in
Table VIII., which is to be used in finding a corrected value of a-,
by the formula used above,
corrected value of r = — I .
yy
If necessary, we may repeat the operation until the assumed and
computed values of J agree ; then we have,
X = sin.3 4 g = sin.2 i (£' — £) ;
from which we easily obtain g or E' — E.
cos. / cos. /
log. JIO/= 5,5680729-}- 2 log. « — 3 log. (—cos./) — J log. {rr') ;
Approx. log. H= log. MM — log. (L — ^) .
With this approximate value of H, find, in Table VIII., the
corresponding approximate value of log. YY, also,
Approx. value 01 x = L — — .
With this approximate value of x, find, in Table IX., the
corresponding approximate value of |, and the corrected value
of H, from the formula,
corrected log. H = log. MM -
-log. (z,-!-^
)•
With this corrected value of H, find a new value of log. YY,
in Table VIII., which is to be used in finding a corrected value of
.r, by the formula used above,
corrected value of a: = i — — - .
If necessary, we may repeat the operation until the assumed and
computed values off agree ; then we have,
X = sin.2 4 g = sin.2 1 (£' _ E) ;
from which we easily obtain g or E' — E.
EXAMPLE.
Given log. r = o,33o764o; log. 7^ = 0,3222239: i' — «) = 2/ = 7'' 34" 53", 73 ; ( = 21'''''", 93391 ; to find ig = E' — E, or rather
i=sin.a ig.
r' log. 0,3222239
r log. 0,3307640
— =tang.4(45''4-«)) log. 9,9914599
45''4-.u) = 44''5i™33' tang. 9,9978650
ft"* .,-.*
ui = — o 27
0,3222239
0,3307640
sum 0,6529879
half 0,3264940
(rr')a log. 0,9794819
arith. co. 9,02o5i8i
/= 3''47™ 26' ,865 COS. arith. co. 0,0009512
è/=i''53"43',4325
sin.2 4/
±i = 0,0010963480
COS./
sin.
same
log.
8,5194986
8,5194986
7,0399484
tang.2 2 to , .
— 2 — = 0,0000242205
COS./
Z = o,ooii2o5685
1 = 0,8333333
2l«^ — 16™ 54' tang. 7,6916163^
same 7,6916163»
/ COS. arith. co. 0,0009512
'!^'i-ii^ = 0,0000242205 log. 5,384i838
COS./
constant log. 5,5680729
J = 2I''')'^9339I log. i,34iii6o
same i,34iii6o
(arith. co. log. cos. /) X 3 o,oo28536
|. log. T r" arith. CO. 9,o2o5i8i
mm log. 7,2736766
Approx. a; = 0,000748018
The correction. Table IX., corresponding to this value of x is insensible, therefore, we may assume this value of x for the true value
of 3in.2 4 g = 0,0007480186.
7 -f I = 0,8344539 log. 9,9214023
■mm log. 7,2736766
Approx. A ^ o,O0225o47 log. 7,3522743
Corresponds in Table VIII., to approx. log. ^y = o,oo2i633
mm log. 7,2736766
yy
= 0,0018685871 log. 7,271 5i33
I = 0,001 i2o5685
PRECEPTS FOR TABLES VIII. AND X., IN A HYPERBOLIC ORBIT.
The process for calculatin;; the elL'iiients of a liyperbolic oiliit, by means of r,r\ v' — r=2/an.l t, varies but very Utile from
that in an elliptical orbit, wliicli we have just explained. Tlie formulas for the computation of «', I, m, L, M, are identically the
sanic. The formulas for h, H, are the same, with the exception of using (" Table X, instead of ç Table IX ; moreover x is changed
into z. For convenience in reference we shall here give the formulas, for the hyperbola, arranged in the same order as lor the
ellipsis,
J =
IVhen f is acute.
4 r'
tang. (45<'-\-w)= v/- ;
sin.2 if j^ tang.2 2 to
COS. / cos. /
log. mm ^ 5,5680729 + 2 log. / — 3 log. cos./ — ^ log. (rr');
approximate log. h = log. mm — log. (I + ^).
With this approximate value of /i, find in Table VIII. the cor-
respoodiog approximate value of log. yy, also
approximate value of z = i .
yy
With this approximate value of ;, find in Table X. the corres-
ponding value of ^, and then the corrected value of h, from the
formula,
corrected log. /i = log. 7ii77i — log. C + ^-t-f)-
With this corrected value of A, find a new value of log. yy, in
Table V'lII., which is to be used in finding a corrected value of z,
by the formula used above, namely,
corrected value 01 z^l — ■ .
yy
If necessary we may repeat the operation, until the assumed
and computed value of ^ agree ; and this must be taken for the true
value of ^, to be used in computing the elements of the orbit, by
(he formulas given in this appendix.
When f is obtuse.
4 r'
tang. (45''-
L = -
sin.2 4/ tang.2 2 to
cos. f cos. /
log. MM = 5,56807394- 2 log. < — 3 log. (— cos. /) — ^ log. (r r') ;
approximate log. //= log. .MJ\I — log. {L — -| ).
With this approximate value of //, find in Table VIII. the cor-
responding approximate value of log. VY, also
approximate value of z = — L .
With this approximate value of 2, find in Table X. the corres-
ponding value of ^, and then the corrected value of H from the
formula,
corrected log. jH'= log. MM— log. (i— | — ^).
With this corrected value of H, find a new value of log. ¥¥,
in Table VIII., which is to be used in finding a corrected value of
z, by the formula given above, namely,
, , , MM
corrected value of z = —mr — L .
IF
If necessary we may repeat the operation, until the assumed and
computed value of f agree ; and this must be taken for the true
value of ^, to be used in computing the elements of the orbit, by
the formulas given in this appendix.
EXAMPLE.
Given log. r = o,o333585; log. r' = o,20o854i ; »' — » = 2/=48'' 12""; « = 5i, '''')" 49788; to find z.
r' log. o,2oo854i
r log. 0,0333585
0,1674956
45''-|-ti• = 47''45'"28^47 tang. 0,0418739
u>= 2 45 28', 47
2tc= 5'' 30"" 56', 94 tang. 8,98483x8
same 8,9848318
/ arith. co. cos. 0,0396081
tang.2 2 u)
cos. f
mm log. 8,7591571
o,20o854i
o,o333585
sum 0,2342126
half 0,1171063
(rr')tlog. o,35i3i89
arith. co. 9,6486811
^0.010215784 log. 8,0092717
constant 5,5680739
^ = 51'^''^', 49788 log. 1,7117894
same 1,7117894
(arith. co. log. cos. /) X 3 0,1188243
^ log. rr' arith. co. 9,6486811
i25
/=24 6™ cos. arith. co. o,o3g6o8i
è/=i2 3
sin. g,3i9658i
same 9,3196581
siD.2 if
COS. f
tang.2 2 w
^0,047744604 log. 8,6789243
COS./
= 0,010215784
I = 0,057960388
1 = 0,8333333
' + 1 = 018913937 log. 9,9500208
mm log. 8,7591671
Approx. ?s^ 0,0644371 log. 8,8091363
Corresponds in Table VIII. to approx. log. i/y = o,o56o848
mm log. 8,7591571
= o,o5o47454 log. 8,7030723
yy
I = o,o57g6o39
Approx. 2 = 0,00748585 =1
Corresponding to this in Table X. is ^= o,ooooo32
Hence, Z -f-|^ -1-^ = 0,8912969 log. 9,9500224
mm log. 8,7591571
corrected A = o,o64436g log. 8,8ogi347
Corresponds in Table VIII. to corrected log. yy=o,o56o846
mm log. 8,7591671
mm
yy
^^o,o5o47456
t = 0,05796039
log. 8,7030725
Corrected z = o,oo748583 which agrees with the assumed value.
TABLE VIII. — For an Ellipsis or Hyperbola.
This table, with Tables IX., X.,ave
,r' ; the included Iieliocentric ai-c v'
AN
for computing the
— 1! = 2/, and the
elements of the orbit, when there are given the two
time ( of describing that arc, expressed in days.
radii
h
H
Log. yy
Los. YY
0,0000
0001
0OO3
ooo3
ooo4
o,ooo5
0006
0007
0008
0009
0,0010
001 1
001 3
ooi3
0014
0,00 1 5
0016
0017
0018
0019
0,0020
0021
0022
0023
0024
0,0025
0026
0037
0028
0029
o,oo3o
oo3i
od32
oo33
oo34
o,oo35
oo36
oo37
oo38
0039
o,oo4o
oo4i
0043
0043
0044
o,oo45
0046
0047
oo48
0049
o,oo5o
oo5i
oo52
oo53
00 54
o,oo55
00 56
0057
oo58
ooSg
0,0060
0,0000000
0965
igSo
2894
3858
0,0004821
5784
M?
7710
8672
0^0009634
10595
1 1 557
I25l7
13478
o,ooi4438
15398
16357
17316
18275
0,0019234
20192
2ii5o
22107
23o64
0,002402
24977
25933
26889
27845
0,0028800
29755
30709
3i663
32617
0,0033570
34523
35476
36428
37381
o,oo38333
39284
40235
4ii86
421 36
o,oo43o86
44o36
44985
45934
46883
0,0047832
48780
49728
50675
51622
o,oo5256g
535i5
54462
55407
56353
0,0057298
965
965
964
964
963
963
963
963
962
962
961
962
960
961
960
960
9^9
959
9^9
959
958
958
957
957
9^7
956
956
956
956
955
955
954
954
954
953
953
953
952
953
95 1
952
951
951
950
950
950
949
94y
9^9
9^9
948
9-18
9^7
947
947
946
947
945
946
945
q45
h
H
0,0060
0061
0063
oo63
0064
o,oo65
0066
0067
0068
0069
0,0070
0071
0073
0073
0074
0,0075
0076
0077
0078
0079
0,0080
0081
0082
oo83
0084
o,oo85
0086
008
0088
0089
0,0090
0091
0093
0093
0094
o,oog5
0096
0097
0098
0099
0,0100
OIOI
0102
oio3
oio4
0,0 io5
0106
0107
0108
0109
0,0110
0111
01 12
oii3
oii4
0,0 1 1 5
0116
0117
0118
0119
0,0120
Log. ijy
Log. YY
0,0057298
58243
59187
601 3 1
61075
0,0062019
62962
63905
64847
65790
0,0066732
67673
68614
69555
70496
0,0071436
73376
73316
74255
75194
0,0076133
77071
78009
78947
79884
0,0080821
81758
82694
83630
84566
o,oo855o2
86437
87372
883û6
89240
0,0090 1 74
91 108
92041
92974
93906
0,0094839
95770
96703
97633
98564
0.0099495
100425
ioi356
102285
io32i5
0,0 1 o4 1 44
io5o73
1 0600 1
106929
107857
0,0108785
109712
1 1 0639
iii565
112491
o,oii34i7
Diff.
945
944
944
944
943
943
942
943
942
941
941
941
941
940
940
940
939
939
939
938
938
938
937
937
937
936
936
936
936
935
935
934
934
934
934
933
933
932
933
932
931
931
931
930
901
9=9
930
929
929
928
928
938
928
957
927
936
936
926
936
A
H
0,0120
0131
0132
0123
0124
0,0135
0126
0127
0128
0139
0,0 i3o
oi3i
Ol32
oi33
oi34
0,01 35
oi36
oi37
01 38
0139
0,0 i4o
oi4i
0142
0143
0144
o,oi45
0146
014?
oi48
0149
0,01 5o
oi5i
0l52
oi53
01 54
0,01 55
01 56
oi57
oi58
0159
0,0160
016
0163
oi63
0164
0,01 65
0166
0167
0168
0169
0,0170
0171
0173
0173
0174
0,0175
0176
0177
0178
0179
0;Oi8o
Log.
Log.
yy
YY
o,oii34i7
114343
1 1 5268
116193
117118
0,0118043
11896'
119890
120814
121737
0,0122660
123582
i245o5
125427
126348
0,0127269
128190
129111
i3oo32
1 30952
0,0131871
132791
133710
134629
135547
0,01 36466
137383
i383oi
139218
i4oi35
o,oi4io52
141968
143884
i438oo
144716
o,oi4563i
146546
147460
148375
149288
0,0l50203
i5i ii5
1 52038
i53g4i
1 53854
0,0154766
155678
1 56589
1 57500
i584ii
0,0159322
160333
161142
162052
162961
0,0163870
164779
165688
166596
167504
o.oi684i3
926
925
925
925
925
924
933
924
923
923
922
933
933
921
921
921
921
921
920
919
920
919
919
gi8
919
917
918
917
917
917
916
gi6
gi6
916
9i5
915
914
gi5
9'3
914
9i3
gi3
9i3
913
912
912
911
9"
9'i
9"
gio
gio
910
909
9°9
909
909
908
go8
908
007
h
H
o,oi8û
0181
0183
oi83
0184
0,01 85
0186
0187
0188
0189
0,0190
0191
0192
0193
0194
0,0195
0196
0197
0198
0199
0,0200
020
0202
0203
0204
0,0205
03o6
0307
0308
0309
0,0210
0211
0312
03l3
03l4
0,03 1 5
0216
0217
0218
02ig
0,0220
0221
0333
0233
023
0,0225
0226
0327
0228
0229
0,033o
023l
0233
0233
0334
0,0335
0236
0337
0238
0239
0,0240
Log. yy
Log. YY
0,0168412
169319
170226
171 i33
172039
0,0172945
i7385i
174757
175662
176567
0,0177471
178376
179280
i8oi83
181087
0,0181990
182893
183796
184698
i856oo
o,oi865oi
187403
i883o4
189205
190105
0,0191005
igigo5
ig28o5
193704
ig46o3
0,0195502
196401
197299
198197
199094
o>oi99992
20088g
201785
202682
203578
0,0204474
205369
206264
207159
2o8o54
0,0208948
209843
210736
2ii63o
212523
0,021 34 1 6
2 1 4309
2l520I
216093
2i6g85
0,0217876
218768
219659
220549
23l44o
0,02 3333o
Diir.
9"7
907
907
906
906
go6
906
9o5
905
go4
905
904
903
904
903
go3
go3
go2
902
901
go2
go I
901
goo
goo
goo
900
899
898
898
897
897
896
897
896
896
895
8g5
8g5
8g5
894
8g5
8g3
894
8g3
893
893
892
892
892
8gi
8g2
891
890
8qo
TABLE VIII. — For an Ellipsis or Hyperbola.
This tabic, with Table IX., X., are for computinE; the elements of the orbit, when there are given the two railii
r, r' ; the iiicliulcd heliocentric arc v' — v= if, and the time t of describing that arc, expressed in days.
y
H
0,0240
0241
0343
0343
0344
0,0345
0246
0347
0248
0349
o,o35o
025l
0252
0253
0254
0,0255
0256
0257
0258
0259
0,0260
0261
0263
0363
0364
0,0265
0266
0267
0268
0269
0,0270
0271
0272
0273
0274
0,0275
0276
0277
0278
0279
0,0280
0281
0282
0283
0284
0,0285
0286
0287
0288
0289
0,0290
0291
0292
0293
02g4
o,02g5
0296
0297
0298
029g
o,o3oo
Log.
Log.
yy
YY
0^333330
223220
224109
224998
225887
0,0236776
227664
228553
22g44o
23o328
0,023 1 21 5
233I03
333988
233875
234761
0,0235647
236532
237417
238303
239187
0,024007 1
24og56
241839
242723
243606
0,024448g
245372
246254
2471 36
248018
o,o348goo
349781
350663
25 1 543
253423
o,o2533o4
254i83
255o63
255942
256822
0,0257700
258579
259457
260335
261213
0,0262090
262g67
263844
364721
2655g7
0,0266473
26734g
268224
269099
269974
0,0270849
271733
373597
273471
274345
0,0375218
890
889
889
889
88g
888
868
888
888
887
887
886
887
886
886
885
885
885
885
884
885
883
884
883
883
883
882
881
880
879
880
879
880
879
878
878
878
877
877
877
877
876
876
876
875
8-5
875
875
874
874
874
874
873
873
y
H
o,o3oo
o3oi
o3o3
o3o3
o3o4
o,o3o5
o3o6
o3o7
o3o8
o3o9
o,o3io
o3ii
03l3
o3i3
o3i4
o,o3 1 5
o3i6
o3i7
o3i8
o3i9
o,o32o
o32i
o322
o323
o324
o,o325
0326
0327
o328
0329
o,o33o
o33i
o332
o333
o334
o,o335
d336
o337
o338
0339
o,o34o
o34i
o342
0343
o344
o,o345
o346
o347
o348
0349
o,o35o
o35
o352
o353
o354
o,o355
o356
0357
o358
o35g
o,o36o
Log. yy
Log. YY
0,02753:8
2760g I
276964
277836
278708
o,027g58o
280452
28i323
283194
283o65
0,0283936
284806
285676
386546
287415
0,0288284
289 r53
290022
290890
391758
0,0292626
293494
294361
395238
396095
0,0396961
397827
2986g3
299559
300424
O,o3oi2go
3o2i54
3o3oig
3o3883
3o4747
o,o3o56ii
306475
307338
308201
3ogo64
0,0309926
310788
3ii65o
3l35l3
3 I 3373
o,o3 14234
3i5og5
3i5g56
3i68i6
3 1 7676
o,o3i8536
3ig3g6
330355
321114
331973
o,o32283
323689
334547
3254o5
326262
0,0327120
Dim
873
S73
872
872
872
872
871
871
871
870
870
869
86g
868
868
868
868
867
867
867
866
866
866
866
865
866
864
865
864
864
864
864
863
863
863
862
862
862
862
861
861
861
861
860
860
860
860
859
859
859
858
858
858
858
857
858
856
H
o,o36o
o36i
o362
o363
o364
o,o365
o366
0367
o368
0369
0,0370
0371
0372
0373
0374
0,0375
0376
o377
0378
0379
o,o38o
o38i
o382
o383
o384
o,o385
o386
0387
o388
0389
0,0390
o3gi
0392
0393
o3g4
o,o3g5
0396
0397
0398
o3gg
o,o4oo
o,o4i
043
043
o44
o,o45
o46
o47
o48
049
o,o5o
o5i
o52
o53
o54
o,o55
o56
o57
o58
059
0,060
Log. yy
Log. YY
0,0327120
327g76
328833
32968g
33o546
o,o33i4oi
332257
333ii2
333967
334822
0,0335677
336531
337385
338239
33gog2
o,o33gg46
340799
34i65i
3425o4
343356
0,0344208
345o59
34591 1
346763
347613
o,o348464
34931
35oi64
35ioi4
35i864
o,o3527i3
353562
35441
355259
356io8
0,0356956
357804
358651
359499
36o346
0,0361192
369646
378075
386478
394856
0,0403209
4ii537
419841
438121
436376
0,0444607
452814
460998
469157
477294
o,o4854o7
493496
5oi563
509607
517628
0,0525626]
Diir.
856
857
856
857
855
856
855
855
855
855
854
854
854
853
854
853
852
853
852
852
85i
852
85i
85i
85i
85o
85o
85u
85û
849
849
849
848
849
847
848
847
846
8454
8429
84o3
8378
8353
8328
83o4
8280
8255
833i
8207
8184
8i5g
8137
8ii3
8067
8044
8021
7998
7976
h
H
0,060
061
062
o63
064
o,o65
o(i6
067
068
069
0,070
071
072
073
074
0,075
076
077
078
079
0,080
081
082
o83
084
0,0
086
087
088
089
0,090
09
09
og3
og4
0,095
096
097
098
099
0,100
101
102
io3
io4
o,io5
106
107
loi
109
0,110
II I
J 13
ii3
ii4
o,ii5
116
117
118
i'9
0,120
Log.
Log.
yy
YY
0,0525626
533602
541 556
549488
557397
o,o565285
573 1 5o
58o9g4
588817
596618
0,0604398
612157
6ig8g5
627612
6353o8
o,o642g84
65o63g
658274
665888
673483
0,0681057
688612
6g6 1 46
70 366 1
711157
0,0718633
736ogo
733527
74og45
748345
0,0755725
763087
770430
777754
785060
0,0792348
799617
806868
8i4ioi
82i3i6
o,o8285i3
8356g3
842854
849999
857125
0,0864335
871327
878401
885459
892500
o,o8gg523
go653o
gi352o
920494
927451
0,0934391
g4i3i5
948223
955114
961990
0,0968849
Difl'.
7976
7954
7932
7909
7888
7865
7844
7823
7801
7780
7759
7738
7717
7696
7676
7655
7635
7614
7595
7574
7555
7534
75i5
7496
7476
7457
7437
7418
7400
7380
7362
7343
7334
73o6
7288
7269
725i
7233
72i5
7>97
7180
7161
7145
7126
7110
7092
7074
7o58
7041
7023
7007
Oygo
6974
6957
6940
6924
6908
68gi
6876
6859
6843
TABLE VIII. — For an Ellipsis or Hyperbola.
elements of the orbit when theie are given tlie two radii
time t of describing that arc, expressed in days.
This table, with Table IX., X., are for computing the
?■, r' ; tlie included heliocentric arc v' — t) := 2/", and the
h
H
123
124
0,125
126
127
128
129
o,i3o
i3i
1 32
i33
1 34
0,1 35
i36
i37
1 38
1 39
o,i4o
i4i
142
J 43
o,i45
i46
147
1 48
'49
o.i5o
)5i
I 52
i53
1 54
o,i55
1 56
1 57
1 58
i59
0,160
161
162
1 63
164
o,i65
:66
167
168
169
0,170
171
172
173
174
0,175
176
177
178
179
0,1
Log. yy
Log. YY
0,0968849
975692
982520
989331
996127
0,1002907
100967a
1016421
io23i54
1029873
0,1036576
1043264
1049936
io56594
1063237
0,1069865
1076478
1083076
1089660
1096229
0,1102783
1109323
1115849
1122360
1128857
o,ii3534o
1141809
1 148264
1 1 54704
ii6ii3i
0,1167544
1173943
1 180329
II 8670 1
1 193059
0,1199404
1205735
1212053
1 218357
1224649
0,1230927
1237192
1243444
1249682
1255908
0,1262121
1268321
1274508
) 280683
1286845
1,1292994
1299131
i3o5255
1 3 1 1 367
1 3 17466
o,i323553
1329628
1335690
1 34 1740
1347778
0,i3538o4
Diff.
6843
6828
6811
6796
6780
6765
674g
6733
6719
6703
6688
6672
6658
6643
6628
66i3
6598
6584
6569
6554
654o
6526
65ii
6497
6483
6469
6455
6440
6427
64i3
6399
6386
6372
6358
6345
633i
63i8
63o4
6292
6278
6265
6252
6238
6226
6213
6200
6187
6175
6162
6149
6i37
6124
6112
6099
6087
6075
6062
6o5o
6o38
6026
6014
h
H
Log. yy
Log. YY
0,180
181
182
i83
184
o,i85
186
187
I ,SH
iSy,
Ojigo
191
192
193
194
0,195
196
197
198
199
0,200
201
20a
2o3
204
o,ao5
206
207
208
209
0,210
211
212
2l3
214
0,2l5
216
217
218
219
0,220
221
22a
223
224
0,225
226
227
228
229
o,23o
23l
232
233
234
0,235
236
237
238
239
0,240
o,i3538o4
1359818
1365821
1371811
1377789
o,i383755
1 3897 10
1395653
i4oi585
1407504
o,i4i34i2
1419309
i425ig4
143 1 068
143693
0,1442782
1448622
1454450
1460268
1466074
0,1471869
1477653
1483427
14891
1494940
o,i5oo68i
1 5o64 1 1
1 5 1 2 1 3o
i5i783S
1523535
0,1529222
1534899
i54o564
1546220
i55i865
0,1557499
1 563123
1568737
1 574340
1579933
o,i5855i6
1591089
1596652
1602204
1607747
o,i6i3279
1618802
i6243i5
1629817
i6353io
0,1640793
1646267
i65i73o
1657184
1662628
o,i66So63
1673488
1678903
i6843og
1689705
0,1695092
Diff.
A
H
6oi4
6oo3
5990
5978
5966
5955
5943
5932
5919
5908
5897
5885
5874
5863
585i
584o
5828
58i8
58o6
5795
5784
5774
5762
5751
5741
5730
5719
5708
5697
5687
5677
5665
5656
5645
5634
5624
56i4
56o3
5593
5583
5573
5563
5552
5543
5532
5523
55i3
55o2
5493
5483
5474
5463
5454
5444
5435
5425
541 5
5406
5396
5387
5378
0,2
241
242
243
244
0,245
246
247
248
249
o,25o
25l
253
253
254
0,255
256
257
258
259
0,260
261
262
263
264
0,265
266
267
268
269
0,270
271
272
273
274
0,275
276
277
278
279
0,280
281
28a
283
284
0,285
286
287
288
289
0,390
291
292
293
294
0,295
296
297
398
299
o,3oo
Log. yij
Log. YY
0,1695092
1 700470
1705838
1711197
1716547
0,1721887
1727218
i73a54o
1737853
I743i56
o, 1748451
1753736
1759013
1764280
1769538
0,1774788
1780029
1785261
1 790483
1795698
o,i8oogo3
1806100
18112
1816467
i82i638
0,1826800
i83ig53
1837098
1842235
1847363
o,i852483
i8575g4
i8626g6
I 8677g I
1872877
o,i877g55
i883o24
1888085
1893138
1898183
0,1903220
190824g
igi326g
igi828i
ig23286
0,1928382
1933271
1938251
1943224
1948188
0,1953145
1958094
1963035
ig67g68
1972894
0,1977811
1982721
1987624
1992518
i9g74o6
0,3002285
Diff.
5378
5368
535g
535o
5340
533 1
5332
53i3
53o3
53g5
5285
5277
5367
5258
535o
5341
5332
5222
52i5
52o5
5197
5i88
5179
5171
5162
5i53
5i45
5i37
5i28
5l20
5iii
5l02
5095
5o86
5078
5069
5o6i
5o53
5o45
5o37
5029
5o3o
5oi3
5oo5
4996
4973
4964
4957
4949
4941
4933
4926
4917
4910
4903
4894
4888
4879
4873
h
H
o,3oo
3oi
3n2
3o3
3o4
o,3o5
3o6
3o7
3o8
3og
0,3 10
3ii
3l2
3i3
3i4
0,3 1 5
3i6
3i7
3i8
3ig
0,320
321
323
333
324
0,325
326
337
338
339
o,33o
33 1
333
333
334
0,335
336
337
338
339
0,340
341
343
343
344
0,345
346
347
348
349
o,35o
35i
352
353
354
0,355
356
357
358
359
o,36o
Log. yy
Log. Y Y
0,2002285
2007157
2012031
2016878
2021727
0,2026569
2o3i4o3
2o3623o
ao4io5o
2045862
0,2050667
2055464
2060254
2o65o37
2069813
0,2074581
2079342
2084096
2088843
2093583
0,3098315
2io3o4o
2107759
21 13470
2 1 1 7 1 74
0,3131871
2126503
2i3i345
2135921
2i4o5gi
0,3145253
2149909
2154558
2i5g3oo
2103835
0,2168464
2173085
2177700
2i833o8
2186910
o,2i9i5o5
3 1 g6og3
2200675
22o525o
>i8
0,2214380
22i8g35
2223483
2338026
2232561
0,2 2370g I
234i6i3
2246l3o
225o64o
2255i43
o,235g64o
3264i3i
3368615
2373og4
3377565
0,3383o3l
Diff.
4872
4864
4857
4842
4834
4827
4830
4812
48o§
4797
479"
4783
4776
4768
4761
4754
4747
473g
4733
4725
4719
4711
4704
4697
4691
4683
4676
4670
4662
4656
4642
4635
4629
4631
46i5
4608
46o3
45g5
4588
4582
4575
4568
4562
4555
4548
4543
4535
453o
4522
4517
45io
45o3
4497
4491
4484
4479
4471
4466
445q
TABLE VIII. — For an Ellipsis or Hyperbola.
This table, witli Table IX., X., arc lor computing the elements of the orbit
»•'; the incliulcil helion-iitric arc r' — !• = :>/, ami the time ( of descril)iii{; th
, when there are given the two radii
at arc, expressed in days.
h
H
o,3tx)
301
3(53
363
364
0,365
360
367
368
369
0,370
37 >
372
373
374
0,375
376
377
378
379
o,38o
38 1
382
383
384
o,385
386
387
388
389
0,390
391
392
393
394
0,395
391-
397
398
399
0,400
4oi
402
4o3
4o4
o,4o5
4o6
407
4o8
409
0,4 10
4ii
4l2
4i3
4i4
o,4i5
Aid
417
4i8
419
0.420
Log. yy
Log. YY
0,2282031
2286490
2290943
2295390
2399831
o,23o4265
2308694
23i3i 16
2317532
2321942
0,2326340
233o'743
2335i35
2339521
2343900
0,2348274
2352642
2357003
236i359
2365709
0,2370053
2374391
2378723
2383o5o
2387370
0,2391685
2395993
2400296
2404594
2408885
0,24i3i7i
2417451
2421725
2425994
2430257
0,24345:4
243876O
2443oi3
2447252
2451487
0,2455716
2459940
2404 1 58
2468371
2472578
0,2476779
2480975
2485i60
2489351
2493531
0,2497705
2501874
25o6o38
251019O
2514349
0,2518496
2522638
2526775
2530906
2535o32
0,2539153
4459
4453
4447
4441
4434
4429
4422
4416
44io
44o4
4397
4392
4386
4379
4374
4368
436 1
4356
435o
4344
4338
4332
4327
4320
43i5
43o8
43o3
4298
4291
4286
4280
4274
42C9
4263
4257
4252
4246
4240
4235
4229
4224
4218
42l3
4207
4201
4196
4I9I
4i85
4180
4174
4169
4i64
4i58
4i53
4i47
4i42
4i37
4i3i
4126
4l2I
4ti6
h
H
0,420
42
422
423
424
0,425
426
427
428
429
o,43o
43 1
432
433
m
0,435
436
437
438
439
0,440
441
442
443
AAA
0,445
449
o,45o
45:
452
453
454
0,455
456
457
458
459
0,460
461
462
463
404
o,465
400
407
468
469
0,470
471
4-2
473
474
0,475
47O
477
478
479
0,480
Log. yy
Log. YY
0,2539153
25433O9
254737g
255i485
2555584
0,2559679
2503769
25O7853
2571932
257600O
0,2580075
2584i39
2588198
2592252
25ge3oo
0,2000344
2O04382
2O08415
2O 1 2444
2G1O4O7
0,2020480
2O24499
2028507
2O32511
26365o9
o,264o5o3
2644492
2648475
2O52454
2650428
o,200o397
2664362
2668321
2672276
2676226
0,2680171
2684111
2O8804O
2691977
2695903
0,2699824
2703741
2707652
2711559
2715462
0,2719360
2723253
2727141
2731025
2734904
0,2738778
2742648
27465 1 3
2750374
2754230
0,2758082
27O1929
2765771
2769609
2773443
0,2777272
4ii6
4iio
4106
4099
4095
4090
4o84
4079
4074
4069
4064
4059
4o54
4o48
4044
4o38
4o33
4029
4023
4019
4oi3
4008
4oo4
3998
3994
3989
3983
3979
3974
3969
3965
3959
3955
3950
3945
3940
3935
3931
392e
3921
3917
391 1
3907
3903
3898
3893
3888
3884
3879
3874
3870
3865
386 1
3850
3852
3847
3842
3838
3834
3829
3834
H
0,480
481
482
483
484
0,485
480
487
0,490
491
492
493
494
0,495
496
497
498
499
o,5oo
5oi
502
5o3
5o4
o,5o5
5oO
5o7
5o8
509
o,5io
5i
5l2
5i3
5i4
o,5i5
5iO
517
5i8
5ig
0,520
521
522
523
524
0,525
526
527
528
529
o,53o
53i
532
533
534
0,535
536
537
538
539
0,540
Log. yy
Log. IT
0,2777272
2781 096
2784916
2788732
2792543
0,2796349
2800152
2803949
2807743
2811532
o,28i53i6
28 1 9096
2822872
2826644
283o4ii
0,2834173
2837932
2841686
2845436
28491 8 1
0,2852923
285O6O0
2800392
2804 1 2 1
2867845
0,2871505
2875281
2878992
2882700
2886403
0,28901
2893797
2897487
2901 174
2go4856
0,2908535
2912209
2915879
2919545
3923207
0,2926864
2930518
2934168
2937813
2g4i455
0,3945092
2948726
2952355
2955981
2959602
0,2963220
29O6833
2970443
2974049
2977650
0,2981248
2984843
2988432
2992018
2995600
0,3999178
Diir.
3824
3820
38iO
38ii
38o6
38o3
3797
3794
3789
3784
3780
3776
3772
3767
3762
3759
3754
3750
3745
3742
3737
3732
3729
3724
3720
371O
371 1
3708
3703
3O99
3695
3O90
3087
3682
3079
3O74
3O70
3000
3002
3057
3654
305o
3045
3642
3637
3634
3629
3626
3621
30i8
36i3
3O10
36o6
3O01
3598
3594
3590
3586
3582
3578
35^4
h
H
0,540
54 1
542
543
544
0,545
540
547
548
549
o,55o
55i
552
553
554
0,555
550
557
558
559
o,50o
56 1
502
503
504
o,565
566
5O7
568
569
0,570
571
572
573
574
0,575
57O
577
578
579
o,58o
58 1
582
583
584
o,585
586
587
588
589
0,590
591
592
593
594
0,595
596
59'
598
599
0.000
Log. yy
Log. YY
0,2999178
3002752
3oo6323
3009888
3oi3452
0,3017011
3o2o566
3024117
3027664
3o3i2o8
o,3o34748
3o38284
3o4i8i6
3045344
3048869
0,3052390
3o55go7
3059420
30O2930
3o66436
0,3069938
3073437
307O931
3o8o422
3083910
0,3087394
3090874
3094350
3og7823
3ioi2g2
o,3io4758
3108220
3iii078
3ii5i33
3i 18584
0,3l2203l
3125475
3i28gi5
3i32352
3 1 35785
o,3i3g2i5
3i43Ô4i
3 1 46064
3i4g483
3152S98
0,3 1 503 10
3i5g7i9
3i03i24
3i66525
3169933
0,3 1733 18
3 1 7O709
318009O
3i8348i
3 1 8086 1
0,3190239
3193612
3196983
32oo35o
3203714
0,3207074
3574
3571
3565
3564
3559
3555
355 1
3547
3544
3540
3530
3532
3528
3525
3521
35i7
35i3
35io
35oe
35o2
34gg
34g4
34gi
3488
3484
3480
3476
3473
3469
3466
3462
3458
3455
345 1
3447
3444
3440
3437
3433
3430
3426
3423
3419
341 5
3412
3409
34o5
3401
3398
3395
3391
3387
3385
338o
3378
3373
3371
3367
3364
336o
a26
TABLE IX. — For an Elliptical Orbit.
This
table is used in connexion witli Ta
Me VIII., in
inding the elements of tl
e orbit, by
means of the two radii
r'. )■ .-
the included heliocentric arc v
-« = 2/,
and tlie time
t of describinn; that arc,
in days.
.
1
Diff.
X
1
UifT.
73
74
76
X
1
Diir.
1 54
X
1
Difl'.
X
1
Diir.
0,000
0,0000000
J
0,060
0,00021 3i
0,120
0,0008845
0,180
0,0020685
244
246
0,340
0,0038289
346
348
35o
352
' 001
0001
I
061
2204
121
8999
i55
181
20929
241
38635
002
0002
062
2278
122
gi54
1 57
i58
182
21175
242
38g83
oo3
ooo5
0
4
o63
2354
123
93 1 1
i83
21422
247
249
243
3g333
004
0009
064
243i
77
124
9469
184
21671
244
39686
5
78
i5g
25l
354
o,oo5
0,0000014
7
7
9
0,065
o,ooo25og
79
81
0,125
0,0009628
161
o,i85
0,0021922
262
264
255
258
0,245
O,oo4oo39
355
358
35g
36 1
006
0021
066
2588
126
09789
162
186
22174
246
4o3g4
007
0028
067
2669
82
127
ogg5i
1 64
i65
187
22428
247
40762
008
0037
068
2761
83
128
ioii5
188
22683
248
4iiii
009
0047
069
2834
12g
10280
189
22g4i
249
41472
10
84
167
258
363
0,010
0,0000057
i3
0,070
0,0002918
86
o,i3o
0,0010447
168
0,190
0,0023 igg
261
262
263
266
0,260
o,oo4i836
364
367
368
371
on
0070
i3
071
3oo4
87
89
i3i
1 061 5
i6g
191
23460
261
42199
012
oo83
i4
16
072
3ogi
1 32
10784
192
23722
262
42666
oi3
0097
073
3i8o
1 33
10955
171
173
193
23g85
253
42934
oi4
oii3
074
326g
1 34
11128
ig4
24261
2 54
433o5
17
91
173
267
372
0,01 5
016
o,ooooi3o
oi48
18
19
0,075
076
o,ooo336o
3453
93
93
95
97
o,i35
1 36
0,001 i3oi
"477
176
o,ig5
1 96
0,0024618
24786
268
0,255
256
0,0043677
44o5i
374
376
377
38o
017
0167
077
3546
137
1 1 654
177
178
180
197
26066
370
257
44427
018
019
0187
0209
22
078
079
364 1
3738
1 38
1 39
ii832
1 20 1 2
198
199
25328
a56o2
272
274
2 58
269
44804
45i84
22
97
181
275
382
0,020
0,000023 1
1/Ï
0,080
o,ooo3835
99
100
o,i4o
0,0012193
i83
184
i85
188
0,200
0,0026877
0,260
0,0045666
383
385
387
3go
021
0255
24
25
26
28
081
3g34
i4i
12376
201
26154
277
261
45g49
022
023
0280
o3o6
082
o83
4o34
4i36
102
io3
142
143
i256o
12745
202
2o3
26433
26713
279
280
282
262
263
46334
46721
024
o334
084
4239
i44
12933
2o4
26996
264
471 1 1
28
io4
188
283
391
0,025
o,oooo362
32
o,o85
0,0004343
io5
o,i45
0,OOl3l2I
0,205
0,0027378
286
287
288
0,265
0,0047602
392
396
397
399
026
o3g2
086
4448
107
108
1 46
i33ii
IQO
206
27664
266
47894
027
0423
087
4555
1 47
i35o3
iga
193
ig5
207
27861
267
4828g
028
0455
34
088
4663
1 10
1 48
13696
208
28139
268
48686
029
0489
089
4773
149
1 389 1
209
28429
290
269
4go85
34
III
ig6
293
4oo
o,o3o
0,0000623
36
37
38
4o
o,ogo
o,ooo4884
112
0,1 5o
0,0014087
198
0,210
0,0028722
293
296
0,270
o,oo4g485
4o3
4o4
407
408
o3i
o559
091
4gg6
ii3
i5i
14285
211
2goi5
271
49888
032
0596
092
5 1 09
ii5
l52
14484
'99
212
2g3i 1
273
60292
o33
0634
093
5224
117
1 53
14684
200
2l3
29608
297
273
60699
o34
0674
094
5341
1 54
14886
202
2l4
29907
299
274
61 107
4o
117
204
300
410
o,o35
0,0000714
42
43
45
45
0,ng5
o,ooo5458
"9
120
o,i55
o,ooi5ogo
205
0,2 1 5
o,oo3o207
302
3o5
3o5
308
0,276
0,oo5i5i7
4i3
4i4
416
418
o36
0756
096
5577
1 56
i52g5
216
30609
276
5ig3o
037
0799
097
5697
122
1 57
i55o2
207
208
217
3o8i4
277
63344
o38
0844
og8
58ig
123
1 58
16710
218
31119
278
62760
o3g
0889
099
5942
159
16920
210
219
31427
279
53178
47
124
211
309
420
o,o4o
0,0000936
48
5i
0,100
0,0006066
126
0,160
o,ooi6i3i
2l3
2l5
216
0,220
o,oo3 1 736
3ii
313
3i5
316
0,280
o,oo635g8
422
04 1
0984
lOI
6192
127
129
i3o
161
i6344
221
32047
281
54020
042
io33
102
63ig
162
i655g
222
3235g
282
54444
424
426
428
043
1084
io3
6448
1 63
16775
223
02674
283
54870
044
ii35
104
6578
164
16992
217
224
32990
284
552g8
53
i3i
219
3i8
43o
0,045
0,0001188
54
56
56
58
o,io5
0,0006709
1 33
o,i65
0,0017211
0,225
o,oo333o8
319
333
333
325
0,285
0,0066728
432
434
436
438
o46
1242
106
6842
1 34
i35
166
17432
221
226
33637
286
66160
047
1298
107
6976
167
17654
222
224
225
227
33g49
387
565g4
048
1 354
108
7rii
1 37
168
17878
228
34272
288
57o3o
049
l4l2
109
7248
169
i8io3
22g
34597
289
57468
59
1 38
227
327
440
o,c5o
0,0001471
61
61
63
64
0,110
0,0007386
i4o
0,170
o.ooi833o
228
o,23o
o,oo34g34
338
33o
332
0,290
0,0067908
442
446
446
448
o5i
o52
1 532
1593
III
113
7526
7667
i4i
142
171
172
18558
18788
23o
232
23l
232
35252
35582
291
292
6835o
587g5
o53
i656
ii3
7809
1 44
173
19020
233
233
35914
334
3g3
59241
o54
1720
ii4
7953
■74
19253
234
36348
2g4
69689
65
i45
234
336
45o
o,o55
0,0001785
67
68
69
71
o,ii5
o,ooo8og8
T 4-!
0,175
0,0019487
237
237
240
24l
243
244
0,235
o,oo36584
337
339
341
343
345
346
0,395
0,0060139
4 53
454
457
458
46 1
o56
i852
116
8245
1 48
176
19724
236
36931
296
60691
o57
1920
117
8393
177
1 996 1
237
37360
297
61045
o58
059
■989
2060
118
"9
8542
86g3
149
i5i
l52
178
179
20201
20442
238
239
37601
37944
298
299
61602
61960
0,060
0,0002 1 3i
71
73
0,120
0,0008845
i54
0,180
o,oo2o685
0,240
0,0038289
o,3oo
0,0062431
TABLE X. — For a Hyperbolic Orbit.
This
tabic is used in connexion with Table VIII., in fimlins; the
elenie
nts of the orbit bv
means of th
c two radii »', r ;
the iiic
viiied heliocentric
arc I)'
-V=7f,
uid tl
e time
t of describ
ing that arc,
in days.
z
^
Ditr.
^
Diff.
~
s"
Dili'.
~
^
Dili'.
178
~
s"
Dill'.
0,000
0,0000000
0,060
0,0001988
66
67
68
68
0,120
0,0007698
, „ /
0,180
0,0016783
o,34o
0,0028989
001
0001
I
c6i
2o54
121
7822
134
126
126
128
181
16960
241
29166
337
2 38
002
0002
I
3
4
062
2121
122
7948
183
'7139
17g
180
181
242
29894
oo3
004
ooo5
0009
o63
064
2189
2257
123
124
8074
8202
1 83
1 84
'73ig
I750Ô
243
244
39638
29852
3 29
239
5
70
128
181
281
o,oo5
0,0000014
6
8
8
o,o65
0,0002327
0,135
o,ooo833o
o,i85
0,0017681
i83
1 83
184
i85
0,245
o,oo3oo83
281
281
283
233
006
0020
066
2398
71
136
8459
139
i3i
i3i
l33
186
I7S64
346
3o3i4
007
008
0028
oo36
067
068
2470
2543
72
73
127
138
8590
8721
187
188
18047
18281
247
34s
3o545
30778
009
0046
10
069
2617
74
139
8853
189
I84I6
249
3ioii
II
74
i33
186
234
0,010
on
0,0000057
0069
12
i3
i4
i5
0,070
071
0,0002691
2767
76
o,i3o
i3i
0,0008986
9120
i34
i35
i35
1 37
0,190
191
0,0018602
1878g
187
187
189
i8g
o,25o
25l
0,0081245
3 1480
335
386
OI3
0082
072
2844
77
l32
9255
193
18976
253
81716
286
287
oi3
0096
073
3922
78
1 33
9390
193
19165
253
31953
oi4
OIII
074
3ooi
79
1 34
9527
194
19354
254
82189
16
80
1 38
190
288
0,01 5
0,0000127
18
0,075
o,ooo3o8i
81
82
83
84
o,i35
0,0009665
1 38
i4o
i4o
i4i
0,195
o,ooig544
0,255
0,0083437
289
289
34l
016
oi45
076
3162
i36
09803
196
.g735
'9'
256
82666
017
0164
'9
077
3244
1 37
09943
197
19926
'9'
ig3
ig3
257
82905
018
oi83
'9
078
3327
138
ioo83
198
20119
258
33i46
oig
0204
21
079
3411
1 39
10334
199
203l2
259
33387
3^1
32
85
142
ig5
24l
0,020
021
0,0000226
0249
23
34
25
0,080
081
0,0003496
3582
86
o,i4o
i4i
o,ooio366
loSog
143
i44
145
i46
0,200
201
0,0020507
20703
195
195
0,360
361
0,0088628
88871
243
243
344
245
022
0273
082
3669
87
88
89
l43
10653
303
30897
363
34114
023
0398
o83
3757
i43
10798
203
2iog4
'97
198
263
34858
024
o325
27
084
3846
144
10944
204
21392
264
84608
27
90
i47
198
345
0,025
o,oooo352
o,o85
0,0003936
0,145
0,0011091
l47
1 49
i4g
o,3o5
0,0021490
0,265
0,0084848
346
247
248
24g
026
o38i
29
086
4027
91
1 46
1 1 338
206
21689
199
266
35094
027
o4io
29
3i
32
087
4119
9'
93
i47
ii387
207
2188g
300
267
35341
028
0441
088
4212
148
1 1 536
208
22ogo
301
368
3558g
029
04-3
089
43o6
94
149
1 1687
IJI
209
222gi
201
269
35888
33
95
i5i
2o3
249
o,o3o
o,oooo5o6
33
36
36
37
0,090
o,ooo44oi
95
97
98
o,i5o
0,001 1838
l53
0,3I0
o,oo224g4
2n3
204
2o5
2o5
0,270
0,0086087
o3i
o32
0539
0575
091
092
44g6
4593
i5i
l53
1 1990
12143
1 53
i53
21 1
212
236g7
33901
271
272
36837
36587
25o
353
352
o33
061 1
093
4691
1 53
13296
i55
2l3
23io6
273
86889
o34
o648
094
4790
99
1 54
i345i
2l4
233ii
274
37091
38
100
1 56
207
253
o,o35
0,0000686
4o
40
4i
43
0,095
o,ooo48go
o,i55
0,0013607
1 56
1 58
1 58
1 59
0,21 5
o,oo235i8
0,275
0,0087844
254
254
255
256
o36
o37
o38
0726
076e
0807
096
097
098
4991
5og2
6195
lOI
lOI
io3
104
1 56
1 57
1 58
12763
I2g2i
1 307g
216
217
218
23725
23932
34142
207
207
210
276
277
378
37598
87852
88107
039
08 5o
099
529g
1 59
1 3338
219
24352
210
279
38363
44
io4
160
210
257
o,o4o
0,0000894
44
46
47
48
0,100
o,ooo54o3
106
0,160
o,ooi33g8
161
0,220
0,0034562
0,280
0,0088620
257
258
35g
360
o4i
0938
101
5509
161
i355g
ifio
231
24774
2 I 2
281
38877
042
0984
102
56i6
107
163
1373 1
1U2
162
164
233
24986
213
2l3
213
283
89135
043
o44
io3i
1079
io3
io4
5723
5832
107
log
1 63
164
1 3883
i4o47
233
224
25199
35413
283
284
39894
89654
49
log
164
2l5
260
0,045
0,0001128
5o
o,io5
o,ooo594i
o,i65
0,00 1 42 1 1
J 66
0,225
0,0025637
2l5
216
0,285
0,0039g i4
261
262
263
263
046
1 178
5i
106
6o53
III
166
14377
166
226
35842
286
40175
047
1229
52
53
107
6i63
II I
167
14543
167
168
227
26o58
387
40437
o48
1281
108
6275
IT2
ii4
168
14710
228
26275
217
218
388
40700
049
1 334
109
6389
i6g
14878
229
36493
289
40968
55
ii4
169
218
264
o,o5o
0,0001389
55
0,110
o,ooo65o3
ii5
116
0,170
o,ooi5o47
169
171
171
172
o,23o
0,0026711
0,390
0,0041337
364
366
366
367
o5i
o53
1 444
i5oo
56
58
III
112
6618
6734
171
173
i53i6
1 5387
33l
233
26931
27l5l
220
2gi
293
41491
41757
o53
o54
1 558
1616
58
ii3
ii4
685 1
696g
"7
118
173
174
i5558
1 5730
233
234
37371
27593
220
222
2g3
294
43038
422go
59
"9
173
223
267
o,o55
0,0001675
61
62
62
64
64
66
0,1 1 5
0,0007088
0,175
0,001 5903
174
175
176
176
178
,78
0,235
0,0027816
223
224
224
236
226
327
o,3g5
0,0042557
269
269
269
271
271
o56
057
o58
o5g
1736
1798
i860
1924
116
117
118
"9
7208
733g
745 1
7574
1 20
121
122
123
124
134
176
177
178
17g
16077
16352
16438
16604
2 36
237
338
239
28039
28363
38487
38713
296
297
298
299
42826
43og5
43864
43635
0,060
0,0001988
0,120
0,0007698
0,180
0,0016783
0,240
0,0038989
o,3oo
0,0048906
PRECEPTS FOR THE USES OF TABLES XI. AND XII.
These Tiibles are inserted for the purpose of changing the arcs of the centesimal division of the quadrant into sexagesimals.
Table XI., is divided into three distinct parts. The first part gives the degrees and minutes, in sexagesimals, for every degree of
the centesimal division, from 0° to .399° ; the tens being in the side column, and the units at the top. Thus we see by inspection, that
260° = 234'' 00'" ; 261° = 2.34'' 54"' ; &c. The second part gives the minutes and seconds in sexagesimals, corresponding to the
centesimal division from 0' to 99'; the tens of minutes being at the side, and the units at the top ; thus 60' = 32'" 24*; 61 ' = .32"' 56 ,4:
&c. The third part gives the seconds and decimals in sexagesimals, corresponding to the centesimal division, from 0" to 99" ; the tens
of seconds being at the side and the units at the top; thus 40" = 12% 960; 41" = 13% 284; &c. The two following examples, show
its use in more complicated cases; they require no particular explanation.
EXAMPLE I .
Change agS 21' 17" into sexagesimals.
Table XI. 293° = 26^ A" 00*
21' = II 20,4
17" = 5,5o8
293''2i'i7" =263''53'"25',9o8
EXAMPLE II.
Change 263 53"" 25', 908 into centesimals.
Table XI. 263"^ 42"'
= 293° 00' 00"
Remainder, ii"" 25 , 908
Table XI. 11'" 20', 4 = o 21 00
Remainder Table XI. 5^, 5o8 =
17
263'^ 53°" 25* 908 =2g3° 21' 17"
Table XII., gives the seconds and decimals, in sexagesimals, for every second of the centesimal division, from 0" to 999"; the tens
being in the side column and tlie units at the top. It is computed by the rule s = 0,324. c; s being the number of sexagesimal seconds
corresponding to c In centesimal seconds. Hence we have by inspection 570" = 184% 680 ; 571" = 185*, 004 ; &c. If we change the
decimal point, three places to the left, we shall get, from the table, by inspection, the value of every thousandth part of 1" from 0",001
to 0",999. Thus we have, by using the same numbers as before, 0",.570 = O", 184680 ; 0",571 = 0',185004 ; &c. In like manner, by
changing the decimal point to the left 6 units we get the values from 0",000001 to 0",000999; &c. We may also change the decimal
point to the right, if larger numbers are wanted.
EXAMPLE III.
Change 327", 345 into se.xagesimals.
327", 000 :=: io5*, 948
,345 111780
327", 345 = 106*, 059780
EXAMPLE V.
Change 327345" into sexagesimals.
327000" = 105948'
345"= in*, 780
327345" = io6o59% 780
EXAMPLE VII.
Change o", 6443o2 into sexagesimals.
o", 644 = o*, 2o8656
o", ooo3o2 = q8
O", 644302 = Q,\ 208754
EXAMPLE IV.
Change 106*, 059780 into centesimal seconds.
Table XII. io5*, 948 = 327", 000
Table XII. 111780= 345
106*, 059780 = 327", 345
EXAMPLE VI.
Change 106059*, 7^° ''^*'' centesimal seconds.
Table XII. 105948* = 327000"
Table XIII. in', 780 = 345"
106059*. 780 = 327345"
EXAMPLE VIII.
Change o", 076897 into sexagesimals,
o", 076 = o', 024624
o", 000897 = 291
o", 076897 = o", 024915
Table XII., has been found very convenient in making the reductions of the planetary inequalities, in this volume, from centesimal
to sexagesimal seconds, to six places of decimals, as in the two last examples. Since it is easy (0 obtain the sum of the two parts of the
fraction, without the trouble of writing them down separately; the last part of the fraction being generally so small that it is easy to
add it to the large tabular number corresponding to the first part. Thus in example VII., the number 98 is easily added lo 0,208656,
to obtain 0,208754, by mere inspection. The numbers given in this volume were in the first place computed from the table, and then
verified by a numerical calculation ; found by putting .s' = 0,3 c and s = *' + 0,08 s'. So that instead of writing down the number c
and then multiplying it by 0,324 ; we may write down, in the first instance 0,3 c ; and then multiply it by 0,08, which gives 0,024 c;
whose sum is s ^0,324 c. This method, applied to the preceding examples VII., VIII., produce the following results:
EXAMPLE IX.
Change 0", 6443o2 into sexagesimals.
0,3 c =; o*, 1932906
Multiply by 0,08 i5463248
o', 208753848
EXAMPLE X .
Change o", 076897 into sexagesimals.
0,3 c = o*, 0230691
Multiply by 0,08 1845528
o*, 024914628
TABLE XI.
To convert ccnte
iiiiKil degrees, minutes, and seconds, into
^exagcsimals.
1.—
To couvert
coiUcàiinul i
ogrees into soxagesimals
Cenic.
0
1
2
3
4
5
6
7
8
9
i m
d m
d m
d m
d ,11
d m
d m
(l m
d m
d m
O
0,00
0,54
1,48
2,42
3,36
4,3o
5,24
6,18
7,12
8,06
I
y, 00
9,54
10,48
11,42
12,36
i3,3o
14,24
i5,i8
16,12
17,06
2
18,00
18,54
19,48
20,42
2 1,36
22,3o
23,24
24,18
25,12
26,06
3
27,00
27,54
28,48
29,42
3o,36
3i,3o
32,24
33.18
34,12
35,06
4
36,oo
36,54
37,48
38,42
39,36
4o,3o
41,24
42,18
43,12
44,06
5
45,00
45,54
46,48
47,42
48,36
49,3o
5o,24
5i,i8
52,12
53,06
6
54,00
54,54
55,48
56,42
57,36
58,3o
59,24
60,18
61,12
62,06
7
63,oo
63,54
64,48
65,42
66,36
67,30
68,24
69,18
70,12
71,06
8
72,00
72.54
73,48
74,42
75,36
76,30
77,24
78,18
79,12
80,06
9
81,00
81,54
82,48
83,42
84,36
85, 3o
86,24
87,18
88,12
89,06
10
90,00
90,54
9', 48
92,42
93,36
g4,3o
95,24
96,18
97,12
98,06
1 1
99>oo
99-54
100,48
101,42
102,36
io3,3o
104,24
io5,i8
106,12
107,06
l:>
ioS,oo
108,54
109,48
110,42
111,36
II2,3o
ii3,24
114,18
Il5,12
1 16,06
l3
117,00
117,54
118,48
119,42
120,36
1 2 1 ,3o
122,24
123,18
124,12
125,06
i4
126,00
126,54
127,48
128,42
129,36
i3o,3o
i3i,24
i32,i8
i33,i2
1 34,06
i5
i35,oo
i35,54
1 36,48
1 37,42
I 38,36
I 39,30
i4o,24
i4i,i8
142,12
143,06
i6
1 44,00
144,54
1 45,48
i46,42
i47,36
i48,3o
149,24
i5o,i8
l5l,12
1 52,06
17
1 53,00
i53,54
1 54,48
i55,42
1 56,36
i57,3o
1 58,24
1 59,18
160,12
161,06
iS
162,00
162,54
i63,48
164,42
i65,36
i66,3o
167,24
168,18
169,12
1 70,06
■9
171,00
171,54
172,48
173,42
174,36
175,30
176,24
177,18
178,12
1 79,06
20
180,00
180,54
181,48
182,42
1 83,36
i84,3o
i85,24
186,18
187,12
188,06
21
iSg,oo
189,54
190,48
191,42
192,36
193,30
194,24
195,18
196,12
197,06
22
198,00
198,54
199,48
200,42
201,36
202,3o
2o3,24
204,18
2o5,12
206,06
23
207,00
207,54
208,48
209,42
2io,36
21 i,3o
212,24
2i3,i8
2l4,i2
21 5,06
24
216,00
216,54
217,48
218,42
2ig,36
220, 3o
221,24
222,18
223,12
224,06
25
225,00
225,54
226,.'î8
227,42
228,36
229,30
23o,24
23i,i8
232,12
233,06
26
234,00
234,54
235,48
236,42
237,36
238,3o
239,24
240,18
241,12
242,06
27
243,00
243,54
244,48
245,42
246,36
247,3o
248,24
249,18
25o,I2
25 1,06
28
252,00
252,54
253,48
254,42
255,36
256,30
257,24
258,i8
259,12
260,06
29
261,00
261,54
262,48
263,42
264,36
265,3o
266,24
267,18
268,12
269,06
3o
270,00
270,54
271,48
272,42
273,36
274,30
275,24
276,18
277,12
278,06
3i
279,00
279,54
280,4s
281,42
282,36
283,3o
284,24
285,18
286,12
287,06
32
288,00
288,54
289,48
290,42
291,36
292,30
293,24
294,18
295,12
296,06
33
297,00
297,54
298,4s
299,42
3oo,36
3oi,3o
302,24
3o3,i8
3o4,i2
3o5,o6
34
3o6,oo
3o6,54
307,48
3o8,42
3o9,36
3io,3o
3ll,24
3i2,i8
3i3,i2
3 14,06
35
3 1 5,00
3 I 5,54
3i6,48
317,42
3 1 8,36
3 19,30
320,24
321,18
322,12
323,06
36
324,00
324,54
325,48
326,42
327,36
328,30
329,24
33o,i8
33l,12
332,06
37
333,00
333,54
334,48
335,42
336,36
337,3o
338,24
339,18
340,12
341,06
38
342,00
342,54
343,48
344,42
345,36
346,3o
347,24
348,18
349,12
35o,o6
39
35 1,00
35 1,54
352,48
353,42
354,36
355, 3o
356,24
357,18
358,12
359,06
II. — T
0 convert c
sntesiinal ini
notes into se
^agesimals.
Centet.
0
1
2
3
4
5
6
7
8
9
m s
7n 5
m s
m s
m s
ïft s
m s
m s
m s
m s
0
0,00,0
0,32,4
1,04,8
1,37,2
2,09.6
2,42,0
3,i4,4
3,46,8
4,19,2
4,5 1,6
I
5,24,0
5,56,4
6,28,8
7,01,2
7,33,6
8,06,0
8,38,4
9,10,8
9,43,2
10,1 5,6
2
10,48,0
11,20,4
11,52,8
12,25,2
12,57,6
i3,3o,o
14,02,4
1 4,34,8
15,07,2
15,39,6
3
16,12,0
16,44,4
17,16,8
17,49,2
18,21,6
18,54,0
19,26,4
19,58,8
20,3l,2
2i,o3,6
4
21,36,0
22,08,4
22,40,8
23,l3,2
23,45,6
24,18,0
24,5o,4
25,22,8
25,55,2
26,27,6
5
27,00,0
27,32,4
28,04,8
28,37,2
29,09,6
29,42,0
3o,i 4,4
3o,46,8
3i,i9,2
3i,5i,6
6
32,24,0
32,56,4
33,28,8
34,01,2
34,33,6
35,06,0
35,38,4
36, 10,8
36,43,2
37,1 5,6
7
37,48,0
38,2o,4
38,52,8
39,25,2
39,57,6
4o,3o,o
41,02,4
41,34,8
42,07,2
42,39,6
8
43,12,0
A^AM
44,16,8
44,49,2
45,21,6
45,54,0
4C\-^<iÀ
46,58,8
47,5i,2
48,o3,6
9
48,36,0
49,08,4
49,40,8
5o,i3,2
5o,45,6
5 1,1 8,0
5i,5o,4
52,22,8
52,55,2
53,27,6
m.— ï
0 convert c
anteaimal sec
onds into so:;
ageainmlâ.
1
Cînto.
0
0
1
2
3
4
5
s
1,620
6
7
8
9
s
0,000
s
0,324
0,648
0,972
s
1,296
1,944
5
2,268
s
2,592
5
2,916
I
3,240
3,564
3,888
4,212
4,536
4,860
5,184
5,5o8
5,832
6,1 56
2
6,48o
6,804
7,128
7,452
7,776
8,100
8,424
8,748
9,072
9,396
3
9,720
10,044
io,368
10,692
11,016
1 1 ,340
11,664
11,988
I2,3l2
12,636
4
12,960
i3,284
1 3,608
13,932
i4,256
i4,58o
14,904
l5,228
i5,552
15,876
5
16,200
16,524
i6,848
17,172
17,496
17,820
i8,i44
18,468
18,792
19,116
6
19,440
19,764
20.088
20,4 12
20,736'
21,060
21,384
2 1 ,708
22,o32
22,356
7
22,680
23,004
23,328
23,652
23,976
24,3oo
24,624
24,948
25,272
25,596
8
25,920
26,244
26.568
26,892
27,216
27,540
27,864
28,188
28,512
28,836
9_
29,160
29,484
29,808
3o,i32
3o,456
30.780
3i,io4
31,428
3 1,752
32,076
a27
TABLE XII.
To convert centesimal seconds into
sexagesimr
Is.
Con(c>.
0
1
2
3
4
5
6
7
8
9
Ccnic.
5
s
S
£
s
5
s
s
£
0
s
0,324
o,648
0,972
1,296
1,620
1,944
2,268
2,592
2,916
0
I
3,240
3,564
3.888
4,212
4,536
4,860
5,184
5,5o8
5,832
6,1 56
I
1
6,480
6,804
7,128
7.452
7.776
8,100
8,424
8,748
9,072
9,396
3
3
9,720
10,044
10, 368
10,693
11,016
1 1 ,340
1 1 ,664
11,988
12,3 12
12,636
3
4
12, 960
i3,284
1 3,608
13,932
14,256
i4,58o
14,904
15,228
i5,552
15,876
4
5
16,200
16,524
16,848
17,172
17,496
17,820
i8,i44
18,468
18,792
19,1 16
5
6
19,440
19,764
20,088
20,4 12
20,736
2 1 ,060
21,384
21,708
22,o32
2 2,356
6
7
22,680
23,oo4
23,328
23,652
23,976
24,3oo
24,624
24,948
25,272
25.596
7
8
25,920
26,244
26,568
26,892
27,216
27,540
27,864
28,188
28,512
28,836
8
9
29,160
29,484
29,808
3o,i32
3o,456
30,780
3i,io4
31,428
3i,752
32,076
9
10
32,4oo
32,724
33,048
33,372
33,696
34,020
34,344
34,668
34,992
35,3i6
10
II
35,640
35,964
36,288
36,6i2
36,936
37,260
37,584
37,908
38,232
38,556
11
12
38,880
39,204
39,528
39,852
40,176
4o,5oo
40.824
4i.i48
4 1,472
41,796
12
i3
42,120
42,444
42,768
43,092
43,416
43,740
44,064
44,388
44,7 > 2
45,o36
i3
i4
45,36o
45,684
46,008
46,332
46,656
46,980
47.3o4
47,628
47,952
48,276
i4
i5
48,600
48,924
49,248
49.572
49,896
5o,220
5o,544
5o,868
51,192
5i,5i6
i5
i6
5 1,840
52,164
52,488
52,812
53,i36
53,460
53,784
54,108
54,432
54,756
16
17
55,080
55,4o4
55,728
56,o52
56,376
56,700
57,024
57,348
57,672
57,996
17
i8
58,320
58,644
58,g68
59,292
59,616
59,940
60,264
6o,588
6o,qi2
6i,236
18
19
6i,56o
6 1,884
62,208
62,532
62,856
63,1 80
63,5o4
63,828
64,1 52
64,476
19
20
64,800
65,124
65,448
65,772
66,096
66,430
66,744
67,068
67,392
67,716
20
21
68,o4o
68,364
68,688
69,0 1 2
69,336
69,660
69,984
70,308
70,632
70,956
21
22
71,280
71,604
71,928
72,252
72,576
72,900
73,224
73,548
73,872
74,196
22
23
74,520
74.844
75,168
75,492
75,816
76,140
76,464
76,788
77,112
77,436
23
24
77,760
78,084
78,408
78,732
79.056
79,380
79,704
80,028
80,352
80,676
24
25
8 1 ,000
81,324
81,648
81.972
82,296
82,620
82,944
83,268
83,5q2
83,916
25
36
84,240
84,564
84,888
85,212
85,536
85,86o
86,184
86,5o8
86,832
87,1 56
26
27
87,480
87,804
88,128
88,452
88,776
89,100
89,424
89,748
90,072
90,396
27
28
90,720
91,044
91,368
91,692
92,016
92,340
92,664
92,988
93,3i2
93,636
28
29
93,960
94,284
94,608
94,932
95,256
95,580
95,904
96,228
96,552
96.876
29
3o
97,200
97,524
97,848
98,172
98,496
98,820
99,144
99,468
99,792
100,116
3o
3i
100,440
100,764
1 0 1 ,088
IOI,4l2
101,786
102,060
102,384
102,708
io3,o32
103,356
3i
32
io3,68o
io4,oo4
104,328
104,652
104,976
io5,3oo
105,624
105,948
106,272
106,596
32
33
106,920
107,244
107,568
107,892
108,316
108,540
108,864
109,188
109,512
109,836
33
34
110,160
110,484
110,808
111,132
111,456
111,780
112,104
112,428
112,752
113,076
34
35
ii3,4oo
113,724
ii4,o48
114,372
114,696
Il 5,020
11 5,344
Il 5,668
115,992
116,3 16
35
36
ii6,64o
1 1 6,964
117,388
117,612
117,936
118,260
118,584
118,908
119,282
119,556
36
37
119,880
120,204
120,528
120,852
121,176
12I,5oO
121,824
123, i48
122,472
123,796
37
38
123,120
123,444
123,768
124,092
124,416
124,740
125,064
125,388
125,712
i26,o36
38
39
126,360
126,684
127,008
127,332
127,656
127,980
128,304
128,628
128,952
129,276
39
40
129,600
129,924
i3o,248
1 30,572
130,896
l3l,220
1 3 1,544
1 3 1,868
133,192
i32,5i6
40
4i
1 3 2,840
i33,i64
I 33,488
i33,8i2
1 34,1 36
1 34,460
134,784
i35,io8
i35,432
135,756
41
42
i36,o8o
i36,4o4
1 36,728
i37,o52
137,376
137,700
i38,o24
138,348
138,672
1 38,996
42
43
139,320
139,644
139,968
140,292
1 40,61 6
1 40,940
141,264
i4i,588
141,912
i42,236
43
44
i42,56o
142,884
143,208
143,532
143,856
144,180
i44,5o4
144,828
i45,i52
145,476
44
45
145,800
i46>i24
146,448
146,772
147.096
147.420
147,744
148,068
148,392
148,716
45
46
1 49,040
149,364
149,688
l5o,012
i5o,336
1 5o,66o
1 50,984
i5i,3o8
1 5 1,632
1 5 1.966
46
47
152,280
i52,6o4
152,928
i53,252
153,576
153,900
i54,224
154,548
154,872
155,196
47
48
i55,52o
155,844
i56,i68
1 56,492
1 56,8 16
i5-7,r4o
157,464
157.788
i58,ii2
158,436
48
49
158,760
159,084
159,408
159,732
i6o,o56
160,380
160,704
161,028
161,352
161,676
49
0
1
2
3
4
5
6
7
8
9
TABLE XII.
Ti
convert centesimal seconds into sexagesima
s.
Centei.
0
1
2
3
4
5
6
7
8
9
Cemci.
5o
5
162,000
s
162,324
162,648
s
162,972
163,296
s
163,620
163,944
164,268
5
164,592
164,916
^0
5i
i65,24o
165,564
i65,888
1(56,212
166,536
166,860
167,184
167,508
167,832
168,1 56
5i
52
168,480
168,804
169,128
169,452
169,776
170,100
170,424
170,748
171,072
171,396
52
53
171,720
172,044
172,368
172,692
173,016
173,340
173,664
173,988
174,312
174,636
53
54
174,960
175,284
175,608
175,932
176,256
176,580
176,904
177,228
177,552
177,876
54
55
178,200
178,524
178,848
179,172
179.496
179,820
180,144
180,468
180,792
181,116
55
56
181,440
181,764
182,088
182,412
182,736
1 83, 060
183,384
183,708
i84,o32
I 84,356
56
57
184,680
i85,oo4
185,328
185,652
185,976
i86,3oo
186,624
186,948
187,272
187,596
57
58
187,920
188,244
188,568
188,892
189,216
189,540
189,864
190,188
190,512
1 go ,836
58
59
191,160
191,484
191,808
192,132
192,456
192,780
193,104
193,428
193,752
194,076
59
6o
194,400
194,724
195,048
195,372
195,696
196,020
196,344
196,668
196,992
197,316
60
6i
197,640
197,964
198,288
198,612
198,936
199,260
199,584
199,908
200,232
200,556
61
62
200,880
201,204
201,528
201,852
202,176
202,5oO
202,824
2o3,i48
203,472
203,796
62
63
204,120
204,444
204,768
205,092
2o5,4i6
2o5,74o
206,064
206,388
206,712
207,036
63
64
207,360
207.684
208,008
208,332
2o8,656
208,980
209,304
209,628
209,952
210,276
64
65
210,600
210,924
211,248
211,572
2 1 1 ,896
212,220
212,544
212,868
213,192
2i3,5i6
65
66
2i3,84o
2i4,i64
214,488
214,812
2i5,i36
21 5,460
215,784
216,108
216,432
3i6,756
66
67
217,080
217,404
217,728
2l8,o52
218,376
218,700
219,024
219,348
219,672
219,996
67
68
220,320
220,644
220,968
221,292
221,616
221,940
222,264
222,588
222,912
223,236
68
69
2 23,56o
223,884
224,208
224,532
224,856
225,180
225,5o4
225,828
226,152
226,476
69
70
226,800
227,124
227,448
227,772
228,096
228,420
228,744
229,068
229,392
229,716
70
71
23o,o4o
230,364
23o,688
23l,OI2
231,336
23 1,660
231,984
232, 3o8
232,632
232,956
71
72
233,280
233,604
233,928
234,252
234,576
234,900
235,224
235,548
235,872
236,196
72
73
236,520
236,844
237,168
237,492
237,816
238, i4o
238,464
238,788
239,112
239,436
73
74
239,760
240,084
240,408
240,732
24 1 ,o56
241, 38o
241,704
242,028
242,352
242,676
74
75
243,000
243,324
243,648
243,972
244,396
244,620
244,944
245,268
245,592
245,916
75
76
246,240
246,564
246,888
247,212
247,536
247,860
248,184
248,5o8
248,832
349,1 56
76
77
249,480
249,804
25o,I2S
25o,452
250,776
25l,IOO
25i,424
25i,748
252,072
252,396
77
78
252,720
253,044
253,368
253,692
254,016
254,340
254,664
254,988
255,3i2
255,636
78
79
255,960
256,284
256,6q8
256,932
257,256
257,580
257,904
258,228
258,552
258,876
79
80
239,200
259,524
259,848
260,172
260,496
260,820
261,144
261,468
261,792
262,116
80
81
262,440
262,764
263,088
263,412
263,736
264,060
264,384
264,708
265,032
265,356
81
82
265,680
266,004
266,328
266,652
266,976
267,300
267,624
267,948
268,272
268,596
82
83
268,920
269,244
269,568
269,892
270,216
270,540
270,864
271,188
271,512
271,836
83
84
272,160
272,484
272,808
273,132
273,456
273,780
274,104
274,428
274,752
275,076
84
85
275,400
275,724
276,048
276,372
276,696
277,020
277,344
277,668
277,992
278,316
85
86
278,640
278,964
279,288
279,612
279,936
280,260
280,584
280,908
281,232
281,556
86
87
281,880
282,204
282,528
282,852
383,176
283,5oo
283,824
284,148
284,472
284,796
87
88
285,120
285,444
185,768
286,092
286,416
286,740
287,064
287,388
287,712
288,036
88
89
288,360
288,684
289,008
289,332
289,656
289,980
290,304
290,628
290,952
291,276
89
90
291,600
291,924
292,248
292,572
292,896
293,220
293,544
293,868
294,192
294,516
90
91
294,840
295,164
295,488
295,812
296,136
296,460
296,784
297,108
297,432
297,756
91
92
298,080
298,404
298,728
299,052
299,376
299,700
3oo,o24
3oo,348
300,672
300,996
92
93
3oi,32o
3oi,644
301,968
302,292
3o2,6i6
3o2,94o
303,264
3o3,588
303,912
3o4,236
93
94
3o4,56o
3o4,884
3o5,2o8
3o5,532
3o5,856
3o6,i8o
3o6,5o4
306,828
3o7,i52
307,476
94
95
307,800
3o8,i24
3o8,448
308,772
309,096
309,420
309,744
3 10,068
310,392
3io,7i6
95
96
3 1 1 ,o4o
3 11,364
3 11,688
3l2,012
312,336
3 12,660
312,984
3i3,3o8
3i 3,632
3i3,956
96
97
3 14,280
3i4,6o4
314,928
3i5,252
315,576
3 15,900
3i6.234
3 16.548
316,872
317,196
97
98
3i7,520
317,844
3i8,i68
318,492
3iS,8i6
319, i4o
3 19,464
319,788
320,112
320,436
98
99
320,760
321,084
321,408
321,732
3
322,o56
322,38o
322,704
323,028
323,352
323,676
99
0
1
2
4
5
6
7
8
9
,,^„^;-,;^,
AUTHOR
La Place.
TITLE
351
L32
cXt^^^-^
31037
Astron
qQB
351
L32
3
31037
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