About this book

My library

Books on Google Play

CHAP. IV.

Page

The Elliptical Elements of a Planet's Orbit determined: its Major

Axis, Eccentricity, Longitude of the Perihelion, Inclination of its

Plane, Longitude of the Node, Epoch of the Passage of the

Perihelion. The Elements of the Orbit considered as the

Arbitrary Constant Quantities introduced by the Integration of

the Differential Equations. Their invariability in the System of

two Bodies. Expression for the Velocity in an Ellipse: in a Circle:

in a Right Line, the Centripetal Force varying inversely as the

Square of the Distance. Modification of the preceding Results,

by considering the Masses of the Revolving and Central Body... 35

CHAP. V.

A third attracting Body introduced into the System of two Bodies.

Its Effects in disturbing the Laws of Motion and the Elements

of that System. Expressions of the Values of the resolved

Parts of the disturbing Force; the Ablatitious; the Addititious:

the Force in the direction of the Radius Vector: the Tangen-

tial Force: Effects of these Forces in altering Kepler's Laws, &c.

Approximate Values of the Forces when the disturbing Body is

very remote. Expressions for the Forces, in the Problem of

the Three Bodies, by means of the Partial Differentials of a Func-

tion of the Body's Parallax, Longitude and Latitude............... 46

CHAP. VI.

The Motion of the Centre of Gravity of two or more Bodies not

affected by their mutual Action: their Centre of Gravity at-

tracted by a distant External Body (the System revolving round

it) by a Force nearly as the Inverse Square of the Distance: it

describes therefore an Ellipse, nearly, round that Body. The

Centre of Gravity of the Earth and Moon, the Centres of

Gravity of Jupiter and his Satellites, of Saturn and his, all de-

scribe, very nearly, Ellipses round the Sun, and Areas propor-

tional to the Times. Values of the Disturbing Forces that

prevent the exact Description. The Moon's Menstrual Motion:

Values of the Perturbations of her Parallax and Longitude by

the Earth's Action: Value of the Menstrual Parallax............ 72

CHAP. VII.

Elimination of dt from the Differential Equations. The Three

Equations that belong to the Theory of the Moon, and the

Problem of the Three Bodies. The Approximate Integration of

these Equations by the Method called the Variation of the

Parameters. Application of that Method to particular In-

stances........

92

CHAP. VIII.

On certain Ambiguities of Analytical Expression that occur in the

Problem of the Three Bodies; their Source and Remedy. A

new Form for the Integral value of u from which the Arcs of

Circles are excluded. Consideration on the Alteration which

certain small Quantities may receive from the Process of In-

tegration. Comparison between the Analytical Formulæ, and

the Results of the Geometrical Method. Observations on the

Ninth Section of the Principia........

CHAP. VIII. *

106

First Solution of the Problem of the Three Bodies under its most

simple Conditions: that is, when the Body, previously to the

Action of the Disturbing Force, is supposed to revolve in an Orbit

without Eccentricity and Inclination; the Orbit, changed by

the Action of the Disturbing Force, not strictly Elliptical........ 124

CHAP. IX.

Continuation of the Solution of the Problem of the Three Bodies:

the Orbit of the disturbed Body is supposed to be Elliptical:

the resulting Value of the Radius Vector thereby augmented

with additional Terms. Clairaut's First Method of determining

the Progression of the Lunar Apogee...............................

CHAP. X.

On the Form of the Differential Equation, when the Approximation

includes Terms that involve e2. The Error, in the Computed

Quantity of the Apogee, the same as before, and very little

lessened by taking account of Terms involving e3......

137

.... 150

CHAP. XI.

On the Corrections due to the Eccentricity of the Solar Orbit, and

to the Inclination of the Plane of the Moon's Orbit. Method of

deriving Corrections. Their Formulæ exhibited in a Table.

The Error in the determination of the Lunar Apogee not removed

by these Corrections. The deduction of Terms on which the

Secular Equations of the Moon's Mean Longitude and of the

Progression of the Apogee depend.........

CHAP. XII.

....... 159

Principle of the Method of correcting the Value of the Radius

Vector, obtained by an Approximate Integration of the Dif

ferential Equation...........

....

184

CHAP. XIII.

The Method of determining the Progression of the Apsides in the

simplest Case of the Problem of the Three Bodies. Clairaut's

Analogous Method for determining the Progression of the Lunar

Apogee. His first Erroneous Result. Its Cause, and the

Means of correcting it. Quantity of the Progression computed

from the Condition of a sole Disturbing Force acting in the

Direction of the Radius Vector. Remarkable Result obtained

by the first Integration of the Differential Equation. Dalembert's

Method of indeterminate Coefficients, for finding the Value of

the Inverse of the Radius Vector, adopted by Thomas Simpson

and Laplace..........

196

CHAP. XIV.

Expression for the Time: first, when the Body revolving in a

Circular Orbit is disturbed by the Action of a very distant

Body. The Mean Longitude expressed in Terms of the True:

the True thence expressed in Terms of the Mean by the Re-

version of Series. The Introduction of Inequalities in the Mean

Motion by the Disturbing Force: the Elliptic Inequality, the

Variation the greatest Value of the latter in an Orbit nearly

Circular. Expression for the Differential of the Time in an

Elliptical Orbit, the Disturbing Body revolving also in an Orbit

of the same kind. The Expression integrated, and the Mean

Longitude expressed in Terms of the True. Expression in this

Case, of the Coefficient or greatest Value of the Variation. The

Secular Equation of the Mean Motion, explanatory of the Acce-

leration of that Motion. Digression concerning the Properties

and Uses of the Formula of Reversion. By means of that

Formula the True Longitude expressed in Terms of the Mean :

the Terms expound Inequalities: the greatest denominated the

Variation, the Evection, the Annual Equation, the Reduction:

Causes of their Magnitude. Lunar Tables, in what manner,

improved by Theory........

213

CHAP. XV.

On the Integration of the Equation on which the Moon's Latitude

depends. Formation of Equations correcting the Latitude. Re-

gression of the Nodes. Secular Equation of the Regression...... 243

CHAP. XVI.

Differential Equation for determining the Radius Vector: Expression

for R: its development into a Series of Cosines of Multiple Arcs.

Conditions on which the Convergency of such Series depends.

Application of the Differential Equation to the Investigation of

the Perturbations in the Radius Vector and Longitude of the

Earth by the Moon's Action.........

CHAP. XVII.

........ 256

On the Development of R in terms of the Cosines of the Mean

Motions of the disturbed and disturbing Planets. On the Method

of Computing the Coefficients of the Development, when the

Radius of the Orbit of the Disturbed Body differs considerably

from that of the Disturbing: Application of the Formula to the

Case of Jupiter disturbing the Earth. New Formulæ necessary

when the Radii of the Orbits of the two Bodies are nearly

Equal......................

273

CHAP. XVIII.

On the Method of determining the Coefficients of the Development

of (r2-2rr cos. w+2)-2 when the Fraction does not

differ much from 1. Application of the Formula to the Mutual

Perturbations of the Earth and Venus........

CHAP. XIX.

On certain Inequalities of Jupiter and Saturn, which depend on the

near Commensurability of their Mean Motions. Five times

Saturn's Mean Motion nearly equal to twice Jupiter's. The

peculiar Inequalities of Jupiter and Saturn expounded by Terms

involving the Cubes of the Eccentricities. The Cause of their

magnitude. Connexion, in the same Term, between the Power

of the Eccentricity and the Form of the Argument. Expres-

sions for the Retardation of Saturn, and the corresponding Ac-

celeration of Jupiter. Agreement of the Results of Computation

and Observation. Period of the Inequality. A similar In-

equality in the Motion of Mercury, &c. &c.......

CHAP. XX.

286

..... 320

Deduction of the Value of R: First, when the Sun, secondly, when

a Satellite, is the disturbing Body. Values of the Inequalities

in Longitude and Parallax of a Satellite. Variation in a

Satellite's Longitude arising from the Sun's disturbing Force.

By reason of the near Commensurability of the Mean Motions of

the Three first Satellites, their Inequalities in Longitude ex-

pressed, each, by a single Term. The Inequalities of the Second

Satellite arising from the Actions of the First and Second

Satellite blended together and expounded by a single Term. The

Period of the Inequalities of the Three first Satellites

= 437 15h 48m 57'. The Elements of the Theory of the

Satellites determined from the Epochs and Durations of

their Eclipses......... ....... 357

CHAP. XXI.

Principle of the Method for determining the Variations of the

Elements of a Planet's Orbit. The Elements viewed as the

Arbitrary Quantities introduced by the Integration of the Dif

ferential Equations of Motion, or as their Functions. Expres-