TABLE OF CONTENTS
ACKNOWLEDGEMENTS xvii
FOREWORD TO THE SECOND EDITION xix
Notes to Foreword xxiii
GLOSSARY OF SYMBOLS & LIST OF IMAGES xxvii
PRELIMINARY DISSERTATION 1
Notes to Preliminary Dissertation 36
INTRODUCTION
PHYSICAL ASTRONOMY 41
Notes to Introduction 42
BOOK I - DYNAMICS
Foreword to Book I 45
The Figure of the Earth 45
The Rotation of the Earth 50
The Sun, Moon, Planets and Satellites 57
BOOK I: CHAPTER I
DEFINITIONS, AXIOMS, &c.
1. Definitions 65
4. Uniform Motion 66
22. Composition and Resolution of Forces 67
39. The General Principles of Equilibrium 71
43. On Pressure 73
12. On the Normal 74
48. Equilibrium of a Particle on a curved Surface 74
51. Virtual Velocities 76
56. Variations 77
Notes to Bk. I, Chap. I 78
BOOK I: CHAPTER II
VARIABLE MOTION
60. Definitions 79
63. Central Force 79
Demonstration: perpetually varying central force 79
68. Demonstration: force and acceleration 80
69. General Equations of the Motions of a Particle of Matter 81
Demonstration: general equations both free and constrained 81
75. Demonstration: resolution of forces 85
79. Principle of Least Action 87
81. Motion of a Particle on a Curved Surface 89
83. Radius of Curvature 90
84. Pressure of a Particle Moving on a Curved Surface 93
Demonstration: tangent and normal components 93
85. Centrifugal Force 93
93. Demonstration: centrifugal force and central force 96
96. Motion of Projectiles 96
97. Demonstration: effect of air resistance 96
99. Theory of Falling Bodies 99
100. Comparison of Centrifugal Force with Gravity 100
100. Simple Pendulum 101
106. Demonstration: impelled initial velocity 101
109. Isochronous Curve 105
113. Curve of Quickest Descent 108
Notes to Bk. I, Chap. II 109
BOOK I: CHAPTER III
ON THE EQUILIBRIUM OF A SYSTEM OF BODIES
114. Definitions and Axioms 111
119. Reaction equal and contrary to Action 111
120. Mass proportional to Weight 112
121. Density 112
122. Mass proportional to the Volume into the Density 112
123. Specific Gravity 112
124. Equilibrium of two Bodies 113
Demonstration: lever 113
125. Equilibrium of a System of Bodies 113
127. Demonstration: equilibrium 113
128. Rotatory Pressure 115
130. On the Lever 116
Demonstration: moments 116
131. Projections of Lines and Surfaces 116
132. Equilibrium of a System of Bodies invariably united 117
Demonstration: equilibrium about a point 117
135. On the Centre of Gravity 120
137. On the Position and Properties of the Centre of Gravity 121
138. Demonstration: centre of gravity displaced from origin of co-ordinates 122
143. Equilibrium of a Solid Body 124
Notes to Bk. I, Chap. III 124
BOOK I: CHAPTER IV
MOTION OF A SYSTEM OF BODIES
151. On the Motion of a System of Bodies in all possible Mathematical relations
between Force and Velocity 129
Demonstration: uniform motion of centre of gravity 129
153. Demonstration: as a consequence of the law of reaction and action 130
158. On the Constancy of Areas 131
Demonstration: general equation of a system of bodies 132
165. Demonstration: general equations for a system moving uniformly 136
168. On the motion of a system in all possible relations between force and velocity 138
Notes to Bk. I, Chap. IV 139
BOOK I: CHAPTER V
THE MOTION OF A SOLID BODY OF ANY FORM WHATEVER
174. Determination of the general Equations of the Motion of the Centre of Gravity
of a Solid in Space 142
176. Rotation of a Solid 143
181. Demonstration: three permanent axes of rotation 145
202. Rotation of a Solid not subject to the action of Disturbing Forces, and at liberty
to revolve freely about a Fixed Point, being its Centre of Gravity, or not 156
216. Rotation of a Solid which turns nearly round one of its principal Axes, as the
Earth and the Planets, but not subject to the action of accelerating Forces 163
222. Compound Pendulums 166
Notes to Bk. I, Chap. V 168
BOOK I: CHAPTER VI
ON THE EQUILIBRIUM OF FLUIDS
225. Definitions, &c. 169
231. Equilibrium of Fluids 169
233. Equation of Equilibrium 170
238. Equations of Condition 171
239. Equilibrium of homogeneous Fluids 172
240. Equilibrium of heterogeneous Fluids 172
Demonstration: relationship between pressure and density 172
246. Equilibrium of Fluids in Rotation 174
Notes to Bk. I, Chap. VI 175
BOOK I: CHAPTER VII
MOTION OF FLUIDS
248. General Equation of the Motion of Fluids 177
250. Equation of Continuity 178
252. Development of the Equation of Continuity 178
257. Second form of the Equation of the Motions of Fluids 181
258. Integration of the Equations of the Motions of Fluids 182
260. Demonstration: integration of equations when exact differential 183
261. Theory of small Undulations of Fluids 184
263. Rotation of a Homogeneous Fluid 185
266. Determination of the Oscillations of a Homogeneous Fluid covering a
Spheroid, the whole in rotation about an axis; supposing the fluid to be slightly
deranged from its state of equilibrium by the action of very small forces 186
267. Action of the Sun and Moon 186
269. Determination of the general Equation of the Oscillation of all parts of the
Fluids covering the Earth 187
272. Equation at the Surface 189
278. Continuity of Fluids 191
280. Oscillations of the Oceans 192
291. On the Atmosphere 197
293. Density of the Atmosphere 197
296. Equilibrium of the Atmosphere 198
302. Oscillations of the Atmosphere 201
306. Oscillations of the Mercury in the Barometer 203
307. Conclusion 204
Notes to Bk. I, Chap. VII 204
BOOK II - UNIVERSAL GRAVITATION
Foreword to Book II 207
Gravitation 207
Elliptical Orbits 210
Perturbations of the Planets 212
BOOK II: CHAPTER I
PROGRESS OF ASTRONOMY
308. Historical Review 219
Notes to Bk. II, Chap. I 223
Art. BOOK II: CHAPTER II Page.
ON THE LAW OF UNIVERSAL GRAVITATION
309. Kepler’s Laws 225
326. On Parallax 223
332. Force of Gravitation at the Moon 236
Notes to Bk. II, Chap. II 240
BOOK II: CHAPTER III
DIFFERENTIAL EQUATIONS OF A SYSTEM OF BODIES
344. Introduction 241
352 Motion of the Centre of Gravity 246
354. Attraction of Spheroids 249
Notes to Bk. II, Chap. III 252
BOOK II: CHAPTER IV
ON THE ELLIPTICAL MOTION OF THE PLANETS
359. Introduction 255
365. Motion of one Body 256
374. Determination of the Elements of Elliptical Motion 261
378. Elements of the Orbit 266
379. Equations of Elliptical Motion 266
386. Determination of the Eccentric Anomaly in functions of the Mean Anomaly 268
387. Determination of the Radius Vector in functions of the Mean Anomaly 270
388. Kepler’s Problem-To find a Value of the true Anomaly in functions of the
Mean Anomaly 272
392. True Longitude and Radius Vector in functions of the Mean Longitude 275
397. Determination of the Position of the Orbit in space 276
398. Projected Longitude in Functions of true Longitude 277
True Longitude in Functions of projected Longitude 277
399. Projected Longitude in Functions of Mean Longitude 277
400. Latitude 278
401. Curtate Distances 278
404. Motion of Comets 279
405. Arbitrary Constant Quantities of Elliptical Motion, or Elements of the Orbits 279
406. Co-ordinates of a Planet 280
407. Determination of the Elements of Elliptical Motion 280
408. Velocity of Bodies moving in Conic Sections 282
Notes to Bk. II, Chap. IV 284
BOOKII: CHAPTER V
THEORY OF THE PERTURBATIONS OF THE PLANETS
Page.
410. Introduction 287
417. Demonstration of Lagrange’s Theorem 288
422. Variation of the Elements, whatever the Eccentricities and Inclinations may be 291
428. Variations of the Elliptical Elements of the Orbits of the Planets 297
452. Determination of the Coefficients of the Series R 317
455. Coefficients of the series R 323
Notes to Bk. II, Chap. V 327
BOOK II: CHAPTER VI
SECULAR INEQUALITIES IN THE ELEMENTS OF THE ORBITS
462 Stability of the Solar System, with regard to the Mean Motions of
The Planets and the greater axes of their Orbits 329
473. Differential Equations of the Secular Inequalities in the Eccentricities,
Inclinations, Longitudes of the Perihelia and Nodes, which are
the annual and sidereal variations of these four elements 336
480. Approximate Values of the Secular Variations in these four Elements
in Series, ascending according to the powers of the Time 340
481 Finite Values of the Differential Equations relative to the eccentricities and
longitudes of the Perihelia. 341
488. Stability of the Solar System with regard to the Form of the Orbits 346
498. Secular Variations in the Inclinations of the Orbits and Longitudes of their
Nodes 351
499. Stability of the Solar System with regard to the Inclination of the Orbits 352
510. Annual and Sidereal Variations in the Elements of the Orbits,
with regard to the variable Plane of the Ecliptic 355
511. Motion of the Orbits of two Planets 357
512. Secular Variations in the Longitude of the Epoch 357
515. Stability of the System, whatever may be the powers of the Disturbing Masses 360
525. The Invariable Plane 365
Notes to Bk. II, Chap. VI 367
BOOK II: CHAPTER V11
PERIODIC VARIATIONS IN THE ELEMENTS OF THE ORBITS
529. Variations depending on the first Powers of the Eccentricities and Inclinations 371
Notes to Bk. II, Chap. VII 375
BOOK II: CHAPTER VIII
PERTURBATIONS OF THE PLANETS
532. Introduction 377
536. Perturbations in the Radius Vector 378
537 The Perturbations in Longitude 379
544. Perturbations in Latitude 384
Notes to Bk. II, Chap. VIII 385
BOOK II: CHAPTER IX
SECOND METHOD OF FINDING THE PERTURBATIONS OF A PLANET
546. Determination of the general Equations 387
552. Perturbations in the Radius Vector 391
555. Perturbations in Longitude 395
558. Perturbations in Latitude 398
559. Perturbations, including the Squares of the Eccentricities and Inclinations 399
563. Perturbations depending on the Cubes and Products of three Dimensions of the
Eccentricities and Inclinations 403
566. Secular Variation of the Elliptical Elements during the periods of the
Inequalities 405
Notes to Bk. II, Chap. IX 408
BOOK II: CHAPTER X
THE THEORY OF JUPITER AND SATURN
571. Introduction 411
572. Periodic Variations in the Elements of the Orbits of Jupiter and Saturn,
depending on the First Powers of the Disturbing Forces 412
578. Periodic Variations in the Elements of the orbits of Jupiter and Saturn,
depending on the Squares of the Disturbing Forces 417
580. Secular Variations in the Elements of the Orbits of Jupiter and Saturn,
depending on the Squares of the Disturbing Forces 421
588. Periodic Perturbations in Jupiter’s Longitude depending on the Squares of the
disturbing Forces 426
Notes to Bk. II, Chap. X 428
BOOK II: CHAPTER XI
INEQUALITIES OCCASIONED BY THE ELLIPTICITY OF THE SUN
592. Discussion 431
BOOK II: CHAPTER XII
PERTURBATIONS OCCASIONED BY THE SATELLITES
594. Introduction 435
Notes to Bk. II, Chap. XII 437
Art. BOOK II: CHAPTER XIII Page.
DATA FOR COMPUTING THE CELESTIAL MOTIONS
596. Introduction 439
587. Masses of the Planets 439
607. Densities of the Planets 444
608. Intensity of Gravitation at the Surfaces of the Sun and Planets 445
611. Mean Distances of the Planets, or Values of a, a¢, a¢, &c. 447
612. Ratio of the Eccentricities to the Mean Distances, or Values of e, e¢, &c. for
1801 448
613. Inclinations of the Orbits on the Plane of the Ecliptic in 1801 448
614. Longitudes of the Perihelia 449
615. Longitudes of the Ascending Nodes 449
618. Halley’s Comet of 1682 450
Encke’s Comet of 1819 450
Claussen and Gambart’s Comet of 1825 451
Notes to Bk. II, Chap. XIII 451
BOOK II: CHAPTER XIV
NUMERICAL VALUES OF THE PERTURBATIONS
619. The French Tables 453
621 Secular Variations of Jupiter and Saturn 456
624. Periodic Inequalities of Jupiter 460
626. Inequalities depending on the Squares of the Eccentricities and Inclinations 464
628. Perturbations depending on the Third Powers and Products of the
Eccentricities and Inclinations 466
629. Inequalities depending on the Squares of the Disturbing Force 469
630. Periodic Inequalities in the Radius Vector, depending on the Third Powers and
Products of the Eccentricities and Inclinations 469
631 Periodic Inequalities in Latitude 469
633. On the Laws, Periods, and Limits of the Variations in the Orbits of Jupiter and
Saturn 471
635. Mercury 476
637. Venus 477
Table of the Transits of Venus 477
642. The Earth 480
646. Secular Inequalities in the Terrestrial Orbit 483
649. Mars 484
650. The New Planets 484
651. Jupiter 485
652. Saturn 485
653. Uranus, or the Georgium Sidus 486
655. On the Atmosphere of the Planets 487
656. The Sun 487
659. Influence of the Fixed Stars in disturbing the Solar System 489
660. Disturbing Effect of the Fixed Stars on the Mean Motions of the Planets 490
661. Construction of Astronomical Tables 492
662. Method of correcting Errors in the Tables 493
Notes to Bk. II, Chap. XIV 495
BOOK III - LUNAR THEORY
Foreword to Book III 499
Rotation of the Moon 499
Lunar Perturbations 501
BOOK III: CHAPTER I
LUNAR THEORY
665. Introduction 509
666. Phases of the Moon 509
667. Mean or Circular Motion of the Moon 510
673. Elliptical Motion of the Moon 511
677. Lunar Inequalities 512
687. Analytical Investigations of the Lunar Inequalities 517
713. The Parallax 531
724. Latitude of the Moon 539
729. The Mean Longitude of the Moon 543
Notes to Bk. III, Chap. I 549
BOOK III: CHAPTER II
NUMERICAL VALUES OF THE COEFFICIENTS
735. Data from Observation 551
747. Secular Inequalities in the Moon’s Motions 559
748. The Acceleration 559
749. Motion of the Moon’s Perigee 560
750. Motion of the Nodes of the Lunar Orbit 561
Notes to Bk. III, Chap. II 570
BOOK III: CHAPTER III
INEQUALITIES FROM THE FORM OF THE EARTH
770 Introduction 573
Notes to Bk. III, Chap. III 578
Art. BOOK III: CHAPTER IV Page.
INEQUALITIES FROM THE ACTION OF THE PLANETS
780. Introduction 579
782. Numerical Values of the Lunar Inequalities occasioned by the Action of the
Planets 584
Notes to Bk. III, Chap. IV 586
BOOK III: CHAPTER V
EFFECTS OF THE SECULAR VARIATIONS IN THE ECLIPTIC
786. Introduction 587
Notes to Bk. III, Chap. V 589
BOOK III: CHAPTER VI
EFFECTS OF AN ETHEREAL MEDIUM ON THE MOON
788. Introduction 591
Notes to Bk. III, Chap. VI 602
BOOK IV - THE SATELLITES
Foreword to Book IV 603
Rotation of the Planets 603
Jupiter’s Satellites 605
Satellites of Jupiter, Saturn, Uranus and Neptune 608
BOOK IV: CHAPTER I
THEORY OF JUPITER’S SATELLITES
798. Introduction 611
818. FIRST APPROXIMATION 618
Perturbations in the Radius Vector and Longitude of m that are Independent of
the Eccentricities 618
829. SECOND APPROXIMATION 626
Inequalities depending on the First Powers of the Eccentricities 626
836. Action of the Sun depending on the Eccentricities 631
838. Inequalities depending on the Eccentricities which become sensible in
consequence of the Divisors they acquire by double integration 633
840. Inequalities depending on the square of the Disturbing Force 635
843. Librations of the three first Satellites 639
Notes to Bk. IV, Chap. I 650
BOOK IV: CHAPTER II
PERTURBATIONS OF THE SATELLITES IN LATITUDE
859. Introduction 651
863. The Effect of the Nutation and Precession of Jupiter’s Satellites 654
868. Inequalities occasioned by the Displacement of Jupiter’s Orbit 657
869. On the Constant Planes 657
871. To determine the Effects of the Displacements of the Equator and Orbit of
Jupiter on the quantities q = QNF , q¢ = QNJ , y, y¢, and L 659
881. Secular Inequalities of the Satellites 665
Notes to Bk. IV, Chap. II 670
BOOK IV: CHAPTER III
NUMERICAL VALUES OF THE PERTURBATIONS
885. Introduction 671
889. Determination of the Masses of the Satellites and the Compression of Jupiter 677
907. Theory of the First Satellite 691
Longitude 691
908. Latitude 692
909. Theory of the Second Satellite 693
910. The Latitude 694
911. Theory of the Third Satellite 695
913 Latitude 697
914. Theory of the Fourth Satellite 698
915. Latitude 699
Notes to Bk. IV, Chap. III 701
BOOK IV: CHAPTER IV
ECLIPSES OF JUPITER’S SATELLITES
918. Introduction 703
956. The Satellites of Saturn 725
958. Satellites of Uranus 726
Notes to Bk. IV, Chap. IV 726
CRITICAL REVIEWS OF MECHANISM OF THE HEAVENS
The Literary Gazette, and Journal of the Belle Lettres, Dec., 1831 729
Charles Buller, The Athenaeum, Jan., 1832 732
Thomas Galloway, Edinburgh Review, April, 1832 736
J. F. W. Herschel, Quarterly Review, July, 1832 753
BASIC BIBLIOGRAPHY 769
SUBJECT INDEX BY ARTICLE 773
NAME INDEX 783
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