# Index:Mécanique céleste Vol 1.djvu

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CONTENTS OF THE FIRST VOLUME.

Preface, plan of the work. xxiii

FIRST BOOK.

ON THE GENERAL LAWS OF EQUILIBRIUM AND MOTION.

CHAPTER I. ON THE EaUILIBRIUM AND COMPOSITION OF FORCES WHICH ACT ON A MATERIAL POINT 1

On motion and force, also on the composition and resolution of forces, [1 — 17] . . . . § 1, 2

Equation of the equilibrium of a point acted upon by any number of forces, in any directions whatever, [18]. Method of determining, when the point is not free, the pressure it exerts upon the surface, or upon the curve to which it is subjected, [19 — ^26]. Theory of the momentum of any force about an axis, [29] § 3

CHAPTER II. ON THE MOTION OF A MATERIAL POINT 23

On the laws of inertia, uniform motion, and velocity, [29"] § 4

Investigation of the relation which exists between force and velocity. In the law of nature they are proportional to each other. Results of this law, [30 — 34'"] § 5j 6

Equations of the motions of a point acted upon by any forces, [37] § 7

General expression of the square of the velocity, [40]. The point describes the curve in which the integral of the product of the velocity, by the element of the curve, is a minimum, [49^]. § 8

Method of computing the pressure which a point, moving upon a surface, or upon a curve, exerts on it, [54]. On the centrifugal force, [54'] § 9

Application of the preceding principles to the motion of a material point, acted upon freely by gravity, in a resisting medium. Investigation of the law of resistance necessary to make the moving body describe a given curve. Particular examination of the case in which the resistance is nothing, [54^—67" ] §10

Application of the same principles to the motion of a heavy body upon a spherical surface. Determination of the time of the oscillations of the moving body. Very small oscillations are isochronal, [67'"— 86] §11

Investigation of the curve which is rigorously isochronal, in a resisting medium ; and particularly if the resistance be proportional to the two first powers of the velocity, [86" — 106]. . . § 12

CHAPTER in. ON THE EaUILIBRIUM OP A SYSTEM OF BODIES 71

Conditions of the equilibrium of two systems of points, which impinge against each other, with directly opposite velocities. Definition of the terms, quantity of motion of a body and similar material points, [106' — 106'"] § 13

On the reciprocal action of material points. Reaction is always equal and contrary to action. Equation of the equilibrium of a system of bodies, from which we may deduce the principle of virtual velocities, [114']. Method of finding the pressures, exerted by bodies, upon the surfaces and curves upon which they are forced to move, [117] §14

Application of these principles, to the case where all the points of the system are rigidly united together, [119] ; conditions of the equilibrium of such a system. On the centre of gravity : method of finding its position ; first, with respect to three fixed rectangular planes, [127] ; second, with respect to three points given in position, [129] § 15

Conditions of equilibrium of a solid body of any figure, [130] §16

CHAPTER IV. ON THE EQUILIBRIUM OF FLUIDS 90

General equations of this equilibrium, [133]. Application to the equilibrium of a homogeneous fluid mass, whose external surface is free, and which covers a fixed solid nucleus, of any figure, [138] §17

CHAPTER V. GENERAL PRINCIPLES OF THE MOTION OF A SYSTEM OF BODIES QQ

General equation of this motion, [142] § 18

Development of the principles comprised in this equation. On the principle of the living force, [144]. It takes place only when the motions of the bodies change by insensible degrees, [145]. Method of estimating the alteration which takes place in the living force, by any sudden change in the motions of the system, [149] § 19

On the principle of the preservation of the motion of the centre of gravity, [155']. It takes place even in those cases, in which the bodies of the system exert on each other, a finite action, in an instant, [159"]. §20

On the principle of the preservation of areas, [167]. It takes place also, like the preceding principle, in the case of a sudden change in the motion of the system, [167'"]. Determination of the system of co-ordinates, in which the sum of the areas described by the projections of the radii vedores, upon two of the rectangular planes, formed by the axes of the co-ordinates, is nothing. This sum is a maximum upon the third rectangular plane ; it is nothing upon every other plane, perpendicular to this third plane, [181"] § 21

The principles of the preservation of the living forces and of the areas take place also, when the origin of the co-ordinates is supposed to have a rectilineal and uniform motion, [182]. In this case, the plane passing constantly through this origin, and upon which the sum of the areas

described by the projection of the radii is a maximum, continues always parallel to itself, [187, &.C.] The principles of the living forces and of the areas, may be reduced to certain relations between the co-ordinates of the mutual distances of the bodies of the system, [189, &c.] Planes passing through each of the bodies of the system, parallel to the invariable plane drawn through the centre of gravity, possess similar properties, [189"] §22

Principle of the least action, [196']. Combined with that of the living forces, it gives the general equation of motion § 23

CHAPTER VI. ON THE LAWS OF MOTION OF A SYSTEM OF BODIES, IN ALL THE RELATIONS MATHEMATICALLY POSSIBLE BETWEEN THE FORCE AND VELOCITY 137

New principles which, in this general case, correspond to those of the preservation of the living forces, of the areas, of the motion of the centre of gravity, and of the least action. In a system which is not acted upon by any external force, we have, Jirst, the sum of the finite forces of the system, resolved parallel to any axis, is constant ; second, the sum of the finite forces, to turn the system about an axis, is constant ; third, the sum of the integrals of the finite forces of the system, multiplied respectively by the elements of their directions, is a minimum : these three sums are nothing in the state of equilibrium, [196'", &c.] § 24

CHAPTER VII. ON THE MOTIONS OF A SOLID BODY OF ANY FIGURE WHATEVER 144

Equations which determine the progressive and rotatory motion of the body, [214 — ^234]. § 25, 26

On the principal axes, [235]. In general a body has but one system of principal axes, [245"]. On the momentum of inertia, [245'"]. The greatest and least of these momenta appertain to the principal axes, [246"], and the least of all the momenta of inertia takes place with respect to one of the three principal axes which pass through the centre of gravity, [248']. Case in which the solid has an infinite number of principal axes, [250, &c.] § 27

Investigation of the momentary axis of rotation of the body, [254"] . The quantities which determine its position relative to the principal axes, give also the velocity of rotation, [260'*]. . . § 28

Equations which determine this position, and that of the principal axes, in functions of the time, [263, &.C,] Application to the case in which the rotatory motion arises from a force which does not pass through the centre of gravity. Formula to determine the distance from this centre to the direction of the primitive force [274]. Examples deduced from the planetary motions, particularly that of the earth [275*] ^29

On the oscillations of a body which turns nearly about one of its principal axes, [278]. The motion is stable about the principal axes, whose momenta of inertia are the greatest and the least ; but it is unstable about the third principal axis, [281'«] § 30

On the motion of a solid body, about a fixed axis, [287]. Determination of a simple pendulum, which oscillates in the same time as this body, [293'] § 31

CHAPTER VIII. ON THE MOTION OF FLUIDS. .... ,n^ Equations of the motions of fluids, [296] ; condition relative to their continuity [303'"]. . § 32

Transformation of these equations ; they are integrable when the density is any function of the pressure, and at the same time, the sum of the velocities parallel to three rectangular axes, each being multiplied by the element of its direction, is an exact variation, [304, &c.] Proof that this condition will be fulfilled at every instant of time, if it is so in any one instant, [316]. . § 33

Application of the preceding principles to the motion of a homogeneous fluid mass, which has a uniform rotatory motion, about one of the axes of the co-ordinates [321] § 34

Determination of the very small oscillations of a homogeneous fluid mass, covering a spheroid, which has a rotatory motion, [324] § 35

Application to the motion of the sea, supposing it to be disturbed from the state of equilibrium, by the action of very small forces, [337] § 36

On the atmosphere of the earth, considered at first in a state of equilibrium, [348]. Its oscillations in a state of motion, noticing only the regular causes which agitate it. The variations which these motions produce in the height of the barometer, [363' v] § 37

SECOND BOOK.

ON THE LAW OP UNIVERSAL GRAVITATION, AND THE MOTIONS OF THE CENTRES OF GRAVITY OF THE HEAVENLY BODIES.

CHAPTER I. THE LAW OF UNIVERSAL GRAVITY DEDUCED FROM OBSERVATION 239

The areas described by the radii vectores of the planets in their motions about the sun, being proportional to the time, the force which acts upon the planets, is directed towards the centre of the sun, [367] ; and reciprocally, if the force be directed towards the sun, the areas described about it, by the planets, will be proportional to the time ^1

The orbits of the planets and comets being conic sections, the force which acts on them, is in the inverse ratio of the square of the distances of the centres of these planets from that of the sun, [SSC]. Reciprocally, if the force follows this ratio, the described curve will be a conic section, [380'^] § 2

The squares of the times of the revolutions of the planets, being proportional to the cubes of the great axes of their orbits ; or, in other words, the areas described in the same time, in different orbits, being proportional to the square roots of their parameters, the force which acts upon the planets and comets, must be the same for all the bodies placed at equal distances from the sun, [385] § 3

The motions of the satellites about their planets exhibiting nearly the same phenomena, as the motions of the planets about the sun ; the satellites must be attracted towards their planets, and towards the sun, by forces reciprocally proportional to the square of the distances, [388] § 4

Investigation of the lunar parallax, from experiments on gravity, supposing gravitation to be in the inverse ratio of the square of the distances, [391]. The result obtained in this manner, being found perfectly conformable to observations, the attractive force of the earth must be of the same nature as that of the heavenly bodies § 5

General reflections on what precedes : they lead us to this general principle, namely, that all the particles of matter attract each other in the direct ratio of the masses^ and in the inverse ratio of the squares of the distances, [391'*', &c] § 6

CHAPTER II. ON THE DIFEKENTIAL EaUATIONS OP THE MOTION OF A SYSTEM OP BODIES, SUBJECTED TO THEIR MUTUAL ATTRACTIONS 261

Differential equations of this motion, [398 — 400] § 7

Development of the integrals of these equations which have already been obtained, and which result from the principles of the preservation of the motions of the centres of gravity, of the areas, and of the living forces, [404 — 410"] § 8

Differential equations of the motions of a system of bodies, subjected to their mutual attractions, about one of them considered as the centre of their motions, [416 — 418]. Development of the rigorous integrals of these equations, which have been obtained, [421 — 442] § 9

The motion of the centre of gravity of the system of a planet and its satellites about the sun, is nearly the same as if all the bodies of this system were united at that point ; and the system acts upon the other bodies nearly as it would in the same hypothesis, [451] § 10

Investigation of the attraction of spheroids : this attraction is given by the partial differentials of the function which expresses the sum of the particles of the spheroid, divided by their distances from the attracted point, [455'"]. Fundamental equation of partial differentials which this function satisfies, [459]. Several transformations of this equation, [465, 466] §11

Application to the case where the attracting body is a spherical stratum, [469] : it follows that a point placed within the stratum is equally attracted in every direction, [469'"] ; and that a point placed without the stratum, is attracted by it, as if the whole mass were collected at its centre, [470^. This result also takes place in globes formed of concentrical strata, of a variable density from the centre to the circumference. Investigation of the laws of attraction, in which these properties exist [484]. In the infinite number of laws which render the attraction very small at great distances, that of nature is the only one in which spheres act upon an external point as if their masses were united at their centres, [485']. This law is also the only one in which the action of a spherical stratum, upon a point placed within it, is nothing, [485"]. § 12

Application of the formulas of § 11 to the case where the attracting body is a cylinder, whose base is an oval curve, and whose length is infinite. When this curve is a circle, the action of a cylinder upon an external point, is inversely proportional to the distance of this point from the axis of the cylinder, [498']. A point placed within a circular cylindrical stratum, of uniform thickness, is equally attracted in every direction, [498"] § 13

Equation of condition relative to the motion of a body, [502] § 14

Several transformations of the differential equations of tlie motion of a system of bodies, submitted to their mutual attractions, [517 — 530] § 15

CHAPTER III- FIRST APPROXIMATION OF THE MOTIONS OF THE HEAVENLY BODIES ; OR THEORY

OF THE ELLIPTICAL MOTION 321

Integration of the differential equations which determine the relative motion of two bodies, attracting each other in the direct ratio of the masses, and the inverse ratio of the square of the distances. The curve described in this motion is a conic section, [534]. Expression of the time, in a converging series of sines of the true motion, [543]. If we neglect the masses of the planets, in comparison with that of the sun, the squares of the times of revolutions will be

as the cubes of the transverse axes of the orbits. This law extends to the motion of the satellites about their primary planets, [544"'^] §16

Second method of integration of the differential equations of the preceding article, [545—558]. § 17

Third method of integration of the same equations ; this method has the advantage of giving the arbitrary constant quantities in functions of the co-ordinates and of their first differentials, [559-597] §18,19

Finite equations of the elliptical motion ; expressions of the mean anomaly, of the radius vector, and of the true anomaly, in functions of the excentric anomaly, [606] § 20

General method of reducing functions into series ; theorems which result from it, [607 — 651]. §21

Application of these theorems to the elliptical motion. Expressions of the excentric anomaly, [657], the true anomaly, [668], and the radius vector of the planets, [659], in converging series of sines and cosines of the mean anomaly. Expressions in converging series, of the longitude, [675], of the latitude, [679], and of the projection of the radius vector, [680], upon a fixed plane but little inclined to that of the orbit § 22

Converging expressions of the radius vector, [683], and of the time, [690], in functions of the true anomaly, in a very excentric orbit. If the orbit be parabolic, the equation between the time and the true anomaly will be an equation of the third degree, [693], which may be resolved by means of the table of the motions of comets. Correction to be made in the true anomaly calculated for the parabola, to obtain the true anomaly corresponding to the same time, in a very excentric ellipsis, [695] § 23

Theory of the hyperbolic motion, [702] §24

Determination of the ratio of the masses of the planets accompanied by satellites, to that of the sun, [709] § 25

CHAPTIIR IV. DETERMINATION OP THE ELEMENTS OP THE ELLIPTICAL MOTION 393

Formulas which give these elements, when the circumstances of the primitive motion are known, [712 — 716']. Expression of the velocity, independent of the excentricity of the orbit, [720]. In the parabola the velocity is inversely proportional to the square root of the radius vector, [720"] §26

Investigation of the relation which exists between the transverse axis of the orbit, the chord of the described arch, the time employed in describing it, and the sum of the extreme radii vectores, [748, 750] § 27

The most convenient method of obtaining by observation the elements of the orbit of a comet, [753", &c.] §28

Formulas for computing, from any number of observations, taken near to each other, the geocentric longitude and latitude of a comet, at any intermediate time, with the first and second differentials of the longitudes and latitudes, [754, &c.] § 29

General method of deducing, from the differential equations of the motion of a system of bodies, the elements of their orbits, supposing the apparent longitudes and latitudes of these bodies, and the first and second differentials of these quantities, to be known, at a given instant, [760, &c.] § 30

Application of this method to the motion of comets, supposing them to be acted upon only by the attraction of the sun: it gives, by an equation of the seventh degree, [784], the distance of the comet from the earth. The mere inspection of three observations, made within very short intervals of each other, will suffice to discover whether the comet is nearer to the sun, or farther from it, than the earth is, [780'*] § 31

Method of finding, as accurately as is necessary, by means of three observations, the geocentric longitudes and latitudes of a comet, and their first and second differentials, divided by the corresponding powers of the element of the time, [787'] § 32

Determination of the elements of the orbit of a comet, when we know, at a given instant, its distance from the earth, and the first differential of this distance divided by the element of the time. Simple method of allowing for the excentricity of the orbit of the earth, [788 — 800]. § 33

When the orbit is a parabola, the greater axis becomes infinite, and this condition furnishes another equation, of the sixth degree, [805], to determine the distance of the comet from the earth. § 34

Hence we may obtain various methods of computing a parabolic orbit. Investigation of the method from which we may expect the most accurate result, and the greatest simplicity in the calculation, [806—811] § 35, 36

This method is divided into two parts : in the first, is given a method of approximation, to find the perihelion distance of the comet, and the time of passing the perihelion, [811" — 820] ; in the second, is given a method of correcting these two elements, by three distant observations, and then deducing from these the other elements, [820'" — 832] § 37

Accurate determination of the orbit, when the comet has been observed in both of its nodes, [833—841] §38

Method of finding the ellipticity of the orbit, when the ellipsis is very excentric, [842 — 849]. § 39

CHAPTER V. GENERAL METHODS OF FINDING THE MOTIONS OF THE HEAVENLY BODIES, BY SUCCESSIVE APPROXIMATIONS 475

Investigation of the alterations which must be made in the integrals of differential equations, to obtain the integrals of the same equations increased by certain terms, [850 — 859]. . . § 40

Hence we derive a simple method of obtaining the rigorous integrals of linear differential equations, when we know how to integrate the same equations deprived of their last terms, [861 — 871"]. § 41

We also obtain an easy method of computing the integrals of differential equations, by successive approximations, [872 — 875] § 42

Method of eliminating the arcs of a circle, which occur in these approximate integrals, when they do not really exist in the rigorous integrals, [876 — 892] § 43

Method of approximation, founded on the variations of the arbitrary constant quantities, [897-912] §45

CHAPTER VI. SECOND APPROXIMATION OP THE CELESTIAL MOTIONS ; OR THEORY OF THEIR PERTURBATIONS 504

Formulas of the motions in longitude and latitude, and of the radius vector in the disturbed orbit. Very simple form under which they appear, when only the first power of the disturbing forces is noticed, [913—932] § 46

Method of finding the perturbations in a series arranged according to the powers and products of the excentricities and of the inclinations of the orbits, [933— 948], .; §47

Development of the function of the mutual distances of the bodies of the system, on which their perturbations depend, in a series. Use of the calculus of finite differences in this development. Reflections upon this series [949 — 963] § 48

Formulas for computing its different terms, [964 — 1008] § 49

General expressions of the perturbations of the motion in longitude and in latitude, and of the radius vector, continuing the approximation to quantities of the order of the excentricities and inclinations, [1009—1034] § 50, 51

Recapitulation of these different results, remarks on farther approximations, [1035 — 1036"]. § 52

CHAPTER VII. ON THE SECULAR INEaUALITIES OF THE MOTIONS OF THE HEAVENLV BODIES. 569 These inequalities arise from the terms which, in the expressions of the perturbations, contain the time without the periodical signs. Differential equations of the elements of the elliptical motion, which make these terms disappear, [1037 — 1051] § 53

In taking notice only of the first power of the disturbing force, the mean motions of the planets will be uniform, and the transverse axes of their orbits constant, [1051' — 1070^^^]. ... § 54

Development of the differential equations relative to the excentricities and to the position of the perihelia, in any system of orbits in which the excentricities and mutual inclinations are small, [1071—1095] §55

Integration of these equations. Determination of the arbitrary constant quantities of the integral, by means of observations, [1096 — 1111] § 56

The system of the orbits of the planets and satellites, is stable, as it respects the excentricities; that is, these excentricities remain always very small, and the system merely oscillates about its mean state of ellipticity, from which it varies but little, [1111'" — 1118] §57

Differential expressions of the secular variations of the excentricity and of the position of the perihelion, [1118^— 1126^] §58

Integration of the differential equations relative to the nodes and inclinations of the orbits. In the motions of a system of orbits, which are very little inclined to each other, the mutual inclinations remain always very small, [1127 — 1139] § 59

Differential expressions of the secular variations of the nodes and of the inclinations of the orbits; first, with respect to a fixed plane; second, with respect to the moveable orbit of one of the bodies of the system, [1140—1146] § 60

General relations between the elliptical elements of a system of orbits, whatever be their excentricities and their mutual inclinations, [1147 — 1161] § 61

Investigation of the invariable plane, or that upon which the sum of the masses of the bodies of the system, multiplied respectively by the projections of the areas described by their radii vectores, in a given time, is a maximum. Determination of the motion of two orbits, inclined to each other by any angle, [1162 — 1167] § 62

CHAPTER VIII. SECOND METHOD OF APPKOXIMATION TO THE MOTIONS OF THE HEAVENLY BODIES. 634

This method is founded on the variations which the elements of the motion, supposed to be elliptical, suffer by means of the periodical and secular inequalities. General method of finding these variations. The finite equations of the elliptical motion, and their first differentials, are the same in the variable as in the invariable ellipsis, [1167' — 1169^"] § 63

Expressions of the elements of the elliptical motion in the disturbed orbit, whatever be its excentricity and its inclination to the plane of the orbits of the disturbing masses, [1170—1194]. . . § 64

Development of these expressions, when the excentricities and the inclinations of the orbits are small. Considering, in the first place, the mean motions and transverse axes ; it is proved, that if we neglect the squares and the products of the disturbing forces, these two elements are subjected only to periodical inequalities, depending on the configuration of the bodies of the system. If the mean motions of the two planets are very nearly commensurable with each other, there may result, in the mean longitude, two very sensible inequalities, affected with contrary signs, and reciprocally proportional to the products of the masses of the bodies, by the square roots of the transverse axes of their orbits. The acceleration of the motion of Jupiter, and the retardation of the mean motion of Saturn, are produced by similar inequalities. Expressions of these inequalities, and of those which the same ratio of the mean motions may render sensible, in the terms depending on the second power of the disturbing masses, [1195—1214] §65

Examination of the case where the most sensible inequalities of the mean motion, occur only among terms of the order of the square of the disturbing masses. This very remarkable circumstance takes place in the system of the satellites of Jupiter, whence has been deduced the two following theorems, T%e mean motion of the first satellite, minus three times that of the second, plus tunce that of the third, is accurately and invariably equal to nothing, [1239'"]. The mean longitude of the first satellite, minu^ three times that of the second, plus ttoice thai of the ihird, is invariably equal to two right angles, [1239^]. These two theorems take place, notwithstanding the alterations which the mean motions of the satellites may suffer, either from a cause similar to that which alters the mean motion of the moon, or from the resistance of a very rare medium. These theorems give rise to an arbitrary inequality, which differs for each of the three satellites, only by its coefficient. This inequality is insensible by observation, [1240 — 1242^] • ... § 66

Diflerential equations which determine the variations of the excentricities and of the perihelia, [1243—1266] §67

Development of these equations. The values of these elements are composed of two parts, the one depending on the mutual configuration of the bodies of the system, which comprises the periodical inequalities ; the other independent of that configuration, which comprises the secular inequalities. This second part is given by the same differential equations as those which we have before considered, [1266', 1279] § 68

A very simple method of obtaining the variations of the excentricities and of the perihelia of the orbits, arising from the ratio of the mean motions being nearly commensurable ; these variations

are connected with the corresponding variationa of the mean motion. They may produce, in the secular expressions of the excentricities, and of the longitudes of the perihelia, sensible terms, depending on the squares and products of the disturbing forces. Determination of these quantities, [1280—1309] § 69

On the variations of the nodes and of the inclinations of the orbits. Equations which determine their periodical and secular values, [1310 — 1327] § 70

Easy method of obtaining the inequalities which arise in these elements from the ratio of the mean motions being nearly commensurable ; they depend on the similar inequalities of the mean motion, [1328—1342] § 71

Investigation of the variation which the longitude of the epoch suffers. It is upon this variation that the secular variation of the moon depends, [1343 — 1345] § 72

Reflections upon the advantages, which the preceding method, founded upon the variation of the parameters of the orbits, presents in several circumstances ; method of deducing the variations

of the longitude, of the latitude^ and of the radius vector, [1345'^», &c.] § 73