E7fl D53 ~ ^ ;^3C| \i - A / ^. '/ M X : - I m^M^: -- r /r-^ C^- i ',T,;^A...-L ^ GRAVITATION. GRAY IT ATION ELEMENTARY EXPLANATION THE PRINCIPAL PERTURBATIONS SOLAR SYSTEM. (WRITTEN FOR THE PENNY CYCLOPEDIAS AND NOW PREVIOUSLY PUBLISHED FOR THE USE OF STUDENTS IN THE UNIVERSITY OF CAMBRIDGE.) BY G. B. AIRY, A.M., LATE FELLOW OF TRINITY COLLEGE ; AND PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE. V t -^ Library. LONDON: CHARLES KNIGHT, 22, LUDGATE STREET. MDCCCXXXIV. ^ i\ LONDON -. PRINTED BY WILLIAM CLOWES, Duke Street, Lambeth. PREFACE. IN laying this work before the public, I think it right to state the object for which it was originally composed, and the circum- stances which have in some degree changed its destination. The treatise was originally designed for a class of readers who might be supposed to possess a moderate acquaintance with the phaenornena and the terms of astro- nomy ; geometrical notions sufficient to en- able them to understand simple inferences from diagrams ; two or three terms of algebra as applied to numbers ; but none of that ele- vated science which has always been used in the investigation of these subjects, and without which scarcely an attempt has been made to explain them. I proposed to my- self, therefore, this general design : to ex- plain the perturbations of the solar system, VI PREFACE. as far as I was able, without introducing- an algebraic symbol. It will readily be believed that, after thus denying myself the use of the most powerful engine of mathematics, I did not expect to proceed very far. In my progress, how- ever, I was surprised to find that a general explanation, perfectly satisfactory, might be offered for almost every inequality recog- nised as sensible in works on Physical Astro- nomy. I now began to conceive it possible that the work, without in the smallest degree ' O departing from the original plan, or giving up the original object, might also be found useful to a body of students, furnished with considerable mathematical powers, and in the habit of applying them to the explana- tion of difficult physical problems. With this idea, the treatise is now printed in a separate form. The utility of a popular explanation of profound physical investigations is not, in my opinion, to be restricted to the instruc- tion of readers who are unable to pursue PREFACE. Vll them with the powers of modern analysis. Much is done when the interest of a good mathematician is excited by seeing, in a form that can be easily understood, results which are important for the comprehension of the system of the universe, and which can be made complete only by the applica- tion of a higher calculus. That such an interest has operated powerfully in our Uni- versities, I have no doubt. How many of our students would have known any thing of the Lunar Theory., if they had not been enjoined to read Newton's eleventh section ? And how many at this time possess the least acquaintance with the curious and compli- cated, but beautiful, theory of Jupiter's satellites, of which no elementary explana- tion is laid before them? But this is not all. The exercise of the mind in understanding a series of propositions, where the last con- clusion is geometrically in close connexion with the first cause, is very different from that which it receives from putting in play the long train of machinery in a profound Vlll PREFACE. analytical process. The degrees of con- viction in the two cases are very different. It is known to every one who has been engaged in the instruction of students at our Universities, that the results of the differential calculus are received by many, rather with the doubts of imperfect faith than with the confidence of rational con- viction. Nor is this to be wondered at ; a clear understanding of many difficult steps, a distinct perception that every connexion of these steps is correct, and a general com- prehension of the relations of the whole series of steps, are necessary for complete confidence. An unusual combination of ta- lents, attainments, and labour, must be re- quired, to appreciate clearly the evidence for a result of deep analysis. I am not un- willing to avow that the simple considera- tions which have been forced upon me in the composition of this treatise, have, in several instances, contributed much to clear up my view of points, which before were obscure, and almost doubtful. To the greater PREFACE. IX number of students, therefore, I conceive a popular geometrical explanation is more useful than an algebraic investigation. But even to those who are able to pursue the investigations with a skilful use of the most powerful methods, I imagine that a popular explanation is not unserviceable. The insight which it gives into the relation of some mechanical causes and geometrical effects, may powerfully, yet imperceptibly, influence their understanding of many others which occur in the prosecution of an alge- braical process. The advanced student who exalts in the progress which the modern calculus enables him to make in the Lunar or Planetary Theories, perhaps, hardly re- flects how much of the power of understand- ing his conclusions has been derived from Newton's general explanations. The utility of such a work being allowed, it cannot, I think, be disputed that there exists a necessity for a new one. The only attempts at popular explanation in general use with which I am acquainted, are New- d 5 PREFACE. ton's eleventh section, and a small part of Sir John Herschel's admirable treatise on Astronomy. The former of these (the most valuable chapter that has ever been written on physical science,) is in some parts very defective. Thus, the explanation of the mo- tion of the line of apses is too general, and enters into particular cases too little, to allow of a numerical calculation being- founded on it. The explanation of evection is extremely defective. The explanation of variation, however, and of alteration of the node and inclination, are probably as com- plete as can be given. The latter treatise, besides expanding some of Newton's reason- ing, alludes to the long inequalities and se- cular disturbances of the planets, but not perhaps with sufficient accuracy of detail to supersede the necessity of further explana- tion. No popular work with which I am acquainted, alludes at all to the peculiarities of the theory of Jupiter's satellites. I have attempted in some degree to sup- ply these defects ; with what success the PREFACE. XI reader must judge. As it was my object to avoid repetition of theorems, which are to be found in treatises on Mechanics and ele- mentary works on Physical Astronomy, and which are fully read and mastered by those who take much interest in these subjects, and which, moreover, do not admit of popu- lar explanation so easily as many of the more advanced propositions, I have omitted no- ticing them any further than the consistency of system seemed to require. Thus, with regard to elliptic motion, Kepler's laws, &c., I have merely stated results ; because the investigations of these are familiar to the higher students, to whom I hope the other explanations may be useful ; and because without great trouble it did not appear pos- sible to put the reasons for these results in the same form as those for other effects of force. I have, however, alluded to some of the dif- ficulties which are apt to embarrass readers in the first instance, as much for the sake of the reasoning contained in the explanation as for the value of the results. The only addi- Xll PREFACE. tions which I have thought it desirable to make for the benefit of readers of Newton, are contained in a few notes referring to one of Newton's constructions. To the reader who may detect faults in the composition of the work, I can merely state in apology, that it has been written in a hurried manner, in the intervals of very pressing employments. I have only to add, that, holding a responsible situation in my University, I have always thought it my duty to promote, as far as I am able, the study of Physical Astronomy ; and that if this treatise shall contribute to extend the knowledge of its phenomena and their rela- tion to their causes, either among the stu- dents of the University, or in that more numerous body for whom it was originally written, I shall hold myself well repaid for the trouble which it has cost me. G. B. AIRY. OBSERVATORY, CAMBRIDGE, March 9, 1834. CONTENTS. Page SECTION I. On ike rules for calculating Attraction, or the law of Gravitation . . . .1 Article 3. Attraction is to be measured by the motion which it produces . . . . .2 5. Attraction does not depend on the mass of the attracted body . . . . . .4 7 Is proportional to the mass of the attracting body . 5 9. Varies inversely as the square of the distance . 5 SECTION II. On the effect of Attraction upon a body which is in motion; and on the Orbital Revolutions of Planets and Satellites . . .8 14. The motion supposed transversal to the direction of the force : a simple case is the motion of a stone thrown from the hand . . . . .9 15. The motion of the stone calculated by the second law of motion . . . . .9 1G. The same rule will apply to a planet, if we restrict the calculation to a very short time . . .11 18. A planet or satellite may move in an ellipse, a parabola, or a hyperbola : in fact, they all move in ellipses dif- fering little from circles . . . .12 21. Distinction between projectile force and attractive force 15 XIV CONTENTS. Article Page 25. The planets, when nearest to the sun, are in no danger of falling to the sun, because their velocity is then necessarily very great . . . .17 27. Explanation of terms . . . .20 29. Law of equable description of areas . . .23 34. The square of the periodic time is proportional to the cube of the mean distance, for all bodies revolving round the same centre . . .26 38. Extension of this rule to bodies revolving round different centres . . . . . .29 40. Further extension of the rule, taking into account the motion of the central body , . .30 SECTION III. General notions of Perturbation; and pertur- bation of the elements of Orbits . . 32 43. Principle of the variation of elements : instantaneous ellipse . . . . . .33 46= Disturbing force directed to the central body, and inde- pendent of time . . . . .35 47. If it varies with the time, the dimensions of the orbit are altered . . . . .37 48. Disturbing force acts in the direction in which the planet is moving : it increases the dimensions of the orbit, and lengthens the time of revolution con- tinually . . . . .37 49. These effects are extremely conspicuous in the alteration of a planet's longitude . . . .39 50. Disturbing force directed to the centre near perihelion, causes the line of apses to progress . . 40 CONTENTS. XV Article Page 53. Near aphelion, it causes the line of apses to regress . 44 56. These effects are greatest when the excentricity is small 47 57. While the planet is moving from perihelion to aphelion, it diminishes the excentricity . . .48 58. The contrary, while the planet is moving from aphelion to perihelion . . . . .50 62. Disturbing force perpendicular to the radius vector, in the direction in which a planet is moving, when the planet is approaching to perihelion, causes the line of apses to regress . . . . .52 63. The contrary after passing perihelion . . 53 64. Before passing aphelion it causes the line of apses to progress, and after passing aphelion it causes the line of apses to regress . . . .54 67. Near perihelion it increases the excentricity, and near aphelion it diminishes the excentricity . 56 SECTION IV. On the nature of the force, disturbing a Planet or Satellite, produced by the attraction of other bodies . . . .58 72. A satellite will revolve round a planet which moves round the sun, nearly in the same manner as if the planet were fixed . . . . .60 73. Disturbing force is the difference of the forces which act on the central and the revolving body . .61 74. General construction for disturbing force . . 62 77. The disturbed body between the disturbing and central bodies : the disturbing force is directed from the cen- tral body, and is great . . .65 78. The disturbed body on the opposite side of the central XVI CONTENTS. Article Page body : the disturbing force is still directed from the central body . . . . .65 79. The latter much less than the former, except the dis- turbing body be distant, when they are nearly equal 66 80. The central and disturbed bodies equally distant from the disturbing body : the disturbing force is directed to the central body . . .68 82. When the disturbing body is distant, these forces are nearly proportional to the distance of the disturbed body from the central body directly . . 69 83. And to the cube of the distance of the disturbing body inversely . . . . . .70 84. Between the situations in which the three bodies are in. the same straight line, and those in which the dis- turbed and central bodies are equidistant from the disturbing body, there is always a force perpendicular to the radius vector, accelerating the disturbed body as it moves from the equidistant points, and retarding is as it moves towards the equidistant points . 71 86. The disturbing body interior to the orbit of the dis- turbed body : different cases depending upon the circumstance of the disturbing body's distance being greater or less than half the disturbed body's dis- tance . . . . . .74 SECTION V. Lunar Theory . . . .76 90. The moon's Annual Equation, produced by the varia- tion in the sun's distance from the earth . . 78 91. The moon's Variation the immediate effect of the prin- cipal part of the disturbing forces . . 79 CONTENTS. XV11 Article Page 94. The Parallactic Inequality, produced by the difference between the disturbing forces at conjunction and those at opposition . . . .84 97. Note. The irregularity produced in an elliptic orbit will be nearly the same as that in a circular orbit . 88 98. The line of apses coinciding with the line of syzygies : the preponderance of the effect of the force at apogee over that at perigee causes the line of apses to pro- gress . . . . . .89 98. Note. If a disturbing force is always proportional to the original force, it may either be considered in com- bination or may be considered separately : conse- quences of the comparison of results obtained in the two ways . . . . .90 99. The force perpendicular to the radius vector produces an effect similar to that in the direction of the radius vector . . . . . .92 99- Note. Instead of supposing the velocity increased or diminished for a time, we may suppose the central force diminished or increased . . .93 101. The line of apses coinciding with the line of quadra- tures, the forces cause the line of apses to regress . 94 104. The regression is less than the progression in the former case, and on the whole the line of apses pro- gresses , . . . .96 105. The progress much greater than might at first be sup- posed, in consequence of the alteration of the time during which the different forces act, produced by the sun's motion . . . . . 97 112. When the line of syzygies is inclined to the line of apses, and the moon passes the latter before passing XV111 CONTENTS, Article Page the former, the excentricity is diminished at every revolution . . . .102 113. When the moon passes the line of syzygies before pass- ing the line of apses, the excentricity is increased at every revolution ..... 105 119. Acceleration of the moon, produced by the alteration which the planets cause in the excentricity of the earth's orbit .... 108 SECTION VI. Theory of Jupiter's Satellites . .110 122. Investigation of the motion of two satellites, one of whose periodic times is very little greater than double the other . . . . . .111 124. The line of conjunctions will regress slowly . .112 128. The inner satellite must revolve in an ellipse whose perijove is turned towards the points of conjunction, supposing it to have no independent excentricity . 115 130. The apojove of the outer satellite must be turned to- wards the points of conjunction . . .116 132. If the periodic time of one were very little less than double the other, the perijoves would be in the posi- tions opposite to those found above . .118 134. Note. If the orbits have independent excentricities, the Sctme inequality (nearly) will be produced as if they have no independent excentricity . .119 135. The investigations above apply to Jupiter's first and second satellite . . , . .121 137. They apply also to the second and third satellite . 123 138. It is found that the regression of the lines of conjunction CONTENTS. XIX Article Page of the second and third is exactly as rapid as that of the first and second . . . .124 139. And that the points of conjunction of the second and third are always opposite to those of the first and second ...... 125 140. In consequence of this, the inequalities, produced by the first and by the third in the motion of the second, are inseparably combined, and produce one great inequality ..... 125 142. The equality of the regressions of the lines of conjunc- tion is not accidental, but a necessary consequence of the mutual action of the satellites, once supposing the rates of regression to be nearly equal .127 151. The excentricity of the fourth satellite's orbit causes a small excentricity of the same kind in the orbit of the third ...... 134 152. The excentricity of the orbit of the third satellite causes an excentricity of the opposite kind in the orbit of the fourth . . . . .136 SECTION VII. Theory of Planets .... 138 157. Some inequalities similar to the Variation, Parallactic Equation, Evection, and Annual Equation, of the moon . . . . . .139 159. The analogy must be applied with great caution . 141 161. General principle of explanation of the sensible per- turbations depending on the excentricities and incli- nations of the orbits . . .142 165. The periodic times of Jupiter and Saturn are in the proportion of 29 : 72 nearly, and hence their conjunc- tions fall near to three equidistant points on their XX CONTENTS. Article Page orbits, advancing slowly at each return to the same point . . . . . .146 170. This advance occasions a small variation of the forces of one planet on the other, which return to their for- mer values only when the places of conjunction have moved through 120 . . . . 149 173. In consequence, the mean distances of the planets are altered by inconsiderable quantities, but these produce considerable effects on the longitude . .151 175. The same causes produce periodical alterations of the excentricities and the places of perihelia . . 153 177' Several similar cases in the solar system . . 155 178. The inequalities are generally greatest when the dif- ference between the two numbers expressing the approximate proportion is small . . . 156 180. The action of one planet upon another does not, in the long run, alter its mean distance . . . 157 182. It will, in the long run, cause the line of apses to pro- gress, (except the orbit of the disturbing planet be very excentric, and the places of their perihelia nearly coincide) . . . . . .162 183. The excentricity of the orbit of the disturbing planet will cause an alteration in that of the disturbed planet, increasing or diminishing it according to the relative position of the line of apses . .163 185. Remarkable relation between the variations of the excentricities of all the planets . . .165 SECTION VIII. Perturbation of Inclination and Place of Node . . . . .166 189. A force which draws the moving body towards a CONTENTS. XXI Article Page plane of reference before it has reached the greatest distance above or below that plane, causes a regression of the line of nodes, and a diminution of the inclina- tion . . . . . .168 191. A similar force, acting after the body has passed the greatest distance, causes a regression of the line of nodes and an increase of the inclination . . 169 195. When the central and revolving body are equidistant from the disturbing body, there is no disturbing force perpendicular to the plane of orbit. When the revolving body is nearer, there is a force directed to that side on which the disturbing body is ; and when farther, a force directed to the opposite side . 171 199. When the moon's line of nodes is in syzygies, there is no alteration of the node or inclination . .174 200. When it is in quadratures, the line of nodes regresses rapidly, but the inclination is not altered . . 174 201. When the moon passes the line of nodes in going from quadrature to syzygy, the inclination is diminished : when she passes it in going from syzygy to quadra- ture, the inclination is increased : in both cases the node regresses . . . . .176 204. Consideration which shows that the regression of the line of nodes will be somewhat less than would be inferred from a first calculation . . .180 207. The inclination and node of an interior planet are affected by a distant exterior planet in nearly the same way in which those of the moon are affected by the sun ...... 182 209. Long inequalities are produced in the inclination and the position of the line of nodes, when the periodic times of two planets are nearly commensurable . 183 XX11 CONTENTS. Article Page 210. In the long run, the action of one planet causes the nodes of another to regress : but the inclination is not altered except one of the orbits is excentric . 184 211. It is supposed in all these propositions that the plane of reference is the orbit of the disturbing planet: if it be any other plane, (as the orbit of another planet) it may happen that the line of nodes will progress . . . . . .185 212. Remarkable relation between the inclinations of several planets disturbing each other . . .186 214. Explanation of the term fundamental plane . . 187 215. Determination of the fundamental planes of Jupiter's satellites ...... 188 220. Their lines of nodes will regress on their fundamental planes, without alteration of inclination to those planes . . . . . . 192 221. An inclination of the orbit of an exterior satellite produces a similar inclination of the orbit of an in- terior satellite . . . . .193 222. An inclination of the orbit of an interior satellite pro- duces an opposite inclination of the orbit of an exterior satellite . . . . . .196 SECTION IX. Effects of the Oblateness of Planets upon the Motions of their Satellites . .198 230. The attraction of an oblate spheroid on a body in the plane of its equator is greater than if its whole mass were collected at its centre ... 200 231. The proportion in which it is greater diminishes as the square of the distance increases . . , 202 CONTENTS. XXlll Article Pag 233. The attraction of an oblate spheroid on a body in the direction of its axis is less than if its whole mass were collected at its centre . . . 203 236. The attraction of an oblate spheroid causes the lines of apses of its satellites to progress ; and more rapidly for the near satellites than for the distant ones . 206 238. If a satellite is not in the plane of the equator, the attraction tends to pull it towards that plane . 208 240. Effect of Saturn's ring on the motions of Saturn's satellites . . . . . .210 242. The moon's fundamental plane is inclined to the ecliptic ...... 212 243. Long inequality in the moon's motion, produced by the earth's oblate ness . . . .213 ON GRAVITATION, SECTION I. On the Rules for calculating Attrac- tion, or, the Law of Gravitation. (1.) THE principle upon which the motions of the earth, moon, and planets are calculated is this : Every particle of matter attracts every other par- ticle. That is, if there were a single body alone, and at rest, then, if a second body were brought near it, the first body would immediately begin to move toward the second body. Just in the same manner, if a needle is at rest on a table, and if a magnet is brought near it, the needle immediately begins to move towards the magnet, and we say that the magnet attracts the needle. But mag- netic attraction belongs only to certain bodies : whereas the attraction of which we speak here be- longs to all bodies of every kind : metals, earths, fluids, and even the air and gases are equally sub- ject to its influence. B 2 GRAVITATION. (2.) The most remarkable experiments which prove that bodies attract each other are a set of experiments made at the end of the last century by Mr. Cavendish. Small leaden balls were sup- ported on the ends of a rod which was suspended at the middle by a slender wire ; and when large leaden balls were brought near to them,, it was found that the wire was immediately twisted by the motion of the balls. But the results of this experiment are striking, principally because they are unusual ; the ordinary force of gravity serves quite as well to prove the existence of some such power. For when we consider that the earth is round, and that, on all parts of it, bodies, as soon as they are at liberty, fall in directions perpen- dicular to its surface, (and therefore fall in oppo- site directions at the places which are diametrically opposite,) w r e are compelled to allow that there is a force such as we call attraction, either directed to the centre of the earth, or produced by a great number of small forces, directed to all the different particles composing the earth. The peculiar value of Cavendish's experiment consists in showing that there is a small force directed to every different particle of the earth. (3.) But it is necessary to state distinctly the TWO EFFECTS OF ATTRACTION. 3 rules by which this attraction is regulated,, and by which it may be calculated; or (as it is techni- cally called) the law of gravitation. Before we- can do this, we must determine which of the effects of attraction we choose to take as its measure. For there are two distinct effects : one is the pressure which it produces upon any obstacle that keeps the body at rest ; the other is the space through which it draws the body in a certain time, if the obstacle is removed and the body set at liberty. Thus,, to take the ordinary force of gravity as an instance: we might measure it by the pressure which is produced on the hand by a lump of lead held in the hand; or we might measure it by the number of inches through which the lump of lead would fall in a second of time after the hand is opened (as the pressure and the fall are both occasioned by gravity). But there is this difference between the two measures ; if we adopted the first, since a large lump of lead weighs more than a small one,, we should find a different measure by the use of every different piece of lead ; whereas, if we adopt the second, since it is well established by careful and accurate experiments that large and small lumps of lead, stones, and even feathers, fall through the same number of inches in a second of B2 4 GRAVITATION. time., (when the resistance of the air, &c., is re- moved,) we shall get the same measure for gravity, whatever body we suppose subject to its influence. The consistence and simplicity of the measure thus obtained incline us to adopt it in every other case; and thus we shall say, Attraction is measured by the space through which it draws a body in one second of time after the body is set at liberty. (4.) Whenever we speak, therefore, of calcu- lating attraction, it must be understood to mean calculating the number of inches, or feet, through which the attraction draws a body in one second of time. (5.) Now the first rule is this : "The attraction of one body upon another body does not depend on the mass of the body which is attracted, but is the same whatever be the mass of the body so attracted, if the distances are the same." (6.) Thus Jupiter attracts the sun, and Jupiter attracts the earth also; but though the sun's mass is three hundred thousand times as great as the earth's, yet the attraction of Jupiter on the sun is exactly equal to his attraction on the earth, when the sun and the earth are equally distant from Ju- piter. In other words, (the attraction being mea- sured in conformity with the definition above,) LAWS OF ATTRACTION. 5 when the sun and the earth are at equal distances from ^Jupiter, the attraction of Jupiter draws the sun through as many inches,, or parts of an inch, in one second of time as it draws the earth in the same time. (7.) The second rule is this : "Attraction is pro- portional to the mass of the body which attracts, if the distances of different attracting bodies be the same." (8.) Thus, suppose that the sun and Jupiter are at equal distances from Saturn ; the sun is about a thousand times as big as Jupiter; then whatever be the number of inches through which Jupiter draws Saturn in one second of time, the sun draws Saturn in the same time through a thou- sand times that number of inches. (9.) The third rule is this : " If the same attracting body act upon several bodies at dif- ferent distances, the attractions are inversely pro- portional to the square of the distances from the attracting body." (10.) Thus the earth attracts the sun, and the earth also attracts the moon ; but the sun is four hundred times as far off as the moon, and there- fore, the earth's attraction on the sun is only TT -oV -co- th part of its attraction on the moon ; or, 6 GRAVITATION. as the earth's attraction draws the moon through about Vth of an inch in one second of time, the earth's attraction draws the sun through ^-g-^.^-.g-g.th of an inch in one second of time. In like man- ner, supposing Saturn ten times as far from the -sun as the earth is, the sun's attraction upon Saturn is only one hundredth part of his attraction on the earth. (11.) The same rule holds in comparing the attractions which one body exerts upon another, when, from moving in different paths, and with different degrees of swiftness, their distance is altered. Thus Mars, in the spring of 1833, was twice as far from the earth as in the autumn of 1832; therefore, in the spring of 1833, the earth's attraction on Mars was only one-fourth of its at- traction on Mars in the autumn of 1832. Jupiter is three times as near to Saturn, when they are on the same side of the sun as when they are on opposite sides; therefore, Jupiter's attraction on Saturn, and Saturn's attraction on Jupiter, are nine times greater when they are on the same side of the sun than when they are on opposite sides. (12.) The reader may ask, How is all this known to be true ? The best answer is, perhaps, the following : We find that the force which the PROOF OF THE TRUTH OF THE THEORY. 7 earth exerts upon the moon bears the same pro- portion to gravity on the earth's surface, which it ought to bear in conformity with the rule just given. For the motions of the planets, calculations are made, which are founded upon these laws, and which will enable us to predict their places with considerable accuracy, if the laws are true, but which would be much in error if the laws were false. The accuracy of astronomical observations is carried to a degree that can scarcely be ima- gined ; and by means of these we can every day compare the observed place of a planet with the place which was calculated beforehand, according to the law of gravitation. It is found that they agree so nearly, as to leave no doubt of the truth of the law. The motion of Jupiter, for instance, is so perfectly calculated, that astronomers have computed ten years beforehand the time at which it will pass the meridian of different places, and we find the predicted time correct within half a second of time. GRAVITATION. SECTION II. On the Effect of Attraction upon a Body which is in motion, and on the Orbital Revolutions of Planets and Satellites. (13.) WE have spoken of the simplest effects of attraction, namely, the production of pressure, if the matter on which the attraction acts is sup- ported, (as when a stone is held in the hand,) and the production of motion if the matter is set at liberty, (as when a stone is dropped from the hand.) And it will easily be understood, that when a body is projected, or thrown, in the same direction in which the force draws it, (as when a stone is thrown downwards,) it will move with a greater velocity than either of these causes sepa- rately would have given it ; and if thrown in the direction opposite to that in which the force draws it, (as when a stone is thrown upw r ards,) its motion will become slower and slower, and will, at last, be turned into a motion in the opposite direction. We have yet to consider a case much more im- portant for astronomy than either of these: Sup- pose that a body is projected in a direction trans- verse to, or crossing, the direction in which the for ce draws it, how will it move ? MOTION OF PROJECTED STONE. 9 (J4.) The simplest instance of this motion that we can imagine is the motion of a stone when it is thrown from the hand in a horizontal direction, or in a direction nearly horizontal. We all know that the stone soon falls to the ground ; and if we observe its motion with the least attention, we see that it does not move in a straight line ; it begins to move in the direction in which it is thrown ; but this direction is speedily changed ; it continues to change gradually and constantly, and the stone strikes the ground, moving at that time in a direc- tion much inclined to the original direction. The most powerful effort that we can make, even when we use artificial means, (as in producing the mo- tion of a bomb or a cannon-ball,) is not sufficient to prevent the body from falling at last. This experiment, therefore, will not enable us imme- diately to judge what will become of a body (as a planet) which is put in motion at a great distance from another body, which attracts it, (as the sun;) but it will assist us much in judging generally what is the nature of motion when a body is pro- jected in a direction transverse to the direction in which the force acts on it. (15.) It appears, then, that the general nature of the motion is this : the body describes a curved B5 10 GRAVITATION. path, of which the first part has the same direc- tion as the line in which it is projected. The cir- cumstances of the motion of the stone may be cal- culated with the utmost accuracy from the follow- ing rule, called the second law of motion, (the accuracy of which has been established by many simple experiments, and many inferences from complicated motion.) If A, fig. 1, is the point from which the stone was thrown, and A B the i. 1. direction in which it was thrown ; and if we wish to know where the stone will be at the end of any particular time, (suppose, for instance, three se- conds,) and if the velocity with which it is thrown would, in three seconds, have carried it to B, sup- posing gravity not to have acted on it ; and if gra- vity would have made it fall from A to C, sup- posing it to have been merely dropped from the hand ; then, at the end of three seconds, the stone really will be at the point D, which is determined by drawing B D parallel and equal to A C ; and it MODIFICATION OF CALCULATION FOR PLANETS. 11 will have reached it by a curved path A D, of which different points can be determined in the same way for different instants of time. (16.) The calculation of the stone's course is easy, because, during the whole motion of the stone, gravity is acting upon it with the same force and in the same direction. The circumstances of the motion of a body attracted by a planet, or by the sun, (where the force, as we have before men- tioned, is inversely proportional to the square of the distance, and therefore varies as the distance alters, and is not the same, either in its amount or in its direction at the point D, as it is at the point C,) cannot be computed by the same simple me- thod. But the same method will apply, provided we restrict the intervals for which we make the cal- culations to times so short, that the alterations in the amount of the force, and in its direction, during each of those times, will be very small. Thus, in the motion of the earth, as affected by the attrac- tion of the sun, if we used the process that we have described, to find where the earth will be at the end of a month from the present time, the place that we should find would be very far wrong ; if we calculated for the end of a week, since the direction of the force (always directed to the sun) 12 GRAVITATION. and its magnitude (always proportional inversely to the square of the distance from the sun) would have been less altered, the circumstances would have been more similar to those of the motion of the stone,, and the error in the place that we should find would be much less than before ; if we calculated by this rule for the end of a day, the error would be so small as to be perceptible only in the nicest observations ; and if we calculated for the end of a minute, the error would be perfectly insensible. (17.) Now a method of calculation has been in- vented, which amounts to the same as making this computation for every successive small portion of time, with the correct value of the attractive force, and the correct direction of force at every particu- lar portion of time, and finding thus the place where the body will be at the end of any time that we may please to fix on, without the smallest error. The rules to which this leads are simple : but the demonstration of the rules requires the artifices of advanced science. We cannot here attempt to give any steps of this demonstration ; but our plan requires us to give the results. (18.) It is demonstrated that if a body (a planet, for instance) is by some force projected from A, PLANETS MOVE IN CONIC SECTIONS. 13 fig. 2, in the direction A B, and if the attraction of the sun, situated at S, begins immediately to act on \B c Fig. 2. G it,, and continues to act on it according to the law that we have mentioned, (that is, being inversely proportional to the square of its distance from S, and always directed to S ;) and if no other force whatever but this attraction acts upon the body ; then the body will move in one of the following curves a circle, an ellipse, a parabola, or a hyper- bola. In every case the curve will, at the point A, have the same direction as the line A B ; or, (to use the language of mathematicians,) A B will be a tan- gent to the curve at A. The curve cannot be a circle unless the line A B is perpendicular to S A, and, moreover, unless the velocity, with which the planet is projected, is nei- ther greater nor less than one particular velocity determined by the length of S A and the mass of the body S. If it differs little from this particular 14 GRAVITATION. velocity, (either greater or less,) the body will move in an ellipse ; but if it is much greater, the body will move in a parabola or a hyperbola. If A B is oblique to S A, and the velocity of pro- jection is small, the body will move in an ellipse ; but if the velocity is great, it may move in a para- bola or hyperbola, but not in a circle. If the body describe a circle, the sun is the centre of the circle. If the body describe an ellipse, the sun is not the centre of the ellipse, but one focus. (The me- thod of describing an ellipse is to fix two pins in a board, as at S and H, fig. 3 ; to fasten a thread S P H to them, and to keep this thread stretched by the point of a pencil, as at P : the pencil will trace out an ellipse, and the places of the pins S and H will be the two focuses.) If the body describe a parabola or hyperbola, the sun is in the focus. (19.) The planets describe ellipses which are very little flattened, and differ very little from circles. EXPLANATION OF PROJECTILE FORCE. 15 Three or four comets describe very long ellipses : and nearly all the others that have been observed are found to move in curves which cannot be dis- tinguished from parabolas. There is reason to think that two or three comets which have been observed move in hyperbolas. But as we do not propose, in this treatise, to enter into a discussion on the motions of comets,, we shall confine our- selves to the consideration of motion in an ellipse. (20.) Every thing that has been said respecting the motion of a planet, or body of any kind, round the sun, in consequence of the sun's attraction ac- cording to the law of gravitation, applies equally well to the motion of a satellite about a planet, since the planet attracts with a force following the same law (though smaller) as the attraction of the sun. Thus the moon describes an ellipse round the earth, the earth being the focus of the ellipse; Jupiter's satellites describe each an ellipse about Jupiter, and Jupiter is in one focus of each of those ellipses ; the same is true of the satellites of Sa- turn and Uranus. (21.) In stating the suppositions on which the calculations of orbits are made, we have spoken of a force of attraction, and a force by which a planet is projected. But the reader must observe that 16 GRAVITATION. the nature of these forces is wholly different. The force of attraction is one which acts constantly and steadily without a moment's intermission, (as we know that gravity to the earth is always acting :) the force by which the body is projected is one which we suppose to be necessary at some past time to account for the planet's motion, but which acts no more. The planets arc in motion, and it is of no consequence to our inquiry how they received this motion, but it is convenient, for the purposes of calculation, to suppose that, at some time, they re- ceived an impulse of the same kind as that which a stone receives when thrown from the hand ; and this is the w r hole meaning of the term "projectile force." (22.) From the same considerations it will ap- pear that, if in any future investigations we should wish to ascertain what is the orbit described by a planet after it leaves a certain point where the velo- city and direction of its motion are known, we may suppose the planet to be projected from that point with that velocity and in that direction. For it is unimportant by what means the planet acquires its velocity, provided it has such a velocity there. (23.) We shall now allude to one of the points which, upon a cursory view, has always appeared PLANETS WILL NOT FALL TO THE SUN. 17 one of the greatest difficulties in the theory of elliptic revolution, but which, when duly considered, will be found to be one of the most simple and na- tural consequences of the law of gravitation. (24.) The force of attraction, we have said, is inversely proportional to the square of the distance, and is therefore greatest when the distance is least. It would seem then, at first sight, that when a planet has approached most nearly to the sun, as the sun's attraction is then greater than at any other time, the planet must inevitably fall to the sun. But we assert that the planet begins then to recede from the sun, and that it attains at length as great a distance as before, and goes on continu- ally retracing the same orbit. How is this re- ceding from the sun to be accounted for ? (25.) The explanation depends on the increase of velocity as the planet approaches to the point where its distance from the sun is least, and on the considerations by which we determine the form of the curve which a certain attracting force will cause a planet to describe. In explaining the motion of a stone thrown from the hand, to which the motion of a planet for a very small time is ex- actly similar, we have seen that the deflection of the stone from the straight line in which it began to 18 GRAVITATION. move is exactly equal to the space through which gravity could have made it fall in the same time from rest, whatever were the velocity with which it was thrown. Consequently, when the stone is thrown with very great velocity, it will have gone a great distance before it is much deflected from the straight line, and therefore its path will be very little curved; a fact familiar to the experience of every one. The same thing holds with regard to the motion of a planet, and thus the curvature of any part of the orbit which a planet describes will not depend simply upon the force of the sun's attraction, but will also depend on the velocity with which the planet is moving. The greater is the velocity of the planet at any point of its orbit, the less will the orbit be curved at that part. Now if we refer to fig. 2, we shall see that, supposing the planet to have passed the point C with so small a velocity that the attraction of the sun bends its path very much, and causes it immediately to be- gin to approach towards the sun ; the sun's attrac- tion will necessarily increase its velocity as it moves through D, E, and F. For the sun's attractive force on the planet, when the planet is at D, is act- ing in the direction D S, and it is plain that (on account of the small inclination of D E to D S) the PLANET'S VELOCITY GREATEST NEAR THE SUN. 19 force pulling in the direction D S, helps the planet along in its path D E, and thereby increases its velocity. Just as when a ball rolls down a sloping bank, the force of gravity (whose direction is not much inclined to the bank) helps the ball down the bank, and thereby increases its velocity. In this manner, the velocity of the planet will be con- tinually increasing as the planet passes through D, E, and F ; and though the sun's attractive force (on account of the planet's nearness) is very much increased, and tends, therefore, to make the orbit more curved, yet the velocity is so much increased that, on that account, the orbit is not more curved than before. Upon making the calculation more accurately, it is found that the planet, after leaving C, approaches to the sun more and more rapidly for about a quarter of its time of revolution ; then for about a quarter of its time of revolution the velocity of its approach is constantly diminishing : arid at half the periodic time after leaving C, the planet is no longer approaching to the sun ; and its velocity is so great, and the curvature of the orbit in consequence so small, (being, in fact, ex- actly the same as at C,) that it begins to recede. After this it recedes from the sun by exactly the same degrees by which it before approached it. 20 GRAVITATION, (26.) The same sort of reasoning will show why, when the planet reaches its greatest distance, where the sun's attraction is least, it does not altogether fly off. As the planet passes along H, K, A, the sun's attraction (which is always directed to the sun) retards the planet in its orbit, just as the force of gravity retards a ball which is bowled up a hill ; and when it has reached C, its velocity is extremely small ; and, therefore, though the sun's attraction at C is small, yet the deflection which it produces in the planet's motion is (on account of the planet's slow- ness there) sufficient to make its path very much curved, and the planet approaches the sun, and goes over the same orbit as before. (27.) The following terms will occur perpe- tually in the rest of this treatise, and it is therefore desirable to explain them now. Let S and H, fig. 4, be the focuses of the ellipse \cp Fig. 4. AEDB; draw the line AB through S and H; EXPLANATION OF TERMS. 21 take C the middle point between S and H, and draw D C E perpendicular to A C B. Let S be that focus which is the place of the sun, (if we are speak- ing of a planet's orbit,) or the place of the planet (if we are speaking of a satellite's orbit.) Then A B is called the major axis of the ellipse. C is the centre. A C or C B is the semi-major axis. This is equal in length to S D ; it is sometimes called the mean distance, because it is half-way between A S (which is the planet's smallest distance from S) and BS, (which is the planet's greatest distance from S.) DE is the minor axis, and D C or C E the semi- minor axis. A is called the perihelion, (if we are speaking of a planet's orbit;) the perigee, (if we are speaking of the orbit described by our moon about the earth;) the perijove, (if we are speaking of the orbit described by one of Jupiter's satellites round Jupiter;) or the perisaturnium, (if we are speak- in & of the orbit described by one of Saturn's satel- o lites about Saturn.) B, in the orbit of a planet, is called the aphe- lion ; in the moon's orbit it is called the apogee ; in the orbit of one of Jupiter's satellites, we shall call it the apojove. 22 GRAVITATION. A and B are both called apses; and the line A B, or the major axis, is sometimes called the line of apses. S C is sometimes called the linear excentricity ; but it is more usual to speak only of the propor- tion which S C bears to A C, and this proportion, expressed by a number, is called the excentricity. Thus, if S C were one-third of A C, we should say, that the excentricity of the orbit was %, or 0'3333. If S cyo is drawn towards a certain point in the heavens, called the first point of Aries, then the angle S A is called the longitude of perihelion, (or of perigee, or of perijove, &c.) If P is the place of the planet in its orbit at any particular time, then the angle PP S P is its longi- tude at that time, and the angle ASP is its true anomaly. (The longitude of the planet is, there- fore, equal to the sum of the longitude of the peri- helion, and the true anomaly of the planet.) The line S P is called the radius vector. In all our diagrams it is to be understood, that the planet, or satellite, moves through its orbit in the direction opposite to the motion of the hands of a watch. This is the direction in which all the planets and satellites would appear to move, if viewed from any place on the north side of the planes of their orbits. CALCULATION OF A PLANET'S PLACE. 23 The time in which the planet moves from any one point of the orbit through the whole orbit, till it comes to the same point again^ is called the planet's periodic time. (28.) If we know the mass of the central body, and if we suppose the revolving body to be pro- jected at a certain place in a known direction with a given velocity, the length of the axis major, the excentricity, the position of the line of apses, and the periodic time, may all be calculated. We can- not point out the methods and formulae used for these, but we may mention one very remarkable result. The length of the axis major depends only upon the velocity of projection, and upon the place of projection, and not at all upon the direction of projection. (29.) We shall proceed to notice the principle on which the motion of a planet, or satellite, in its orbit is calculated. It is plain that this is not a very easy business. We have already explained, that the velocity of the planet in its orbit is not uniform, (being greatest when the planet's distance from the sun is least, or when the planet is at perihelion ;) and it is ob- vious, that the longitude of the planet increases very irregularly ; since, when the planet is near to 24 GRAVITATION. the sun, its actual motion is very rapid, and, there- fore, the increase of longitude is extremely rapid ; and when the planet is far from the sun, its actual motion is slow, and, therefore, the increase of longi- tude is extremely slow. The rule which is demon- strated by theory, and which is found to apply precisely in observation, is this : The areas de- scribed by the radius vector are equal in equal times. This is true, whether the force be inversely as the square of the distance from the central body, or be in any other proportion, provided that it is directed to the central body. (30.) Thus, if in one day a planet, or a satellite, moves from A to a, fig, 5 ; in the next day it will Fig. 5. move from a to />, making the area a S b equal to A S a ; in the third day it will move from b to c, making the area b S c equal to A S a or a S 6, and so on. (31.) Upon this principle mathematicians have invented methods of calculating the place of a pla- net, or satellite, at any time for which it may be EQUATION OF THE CENTRE. 25 required. These methods are too troublesome for us to explain here ; but we may point out the mean- ing of two terms which are frequently used in these computations. Suppose, for instance, as in the figure, that the planet, or satellite,, occupies ten days in describing the half of its orbit, Aabcdefghi B, or twenty days in describing the whole orbit ; and suppose that we wished to find its place at the end of three days after leaving the perihelion. If the orbit were a circle, the planet would in three days have moved through an angle of 54 degrees. If the excentricity of the orbit were small, (that is, if the orbit did not differ much from a circle,) the angle through which the planet would have moved would not differ much from fifty-four degrees. The excentricities of all the orbits of the planets are small ; and it is convenient, therefore, to begin with the angle 54 as one which is not very erro- neous, but which will require some correction. This angle (as 54), which is proportional to the time, is called the mean anomaly ; and the cor- rection which it requires, in order to produce the true anomaly, is called the equation of the centre. If we examine the nature of the motion, while the planet moves from A to B, it will readily be seen, that, during the whole of that time, the angle really c 26 GRAVITATION. described by the planet is greater than the angle .which is proportional to the time, or the equation of the centre is to be added to the mean anomaly, in order to produce the true anomaly ; but while the planet moves in the other half of the orbit, from B to A, the angle really described by the planet is less than the angle which is proportional to the time, or the equation of the centre is to be subtracted from the mean anomaly, in order to pro- duce the true anomaly. (32.) The sum of the mean anomaly and the longitude of perihelion is called the mean longitude of the planet. It is evident, that if we add the equation of the centre to the mean longitude, while the planet is moving from A to B, or sub- tract it from the mean longitude, while the planet is moving from B to A, as in (31.), we shall form the true longitude. (33.) The reader will see, that when the planet's true anomaly is calculated, the length of the radius vector can be computed from a know- ledge of the properties of the ellipse. Thus the place of the planet, for any time, is perfectly known. This problem has acquired considerable celebrity under the name of Kepler *s problem. (34.) There remains only one point to be ex- PERIODIC TIME DEPENDS ON MEAN DISTANCE. 27 plained regarding the undisturbed motion of planets and satellites; namely, the relation be- tween a planet 's periodic time and the dimensions of the orbit in which it moves. Now, on the law of gravitation it has been de- monstrated from theory, and it is fully confirmed by observation, that the periodic time does not depend on the excentricity, or on the perihelion distance, or on the aphelion distance, or on any element except the mean distance or semi-major axis. So that if two planets moved round the sun, one in a circle, or in an orbit nearly circu- lar, and the other in a very flat ellipse ; provided their mean distances were equal, their periodic times would be equal. It is demonstrated also, that for planets at different distances, the relation between the periodic times and the mean distances is the following: The squares of the numbers of days (or hours, or minutes, &c.) in the periodic times have the same proportion as the cubes of the numbers of miles, (or feet, &c.) in the mean distances. (35.) Thus the periodic time of Jupiter round the sun is 4332 -J days, and that of Saturn is 10759-2 days ; the squares of these numbers are 18772289 and 1 15760385. The mean distance of c 2 28 GRAVITATION. Jupiter from the sun is about 487491000 miles, and that of Saturn is about 893955000 miles ; the cubes of these numbers are 1158496 (20 ci- phers), and 7144088 (20 ciphers). On trial it will be found, that 18772289 and 115760385 are in almost exactly the same proportion as 1158496 and 7144088. (36.) In like manner, the periodic times of Jupiter's third and fourth satellites round Jupiter are 7-15455 and 16-68877 days; the squares of these numbers are 51-1876 and 278-515. Their mean distances from Jupiter are 670080 and 1178560 miles; the cubes of these numbers are 300866 (12 ciphers), and 1637029 (12 ciphers), and the proportion of 51-1876 to 278-515 is almost exactly the same as the proportion of 300866 to 1637029. (37.) It must, however, be observed that this rule applies in comparing the periodic times and mean distances, only of bodies which revolve round the same central body. Thus the rules applies in comparing the periodic times and mean distances of Jupiter arid Saturn, because they both revolve round the sun; it applies in comparing the pe- riodic times and mean distances of Jupiter's third and fourth satellites, because they both revolve PERIODIC TIMES ROUND DIFFERENT CENTRES. 29 round Jupiter ; but it would not apply in com- paring the periodic time and mean distance of Saturn revolving round the sun with that of Jupi- ter's third satellite revolving round Jupiter. (38.) In comparing the orbits described by different planets, or satellites, round different cen- tres of force, theory gives us the following law : The cubes of the mean distances are in the same proportion as the products of the mass by the square of the periodic time. Thus, for instance, the mean distance of Jupiter's fourth satellite from Jupiter is 1 1 78560 miles ; its periodic time round Jupiter is 16-68877 days; the mean distance of the earth from the sun is 93726900 miles ; its pe- riodic time round the sun is 365 % 2564 days; also the mass of Jupiter is -jVy^-th the sun's mass. The cubes of the mean distances are respectively 1637029 (12 ciphers), and 823365 (18 ciphers); the products of the squares of the times by the masses are respectively 0-265252 and 133412; and these numbers are in the same proportion as 1637029 (12 ciphers), and 823365 (18 ciphers). (39.) The three rules, that planets move in ellipses ; that the radius vector in each orbit passes over areas proportional to the times, and that the squares of the periodic times are proportional to 30 GRAVITATION. the cubes of the mean distances, are commonly called Kepler's laws. They were discovered by Kepler from observation, before the theory of gra- vitation was invented; they were first explained from the theory by Newton, about A.D. 1680. (40.) The last of these is not strictly true, un- less we suppose that the central body is absolutely immoveable. This, however, is evidently incon- sistent with the principles which we have laid down in Section I. In considering the motion, for instance, of Jupiter round the sun, it is necessary to consider, that, while the sun attracts Jupiter, Jupi- ter is also attracting the sun. But the planets are so small in comparison with the sun, (the largest of them, Jupiter, having less than one-thousandth part of the matter contained in the sun,) that in common illustrations there is no need to take this consideration into account. For nice astronomical purposes it is taken into account in the following manner: The motion which the attraction of Jupiter produces in the sun is less than the mo- tion which the attraction of the sun produces in Jupiter, in the same proportion in which Jupiter is smaller than the sun. If the sun and Jupiter were allowed to approach one another, their rate of approach would be the sum of the motions 31 of the sun and Jupiter, and would, therefore, be greater than their rate of approach, if the sun were not moveable, in the same proportion in which the sum of the masses of the sun and Ju- piter is greater than the sun's mass. That is, the rate of approach of the sun and Jupiter, both being free, is the same as the rate of approach would be if the sun were fixed, provided the sun's mass were increased by adding Jupiter's mass to it. Consequently, in comparing the orbits de- scribed by different planets round the sun, we must use the rule just laid down, supposing the central force to be the attraction of a mass equal to the sum of the sun and the planet ; and thus we get a proportion which is rigorously true :, for different planets, or even for different bodies, revolving round different centres of force, the cubes of the mean distances are in the same pro- portion as the products of the square of the pe- riodic time by the sum of the masses of the attract- ing and attracted body. 32 GRAVITATION. Section III. General Notions of Perturbation ; and Perturbation of the Elements of Orbits. (41.) WE have spoken of the motion of two bodies (as the sun and a planet) as if no other attracting body existed. But, as we have stated in Section I., every planet and every satellite attracts the sun and every other planet and satellite. It is plain now that, as each planet is attracted very differ- ently at different times by the other planets whose position is perpetually varying, the motion is no longer the same as if it was only attracted by the sun. The planets, therefore, do not move exactly in ellipses ; the radius vector of each planet does not pass over areas exactly proportional to the times ; and the proportion of the cube of the mean distance to the product of the square of the periodic time by the sum of the masses of the sun and the planet, is not strictly the same for all. Still the disturbing forces of the other planets are so small in com- parison with the attraction of the sun, that these laws are very nearly true; and (except for our moon and the other satellites) it is only by accu- rate observation, continued for some years^ that the effects of perturbation can be made sensible. VARIATION OF ELEMENTS. 33 (42.) The investigation of the effects of the dis- turbing forces will consist of two parts: the ex- amination into the effects of disturbing forces generally upon the motion of a planet, and the examination into the kind of disturbing force which the attraction of another planet produces. We shall commence with the former ; we shall suppose that a planet is revolving round the sun, the sun being fixed, (a supposition made only for present convenience,) and that some force acts on the planet without acting on the sun, (a restriction introduced only for convenience, and which we shall hereafter get rid of.) (43.) The principle upon which we shall ex- plain the effect of this force is that known to ma- thematicians by the name of variation of elements. The planet, as we have said, describes some curve which is not strictly an ellipse, or, indeed, any regularly formed curve. It will not even describe the same curve in successive revolutions. Yet its motion may be represented by supposing it to have moved in an ellipse, provided we suppose the ele- ments of the ellipse to have been perpetually alter- ing. It is plain that by this contrivance any mo- tion whatever may be represented. By altering the c5 34 . - GRAVITATION. major axis, the excentricity, and the longitude of perihelion, we may in many different ways make an ellipse that will pass through any place of the planet ; and by altering them in some particular proportions, we may, in several ways, make an ellipse in which the direction of motion at the place of the planet shall be the same as the direction of the planet's motion. But there is only one ellipse which will pass exactly through a place of the planet, in which the direction of the motion at that place shall be exactly the same as the direction of the planet's motion, and in which the velocity (in order that a body may revolve in that ellipse round the sun) will be the same as the planet's real velo- city. The dimensions and position of this ellipse may be conceived as follows : if at any instant we suppose the disturbing force to cease, and conceive the planet to be as it were projected with the velo- city which it happens to have at that instant, the attraction of the sun or central body will cause it to describe the ellipse of which we are speaking. We shall in future mention this by the name of the instantaneous ellipse. (44.) If the disturbing force ceases, the planet continues to revolve in the same ellipse, and the DISTURBING FORCE DIRECTED TO CENTRE. 35 permanent ellipse coincides with the instantaneous ellipse corresponding to the instant when the dis- turbing force ceases. (45.) If the disturbing force continues to act, the dimensions of the instantaneous ellipse are continually changing ; but in the course of a single revolution, (even for our moon,) the dimensions alter so little, that the motion in the instantaneous ellipse corresponding to any instant during that revolution will very nearly agree with the real motion during that revolution. We shall now consider the effects of particular forces in altering the elements. (46.) (I.) Suppose that the disturbing force is always directed to the central body. The effect of this would be nearly the same as if the attrac- tion or the mass of the central body was in- creased. The result of this on the dimensions of the orbit will be different according to the part of the orbit where it begins to act, and may be gathered from the cases to be mentioned sepa- rately hereafter, (we do not insist on it at pre- sent, as there is no instance in the planetary system of such sudden commencement of force.) But at all events the relation between the mean 36 GRAVITATION. distance and the periodic time will not be the same as before ; the time will be less for the same mean distance, or the mean distance greater for the same periodic time, than if the disturbing force did not act (38.). If the dis- turbing force is always directed from the central body, the effect will be exactly opposite. If the disturbing force does not alter, except with the planet's distance, the planet will at every suc- cessive revolution describe an orbit of the same size. For, as we have stated, (29.) the radius vector will in equal times pass over equal areas, and mathematicians have proved that, if the variation of force depends only on the distance, the velocity of the planet will depend only on the distance ; and the consideration which deter- mines the greatest or least distance of the planet is, that the planet, moving with the velocity which is proper to the distance, cannot describe the proper area in a short time, unless it move in the direction perpendicular to the radius vec- tor. This consideration will evidently give the same values for the greatest and least distances at every revolution. It may happen that all the greatest distances will not be at the same place ; the body may describe such an orbit as that in fig. 6. DISTURBING FORCE DIRECTED TO CENTRE. 37 F*g. 6. (47.) (II.) If, however, the disturbing force di- rected to the central body increases gradually and constantly during many revolutions, there is no difficulty in seeing that the planet will at every revolution be drawn nearer to the central body, and thus it will move, at every succeeding revolution, in a smaller orbit than at the preced- ing one ; and will consequently perform its revo- lution in a shorter time. If the disturbing force directed to the central body diminishes, the orbit will become larger, and the periodic time longer. In the same manner, if the disturbing force is directed from the central body, a gradual in- crease of the disturbing force will increase the dimensions of the orbit and the periodic time, and a gradual diminution of the disturbing force will diminish the dimensions of the orbit and the periodic time. (48.) (III.) Suppose that the disturbing force acts always in the direction in which the planet is 40 GRAVITATION. distance has been altered in the proportion of 10000: 10001, the periodic time will have been altered in the proportion of 10000 : 10001 J nearly, or the mean motion will have been altered in the proportion of 10001 J to 10000 or 1 : 0*99985 nearly. If this alteration has gone on uniformly, we may suppose the whole motion in the 100 revolutions to have been nearly the same as if the planet had moved with a mean motion, whose value is half way between the values of the first and the last, or 0-999925 x the original mean motion. Therefore, at the time when we should expect the planet to have made 100 revolutions, it will only have made 99-9925 revolutions, or will be behind the place where we expected to see it by 0*0075 revolution, or nearly three degrees; a quantity which could not fail to be noticed by the coarsest observer. To use a borrowed illustration, the alteration of the mean distance in an orbit produces the same kind of effect as the alteration of the length of a clock pendulum : which, though so small as to be insensible to the eye, will, in a few days, produce a very great effect on the time shown by the clock. (50.) (V.) Now suppose the orbit of the planet FORCE DIRECTED TO CENTRE NEAR PERIHELION. 41 or satellite to be an ellipse ; and suppose a dis- turbing force directed to the central body to act upon the planet, &c. only when it is near its perihelion or perigee, &c. In fig. 7, let A B be .-"^^c -.. d the curve in which the planet is moving, and let the dotted line B C D A represent the orbit in which it would have moved if no disturbing force had acted, C being the place of perihelion. At B let the disturbing force, directed towards S, begin to act, and let it act for a little while and then cease. The planet is at that place approaching toward the sun, and the direction of its motion makes an acute angle with SB. It is evident that the disturbing force, which draws the planet more rapidly towards the sun without otherwise affecting its motion, will cause it to move in a direction that makes a more acute angle with S B. The part of the new path, therefore, which is nearest to the sun (that is, the new perihelion) will be farther from B than the perihelion C of the orbit in which the planet would have moved. 40 GRAVITATION. distance has been altered in the proportion of 10000:10001, the periodic time will have been altered in the proportion of 10000 : 10001J nearly, or the mean motion will have been altered in the proportion of 10001 J to 10000 or 1 : 0'99985 nearly. If this alteration has gone on uniformly, we may suppose the whole motion in the 100 revolutions to have been nearly the same as if the planet had moved with a mean motion, whose value is half way between the values of the first and the last, or 0-999925 x the original mean motion. Therefore, at the time when we should expect the planet to have made 100 revolutions, it will only have made 99*9925 revolutions, or will be behind the place where we expected to see it by 0*0075 revolution, or nearly three degrees; a quantity which could not fail to be noticed by the coarsest observer. To use a borrowed illustration, the alteration of the mean distance in an orbit produces the same kind of effect as the alteration of the length of a clock pendulum : which, though so small as to be insensible to the eye, will, in a few days, produce a very great effect on the time shown by the clock. (50.) (V.) Now suppose the orbit of the planet FORCE DIRECTED TO CENTRE NEAR PERIHELION. 41 or satellite to be an ellipse ; and suppose a dis- turbing force directed to the central body to act upon the planet, &c. only when it is near its perihelion or perigee, &c. In/ or 70 GRAVITATION. nearly ilo-th part of the sun's attraction on the earth. Thus, on doubling the moon's distance from the earth, the disturbing force is nearly doubled : and in the same manner, on altering the distance in any other proportion, we should find that the disturbing force is altered in nearly the same proportion. (83.) VI. If, while the moon's distance from the earth is not sensibly altered, the earth's dis- tance from the sun is altered, the disturbing force is diminished very nearly in the same ratio in which the cube of the sun's distance is increased. For if the sun's distance is 400 times the moon's distance, and the moon between the earth and the sun, we have seen that the dis- turbing force is nearly. Tiro-th part of the sun's attraction on the earth at that distance of the sun. Now, suppose the sun's distance from the earth to be made 800 times the moon's distance, or twice the former distance : the sun's distance from the moon will be 799 times the moon's dis- tance, or Iff parts of the sun's former distance from the earth ; the attractions on the earth and moon respectively will be ^ and -^-glfr parts of the former attraction on the earth : and the dis- turbing force, or the difference between these, will FORCE PERPENDICULAR TO RADIUS. 71 be TTTTO-T^ or nearly -nnrirth part of the former attraction of the earth. Thus, on doubling the sun's distance, the disturbing force is diminished to 4-th part of its former value ; and a similar proposition would be found to be true if the sun's distance were altered in any other pro- portion. (84.) VII. Suppose B to have moved from that part of its orbit where its distance from C is equal to A's distance from C, towards the part where it is between A and C. Since at the point where B's distance from C is equal to A's dis- tance from C, the disturbing force is in the di- rection of the radius vector, and directed towards A, and since at the point where B is 'between A and C, the disturbing force is in the direction of the radius vector, but directed/rom A, it is plain that there is some situation of B, between these two points, in which there is no disturbing force at all in the direction of the radius vector. On this we shall not at present speak further : but we shall remark that there is a disturbing force perpendicular to the radius vector, at every such intermediate point. This will be easily seen from the second case of fig. 17. On going through the reasoning in that place it will ap- 72 GRAVITATION. pear that, between the two points that we have mentioned, there is always a disturbing force c? 2 e z perpendicular to the radius vector, and in the same direction in which the body is going. If now we construct a similar figure for the situa- tion fS^fig. 22., in which B is moving from the C a A Fig. 22. dl B4 point between C and A to the other point whose distance from C is equal to A's distance from C, we shall find that there is a disturbing force d 3 e 3 perpendicular to the radius vector, in the direc- tion opposite to that in which B is going. If we construct a figure for the situation B 4 in which B is moving from the point of equal distances, to the point where B is on the side of A opposite to C, we shall see that there is a disturbing force perpendicular to the radius vector, in the same direction in which B is going ; and in the same manner, for the situation B t in fig. 17. where B is moving from the point on the side of A oppo- site C to the next point of equal distances, there is a disturbing force perpendicular to the radius GENERAL VIEW OF RESULTS. 73 vector, in the direction opposite to that in which B is going. (85.) The results of all these cases may be col- lected thus. The disturbing body being exterior to the orbit of the revolving body, there is a dis- turbing force in the direction of the radius vector only, directed from the central body, at the points where the revolving body is on the same side of the central body as the disturbing body, or on the opposite side, (the force in the former case being the greater,) and directed to the central body, at each of the places where the distance from the dis- turbing body is equal to the distance of the central body from the disturbing body. The force directed to the central body at the latter points, is however much less than the force directed from it at the former. Between the adjacent pairs of these four points there are four other points, at which the disturbing force in the direction of the radius vector is nothing. But while the revolving body is moving from one of the points, where it is on the same side of the central body as the disturbing body, or on the opposite side, to one of the equi- distant points, there is always a disturbing force perpendicular to the radius vector tending to re- 74 GRAVITATION. tarcl it ; and while it is moving from one of the equi-distant points to one of the points on the same side of the central body as the disturbing body, or the opposite, there is a disturbing force perpendi- cular to the radius vector tending to accelerate it. (86.) VIII. Now, let the disturbing body be supposed interior to the orbit of the revolving body, (as, for instance, when Venus disturbs the motion of the earth.) If B is in the situation &i,fig. 23, the attraction of C draws A strongly towards B : , and Bi strongly towards A, and, Fig. 23. / \ B1 therefore, there is a very powerful disturbing force drawing B.,, towards A. If B is in the situation B 3 , the attraction of C draws A strongly from B 3 , and draws B 3 feebly towards A; therefore, there is a small disturbing force drawing B 3 from A. At some intermediate points the disturbing force in the direction of the radius vector is nothing. With regard to DISTURBING BODY INTERIOR TO THE ORBIT. 75 the disturbing force perpendicular to the radius vector: if A C is greater than ^AB X , it will be possible to find two points,, B 2 and B 4 , whose distance from C is equal to the distance of A from C, and there the disturbing force per- pendicular to the radius vector is nothing (or the whole disturbing force is in the direction of the radius vector). While B moves from the position B x to B , it will be seen by such reason- ing as that of (75.) and (84.), that the disturb- ing force, perpendicular to the radius vector, retards B's motion ; while B moves from B to B 3 , it accelerates B's motion; while B moves from B 3 to B 4 it retards B's motion; and while B moves from B 4 to B 1 , it accelerates B's mo- tion. But if AC is less than \ A B 1? there are no such points, B 2 B 4 , as we have spoken of; and the disturbing force, perpendicular to the radius vector, accelerates B as it moves from B l to B 2 , and retards B as it moves from B.toB,. We shall now proceed to apply these general prin- ciples to particular cases. 76 GRAVITATION. SECTION V. Lunar Theory. (87.) THE distinguishing feature in the Lunar Theory is the general simplicity occasioned by the great distance of the disturbing body (the sun alone producing any sensible disturbance), in pro- portion to the moon's distance from the earth. The magnitude of the disturbing body renders these disturbances very much more conspicuous than any others in the solar system ; and, on this account, as well as for the accuracy with which they can be observed, these disturbances have, since the invention of the Theory of Gravitation, been considered the best tests of the truth of the theory. Some of the disturbances are independent of the excentricity of the moon's orbit; others depend, in a very remarkable manner, upon the excentricity. We shall commence with the former. (88.) The general nature of the disturbing force on the moon may be thus stated. (See (77.) to (86.) ) When the moon is either at the point be- tween the earth and sun, or at that opposite to the sun (both which points are called syzygies), the force is entirely in the direction of the radius vec- MOON'S MEAN DISTANCE IS DIMINISHED. 77 tor, and directed from the earth. When the moon is (very nearly) in the situations at which the radius vector is perpendicular to the line joining the earth and sun (both which points are called quadra- tures), the force is entirely in the direction of the radius vector, and directed to the earth. At cer- tain intermediate points there is no disturbing force in the direction of the radius vector. Except at syzygies and quadratures, there is always a force perpendicular to the radius vector, such as to retard the moon while she goes from syzygy to quadrature, and to accelerate her while she goes from quadrature to syzygy. (89.) I. As the disturbing force, in the direction of the radius vector, directed from the earth, is greater than that directed to the earth, we may consider that, upon the whole, the effect of the disturbing force is to dimmish the earth's attrac- tion. Thus the moon's mean distance from the earth is less (see (46.) ) than it would have been with the same periodic time, if the sun had not disturbed it. The force perpendicular to the radius vector sometimes accelerates the moon, and sometimes retards it, and, therefore, pro- duces no permanent effect. 78 GRAVITATION. (90.) II. But the sun's distance from the earth is subject to alteration, because the earth revolves in an elliptic orbit round the sun. Now, we have seen (83.) that the magnitude of the dis- turbing force is inversely proportional to the cube of the sun's distance ; and, consequently, it is sensibly greater when the earth is at perihe- lion than when at aphelion. Therefore, while the earth moves from perihelion to aphelion, the disturbing force is continually diminishing; and while it moves from aphelion to perihelion, the disturbing force is constantly increasing. Re- ferring then to (47.) it will be seen, that in the former of these times the moon's orbit is gra- dually diminishing, and that in the latter it is gradually enlarging. And though this altera- tion is not great (the whole variation of dimen- sions, from greatest to least, being less than -g^Vg-), the effect on the angular motion (see (49.) ) is very considerable ; the angular velocity becoming quicker in the former time and slower in the latter; so that while the earth moves from perihelion to aphelion, the moon's angular motion is constantly becoming quicker, and while the earth moves from aphelion to perihelion the moon's angular motion is constantly becoming- MOON'S ANNUAL EQUATION. 79 slower. Now, if the moon's mean motion is determined by comparing two places observed at the interval of many years, the angular motion so found is a mean between the greatest and least. Therefore, when the earth is at perihe- lion, the moon's angular motion is slower than its mean motion ; and when the earth is at aphelion, the moon's angular motion is quicker than its mean motion. Consequently, while the earth is going from perihelion to aphelion, the moon's true place is always behind its mean place (as during the first half of that period the moon's true place is dropping behind the mean place, and during the latter half is gaining again the quantity which it had dropped behind) ; and while the earth is going from aphelion to perihelion, the moon's true place is always be- fore its mean place. This inequality is called the moon's annual equation ; it was discovered by Tycho Brahe from observation, about A.D. 1590; and its greatest value is about 10', by which the true place is sometimes before and sometimes behind the mean place. (91.) III. The disturbances which are periodical in every revolution of the moon, and are inde- pendent of excentricity, may thus be investi- 80 GRAVITATION. gated. Suppose the sun to stand still for a few revolutions of the moon (or rather suppose the earth to be stationary,) and let us inquire in what kind of orbit, symmetrical on opposite sides, the sun can move. It cannot move in a circle : for the force perpendicular to the radius vector retards the moon as it goes from B A to B , fig. 24, and its velocity is, therefore, less at B 2 than at B x , and on this account (sup- B4 Fig. 24. s S. 2 B/ jBa Ba posing the force directed to A at B equal to the force directed to A at B,,) the orbit would be more curved at B 2 than at B 1 . But the force directed to A at B is much greater than that at Bj (see (88.) ) ; and on this account the orbit would be still more curved at B 2 than at B 1 ; whereas, in a circle, the curvature is every where the same. The orbit cannot, therefore, be cir- cular. Neither can it be an oval with the earth in its centre, and with its longer axis passing through the sun, as fig. 25 ; for the velocity being small at B (in consequence of the dis- turbing force perpendicular ta the radius vector MOON'S VARIATION. 81 having retarded it,) while the earth's attrac- tion is great (in consequence of the nearness of Fig. 25. B B 3 Ba B ), and increased by the disturbing force in the radius vector directed towards the earth, the curvature at B 2 ought to be much greater than at Bj, where the velocity is great, the moon far off, and the disturbing force directed from the earth. But, on the contrary, the curvature at B 2 is much less than at B x ; therefore, this form of orbit is not the true one. But if the orbit be supposed to be oval, with its shorter axis directed towards the sun, as in fig. 26, all the conditions will be satisfied. For the velocity at B 2 is diminished by the disturbing force having B4 Fi ff .2G. f^\ 2 Bit * JBs B2 acted perpendicularly to the radius vector, while the moon goes from B 1 to B 2 ; and though the distance from A being greater, the earth's at- traction at B 2 will be less than the attraction at Bj ; yet, when increased by the disturbing E5 82 GRAVITATION. force, directed to A at B 2 , it will be very little less than the attraction diminished by the disturb- ing force at B x . The diminution of velocity then at B being considerable, and the diminution of force small, the curvature will be increased ; and this increase of curvature, by proper choice of the proportions of the oval, may be precisely such as corresponds to the real difference of cur- vature in the different parts of the oval. Hence, such an oval may be described by the moon without alteration in successive revolutions. (92.) We have here supposed the earth to be stationary with respect to the sun. If, however, we take the true case of the earth moving round the sun, or the sun appearing to move round the earth, we have only to suppose that the oval twists round after the sun, and the same reasoning applies. The curve described by the moon is then such as is represented in fig. 27. As the disturb- Fi. 27. ing force, perpendicular to the radius vector, acts in the same direction for a longer time than in the S3 former case,, the difference in the velocity at syzygies and at quadratures is greater than in the former case, and this will require the oval to differ from a circle, rather more than if the sun be supposed to stand still. (93.) If, now, in such an orbit as we have men- tioned, the law of uniform description of areas by the radius vector were followed, as it would be if there were no force perpendicular to the radius vector, the angular motion of the moon near B tT and B^fig. 26, would be much less than that near B x and B . But in consequence of the disturbing force, perpendicular to the radius vector, (which retards the moon from B L to B 2 , and from B 3 to B 4 , and accelerates it from B 2 to B 3 , and from B 4 to B 1 ,) the angular motion is still less at B^ and B 4 , and still greater at B[ and B 3 . The angular motion, therefore, diminishes considerably while the moon moves from B t to B 2 , and in- creases considerably while it moves from B to B 3 , &c. The mean angular motion, determined by observation, is less than the former and greater than the latter. Consequently, the angular motion at Bj is greater than the mean, and that at B 2 is less than the mean; and, therefore, (as in (90.),) from B x to B 2 the moon's true place is before the 84 GRAVITATION. mean; from B 2 to B 3 the true place is behind the mean; from B 3 to B 4 the true place is before the mean; and from B 4 to B t the true place is be- hind the mean. This inequality is called the moon's variation ; it amounts to about 32', by which the moon's true place is sometimes before and sometimes behind the mean place. It was discovered by Tycho, from observation, about A.D. 1590. (94.) We have, however, mentioned, in (79.), that the disturbing forces are not exactly equal on the side of the orbit which is next the sun, and on that which is farthest from the sun; the former being rather greater. To take account of the effects of this difference, let us suppose, that in the investigation just finished, we use a mean value of the disturbing force. Then we must, to represent the real case, suppose the disturbing force near con- junction to be increased, and that near opposition to be diminished. Observing what the nature of these forces is, (77.), (78.), and (84.), this amounts to supposing that near conjunction the force ne- cessary to make up the difference is a force acting in the radius vector, and directed from the earth, and a force perpendicular to the radius vector, accelerating the moon before conjunction, and MOON'S PARALLACTIC EQUATION. 85 retarding her after it, and that near opposition the forces are exactly of the contrary kind. Let us then lay aside the consideration of all other dis- turbing forces, and consider the inequality which these forces alone will produce. As they are very small., they will not in one revolution alter the orbit sensibly from an elliptic form. What then must be the excentricity, and what the position of the line of apses that, with these disturbing forces only, the same kind of orbit may always be de- scribed ? A very little consideration of (57.), (58.), and (68.), will show, that unless the line of apses pass through the sun, the excentricity will either be increasing or diminishing from the action of these forces. We must assume, therefore, as our orbit is to have the same excentricity at each revo- lution, that the line of apses passes through the sun. But is the perigee or the apogee to be turned towards the sun ? To answer this question we have only to observe, that the line of apses must progress as fast as the sun appears to pro- gress, and we must, therefore, choose that position in which the forces will cause progression of the line of apses. If the perigee be directed to the sun, then the forces at both parts of the orbit will, by (51.), (54.), (65.), and (66.), cause the line 86 GRAVITATION. of apses to regress. This supposition,, then, can- not be admitted. But if the apogee be directed to the sun, the forces at both parts of the orbit will cause it to progress; and by (56.), if a proper value is given to the excentricity it will progress exactly as fast as the sun appears to progress. The effect., then, of the difference of forces, of which we have spoken, is to elongate the orbit towards the sun, and to compress it on the op- posite side. This irregularity is called the paral- lactic inequality. We shall shortly show, that if the moon revolved in such an elliptic orbit as we have mentioned, the effect of the other disturbing forces (inde- pendent of that discussed here) would be to make its line of apses progress with a considerable velo- city. The force considered here, therefore, will merely have to cause a progression which, added to that just mentioned, will equal the sun's appa- rent motion round the earth. The excentricity of the ellipse, in which it could produce this smaller motion, will (56.) be greater than that of the ellipse in which the same force could produce the whole motion. Thus the magnitude of the paral- lactic inequality is considerably increased by the indirect effect of the other disturbing forces. MOON'S PARALLACTIC EQUATION. 87 (05.) The magnitude of the forces concerned here is about yl-g-th of those concerned in (91.), &c. ; but the effect is about ^th of their effect. This is a striking instance of the difference of propor- tions in forces,, and the effects that they produce, depending on the difference in their modes of action. The inequality here discussed is a very interesting one, from the circumstance that it enables us to determine with considerable accu- racy the proportion of the sun's distance to the moon's distance, which none of the others will do, as it is found upon calculation, that their magni- tude depends upon nothing but the excentricities and the proportion of the periodic times, all which are known without knowing the proportion of distances. (96.) The effect of this, it will be readily un- derstood, is to be combined with that already found. See the note to (134.) The moon's orbit, therefore, is more flattened on the side farthest from the sun, and less flattened on the side next the sun, than we found in (91.) and (92.) The equable description of areas is scarcely affected by these forces. The moon's variation, therefore, is somewhat diminished near conjunction, and is somewhat increased near opposition. 88 GRAVITATION. (97.) It will easily be imagined, that if there is an excentricity in the moon's orbit, the effect of the variation upon that orbit will be almost ex- actly the same as if there were no excentricity*. * As this general proportion is of considerable importance, we shall point out the nature of the reasoning by which (with proper alteration for different cases,) the reader may satisfy himself of its correctness. The reason why, infg. 29, the moon cannot de- scribe the circle B 1? b 2 , B 3 , 6 4 , though it touches at Bj and B 3 , and the reason that it will describe the oval B,, B 2 , B 3 , B 4 , is, Fig. 29. Pz that the disturbing force makes the forces at B x and B 3 less than they would otherwise have been, and greater at B 2 and B 4 than they would otherwise have been ; and the velocity is, by that part of the force perpendicular to the radius vector, made less atB 2 , than it would otherwise have been. So that, unless we supposed it moving at B x with a greater velocity than it would have had, un- disturbed, in the circle B x , b 2 , B 3 , 5 4 , the great curvature pro- duced by the great force, and diminished velocity at B 2 , would have brought it much nearer to A than the point B 3 ; but with this large velocity at B lt it will go out farther at B 2 , and then the great curvature may make it pass exactly through B 3 . In like manner, in Jig. 30, if the velocity at B x were not B4 Fig. 30. B2 greater than it would have had, undisturbed, in the ellipse B x , 6 2 , B 3 , 6 4 , the increased curvature at B 2 , produced by the VARIATION COMBINED WITH EXCENTRICITY. 89 Thus,, supposing that the orbit without the dis- turbing force had such a form as the dark line in fig. 28, it will, with the disturbing force, have such Fig. 28. a form as the dotted line in that figure. The same must be understood in many other cases of dif- ferent inequalities which affect the motion of the same body. (98.) IV. We now proceed with the disturb- ances dependent on the excentricity : and, first, with the motion of the moon's perigee. In the first place, suppose that the perigee is on the same side as the sun. While the moon is near B u fig. 31, that is near perigee, the disturbing force increased force and diminished velocity there, would have brought it much near to A than the point B 3 ; but with a large velocity at B! it will go out at B further than it would otherwise have gone out, and then the increased force and diminished velocity will curve its course so much, that it may touch the elliptic orbit at B 3 ; and so on. The whole explanation, in one case as much as in the other, depends entirely upon the difference of the forces in the actual case, from the forces, if the moon were not disturbed. 90 GRAVITATION. is directed from A; and, consequently, by (51.), the line of apses regresses. While the moon is Fig. 3}. B 4 ~ B2 near B 3 , that is near apogee, the disturbing force is also directed from A, and, consequently, by (54.), the line of apses progresses. The question, then, now is, which is the greater, the regress, when the moon is near B t , or the pro- gress, when it is near B 3 ? To answer this, we will remark, that if the disturbing force directed from A, were inversely proportional to the square of the distance (and, consequently, less at B 3 than at B t ,) it would amount to exactly the same as if the attraction of A were altered in a given proportion* ; and in that case B would * The reasoning in the text may he more fully stated thus : If with the original attractive force of the earth there be combined another force, directed from the earth, and always bearing the same proportion to the earth's original attraction, this combined force may be considered in two ways : 1st, As a smaller attraction, always proportional to the original attraction, or inversely propor- tional to the square of the distance. 2d, As the original attrac- tion, with a force superadded, which may be treated as a disturb- ing force. The result of the first mode of consideration will be, that the moon will describe an ellipse, whose line of apses does not move. The result of the second mode of consideration will be, APSES IN SYZYGIES PROGRESS. 91 describe round A an ellipse, whose line of apses was invariable ; or the progression produced at B 3 would be equal to the regression produced at Bj. But, in fact, the disturbing force at B 3 is to that at B x in the same proportion as AB 3 to ABj, by (82.), and, therefore, the disturbing force at B 3 is greater than that at B l ; and, consequently, much greater than that which would produce a progression equal to the regression produced at B 1 ; and, therefore, tho effects of the disturbing force at B 3 predomi- nate, and the line of apses progresses. The that the instantaneous ellipse (in which the moon would proceed to move, if the additional force should cease) will have its line of apses regressing, while the moon is near perigee, and progressing wh-ile she is near apogee. There is, however, no incongruity be- tween the immobility of the line of apses in the first mode of consideration, and the progress or regress in the second; because the line of apses of the instantaneous ellipse in the second case, is an imaginary line, determined by supposing the disturbing force to cease, and the moon to move undisturbed. At the apses, however, the line of apses must be the same in both methods of consideration; since, whether the disturbing force cease or not, the perpendicularity of the direction of the motion to the radius vector determines the place of an apse. Consequently, while the moon moves from one apse to the other, the motions of the line of apses in the second mode of consideration, must be such as to pro- duce the same effect on the position of the line of apses as in the first mode of consideration ; that is, they must not have altered its place ; and hence the progression at one time must be exactly ecwal to the regression at the other time, 92 GRAVITATION. disturbing force directed to A in the neighbour- hood of B 2 and B 4 scarcely produces any effect, as on one side of each of those points the effect is of one kind, and on the other side it is of the op- posite kind, (55.) (99.) The disturbing force directed from A, though the only one at B! and B 3 , is not, however, the only one in the neighbourhood of B t and B 3 . While the moon is approaching to B l? the force perpen- dicular to the radius vector accelerates the moon, and therefore, by (65.), as B t is the place of perigee, the line of apses regresses ; when the moon has passed B!, the force retards the moon, and, there- fore, by (66.), the line of apses still regresses. But when the moon is approaching B 3 the force perpen- dicular to the radius vector accelerates the moon, and therefore, by (65.) and (66.) as B 3 is the place of apogee, the line of apses progresses : when the moon has passed B 3 the force retards the moon and the line of apses still progresses. The ques- tion now is, whether the progression produced by the force perpendicular to the radius vector near B a , will or will not exceed the regression produced near B x ? To answer this we must observe, that the rate of this progress or regress depends APSES IN SYZYGIES PROGRESS. 93 entirely upon the proportion * which the velocity produced by the disturbing force bears to the velo- city of the moon ; and since from B 2 to B b and from * Suppose, for facility of conception, that the force perpendicular to the radius vector, acts in only one place in each quadrant be- tween syzygies and quadratures. The portions of the orbit which are bisected by the line of syzygies will be described with greater velocity in consequence of this disturbance (abstracting all other causes) than the other portions. Now the curvature of any part of an orbit does not depend on the central force simply, or on the velocity, but on the relation between them ; so that the same curve may be described either by leaving the central force unaltered and increasing the velocity in a given proportion, or by diminishing the central force in a corresponding proportion, and leaving the velocity unaltered. Consequently, in the case before us. the same curve will be described as if, without alteration of velocity, the cen- tral force were diminished, while the moon passed through the portions bisected by the line of syzygies. If now the imaginary di- minution of central force were in the same proportion (that is, if the real increase of velocity were in the same proportion) at both syzy- gies, which here coincide with the apses, the regression of the line of apses produced at perigee, would be equal to the progression pro- duced at apogee. But the increase of velocity produced by the force perpendicular to the radius vector near apogee, is much greater than that near perigee. First, because the force is greater, in pro- portion to the distance. Second, because the time of describing a given small angle is greater in proportion to the square of the dis- tance ; so that the acceleration produced while the moon passes through a given angle, is proportional to the cube of the distance. Third, because the velocity, which is increased by this acceleration, is inversely proportional to the distance ; so that the ratio in which the velocity is increased is proportional to the fourth power of the distance. The effect at the greater distance, therefore, predomi- nates over that at the smaller distance ; and therefore, on the whole, the force perpendicular to the radius vector produces an effect similar to its apogeal effect ; that is, it causes the line of apses to progress. 94 GRAVITATION. B 3 to B 4 the disturbing force is greater than that from B 4 to B u and from Bj to B 2 , and acts for a longer time (as by the law of equable description of areas, the moon is longer moving from B 2 to B 3 and B 4 , than from B 4 to B! and B 2 ), and since the moon's velocity in passing through B 2 , B 3 , B 4 , is less than her velocity in passing through B 4 , B u B 2 , it follows that the effect in passing through B 2 , B 3 , B 4 , is much greater than that in passing through B 4 , B!, arid B 2 . Consequently, the effect of this force also is to make the line of apses pro- gress. (100.) On the whole, therefore, when the peri- gee is turned towards the sun, the line of apses progresses rapidly. And the same reasoning ap- plies in every respect when the perigee is turned from the sun. (101.) In the second place, suppose that the line of apses is perpendicular to the line joining the earth and sun. The disturbing force at both apses is now directed to the earth, and conse- quently, by (50.) and (53.), while the moon is near perigee, the disturbing force causes the line of apses to progress, and while the moon is near apogee the disturbing force causes the line of apses to regress. Here, as in the last article, the effects APSES IN QUADRATURES REGRESS. 95 at perigee and at apogee would balance if the dis- turbing force were inversely proportional to the square of the distance from the earth. But the disturbing force is really proportional to the dis- tance from the earth : and,, therefore, as in the last article, the effect of the disturbing force while the moon is at apogee preponderates over the other ; and therefore, the force directed to the centre causes the line of apses to regress. (102.) We must also consider the force perpen- dicular to the radius vector. In this instance, that force retards the moon while she is approaching to each apse, and accelerates her as she recedes from it. The effect is, that when the moon is near peri- gee the force causes the line of apses to progress, and when near apogee it causes the line of apses to regress (65.) and (66.) The latter is found to preponderate, by the same reasoning as that in (99.) From the effect, then, of both causes the line of apses regresses rapidly in this position of the line of apses. (103.) It is important to observe here, that the motion of the line of apses would not, as in (56.), be greater if the excentricity of the orbit were smaller. For though the motion of the line of apses is greater in proportion to the force which 96 GRAVITATION. causes it when the excentricity is smaller; yet, in the present instance, the force which causes it is itself proportional to the excentricity (inasmuch as it is the difference of the forces at perigee and apogee, which would be equal if there were no excentricity) : so that if the excentricity were made less, the force which causes the motion of the line of apses would also be made less, and the motion of the line of apses would be nearly the same as before. (104.) It appears then, that when the line of apses passes through the sun, the disturbing force causes that line to progress ; when the earth has moved round the sun, or the sun has appeared to move round the earth, so far that the line of apses is perpendicular to the line joining the sun and the earth, the line of apses regresses from the effect of the disturbing force; and at some intermediate position, it may easily be imagined that the force produces no effect on it. It becomes now a matter of great interest to inquire, whether upon the whole the progression exceeds the regression. Now the force perpendicular to the radius vector, considered in (99.), is almost, exactly equal to that considered in (102.); so that the progression produced by that force when the line of apses passes through the sun, is almost exactly equal to the regression which PROGRESSION OF THE MOON'S APSE. 97 it produces when the line of apses is perpendicular to the line joining the earth and sun ; and this force may, therefore, be considered as producing no effect (except indirectly, as will be hereafter mentioned.) But the force in the direction of the radius vector, tending from the earth in (98.), is, as we have mentioned in (80.), almost exactly double of that tending to the earth in (101.), and, therefore, its effect predominates : and, therefore, on the whole, the line of apses progresses. In fact, the progress, when the line of apses passes through the sun, is about 11 in each revolution of the moon; the regress, when the line of apses is per- pendicular to the line joining the earth and sun, is about 9 in each revolution of the moon. (105.) The progression of the line of apses of the moon is considerably greater than the first consideration would lead us to think, for the fol- lowing reasons. (106.) Firstly. The earth is revolving round the sun, or the sun appears to move round the earth, in the same direction in which the moon is going. This lengthens the time for which the sun acts in any one manner upon the moon, F 98 GRAVITATION. but it lengthens it more for the time in which the moon is moving slowly, than for that in which it is moving quickly. Thus ; suppose that the moon's angular motion when she is near perigee is fourteen times the sun's angular mo- tion : and when near apogee, only ten times the sun's motion. Then she passes the sun at the former time, (as seen from the earth,) with Ifths of her whole motion, but at the latter with only T Vths ; consequently, when near peri- gee, the time in which the moon passes through a given angle from the moving line of syzygies, (or the time in which the angle between the sun and moon increases by a given quantity,) is -rHhs of the time in which it would have passed through the same angle, had the sun been sta- tionary ; when near apogee, the number express- ing the proportion is Vths. The latter number is greater than the former ; and, therefore, the effect of the forces acting near apogee is in- creased in a greater proportion than that of the forces acting near perigee. And as the effective motion of the line of apses is produced by the excess of the apogeal effect above the perigeal effect, a very small addition to the former will 99 bear a considerable proportion to the effective motion previously found; and thus the effective motion will be sensibly increased. (107.) Secondly. When the line of apses is di- rected toward the sun, the whole effect of the force is to make it progress^ that is, to move in the same direction as the sun : the sun passes through about 27 in one revolution of the moon, and, therefore, departs only 16 from the line of apses ; and therefore the apse continues a long time near the sun. When at right angles to the line joining the earth and sun, the whole effect of the force is to make it regress, and therefore, moving in the direction opposite to the sun's motion, the angle between the sun and the line of apses is increased by 36 in each revolution, and the line of apses soon escapes from this position. The effect of the former force is there- fore increased, while that of the latter is dimi- nished : and the preponderance of the former is much increased. It is in increasing the ra- pidity of progress at one time, and the rapidity of regress at another, that the force perpendi- cular to the radius vector indirectly increases the effect of the former in the manner just de- scribed. F 2 100 GRAVITATION. (108.) From the combined effect of these two causes the actual progression of the line of apses is nearly doubled. (109.) The line of apses upon the whole, there- fore, progresses ; and (as calculation and observa- tion agree in showing) with an angular velocity that makes it (on the average) describe 3 in each revolution of the moon, and that carries it completely round in nearly nine years. But as it sometimes progresses and sometimes regresses for several months together, its motion is extremely irregular. The general motion of the line of apses has been known from the earliest ages of astro- nomy. (110.) V. For the alteration of the excentricity of the moon's orbit : first, let us consider the orbit in the position in which the line of apses passes through the sun, fig. 31. While the moon moves from B l (the perigee,) to B 3 , (the apogee,) the force in the direction of the radius vector is sometimes directed to the earth, and sometimes from the earth, and therefore, by (57.) and (59.), it sometimes diminishes the excentricity and sometimes increases it. But while the moon moves from B 3 to Bj, there are VARIATION OF MOON'S EXCENTRICITY. 101 exactly equal forces acting in the same manner at corresponding parts of the half-orbit, and these, by (58.), will produce effects exactly opposite. On the whole, therefore, the disturb- ing force in the direction of the radius vector produces no effect on the excentricity. The force perpendicular to the radius vector increases the moon's velocity when moving from B 4 to B lf and diminishes it when moving from E l to B 2 ; in moving, therefore, from B 4 to B w the excen- tricity is increased (65.), and in moving from B v to B 2 , it is as much diminished (66.). Simi- larly in moving from B a to B 3 , the excentricity is diminished, arid in moving from B 3 to B 4 , it is as much increased. This force, therefore, pro- duces no effect on the excentricity. On the whole, therefore, while the line of apses passes through the sun, the disturbing forces produce no effect on the excentricity of the moon's orbit. (111.) When the line of apses is perpendicular to the line joining the earth and sun, the same thing is true. Though the forces near perigee and near apogee are not now the same as in the last case, their effects on different sides of perihelion 102 GRAVITATION. and aphelion balance each other in the same way. (1 12.) But if the line of apses is inclined to the line joining the earth and sun,, as in fig. 32., the Fig. 32. effects of the forces do not balance. While the moon is near B 4 and near B 2 , the disturbing force in the radius vector is directed to the earth ; at B 4 therefore, (58.), as the moon is moving towards perigee, the excentricity is increased ; and at B 2 , as the moon is moving from perigee, the excen- tricity is diminished. From the slowness of the motion at B 2 , (which gives the disturbing force more time to produce its effects,) and the great- ness of the force, the effect at B 2 will preponder- ate, and the combined effects at B 2 and B 4 will di- minish the excentricity. This will appear from reasoning of the same kind as that in (98.). At B x and B 3 , the force in the radius vector is directed from the earth: at "B lt therefore, by (59.), as the moon is moving from perigee, the excentricity is increased, and at B 3 it is diminished: but from VARIATION OF MOON' S EXCENTRICITY. 103 the slowness of the motion at B 8 and the mag- nitude of the force, the effect at B 3 will prepon- derate, and the combined effects at BI and B 3 will diminish the excentricity. On the whole, there- fore, the force in the direction of the radius vector diminishes the excentricity. The force perpen- dicular to the radius vector retards the moon from B! to B 2 , but the first part of this motion may be considered near perigee, and the second near apo- gee, and, therefore, in the first part, it diminishes the excentricity, and in the second increases it ; and the whole effect from B t to B 2 is very small. Similarly the whole effect from B 3 to B 4 is very small. But from B 4 to B^ the force accelerates the moon, and therefore, by (68.), (the moon being near perigee) increases the excentricity; and from B 2 to B 3 , the force also accelerates the moon, arid by (68.) (the moon being near apogee) diminishes the excentricity ; and the effect is much * greater * To the reader who- is acquainted with Newton's 3rd section, the following demonstration of this point will be sufficient. Four times the reciprocal of the latus rectum is equal to the sum of the reciprocals of the apogeal and perigeal distances. The effect of an increase of velocity at perigee in a given proportion is to alter .the area described in a given time in the same proportion, and therefore, to alter the latus rectum in a corresponding proportion. Consequently an increase of velocity at perigee in a given propor- tion alters the reciprocal of the apogeal distance by a given quan- tity, and, therefore, alters the apogeal distance by a quantity nearly 104 GRAVITATION. (from the slowness of the moon and the greatness of the force) between B 2 and B 3 than between B 4 and BI, and therefore the combined effect of the forces in these two quadrants is to diminish the excentricity. On the whole, therefore, when the line of apses is inclined to the line joining the earth and sun, in such a manner that the moon passes the line of apses before passing the line joining the earth and proportional to the square of the apogeal distance ; and, therefore, the ratio of the alteration of apogeal distance to apogeal distance (on which the alteration of excentricity depends) is nearly propor- tional to the apogeal distance. Similarly, if the velocity at apogee is increased in a given proportion, the ratio of the alteration of peri- geal distance to perigeal distance (on which the alteration of excen- tricity depends) is nearly proportional to the perigeal distance. Thus if the velocity were increased in the same proportion at peri- gee and at apogee, the increase of excentricity at the former would be greater than the diminution at the latter, in the proportion of apogeal distance to perigeal distance. But in the case before us, the proportion of increase of velocity is much greater at apogee than at perigee. First, because the force is greater, (being in the same proportion as the distance.) Second, because the time in which the moon describes a given angle is greater, (being in the same pro- portion as the square of the distance,) so that the increase of velocity is in the proportion of the cube of the distance. Third, be- cause the actual velocity is less, (being inversely as the distance,) so that the ratio of the increase to the actual velocity is propor- tional to the fourth power of the distance. Combining this propor- tion with that above, the alterations of excentricity in the case be- fore us, produced by the forces acting at apogee and at perigee, are in the proportion of the cubes of the apogeal and perigeal distances respectively. VARIATION OF MOON'S EXCENTRICITY. 105 sun, the excentricity is diminished at every revo- lution of the moon. (113.) In the same manner it will appear that if the line of apses is so inclined that the moon passes the line of apses after passing the line join- ing the earth and sun, the excentricity is increased at every revolution of the moon. Here the force in the radius vector is directed to the earth, as the moon moves from perigee and from apogee : and is directed from the earth as the moon moves to peri- gee and to apogee ; which directions are just oppo- site to those in the case already considered. Also the force perpendicular to the radius vector retards the moon both near perigee and near apogee ; and this is opposite to the direction in the case already considered. On the whole, therefore, the excen- tricity is increased at every revolution of the moon. (114.) In every one of these cases the effect is exactly the same if the sun be supposed on the side of the moon's orbit, opposite to that repre- sented in the figure. F5 106 GRAVITATION. (115.) Now the earth moves round the sun, and the sun therefore appears to move round the earth in the order successively represented by the figs. 31, 32, and 33. Hence then; when the sun is in the line of the moon's apses, the excentricity does not alter (110.) ; after this it diminishes till the sun is seen at right angles to the line of apses (112.); then it does not alter (111.) : and after this it increases .till the sun reaches the line of apses on the other side. Consequently, the excen- tricity is greatest when the line of apses passes through the sun ; and is least when the line of apses is perpendicular to the line joining the earth and sun. The amount of this alteration in the excentricity of the moon's orbit is more than ^-th of the mean value of the excentricity; the excentricity being sometimes increased by this part, and sometimes as much diminished; so that the greatest and least excentricities are nearly in the proportion of 6 : 4 or 3 : 2. (116.) The principal inequalities in the moon's motion may therefore be stated thus : 1st. The elliptic inequality, or equation of the centre (31.), which would exist if it were not -disturbed... GENERAL VIEW .OF LUNAR INEQUALITIES. 107 2nd. The annual equation (90.), depending on the position of the earth in the earth's orbit. 3rd. The variation (93.), and parallactic inequality (94.), depending on the position of the moon with respect to the sun. 4th. The general progression of the moons perigee (104.) 5th. The irregularity in the motion of the perigee, depending on the position of the perigee with respect to the sun (109.) 6th. The alternate increase and diminution of the eccentricity, depending on the position of the perigee with respect to the sun (115.) These inequalities were first explained (some im- perfectly) by Newton, about A. D. 1680. (117.) The effects of the two last are combined into one called the evection. This is by far the largest of the inequalities affecting the moon's place : the moon's longitude is sometimes increased 1 15' and sometimes diminished as much by this inequality. It was discovered by Ptolemy, from observation, about A. D. 140. (118.) It will easily be imagined that we have here taken only the principal inequalities. There are many others, arising chiefly from small errors 108 GRAVITATION. in the suppositions that we have made. Some of these, it may easily be seen, will arise from varia- tions of force which we have already explained. Thus the difference of disturbing forces at conjunc- tion and at opposition, whose principal effect was discussed in (94.), will also produce a sensible inequality in the rate of progression of the line of apses, and in the dimensions of the moon's orbit. The alteration of disturbing force depending on the excentricity of the earth's orbit will cause an alteration in the magnitude of the variation and the ejection. The alteration of that part mentioned in (94.) produces a sensible effect depending on the angle made by the moon's radius vector with the earth's line of apses. All these, however, are very small : yet not so small but that, for astrono- mical purposes, it is necessary to take account of thirty or forty. (119.) There is, however, one inequality of great historical interest, affecting the moon's mo- tion, of which we may be able to give the reader a general idea. We have stated in (89.) that the effect of the disturbing force is, upon the whole, to diminish the moon's gravity to the earth : and in (90.) we have mentioned that this effect is greater when the earth is near perihelion, than when the ACCELERATION OF MOON'S MEAN MOTION. 109 earth is near aphelion. It is found, upon accurate investigation, that half the sum of the effects at perihelion and at aphelion is greater than the effect at mean distance, by a small quantity de- pending on the excentricity of the earth's orbit: and, consequently, the greater the excentricity (the mean distance being unaltered) the greater is the effect of the sun's disturbing force. Now, in the lapse of ages, the earth's mean distance is not sensibly altered by the disturbances which the planets produce in its motion ; but the excentricity of the earth's orbit is sensibly diminished, and has been diminishing for thousands of years. Conse- quently the effect of the sun in disturbing the moon has been gradually diminishing, and the gravity to the earth has therefore, on the whole, been gra- dually increasing. The size of the moon's orbit has therefore, gradually, (but insensibly,) dimi- nished (47.) : but the moon's place in its orbit has sensibly altered (49 ),-and the moon's angular motion has appeared to be perpetually quickened. This phenomenon was known to astronomers by the name of the acceleration of the moon's mean motion, before it was theoretically explained in 1787, by Laplace: on taking it into account, the oldest and the newest observations are equally well 110 GRAVITATION. represented by theory. The rate of progress of the moon's line of apses has, from the same cause, been somewhat diminished. SECTION VI. Theory of Jupiter's Satellites. (120.) JUPITER has four satellites revolving round him in the same manner in which the moon re- volves round the earth; and it might seem,, there- fore, that the theory of the irregularities in the motion of these satellites is similar to the theory of the irregularities in the moon's motion. But the fact is, that they are entirely different. The fourth satellite (or that revolving in the largest orbit) has a small irregularity analogous to the moon's variation, a small one similar to the evec- tion, and one similar to the annual equation : but the last of these amounts only to about two mi- nutes, and the other two are very much less. The corresponding inequalities in the motion of the other satellites are still smaller. But these satel- lites disturb each other's motions, to an amount and in a manner of which there is no other ex- ample in the solar system; and (as we shall after- wards mention) their motions are affected in a most remarkable degree by the shape of Jupiter. PERIODIC TIMES NEARLY AS ONE TO TWO. Ill (121.) The theory, however, of these satellites is much simplified by the following circumstances : First, that the disturbances produced by the sun may, except for the most accurate computations, be wholly neglected. Secondly, that the orbits of the two inner satellites have no excentricity in- dependent of perturbation. Thirdly, that a very remarkable relation exists (and, as we shall show, necessarily exists) between the motions of the three first satellites. Before proceeding with the theory of the first three satellites, we shall consider a general pro- position which applies to each of them. (122.) Suppose that two small satellites re- volve round the same planet ; and that the periodic time of the second is a very little greater than double the periodic time of the first ; what is the, form of the orbit in which each can revolve, de- scribing a curve of the same form at every revo- lution ? (123.) The orbits will be sensibly elliptical, as the perturbation produced by a small satellite in one revolution will not sensibly alter the form of the orbit. The same form being supposed to be described each time, the major axis and the ex- centricity are supposed invariable, and the posi- 112 GRAVITATION. tion of the line of apses only is assumed to be variable. The question then becomes, What is the excentricity of each orbit, and what the varia- tion of the position of the line of apses, in order that a curve of the same kind may be described at every revolution ? (124.) In fig. 34. let B 3 , B,, B 2 , represent (he orbit of the first, and C 3 , C 19 C 2 , the orbit of the second. Suppose that when B was at B x , C Fig. 34. was at C , so that A, B 19 C x , were in the same straight line, or that B and C were in conjunction at these points. If the periodic time of C were exactly double of the periodic time of B, B would have made exactly two revolutions, while C made exactly one ; and, therefore, B and C would again be in conjunction at B and C x . But as the periodic time of C is a little longer than double that of B, or the angular motion of C rather slower than is supposed, B will have come up to it (in respect of longitude as seen from A) at some line B 2 C 2 , which it reaches before reaching the PERIODIC TIMES NEARLY AS ONE TO TWO. 113 former line of conjunction B 1 C 1 . And it is plain that there has been no other conjunction since that with which we started, as the successive conjunc- tions can take place only when one satellite has gained a whole revolution on the other. The first conjunction then being in the line A B x C t , the next will be in the line A B 2 C 3 , the next in a line A B 3 C 3 , still farther from the first, &c. ; so that the line of conjunction will regress slowly ; and the more nearly the periodic time of one satel- lite is double that of the other, the more slowly will the line of conjunction regress. (125.) As the principal part of the perturba- tion is produced when the satellites are near con- junction, (in consequence of the smallness of their distance at that time,) it is sufficiently clear that the position of the line of apses, as influenced by the perturbation, must depend on the position of the line of conjunction ; and, therefore, that the motion of the line of apses must be the same as the motion of the line of conjunction. Our ques- tion now becomes this : What must be the excen- tricities of the orbits, and what the positions of the perijoves, in order that the motions of the lines of apses, produced by the perturbation, may be the same as the motion of the line of conjunction ? 114 GRAVITATION. (126.) If the line of apses of the first satellite does not coincide with the line of conjunction, the first satellite at the time of conjunction will either be moving from perijove towards apojove, or from apojove towards perijove. If the former,, the dis- turbing force,, which is directed from the central body, will, by (59.), cause the excentricity to increase ; if the latter, it will cause it to decrease. As we have started with the supposition, that the excentricity is to be supposed invariable, neither of these consequences can be allowed, and, there- fore, the line of apses must coincide with the line of conjunction. , (127.) If the apojove of the first satellite were in the direction of the points of conjunction, the disturbing force in the direction of the radius vector, being directed from the central body, would, by (54.), cause the line of apses to pro- gress. Also the force perpendicular to the radius vector, before the first satellite has reached con- junction, (and when the second satellite, which moves more slowly, is nearer to the point of con- junction than the first,) tends to accelerate the first satellite ; and that which acts after the satel- lites have passed conjunction, tends to retard the first satellite; and both these, by (65.) and (66.), ORBIT OF INTERIOR SATELLITE. 115 cause the line of apses to progress. But we have assumed, that the line of apses shall move in the same direction as the line of conjunction,, that is, shall regress ; therefore, the apojove of the first satellite cannot be in the direction of the points of conjunction. (128.) But if we suppose the perijove of the first satellite to be in the direction of the points of conjunction, every thing becomes consistent. The disturbing force, in the direction /10l ; and, therefore, the effective attraction of D, estimated by the space through which it draws the satellite towards A, must be called =. And 101 xV 101 this is the whole effect which its attraction pro- duces ; for though the attraction of D alone tends to draw the satellite above A B, yet the attraction of E will tend to draw it as much below A B ; and thus the parts of the force which act perpendi- cular to A B will destroy each other. We have, then : the attraction of the lump D, if placed at A, would be represented by y^=iO-01 ; but as placed at D, its effective attraction is represented K5 202 GRAVITATION. 10 by - -=_ =0-0098518. The difference is 101 x V101 0-0001482, or nearly 1QQQOQ th of the whole at- traction of D, and the same for E. Consequently, the lumps at D and E produce a smaller effective attraction on B than if they were collected at A ; but the whole sphere produces the same effect as if its whole mass were collected at A ; and, there- fore, the part left after cutting away the lumps at D and E produces a greater attraction than if its whole mass were collected at A. (231.) But it is important to inquire, whether this attraction is greater than if the matter of the spheroid were collected at the centre, in the same proportion at all distances of the satellite. For this purpose, suppose the distance of the satellite to be 20. The same reasoning would show, that the attraction of the lump D, if placed at A, must now be represented by = 0-0025 ; but that, if placed at D, its effective attraction is represented by 20 __. __ 0-002490653. The difference now 401 is 0-000009347, or nearly of the whole at- EQUATORIAL ATTRACTION INCREASED. 203 traction of D. Consequently by removing the satellite to twice the distance from A, the difference between the effective attraction of the lump at A and at D, bears to the whole attraction of the lump at A, a proportion four times smaller than before. And, therefore, the attraction of the sphe- roid, though still greater than if its whole matter were collected at A, differs from that by a quan- tity, whose proportion to the whole attraction is only one-fourth of what it was before. If we tried different distances in the same manner, we should find, as a general law, that the proportion which the difference (of the actual attraction, and the attrac- tion supposing all the matter collected at the centre) bears to the latter, diminishes as the square of the distance from A increases. (232.) The attraction of an oblate spheroid upon a satellite, or other body, in the plane of its equator, may, therefore, be stated thus : There is the same force as if all the matter of the spheroid were collected at its centre, and, besides this, there is an additional force, depending upon the oblate - ness, whose proportion to the other force dimi- nishes as the square of the distance of the satellite is increased. (233.) Now, let us investigate the law accord- 204 GRAVITATION. ing to which an oblate spheroid attracts a body, situate in the direction of its axis. Proceeding in the same manner as before, and supposing the distance A B to be 10, the attraction of the lump, which at A would be represented by f -o-o-j will at D be represented by ^ T) and will at E be represented by -^-^ (since the distances of D and E from B are respectively 9 and 11.) Hence, if the two equal lumps, D and E, were collected Fig. 49. at the centre, their attraction on B would be loo + Too = Bo = ' 02 ' In the P sitions D and E, the sum of their attractions on B is 81 + I2l ~ ' 0206100 ' The difference is 0-0006100, by which the attraction in the latter case is the greater. Consequently, the attraction of the lumps in the positions D and E is greater POLAR ATTRACTION DIMINISHED. 205 than if they were collected at the centre by nearly yfyth of their whole attraction; but the attraction of the whole sphere is the same as if all the matter of the sphere were collected at the centre ; therefore, when these parts are removed, they must leave a mass, whose attraction is less than if its whole matter were collected in the centre. With regard to the alteration depending on the distance of B, it would be found, on trial, to follow the same law as before. (234.) The attraction of a spheroid on a body in the direction of its axis may, therefore, be repre- sented, by supposing the whole matter collected at the centre, and then supposing the attraction to be diminished by a force depending on the oblateness, whose proportion to the whole force diminishes as the square of the distance of the body is in- creased. (235.) Since the attraction on a body, in the plane of the equator, is greater than if the mass of the spheroid were collected at its centre, and the attraction on a body in the direction of the axis is less, it will readily be understood, that in taking directions, successively more and more inclined to the equator, on both sides, the attraction succes- sively diminishes. And there is one inclination, at 206 GRAVITATION. which the attraction is exactly the same as if the whole mass of the spheroid were collected at its centre. (236.) Now, suppose that a satellite revolves in an orbit, which coincides with the plane of the equator, or makes a small angle with it ; what will be the nature of its orbit ? For this investigation o we have only to consider, that there is acting upon the satellite a force, the same as if all the matter of the spheroid were collected at its centre, and, consequently, proportional inversely to the square of the distance, and that, with this force only, the satellite would move in an ellipse, whose focus coincided with the centre of the spheroid. But besides this, there is a force always directed to the centre, depending on the oblateness. One effect of it will be, that the periodic time will be shorter with the same mean distance, or the mean dis- tance greater with the same periodic time, than if the former were the only force. (46.) Another effect will be, that when the satellite is at its greatest distance, this force will cause the line of apses to regress, and when at its smallest distance, this force will cause the line of apses to progress. (50.) and (53.). If this force, at different distances, were in the same proportion as the other attractive SATELLITES' APSES PROGRESS. 207 force, it would,, on the whole, cause no alteration in the position of the line of apses, (for it would amount to the same as increasing the central mass in a certain proportion, in which case an ellipse, with invariable line of apses, would be described ; that is, the regression at the greatest distance would be equal to the progression at the least dis- tance. See the note to (98.) ). But (231.) the proportion of this force to the other diminishes as the distance is increased. Consequently, the re- gression at the greatest distance is less than the progression at the least distance, and, therefore, on the whole, the line of apses progresses. Also, the nearer the satellite is to the planet, the greater is the proportion of this force to the other attraction ; and, therefore, the more rapid is the progression of the line of apses at every revolution. The pro- gression of the line of apses of the moon's orbit, produced by the earth's oblateness, is so small in comparison with that produced by the sun's dis- turbing force, that it can hardly be discovered ; but the progression of the lines of apses in the orbits of Jupiter's satellites, produced by the oblateness of Jupiter, is so rapid, especially for the nearest satellites, that the part produced by the sun's disturbing force is small in comparison with it. 208 GRAVITATION. (237.) We shall now proceed with the investi- gation of the disturbance in a satellite's latitude, produced by the oblateness of a planet. (238.) First, It is evident that if the satellite's orbit coincides with the plane of the planet's equator, there will be no force tending to pull it up or down from that plane; and, therefore, it will continue to revolve in that plane. In this case, then, there is no disturbance in latitude; we must, therefore, in the following investigation, suppose the orbit inclined to the plane of the equator. In fig. 50., then, let us consider (as before) the effect of the attractions of the two lumps at D and E, in pulling the satellite B perpendicularly to the line A B. Now D is nearer to B than E is ; also the line D B is more inclined than E B to A B. Fig. 50. If the attraction of D alone acted, it would in a certain time draw the satellite to d; audfd would FORCE TOWARDS PLANE OF PLANET'S EQUATOR. 209 be the part of the motion of B, which is perpendi- cular to A B ; and this motion is upwards. Tn like manner, if the attraction of E alone drew B to e in the same time, g e would be the motion perpendicular to A B, and this motion is down- wards. When both attractions act, these effects are combined; the question then is, which is greater, fd or g e ? Now, since D is nearer than E, the attraction of D is greater than that of E, therefore B d is greater than Be; also B d is more inclined than Be to B A ; therefore df is much greater than g e. Hence, the force which tends to draw B upwards is the preponderating force ; and therefore, on the whole, the combined attractions of D and E will tend to draw the satellite upwards from the line B A. But the attraction of the whole sphere would tend to draw it along the line B A. Therefore, when D and E are removed, the attrac- tion of the remaining mass (that is, the oblate spheroid) will tend to draw B below the line B A. In estimating the attraction of an oblate spheroid, therefore, we must consider, that besides the force directed to the centre of the spheroid, there is always a force perpendicular to the radius vector directed towards the plane of the equator, or tending to draw a satellite from the plane of 210 GRAVITATION. its orbit towards the plane of the planet's equator. If the satellite is near to the planet, the inequality of the proportion of the distances D B and E B is increased, and the inequality of the inclinations to B A is also increased ; and the disturbance is, therefore, much greater for a near satellite than for a distant one. (239.) We have seen (215.) the effect of this disturbing force in determining the fundamental planes of the orbits of Jupiter's satellites. And from (192.), &c., we can infer, at once, that this force will cause the line of nodes to regress, if the orbit is inclined to the fundamental plane, and the more rapidly as the satellite is nearer to the planet. If there were no other disturbing force, the inclination of those orbits to the plane of Jupiter's equator would be invariable, and their nodes would regress with different velocities, those of the near satellites regressing the quicker. In point of fact, the circumstances of the inner satel- lites are very nearly the same as if no other dis- turbing force existed, so great is the effect produced by Jupiter's oblateness. (240.) The figure of Saturn, including in our consideration the ring which surrounds him, is different from that of Jupiter ; but the same prin- ATTRACTION OF SATURN^S RING. 211 ciples will apply to the general explanation of its effects on the motion of its satellites. The body of Saturn is oblate, and the forces which it produces are exactly similar to those produced by Jupiter. The effect of the ring may be thus conceived : -If we inscribe a spherical surface in an oblate sphe- roid, touching its surface at the two poles, the spheroid will be divided into two parts ; a sphere whose attraction is the same as if all its matter were collected at its centre, and an equatorial pro- tuberance analogous in form to a ring. The whole irregularity in the attraction of the spheroid is evi- dently due to the attraction of this ring-like pro- tuberance, since there is no such irregularity in the attraction of the sphere. We infer^ therefore, that the irregularity in the attraction of a ring is of the same kind as the irregularity in the attraction of a spheroid, but that it bears a much greater propor- tion to the whole attraction for the ring than for the spheroid, since the ring produces all the irregu- larity without the whole attraction. Now, the plane of Saturn's ring coincides with the plane of Saturn's equator, so that the effect of the body and ring together is found by simply adding effects of the same kind, and is the same as if Saturn were very oblate. The rate of progression of the perisa- 212 GRAVITATION. turnium of any satellite, and the rate of regression of its node, will, therefore, be rapid. In other re- spects it is probable, that, the theory of these satel- lites would be very simple, since all (except the sixth) appear to be very small, and the sun's dis- turbing force is too small to produce any sensible effects. (241.) The satellites of Saturn, except the sixth, have been observed so little, that no materials exist upon which a theory can be founded. A careful series of observations on the sixth satellite has lately been made by Bessel, from which, by com- paring the observed progress of the perisaturnium and regression of the node, with those calculated on an assumed mass of the ring, the real mass of the ring has been found. It appears, thus, that the mass of the ring (supposing the whole effect due to the ring) is about T T^th of the mass of the planet. (242.) The effect of the earth's oblateness in increasing the rapidity of regression of the moon's nodes is so small, that it cannot be discovered from observation. But the effect on the position of the fundamental plane is discoverable. We have seen (204.) that the moon's line of nodes regresses com- pletely round in 19J years. The plane of the earth's equator. is inclined 23% to the earth's orbit, EFFECTS OF EARTH'S OBLATENESS. 213 and the line of intersection alters very slowly. At some time, therefore, the line of nodes coincides with the intersection of the plane of the earth's equator and the plane of the earth's orbit, so that the plane of the moon's orbit lies between those two planes ; and 9J years later, the line of nodes again coincides with the same line, but the orbit is in- clined the other way, so that the plane of the moon's orbit is more inclined than the plane of the earth's orbit to the plane of the earth's equator. Now it is found, that in the former case the incli- nation of the moon's orbit to the earth's orbit is greater than in the latter by about 16", and this shows, that the plane to which the inclination has been uniform, is neither the plane of the earth's equator, nor that of the earth's orbit, but makes with the latter an angle of about 8", and is inclined towards the former. (243.) There is another effect of the earth's oblateness (the only other effect on the moon which is sensible) that deserves notice. The incli- nation of the moon's orbit to the earth's orbit is less than 5, and the inclination of the earth's equator to the earth's orbit is 23^. Conse- quently, when the moon's orbit lies between these 214 GRAVITATION. two planes, the inclination of the moon's orbit to the earth's equator is about 19; and when the line of nodes is again in the same position, but the orbit is inclined the other way, the inclination of the moon's orbit to the earth's equator is about 28. At the latter time, therefore, in consequence of the earth's oblateness, the moon, when farthest from its node, will, by (235.), experience a smaller attraction to the earth than at the former time when farthest from its node. When in the line of nodes, the attractions in the two cases will be equal. On the whole, therefore, the attraction to the earth will be less at the latter time than at the former. For the period of 9| years, therefore, the earth's attraction on the moon is gradually dimi- nished, and then is gradually increased for the same time. The moon's orbit (47.) becomes gra- dually larger in the first of these times, and smaller in the second. The change is very minute, but, as explained in (49.), the alteration in the longitude may be sensible. It is found by observation to amount to about 8", by which the moon is sometimes before her mean place, and sometimes behind it. If the earth's flattening at each pole were more or less than -3-wth of the semi-diameter, the effects on EFFECTS OF EARTH'S OBLATENESS. 215 the moon, both in altering the position of the fun- damental plane, and in producing this inequality in the longitude, would be greater or less than the quantities that we have mentioned ; and thus we are led to the very remarkable conclusion, that by observing the moon we can discover the amount of the earth's oblateness, supposing the theory to be true. This has been done; and the agreement of the result thus obtained, with that obtained from direct measures of the earth, is one of the most striking proofs of the correctness of the Theory of Universal Gravitation. THE END. LONDON : Printed by WILLIAM CLOWES, Duke-street, Lambeth. JT' \ "' ' ~- V>Y RETURN 14 DAY USE TO DESK FROMJB7HJOI BpRR LOAN DLPT. .OWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 12Nov'57Pt RECD LD nr.r an 19 H l -^> ; : : . REC'D LD MAR 19 '65 -3PM ->- fe m am LD 21-100m-6,'56 i^LV: (B9311slO)476 General Library University of California m re 16982 fiiitiit? ^m^^m^ -x UNIVERSITY OF CALIFORNIA IvIBRARY .Jva^.-C'-Tf , .^/^v.^ !;,?.>.< f'-jT' ---. V 'v.-,-^- . vV> m