NRLF B M SSO 615 1 m* myS$.....-. .... ,'= .. 148 211. 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 Hue of nodes will progress . . . 149 XVI CONTENTS. Article PAGE 212. Kemarkable relation between the inclinations of several planets disturbing each other . . . . .150 214. Explanation of the term fundamental plane . . . 151 215. Determination of the fundamental planes of Jupiter's satellites . . .151 220. Their lines of nodes will regress on their fundamental planes, without alteration or inclination to those planes. 155 221. An inclination of the orbit of an exterior satellite produces a similar inclination of the orbit of an interior satellite . 156 222. An inclination of the orbit of an interior satellite produces an opposite inclination of the orbit of an exterior satellite 158 SECTION IX. Effects of the Oblateness of Planets upon the Motions of their Satellites . . . .160 230. The attraction of an oblate spheroid on a body in the plane of its equator is greater than if its whole mass were col- lected at its centre -. . . . . . .161 231. The proportion in which it is greater diminishes as the square of the distance increases . . - . . . 163 233. The attraction of an oblate spheroid on a body in the direc- tion of its axis is less than if its whole mass were col- lected at its centre * . , , . . . . 164 236. 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 . . . 166 238. If a satellite is not in the plane of the equator, the attraction tends to pull it towards that plane . . . 167 240. Effect of Saturn's ring on the motions of Saturn's satellites . 170 242. The moon's fundamental plane is inclined to the ecliptic . 171 243. Long inequality in the moon's motions, produced by the earth's oblateness 172 ON GEAVITATION. SECTION I. On the Rules for calculating Attraction, 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 particle. 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 imme- diately begin to move towards 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 imme- diately begins to move towards the magnet, and we say that the magnet attracts the needle. But magnetic attraction belongs only to certain bodies ; whereas the attraction of which we speak here belongs to all bodies of every kind : metals, earths, fluids, and even the air and gases are equally subject to its influence. (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 supported on the ends of a rod which was suspended at the middle by a slender B 2 GRAVITATION. wire ; and when large leaden balls were brought near 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 perpendicular to its surface (and there- fore fall in opposite directions at the places which are diametrically opposite), we 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 num- ber 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 rules by which this attraction is regulated, and by which it may be calculated; or (as it is technically 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 in- stance : we might measure it by the pressure which is produced on the hand by a lump of lead held in the TWO EFFECTS OF ATTRACTION. 3 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 pres- sure .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 mea- sure 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 time (when the resistance of the air, &c., is removed), we shall get the same measure for gravity, whatever body we sup- pose 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 ly 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 calculating attraction, it must be understood to mean calculating the number of inches, or feet, through which the attrac- tion draws the 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 B 2 4 GRAVITATION. 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 Jupiter. In other words (the attraction being measured in con- formity with the definition above), 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 thousand times that number of inches. (9.) The third rule is this : " If the same attracting body act upon several bodies at different distances, the attractions are inversely proportional 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 therefore, the earth's attraction on the sun is only I6o l 00 O th part of its attraction on the moon; or, as the earth's attraction draws the moon through about -^th of an inch in one second of time, the earth's attraction draws the sun LAWS OF ATTRACTION. through 3/ g o oyo o O th f an inch i n one second -of time. In like manner, 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 attrac- tions 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 attraction 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 earth exerts upon the moon bears the same proportion 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 b - GRAVITATION. carried to a degree that caii scarcely be imagined ; 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 gravita- tion. 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. 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 attrac- tion, namely, the production of pressure, if the matter on which the attraction acts is supported (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 under- stood, 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 separately 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 upwards), its motion will become slower and slower, and will, at last, be turned into a motion in the MOTION OF PROJECTED STONE. 7 opposite direction. We have yet to consider^ a case much more important for astronomy than either of these : Suppose that a body is projected, in a direction transverse to, or crossing, the direction in which the force draws it, how will it move ? (14.) 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 con- stantly, and the stone strikes the ground, moving at that time in a direction much inclined to the original direc- tion. The most powerful effort that we can make, even when we use artificial means (as in producing the motion 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 immediately 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 projected in a direction transverse to the direc- tion 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 path, of which the first part has the same direction as the line 8 GRAVITATION. in which it is projected. The circumstances of the motion of the stone may be calculated with the utmost accuracy from the following 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 direction in FIG. 1. which it was thrown ; and if we wish to know where the stone will be at the end of any particular time (sup- pose, for instance, three seconds), and if the velocity with which it is thrown would, in three seconds, have carried it to B, supposing gravity not to have acted on it ; and if gravity would have made it fall from A to C, supposing 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 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 MODIFICATION OF CALCULATION FOR PLANETS. 9 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 mentioned, is inversely propor- tional 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 method. But the same method will apply, provided we restrict the intervals for which we make the calcu- lations 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 attraction 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) 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 percep- tible only in the nicest observations ; and if we calcu- lated for the end of a minute, the error would be per- fectly insensible. (17.) Now a method of calculation has been invented, which amounts to the same as making this computation 10 GRAVITATION. for every successive small portion of time, with the correct value of the attractive force, and the correct direction of force at every particular 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 with- out the smallest error. The rules to which this leads are simple : but the demonstration of the rules re- quires the artifices of advanced science. "We cannot here attempt to give any steps of this demon stration ; 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, fig. 2, FIG. 2. in the direction A B, and if the attraction of the sun, situated at S, begins immediately to act on it, and con- tinues 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 hyperbola. In every case the curve will, at the point A, have the same direction as the line A B ; or (to use the language PLANETS MOVE IN CONIG SECTIONS. 11 of mathematicians), A B will be a tangent to #ie 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 neither 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 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 AB is oblique to SA, 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 parabola 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 method of describing an ellipse is to fix two pins in a board, as at S and H, fig. 3 ; to fasten a thread SPH to them, and FIG. 3. 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.) 12 GRAVITATION. 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. 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 distinguished from parabolas. There is reason to think that two or three comets which have been observed move in hyperbolas. But as we do not purpose, in this treatise, to enter into a discussion on the motions of comets, we shall confine ourselves to the consideration of motion in an ellipse. (20.) Everything 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 according 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 Saturn and Uranus. (21.) In stating the suppositions on which the cal- culations 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 the nature of these forces is wholly different. The force of attraction is one which acts constantly and steadily without a moment's EXPLANATION OF PEOJECTILE FORCE. 13 intermission (as we know that gravity to thejearth is always acting): the force by which the body is pro- jected 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 are in motion, and it is of no consequence to our inquiry how they received this motion, but it is convenient, for the purposes of calcula- tion, to suppose that, at some time, they received an impulse of the same kind as that which a stone receives when thrown from the hand; and this is the whole meaning of the term " projectile force." (22.) From the same considerations it will appear 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 velocity 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 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 natural consequences of the law of gravitation. (24.) The force of attraction, we have said, is in- versely 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 14 GRAVITATION. approached most nearly to the sun, as the sun's attrac- tion 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 continually retracing the same orbit. How is this receding 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 exactly similar, we have seen that the deflection of the stone from the straight line in which it began to 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 tiling 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 PLANET'S VELOCITY GREATEST NEAR THE SUN. 15 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 begin to approach towards the sun ; the sun's attraction will necessarily increase its velocity as it moves through D, E, and F. For the $un's attractive force on the planet, when the planet is at D, is acting in the direction D S, and it is plain that (on account of the small inclination of DE to D S) the force pulling in the direction DS helps the planet along in its path DE, and thereby increases its velo- city. 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 continually increasing as the planet passes through D, E, and F ; and though the sun's attractive force (on account of the planet's near- ness) 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 : and at half the periodic time after leaving C, the planet is no longer approaching to the sun ; and its velocity is so great, and 16 GKAVITATION. the curvature of the orbit in consequence so small (being, in fact, exactly 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 (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 slowness 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 perpetually 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 FIG. 4. A E D B ; draw the line A B through S and H ; take C, the middle point between S and H, and draw DCE EXPLANATION QF TERMS. 17 perpendicular to A C B. Let S be that focus^which is the place of the sun (if we are speaking 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. AC 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 B S (which is the planet's greatest distance from S). D E 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) ; perijove (if we are speaking of the orbit described by one of Jupiter's satellites round Jupiter) ; or the perisaturnium (if we are speaking of the orbit described by one of Saturn's satellites about Saturn). B, in the orbit of a planet, is called the aphelion ; 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. 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 proportion which S C bears to A C, and this proportion, expressed by a number, is called the excentricity. Thus, if S C were c 18 GRAVITATION. one-third of AC, we should say, that the excentricity of the orbit was , or 0'3333. If S<> 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 qp S P is its longitude at that time, and the angle A S P is its true anomaly. (The longitude of the planet is, therefore, equal to the sum of the longitude of the perihelion, 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. 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 projected at a certain place in a known direction with a given velocity, the length of the axis major, the excentricity, the posi- tion of the line of apses, and the periodic time, may all be calculated. We cannot point out the methods and CALCULATION OF A PLANET'S PLACE. 19 formulae used for these, but we may mention jone 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 obvious that the longi- tude of the planet increases very irregularly, since, when the planet is near to the sun, its actual motion is very rapid, and, therefore, the increase of longitude is ex- tremely rapid ; and when the planet is far from the sun, its actual motion is slow, and, therefore, the increase of longitude is extremely slow. The rule, which is demonstrated by theory, and which is found to apply precisely in observation, is this : The areas described 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 move from a to I, making the area a S & equal to A S a ; in c 2 20 GRAVITATION. the third day it will move from 6 to c, making the area 5 S c equal to A S a or a S b, and so on. FIG. 5. (31.) Upon this principle mathematicians have in- vented methods of calculating the place of a planet, or satellite, at any time for which it may be required. These methods are too troublesome for us to explain here ; but we may point out the meaning of two terms which are frequently used in these computatibns. Sup- pose, for instance, as in the figure, that the planet, or satellite, occupies ten days in describing the half of its orbit, Aa'bcdef(/hi~B ) OT 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 peri- helion. 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 54 degrees. The excentricities of all the orbits of the planets are small ; and it is convenient, therefore, to begin with the angle of 54 degrees as one which is not very erroneous, but which will require some correction. This angle (as 54 degrees), which is propor- EQUATION OF THE CENTRE. 21 tional to the time, is called the mean anomaly; and the correction 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 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 sub- tracted from the mean anomaly, in order to produce the true anomaly. (32.) The sum of the mean anomaly and the longi- tude 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 subtract 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 knowledge 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 explained 22 GRAVITATION. regarding the undisturbed motion of planets and satel- lites ; namely, the relation between a planet's periodic time and the dimensions of the orbit in which it moves. Now, on the law of gravitation it has been demon- strated from theory, and it is fully confirmed by obser- vation, 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 circular, 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 follow- ing : The squares of the numbers of days (or hours, or minutes, &c.) in the periodic times have the same pro- portion as the cubes of the numbers of miles (or feet, &e.) in the mean distances. (35.) Thus the periodic time of Jupiter round the sun is 43327 days, and that of Saturn is 10759-2 days; the squares of these numbers are 18772289 and 115760385. The mean distance of Jupiter from the sun is about 487491000 miles, and that of Saturn is about 893955000 miles; the cubes of these numbers are 1158496 (20 ciphers), 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. PERIODIC TIMES ROUND DIFFERENT CENTRES. 23 (36.) In like manner, the periodic times of* Jupiter's third and fourth satellites round Jupiter are 7"! 545 5 and 16-68877 days; the squares of these numbers are 511876 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 rule applies in comparing the periodic times and mean distances of Jupiter and Saturn, because they both revolve round the sun; it applies in comparing the periodic times and mean distances of Jupiter's third and fourth satellites, be- cause they both revolve round Jupiter; but it would not apply in comparing the periodic time and mean distance of Saturn revolving round the sun with that of Jupiter's third satellite revolving round Jupiter. (38.) In comparing the orbits described by different planets, or satellites, round different centres of force, theory gives us the following law : The cubes of the mean distances are in the same proportion as the pro- ducts of the mass by the square of the periodic time. Thus, for instance, the mean distance of Jupiter's fourth satellite from Jupiter is 1178560 miles; its periodic time round Jupiter is 16'68877 days; the mean dis- tance of the earth from the sun is 93726900 miles ; its 24 GRAVITATION. periodic time round the sun is 365*2564 days; also the mass of Jupiter is i O ' a O 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 the cubes of the mean distances, are commonly called Kepler's laws. They were discovered by Kepler from observation, before the theory of gravitation was invented; they were first explained from the theory by Newton, about A.D. 1680. (40.) The last of these is not strictly true, unless we suppose that the central body is absolutely immovable. This, however, is evidently inconsistent with the prin- ciples 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, Jupiter 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 LAW ALTERED BY PLANET'S ATTRACTION. 25 manner : The motion which the attraction of Jupiter produces in the sun is less than the motion 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 of the sun and Jupiter, and would, therefore, be greater than their rate of approach, if the sun were not movable, in the same proportion in which the sum of the masses of the sun and Jupiter 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 described 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 proportion as the products of the square of the periodic time by the sum of the masses of the attracting and attracted body. 26 GEAVITATION. 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 differently at different times by the other planets whose position is per- petually 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 comparison 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 accurate observation, continued for some years, that the effects of perturbation can be made sensible. (42.) The investigation of the effects of the disturb- ing forces will consist of two parts : the examination 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 VARIATION OF ELEMENTS. 27 shall suppose that a planet is revolving round the sun, the sun being fixed (a supposition made only for pre- sent convenience), and that some force acts on the planet without acting on the sun (a restriction intro- duced only for convenience, and which we shall here- after get rid of). (43.) The principle upon which we shall explain the effect of this force is that known to mathematicians 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 revolu- tions. Yet its motion may be represented by supposing it to have moved in an ellipse, provided we suppose the elements of the ellipse to have been perpetually alter- ing. It is plain that by this contrivance any motion whatever may be represented. By altering the 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 alter- ing 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 velocity. The dimen- 28 GRAVITATION. sions 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 pro- jected with the velocity 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 con- tinues to revolve in the same ellipse, and the permanent ellipse coincides with the instantaneous ellipse cor- responding to the instant when the disturbing 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 attraction or the mass of the central body was increased. 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 men- DISTURBING FORCE DIRECTED TO CENTRE. 29 tioned separately hereafter (we do not insjst on it at present, as there is no instance in the planetary system of such sudden commencement of force). But at all events the relation between the mean 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 disturbing force is always directed from the central body, the effect will be exactly opposite. If the dis- turbing force does not alter, except with the planet's distance, the planet will at every successive revolu- tion 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 determines 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 vector. This considera- tion 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. (47.) (II.) If, however, the disturbing force directed to the central body increases gradually and constantly 30 GRAVITATION. 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 FIG. 6. move, at every succeeding revolution, in a smaller orbit than at the preceding one ; and will conse- quently perform its revolution 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 dis- turbing force is directed from the central body, a gradual increase 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 moving. The reader might imagine, at first sight, that this would shorten the time of revolution. The effect, however, is exactly opposite. For in fig. 2, if the planet be projected from A, the reason that the . sun's attraction is able to pull the planet in at C and make it approach to itself is, that the velocity of the EFFECT OF ALTERATION OF MEAN DISTANCE. 31 planet is so small as to allow the force tg curve the orbit much. If the velocity were greater, the orbit as we have said in (25.) would be less curved in every part, and would therefore pass on the outside of the orbit A C D E F. The effect then of a force in the direction of the planet's motion, which increases the planet's velocity, is to increase the size of its orbit ; and the bigger the orbit is, the longer is the time of revolution. If the force acts continually, the time of revolution lengthens continually. If the dis- turbing force acts in the direction opposite to that in which the planet is moving, the effect is to make the orbit smaller, and to make the time of revolution shorter. The retardation produced by motion through extremely thin air is of this kind. It is found that a comet (known by the name of Encke's comet) which moves in an ellipse, whose length is not much greater than the diameter of Mars' orbit, performs every new revolution in a shorter time than the preceding one; and we infer from this circumstance that it, experi- ences some resistance in its motion. (49.) (IV.) There is one consideration of great im- portance in the estimation of the effects mentioned in (II.) and (III.) The alteration of the dimensions of the orbit produces an alteration in the periodic time, and this alters the planet's mean motion, or the number of degrees by which the mean longitude is increased in a given time (suppose one year). The effects of these alterations are added together at every successive revolution, and thus may produce 32 GRAVITATION. an alteration in the planet's mean longitude (which differs from the true longitude only by the equation of the centre) that is vastly more conspicuous than the alteration in the dimensions of the orbit. Sup- pose, for instance, that a disturbing force acted on a planet (either a constant force in the direction of its motion, or a variable force in the direction of the radius vector), such as to increase the mean distance by oi-o-o-th P ar ^ i n 100 revolutions of the planet. This alteration of the planet's distance from the sun could hardly be discovered by the nicest observations. But as the mean 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 1000 1J 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 revolu- tion, 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 FORCE DIRECTED TO CENTRE NEAR PERIHELION. 33 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 or satellite to be an ellipse; and suppose a disturbing 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 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 S B. 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 D 34 GRAVITATION. which the planet would have moved. The reader's conception of this will be facilitated by supposing the orbit instead of a curve to be a straight line, as B C in fig. 7*, and the place of perihelion to be determined by letting fall a perpendicular S C from the sun upon the line ; when it will be seen that, if the disturbing force, acting towards S for a short time at B, changes the path of the planet from the direction B C to B c, the distance of the foot of the perpendicular S c from B is greater than that of S C.* With a curved orbit the result is just the same. In other words, the planet, instead of describing BC, fig. 7, will, in consequence of the action of the disturbing force, describe Be; and the place of perihelion, instead of C, will be c, a point more distant from B than C is. Now, if the disturbing force should not act again, the planet would move in an ellipse c d b, and the line of apses, instead of CSD, would be cSd. The line of apses has therefore twisted round in the same angular direction as that in which the planet was going ; and this is expressed by saying that the line of apses progresses. If, after passing c, the dis- turbing force should again act for a little while, at e.for instance, the recess of the planet from the sun would be diminished, its path would be more PROGRESSION OF LINE OF APSES. 35 nearly perpendicular to the radius vector, and there- fore the inclination of the path would be such as corresponds to a smaller distance from perihelion than the planet really has. That is, when the planet leaves , the inclination of its path to the radius vector is greater than it would have been if the planet had continued to move in the orbit c d b, but is the same as if its perihelion had been at some such situation as /, supposing no disturbing force to act. Now let the disturbing force cease entirely to act ; and the planet, which at e is moving as if it had come from the perihelion /, will continue to move as if it had come from the perihelion /; it will proceed, therefore, to describe an elliptic orbit in which fSg is the line of apses: the line of apses has been twisted round in the same direction as before, or the line of apses has still progressed. The effect then of a disturbing force directed to the central body before and after passing the perihelion, is to make the line of apses progress.* * This result, and those which follow immediately, may be inferred from the construction in Newton's ' Principia,' book i. sect. 3. prop, xvii. If we assume (as we suppose in all these investigations) the excentricity to be small, the disturbing force directed to the sun will not sensibly alter the planet's velocity, but will change the direction of its path at P, the place of action (in Newton's figure) ; the length of P H, therefore, will not be altered (since that length depends only on the velocity), but its position will be altered, the position of P H being determined by making the angle E P H equal to the supple- ment of K P S. On trying the effects of this in different positions of P, and observing that the immediate effect of a disturbing force directed to the centre is to increase the rate of approach, or to dimmish the rate of receding, and that the effect of a force directed from the centre is the opposite, all the cases in the text will be fully explained. D 2 36 GRAVITATION. (51.) In the same manner it will be seen, that the effect of a disturbing force, directed from the central body before and after passing the perihelion, is to make the line of apses regress. (52.) The motion of the planet, subject to such forces as we have mentioned, would be nearly the same as if it was revolving in an elliptic orbit, and this elliptic orbit was at the same time revolving round its focus, turning in the same direction as that in which the planet goes round, and always carrying it on its circumference. And this is the easiest way of representing to the mind the general effect of this motion ; the physical cause is to be sought in such explanations as that above. (53.) (VI.) Suppose a disturbing force directed to the centre, to act upon the planet when it is near aphe- lion. As the planet is going towards aphelion it is receding from the sun. The effect of the disturbing force is to diminish the rate of recess from the sun ; and, therefore, to increase the inclination of the planet's path to the radius vector. The aphelion is the place where the planet's path is perpendicular to the radius vector. The effect of the disturbing force, then, which increases the inclination of the planet's path to the radius vector, will be to make that path perpen- dicular to the radius vector sooner than if the dis- turbing force had not acted. That is, the planet will be at aphelion sooner than it would have been if no disturbing force had acted. The aphelion has as it were, gone backwards to meet the planet. If the FORCE DIRECTED TO CENTRE NEAR APHELION. 37 disturbing force should entirely cease, ^the planet will move in an elliptic orbit, of which this new aphe- lion would be the permanent aphelion. The line passing through the aphelion has, therefore, twisted in a direction opposite to the planet's motion, or the line of apses has regressed. After passing aphelion, if the disturbing force still continues to act, the planet's approach to the sun will be quickened by the dis- turbing force, and, therefore, after some time, the planet's rate of approach will be greater than that corresponding in an undisturbed orbit to its actual distance from aphelion, and will be equal to that cor- responding in an undisturbed orbit to a greater dis- tance from aphelion. If, now, the disturbing force ceases, the planet, moving as if it came in an un- disturbed orbit from an imaginary aphelion, will continue to move as if it came from that imaginary aphelion ; and that imaginary aphelion having been at a greater distance behind the planet than the real aphelion, its place will be represented by saying that the line of apses has still regressed. The effect, then, of a disturbing force directed to the central body, before and after passing the aphelion, is to make the line of apses regress. (54.) In the same manner it will be seen, that the effect of a disturbing force, directed from the central body, before and after passing the aphelion, is to make the line of apses progress. (55.) (VII.) Since a disturbing force, directed to the 38 GRAVITATION. central body, or one directed from the central body, produces opposite effects with regard to the motion of the line of apses, according as it acts near perihelion, or near aphelion, it is easy to perceive that there must -be some place between perihelion and aphelion, where the disturbing force, directed to the central body, will produce no effect on the position of the line of apses. It is found by accurate investigation, that this point is the place where the radius vector is perpendicular to the line of apses.* (56,) (VIII.) The effects mentioned above are greatest when the excentricity is small. Thus, if we compare two orbits, as figures 8 and 9, in one of which the excentricity was great, and in the other small ; and if (for instance) we supposed the disturbing force to act for a short time at the perihelion C, and supposed the forces in the two orbits to be such as to deflect the new paths from the old orbits by equal angles in the two cases : it is plain that in fig. 8, in consequence * To the reader who is familiar with Newton's 'Principia,' sect. 8, the following demonstration will be sufficient : The disturbing force, which is entirely in the direction of the radius vector, will not alter the area described in a given time, and, therefore, will not alter the latus rectum (to the square root of which the area is proportional). But half the latus rectum of the undisturbed orbit is the radius vector at the supposed place of action of the disturbing force (since that radius vector is supposed perpendicular to the major axis). Therefore, half the latus rectum of the new orbit is the radius vector at the point in question ; and, consequently, the radius vector, at the point in question, is perpendicular to the major axis in the new orbit ; but it was so in the undisturbed orbit; and, therefore, the major axes in the new orbit and the undisturbed orbit coincide. MOTION OF APSES. 39 of the curvature at C differing much from that of a circle whose centre is S, we should find the new peri- helion c at a small distance from C ; whereas in fig. 9, where the orbit does not differ much from the circle whose centre is S, c would be far removed from C. In fact, c would in both cases bisect the part of the orbit lying within that circle ; and it is evident, that the angle at C being the same in both, the length of the part lying within the circle would be much less in fig. 8, where the orbit is almost a straight line, than in fig. 9, where the curvature of the orbit differs FIG. 9. c little from that of the circle. Or we may state it thus: The alteration of the place of perihelion, or aphelion, depends on the proportion which the alteration in the approach or recess produced by the disturbing force bears to the whole approach or recess; and is, therefore, greatest when the whole approach or recess is least ; that is, when the orbit is little excentric. (57.) (IX.) To judge of the effect which a disturbing force, directed to the sun, will produce on the excen- tricity of a planet's orbit, let us suppose the planet 40 GRAVITATION. to have left its perihelion, and to be moving towards aphelion, and, consequently, to be receding from the sun, and now let the disturbing force act for a short time. This will cause it to recede from the sun more slowly than it would have receded without the action of the disturbing force ; and, consequently, the planet, without any material alteration in its velocity (and, therefore, without any material alteration in the major axis of its orbit (28.) ), will be moving in a path more inclined to the radius vector than if the disturbing force had not acted. The planet may, therefore, be considered as projected from the point A, fig. 10, in FIG. 10. the direction A b instead of A B, in which it was moving; and, therefore, instead of describing the orbit A C Gr, in which it was moving before, it will describe an orbit Ac#, more resembling a circle, or less excentric than before. The effect, therefore, of a disturbing force directed to the centre, while a planet is moving from perihelion to aphelion, is to diminish the excentricity of the orbit. (58.) If we suppose the planet to be moving from ALTERATION OF EXCENTRICITY. 41 aphelion to perihelion, it is approaching to th& sun ; the disturbing force directed to the sun makes it approach more rapidly ; its path is, therefore, less inclined to the radius vector than it would have been without the dis- turbing force; and this effect may be represented by supposing that at E, fig. 11, instead of moving in the FIG. 11. direction E F in which it was moving, the planet is pro- jected in the direction E /. Instead, therefore, of de- scribing the ellipse E G H, in which it was moving before, it will describe such an ellipse as ~&gh, which is more excentric than the former. The effect, therefore, of a disturbing force directed to the centre, while a planet is moving from aphelion to perihelion, is to increase the excentricity of the orbit. (59.) In a similar manner it will appear, that the effect of a disturbing force, directed from the centre, is to increase the excentricity as the planet is moving from perihelion to aphelion, and to diminish it as the planet moves from aphelion to perihelion. (60.) (X.) Let us now lay aside the consideration of a force acting in the direction of the radius vector, and consider the effect of a force acting perpendicularly to 42 GRAVITATION. the radius vector, in the direction in which the planet is moving.. And first, its effect on the position of the line of apses. (61.) If such a force act at one of the apses, either perihelion or aphelion, for a short time, it is clear that its effect will be represented by supposing that the velocity at that apse is suddenly increased, or that the velocity with which the planet is projected from peri- helion is greater than the velocity with which it would have been projected if no disturbing force had acted. This will make no difference in the position of the line of apses ; for with whatever velocity the planet is pro- jected, if it is projected in a direction perpendicular to the radius vector (which is implied in our supposition, that the place where the force acts was an apse in the old orbit), the place of projection will infallibly be an apse in the new orbit ; and the line of apses, which is the line drawn from that point through the centre, will be the same as before. (62.) But if the force act for a short time before the planet reaches the perihelion, its principal * effect will be to increase its velocity ; the sun's attraction will, therefore, have less power to curve its path (25.) ; the new orbit will be, in that part, exterior to the old one. In fig. 12, we must, therefore, suppose that the planet, after leaving A, where the force has acted to accelerate * It is supposed here, and in all our investigations, that the excen- tricity of the orbit is small, and, consequently, that a force perpen- dicular to the radius vector produces nearly the same effect as a force acting in the direction of a tangent to the ellipse. ALTERATION OF THE LINE OF APSES. 43 its motion, instead of describing the orbit A>C G, pro- ceeds to describe the orbit Acd, which at A has the same direction (or has the same tangent AB) as the orbit A C G. It is plain now that c is the part nearest FIG. to the sun, or c is the perihelion : and it is evident here, that the line of apses has altered its position from S C to Sc, or has twisted in a direction opposite to the angular motion of the planet, or has regressed. (63.) If the force act for a short time after the planet has passed perihelion, as at D in fig. 13, the planet's velocity is increased there, and the path described by the planet is D/, instead of DF, having the same direction at D (or having the same tangent D E), but less curved, and, therefore, exterior to D F. If now we conceive the planet to have received the actual velocity with which it is moving in D/, from moving without disturbance in an elliptic orbit c D/ (which is the orbit that it will now proceed to describe, if no disturbing force continues to act), it is evident that the part c D must be described with a greater velocity than C D, inasmuch as the velocity at D from moving in c D is 44 GRAVITATION. greater than the velocity from moving in CD; c D is, therefore, less curved , than C D, and, therefore, exterior to it (since it has the same direction at D) ; and then FIG. 13. the perihelion is some point in the position of c, and the line of apses has changed its direction from S C to S c, or has twisted round in the same direction in which the planet is moving, or has progressed. (64.) If the force act for a short time before passing aphelion, it will be seen in the same manner that the line of apses is made to progress. It is only necessary to consider that (as before) the new orbit has the same direction at the point H, fig. 14, where the force has FIG. 14. acted as the old one, but is less curved, and, therefore, exterior to it ; and the aphelion, or point most distant from the sun, is g instead of G, and the position of the ALTERATION OF THE LINE OF APSES. 45 line of apses has shifted from SG to $g. I.the force act after the planet has passed aphelion, as at K, fig. 15, the orbit in which we must conceive the planet to have come, in order to have the increased velocity, must be g K exterior to G K, the point most distant from the sun must be g instead of G, and the line of apses must have changed from S G to S#, or must have regressed. (65.) Collecting these conclusions,* we see that, if a disturbing force act perpendicularly to the radius vector, in the direction in which the planet is moving, its action, while the planet passes from perihelion to aphelion, causes the line of apses to progress ; and its action, while the planet passes from aphelion to peri- helion, causes the apses to regress. (66.) By similar reasoning, if the direction of the disturbing force is opposite to that in which the planet is moving, its action, while the planet passes from perihelion to aphelion, causes the line of apses to * These conclusions, and those that follow, will be easily inferred from Newton's construction, Prop. XVII., by observing, that an increase of the velocity increases the length of P H in Newton's figure without altering its position. 46 GRAVITATION. regress, and while the planet passes from aphelion to perihelion causes the apses to progress. (67.) (XL) For the effect on the excentricity ; suppose the disturbing force, increasing the velocity, to act for a short time at perihelion ; the effect is the same as if the planet were projected from perihelion with a greater velocity than that which would cause it to describe the old orbit. The sun's attraction, there- fore, will not be able to pull it into so small a com- pass as before ; and at the opposite part of its orbit, that is, at aphelion, it will go off to a greater distance than before ; but as it is moving without disturbance, and, therefore, in an ellipse, it will return to the same perihelion. The perihelion distance, therefore, remaining the same, and the aphelion distance being- increased, the inequality of these distances is in- creased, and the orbit, therefore, is made more excen- tric. Now, suppose the force increasing the velocity to act at aphelion. Just as before, the sun's attrac- tion will be unable to make the planet describe an orbit so small as its old orbit, and the distance at the opposite point (that is, at perihelion) will be increased ; but the planet will return to the same aphelion dis- tance as before. Here, then, the inequality of distances is diminished, and the excentricity is diminished. (68.) Thus we see that a disturbing force, acting perpendicularly to the radius vector, in the direction of ALTERATION OF THE EXCENTRICITY. 47 the planet's motion, increases the excentricity if it acts on the planet near perihelion, and diminishes the excentricity if it acts on the planet near aphelion. And, similarly, if the force acts in the direction oppo- site to that of the planet's motion, it diminishes the excentricity by acting near perihelion, and increases it by acting near aphelion. (69.) (XII.) In all these investigations, it is supposed that the disturbing force acts for a very short time, and then ceases. In future, we shall have to con- sider the effect of forces, which act for a long time, changing in intensity, but not ceasing. To estimate their effect we must suppose the long time divided into a great number of short times ; we must then infer, from the preceding theorems, how the elements of the instantaneous ellipse (43.) are changed in each of these short times by the action of the force, which is then disturbing the motion ; and we must then recollect, that the instantaneous ellipse, at the end of the long time under consideration, will be the same as the permanent ellipse in which the planet will move, if the disturbing force then ceases to act (43.), and that it will, at all events, differ very little from the curve described in the next revolution of the planet, even if the disturbing force continue to act (41.). 48 GRAVITATION. SECTION IV. On the Nature of the lorce disturbing a ' Planet or Satellite, produced ly the Attraction of other Bodies. (70.) HAVING examined the effects of disturbing forces upon the elements of a planet's or satellite's orbit, we have now to inquire into the kind of the disturbing force which the attraction of another body produces. The inquiry is much simpler than might at first sight be expected; and this simplicity arises, in part, from the circumstance that (as we have mentioned in (6.) ) the attraction of a planet upon the sun is the same as its attraction upon another planet, when the sun and the attracted planet are equally distant from the attracting planet. (71.) First, then, we have to remark, that the dis- turbing force is not the whole attraction. The sun, for instance, attracts the moon, and disturbs its elliptic motion round the earth ; yet the force which disturbs the moon's motion is not the whole attraction of the sun upon the moon. Tor the effect of the attraction is to move the moon from the place where it would other- wise have been ; but the sun's attraction upon the earth also moves the earth from the place where it would otherwise have been ; and if the alteration of the earth's place is exactly the same as the alteration of the moon's place, the relative situation of the earth and moon will be the same as before. Thus if, in fig. 16, any attrac- tion carries the earth from E to e, and carries the moon from M to m, and if E e is equal and parallel to M m, UNEQUAL FORCES CAUSE DISTURBANCE. 49 then e m, which is the distance of the earth, and moon, on the supposition that the attraction acts on both, is equal to E M, which is their distance, on the supposition that the attraction acts on neither ; and the line e m, FIG. 16. which represents the direction in which the moon is seen from the earth, if the attraction acts on both, is parallel to E M, which represents the direction in which the moon is seen from the earth, if the attraction acts on neither. The distance, therefore, of the earth and moon, and the direction in which the moon is seen from the earth, being unaltered by such a force, their relative situation is unaltered. An attraction, therefore, which acts equally, and in the same direction, on both bodies, does not disturb their relative motions. From this we draw the two following important con- clusions : (72.) Firstly. A planet may revolve round the sun, carrying with it a satellite, and the satellite may revolve round the planet in nearly the same manner as if the planet was at rest. For the attraction of the sun on the planet is nearly the same as the attraction of the sun on the satellite. It is true that they are. not exactly the same, and the effects of the difference E 50 GRAVITATION. will soon form an important subject of inquiry ; but they are, upon the whole, very nearly the same. The moon is sometimes nearer to the sun than the earth is, and sometimes farther from the sun ; and, therefore, the sun's attraction on the moon is sometimes greater than its attraction on the earth, and sometimes less ; but, upon the whole, the inequality of attractions is very small. It is owing to this that we may consider a satellite as revolving round a planet in very nearly the same manner (in respect of relative motion) as if there existed no such body as the sun. (73.) Secondly. The force which disturbs the motion of a satellite, or a planet, is the difference of the forces (measured as in (4.), by the spaces through which the forces draw the bodies respectively) which act on the central and the revolving body. Thus, if the moon is between the sun and the earth, and if the sun's attraction in a certain time draws the earth 200 inches, and in the same time draws the moon 201 inches, then the real disturbing force is the force which would produce in the moon a motion of one inch from the earth. (74.) In illustrating the second remark, we have taken the simplest case that can well be imagined. If, however, the moon is in any other situation with respect to the earth, some complication is introduced. Not only is the moon's distance from the sun different from the earth's distance (which according to (9.) produces an inequality in the attractions upon the earth and, moon), CONSTRUCTION FOR DISTURBING FORCE. 51 but also the direction in which the attraction acts on the earth is different from the direction in which it acts on the moon (inasmuch as the attraction always acts in the direction of the line drawn from the attracted body to the attracting body ; and the lines so drawn from the earth and moon to the sun are in different direc- tions). The same applies in every respect to the pertur- bation which one planet produces in the motion of a second planet round the sun, and which depends upon the difference in the first planet's attractions upon the sun and upon the second planet. To overcome this diffi- culty, we must have recourse to geometrical considera- tions. In fig. 17, let B! be a body revolving about A, PIG. 17. ''" and let C be another body whose attraction disturbs the motion of Bj round A. The attraction of C will in a certain time draw A to a ; it will in the same time draw B! to bi. Make Bj d l equal and parallel to A a ; then a d^ will be equal and parallel to A B x . Now if the force upon B x were such as to draw it to d l} the motion of B! round A would not be disturbed by that force. But the force upon B! is really such as to draw it to b^ The real disturbing force then may be represented as a force which draws the revolving body from d t to b^ If, instead of supposing the revolving body to be at B 1? we E 2 52 GRAVITATION. suppose it at B 2 , and if the attraction of C would draw it through B 2 b 2 while it draws A through A a, then (in the same manner, making B 2 d 2 equal and parallel to A a) the real disturbing force may be represented by a force which in the same time would draw B 2 through d 2 b 2 . (75.) Both the magnitude and the direction of this force are continually varying, and we must, if possible, find a convenient way of representing it. We shall have recourse here to the " composition of motion." In fig. 18, if d b represent the space through which a force FIG. 18. <1 has drawn a body in a certain time, the same effect may be produced by two forces of which one would in the same time draw the body from d to e, and the other would in the same time draw the body from e to b. And this is true whatever be the directions and lengths of d e and e b, provided that with d b they form a tri- angle. To accommodate the investigations of this Section to those of Section III., we will suppose de perpendicular to the radius vector, and eb parallel to the radius vector. In fig. 17, draw de perpendicular to A B or a d, and e b parallel to A B or a d ; and now we can say : the disturbing force produced by the attraction of C is a force represented by d e perpendicular to the radius vector, and a force represented by eb in the direction of the radius vector. (76.) We now want nothing but estimations of the magnitudes of these forces in order to apply the investi- DISTURBING FORCE AT CONJUNCTION. 53 gations of Section III. For the present we'shall con- tent ourselves with pointing out some of the most interesting cases. (77.) I. Let the disturbing body be exterior to the orbit of the disturbed body (this applies to the disturbance of the moon's motion produced by the sun's attrac- tion, the disturbance of the earth's motion by Jupi- ter's attraction, the disturbance of the motion of Venus by the earth's attraction, &c.) : and first, let the revolving body B be between the disturbing body C and the central body A (as in fig. 19). If the FIG. 19. c b d B A attraction of C will in a certain time draw A to a, it will in the same time draw B to &, where B 6 is much greater than A a. Take B d equal to A a, then d b is the effect of the disturbing force, which tends to draw B further from A. In this case, then, the dis- turbing force is entirely in the direction of the radius vector, and directed from the central body. This is the greatest disturbing force that can be produced by C. (78.) II. Let CAB (fig. 20) be in the same straight line, but let B be on the side of A, opposite to C. In this case B b is less than A a ; and if B c? is taken FIG. 20. c a A dE equal to A a, the disturbing force represented by d b will be entirely in the direction of the radius vector, and directed from the central body. This case is 54 GRAVITATION particularly deserving of the reader's consideration, as the effectual disturbing force is exactly opposite to the attraction which actually exerts upon B. (79.) III. The disturbing force, in the case represented in fig. 19, is much greater than that in the case of fig. 20, except C be very distant. Thus, suppose A B to be half of AC. In the first case, the attraction upon B (by the law of gravitation) is four times as great as the attraction upon A, and therefore the dis- turbing force (which is the difference of the forces on A and B) is three times as great as the attraction upon A. In the second case, the distance of B is f of the distance of A, and therefore the attraction upon B is |- of the attraction upon A, and the disturbing force is -f- of the attraction upon A. The disturbing force in the first case is, therefore, greater than in the second case, in the proportion of 3 to -|, or 27 to 5. This remark applies to nearly all the cases of pla- netary disturbance where the disturbing planet is exterior to the orbit of the disturbed planet, the ratio between these distances from the sun being a ratio of not very great inequality. But it scarcely applies to the moon. For the sun's distance from the earth is nearly 400 times the moon's distance : consequently when the moon is between the sun and the earth, the attraction of the sun on the moon is (|-f f ) 2 X the attraction of the sun on the earth, or 44linrT parts of the sun's attraction on the earth, and the disturbing force therefore is T3 z g s^ 5T parts of the sun's attraction on the earth: but when the moon is on that side farthest from the sun, the sun's attraction on the GREATER THAN AT OPPOSITION. 55 moon is (fg-r) 2 or 1 g g g g ? P arts f tne sun ^ s attraction on the earth, and the disturbing force is \ / 8 T 1 parts of the sun's attraction on the earth, which is very little less than the former. The effects of the dif- ference are, however, sensible. FIG. 21. a A I (80.) IV. Suppose B, fig. 26, to be in that part of its orbit which is at the same distance from C as the distance of A from C. The attraction of C upon the two other bodies, whose distances are equal, will be equal, but not in the same direction. B &, therefore, will be equal to A a. But since C B is also equal to C A, it is evident that a b will be parallel to A B, and therefore b will be in the line a d. Consequently in this case also the disturbing force will be entirely in the direction of the radius vector ; but here, unlike the other cases, the disturbing force is directed towards the central body. The magnitude of the disturbing force bears the same proportion to the whole attraction on A which b d bears to B b, or A B to A C. Thus, in the first numerical instance taken above, the disturbing force in this part of the orbit is -i- of the attraction on A : and in the second nume- rical instance, the disturbing force is -p^- of the attraction on A. It is important to observe that in both instances the disturbing force, when wholly directed to the centre, is much less than either value 56 GRAVITATION of the disturbing force when wholly directed from the centre : in the latter instance it is almost exactly one-half. (81.) When the disturbing body is distant, the point of the orbit which we have here considered is very nearly that determined by drawing A B perpendicular toCA. (82.) Y. When C is distant (as in the case of the moon disturbed by the sun), the disturbing forces mentioned in (III.) and (IV.) are nearly propor- tional to the distance of the moon from the earth. For the force mentioned in (IV.) this is exactly true, whether C be near or distant, because (as we have found) the disturbing force bears the same propor- tion to the whole attraction on A which A B bears to A C. With regard to the force mentioned in (III.) ; if we suppose the moon's distance from the earth to be 4-^-0 of the sun's distance, the disturbing force when the moon is between the earth and the sun is TsWor P arts f tne sun ' s attraction on the earth, or nearly -5^-5 th part. But if we suppose the moon's distance from the earth to be -g^th of the sun's distance, the attraction on the moon (when between the earth and the sun) would be (-rf-) 2 or -f-JHnrr parts of the attraction on the earth ; the disturbing force, or the difference of attractions on the earth and moon, would be T | f-| T , or nearly -r^th part of the sun's attraction on the earth. Thus, on doubling the moon's distance from the earth, the disturbing DEPENDS ON BODIES' DISTANCES. 57 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 distance 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 disturbing force is nearly -g-i^- 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 - parts of the sun's former distance from the earth; the attractions on the earth and moon respectively will be and j jJSSoi parts of the former attraction on the earth: and the disturb- ing force, or the difference between these, will be aaVsVo-tJ or nearly TB i 5 ^.th part of the former attrac- tion of the earth. Thus, on doubling the sun's distance, the disturbing force is diminished to ^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 proportion. (84.) VII. Suppose B to have moved from that part of its orbit where its distance from C is equal to A's 58 GRAVITATION. 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 distance from C, the disturbing force is in the direction 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 from 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 appear that, between the two points that we have mentioned, there is always a disturbing force d 2 e 2 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 situation B 3 , fig. 22, in which B is moving from the 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 GENERAL VIEW OF RESULTS. 59 vector, in the direction 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 dis- tances, 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!, in fig. 17, where B is moving from the point on the side of A opposite C to the next point of equal distances, there is a dis- turbing force perpendicular to the radius vector, in the direction opposite to that in which B is going. (85.) The results of all these cases may be collected thus. The disturbing body being exterior to the orbit of the revolving body, there is a disturbing 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 disturb- ing 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 disturbing 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 60 GRAVITATION. disturbing body, or on the opposite side, to one of the equidistant points, there is always a disturbing force perpendicular to the radius vector tending to retard it ; and while it is moving from one of the equidistant points to one of the points on the same side of the cen- tral body as the disturbing body, or the opposite, there is a disturbing force perpendicular 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 B 1? fig. 23, the FIG. 23. / attraction of C draws A strongly towards B b and Bj strongly towards A, and, therefore, there is a very powerful disturbing force drawing B x 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 the disturbing force perpendicular to the radius vector : If A C is greater than A Bj, it will be possible to find two points, B 2 and B 4 , whose dis- tance from C is equal to the distance of A from C, DISTURBING BODY INTERIOR TO THE ORBIT. 61 and there the disturbing force perpendicular 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 2 , it will be seen by such reasoning as that of (75.) and (84.), that the disturbing force, perpendicular to the radius vector, retards B's motion ; while B moves from B 2 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!, it accelerates B's motion. But if A C is less than 4- A B x , 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! to B 3 , and retards B as it moves from B 3 to B!. We shall now proceed to apply these general principles to particular cases. SECTION V. Lunar Theory. (87.) THE distinguishing feature in the Lunar Theory is the general simplicity occasioned by the great dis- tance of the disturbing body (the sun alone producing any sensible disturbance), in proportion to the moon's distance from the earth. The magnitude of the dis- turbing 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. 62 GRAVITATION. 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 between 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 vector, 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 quadratures), the force is entirely in the direction of the radius vector, and directed to the earth. At certain 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 diminish the earth's attraction. Thus the moon's mean distance from the earth is less (see (46.) ) than it would have been with the same MOON'S ANNUAL EQUATION. 63 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, produces no permanent effect. (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 disturbing force is inversely proportional to the cube of the sun's dis- tance ; and, consequently, it is sensibly greater when the earth is at perihelion than when at aphelion. Therefore, while the earth moves from perihelion to aphelion, the disturbing force is continually dimi- nishing ; and while it moves from aphelion to peri- helion, the disturbing force is constantly increasing. Eeferring then to (47.) it will be seen that in the former of these times the moon's orbit is gradually diminishing, and that in the latter it is gradually enlarging. And though this alteration is not great (the whole variation of dimensions, from greatest to least, being less than -3~oVo)> 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 slower. Now, if the moon's mean motion is determined by comparing two places observed at the interval of 64 GRAVITATION. many years, the angular motion so found is a mean between the greatest and least. Therefore, when the earth is at perihelion, 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 before its mean place. This inequality is called the moon's annual equation; it was disco- vered 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 independent of excentricity, may thus be investigated. 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^SuP can move. It cannot move in a circle : for the force perpendicular to the radius vector retards the moon as it goes from B^ to B 2 , fig. 24, and its velocity is, therefore, less at B 2 than at Bj, and on this account (supposing the force MOON'S VARIATION. 65 directed to A at B 2 equal to the force directed to A at B x ), the orbit would be more curved at B 2 than at B,. But the force directed to A at B 2 is much greater than that at B x (see (88.) ) ; and on this account the FIG. 24. orbit would be still more curved at B 2 than at B! ; whereas, in a circle, the curvature is everywhere the same. The orbit cannot, therefore, be circular 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 FIG. 25. B 4 c B (in consequence of the disturbing force perpendicular to the radius vector having retarded it), while the earth's attraction is great (in consequence of the nearness of B 2 ), 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 B D where the velocity is great, the moon far oif, and the disturbing force directed from the earth. But, on the contrary, the curvature at B 2 is much less than at Bj ; therefore, this form of orbit is not the true one. But if the orbit be supposed to be oval, with F 66 GRAVITATION. 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 FIG. 26. force having acted perpendicularly to the radius vector, while the moon goes from B x to B 2 ; and though the distance from A being greater, the earth's attraction at B 2 will be less than the attraction at BI; yet, when increased by the disturbing force, directed to A at B 2 , it will be very little less than the attraction diminished by the disturbing force at B! The dimunition of velocity then at B 2 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 curvature 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 sta- tionary 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, WQ have only to suppose that the oval twists round after the sun, and the same reasoning applies. The curve described by MOON'S VARIATION. 67 the moon is then such as is represented in fi#. 27. As the disturbing force, perpendicular to the radius vector, acts in the same direction for a longer time than in the FIG. 27. 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 mentioned, the law of uniform description of areas ly 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 2 and B^ fig. 26, would be much less than that near B x and B 3 . But in conse- quence of the disturbing force, perpendicular to the radius vector (which retards the moon from Bj 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^, the angular motion is still less at B 2 and B 4 , and still greater at B x and B 3 . The angular motion, therefore, diminishes considerably while the moon moves from B! to B 2 , and increases considerably while it moves from B 2 to B 3 , &c. The mean angular motion, deter- mined by observation, is less than the former and greater than the latter. Consequently, the angular motion at B! is greater than the mean, and that at B 2 is less than the mean ; and therefore (as in (90.) ), from B x to F 2 68 GRAVITATION. B 2 the moon's true place is before the 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! the true place is behind 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 some- times 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 conjunction to be increased, and that near opposi- tion to be diminished. Observing what the nature of these forces is (77.), (78.), and (84.), this amounts to supposing that near conjunction the force necessary to make up the difference is a force acting in the radius vector, and directed from the earth, and a force per- pendicular to the radius vector, accelerating the moon before conjunction, and 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 disturbing 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 MOON'S PARALLACTIC EQUATION. 69 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 described ? A very little consideration of (57.), (58.), and (68.), will show, that unless the line of apses pass through the sun, the excen- tricity 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 progress, and we must, therefore, choose that position in which the forces will cause pro- gression 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 of apses to regress. This supposition, then, cannot 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 pro- gress. 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 opposite side. This irregularity is called the parcdlactic 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 (independent of that dis- cussed here) would be to make its line of apses progress 70 GRAVITATION. with a considerable velocity. 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 parallactic inequality is consider- ably increased by the indirect effect of the other dis- turbing forces. (95.) The magnitude of the forces concerned here is about 1-5-0 th of those concerned in (91.), &c. ; but the effect is about -^-ih of their effect. This is a striking instance of the difference of proportions in forces, and the effects that they produce, depending on the differ- ence in their modes of action. The inequality here dis- cussed 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 magnitude 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 understood, 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 MOONS PARALLACTIC EQUATION. 71 forces. The moon's variation,^ therefore, is somewhat diminished near conjunction, and is somewhat increased near opposition. (97.) It will easily be imagined, that if there is an excentricity in the moon's orbit, the effect of the varia- tion upon that orbit will be almost exactly the same is if there were no excentricity.* Thus, supposing that * As this general proposition 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, in fig. 28, the moon cannot describe the circle Bj, & 2 , FIG. 28. C B 3 , & 4 , though it touches at Bj and B 3 , and the reason that it will describe the oval B 1? B 2 , B 3 , B 4 , is, that the disturbing force makes the forces at Bj 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 at B 2 , than it would otherwise have been. So that, unless we supposed it moving at B v with a greater velocity than it would have had, undisturbed, in the circle B 1} & 2 , B 3 , & 4 , the great curvature produced 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 1? 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 FIG. 29. fig. 29, if the velocity at B, were not greater than it would have had, undisturbed, in the ellipse B t , & 2 , B 3 , & 4 , the increased curvature at B 2 , produced by the increased force and diminished velocity there, would 72 -GRAVITATION. the orbit without the disturbing force, had such a form as the dark line in fig. 30, it will, with the disturbing FIG. 30. force, have such a form as the dotted line in that figure. The same must be understood in many other cases of different inequalities which affect the motion of the same body. (98.) IV. We now proceed with the disturbances 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 l} fig. 31, that is near FIG. 31. B 4 perigee, the disturbing force is directed from A ; and consequently, by (51.), the line of apses regresses. While the moon is near B 3 , that is, near apogee, the have brought it much nearer to A than the point B 3 ; but with a large velocity at Bj it will go out at B 2 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. VARIATION COMBINED WITH EXCENTRICITY. 73 disturbing force is* also directed from A, and, conse- quently, by (54.), the line of apses progresses. The question, then, now is, which is the greater, the regress, when the moon is near B 1? or the progress, when it is near B 3 ? To answer this, we will remark that if the disturbing force, directed from A, were in- versely proportional to the square of the distance (and, consequently, less at B 3 than at B x ), it would amount to exactly the same as if the attraction of A were altered in a given proportion ;* and in that case * The reasoning in the text may be 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 proportional to the square of the distance. 2nd, as the original attraction, with a force superadded, which may be treated as a disturbing 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, 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 while she is near apogee. There is, however, no incon- gruity between 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 consider- ation, must be such as to produce 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 equal to the regression at the other time. 74 GRAVITATION. B would 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 B t . But, in fact, the disturbing force at B 3 is to that at B t in the same proportion as A B 3 to A B 1? by (82.), and, therefore, the disturbing force at B 3 is greater than that at B! ; and, consequently, much greater than that which would produce a progression equal to the regression produced at B x ; and, therefore, the effects of the disturbing force at B 3 , predominate, and the line of apses progresses. The disturbing force directed to A in the neighbourhood 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 opposite kind (55.). (99.) The disturbing force directed from A, though the only one at B x and B 3 , is not, however, the only one in the neighbourhood of B jand B 3 . While the moon is approaching to B l5 the force perpendicular to the radius vector accelerates the moon, and therefore by (65.), as B! is the place of perigee, the line of apses regresses ; when the moon has passed B 1? the force retards the moon, and therefore, by (66.), the line of apses still regresses. But when the moon is approaching B 3 , the force perpendicular 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 question now is, APSES IN SYZYGIES PROGRESS. 75 whether the progression produced by the force perpen- dicular to the radius vector near B 3 , 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 entirely upon the proportion * which the velocity produced by the disturbing force bears to the * Suppose, for facility of conception, that the force, perpendicular to the radius vector, acts in only one place in each quadrant between syzygies and quadratures. The portions of the orbit which are bisected by the line of syzygies will be described with greater velocity in conse- quence 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 central force were diminished, while the moon passed through the portions bisected by the line of syzygies. If now the imaginary diminution of central force were in the same proportion (that is, if the real increase of velocity were in the same proportion) at both syzygies, which here coincide with the apses, the regression of the line of apses produced at perigee, would be equal to the progression produced at apogee. But the increase of velocity pro- duced by the force perpendicular to the radius vector near apogee, is much greater than that near perigee. First, because the force is greater, in proportion to the distance. Second, because the time of describing a given small angle is greater in proportion to the square of the distance ; 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, predominates 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. 76 GRAVITATION. velocity of the moon ; and since from B 2 to B 3 , and from B 3 to B 4 , the disturbing force is greater than that from B 4 to Bj, and from B x 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 Bj 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 x , B^ it follows, that the effect in passing through B 2 , B 3 , B 4 , is much greater than that in passing through B^ B 1} and B 2 . Con- sequently, the effect of this force also is to make the line of apses progress. (100.) On the whole, therefore, when the perigee is turned towards the sun, the line of apses progresses rapidly. And the same reasoning applies 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 consequently, 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 at perigee and at apogee would balance if the disturbing force were inversely proportional to the square of the distance from the earth. But the disturbing force is really pro- portional to the distance from the earth: and, there- fore, as in (98.), the effect of the disturbing force while the moon is at apogee preponderates over the other ; and APSES IN QUADRATURES REGRESS. 77 therefore the force directed to the centre causes the line of apses to regress. (102.) We must also consider the force perpendicular 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 perigee 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 causes it when the excen- tricity is smaller ; yet, in the present instance, the force which causes it is itself proportional to the ex- centricity (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 78 GRAVITATION. 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, con- sidered in (99.), is almost exactly equal to that con- sidered 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 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 perpendicular 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 following reasons. t PROGRESS OF THE MOON'S APSE. 79 (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, 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 -}-fths of her whole motion, but at the latter with only -^-ths; conse- quently, when near perigee, 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 quan- tity), is -pf-ths of the time in which it would have passed through the same angle had the sun been stationary ; when near apogee, the number expressing the proportion is yths. The latter number is greater than the former ; and, therefore, the eifect of the forces acting near apogee is increased 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 bear a considerable proportion to the effective motion previously found ; and thus the effec- tive motion will be sensibly increased. 80 GRAVITATION. (107.) Secondly. When the line of apses is directed 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 therefore increased, while that of the latter is diminished : and the preponderance of the former is much increased. It is in increasing the rapidity of progress at one time, and the rapidity of regress at another, that the force perpendicular to the radius vector indirectly increases the effect of the former in the manner just described. (108.) From the combined effect of these two causes the actual progression of the line of apses is nearly double of what it would have been if, in different revo- lutions of the moon, different parts of its orbit had been equally subjected to the disturbing force of the sun. (109.) The line of apses upon the whole, therefore, progresses ; and (as calculation and observation agree in showing) with an angular velocity that makes it (on the average) describe 3 in each revolution of the moon, and VARIATION OF MOON'S EXCENTRICITY. 81 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 astronomy. (110.) Y. 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 Bj (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 excen- tricity and sometimes increases it. But while the moon moves from B 3 to B 1? there are exactly equal forces acting in the same manner at corresponding parts of the half-orbit, and these, by (58.), will pro- duce effects exactly opposite. On the whole, there-? fore, the disturbing 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 1? and diminishes it when moving from B x to B 2 ; in moving, therefore, from B 4 to B lf the excentricity is increased (65.), and in moving from B x to B 2 , it is as much diminished (66.). Similarly in moving from B 2 to B 3 , the excentricity is diminished, and in moving from B 3 to B 4 , it is as much increased. This force, therefore, produces no effect on the excentricity. G 82 GRAVITATION. 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 dif- ferent sides of perigee and apogee balance each other in the same way. (112.) But if the line of apses is inclined to the line joining the earth and sun, as in fig. 32, the effects of the FIG. 32. 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 excentricity is diminished. From the slowness of the motion at B 2 (which gives the disturbing force more time to produce its effects), and the greatness of the force, the effect at B 2 will preponderate, and the com- bined effects at B 2 and B 4 will diminish the excen- tricity. This will appear from reasoning of the same kind as that in (98.). At B! and B 3 , the force in the VARIATION OF MOON'S EXCENTRICITY. 83 radius vector is directed from the earth : at B 1} there- fore, by (59.), as the moon is moving from perigee, the excentricity is increased, and at B 3 it is diminished: but from the slowness of the motion at B 3 and the mag- nitude of the force, the effect at B 3 will preponderate, and the combined effects at Bj and B 3 will diminish the excentricity. On the whole, therefore, the force in the direction of the radius vector diminishes the excen- tricity. The force perpendicular to the radius vector retards the moon from B x to B 2 , but the first part of this motion may be considered near perigee, and the second near apogee, and, therefore, in the first part, it di- minishes the excentricity, and in the second increases it; and the whole effect from B! 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 x , the force accelerates the moon, and therefore, by (68.) (the moon being near perigee), in- creases the excentricity ; and from B 2 to B 3 , the force also accelerates the moon, and by (68.) (the moon being near apogee), diminishes the excentricity ; and the effect is much * greater (from the slowness of the moon and * 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 proportion alters the reciprocal of the apogeal distance by a given quantity, and, therefore, alters the apoge&l distance by a quantity nearly 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) G 2 84 GRAVITATION. the greatness of the force) between B 2 and B 3 than between B 4 and B 1? 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 sun, the excen- tricity is diminished at every revolution 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 joining 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 is nearly proportional to the apogeal distance. Similarly, if the velocity at apogee is increased in a given proportion, the ratio of the alteration of perigeal distance to perigeal distance (on which the alteration of excentricity depends) is nearly proportional to the peri- geal distance. Thus if the velocity were increased in the same proportion at perigee 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 proportion as the square of the distance), so that the increase of velocity is in the proportion of the cube of the distance. Third, because the actual velocity is less (being inversely as the distance), BO that the ratio of the increase to the actual velocity is proportional to the fourth power of the distance. Combining this proportion with that above, the alterations of excentricity in the case before us, produced by the forces acting at apogee and at perigee, are in the proportion of the cubes of the apogeal and perigeal distances respec- tively. VARIATION OF MOON'S EXCENTRICITY. 85 and from apogee : and is directed from the earth as the moon moves to perigee and to apogee ; which directions are just opposite to those in the case already con- sidered. Also the force perpendicular to the radius FIG. 33. 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 excentricity 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 represented in the figure. (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 excentricity is greatest when the line of apses passes through the sun; and is least when the line 86 GEAYITATIOX. 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 dimi- nished ; 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. 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 moon's perigee (104.). 5th. The irregularity in the motion of the perigee, de- pending on the position of the perigee with respect to the sun (109.). 6th. The alternate increase and diminution of the excen- tricity, depending on the position of the perigee with respect to the sun (115.). These inequalities were first explained (some imper- fectly) 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 GENERAL VIEW OF LUNAR INEQUALITIES, 87 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 in the suppo- sitions that we have made. Some of these, it may easily be seen, will arise from variations of force which we have already explained. Thus the difference of disturbing forces at conjunction 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 de- pending 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 astronomical 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 motion, 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 88 GEAVITATION. to the earth : and in (90.) we have mentioned that this effect is greater when the earth is near perihelion, than when the 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 depending 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 disturb- ing 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. Con- sequently 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 gradually in- creasing. The size of the moon's orbit has therefore, gradually (but insensibly), diminished (47.) : but the moon's place in its orbit has sensibly altered (49.), and the moon's angular motion has appeared to be per- petually quickened. This phenomenon was known to astronomers by the name of the acceleration of the moon's mean motion, before it was theoretically ex- plained in 1787, by Laplace : on taking it into account, the oldest and the newest observations are equally well represented by theory. The rate of progress of the moon's line of apses has, from the same cause, been somewhat diminished. PERIODIC TIMES NEARLY AS ONE TO TWO. 89 SECTION VI. Theory of Jupiter's Satellites. (120.) JUPITER has four satellites revolving round him in the same manner in which the moon revolves round the earth ; and it might seem, therefore, 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 dif- ferent. The fourth satellite (or that revolving in the largest orbit) has a small irregularity analogous tq 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 minutes, and the other two are very much less. The corresponding inequalities in the motion of the other satellites are still smaller. But these satellites disturb each other's motions, to an amount and in a manner of which there is no other example in the solar system; and (as we shall afterwards mention) their motions are affected in a most remarkable degree by the shape of Jupiter. (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 independent of pertur- bation. Thirdly, that a very remarkable relation exists (and, as we shall show, necessarily exists) between the motions of the first three satellites. Before proceeding with the theory of the first three 90 GRAVITATION. satellites, we shall consider a general proposition which applies to each of them. (122.) Suppose that two small satellites revolve 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, describing a curve of the same form at every revolution ? (123.) The orbits will be sensibly elliptical, as the perturbation produced by a small satellite in one revo- lution will not sensibly alter the form of the orbit. The same form being supposed to be described each time, the major axis and the excentricity are supposed invariable, and the position 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 variation 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 b B 2 , represent the orbit of FIG. 34. the first, and C 3 , Cj, C 2 , the orbit of the second. Sup- pose that when B was at B 1? C was at C b so that A, B 1} G!, 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 PERIODIC TIMES NEARLY AS ONE TO TWO. 91 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 Bj and C^ 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 former line of conjunction BX GX. And it is plain that there has been no other conjunction since that with which we started, as the successive conjunctions 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! C ly the next will be in the line A Ba C& 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 satellite is double that of the other, the more slowly will the line of conjunction regress. (125.) As the principal part of the perturbation is produced when the satellites are near conjunction (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 question now becomes this : What must be the excen- tricities of the orbits, and what the positions of the 92 GRAVITATION* 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 ? (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 disturbing 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 al- lowed, and, therefore, 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 progress. Also the force perpen- dicular to the radius vector, before the first satellite has reached conjunction (and when the second satellite, which moves more slowly, is nearer to the point" of conjunction than the first), tends to accelerate the first satellite ; and that which acts after the satellites have passed conjunction, tends to retard the first satellite; and both these, by (65.) and (66.), 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 ORBIT OF INTERIOR SATELLITE. 93 conjunction, that is, shall regress; therefore, the apo- jove 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 conjunc- tion, everything becomes consistent. The disturbing force, in the direction of the radius vector, from the central body, will, by (51.), cause the line of apses to regress. The force perpendicular to the radius vector, which accelerates the first satellite before it has reached conjunction, that is, before it has reached the perijove, and retards it after that time, will also, by (65.) and (66.), cause the line of apses to regress. Also, as in (56.), this regression will be greater as the excentricity of the orbit is less, because the disturbing force, which acts here, does not depend on the excentricity. By proper choice, therefore, of a value of the excentricity, we can make an orbit, whose line of apses will always regress exactly as fast as the line of conjunction, and will, therefore, always coincide with it ; whose excentri- city, in consequence, will never alter, by (59.) and (68.) ; and whose general shape, therefore, will be the same at every successive revolution. (129.) We shall mention hereafter, that the form of Jupiter is such as would cause the perijove of the first satellite, if it were not disturbed by the second satellite, to progress with a velocity not depending upon the ex- centricity of the orbit. The only alteration which this makes in our conclusions is, that the excentricity of the orbit must be so chosen, that the perturbation of which 94 GRAVITATION. we have spoken will cause a regression equal to the sum of the progression which Jupiter's shape would occa- sion, and the regression of the line of conjunction. As this is greater than the regression of the line of con- junction alone, the excentricity of the orbit must be less. So that the only effect of Jupiter's shape is to diminish, in some degree, the excentricity of the orbit. (130.) Now let us inquire what must be the form and position of the orbit of the second satellite. As before, the principal part of the perturbation is near conjunction. At and near the conjunction, the disturb- ing force, in the direction of the radius vector, is directed to the central body. Before conjunction, when the first satellite is less advanced than the second, the disturbing force, perpendicular to the radius vector, retards the second, by (86.). For, the periodic time of the second being nearly double that of the first, the mean distances from the planet will be nearly in the proportion of 7 to 11 (as the proportion of the cube of 7 to the cube of 11 is nearly the same as the proportion of the square of 1 to the square of 2, see (34) ), and, therefore, near con- junction, the distance of the first from the second is less than the distance of the first from the central body. After conjunction, the disturbing force accelerates the second body. Now, without going through several cases as before, which the reader will find no trouble in doing for himself, we shall remark, at once, that if the apojove of the second satellite is in the direction of the points of conjunction, both the disturbing force, directed to the central body at apojove, and that perpendicular to the ORBIT OF EXTERIOR SATELLITE. 95 radius vector, retarding it before it reaches apojove, and accelerating it afterwards, by (53.), (65.), and (66.), will cause the line of apses to regress ; and that, by proper choice of excentricity, the regression of the line of apses may be made exactly equal to the regression of the line of conjunction. (131.) Our conclusion, therefore, is : If two satellites revolve round a primary, and if the periodic time of one is very little greater than double the periodic time of the other, and if we assume that the orbits described have always the same form (that is, if they have no excentricity independent of perturbation) ; then the orbits will not sensibly differ from ellipses, the lines., of apses of both orbits must always coincide with the line of conjunctions, and the perijove of the first orbit, and the apojove of the second, must always be turned towards the points of conjunction. It appears also, that these conditions are sufficient, inasmuch as the rate of regress of the lines of apses will (with proper values for the excentricities) be the same as the rate of regress of the line of conjunctions, and the excentricities then will not change. The excentricities of the orbits will be greater as the regresa of the line of conjunctions is slower, or as the proportion of the periodic times approaches more exactly to the proportion of 1 : 2. (132.) In the same manner it would be found, that if the periodic time of one satellite were very little less than double that of the other, the lines of apses (in order that similar orbits may be traced out at each revolution) must always coincide with the line of conjunction, and 96 GRAVITATION. the apojove of the first satellite and the perijove of the second must always be turned towards the points of conjunction; and the excentricities of the orbits must be greater, as the proportion of the periodic times approaches more exactly to the proportion of 1 : 2. (133.) The same thing exactly would hold, if the periodic times were very nearly in the ratio of 2 : 3, or of 3:4, &c., but these suppositions do not apply to Jupiter's satellites. (134) Having thus found the distortion produced by the disturbing force in orbits which have no excen- tricity, independent of perturbation, it will easily be imagined, that the same kind of distortion will be pro- duced if the orbits have an original excentricity. If we make, in an elliptic orbit, the same kind of alteration which must be made in a circular orbit, in order to form the figure found above, we shall have nearly the orbit that will be described from the combined effects of pertur- bation and of excentricity independent of perturbation.* * The truth of this proposition may be shown more fully in the following manner : Let A, fig. 35, be the place of the primary, A C the line of conjunctions ef the first and second satellites, B D E the C ^ZZ^Z^H * FIG. 35. elliptic orbit, in which the first satellite would move if undisturbed, D its perijove. Suppose (to simplify the figure) that the attraction of the second satellite acts only for a limited space; for instance, APPLICATION TO JUPITER'S SATELLITES. 97 We shall now proceed with the application of these conclusions to Jupiter's first three satellites. while the first satellite passes from F to H. Then the result of the investigations from (122.) to (131.) is, that the first satellite will be drawn outwards from the orbit in which it would have moved, so as to describe a curve F G H ; and when the disturbing force ceases at H, it will proceed to describe an ellipse, H e b d, similar to B D E, but with this difference, that the perijove is at d instead of D. The conclusion, however, now that it has been securely obtained from the reasoning above, may be stated as the result of the following reasoning : In consequence of the disturbing force, which has drawn the first satellite outwards, without, upon the whole, altering its velocity (accelerating it before conjunction, and retarding it afterwards), the satellite has moved in a curve, F G H, external to the ellipse F D, in which it would have moved ; and after the disturbing force has ceased at H, the satellite (which is moving in a path inclined externally from the old orbit) continues to recede from the old orbit till the diminution of velocity (26.) allows its path to be so much curved, that at e it begins to approach, and at L the new orbit intersects the old one ; and after this, the path is inclined internally from the old orbit, till the increase of velocity (25.) makes its path so little curved that it approaches the old orbit again, and again crosses it between d and D. In like manner, if, as in fig. 36, the orbit B F E have an excentricity independent of perturbation (the perijove being at any point D 1 ), nevertheless, we may state that, in consequence of the disturbing force, the satellite will move in a curve F G H external to F E ; but when the disturbing force ceases at H, the satellite (which is moving in a path inclined externally from the old orbit) continues to recede from the old orbit till the diminution of velocity (26.) allows its path to be so much curved, that it begins to approach at some point e ; that at some point L, nearly opposite to C, the new orbit intersects the old H 98 GRAVITATION. (135.) The periodic time of Jupiter's first satellite is, 1 day, 18 hours, 27 minutes, and 34 seconds ; that of one ; and that, after this, the path is inclined internally from the old orbit, till the increase of velocity (25.) makes its path so little curved that it approaches the old orbit again, and again crosses it between F and H. Thus, the alteration of the radius vector, drawn in any given direction, as A K (which in the new orbit is altered to A fc) is nearly the same in the second case as in the first. This, however, is the alteration produced in a single revolution of the satellite ; but as the same applies to every successive revolution, it follows that the inequality or variation of the radius vector in the second case is nearly the same as in the first case ; and thus the proposition of the text is proved. The inequality of the radius vector would be somewhat different if the excentricity of the orbit in the second case were considerable, partly because the places of conjunction would not be at equal angular distances, partly because the disturbing forces would be different (as the distance between the satellites in conjunction would not always be the same), and partly because the effect of a given force is really different, according to the part of the orbit at which it acts. But where the excentricity is so small, as in the orbit of Jupiter's third satellite, or in those of the old planets, the alteration of the inequality of the radius vector produced by these differences is hardly sensible. The reasoning of this note may be applied, with the proper altera- tions, to every case of perturbation, produced by a disturbing force which is nearly independent of the form of the orbit ; and as this will apply successively to each of the causes producing disturbance, we shall at last arrive at the following general proposition : If several disturbing forces act on a planet or satellite, and if we estimate the inequality in the radius vector, which each of these would produce, supposing the orbit to have no excentricity independent of pertur- bation; then the inequality really produced, supposing the orbit to have an independent excentricity, will be nearly the same as the sum of all the inequalities so estimated. It is to be remarked, that if an orbit have an independent excen- tricity, and if the orbit receive an alteration similar to an elliptic inequality (that is, if it be elongated on one side and flattened on the other), the orbit is still sensibly an ellipse, of which the original focus is still the focus. Thus, in the instance occupying the first part of this note, as the inequality impressed on the elliptic orbit in the second case is the same as the inequality in the first case, that is, is ORBITS OF FIRST AND SECOND SATELLITES. 99 the second satellite is 3 days, 13 hours, 13 minutes, and 42 seconds : that of the third satellite is, 7 days, 3 hours, 42 minutes, and 32 seconds. The periodic time of the second satellite exceeds, by a small quantity, double that of the first, so that the preceding investigations apply to the motion of these two satellites. In fact, 275 revolutions of the first satellite are finished in almost exactly the same time as 137 revolutions of the second. If then, at a certain time, these two satellites start from conjunction, they will be in con- junction near the same place at every revolution of the second satellite, or at every second revolution of the first satellite : but the line of conjunction will regress slowly ; and when the first satellite has finished 275 revolutions, or one revolution more than double the number made by the second satellite, they will again be in conjunction in the same place as before, the line of conjunction having regressed till it has again reached the same position : this takes place in 486J days. (136.) From the preceding investigation then it ap- pears that, as these orbits have no excentricity inde- pendent of perturbation, they will be elliptic, and the line of apses of each orbit will regress so as to turn completely round in 48 6J days ; and that when in con- junction, the first satellite will always be in perijove, and the second satellite will always be in apojove. similar to an elliptic inequality, the orbit so altered will still be an ellipse, whose excentricity and line of apses are altered. We might, therefore, have obtained our results by at once investigating the alterations of the excentricity and line of apses produced by the dis- turbing forces ; but the method adopted in the text is simpler. H 2 100 GRAVITATION. (137.) But the periodic time of the third satellite is almost exactly double that of the second satellite, ex- ceeding the double by a small quantity; and on this account the orbit of the second satellite will be distorted from the form which otherwise it would have had, by an inequality similar to that just investigated. In a word, the line of conjunction of the second and third satellites will slowly regress, and the orbit of the second satellite will always be compressed on the side next the points of conjunction, and elongated on the opposite side ; and the orbit of the third satellite will always be elongated on the side next the points of conjunction, and compressed on the opposite side. (138.) Now we come to the most extraordinary part of this theory. We have remarked that 275 revolutions of the first satellite are finished in almost exactly the same time as 137 revolutions of the second ; but it will also be found that 137 revolutions of the second are finished in almost exactly the same time as 68 revolu- tions of the third : all these revolutions occupying 486 j- days. Because 275 exceeds the double of 137 by 1, we have inferred that the line of conjunctions of the first and second satellites regresses completely round in 275 revolutions of the first satellite, or in 486-J days. In like manner, because 137 exceeds the double of 68 by 1, we infer that the line of conjunctions of the second and third satellites regresses completely round in 137 revolutions of the second satellite, or in 48 6 days. Hence we have this remarkable fact : the regression of the line of conjunction of the second and third satellites KELATION BETWEEN LINES OF CONJUNCTION. 101 is exactly as rapid as the regression of the line of conjunc^ tion of the first and second satellites, So accurate Is this law, that in the thousands of revolutions of the satellites which have taken place since they were dis- covered, not the smallest deviation from it (except what depends upon the elliptic form of the orbit of the third satellite) has ever been discovered. (139.) Singular as this may appear, the following law is not less so :-~-The line of conjunction of the second and third satellites always coincides with the line of conjunction of the first and second satellites produced backwards, the conjunctions of the second and third satel- lites always taking place on the side opposite to that on which the conjunctions of the first and second take place. This defines the relative position of the lines of conjunc- tion, which (by the law of the last article) is invariable. Like that law, it has been found, as far as observation goes, to be accurately true in every revolution since the satellites were discovered, (140.) The most striking effect of these laws in the perturbations of the satellites is found in the motions of the second satellite. In consequence of the disturbing force of the first satellite, the orbit of the second satel- lite will be elongated towards the points of conjunction of the first and second (130.), and consequently compressed on the opposite side. In consequence of the disturbing force of the third satellite, the orbit of the second satel- lite will be compressed on the side next the points of conjunction of the second and third (128.). And be- cause the points of conjunction of the second and third 102 GRAVITATION. are always opposite to the points of conjunction of the first and second, the place of compression from one cause will always coincide with the place of com- pression from the other cause ; and therefore, the orbit of the second satellite will be very much compressed on that side, and consequently very much elongated on the other side. The excentricity of the orbit, depending thus entirely on perturbation, exceeds considerably the excentricity of the orbit of Venus. The inequalities in the motions of the satellites, produced by these excen- tricities, were first discovered (from observation) by Bradley about A.D. 1740, and first explained from theory by Lagrange, in 1766. (141.) The singularity of these laws, and the accuracy with which they are followed, lead us to suppose that they do not depend entirely on chance. It seems natural to inquire whether some reason may not be found in the mutual disturbances of the satellites, for the preservation of such simple relations. Now we are able to show that, supposing the satellites put in motion at any one time, nearly in conformity with these laws, their mutual attraction would always tend to make their motions follow these laws exactly. We shall show this by supposing a small departure from the law, and inves- tigating the nature of the forces which will follow as a consequence of that departure. (142.) Suppose, for instance, that the third satellite lags behind the place defined by this law ; that is, suppose that, when the second satellite is at the most compressed part of its ellipse (as produced by the THIS RELATION IS NECESSARY. 103 action of the first satellite), the third satellite is behind that place. The conjunction then of the second and third satellites will happen before reaching the line of apses of the orbit of the second, as produced by the action of the first. Now in the following estimation of the forces which act on the third satellite, and of their vari- ation depending on the variation of the positions of the lines of conjunction, there is no need to consider the influence which the ellipticity of the orbit of the second as produced by the third, or that of the third as pro- duced by the second, exerts upon the third satellite; because the flattening arising from the action of the third, and the elongation arising from the action of the second, will always be turned towards the place of con- junction of the second and third, and the modification of the action produced by this flattening and elongation will always be the same, whether the lines of con- junction coincide or not. In fig. 37, let C be the peri- FIG. 37. jove of the orbit of the second satellite (as produced by the action of the first satellite alone), D the point of the orbit of the third which is in the line A C pro- duced. If the third satellite is at D when the second is at C, the force produced by the second perpendicular 104 GRAVITATION. to the radius vector, retards the third before it reaches D, and accelerates it after it has passed D, by equal quantities. But if, as in the supposition which we have made, the conjunction takes place in the line A C^ D x , the retardation of the third satellite before conjunction is produced by the attraction of the second satellite before it arrives at perijove, when it is near to the orbit of the third satellite (and therefore acts powerfully), and moves slowly (and therefore acts for a long time) ; while the acceleration after conjunction is produced by the second satellite near its perijove, when it is far from the orbit of the third satellite (and therefore acts weakly), and moves rapidly (and therefore acts for a short time). The retardation therefore exceeds the acceleration ; and the consequence is, by (48.), that the periodic time of the third satellite is shortened, and therefore its angular motion is quickened; and therefore at the next con- junction, it will have gone further forward before the second satellite can come up with it, or the line of con- junction will be nearer to the place of perijove of the second satellite, depending on the action of the first. In the same manner, if we supposed the third satellite moving rather quicker than it ought in conformity with the law, the tendency of the forces -would be to ac- celerate it, to make its periodic time longer, and thus to make its angular motion slower. By the same kind of reasoning it will be seen that there are forces acting on the first satellite, produced by the elliptic inequality which the third impresses on the orbit of the second, tending to accelerate the angular motion of the first EXCENTRICITY OF THIRD SATELLITE. 105 satellite in the first case, and to retard it in the second. The same reasoning will also show that both the first and third satellites exert forces on the second, tending to retard its angular motion in the first case, and to ac- celerate it in the second. All these actions tend to preserve the law ; in the first case by making the line of conjunctions of the first and second satellites regress, and that of the second and third progress, till they coincide ; and in the second case, by altering them in the opposite way, till they coincide. (143.) Perhaps there is no theoretical permanence of elements on which we can depend with so great cer- tainty as on the continuance of this law. The greatest and most irregular perturbations of Jupiter or of his satellites, provided they come on gradually, will not alter the relation between their motions ; the effect of a resisting medium will not alter it ; though each of these causes would alter the motions of all the satellites ; and though similar causes would wholly destroy the conclu- sions which mathematicians have drawn as to the stability of the solar system, with regard to the elements of the planetary orbits. The physical explanation of this law was first given by Laplace, in A.D. 1784. (144.) We have terminated now the most remark- able part of the theory of these satellites. There are, however, some other points which are worth attending to, partly for their own sake, and partly as an intro- duction to the theory of the planets. (145.) The orbit of the third satellite, as we have mentioned, has a small excentricity independent of per- 106 GRAVITATION. turbation. Consequently, when the conjunction with the second takes place near the independent perijove of the third, the effect of the disturbance on the second is rather greater than at any other time ; and this pro- duces an irregularity in the excentricity of the second, and in the motion of its apses, depending on the dis- tance of the line of conjunction from the independent perijove of the third. The departure from uniformity in the angular motion of the third, also produces a departure from uniformity in the regression of the line of conjunction, and this contributes to the same irregularity. (146.) The disturbing force in the direction of the radius vector, produced by an inner satellite, is some- times directed to the central body and sometimes from it, but, on the whole, the former exceeds the latter (86.). Now the principal part of the effect really takes place when the satellites are near conjunction ; consequently, when the line of conjunction passes near the inde- pendent perijove of the third satellite, the force by which the third satellite is urged to the planet is greater than at any other time ; and as the line of con- junction revolves, the force alternately increases and diminishes. This produces an irregularity in the major axis, and consequently in the motion of the third satel- lite (47.), depending on the distance of the line of conjunction from the perijove of the third. (147.) The disturbing force in the direction of the radius vector produced by an outer satellite is some- times directed to the central body, and sometimes from EXCENTRICITY OF FOURTH SATELLITE. 107 it, but, on the whole, the latter exceeds the former (80.). For the reasons, therefore, in the last article, there is in the motion of the second satellite an irregularity depending on the distance of the line of conjunction from the independent perijove of the third, but opposite in its nature to that of the third satellite. (148.) Each of these irregularities in the motion of one of these satellites produces an irregularity in the motion of the others ; and thus the whole theory becomes very complicated when we attempt to take the minute irregularities into account. (149.) The motion of the fourth satellite is not related to the others in the same way in which they are related among themselves. Its periodic time is to the periodic time of the third nearly in the proportion of 7:3. Some of the irregularities then which it ex- periences and which it occasions are nearly similar to those in the motions of the planets. These, however, are small ; the most important are those depending on the changes in the elements which require many revo- lutions of the satellites to go through all their various states, but which, nevertheless, have been observed since the satellites were discovered. We shall proceed with these. . (150.) First, let us suppose that the third satellite has no excentricity independent of perturbation, and that the fourth satellite has a sensible excentricity, its line of apses progressing very slowly, in conseqneuce principally of the shape of Jupiter (so slowly as not to have gone completely round in eleven thousand revo- 108 GRAVITATION. lutions of the satellite). When each of the satellites has revolved a few hundred times round Jupiter, their conjunctions will have taken place almost indifferently in every part of their orbits. If the orbit of the fourth as well as that of the third had no independent ellip- ticity, there would be no remarkable change of shape produced by perturbation, as the action of one satellite upon the other would be the same when in conjunction in all the different parts of the orbit. But the orbit of the fourth being excentric, the action of each satellite on the other is greatest when the conjunction happens near the perijove of the fourth satellite. We may con- sider then that the preponderating force takes place at this part of the orbits ; and we have to inquire what form the orbit of the third satellite must have, to pre- serve the same excentricity at every revolution. It must be remembered here that the effect of Jupiter's shape is to cause a more rapid progress of the line of apses of the third satellite, if its orbit be excentric, than of the line of apses of the fourth. (151.) Considering, then, that the preponderating force on the third satellite in the direction of the radius vector is directed from the central body towards the perijove of the fourth, and that the preponderating force perpendicular to the radius vector accelerates it as it approaches that part, and retards it afterwards, it is plain from (51.), (65.), and (66.), that, if the perijove of the third satellite were in that position, the forces would cause the line of apses to regress ; and this regression, if the excentricity of the third be small, may be con- EXCENTRICITY OF FOURTH SATELLITE. 109 siderable (though the preponderance of force which causes it is extremely small), and may overcome so much of the progression caused by Jupiter's shape, as to make the real motion of the line of apses as nearly equal as we please to the motion of the line of apses of the fourth. But the motion of the line of apses of the fourth will itself be affected (though very little) by the greater action of the third satellite on it at the same place ; and the part in the radius vector being directed at its perijove to the central body, and the part per- pendicular to the radius vector retarding it before it reaches the perijove, and accelerating it afterwards, will cause a small increase of progression of its apse. The state of things will be permanent, so far as depends on these forces, when the increased progression of the apse of the fourth satellite is equal to the diminished progression of the apse of the third ; and thus the pro- gression of the apse of the fourth will be somewhat increased, and the third satellite's orbit will have a compression corresponding in direction to the perijove of the fourth, and an elongation in the same direction as the apojove of the fourth. This would be the case if the third satellite had no excentricity independent of perturbation ; but we may, as in other cases, consider that the same kind of distortion will be produced in the orbit if it has an independent excentricity. (152.) Now let us suppose the fourth satellite to have no excentricity independent of perturbation, and the third satellite to have an independent excentricity. The greatest action will now be at the apojove of the 110 GRAVITATION. third satellite, and this will (though in a small degree) cause the line of apses of the third satellite to progress ; that is, it will increase the rapidity of progression which Jupiter's shape gives it. If, now, we wish to dis- cover the form of orbit of the fourth satellite which will at every revolution preserve the same excentricity, and have its line of apses always corresponding with that of the third satellite, and therefore progressing more rapidly than the shape of Jupiter alone would make it progress, we must evidently suppose the perijove of the fourth satellite turned towards the apojove of the third, and, by supposing the excentricity small enough, the progression may be made as rapid as we please. Thus the effect of excentricity in the orbit of the third satel- lite is, that its line of apses is made to progress rather more rapidly, and that the orbit of the fourth satellite is compressed on the side next the apojove of the third satellite, and elongated on the opposite side. We have supposed for this investigation that the fourth satellite had no excentricity independent of perturbation, but the conclusion as to the distortion of the orbit may be applied if we suppose it to have independent excentricity. (153.) In fact, the orbits of both the third and fourth satellites have independent excentricities, and both our conclusions apply to them. The fourth satellite, besides its independent excentricity, has an excentricity im- pressed upon it, opposite in kind to that of the third ; and the third satellite, besides its independent excen- tricity, has an excentricity impressed upon it of the ORBITS OF THIRD AND FOURTH SATELLITES. Ill same kind as that of the fourth. In the same manner, the orbits of the first and second satellites have small excentricities impressed on them, similar in their kind to those of the third and fourth. (154.) It will readily be conceived that the excen- tricities of the orbit of the third satellite will affect the great inequality (137.) which it produces in the motion of the second ; and on the contrary, that the inequality in the motion of the third produced by the attraction of the second, will influence the effect of the third on the fourth. "We shall not, however, notice these further than to state that their effects are small. (155.) We have now gone over the principal inequa- lities of the motions of Jupiter's satellites. They are so much connected, and (as we may say) so completely entangled, that though they may be explained in the way in which we have considered them, it would hardly be possible to calculate them in that way. A mathe- matical process of the most abstruse kind, which will at the same time embrace the motions of all, is alone com- petent to this object. We shall, however, have attained our end if we have given the reader a general idea of the explanation of disturbances in the most curious and complicated system that has ever been reduced to calculation. SECTION VII. Theory of Planets. (156.) The theory of the planets may be considered as holding a middle place between that of our moon and 112 GEAVITATION. that of Jupiter's satellites. In our moon, the principal inequalities are those that exhibit themselves in nearly the same order at every revolution, or, at longest, in the earth's revolution round the sun, depending entirely upon the relative position of the moon, the sun, and the lines of apses. In Jupiter's satellites, some of the prin- cipal inequalities (as those of the third and fourth satel- lites) do not depend at all upon the relative position of the bodies, but depend on the position of the lines of apses, whose revolutions, though slow, may yet be completely observed. But in the planets, the terms analogous to those which we have mentioned in the moon's motions are small : the changes of elements are so slow, that, though they may be in some degree observed, many thousands of years would be necessary to observe them completely. The most remarkable irregularities are those produced by changes in the elements occupying several revolutions of the planets, and more nearly analogous to the mutual perturbations of the first three satellites of Jupiter than to any other that we have seen ; differing from them, however, in this respect, that for most of them independent excen- tricities are quite essential. (157.) There are, however, some terms very nearly similar to those mentioned in the theory of the moon. Suppose, for instance, we consider the perturbations of Mercury by Jupiter (whose distance from the sun is more than thirteen times as great). This case is almost exactly analogous to the case of the moon disturbed by the sun. And in consequence, Mercury's orbit is flat- PLANETARY PERTURBATIONS. 113 tened a little on the sides nearest to and farthest from Jupiter ; but this effect is much disguised by the effect of forces analogous to those mentioned in (94), which here preponderates greatly : his line -of apses progresses a little at every revolution, when Jupiter is nearly in that line, and regresses a little when Jupiter is in the line perpendicular to it : his orbit is a little more excen- tric in the former case, and a little less so in the latter ; and his orbit is a little larger when Jupiter is at peri- helion than when at aphelion. The same thing applies very nearly to the disturbances of Venus, the Earth, and Mars, produced by Jupiter. (158.) The instance taken above is almost an extreme one. When we consider the perturbations of two planets which are nearer to each other, we are obliged to alter our conclusions considerably. The disturbing force becomes so much greater where the planets are near conjunction than at any other part, that the orbit is much more changed there than at any other part. How- ever, the reasoning upon which, in (91.), we determined the form of the moon's orbit, laying aside the considera- tion of independent excentricity, will, to a certain extent, apply here. The orbit in several cases will be flattened on the side where conjunction takes place, and on the opposite side, but generally most so on the latter ; and will be made protuberant at the parts where the disturbing force tends wholly to increase the gravitation towards the sun. The same general reasoning will, in many cases, help us to find the form of the orbit which is influenced by the attraction of an interior planet. I 114 GRAVITATION. (159.) A consideration, however, of particular cases will show how cautious we must be in applying this conclusion. Suppose, for instance, we consider the reciprocal perturbations of the Earth and Mars. The periodic time of Mars is nearly double that of the Earth. Here, then, we fall upon an inequality of such a kind as that discussed in (122.), &c., for the satellites of Jupiter. And though the periodic time of Mars is not very nearly double that of the Earth, so that the distor- tions produced in the orbits of the Earth and Mars are not very striking ; still they are the greatest (of those depending only on the position of the planets) which these two bodies produce in each other's motions. Here, then, the disturbance, which on a hasty view we might suppose analogous to the variation of the Moon, becomes, from the small disproportion of dis- tances, and the near commensurability of the periodic times, much more nearly similar to the slow variation of the elements of orbits. (160.) It seems quite hopeless to attempt to give a notion of the calculations by which, in all the dif- ferent cases, the disturbances independent of the excentricities can be computed. It is sufficient to state, that the same methods apply to all, and that they are much more simple than those relating to other points, of which an idea may be given by general explanation. (161.) Let us now consider the inequalities of motion which depend on the excentricities and inclinations of the planets' orbits. The idea that will probably first INEQUALITIES OF LONG PERIOD. 115 occur to the reader is this : " If the disturbances of the planets, supposing their orbits to have no inde- pendent excentricities, amount only to a few seconds, how is it likely that the small alterations of place, which are produced by the trifling excentricities and inclinations of their orbits, will so far alter their forces upon each other as to produce any sensible difference in the magnitude of the irregularities ? " In answer to this we must say, " It is true that these forces, or altera- tions of forces, are exceedingly small, and those parts of them which act in the same direction for a short time only (as for a fraction of the periodic time of a planet) do not produce any sensible effect. But we can find some parts of them which act in the same manner during many revolutions : the effects of these may grow up in time to be sensible ; and those in parti- cular which alter the mean distance and the periodic time may produce in time an effect on the longitude of the planet (49.), very much more conspicuous than that in the alteration of the orbit's dimensions." (162.) In this consideration is contained the whole general theory of those inequalities known by the name of inequalities of long period. They are the only ones depending on the excentricities (besides those similar to the moon's evection) which ever become important. (163.) To enter more minutely into the explanation, let us take the instance of the long inequality of Jupiter and Saturn : the most remarkable for its mag- nitude, and for the length of time in which the forces act in the same manner, as well as for the difficulty I 2 116 GRAVITATION. which it had given to astronomers before it was ex- plained by theory, that has been noticed since the first explanation of the Moon's irregularities. (164.) The periodic times of Jupiter and Saturn are very nearly in the proportion of 2 to 5 (the periodic times being 4332 days, 17 hours, and 10,759 days, 5 hours), or the number of degrees of longitude that they will describe in the same time, omitting all notice of their excentricities, will be in the proportion of 5 to 2 nearly. Suppose, now, that they were exactly in the proportion of 2 to 5 ; and suppose that Jupiter and Saturn started from conjunction; when Saturn has described 240 degrees, Jupiter wjil have described 600 degrees (as these numbers are in the proportion of 2 to 5) : but as 360 degrees are the circumference, Jupiter will have gone once round, and will besides have described 240 degrees. It will, therefore, again be in conjunction with Saturn. When Saturn has again described 240 degrees, that is, when Saturn has described in all 480 degrees, or has gone once round and has described 120 degrees more, Jupiter will have described 1200 degrees, or will have gone three times round and described 120 degrees more, and, therefore, will again be in conjunction with Saturn. When Saturn has again described 240 degrees, that is, when it has gone exactly twice round, Jupiter will have gone exactly five times round, and they will again be in conjunction. So that, if the periodic times were exactly in the proportion of 2 to 5, there would be a continual succession of conjunctions at the points LONG INEQUALITY OF JUPITER AND SATURN. 117 whose longitudes exceeded the longitude of the first place of conjunction by 240, 120, 0, 240, 120, 0, &c. Thus, in fig. 38, if B x is the place of Jupiter at FIG. 38. first, and C x that of Saturn, Jupiter will have gone quite round, and also as far in the next revolution as B 2 , while Saturn has described part of a revolution only to C 2 : then Jupiter will again have gone quite round, and also as far in the next revolution as B 3 , while Saturn has described part of a revolution to C 3 : then Jupiter will have performed a whole revolution, and part of another to B lf while Saturn has performed part of a revolution to C^: and then the same order of conjunctions will go on again. If, then, the peri- odic times were exactly in the proportion of 2 to 5, the conjunctions would continually take place in the same three points of the orbits. This conclusion will not be altered by supposing the orbits excentric : for though the places of conjunction may then be some- what altered, the conjunctions, after the third (when Saturn has gone round exactly twice, and Jupiter exactly five times), will go on in the same order, and happen at the same places as before. (165.) But the periodic times are not exactly in the 118 GRAVITATION. proportion of 2 to 5, but much more nearly in the pro- portion of 29 : 72. This alters the distance of the places of conjunction. We must now suppose Saturn to move through 242'79, and Jupiter (by the propor- tion just mentioned) will then have moved through 602-79, or through a whole circumference and 242'79, and they will be in conjunction again. The next conjunction will take place when Saturn has moved through double this angle, or 485*58, or when Saturn has performed a whole revolution, and 125*58 of the next revolution: and the following conjunction will take place when Saturn has moved through 728*37, or when Saturn has gone twice round, and has described 8'37 more. Now, then, the same order of conjunctions will not go on again at the same places as before, but the next three after this will be shifted 8'37 before the former places, the three following the last-mentioned three will be again shifted 8'37, and so on. The places of successive conjunction, in fig. 38, will be at B 1; C lf b 2 c 2 , b 3 c 3 , ~b c 4 , & 5 c 5 , 1 6 c 6 , &c. The shifting of the places of conjunction will take place in nearly the same manner, whether the orbits are excentric or not. (166.) From this the following points are evident: First. In consequence of the periodic times being nearly in the proportion of 2 to 5, many successive con- junctions happen near to three equidistant points on the orbits. Secondly. In consequence of the proportion being not exactly that of 2 : 5, but one of rather less inequality, the points of conjunction shift forward, so that each sue- LONG INEQUALITY OF JUPITER AND SATURN. 119 cessive set of conjunctions is at points of the orbits more advanced, by 8'37, than the preceding one. (167.) Let us now inquire how long it will be before the conjunctions happen at the same parts of the orbits as at first. This will be when the series of points 6 4 , & 7 , & 10 , &c., extends to B 3 . For then the series & 5 , & 8 , & u , &c., will extend to B x , and the series 6 3 , & 6 , 6 9 , &c., will extend to B 2 . The time necessary for this will be gathered from the consideration, that in three conjunctions the points are shifted 8'37 ; and that the points must shift 120 from Bj, before they reach B 3 : and that we may, there- fore, use the proportion, As &6*37 is to 3, so is 120 to 43 nearly, the number of conjunctions that must have passed before the points of conjunction are again the same. And as Saturn advances 242- 7 9 between any conjunction and the next, he will, at the forty-third con- junction from the first, have described 10440, or 29 cir- cumferences ; and Jupiter, therefore (by the proportion of their periodic times), will have described 72 circum- ferences. The time, then, in which the conjunctions return to the same points is twenty-nine times Saturn's periodic time, or seventy-two times Jupiter's periodic time, or about 855 years.* (168.) Now let us examine into the effects of this slow motion of the points of conjunction upon the forces which one body exerts to disturb the other. (169.) If the orbits had no independent excentricity, * These numbers are not quite exact : the proportion of 29 : 72 not being quite accurate. 120 GRAVITATION. it would affect them no further than by the periodical distortion which would take place at every conjunction. There would be nothing in one set of conjunctions, more than in another, which could affect the dimensions of the orbits. (170.) But if the orbits are not circular, this is no longer true. It is not the same thing whether the con- junctions take place at B x C lf B 2 C 2 , and B 3 C 3 , fig. 39, FIG. 39. or at Z>! Cj, & 2 c 2 , and 6 3 c 3 . The distances of the planets are not the same, and consequently the forces which they exert on each other are not the same ; also their velocities are different in different parts of their orbits, or at different points of conjunction, and therefore the times during which they can act on each other are not the same. It is true that, in the figure, the distance at & 2 c 2 is less than at B 2 C 2 , while that at b 3 c 3 is greater than at B 3 C 3 ; and thus there is a partial compensation in the changes of the effects produced in different points of the orbit. But it can be discovered only by very complete calculations, whether the compensation is per- fect or not. The calculations necessary for this purpose are probably the most complicated that physical science has ever given occasion for ; and the reader must not LONG INEQUALITY OF JUPITER AND SATUEN. 121 here expect the smallest account of them. This only can be stated as a result, that in no instance in the planetary system is the compensation perfect, and that the chances for its being perfect in any case are infi- nitely small. (171.) We have here considered the varying influence of one body on the other at conjunction, as depending entirely on the excentricities of the two orbits. But there is another circumstance which may also cause the influence to vary. The orbits may be inclined, and this will affect both the distance of the bodies and the direction in which they attract each other. (172.) In the case, then, of Jupiter and Saturn, we have the two planets acting on each other with forces which are nearly the same at every third conjunction, but are not exactly the same, and whose variations occupy a period of 850 years. Of these forces, parts are in the direction of the radius vector, and these tend directly to affect the major axes of the orbit described : other parts are perpendicular to the radius vector, some- times accelerating and sometimes retarding ; and these tend (though in opposite ways) to affect the major axes of the orbits. There are, therefore, forces tending to alter the major axes of the orbits, which go through all their changes only in 850 years. During half of this time they tend to make the major axis of Jupiter's orbit less, and that of Saturn's orbit greater ; and during the other half they tend to make the major axis of Jupiter's orbit greater, and that of Saturn's orbit less. This coin- cidence, in time, of the increase of one major axis with 122 GRAVITATION. the decrease of the other, is the result of investigation, that we cannot explain here. (173.) After the partial compensation that we have mentioned, it will readily be understood that the varying force which produces these effects is small. So small, indeed, is it, that after acting more than 400 years, it has increased (or diminished) the major axis of Saturn's orbit only by -rgVoth part, and diminished (or increased) that of Jupiter's orbit only by -oVotii part. These alterations would hardly be discoverable with our best instruments. But during 400 years the major axis of each orbit differs from the major axis during the next 400 years by a part of these quantities : the planet's rate of annual angular motion is, for 400 years, constantly less than its average rate ; and for the next 400 years it is constantly greater than its average rate : and in this length of time the inequality in longitude may (49.) grow up into a most formidable quantity. In fact, the inequality thus produced in Saturn's longitude amounts to about 48', by which its true place is sometimes before and sometimes behind its mean place : that in Jupiter's longitude amounts to about 21'. (The greatest inequa- lity of any other planet does not exceed 3' and the greatest of the planets inferior to Jupiter does not exceed 25".) The theoretical explanation of these inequalities was first given by Laplace in 1785. (174.) The magnitude of these inequalities in the motions of Jupiter and Saturn, as we have seen, depends principally on the length of time during which the forces act in the same manner ; first, because in this long time LONG INEQUALITY IN EXCENTRICITIES. 123 they can produce a sensible alteration in the major axis and annual angular motion ; secondly, because the two planets move for so long a time with this altered angular motion. But it must also be borne in mind, that these two planets are by far the largest in the system ; the mass of Jupiter being 300 times that of the earth, and the mass of Saturn being 100 times that of the earth (the next of the planets in the order of magnitude, except Uranus). (175.) The same general reasoning, by which we have shown that there is a periodical inequality of the major axis of either of these orbits, will also show that there is a periodical inequality in the excentricity and in the place of the perihelion. It will also appear, in the same way, that these effects are the remainder, after partial compensation of effects in different parts of the orbit. Thus, if one conjunction happen when Jupiter is going towards aphelion, the effect of Saturn's disturb- ing force is to pull Jupiter from the sun ; and therefore, by (59.), to increase the excentricity of Jupiter's orbit. But it is then perfectly certain that either the next con- junction, or the next but one, or perhaps both these, will happen at a part where Jupiter is going towards perihelion ; and then by (59.), the excentricity of Jupiter's orbit is diminished. Similar reasoning applies to the ex- centricity of Saturn's orbit. It becomes, then, a matter of calculation, whether the compensation is perfect or not. Now it appears, upon investigation, that the com- pensation is not perfect, but that, while the points of conjunction shift through 120, the effect of the uncom- 124 GRAVITATION. pensated part is, for half the time, to increase the excentricity, and for half the time to diminish it. It appears, also, that there is no necessary connexion between the time at which the excentricity is greatest or least, and that when the major axis is greatest or least ; so that we cannot assert that when the major axis is greatest the excentricity is greatest, or the contrary ; or that the excentricity of one is greatest when that of the other is greatest ; all that we can assert is, that the excentricity of each orbit occupies the same time in *^oing through its changes from greatest to least, as the major axis occupies in going through its change from greatest to least. The effect on the planet's distance from the sun, produced by the change of excentricity' is much more considerable than that from the change in the major axis ; being for Jupiter y^Vo- f h* 8 whole distance, and for Saturn -g-^ of his whole distance. (176.) Similar remarks apply, in every respect, to the motion of the perihelion of each orbit. Each is made to progress during 425 years, and to regress during 425 years ; but there is no necessary relation between the time when one has progressed furthest, and the time when the other has progressed furthest. There is, however, a necessary relation between the change of excentricity and the motion of the perihelion of each orbit: the excentricity of either orbit has its mean value when the perihelion of that orbit has progressed furthest or regressed furthest; and when the excen- tricity is either greatest or least, the perihelion is at its mean place. LONG INEQUALITIES OF OTHER PLANETS. 125 (177.) We have taken the long inequality of Jupiter and Saturn as the most imposing by its magnitude, and the most celebrated for its history (as, before it was explained theoretically, astronomers were completely bewildered by the strange irregularity in the motion of these planets). But there are several others which, in theory, are as curious. Eight times the periodic time of the earth is very nearly equal to thirteen times the periodic time of Venus ; and this produces, in the motions of the earth and Venus, a small inequality, which goes through all its changes in 239 years. Four times the periodic time of Mercury is nearly equal to the periodic time of the earth, and this produces an inequality whose period is nearly 7 years. The periodic time of Mars is nearly double of the earth's, and this produces a consider- able inequality, depending on the excentricities, &c., besides that mentioned in (159.), which was independent of the excentricities. Twice the periodic time of Venus is nearly equal to five times that of Mercury; three times the periodic time of Venus is nearly equal to that of Mars ; three times the periodic time of Saturn does not much differ from that of Uranus. Each of these approximations to equality gives rise to an equation of sensible magnitude, and of long period, in the motion of both planets. (178.) But it will easily be seen that the defect of compensation, on which the effects depend, is much greater in some cases than in others. The conjunctions of the earth and Mars take place at only one point, and the points near it, for several revolutions: those of 126 GRAVITATION. Venus and Mars take place only at two opposite points and their neighbourhood (as each successive conjunction takes place when Mars has described half a revolution, and Venus 1-^ revolutions) ; those of Jupiter and Saturn, as we have seen, at three points ; those of Venus and the earth at five points. It is evident that, in the first of these, the whole effect of the change of one point of conjunction has its influence in altering the orbit's dimensions ; that in the second there is only the dif- ference between two effects; that in the third there is the mixture of three, which tend to balance; that in the next there is the mixture of five in the same way. The smaller, then, is this number of points, the more favourable are the circumstances (supposing the same length of period for the inequality) for producing a large inequality. This number of points is always the same as the difference between the two least numbers, expressing nearly the proportion of the periodic times. Thus we may expect to find a large inequality when the periodic times of two planets are very nearly in the same proportion as two numbers whose difference is small. (179.) We shall now proceed to mention the secular variations of the elements of the orbits of planets. By this term is meant those variations which do not depend upon the positions of the planets in their orbits, or the places of conjunction, but merely upon their relative distances and excentricities, and the posi- tions of their lines of apses. They are, therefore, the variations which depend upon the mean or average SECULAR VARIATION OF ELEMENTS. 127 action of one planet upon another in the long run : all the sensible departures from the secular variation, pro- duced by the irregularity of the action of one planet upon another, being supposed to be contained in the inequalities already discussed. (180.) First, then, with regard to the mean distance of a planet. If we consider an exterior planet disturb- ing an interior one (as Saturn disturbing Jupiter), the disturbing force in the direction of the radius vector, by (77.), &c., tends sometimes to draw it from the sun, sometimes to draw it towards the sun, but the former is the greater, and we may therefore consider the force as, upon the whole, diminishing the sun's attraction. This, by (46.), alters the relation between the periodic time and the mean distance, so that the mean distance is less than it would have been with the same periodic time, had there been no disturbance. If we consider an interior planet disturbing an exterior one (as Jupiter disturbing Saturn), the disturbing force tend- ing to draw it to the sun isf greatest; and here the mean distance is greater than it would have been with the same periodic time, had there been no disturbance. But so long as these general effects in the force directed to the sun continue unaltered, the mean dis- tances will not alter (46.), &c. Now, upon taking a very long period (as several thousand years), it is easy to see that, if we divide that period into two or three parts, the two planets have in each of those parts been in conjunction indifferently in all parts of their orbits; that they have had every possible rela- 128 GRAVITATION. live position in every part ; and that (if we make the periods long enough) the force which one planet has sustained in any one point will be accurately the mean of all which it would sustain, if we estimated all those that it could suffer from supposing the other planet to go with its usual motion through the whole of its orbit. As this mean will be the same for each of the periods, there will, in the long run, be no alteration of the force in the direction of the radius vector, and we may assert at once that the mean distance cannot be altered by it. (181.) But with regard to the disturbing force acting perpendicularly to the radius vector, the circumstances are different. The mere existence of such a force, without variation, causes an alteration in the mean distance (48.) ; and it is necessary to show that the nature and variations of the force are such that, in the long run, the velocity of the disturbed planet is not affected by it. For this purpose, instead of consider- ing merely the disturbing force perpendicular to the radius vector, we will consider separately the whole force which the disturbing planet exerts on the sun, and the whole force which it exerts on the disturbed planet. Now, the force which it exerts on the sun tends to pull the sun sometimes in one direction and sometimes in another, but, on the whole, produces no permanent displacement: this force, then, may at once be neglected. The force which one planet has exerted on the other has acted when, for any arbitrary position of the disturbing planet, the disturbed planet has been at every point of its orbit. Since the whole SECULAR VARIATION OF MEAN DISTANCE. 129 acceleration produced in a long time is the sum of all the accelerations diminished by the sum of all the retardations, we may divide them into groups as we please, and sum each group. Let us, then, group together all the accelerations and retardations pro- duced in one position of the disturbing planet. The disturbed planet having been in every small part of its orbit, during a time proportional to the time which it would occupy in passing through that small part in any one revolution, the various accelerations and retardations will bear the same proportion as if the disturbed planet had made one complete revolution, and the disturbing planet had been fixed. Now, it is a well-known theorem of mechanics, that when a body moves through any curve, acted on by the attrac- tions of any fixed bodies, its velocity, when it reaches the point from which it started, is precisely the same as when it started : the accelerations and retardations having exactly balanced. Consequently, in the case before us, if the disturbing planet had been fixed, and the disturbed planet had made one complete revolu- tion, the latter would, on the whole, have been neither accelerated nor retarded; and, therefore, in the long run, all the accelerations and retardations of the dis- turbed planet, produced in any arbitrary position of the disturbing planet, will exactly balance. The same may be shown for every position of the disturbing planet ; and thus, on the whole, there is no alteration of velocity. Since, then, in the long run, the planet's velocity is not altered, and since (180.) the force K 130 GRAVITATION. directed to the sun is not altered, the planet's mean distance will not be altered. This reasoning does not prevent the increase or diminution of the velocity at particular parts of the orbit, and therefore the excen- tricity and the line of apses may vary ; but it shows that, if there is an increase at one part, there is a dimi- nution that balances it at another; and at the point where the orbit at the beginning of a long time, and the orbit at the end of that time intersect (which will be at mean distance nearly), the velocity will not be altered. Our demonstration supposes that the portions of the curves described in different revolutions, for the same position of the disturbing planet, are parts of one orbit, and therefore does not take account of the alteration in the magnitude of the disturbing force produced by the alteration of place which that force has previously caused. This has been taken into account, to a certain degree, by several mathematicians, and it appears, as far as they have gone, that this produces no alteration in' the conclusion. (182.) Secondly, as to the place of perihelion, or the position of the line of apses. The motion of this will depend essentially on the excentricity of the orbit of the disturbing planet. Suppose, for instance, that the orbit of Yenus was elliptical and the earth's orbit circular ; as the distance of these planets in conjunction is little more than th of the earth's distance from the sun, the ellipticity of the orbit of Venus would bring that planet at aphelion so much nearer to the earth's orbit, that by far the greatest effect would take place when in con- SECULAR VARIATION OF EXCENTRICITY. 131 junction there ; and this, by (54), would make Venus' line of apses progress. But if the earth's orbit were more elliptic than that of Venus, and if the earth's perihelion were on the same side of the sun as the perihelion of Venus, it might happen that the principal action would take place at perihelion, and then, by (51.), the line of apses would regress. These effects would continue to go on, while the relative position of the lines of apses, and the proportion of the excentricities, remained nearly the same. As, in the long run, con- junctions would happen everywhere, the preponderating effect would be similar to the greatest effect ; and thus, the secular motion of the line of apses will be constant (till the positions of the lines of apses, &c., shall have changed considerably) ; its magnitude and direction will depend on the excentricities of both orbits ; but if the disturbed planet is the interior, and if the orbit of the other be not excentric, the line of apses will progress. The same is true, if the disturbed planet is exterior (the greatest action being then at the perihelion, jfc the interior orbit have no excentricity, and being directed to the sun). (183.) Thirdly, as to the excentricity. If the orbit of the disturbing planet were circular, the effect on the excentricity produced by conjunction at the place where the orbits are nearest, would be of one kind before con- junction, and of the opposite kind after conjunction, from the disturbing force in the radius vector, as well as from that perpendicular to the radius vector; and thus the excentricity would not be altered. The same K 2 132 GRAVITATION. would happen if both orbits were excentric, provided their lines of apses coincided. Thus it appears that there is no variation of excentricity, except the orbit of the disturbing planet is excentric, and its line of apses does not coincide with that of the disturbed planet. When these conditions hold (as they do in every planetary orbit), a general idea of the effect may be obtained by finding where the orbits approach nearest ; then, if we consider the disturbance of the interior planet, since the force draws it from the sun, the excen- tricity will be increased if it is moving from perihelion, or diminished if it is moving towards perihelion. For the exterior planet, as the force draws it towards the sun, the conclusion will be of the opposite kind. These effects are constant, till the excentricities and the posi- tions of the lines of apses have changed sensibly. The place where the force at conjunction produces the greatest effect on the excentricity may not be strictly the place where the orbits are nearest, but probably will not be far removed from that place. At the place where the orbits approach nearest, both planets in general are moving from perihelion, or both towards perihelion, so that when one excentricity is increased, the other is diminished. (184) For the general stability of the planetary system, the positions of the lines of apses are not im- portant, but the permanency of the major axes and the excentricities are of the greatest importance. The con- clusion which we have mentioned as to the absence of secular variation of the major axis, from the action of RELATION OF VARIATIONS OF EXCENTRICITIES. 133 one planet, applies also to the disturbances produced by any number of planets, and thus we can assert that the major axes of the orbits of the planets are not subject to any secular variation. The excentricities are subject to secular variation, but even this corrects itself in a very long time : when the investigation is fully pursued, it is found that each of the excentricities is expressed by a number of periodic terms, the period of each being many thousands of years. Thus the major axis of the earth's orbit, notwithstanding its small and frequent variations, has not sensibly altered in many thousands of years, and will not sensibly alter ; the excentricity, besides suffering many small variations, has steadily diminished for many thousands of years, and will di- mmish for thousands of years longer, after which it will again increase. (185.) A remarkable relation exists between the varia- tion of the excentricities (of which that mentioned in (183.) is a simple instance), the result of which, as to the state of the excentricities at any time, is given thus : The sum of the products of the square of each excen- tricity by the mass of the planet, and by the square root of the major axis, is always the same. SECTION \ 7 III. Perturbation of Inclination and Place of Node. (186.) WE have hitherto proceeded as if the sun, the moon, and all the planets, revolved in the same plane as if, for instance, the sun were fixed in the centre of a 134 GRAVITATION. table, and all the planets, with their satellites, revolved on the surface of the table. But this supposition is not true. If we suppose the earth to revolve on the surface of the table, the moon will, in half her revolution (we mean while she describes 180, not necessarily in half her periodic time), rise above the table, and in the other half she will go below it, crossing the surface at two points which, as seen from the earth, are exactly oppo- site. Venus will, in half her revolution, rise above the table, and in half will sink below it, crossing the table at two points which, as seen from the sun, are exactly opposite ; each of the other planets and satellites in like manner crosses the plane at points which, as seen from the central body, are exactly opposite. In different investigations it is necessary to consider the inclination of the plane of revolution or the plane of the orbit to different planes of reference : the line in which the plane of revolution crosses the plane of reference is called the line of nodes on that plane; and the angle which the plane of revolution makes with the plane of reference is called the inclination to that plane. The plane of refer- ence must always be supposed to pass through the central body. (187.) The inclinations of all the orbits, except those of the small planets, are so trifling (the largest namely, that of the moon's orbit to the earth's orbit being, at its mean state, only 5), that they may in general be wholly neglected in estimating the disturbance which one planet produces in the motion of another in its own plane. In some cases, however, as in the inequalities of INCLINATIONS ARE GENERALLY SMALL. 135 long period, where the effective force is only the small part which remains after a compensation more or less perfect, no alteration of the forces must be neglected ; and here, as we have hinted in (171.), the inclinations must be taken into account. (188.) But though the alteration which the inclina- tion produces in the forces that tend to disturb the body's motion in its plane may, in most cases, be neglected, yet the force which tends to pull the body above the plane or below the plane cannot be neglected. In almost every case this force will be less than the force tending to disturb the motion in the plane, yet it will be much greater than the alteration which the inclination produces in that force. It is our object in this section to show the nature of the alteration which is produced by the force tending to pull the body from the plane, (189.) First, then, as to the effect of a force generally which acts perpendicularly to the plane of revolution. (We shall confine ourselves at present to forces which act perpendicularly to the plane, because it is evident that forces which act in, or parallel to, the plane of the orbit, whether in the radius vector or perpendicularly to it, will not cause the planet to depart from that plane.) Let fig. 40 be a perspective representation of an orbit, and a plane of reference. Suppose M A N to be the line of nodes at which the plane of the orbit N B x B 2 crosses the plane of reference D E ; the central body A being in the line of nodes, and the part of the orbit marked by a dark line being above the plane, and that marked by a 136 GEAVITATION. dotted line being below it. Suppose that the planet has moved from N to B x , and that at B x , before it reached the point highest above the plane D E, a force pulls it down towards the plane. After a short time, instead of going to B 2 , where it would have been if no force had disturbed it, it will be found at & 2 , having described B! & 2 , instead of B x B 2 . It is plain that the orbit in which the planet must have moved without a disturbing force, in order to describe B x b 2 now, could not be N B 1; but must be such a curve as n B b crossing the plane D E at a point in the situation of the point n. There- fore, if no more disturbing forces act, the planet, which has described B x & 2 as if it came without disturbance from n, will go on to describe an orbit as if it had come without disturbance from n, and will therefore describe an orbit n B L & 2 m, crossing the plane D E in the points n and m. The line of nodes is changed from M A N to mAn. (190.) Here the line of nodes has twisted in a direc- tion opposite to the planet's motion, or has regressed. The inclination of the new plane is evidently less than that of the old one, since it passes through the same point B, and cuts the plane of reference in a line more CHANGE OF NODE AND INCLINATION. 137 V distant from B than the line in which the old one cut it, or the inclination is diminished. (191.) Now, if we conceive that at B 3 , fig. 41, after the planet has passed the point highest above the plane, FIG. 41. D a force tends to pull it towards the plane, the planet, instead of going to B 4 , will go to & 4 , and instead of crossing the plane D E at M, will cross it at m ; and then, if it is not disturbed again, will proceed in an orbit of which B 3 Z> 4 ?7i is a part, and which will cross the plane D E at the points m and n. The new line of nodes has twisted here also in the direction opposite to the direc- tion of the planet's motion, or has regressed. But the inclination of the new plane is greater than that of the old one, since it passes through the same point B 3 , and cuts the plane of reference in a line less distant from B 3 than the line in which the old one cut it, or the inclination is increased. (192.) We have, then, this general result : If a force acting perpendicularly to the orbit tends to draw the planet towards the plane of reference, it always causes the line of nodes on that plane to regress; while the planet is moving from a node to the point highest above the plane of reference, it diminishes the in- clination to that plane ; and while the planet is moving 138 GRAVITATION. from the highest point to a node, it increases the in- clination. (193.) In the same manner, if the force tends to draw the planet from the plane of reference, it always causes the line of nodes to progress. While the planet is moving from a node to the point highest above the plane, it increases the inclination ; and while the planet is moving from the highest point to a node, it diminishes the inclination. (194.) Similar results would have been obtained if we had considered the action of the force while the planet is in that part of its orbit, which is on the other side of the plane D E. We shall now proceed with the consideration of the force perpendicular to the orbit, which is produced by the attraction of a disturbing body, (195.) First: It is plain that, if the disturbing body is in the plane of the orbit (produced, if necessary), it will not tend to draw either the central body or the planet out of that plane, and therefore will produce no disturbing force perpendicular to the plane of the orbit. Proceeding, then, with the supposition that the dis- turbing body is not in the plane of the orbit; and supposing fig. 42 to be a perspective view of an orbit B 2 B! B 3 (which, to assist our ideas, may be conceived to differ little from a circle) with the disturbing body C out of the plane of the orbit, let us take three points, B! B 2 B 3 , of which B x is at the the same distance as A from C, B 2 is nearer to C, and B 3 farther from C than A is. Suppose that the attraction of C draws A in a FORCE PERPENDICULAR TO PLANE OF ORBIT. 139 certain small time through the space A a, and that when the planet is at B D or B 2 , or B 3 , the attraction draws the planet in the same time through B! b l} or B 2 5 2 , or B 3 5 3 , respectively. Then (as in (71.) ) the attraction of C FIG. 42. upon the two bodies A and B would produce no disturb- ance in their relative motions, if it drew them through equal spaces in the same direction. Draw B! d l} B 2 d 2 , B 3 d 3 , each equal and parallel to A a ; then, if the attrac- tion had drawn B! to d l} there would have been no dis- turbance, and consequently the real disturbance at B! is represented by a force which would have drawn the planet from d 1 to fy. Similarly, the real disturbances at B 2 and B 3 , are represented by forces which would have drawn the planet from d 2 to > 2 , and from d 3 to 6 3 , respec- tively. Now, since C Bj is equal to C A, the forces of C upon A and B x are equal, and therefore Bj & x is equal to A a, and therefore a &j is parallel to A B : , and therefore is in the same straight line with & x d l ; and consequently at B! the whole disturbing force is parallel to the radius vector, and there is no part perpendicular to the plane of the orbit. But at B 2 the planet is nearer to C, the force therefore on the planet is greater, and B 2 6 2 is therefore greater than A a or B 2 d 2 ; also it is more nearly perpendicular to the plane of the orbit than B 2 d 2 ; and 140 GRAVITATION. consequently Z> 2 is farther from the plane of the orbit than d 2 ; and therefore the disturbing force d 2 t> 2 is directed from the plane of the orbit towards the side on which C is. On the contrary, at B 3 the planet is farther from C ; the force on the planet is therefore less ; and B 3 b 3 is therefore less than A a or B 3 d 3 ; moreover, it is inclined more to the perpendicular than B 3 d 3 , and con- sequently b 3 is nearer to the plane of the orbit than d 3 ; and therefore the disturbing force d 3 & 3 is directed from the side on which C is. Thus we find, (196.) When the central and revolving bodies are equally distant from the disturbing body, there is no disturbing force perpendicular to the plane of the orbit (197.) When the revolving body is nearer the dis- turbing body than the central body is, the disturbing force perpendicular to the plane tends to draw the revolving body out of the plane to that side on which the disturbing body is. (198.) When the revolving body is farther from the disturbing body than the central body is, the disturb- ing force perpendicular to the plane tends to draw the revolving body out of the plane to the side opposite the disturbing body. We may now apply these conclusions to the altera- tion of the node and inclination of the moon's orbit produced by the sun's attraction. The plane of refer- ence is here supposed to be the plane of the earth's orbit. (199.) First : suppose the line of nodes of the moon's orbit to be in syzygies, or to pass through the sun. MOON'S NODES IN QUADRATURES. 141 Here the sun is in the moon's orbit produced, and therefore by (189.), there is no disturbing force perpen- dicular to the moon's orbit. (200.) Secondly : suppose the line of nodes to be in quadratures, or to be perpendicular to the line drawn from the earth to the sun, as in fig. 43. The sun, in FIG. 43. the figure, may be considered as being below the plane of the moon's orbit. Also, the moon's distance from the earth being small, the points, at which the moon's distance from the sun is the same as the earth's, are very nearly the same as the points of quadrature, or (in the case before us) they are very nearly the same as the nodes. Consequently, while the moon moves from B 4 through B x to B 2 , she is nearer to the sun than the earth is, and therefore the disturbing force, by (197.), tends to pull her downwards from the plane of her orbit : while the moon moves from B 2 , through B 3 , to B 4 , she is farther from the sun than the earth is, and therefore the disturbing force tends to pull her upwards from the plane of her orbit. In the case before us, then, the disturbing force is always directed towards the plane of reference. Consequently, by (192.), while the moon moves from B 4 to B!, the line of nodes is made to regress, and the inclination is diminished; while the moon moves from Bj to B^ the line of nodes 142 GRAVITATION. regresses, and the inclination is increased; while the moon moves from B 2 to B 3 , the line of nodes regresses, and the inclination is diminished : and while the moon moves from B 3 to B 4 , the line of nodes regresses, and the inclination is increased. The inclination, there- fore, is not sensibly altered in a whole revolution, but the line of nodes regresses during the whole of the revolution. (201.) Thirdly: suppose the line of nodes to be in such a position that the moon passes the line of nodes in going from quadrature to syzygy, as in fig. 44. Here the sun is to be considered as below the moon's orbit, and, therefore, while the moon moves from B 4 , through B b to B 2 , the disturbing force tends to pull her down from the plane of the orbit, and while she moves from Ba, through B 3 to B 4 , the force tends to pull her up from the plane of her orbit. Therefore, in going from B 4 to N, the force pulls the moon from the plane of reference; and causes thereby a progression of the line of nodes and a diminution of the inclination (193.) ; in going from N to the highest point 0, the force pulls the moon towards the plane of reference; and, therefore, causes the nodes to regress, and the inclination to diminish (192.) ; in going from the highest point to B 2 , the force still pulls the moon towards the plane of reference; and, therefore, still causes the nodes to regress, but causes the inclination MOON'S NODES IN ANY POSITION. 143 to increase. Thus while the moon moves from B 4 to N, the force causes the line of nodes to progress, and while she moves from N to B 2 , it causes the line of nodes to regress ; and, similarly, while she moves from B 2 to M, the force causes the line of nodes to progress ; and while she moves from M to B 4 , it causes the line of nodes to regress. On the whole, therefore, the line of nodes regresses, but not so rapidly as in the second case. Also, while the moon moves from B 4 to 0, the inclination is diminished, and while she moves from O to B 2 the inclination is increased; and, similarly, while she moves from B 2 to P the inclination is dimi- nished ; and while she moves from P to B 4 the incli- nation is increased. On the whole, therefore, the inclination is diminished. (202.) Fourthly: suppose the line of nodes to be in such position that the moon passes it in going from syzygy to quadrature, as in fig. 45. Here, also, the sun FIG. 45. is below the plane of the orbit produced ; and, therefore, from B 4 to B 2 the force tends to pull the moon down from her orbit ; and from B 2 to B 4 it tends to pull her up from it. As in the last case it would be seen, that while the moon moves from B 4 to M, the line of nodes regresses, while from M to B 2 , the line of nodes pro- gresses ; while from B 2 to N, the line of nodes regresses ; and while from N to B 4 , the line of nodes progresses. 144 GRAVITATION. On the whole, therefore, the line of nodes regresses. Also, it will be seen, that while the moon moves from B 4 to 0, the inclination is diminished ; while from to B 2 the inclination is increased ; while from B 2 to P, the inclination is diminished ; and while from P to B 4 , the inclination is increased. On the whole, therefore, the inclination is increased. The same reasoning would apply, and lead to the same conclusions in every respect, if we supposed the moon's orbit inclined in the opposite direction. (203.) Now the earth moves round the sun, and, therefore, the sun appears to move round the earth, in the same direction in which the moon moves round the earth. If then we begin with the state in which the line of nodes is passing through the sun (and in which neither the node nor the inclination undergoes any change, by the first case), we come next to the state in which the moon passes the line of nodes in going from quadrature to syzygy (in which the node regresses and the inclination diminishes, by the third case) ; then we come to the state in which the line of nodes coincides with the line of quadratures (in which the node regresses rapidly, and the inclination is not altered, by the second case) ; then we come to the state in which the moon passes the line of nodes in going from syzygy to quad- rature (in which the node regresses and the inclination is increased, by the fourth case) ; and then we come to the state in which the line of nodes again passes through the sun. This is when the sun has described, appa- rently, half a revolution round the earth (or rather less, SUCCESSIVE POSITIONS OF MOON'S NODES. 145 in consequence of the regression of the node), and in the other half revolution, the same changes in every respect take place in the same order. The inclination, therefore, is greatest when the line of nodes passes through the sun, or coincides with the line of syzygy ; and is least when the line of nodes coincides with the line of quadratures ; since it is constantly diminishing while we are going from the former state to the latter, and constantly increasing while we are going from the latter state to the former. This is the principal irregu- larity in the inclination of the moon's orbit ; all the others are very small. (204.) The line of nodes is constantly regressing at every revolution of the moon, except when the line of nodes passes through the sun. The annual motion which we might at first expect it to have, is somewhat diminished by the circumstance, that the rapid regres- sion of the line of nodes, when in the position in which the greatest effect is produced, carries it from the line of quadratures more swiftly than the sun's progressive motion only, by making the line of quadratures to pro- gress, would separate them. But as the line of nodes never progresses, the diminution of the motion of the line of nodes occasioned thus, is very much less than the increase of the motion of the line of apses (107.). Also, as the force acting on opposite points of the orbit, tends to produce effects of the same kind, there is no irregu- larity similar to that explained in (106.). Hence the actual regression of the line of nodes, though a little less than might at first be expected, differs from that regres- L 146 GRAVITATION. sion by a much smaller quantity than that, by which the actual motion of the line of apses differs from the motion which at first we might expect it to have. The line of nodes revolves completely round in something more than nineteen years. (205.) The effect of the irregularity in the regression of the nodes, and the effect of the alternate increase and diminution of the inclination, are blended into one inequality of latitude, which depends on the sun's longitude, the longitude of the moon's node, and the moon's longitude. This inequality was discovered (from observation) by Tycho Brahe, about A.D. 1590. It may be considered to bear the same relation to the inclina- tion which the evection bears to the excentricity ; and, like the evection in longitude, it is the greatest of the inequalities in latitude. It is, however, much less than the evection ; its greatest effect on the moon's latitude being about 8', by which the mean inclination is some- times increased and sometimes diminished. (206.) There are other small inequalities in the moon's latitude, arising partly from the changes in the node and inclination, which take place several times in the course of each revolution (200.), &c. ; partly from the excentricity of the orbits of the moon and the earth, partly from the distortion accompanying the variation, and partly from the variability of the inclination itself. We shall not, however, delay ourselves with the expla- nation of all these terms. (207.) We shall now proceed with the disturbance of the planets in latitude. MOTION OF PLANETS' NODES. 147 In this inquiry it is always best to take the orbit of the disturbing planet for the plane of reference. Now let us first consider the case of Mercury or Venus dis- turbed by Jupiter. In this case, Jupiter revolving in a long time round the sun, which is the central body to Mercury or Venus, produces exactly the same effect as the sun revolving (or appearing to revolve) round the earth, which is the central body to the moon. The dis- turbing force of Jupiter, therefore, produces a regression of the nodes of the orbits of Mercury and Venus on Jupiter's orbit; and an irregularity in the motion of each node, and an alteration in the inclination, whose effects might be combined into one: and this is the only inequality in their latitude, produced by Jupiter, whose effects are sensible. (208.) The other inequalities in latitude depending on the relative position of the planets, possess no par- ticular interest; and a general notion of them may be formed from the remarks in the discussion of the motion of the moon's node. One case, however, may be easily understood. When an exterior planet is dis- turbed by the attraction of an interior planet, "whose distance from the sun is less than half the distance of the exterior planet, and whose periodic time is much shorter, then the exterior planet is always farther from the interior planet than the sun is, and therefore, by (195.), there is a disturbing force urging it from the plane of reference when the planets are in conjunction, and to it when they are in opposition ; and thus the exterior planet is pushed up and down for every con- L 2 148 GKAVITATION. junction of the two planets. The disturbance, however, is nothing when the exterior planet is at the line of its nodes (195.). (209.) The near commensurability of periodic times, which so strikingly affects the major axis, the excen- tricity, and the place of perihelion, produces also con- siderable effects on the node and the inclination. The reasoning of (175.) and (176.) will in every respect apply to this case : the greatest effect is produced, both on the motion of the node and on the change of inclination, when the planets are in conjunction : the gradual alteration of the point of conjunction produces a gra- dual alteration of these effects, which, however (in such a case as that of Jupiter and Saturn), is partially counteracted by the gradual change of the other points of conjunction : the uncompensated part, however, may, in many years, produce a very sensible irregularity in the elements. If we put the words line of nodes for line of apses, and inclination for excentricity, the whole of the reasoning in (175.), &c., will apply almost without alteration. (210.) For the secular variation of the position of the orbit, the following considerations seem sufficient. In the long run, the disturbed planet has been at every one point of its orbit a great number of times, while the disturbing planet has been at almost every part of its orbit. The disturbing force is always the differ- ence of the forces which act on the sun and on the disturbed planet. As the disturbing planet, in these various positions, acts upon the sun in all directions in NODES ALWAYS REGRESS. 149 the plane of its orbit, its effect on the sun may be wholly neglected; and then it is easy to see that, whether the disturbing planet be exterior or interior to the other, the combined effect of the forces in all these points on the disturbed planet at one point, is to pull it from its orbit towards the plane of the disturbing planet's orbit. (This depends upon the circumstance that the force is greatest when the disturbing planet is nearest.) Consequently, by (192.), the line of nodes of the disturbed planet's orbit on the disturbing planet's orbit, in the long run, always regresses. If the orbits are circular, there is no alteration of the inclination, since, at points equally distant from the highest point, there is the same force on the disturbed planet ; and, therefore, by (192.), the inclination is increased at one time, and diminished as much at another. If the orbits are elliptic, one point may be found where the effect of the force on the inclination is greater than at any other, and the whole effect on the inclination will be similar to that. (211.) In stating that the nodes always regress in the long run, the reader must be careful to restrict this expression to the sense of regressing on the orbit of the disturbing planet. It may happen that on another orbit they will appear to progress. Thus the nodes of Jupiter's orbit are made to regress on Saturn's orbit by Saturn's disturbing force. The nodes of these orbits on the earth's orbit are not very widely separated : but the inclination of Saturn's orbit is greater than that of Jupiter's. If we trace these on a celestial globe, we 150 GRAVITATION. shall have such a figure as fig. 46 ; where E C represents the plane of the earth's orbit, J E the orbit of Jupiter, and S T that of Saturn. The orbit of Jupiter, by re- gressing on Saturn's orbit, assumes the position of the dotted line j e ; but it is plain that the intersection of FIG. 46. this orbit with the earth's orbit has gone in the opposite direction, or has progressed. If the motion of the node on Saturn's orbit from J to j is regression, the motion of the node on the earth's orbit from E to e must be progression. (212.) There is a remarkable relation between the inclination of all the orbits of the planetary system to a fixed plane, existing through all their secular varia- tions, similar to that between their excentricities. The sum of the products of each mass, by the square root of the major axis of its orbit, and by the square of the inclination to a fixed plane, is invariable. (213.) The disturbance of Jupiter's satellites in latitude presents circumstances not less worthy of re- mark than the disturbance in longitude. The masses are so small, and their orbits so little inclined to each other, that the small inequalities produced in a revolu- tion may be neglected. Even that depending on the slow revolution of the line of conjunctions of the first three satellites, so small is the mutual inclination of FUNDAMENTAL PLANE. 151 their orbits, does not amount to a sensible quantity. We shall, therefore, consider only those alterations in the position of the planes of the orbits which do not vary sensibly in a small number of revolutions. For this purpose, we must introduce a term which has not been introduced before. (214.) If the moon revolved round the earth in the same plane in which the earth revolves round the sun, the sun's attraction would never tend to draw the moon out of that plane. But (taking the circumstances as they really exist), the moon revolves round the earth in a plane inclined to the plane in which the earth revolves round the sun; and the consequence, as we have seen, is that the line of nodes upon the latter plane regresses, and the inclination of the orbit to the latter plane remains, on the whole, unaltered. The plane of the earth's orbit, then, may be considered a fundamental plane to the moon's motion ; by which term we mean to express, that if the moon moved in that plane, the disturbing force would never draw her out of it ; and that if she moved in an orbit inclined to it, the orbit would always be inclined at nearly the same angle to that plane, though its line of nodes had sensibly altered. The latter condition will, in general, be a consequence of the former. (215.) In order to discover what will be the funda- mental plane for one of Jupiter's satellites, we must consider that, besides the sun's attraction, there is another and more powerful disturbing force acting on these bodies, namely, the irregularity of attraction 152 GRAVITATION. produced by Jupiter's flatness. The effect of this (as we shall show) is always to pull the satellites towards the plane of Jupiter's equator. If Jupiter were spherical, the only disturbing force would be the sun's attraction, tending on the whole to draw the satellite towards the plane of Jupiter's orbit, and that plane would be the fundamental plane of the satellite. If Jupiter were flattened, and if the sun did not disturb the satellite, the irregularity in Jupiter's shape would always tend to draw the satellite towards the plane of his equator, and the plane of his equator would be the fundamental plane of the satellite. As both causes exist, the posi- tion of the actual fundamental plane must be found by the following consideration. We must discover the position of a plane from which the sun's disturbing force tends, on the whole, to draw the satellite down- wards, and the disturbing force depending upon Jupiter's shape, tends to draw it upwards (or vice versa) by equal quantities ; and that plane will be the fundamental plane. This plane must lie between the planes of Jupiter's orbit and Jupiter's equator, because thus only can the disturbing forces act in opposite ways, and therefore balance each other ; and it must pass through their intersection, as otherwise it would at that part be above both or below both, and the forces depending on both causes would act the same way. (216.) The disturbing force of the sun, as we have seen (82.), &c., is greater as the satellite is more distant ; the disturbing force depending on Jupiter's shape is then less, as we shall mention hereafter. Consequently, ORBITS OF JUPITER'S SATELLITES. 153 as the satellite is more distant, the effect of the sun's disturbing force is much greater in proportion to that depending on Jupiter's shape. Thus, if there were a single satellite at the distance of Jupiter's first satellite, its fundamental plane would nearly coincide with the plane of Jupiter's equator ; if at the distance of Jupiter's second satellite, its fundamental plane would depart a little farther from coincidence with the plane of the equator ; and so on for other distances ; and if the dis- tance were very great, it would nearly coincide with the plane of Jupiter's orbit. If, then, Jupiter's four satellites did not disturb each other, each of them would have a separate fundamental plane, and the posi- tions of these planes would depend only upon each satellite's distance from Jupiter. (217.) In fact, the satellites do disturb each other. In speaking of the planets (210.), we have observed that the effect of the attraction of one planet upon another, in the long run, is to exert a disturbing force tending to draw that other planet (at any part of its orbit) towards the plane of the first planet's orbit. The same thing is true of Jupiter's satellites. Now, though each of them moves generally in an orbit inclined to its fundamental plane, yet in the long run (when the nodes of the orbit have regressed many times round), we may consider the motion of each satellite as taking place in its funda- mental plane. The question, therefore, must now be stated thus. The four satellites are revolving in four different fundamental planes ; and the position of each of these planes is to be determined by the consideration 154 GRAVITATION. that the satellite in that plane is drawn towards the plane of Jupiter's orbit by the sun's disturbing force, towards the plane of Jupiter's equator by the force de- pending on Jupiter's shape, and towards the plane of each of the other three satellites, by the disturbing force produced by each satellite: and these forces must balance in the long run. (218.) The determination of these planes is not very difficult, when general algebraical expressions have been investigated for the magnitude of each of the forces. The general nature of the results will be easily seen ; the several fundamental planes will be drawn nearer together (that of the first satellite, that of the second, and that of the third, will be drawn nearer to Jupiter's orbit, while that of the fourth will be drawn nearer to Jupiter's equator). The four planes will still pass through the intersection of the plane of Jupiter's equator with that of Jupiter's orbit. Thus, if we conceive the eye to be placed at a great distance, in the intersection of the planes of Jupiter's orbit and Jupiter's equator, and if the dotted lines in fig. 47 represent the appearance of FIG. 47. the fundamental planes which would exist if the satel- lites did not disturb each other, then the dark lines will represent the positions of these planes as affected by ORBITS OF JUPITER'S SATELLITES. 155 the mutual disturbances. The inelination of Jupiter's equator to Jupiter's orbit is about 3 5' ; and so great is the effect of his shape, that the fundamental plane of the first satellite is inclined to his equator by only 7" ; that of the second satellite by about 1' ; that of the third by about 5' ; and that of the fourth by about 24'J. Without mutual perturbation, the inclinations to Jupi- ter's equator would have been about 2", 20", 4,' and 48'. (219.) Having considered the positions of the funda- mental planes, we shall now consider the motion of a satellite, when moving in an orbit inclined to its funda- mental plane. (220.) The general effect will be of the same kind as that for the moon. Since the disturbing force which then tends to pull it from the plane of its orbit, tends to pull it towards the fundamental plane (as, supposing the satellite to be on that side of the fundamental plane next the plane of Jupiter's equator, the sun's disturbing force towards Jupiter's orbit is increased, that towards Jupiter's equator is diminished, and so for the others), the line of nodes will regress on the fundamental plane. The inclination on the whole will not be altered. That part of the regression of the nodes which depends on the sun's disturbing force will be greater for the distant satellites than for the near ones ; but that which depends on the shape of Jupiter (and which is much more important) will be greater for the near satellites than for the distant ones. On the whole, therefore, the lines of nodes of the interior satellites, will regress more rapidly than those of the exterior ones. Their annual 156 GRAVITATION. regressions (beginning with the second) are, in fact, 12, 2 32', and 41'. (221.) But the disturbing force of one satellite upon the others will be altered by the circumstance of its orbit not coinciding with its fundamental plane; and the orbit remains long enough in nearly the same position to produce a very sensible irregularity. To discover the nature of this, we must observe that the force of one satellite, perpendicular to the orbit of another, depends wholly upon the inclination of the two orbits, so that, upon increasing the inclination, the disturbing force is affected. Suppose now, to fix our ideas, the second satellite moves in an orbit inclined to its fundamental plane ; what is the kind of disturbance that it will produce in the latitude of the first satellite ? First, it must be observed, that when moving in the fundamental planes, the forces depending upon the inclination of those planes were taken into account in determining the position of those planes ; so that here we have to consider only the alteration produced by the alteration in the second satellite's place. Next, we shall proceed in the same manner as in several pre- ceding instances, by finding what is the motion of the first satellite, related to the motion of the second satel- lite, which can exist permanently with this inclination of the second satellite. Now, in whatever part the actual orbit of the second is higher above, or less depressed below, the orbit of the first, than the funda- mental plane of the second was, at that part there will be a greater force drawing the first satellite up, or a ORBITS OF JUPITER'S SATELLITES. 157 smaller force drawing it down (in the conjunctions at that part), than before. The alteration of force, then, will be generally represented by supposing a force to act on the first satellite, at different points of its orbit, towards the same side of its plane as the side on which the second satellite's orbit is there removed from its fundamental plane, and proportional to the magnitude of that removal. Now, conceiving the inequality intro- duced into the motion of the first satellite to be a small inclination of its orbit to its fundamental plane (which is the only inequality of Jupiter's satellites that we consider), the nodes of this orbit cannot correspond to the places where the second satellite is furthest from its fundamental plane ; for then, at one node of the first satellite, the disturbing force, before and after passing that node, being great, and not changing its direction, would not alter the place of the node, but would greatly alter the inclination : and at the opposite node, the force acting in the opposite direction would produce the same effect ; and thus the permanency of the inequality would be destroyed. We must then suppose the nodes of the orbit of the first satellite on its fundamental plane to coincide with those of the orbit of the second satellite on its fundamental plane. But is the inclina- tion to be the same way, or the opposite way ? To answer this, we must consider that the action of Jupi- ter's shape would tend to make the nodes of the first satellite regress much more rapidly than those of the second ; but as our orbit of the first satellite is assumed to accompany the second in its revolution, the dis- 158 GRAVITATION. turbing force depending on the second must be such as to destroy a part of this regression, or to produce (sepa- rately) a progression of the nodes of the first ; conse- quently, the disturbing force produced by the second must tend to draw the first from its fundamental plane (193.). But the disturbing force produced by the second is in the same direction as the distance of the second from the fundamental plane of the second ; con- sequently, the orbit of the first must lie in the same position, with regard to the fundamental plane of the first, in which the orbit of the second lies with regard to the fundamental plane of the second. The same reasoning applies to every other case of an interior satellite disturbed by an exterior; and thus we have the conclusion : If the orbit of one of Jupiter's satel- lites is inclined to its fundamental plane, it affects the orbit of each of the satellites interior to it with an inclination of the same kind, and with the same nodes. (222.) Let us now inquire what will be the nature of the inequality produced in the latitude of the third satellite. The same reasoning and the same words may, in every part, be adopted, except that the regression of the nodes of the third satellite, as produced by Jupiter's shape, will be slower than that of the second satellite, and therefore the disturbing force which acts on the third, must now be such as to quicken the regression of its nodes, and must therefore be directed towards its fundamental plane. From this consideration we find, as a general conclusion, if the orbit of one of Jupiter's satellites is inclined to its fundamental plane, it affects ORBITS OF JUPITER'S SATELLITES. 159 the orbit of each of the satellites exterior to it, with an inclination of the opposite kind, but with the same nodes. (223.) The first satellite's orbit appears to have no sensible inclination to its fundamental plane ; but those of the second, third, and fourth are inclined to their fundamental planes (the second 25', and the third and fourth about 12'), and these are found to produce in the others inequalities such as we have investigated. (224.) It is only necessary to add, that the disturbance of the first satellite by the second produces an altera- tion in the action of the first on the second, tending to draw the second from its fundamental plane, and there- fore to diminish, by a small quantity, the regression of its nodes. In the same manner, the altered action of the third on the second tends to draw the second towards its fundamental plane, and therefore to increase, by a small quantity, the regression of its nodes. There is exactly the same kind of complication with regard to the disturbances of those bodies in latitude as with regard to those in longitude, explained in (150.), &c. (225.) The only other inequality in latitude, which is sensible, is that depending on the position of the sun, with regard to the nodes of the orbits on the plane of Jupiter's orbit (that is, with regard to the node of Jupiter's equator on Jupiter's orbit), and this amounts to only a few seconds. It is exactly analogous to that of the moon, explained in (205.). 160 GRAVITATION. SECTION IX. Effects of the OUateness of Planets upon the Motions of their Satellites. (226.) IN the investigations of motion about a central body, we have supposed that central body to be a spherical ball. This makes the investigation remark- ably simple ; for it is demonstrated by mathematicians, that the spherical form possesses the following pro- perty : the attraction of all the matter in a sphere upon another body at any distance external to it is exactly the same as if all the matter of the sphere were col- lected at the centre of the sphere. In the investigation of motion about a centre, we may therefore lay aside (as we have usually done) all consideration of the size of the attracting body, if that body is spherical. (227.) But the planets are not spherical. Whether or not they have ever been fluid, still ,they have (at least, the earth has) a great extent of fluid on its sur- face, and the form of this fluid will be affected by the rotation of the planet. The fluid will spread out most where the whirling motion is most rapid, that is, at the equator. Thus it appears from theory, and it is also found from measures, that the earth is not a sphere, but a spheroid, flattened at the north and south poles, and protuberant at the equator. The proportion of the axes differs little from the proportion of 299 : 300 ; so that a line drawn through the earth's centre, and passing through the equator, is longer than one passing through the poles, by 27 miles. (228.) The flattening of Jupiter is still more remark- OBLATENESS OF PLANETS. 161 able. The proportion of his axes differs little from that of 13 : 14, and thus the difference of his diameters is nearly 6000 miles. In fact, the eye is immediately caught by the elliptic appearance of Jupiter, on viewing him for a moment in a telescope. (229.) It is our business, in the present section, to point out the general effects of this shape upon the motion of satellites. The agreement of observation with calculation on this point is certainly one of the most striking proofs of the correctness of the theory, " that every particle of matter attracts every other par- ticle, according to the law of Universal Gravitation." (230.) We will begin with explaining the law according to which an oblate planet attracts a satellite in the plane of its equator. The spheroid represented by the dark line in fig. 48 FIG. 48. may be supposed to be formed from the sphere repre- sented by the dotted line, by cutting off a quantity of matter from each pole. . To simplify our conception, let us suppose that all the matter cut off was in one lump at each pole; that is, at the points D and E. The attraction of the whole sphere on the satellite B, as we have remarked, is the same as if all the matter of the sphere were collected at A. But the attraction of the part cut off is not the same as if it were col- lected at A, inasmuch as its distance from B is greater, M 162 GRAVITATION. and as the direction of the attraction to D, or to E, is not the same as that to A. Thus, suppose AD is called 1, and A B is called 10. Since the forces are inversely as the squares of the distances at which the attracting mass is situate, the attraction of the lump D, if at the point A, where its distance from B is 10, may be called -pi-^ ; but if at D, it must be called -j-^-j-, since the square of B D is equal to the sum of the squares of B A and A D, that is, to the sum of 100 and 1. Also the direction of attraction is not the same ; for, if the attraction of D should draw the satellite through B b, and if b c be drawn perpendicular to A B, the only effec- tive approach to A is the distance B c, which is less than B & in the proportion of B A to B D, or of 10 to -y/101 ; and, therefore, the effective attraction of D, estimated by the space through which it draws the 10 satellite towards A, must be called - - . And 101 x -x/101 this is the whole effect which its attraction produces ; 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 perpendicular to A B will destroy each other. We have, then : the attraction of the lump D, if placed at A, would be represented by - = O'Ol ; but as placed at D, its effective attraction is repre- 10 sented by - =_ = 0'0098518. The difference 101 x x/101 is 0-0001482, or nearly ^ of the whole attrac- EQUATORIAL ATTRACTION INCREASED. 163 tion of D, and the same for E. Consequently, the lumps at D and E produce a smaller effective attrac- tion 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, therefore, 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 in 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 at- 20 traction is represented by =. = 0*002490653. 401 x x/401 QI7K The difference now is 0*000009347, or nearly 100000 of the whole attraction of D. Consequently by re- moving 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 spheroid, though still greater than if its whole matter were collected at A, differs from that by a quantity, whose proportion to the whole attraction is only one-fourth of M 2 164 GRAVITATION. 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 attraction 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 oblateness, whose proportion to the other force diminishes as the square of the distance of the satellite is increased. (233.) Now, let us investigate the law according to which, an oblate spheroid attracts a body, situate in the direction of its axis. Proceeding in the same manner as before, and sup- posing the distance A B to be 10, the attraction of the FIG. 49. lump, which at A would be represented by -j-^-, will at D be represented by -Q I T) and will at E be represented POLAR ATTRACTION DIMINISHED. 165 by i-j. (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 at the centre, their attraction on B would be + r = -~ == 0*02. In the positions D and E, the sum of their attrac- tions on B is gj + j^j = 0-0206100. 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 than if they were collected at the centre by nearly -i-f-g-th 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 col- lected 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 represented, 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 increased. (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 attrac- 166 GRAVITATION. tion on a body in the direction of the axis is less, it will readily be understood, that in taking directions, succes- sively more and more inclined to the equator, on both sides, the attraction successively diminishes. And there is one inclination, at 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 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 col- lected 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 distance 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 force, it would, on the whole, cause no alteration in the position of the line of apses (for it SATELLITES' APSES PROGRESS. 167 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 distance. See the note to (98.) ). But (231.) the proportion of this force to the other diminishes as the distance is increased. Consequently, the regression 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 pro- gression of the line of apses at every revolution. The progression 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 disturbing force, that it can hardly be discovered; but the pro- gression 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. (237.) We shall now proceed with the investigation of the disturbances 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 168 GRAVITATION. that plane. In this case, then, there is no disturbance in latitude ; we must, therefore, in the following inves- tigation, 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. If the attraction of D alone acted, it would in a certain time draw the satellite to d; and fd wouM be the part of the motion of B, which is perpendicular to A B ; and this motion is upwards. In 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 prepon- derating force ; and therefore, on the whole, the com- FORCE TOWARDS PLANE OF PLANET'S EQUATOR. 169 bined attractions of D and E will tend to draw the satellite upwards from the line BA. But the attrac- tion of the whole sphere would tend to draw it along the line B A. Therefore, when D and E are removed, the attraction 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 sphe- roid, 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 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 DB and EB is increased, and the inequality of the inclinations to BA is also increased; and the dis- turbance is, therefore, much greater for a near satellite than for a distant one. (239.) We have seen (215.) the effect of this dis- turbing 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 satellites 170 GRAVITATION. are very nearly the same as if no other disturbing force existed, so great is the effect produced by Jupiter's oblateness. (240.) The figure of Saturn, including in our con- sideration the ring which surrounds him, is different from that of Jupiter; but the same principles 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 spheroid, 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 protuberance analogous inform to a ring. The whole irregularity in the attraction of the spheroid is evi- dently due to the attraction of this ring-like protu- berance, 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 proportion to the whole attraction for the ring than for the spheroid, since the ring produces all the irregularity 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 ATTRACTION OF SATURN'S RING. 171 progression of the perisaturnium of any satellite, and the rate of regression of its node, will, therefore, be rapid. In other respects, it is probable, that the theory of these satellites would be very simple, since all (except the sixth) appear to be very small, and the sun's disturbing 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 comparing 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 -firth of the mass of the planet. (242.) The effect of the earth's oblateness in in- creasing the rapidity of regression of the moon's nodes is so small, that it cannot be discovered from observa- tion. But the effect on the position of the fundamental plane is discoverable. We have seen (204.) that the moon's line of nodes regresses completely round in 19 J years. The plane of the earth's equator is inclined 23 J to the earth's orbit, 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 172 GRAVITATION. two planes ; and 9f years later, the line of nodes again coincides with the same line, but the orbit is inclined 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 inclination 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 incli- nation 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 oblate- ness (the only other effect on the moon which is sen- sible) that deserves notice. The inclination 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 J. Consequently, when the moon's orbit lies between these 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. EFFECTS OF EARTH'S OBLATENESS. 173 For the period of 9| years, therefore, the earth's attrac- tion on the moon is gradually diminished, and then is gradually increased for the same time. The moon's orbit (47.) becomes gradually 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 observa- tion to amount to about 8", by which the moon is sometimes before her mean place, and sometimes be- hind it. If the earth's flattening at each pole were more or less than -5-5-0 th of the semi-diameter, the effects on the moon, both in altering the position of the funda- mental 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: FEINTED BY WILLIAM CLOWES AND SONS, LIMITED, STAMFORD STREET AND CHARING CROSS. ETURN CIRCULATION DEPARTMENT O ""i^ 202 Ma i n L I bra ry 642-3403 DAN PERIOD 1 HOME USE 2 3 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1 -month loans may be renewed by calling 642-3405 -month loans may be recharged by bringing books to Circulation De; Renewals and recharges may be made 4 days prior to due date DUE AS STAMPED BELOW JAN 3 3 1977 WftSEP 9 77 978 ORM NO. DD 6, 40m, 6V6 UNIVERSITY OF CALIFORNIA, BERKELE BERKELEY, CA 94720 CDbl3S73b3 1L titiir in $i |;^ fab ii