the two cases; with centre s describe the circle cde. Then it is evident that the very excentric orbit c B in fig. 8 is widely separated from the circle cde, and therefore, when it is bent through a given angle to the position cd, it will intersect the circle at a point d not distant from c. In fig. 9, on the contrary, the orbit CB is not widely separated from the circle, and therefore, when it is bent through a given angle, its intersection d will be distant from c. Now the new perihelion c will be found, in both cases, by bisecting cd; and, therefore, its change of position in fig. 8, where the orbit is very excentric, is much less than in fig. 9, where the excentricity is small. 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 excentricity of a planet's orbit, let us suppose the planet 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 the

where the force has acted to accelerate its motion, instead of describing the orbit A CG, proceeds to describe the orbit aed, which at a has the same direction (or has the same tangent A B) as the orbit A C G. It is plain now that c is the part nearest to the sun, or c is the perihelion : and it is evident here, that the line of apses has altered its position from sc 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,

G

g

direction ab instead of a B, in which it was moving; and, therefore, instead of describing the orbit a CG, in which it was moving before, it will describe an orbit a cg, 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 aphelion to perihelion, it is approaching to the 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 disturbing force; and this effect may be represented by supposing that at E, fig. 11, instead of moving in the direction EF in which it was

moving, the planet is projected in the direction Ef. Instead therefore of describing the ellipse E GH, in which it was moving before, it will describe such an ellipse as Egh, 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 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 perihelion 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

S

and the path described by the planet is Df, instead of DF, having the same direction at D (or having the same tangent DE), 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 Df, from moving without disturbance in an elliptic orbit cDf (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 greater than the velocity from moving in c D; C D is therefore less curved than c D, and therefore exterior to it (since it has the same direction at D); and then the perihelion is some point in the position of c, and the line of apses has changed its direction from sc to sc, 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 the 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 н, fig. 14, where the force has 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 line of apses has shifted from 8 G to sg. If the force act after the planet has passed the 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 GK; the point most

* It is supposed here, and in all our investigations, that the excentricity of the orbit is small, and, consequently, that a force perpendicular to the radius vector produces nearly the same effect as a force acting in the direction of a tangent to the ellipse.

༡

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 attraction carries the earth from E to e, and carries the

(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 perihelion, 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 regress, and while the planet passes from aphelion to perihelion causes the apses to progress.

(67.) (XI.) 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 therefore will not be able to pull it into so small a compass 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 increased, and the orbit therefore is made more excentric. Now, suppose the force increasing the velocity to act at aphelion. Just as before, the sun's attraction 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 distance 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 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 opposite 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. we shall have to consider the effect of forces, which act for a long In future, 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.)

SECTION IV. On the Nature of the Force disturbing a Planet or Satellite, produced by the Attraction of other Bodies.

Fig. 16.

E

m M

moon from м to m, and if E e is equal and parallel to м m, then em, 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, 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.

(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 disturbing 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. For the effect of the attraction is to move the moon from the place where it would otherwise have been; but the sun's attraction upon the earth also moves the earth from the place where it would

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 PH in Newton's figure without altering its position.

From this we draw the two following important conclusions:(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 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 aet 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 situation with respect to the earth, some complication is introduced. case that can well be imagined. If, however, the moon is in any other earth's distance, (which according to (9) produces an inequality in the Not only is the moon's distance from the sun different from the attractions upon the earth and moon,) 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 perturbation which one planet produces in the motion of a second are in different directions.) The same applies in every respect to the 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 difficulty we must have recourse to geometrical considerations. In fig. 17, let в be a body revolving about A, and let c be Fig. 17.

Ва

d1

The attraction of c will in a certain time draw A to a; it will in the another body whose attraction disturbs the motion of B, round A. same time draw B,to b,. Make в, d, equal and parallel to ▲ a; then a d, will be equal and parallel to A B1. Now if the force upon B, were such as to draw it to d,, 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, to b,. If, instead of supposing the revolving body to be at B, we suppose it at B, and if the attraction of c would draw it through B, b, while it draws a through A a, then (in the same manner, making B, d, equal and parallel to ▲ a) the real disturbing force may be represented by a force which in the same time would draw B, through dy b

motion." In fig. 18, if d b represent the space through which a force 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 de and eb, provided that with db they form a triangle. 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 d e perpendicular to A B or a d, and e b parallel to ▲ B or a d; and now we can say: the disturbing force produced by the attraction of c is a force represented by de perpendicular to the radius vector, and a force represented by e b 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 investigations of Section III. For the present we shall content 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 attraction, the disturbance of the earth's motion by Jupiter's attraction, the disturbance of the motion Fig. 19. a A

с

bd B

• •

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 attraction of c will in a certain time draw a to a, it will in the same time draw в to b, where в b is much greater than A a. Take Bd equal to ▲ a, then db is the effect of the disturbing force, which tends to draw B further from A. In this case then, the disturbing 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 C A B (fig. 20) be in the same straight line, but let в be Fig. 20. c

a A

d B b

...

on the side of A, opposite to c. In this case Bb is less than A ɑ; and if Bd is taken equal to a a, the disturbing force represented by db will be entirely in the direction of the radius vector, and directed from the central body. This case is particularly deserving of the reader's consideration, as the effectual disturbing force is exactly opposite to the attraction which c 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 A C. 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 disturbing 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 of the distance of A, and therefore the attraction upon B is of the attraction upon A, and the disturbing force is 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 planetary 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 (3) × the attraction of the sun on the earth, or 1600 parts of the sun's attraction on the earth, and the disturbing force therefore is 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 moon is (100) or parts of the sun's attraction on the earth, and the disturbing force is 801 parts of the sun's attraction on the earth, which is very little less than the former. The effects of the difference are, however, sensible.

799

equal to A a. But since C B is also equal to CA, it is evident that ab Conwill be parallel to A B, and therefore 6 will be in the line a d. sequently 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 bd bears to Bb, or A B to A 0. Thus, in the first numerical instance taken above, the disturbing force in this part of the orbit is of the attraction on A: and in the second numerical instance, the disturbing force is 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 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 to c a.

(82.) V. 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 proportional 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 proportion to the whole attraction on A which A B bears to A C. With regard to he force mentioned in (III.); if we suppose the moon's distance from the earth to be of the sun's distance, the disturbing force when the moon is between the earth and the sun is parts of the sun's attraction on the earth, or nearlyth part. But if we suppose the moon's distance from the earth to beth of the sun's distance, the attraction on the moon (when between the earth and the sun) would be (9) or 000 parts of the attraction on the earth; the disturbing force or the difference of attractions on the earth and moon, would be or nearly th 30001, part of the sun's attraction on the earth. Thus, on doubling the moon's distance from the earth, the disturbing force is nearly doubled and in the same manner, on altering the distance in any other proportion, we should find that the disturbing force is altered in nearly the same proportion.

159201

(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 nearlyth 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 distance, or 188 parts of the sun's former distance from the earth; the attractions on the earth and moon respectively will be and 1800 parts of the former attraction on the earth: and the disturbing force, or the difference between these, will be 15, or nearlyth part of the former attraction 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.

400

(84.) VII. Suppose B to have moved from that part of its orbit where its distance from c is equal to a's distance from c, towards the part where it is between A and c. Since at the point where B's distance from c is equal to A's 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 de, 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, fig. 22, in A

C

a

Fig. 21.

d B

da

B1

a A

(80.) IV. Suppose B, fig. 21, 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. Bb, therefore, will be

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 de, perpendicular to the radius vector, in the direction opposite to that in which B is going. If we construct a figure for the situation B., in which B is moving from the point of equal distances to the point where B is on the side of a opposite to C, we shall see that there is a disturbing force perpen

dicular 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 disturbing 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 disturbing body, or on the opposite side (the force in the former case being the greater), and directed to the central body, at each of the places where the distance from the 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 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 central 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 â, fig. 23, the

attraction of c draws A strongly towards B,, and B, strongly towards A, and, therefore, there is a very powerful disturbing force drawing B, towards A. If B is in the situation B,, the attraction of c draws A strongly from B,, and draws B, feebly towards A; therefore, there is a small disturbing force drawing B, 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 AB,, it will be possible to find two points, B, and B,, whose distance from c is equal to the distance of a from c, 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 в moves from the position B, to B, 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 в moves from B, to B, it accelerates B's motion; while B moves from B, to B, it retards B's motion; and while B moves from B, to B1, it accelerates B's motion. But if A c is less than AB,, there are no such points, B, B,, as we have spoken of; and the disturbing force perpendicular to the radius vector, accelerates B as it moves from B1 to B, and retards B as it moves from B, to B1.

C

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 distance 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 disturbing body renders these disturbances very much more conspicuous than any others in the solar system; and, on this account, as well as for the accuracy with which they can be observed, these disturbances have, since the invention of the Theory of Gravitation, been considered the best tests of the truth of the theory.

Some of the disturbances are independent of the excentricity of the moon's orbit; others depend, in a very remarkable manner, upon the excentricity. We shall commence with the former.

(88.) The general nature of the disturbing force on the moon may be thus stated. (See (77) to (86).) When the moon is either at the point 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 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 distance; 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 diminishing; and while it moves from aphelion to perihelion, the disturbing force is constantly increasing. Referring 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 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 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 discovered by Tycho Brahé from observation, about 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 sun It cannot move in a circle: for the force perpendicular to the radius vector retards the moon as it goes from B, to B2, fig. 24,

can move.

and its velocity is therefore less at B, than at B,, and on this account (supposing the force directed to A at B, equal to the force directed to A at B), the orbit would be more curved at B, than at B. But the force directed to A at B, is much greater than at B, (see (88)); and on this account the orbit would be still more curved at B, 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. (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), and increased by the disturbing force in the radius vector directed towards the earth, the curvature at B, ought to be much greater than at B1, where the velocity is great, the moon far off, and the disturbing force directed from the earth. But, on the contrary, the curvature at B, is much less than at B,; therefore, this form of orbit is not the true one. But if the orbit be supposed

to be oval, with its shorter axis directed towards the sun, as in

the radius vector, while the moon goes from B, to B; and though, the distance from a being greater, the earth's attraction at B, will be less than the attraction at B, ; yet, when increased by the disturbing force, directed to a at B,, it will be very little less than the attraction diminished by the disturbing force at B. The diminution of velocity then at B, being considerable, and the diminution of force small, the curvature will be increased; and this increase of curvature, by proper choice of the proportions of the oval, may be precisely such as corresponds to the real difference of 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 stationary with respect to the sun. If however we take the true case of the earth moving round the sun, or the sun appearing to move round the earth, we have only to suppose that the oval twists round after the sun, and the same reasoning applies. The curve described by the moon is then such as is represented in fig. 27. As the disturbing force, perpendicular to the

Fig. 27.

radius vector, acts in the same direction for a longer time than in the 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 by the radius vector were followed, as it would be if there were no force perpendicular to the radius vector, the angular motion of the moon near B, and B,, fig. 26, would be much less than that near B, and B. But in consequence of the disturbing force, perpendicular to the radius vector (which retards the moon from B to B, and from B, to B, and accelerates it from B, to B,, and from B to B), the angular motion is still less at B, and B,, and still greater at B, and B. The angular motion therefore diminishes considerably while the moon moves from B, to B, and increases considerably while it moves from B2 to B, &c. The mean angular motion, determined 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, is less than the mean; and therefore (as in (90),) from B1 to B, the moon's true place is before the mean; from B2 to B, the true place is behind the mean; from B, to B, the true place is before the mean; and from B, 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 sometimes behind the mean place. It was discovered by Tycho Brahé, from observation about 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 opposition to be diminished. Observing what the nature of these forces is (77), (78), and (84), this amounts to supposing that near conjunction the force necessary to make up the difference is a force acting in the radius vector, and directed from the earth, and a force perpendicular to the radius vector, accelerating the moon before conjunction, and 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 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 excentricity will either be increasing or diminishing from the action of these forces. We must assume therefore, as our orbit is to have the same excentricity at each revolution, 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 lines of apses must progress as fast as the sun appears to pro

gress, and we must therefore choose that position in which the forces will cause progression of the line of apses. If the perigee be directed to the sun, then the forces at both parts of the orbit will, by (51), (54), (65), and (66), cause the line 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 progress. The effect, then, of the difference of forces of which we have spoken, is to elongate the orbit towards the sun, and to compress it on the opposite side. This irregu larity is called the parallactic 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 discussed here) would be to make its line of apses progress 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 apparent 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 considerably increased by the indirect effect of the other disturbing forces.

(95.) The magnitude of the forces concerned here is about th of those concerned in (91), &c.; but the effect is about th of their effect. This is a striking instance of the difference of proportions, in forces and the effects that they produce, depending on the difference in their modes of action. The inequality here discussed is a very interesting one, from the circumstance that it enables us to determine with considerable accuracy 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 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 variation upon that orbit will be almost exactly the same as if there were no excentricity.* Thus, supposing that the orbit without the disturbing force had such a form as the dark line in fig. 28, it will, with the disturbing force, have such

that the disturbing force makes the forces at B, and B, less than they would otherwise have been, and greater at B, and B, 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, than it would otherwise have been. So that, unless we supposed it moving at B, with a greater velocity than it would have had undisturbed in the circle B1, b2, Bg, b4, the great curvature produced by the great force and diminished velocity at B would have brought it much nearer to A than the point B.; but with this large velocity at B1, it will go out further at B, and then the great curvature may make it pass exactly through B3. like manner, in fig. 30, if the velocity at B1 were not greater than it would have had undisturbed in the ellipse B1, bg, Bg, b, the increased curvature at B2, produced by the increased force and diminished velocity there, would have brought it much nearer to a than the point B.; but with a large velocity at B1 it will go out at B, further than it would otherwise have gone out, and then the increased force and diminished velocity will curve its course so much, that it may touch the elliptic orbit at B3; and so on. The whole explanation, in

In