directed to A at B2 equal to the force directed to A at B1), the orbit would be more curved at B2 than at B1. But the force directed to A at B2 is much greater than that at B, (see (88.)); and on this account the

orbit would be still more curved at B2 than at B1; 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 B2), and increased by the disturbing force in the radius vector directed towards the earth, the curvature at B2 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 B2 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

F

its shorter axis directed towards the sun, as in fig. 26, all the conditions will be satisfied. For the velocity at B2 is diminished by the disturbing

2

force having acted perpendicularly to the radius vector, while the moon goes from B1 to B2; and though the distance from A being greater, the earth's attraction at B2 will be less than the attraction at B1; yet, when increased by the disturbing force, directed to A at B2, it will be very little less than the attraction diminished by the disturbing force at B1. The dimunition of velocity then at B2 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 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 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 B2 and B, fig. 26, would be much less than that near B1 and B. But in consequence of the disturbing force, perpendicular to the radius vector (which retards the moon from B1 to B2, and from B, to B4, and accelerates it from B2 to B, and from B4 to B1), the angular motion is still less at B2 and B1, and still greater at B1 and B. The angular motion, therefore, diminishes considerably while the moon moves from B1 to B2, 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 B2 is less than the mean; and therefore (as in (90.) ), from B1 to

4

2

B2 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, 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 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 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 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 irregularity 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