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gravity to the earth for a given distance of the earth from the sun. Hence, as the distance a varies with the time in the year, or with the sun's anomaly, the mean diminution of the moon's gravity must also vary. This variation causes a change in the moon's distance from the earth, and, consequently, in her velocity. The change in her place resulting from this change of velocity, is the annual equation.
406. The Evection is produced by an inequality in the sun's disturbing force, depending on the position of the line of the apsides of the moon's orbit with regard to the line of the syzygies.
Let R and r denote the distances of the moon from the earth, in apogee and perigee, when the line of the apsides coincides with the line of the syzygies, X and x, the distances at which the moon would be from the earth, in apogee and perigee, if she was not acted on by the sun, and G and g, the perigean and apogean graf ass
and supposing the earth's
vities in that case. Also, put n
sun to remain
distance from the
X2x2:: G: g,
and R2 : r2 :: G — 2rn : g
G - 2rn
Now, as G is greater than g, and 2rn less than 2Rn, it is evident that
is greater than
is greater than
It therefore follows, that, when the line of the apsides coincides with the line of the syzygies, the ratio of the apogean distance of the moon to the perigean distance, and, consequently, the eccentricity of the orbit, is increased by the action of the sun. In like manner it may be shown, that, when the line of the apsides coincides with the line of the quadratures, the sun's action diminishes the eccentricity of the orbit. The change in the eccentricity of the orbit, produces a change in the equation of the centre; and this change is the Evection.
407. The Variation is produced by the resolved part of the sun's disturbing force that acts in the direction of a tangent to the moon's orbit.
It has been shown (404 I), that MP, the part of the sun's disturbing force that acts in the direction of a tangent to the moon's orbit, and therefore changes her motion in her orbit, is equal to 3 fr
sin 2x. Hence, supposing the earth's distance from the sun, 2a3 and the moon's distance from the earth, to remain constant, this force is proportional to sin 2x; that is, to the sine of twice the distance of the moon from the quadratures. It is, therefore, greatest in the octants, and is nothing in the syzygies and quadratures. The inequality in the moon's motion thus produced is the Variation.
408. The motion of the Apsides of the moon's orbit is produced by the action of the sun in diminishing the moon's gravity to the earth.
If the moon was only acted on by the earth's attraction, she would describe an ellipse, and her angular motion, would be just 180°, from one apsis to the other; or, which is the same, from one place where the orbit cuts the radius vector at right angles, to the other. But, in consequence of the change produced in the moon's gravity to the earth, by the action of the sun, the moon's path is not truly an ellipse. When the effect of the sun's action is a diminution of the moon's gravity, she will continually recede from the ellipse that would otherwise be described, her path will be less curved, and she must move through a greater distance before the
radius vector intersects the path at right angles. She must, therefore, move through a greater angular distance than 180°, in going from one apsis to the other, and, consequently, the apsides will advance. On the contrary, when the gravity is increased by the sun's action, the moon's path will fall within the ellipse which she would otherwise describe, its curvature will be increased, and the distance through which she must move before the radius vector intersects her path at right angles, will be less. The apsides will, therefore, move backwards. Now, it has been shown (404), that the sun's action alternately diminishes and increases the moon's gravity to the earth. The motion of the apsides will, therefore, be alternately direct and retrograde. But, as the diminution has place during a much longer part of the moon's revolution, and is besides greater than the increase, the direct motion will exceed the retrograde. Consequently, in an entire revolution of the moon, the apsides have a progressive motion.
409. Motion in the moon's nodes and change in the inclination of her orbit. The direction in which the sun's disturbing force acts on the moon, does not, except in some particular cases, coincide with the plane of her orbit. This force, therefore, causes the moon to leave the plane of her orbit, or, which is equivalent, causes this plane itself to change its position, varying both the line in which it intersects the plane of the ecliptic and the angle it makes with that plane. By a simple but tedious investigation, it may be shown, that, in consequence of the sun's action, the nodes must, during each synodic revolution of the moon, move alternately backwards and forwards; the backward motion being, however, the greater, so that, on the whole, they must have a retrograde motion. It may also be shown, that the inclination of the orbit must alternately increase and diminish, vibrating thus, about its mean value, from which it never widely deviates.
410. Stability of the solar system. The mutual actions of the planets and satellites, and the inequality of the sun's action on a planet and its satellite in different positions, produce continual changes in the motions of the bodies, and in the eccentricities and inclinations of their orbits. Although some of these changes are
ascertained from observations to be periodical, and it is found that the quantities subject to them, alternately increase and decrease, so that their mean or average values remain the same, yet there are others which have always been accumulating from the period of the earliest observations to the present time. One of these, the acceleration of the mean motion of the moon (193), has long attracted attention. If this acceleration of her motion, and the consequent diminution of her distance, were perpetually to continue, it would follow that she would eventually be precipitated to the earth. Such a result, if it were a necessary consequence of the structure and working of the system, would seem to imply some imperfection in the works of the all-wise Creator of the universe. But the profound investigations of Lagrange and Laplace have shown, that, with the system constituted as it is, no such result can have place; that not only some, but all, the changes produced in the motions and orbits, by the mutual attractions of the bodies, must be periodical; and that, though some of the quantities in which these changes are produced must continually increase, or continually decrease, for many thousands of years, they cannot perpetually do so. Through the operation of the very same causes, the quantities that are now increasing, must in process of time decrease, and those that are decreasing, must increase. None of them can ever widely deviate from their average values. Thus, notwithstanding the many perturbations and seeming irregularities, the stability of the system is preserved.
411. Tables relative to the planets and satellites. The following tables contain the elements of the orbits of the planets, and their masses and densities as far as they are known. The longitudes are reckoned from the mean equinox of the epoch.
The fourth table, page 218, contains the elements of the first twenty-seven asteroids, which have been collected from the most reliable sources.
ELEMENTS OF THE PRINCIPAL PLANETS.
4332.584821 10759.219817 30686.820830 60125.483
Longitude of the
48.6 74° 21′ 46."9+
43 53. 1
30 5. 0
26 21 1
4 28 19 41
56. 6+ 26 22
8 34. 6
Inclination to the
9 29. 8
7° 0/ 9./1
32 17 4 4 0
+18./1 4. 5
15. 5 3. 1
in Mean Solar Days. from the Sun, or Eccentricity. Secular Variation. Semi-major Axis.