when her ascending node is iu Aries, the angle which her orbit makes with the horizon will be 5 less than that which the ecliptic makes with the horizon, and the difference of time between her risings on two successive evenings will be less than 17 minutes, which would have been the time had her orbit coincided with the ecliptic. On the contrary, when the descending node comes to Aries, the angle which her orbit makes with the horizon will be greater by 5, and consequently the difference of the tiaies of her successive risings will be greater than if she moved in the plane of the ecliptic. If when the full moon is in Pisces or Aries the ascending node of her orbit is also in one of those signs, the difference of the times of her rising will not exceed 1 hour and 40 minutes during a whole week ; but when her nodes are differently situated, the difference in the time of her rising in the same signs may amount to 3 hours in the space of a week. In the former case the harvest moons are the most beneficial, in the latter the least beneficial to the husbandman. All the variations in the intervals between the consecutive risings or settings take place within the period in which the line of the nodes makes a
complete revolution.
The moon s orbit at any moment is an ellipse, having the earth at one of the foci ; but this orbit is continually changing in form and position, the eccentricity alternately increasing and decreasing between the limits O OGG and 044, and the perihelion sometimes advancing and some times retrograding, but on the whole advancing at an average rate of 40$ per annum, so as to complete a sidereal revolution in 323 2 5 75 mean solar days. See also table of Lunar Elements. These changes, like those which aS ect the inclination of the orbit and the position of the nodes, are due to the perturbing influence of the sun on the moon s motions relatively to the earth. The consideration of these perturbations, whether as affecting the figure and position of the moon s orbit, or her motions in that orbit, constitutes what is called the Lunar Theory. The following are the chief peculiarities of the lunar movements:—
In the first place, the moon s motion differs from uniform motion around the earth as centre, because the moon s orbit is elliptic, so that an equation of the centre corresponding to that already described in dealing with the sun s motion, but greater in amount, has to be applied to the mean motion.
Secondly, the sun lying constantly far outside the moon's orbit relatively to the earth, his influence on the whole must tend to diminish the earth s influence. It is true that when the moon is iu quadrature the sun s attraction tends to draw her towards the earth ; but it is manifest that this influence is small compared with the action of the sun in drawing the moon from the earth when she is in conjunction with him, and in drawing the earth from the moon when she is in opposition. The balance of effects during a single lunation must correspond to a diminution of the earth s influence, or in other words, to an increase in the moon s period. Now, if the earth and moon, or their common centre of gravity, were always at the same dis tance from the sun, this action of his would be uniform all the year round. But as he is nearer in perihelion than iu aphelion, he exerts a greater influence in the former than in the latter position ; in other words, the lunar month in winter (when the sun is near perihelion) is lengthened to a greater degree by the sun s perturbing action than in summer (when the sun is in aphelion). Accordingly, on the whole, the moon s motion in longitude is less in winter than in summer. She lags behind her mean place more and more throughout the whole time that the sun is at a distance exceeding his mean distance ; she is therefore at her maximum displacement, behind her mean place, near the time that the sun is at his mean distance after perihelion passage ; while she is at her maximum displacement, in advance of her mean place, when the sun is at his mean distance after aphelion passage. The greatest amount by which, so far as this cause is considered, she gets in advance of or behind her mean place, is about 11 12*; and this displacement, because of its period, is called the Annual Equation.
Since the sun acts to diminish the earth s influence when the moon is in syzygies, and to increase that influence when the moon is in quadratures, the motion of the moon is retarded in the former case and accelerated in the latter, and at the octants there is neither acceleration nor retarda tion. Hence arises an inequality called the Variation. It was discovered by Tycho Brahe, who found that the moon s place, calculated from her mean motion, the equation of the centre, and the evection, does not always agree with her true place, and that the variations are greatest in the octants, or when the line of the apsides makes an angle of 45 with that of the syzygies and quadratures. Having observed the moon at different points of her orbit, he found that this correction has no dependence on the position of the apsides, but only on the moon s elongation from the sun. Its maximum value is additive in the octants which come immediately after the syzygies, and subtractive in the octants which precede the syzygies. It vanishes altogether in the syzygies and quadratures, and on this account was not perceived by the ancient astronomers, who only observed the sun in those positions. Its maximum value is 35 42".
The next inequality to be mentioned is the largest of all, except, of course, the equation of the centre. It is called the evection, and was discovered by Hipparchus, but Ptolemy first recognised the law of its effects. These are to diminish the equation of the centre when the line of the apsides lies in syzygy, and to augment it when the same line lies in quadratures. Thus, supposing the apsides to lie in syzygy, and that it is sought to compute the moon s true longitude about seven days after she has left the perigee, by adding the equation of the centre to the mean anomaly, the resulting longitude will be found to be above 80 less than that which is given by observation. But if the line of the apsides lies in quadratures, the place of the moon at about the same distance, that is, 90 3 from the perigee, computed in the same manner, will be found to be before the observed place by above 80 ; that is, the com puted will be greater than the observed longitude by more than 80 . The maximum value of the evection is 1 20 29" 9. It is occasioned by the sun s action in modifying the form of the lunar orbit, and so causing the equation of the centre to vary in amount.
The lunar inequalities which we have as yet considered are all of a periodic nature. But there are others of a different kind, the periods of which are so long that, with reference to the duration of human life, they may be con sidered as permanently affecting the elements of the lunar orbit. These are the Secular Inequalities, the most remarkable of which is the acceleration of the moon's mean, motion.
On comparing the lunar observations made within the last two centuries with one another, there results a mean secular motion greater than that which is given by comparing them with those made by Ibu-Junis, near Cairo, towards the end of the 10th century, and greater still than that which is given by comparing them with observations of eclipses made at Babylon in the years 719, 720, and 721 before our era, and preserved by Ptolemy in the Almagest. This acceleration of the moon s mean motion was first remarked by Dr Halley, and was fully confirmed by Dunthorne, who was led, by the discussion of a great number of ancient observations of eclipses, to suppose that it
proceeded uniformly at the rate of 10* in 100 years. This