was the first attempt to estimate the value of the secular equation, which had hitherto been confounded with the mean secular motion. Mayer was led to the value 7", which he advanced to 9" in his last tables published in 1770. Lalande found it to amount to 9" 8S6. In 1786 Laplace demonstrated that the acceleration is one of the effects of the attraction of the sun, and connected with the variations of the eccentricity of the earth s orbit, in such a manner that the moon will continue to be accelerated while the eccentricity diminishes, but cease to be accelerated when the eccentricity has reached its maximum value ; and when that element begins to increase, the mean motion of the moon will be retarded. Professor Adams, however, has recently shown that though a portion of the acceleration is undoubtedly due to the cause assigned by Laplace, in reality one-half or thereabouts remains unexplained by that cause. The researches of Delaun ay confirm this view. The theory is entertained that the unexplained part of the acceleration is only apparent, the real cause in operation
being a retardation of the earth s motion of rotation.The same cause which gives rise to the acceleration of the mean motion, namely, the diminution of the eccentricity of the earth s orbit, also occasions secular inequalities in the motion of the perigee and nodes of the orbit of the moon. These two inequalities are, however, affected with opposite signs to that of the former, that is, while the mean motion 1 of the moon is accelerated, the motion of her perigee and that of her nodes are retarded. By push ing the approximations to a great length MM. Plana and Carlini, and M. Damoiseau, in Memoirs which obtained the prize of the Academy of Sciences for 1820, found different numbers; those of Damoiseau are 1, 4 702, and 0-612.
The three secular inequalities which have been pointed out will obviously occasion others, for all quantities de pending on the mean motion, the motion of the perigee, or of the nodes, mnst be in some degree modified by them. They can only be developed by the complete integration of the differential equations of motion. What is most essential is to select, among the multitude of terms, such as may possibly acquire considerable co-efficients by integration.
Understanding by the term month the time which the moon employs to make an entire revolution relatively to any given point, movable or fixed, we have as many different species of months as there are different motions with which that of the moon can be compared. For example, if we estimate her revolution relatively to the sun, the month will be the time which elapses between two consecutive conjunctions or oppositions. This is called the synodic month, lunar month, or lunation. If we con sider her revolution as completed when she has gone through 360 of longitude counted from the movable equinox, we shall have the tropical or periodic month. The interval between two successive conjunctions with the same fixed star is the sidereal month. A revolution with regard to the apsides of her orbit, that is to say, the time in which she returns to her perigee or apogee, gives the anomalistic month. And, finally, the revolution with regard to the nodes is the nodical month.
It is clear that, taking the sidereal lunar month as a standard of reference, any other month will be greater or less according as the point which defines it moves in the same direction around the star-sphere as the moon or in the reverse direction, and that the excess or defect will be greater or less as such motion is more or less rapid. Thus, as the sun advances with considerable rapidity, the synodical month will much exceed the sidereal month; as the point V retrogrades very slowly, the tropical month is very slightly less than the sidereal month ; as the apsides advance, on the whole completing a revolution in 8 85 years, the anomalistic month will exceed the sidereal, but only by about v-
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Synodical month 29-53059 days. Sidereal ,, 27 32166 ,, Tropical ,, 27 32156 Anomalistic,, 27 55460 ,, Nodical .. 27 21222 . Differs from Sidereal Month. + 2-20893 days. o-ooooo -0-00010 + 0-23294 ,, -0-10944 ,
The moon at all times presents very nearly the same face to the earth. If this were rigorously the case, it would follow that the moon revolves about an axis per pendicular to the plane of her orbit in the same time in which she completes a sidereal revolution about the earth, and that the angular velocities of the two motions are exactly equal. It is, however, proved by observation, that Librat , there are some variations in the apparent position of the spots on the lunar disk. Those which are situated very near the border of the disk alternately disappear and become visible, making stated periodical oscillations, which indicate a sort of vibratory motion of the lunar globe (apparent only), which is known by the appellation of the Libration.
The rotation of the moon is sensibly uniform, while the motion of revolution is variable. The apparent rotation occasioned by the revolution of the moon round the earth is, consequently, not exactly counterbalanced by the real rotation, which remains constantly the same. Hence the different points of the lunar globe must appear to turn about her centre, sometimes in one direction, and sometimes in the contrary, and the same appearances be produced as would result from a small oscillation of the moon, in the plane of her orbit, about the radius vector drawn from her centre to the earth. The spots near the eastern or western edge of her disk disappear according as her motion in her orbit is more or less rapid than her mean motion. This is called the Libration in Longitude. Its maximum value corresponds to a rotation through 7 45.
Further, the axis of rotation of the moon is not exactly perpendicular to the plane of her orbit ; hence the two poles of rotation, and those parts of her surface which are near these poles, are alternately visible from the earth. This is the Libration in Latitude. Its maximum value amounts to 6 44 .
Again, the observer is not placed at the centre of the earth, but at its surface. Thus in the course of a day the moon appears to oscillate about her radius vector because of the earth s rotation. This phenomenon constitutes what is called the Diurnal Libration, and is evidently the effect of the lunar parallax, and corresponds to it in amount, measured in minutes of arc. It therefore never exceeds 1 28"-8.
The libration in latitude and the diurnal libration were discovered by Galileo soon after the invention of the tele scope. It was Hevelius who discovered and first explained the libration in longitude. Regarding libration in gene ral, it remains to be stated that, instead of one-half of the moon remaining invisible, about 4111 parts out of 10,000 are absolutely and at all times unseen. If diurnal .ibration be neglected, about 4198 parts out of 10,000 may oe regarded as altogether unseen.
