The following are all the periodic terms in the moon's longi tude as given in Brown's Tables with a coefficient over The parallactic term is added because of the attention it has attracted. The largest of the planetary terms, due to Venus, has a coefficient of 14"•27 and a period of 273 years. The next largest planetary terms are two or three of the order of i". The planetary terms usually have periods of about a month or a few years. The notation is as follows : 1, l' are the mean anomalies of the moon and sun, D the excess of the moon's mean longitude over the sun's, F the argument of latitude.
+22639 1500 sin/ Largest terms in equation of centre.
± 769.o16 sin2/ — 4586 .426 sin(/— 2D) Evection.
— I 25 . 154 sinD Parallactic inequality.
— 668 . II I sin/' Annual equation.
— 411.6o8 sin2F Reduction to ecliptic.
In compiling the constants which could be derived directly from observation or by theory from other astronomical constants, the latter method generally has been followed. The term 7".14 in the mean longitude was derived from pure theory, but it was known to be too small to fit ancient eclipse observations. The existence of a secular term in the moon's motion had been dis covered by Halley. No explanation could then be given as to its origin. Lunar theory then took no account of planetary pertur bations and gave no inequality in the moon's motion with a period exceeding 18.6 years. The first explanation of the secular term was given by Laplace who, from the secular change in the earth's orbit, computed a secular term with a coefficient of 1o" in the moon's motion. This agreed with the results derived from ancient eclipses. Laplace's immediate successors, amongst whom were Hansen, Plana and Pontecoulant, found a larger value, Hansen obtaining the value 12".5. This value was found by himself and Airy to represent fairly well several ancient eclipses. But in 1853 Adams showed that these calculations were only a rough first approximation and that the rigorous theoretical value was only half as large. For some time there was considerable controversy on the subject but finally Adams's result was fully confirmed by Delaunay. The question then arose again as to the reason for the difference between observation and theory. It was pointed out independently by W. Ferrel in 1856 and a few years later by Delaunay that the attraction of the moon on the tidal wave must produce a frictional force tending to slow down the earth and apparently to accelerate the moon.
It will be seen that Brown has introduced one empirical term with a period of about 26o years into his tables in order to make them fit the observations from 1750 onwards more accurately than they otherwise would. When this is done there still remain residuals amounting to These cannot be satisfied by one additional periodic term, hut they can be satisfied very closely by two periodic terms in a variety of ways. According to the Monthly Notices of the Royal Astronomical Society (vol. lxxxiii., p. 359) Sir Frank Dyson and Dr. A. C. D. Crommelin analysed the residuals and found terms with coefficients of 3".09 and i".66 and periods of about 7o years and 59 years respectively. It is not supposed that these periodic terms will succeed in predicting accurately the moon's motion in the future.
It thus appears that the gravitational theory does not exactly explain the moon's motion. Searches for inequalities of long period produced by the action of planets have been carried out very thoroughly. It is now believed that the cause of the differ
ence between prediction and observation lies in a variation of the earth's rate of rotation. If the earth rotates more slowly, the moon appears to go faster. Careful comparison of the ob served and predicted positions of the other bodies in the solar system seem to indicate similar irregularities. The cause of a variable rotation of the earth has not yet been cleared up. If due to the action of the moon it will affect the mean motion of the moon so that other bodies in the solar system will not show the same proportional change as the moon; if due to a change in the earth's moment of inertia all bodies will show the same pro portional change.
The following are amongst the works relating to the motion of the moon which are of historic importance or present interest: Clairaut, Theorie de la lune (2nd ed., 1765) ; L. Euler, rheoria motuum lunae nova methodo pertractata (Petropolis, 1772) ; G. Plana, Theorie du mouvement de la lune (3 vols., Turin, 1832) ; P. A. Hansen, Funda menta nova investigationis orbitae verae quam luna perlustrat (Gotha, 1838) ; Darlegung der theoretischen Berechnung der in den Mondta feln angewandten Storungen (Leipzig, 1862) ; Tables de la lune (Lon don, 1857) (Admiralty publication) ; C. E. Delaunay, Theorie du mouvement de la lune ; F. F. Tisserand, Traite de mecan ique celeste, tome iii., Exposé de l'ensemble des theories relatives an mouvement de la lune (1894) ; S. Newcomb, "Researches on the Motion of the Moon" (Appendix to Washington Observations for 1875, discussion of the moon's mean motion) ; "Transformation of Hansen's Lunar Theory," Ast. Papers of the Amer. Ephemeris, vol. i.; "Action of the Planets on the Moon," Ast. Papers of the Amer. Ephemeris, vol. v. pt. 3 ; R. Radau, "Inegalites planetaires du mouve ment de la lune" (Annales, Paris observatory, vol. xxi.) ; Tables de la lune (Annales du Bureau des Longitudes, vol. vii.) ; E. W. Brown, "Theory of the Motion of the Moon," Memoirs of the Royal Astr. Soc. (1901-1908) ; "Constants for the Tables," Monthly Notices (1913 1915) ; Tables of the Moon, Yale University Press (1919). See also Brown's Introductory Treatise on the Lunar Theory (Cambridge Uni versity Press, 1896) and Inequalities in the Moon's Motion produced by the Action of Planets (Cambridge, 1907). (J.