(3) The third method seeks to avoid the difficulty by using the numerical values of the elements instead of their algebraic symbols. This method has the advantage of leading to a more rapid and certain determination of the numerical quantities re quired. It has the disadvantage of giving the solution of the problem only for a particular case, and of being inapplicable in researches in which the general equations of dynamics have to be applied. It leads, however, to the most accurate results for the motion of the moon.
Amongst the applications of the third method an outstanding place is occupied by the researches of P. A. Hansen. His first work, Fundamenta nova appeared in 5838 and contained an expo sition of his methods. During the twenty years following he de voted a large part of his energies to the numerical computation of the lunar inequalities, the re-determination of the constants of the motion and the preparation of new tables for computing the moon's position. The importance of these tables was im pressed upon the British Government by Airy and they were pub lished by the Admiralty in 1857. The theory was published in "Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Storungen." A peculiarity of Hansen's method is that the angular perturbations in the plane of the orbit are added to the mean anomaly in an auxiliary ellipse. The tables were brought into use for computing the positions of the moon for the Nautical Almanac in 1862 when they replaced Burckhardt's Tables (Paris, 1812). They remained in use till 1922, but Newcomb's corrections were introduced in 1883. These corrections were partly theoretical and partly observational. Amongst the general theories of the second class the most noteworthy is that due to C. E. Delaunay.
A completely new theory and set of tables have been con structed by Professor E. W. Brown of Yale during the present century. The method employed may be briefly indicated. The four small quantities, e, é, y and m, have been mentioned. The moon's co-ordinates have to be expressed in terms of the powers and products of these. Euler conceived the idea of starting with a preliminary solution of the problem in which the orbit of the moon was supposed to lie in the ecliptic and to have no eccentricity; while that of the sun was circular. This solution being reached, the additional terms were found which depended on the first power of the eccentricities and of the inclination.
Then the terms of the second order were found, and so on to any extent. In a series of remarkable papers published in 1877-1888 G. W. Hill improved Euler's method, and worked it out with much more rigour and detail than Euler had been able to do. His most important contribution to the subject consisted in work ing out by extremely elegant mathematical processes the method of determining the motion of the perigee. J. C. Adams determined the motion of the node in a similar way. The numerical compu tations were worked out by Hill only to the first approximation. The work of constructing a complete lunar theory by this method was undertaken by Brown who published his results in the Memoirs of the Royal Astronomical Society during the years 1901 to 1908. Thereafter he determined the numerical constants by comparison with the Greenwich meridian observations for the years 1750 to 1910 as worked up by G. B. Airy and P. H. Cowell. The values finally adopted were published in a series of papers published in the Monthly Notices of the Royal Astronomical Society during the years 1913 to 1915. Professor Brown, assisted by H. B. Hedrick, then proceeded with the calculation of the tables which were printed by the Cambridge University Press and pub lished by Yale University Press in three volumes in 1919. By the use of various devices the tabulation was simplified so that although the number of terms included is about 1,500 or about five times that of Hansen's Tables the task of calculating an annual ephemeris is no greater. The effect of every known sensible term is included and many terms are also included which must be classed as insensible in comparison with the best modern obser vations. The principal constants determined by Brown (Monthly Notices of the Royal Astronomical Society, vol. lxxv., p. sto) are given below. The epoch is 1899, Dec. 31, Greenwich mean noon, from which T is computed in Julian centuries of 36,525 days; r denotes a revolution or 36o°.
Eccentricity .054900489, corresponds to the coefficient 22639".550 in the principal elliptic term; and sine of half the inclination .044886967, corresponds to the coefficient 18461"•400 in the principal term in latitude.