MOON 801 aa! cos A sin A. But we have cos A sin A = % sin (A + A) + i sin (A 1 - A). Hence the product -X" Y will be of the form X Y= ^ aa sin (A 1 + A ) + $ oa sin (A 1 - A) + ab sin (A + + ),&c., which is another series of the same general form. Moreover, if we suppose the angles A, B, &c., to increase uniformly with the time that is, to admit of expression in the form A = a + mt, A = a + m t, &c we shall have, by integrating, 2 XYdt=- m + m cos (A - A], &c., which, again, is a trigonometric series of the same general form, which admits of being manipulated at pleasure in the same way as the original expressions X and Y. This property does not belong to the elliptic functions, and in consequence, notwith standing the great length of the trigonometric series, no attempt to supersede them has been successful. The efforts to express the moon s motion by integrating the differential equations of the dynamical theory may be divided into three classes. (1) Laplace and his immediate successors found the problem so complex that they sought to simplify it by reversing its form ; instead of trying from the beginning to express the moon s coordinates in terms of the time, they effected the integration by expressing the time in terms of the moon s true longitude. Then, by a reversal of the series, the longitude was expressed in terms of the time. Although it would be hazardous to say that this method is unworthy of further consideration, we must admit that its essential inelegance is such as to repel rather than attract study, and that it holds out no promise of further development. (2) By the second general method the moon s coordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining the algebraic symbols which express the values of the various elements. Most of the elements are small numerical fractions : e, the eccentricity of the moon s orbit, about 055 ; e the eccentricity of the earth s orbit, about O Ol 7 ; y, the sine of half the inclination of the moon s orbit, about 046 ; m, the ratio of the mean motions of the moon and earth, about - 075 ; and the expressions for the longitude, latitude, and parallax appear as an infinite trigonometric series, in which the coefficients of the sines and cosines are themselves infinite series proceeding accord ing to the powers of the above small numbers. This method was applied with success by Pontecoulant and Sir John W. Lubbock, and afterwards by Delaunay. It should be remarked that the solution by the first method appears in the same form as by this one after the true longitude is expressed in terms of the mean longitude. (3) By the method just mentioned the series converge so slowly, and the final expressions for the moon s longitude are so long and complicated, that the series has never been carried far enough to insure the accuracy of all the terms. This is especially the case with the development in powers of m, the convergence of which has often been questioned. Hence, when numerical precision alone is aimed at, it has been found best to avoid this difficulty by using the numerical values of the elements instead of their algebraic symbols. This method has the advantage of leading to the more rapid and certain determination of the numerical values of the several coefficients of sines and cosines. 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 has been employed by Damoiseau, Hansen, and Airy. The methods of the second general class are those most worthy of study. And among these we must assign the first rank to the method of Delaunay, developed in his Theorie du Mouvement de la Lune, because it contains a germ which may yet develop into the great desideratum of a general method in celestial mechanics. To explain it, we must call to mind the general method of " variation of elements," due to Lagrangc. This method is applicable to cases in which a problem of dynamics can be completely solved when any small forces which come into play are left out, but which does not admit of direct solution when these forces are included. Omitting the small forces, commonly called " disturbing forces," let us suppose the problem of the motion of a body under the influence of the "principal forces" completely solved. This will mean that we have found algebraic expressions for the coordi nates which determine the position of the body in terms of the time, and (in the case of a material point) of six constant quantities, to which we may assign values at pleasure. Then Lagrange showed how, by supposing these constant quantities to become variable, the same expressions could be used for the case in which the effect of the disturbing forces was included. In other words, the effect of the disturbing forces could be determined by assuming them to change the constants of the first approxi mate solution into very slowly varying elements. In the researches on the lunar theory before Delaunay the principal force was taken to be the attraction of the earth upon the moon, and the disturbing force Avas that due to the sun s attraction. When the action of the earth alone was included the moon would move in an ellipse, in accord ance with Kepler s laws. The effect of the sun s action could be allowed for by supposing this ellipse to be mov able and variable. But when it was required to express this variation the problem became excessively complicated, owing to the great number of terms required to express the sun s disturbing force. Now, instead of passing from the elliptic to the disturbed motion by one single difficult step, Delaunay effected the passage by a great number of easy steps. Out of several hundred periodic terms, the sum of which expressed the disturbing force of the sun, he first took one only, and determined the variations of the Keplerian ellipse on the supposition that this term was the only one. In the solution the variable elements of the ellipse would be expressed in terms of six new con stants. He then showed how these new constants could be taken as variables instead of the elements of the original ellipse. Taking a second term of the disturbing force, he expressed the new constants in terms of a third set of con stants, and so repeated the process until all the terms of the disturbing force were disposed of. Among applications of the third or numerical method, the most successful yet completed is that of Hansen. His first work appeared in 1838, under the title Funda- menta nova investigationis orbitx verse quam luna perlustrat, and contained an exposition of his ingenious and peculiar methods of computation. During the twenty years follow ing he devoted a large part of his energies to the numerical computation of the lunar inequalities, the re-determination of the elements of motion, and the preparation of new tables for computing the moon s position. In the latter branch of the work, he received material aid from the British Government which published his tables on their completion in 1857. The computations of Hansen were published some seven years later by the Saxon Royal Society of Sciences. It is found on comparing the results of Hansen and Delaunay that there are some outstanding discrepancies, which, though too small to be of great practical importance, are of sufficient magnitude to demand the attention of those interested in the mathematical theory of the subject. It is therefore desirable that the numerical inequalities should be again determined by an entirely different method. This is the object of Sir G. B. Airy s Numerical Lunar Theory, which is not yet completely published, but is sufficiently far advanced to give hopes of an early comple tion. The essence of Sir George s method consists in
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