Elementary Treatise On The Lunar Theory

AN ELEMENTARY TREATISE ON THE LUNAR THEORY, A BRIEF SKETCH OF THE HISTORY OF THE PROBLEM BEFORE NEWTON. BY HUGH GODFRAY, MA. OF ST. JOKNS OOLLKUII, CAMBRIDGE, LATE MATHEMATICAL LECTURER AT PEMBROKE COLLEGE. FOURTH EDITION. PREFACE TO THE FIRST EDITION. OF all the celestial bodies whose motions have formed the subject of the investigations of astronomers, the Moon has always been regarded as that which presents the greatest difficulties, on account of the number of inequalities to which it is subject but the frequent and important applications of the results render the Lunar Problem one of the highest interest, and we find that it has occupied the attention of the most celebrated astronomers from the earliest times. Newtons discovery of Universal Gravitation, suggested, it is supposed, by a rough consideration of the motions of the Moon, led him naturally to examine its application to a more severe explanation of her disturbances and his Eleventh Section is the first attempt at a-theoretical investigation of the Lunar inequalities. The results he obtained were found to agree very nearly with those determined by observation, and afforded a remarkable confirmation of the truth of his great principle but the geometrical methods which he had adopted seem inadequate to so complicated a theory, and recourse has been had to analysis for a complete determination of the disturbances, and for a knowledge of the true orbit. VI PEFACE. The following pages will, it is hoped, form a proper introduction to more recondite works on the subject the difficulties which a person entering upon this study is most likely to stumble at have been dwelt considerable length, and though different methods ofinvestigation have been employed by different astronomers, the difficulties met with are nearly the same, and the principle of successive approximation is common to all. In the present work, the approximation is carried to the second order of small quantities, and this, though far from giving accurate values, is amply sufficient for the elucidation of the method. The differences in the analytical solutions arise from the various ways in which the position of the moon may be indicated by altering the system of coordinates to which it is referred, or again, in the same system, by choosing different quantities for independent variables. DAlembert and Clairaut chose for coordinates the projection of the radius vector on the plane of the ecliptic and the longitude of this projection. To form the differential equations, the true longitude was taken for independent variable. To determine the latitude, they, by analogy to Newtons method, employed the differential variations of the motion of the node aad of the Inclination of the orbit. Laplace, Damoiseau, Plana, and also Herschel and Airy, in their more elementary works, have found it more con venient to express the variations of the latitude directly, by an equation of the same form as that of the radius vector. Lubbock and Pontocoulant, taking the same coordinates of the Moons position, make the time the independent LUNAR THEORY. CHAPTER I. INTRODUCTION. BEFORE proceeding to the consideration of the moons motion, it will be desirable to say a few words on the law of attractions, and on the peculiar circumstances which enable us to simplify the present investigation. 1...