388 ASTROMETRY 



development of celestial mechanics during the eighteenth century 

 and furnishes at the same time the starting-point for the researches 

 of the nineteenth. We recapitulate, therefore, some of the principal 

 points of the same in order that the discussion may be more easily 

 understood. The coordinates and the elements of the planets and 

 satellites were expressed in series containing: (1) periodic terms 

 (sines and cosines of multiples of the mean and true longitudes); 

 (2) non-periodic terms involving powers of the time, i. e., the so- 

 called secular terms; (3) semi-secular terms, that is, products of the 

 time or the angle into sine and cosine functions; the development 

 being made in powers of the eccentricities and inclinations con- 

 sidered as small quantities. The appearance of the time or the angle 

 explicitly, outside the sine and cosine functions, was considered, 

 at least in part, both by Laplace and Lagrange as the result of 

 incomplete operations. But on the other hand, from the standpoint 

 of astronomy, it was considered entirely useless to complicate the 

 expressions by introducing trigonometrical series in place of angles. 

 The constants of integration, i. e., the elements, were determined 

 numerically for each of the planets then known, and the numerical 

 values of the coefficients of the individual terms of the series were 

 derived therefrom. It was then sufficient in most cases to consider 

 only the lowest powers of the eccentricities in order to obtain the 

 places of the planets with an accuracy corresponding with that of 

 the observations. As a result astronomers were enabled to explain 

 all the observed inequalities; for example, the great inequality in 

 the motions of Jupiter and Saturn, the inequality in the motion of 

 Jupiter's satellites discovered by Wargentin, etc. With the aid 

 of his epoch-making theory of the variation of constants, Lagrange 

 proved the famous theory that the expression for the major axis 

 contains only periodic terms, when powers of the mass higher than 

 the first are neglected. Both Lagrange and Laplace had found that 

 in the first approximation the eccentricities and inclinations may 

 be considered as long-period functions of the time. Through the 

 researches of Laplace the theory of the moon, the motion of the 

 earth about its centre of gravity, and the theory of the figures of 

 the planets, were so developed as completely to satisfy the corre- 

 sponding observations. 



These brief statements of some of the principal points may be 

 sufficient. 



The nineteenth century was introduced by two researches of 

 Poisson of remarkable value to celestial mechanics. One was the 

 extension of Lagrange's theory in regard to the major axis to the 

 second power of the mass; the other and far more important was 

 his classic theory of the motion of the earth about its centre of 

 gravity, which was built up by the aid of the method of the varia- 



