INTRODUCTION. i x 



by all the members of the solar system, it follows that the orbits of the planets 

 will never be very eccentrical or much inclined to each other by reason of their 

 mutual attraction. The important truths in relation to the forms and positions of 

 the planetary orbits are embodied in the two following theorems by the author of 

 the Mecanique Celeste: I. If the mass of each planet he multiplied hy the product of 

 the square of the eccentricity and square root of the mean distance, the sum of all 

 these products will ahoays retain the same magnitude. II. If the mass of each 

 planet he multiplied by the product of the square of the inclination of the orbit and 

 the square root of the mean distance, the sum of these products will always remain 

 invariable. Now, these quantities being computed for a given epoch, if their sum 

 is found to be small, it follows from the preceding theorems that they will always 

 remain so ; consequently the eccentricities and inclinations cannot increase inde- 

 finitely, but will always be confined within narrow limits. 



In order to calculate the limits of the variations of the elements with precision, 

 it is necessary to know the correct values of the masses of all the planets. Unfor- 

 tunately, this knowledge has not yet been attained. The masses of several of the 

 planets are found to be considerably different from the values employed by La 

 Grange in his investigations. Besides, he only took into account the action of the 

 six principal planets which are within the orbit of Uranus. Consequently, his solu- 

 tion afforded only a first approximation to the limits of the secular variations of 

 the elements. 



The person who next undertook the computation of the secular inequalities was 

 Pontecoulant, who, about the year 1834, published the third volume of his Theorie 

 Analytique du Systeme du Monde. In this work he has given the results of his 

 solution of this intricate problem. But the numerical values of the constants 

 which he obtained are totally erroneous on account of his failure to employ a suf- 

 ficient number of decimals in his computation. Our knowledge of the secular 

 variations of the planetary orbits was, therefore, not increased by his researches. 



In 1839 Le Verrier had completed his computation of the secular inequalities 

 of the seven principal planets. This mathematician has given a new and accurate 

 determination of the constants on which the amount of the secular inequalities 

 depends ; and has also given the coefficients for correcting the values of the con- 

 stants for differential variations of the masses of the different planets. This 

 investigation of Le Verrier's has been used as the groundwork of most of the sub- 

 sequent corrections of the planetary elements and masses, and deservedly holds 

 the first rank as authority concerning the secular variations of the planetary orbits. 

 But Le Verrier's researches were far from being exhaustive, and he failed to notice 

 some curious and interesting relations of a permanent character in the secular 

 variations of the orbits of Jupiter, Saturn, and Uranus. Besides, the planet Nep- 

 tune had not then been discovered ; and the action of this planet considerably 

 modifies the secular inequalities which would otherwise take place. We will now 

 briefly glance at the results of our own labors on the subject. 



On the first examination of the works of those authors who had investigated 

 this problem, we perceived that the methods of reducing to numbers those 

 analytical integrals which determine the secular variations of the elements were 



B June, 1872. 



