264 RECENT EESEARCIIES ON THE 



masses of the different planets. This investigation of Le Verrier's has 

 been used as tlie groundwork of most of the subsequent 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 Neptune had not then been discovered ; 

 and the action of this planet considerably modifies the secular inequali- 

 ties which would otherwise take place. We will now briefly glance at 

 the results of our own Inbors on the subject. 



On the first examination of the works of those authors who had investi- 

 gated this problem, we perceived that the methods of reducing to num- 

 bers those analytical integrals which determine the secular variations 

 of the elements, were far from possessing that elegance and symmetry 

 of form which usually characterizes the formulas of astrouomj'. The 

 first step, therefore, was to devise a system of algebraic equations, by 

 means of which we should be enabled to obtain the values of the unknown 

 (juantities with the smallest amount of labor. It was soon found to be 

 impracticable to deduce algebraic formulas for the constants, by the 

 elimination of eight unknown quantities from as many linear symmet- 

 rical equations, of sufiicient simplicity to be used in the deduction of 

 exact results. It therefore became necessary to abandon the idea of a 

 direct solution of the equations, and to seek for the best approximative 

 method of obtaining rigorous values of the unknown quantities. This 

 we have accomplished as completely as could be desired, and by means 

 of the formulas which we have obtained, it is now possible to determine 

 the secular variations of the planetary elements, with less labor, perhaps, 

 than would be necessary for the accurate determination of a comet's 

 orbit. The method and formulas are given in detail in a Memoir on the 

 Secular Variations of the Elements of the Orbits of the Eight Princijml 

 Flancts, now being published in vol. XVIII, of the iSmithsoniau Contri- 

 butions to Knoiclcdge. 



After computing anew the numerical coefficients of the differential 

 equations of the elements, we have substituted them in these equations, 

 and have obtained, by means of successive approximations, the rigorous 

 values of the constants corresponding to the assumed masses and ele- 

 ments. The details of the computation are given in the memoir referred 

 to, and it is unnecessary to speak of them here. We shall, therefore, 

 only briefly allude to some of the conclusions to which our computa- 

 tions legitimatelj- lead. 



The object of our investigation has been the determination of the 

 numerical values of the secular changes of the elements of the planet- 

 ary orbits. These elements are four in number, viz : the eccentricities 

 and inclinations of the orbits, and the longitudes of the nodes and 

 perihelia. The questions that may legitimately arise in regard to the 



