x INTRODUCTION. 



far from possessing that elegance and symmetry of form Avhich usually charac- 

 terize the formulas of astronomy. 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 quantities 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 symmetrical 

 equations, of sufficient 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 approximate 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 methods and formulas are given in detail in the following Memoir. 



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 corre- 

 sponding to the assumed masses and elements. 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 

 computations legitimately lead. 



The object of our investigation has been the determination of the numerical 

 values of the secular changes of .the elements of the planetary 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 eccentricities and inclinations relate chiefly to their magni- 

 tude at any time ; but we may also desire to know their rates of change at any 

 time, and the limits within which they will perpetually oscillate. In regard to the 

 nodes and perihelia, it is sometimes necessary to know their relative positions 

 when referred to any plane and origin of coordinates ; and also their mean motions, 

 together with the amount by which their actual places can differ from their mean 

 places. With respect to the magnitudes and positions of the elements, together 

 with their rates of change, we may observe that our equations will give them for 

 any required epoch, by merely substituting in the formulas the interval of time 

 between the epoch required and that of the formulas, which is the beginning of 

 the year 1850. A tabulation of the various planetary elements, of sufficient extent 

 to supply the needs of the astronomer, is given at the close of the work. A 

 similar tabulation of the elements of the earth's orbit of sufficient extent to be 

 useful in geological investigations, does not come within the scope of our work ; and 

 we leave the computation of the elements for special epochs to those investigators 

 who may need them in their researches. We shall here give the limits between 

 which the eccentricities and inclinations will always oscillate, together with the 

 mean motions of the perihelia and nodes on the fixed ecliptic of 1850; and shall 

 also give the inclinations and nodes referred to the invariable plane of the plane- 

 tary system. 



