Astronomy 585 



ties, and in most researches it is sufficient to suppose that the 

 latter sort vary uniformly with the time. 



The demonstration that the secular inequalities of the 

 planets are not indefinitely progressive, but are, in fact, 

 themselves periodic, is due to Laplace, who showed that the 

 elements of the planets will perpetually oscillate about their 

 mean values, provided only that all the planets revolve round 

 the sun in the same direction — as they do. 



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. 



The calculations of Lagrange (on the six planets known to 

 him) were based on erroneous values of their masses. Ponte- 

 coulant's investigations (1834) did not improve our knowledge 

 in this respect. Leverrier's researches (1839) on the seven 

 planets then known are far from being exhaustive. The ob- 

 ject of the present paper is to make as complete a determina- 

 tion of the variations of the elements of the eight planets of 

 the system as is possible. 



By the introduction of new methods of research the author 

 has solved the problem in hand, and has obtained formulae 

 which will determine the secular variations of the planetary 

 elements with less labor than would be necessary for the 

 accurate determination of a comet's orbit. 



Some of the conclusions derived may be briefly alluded to. 

 The object of the investigation is to determine the numerical 

 values of the secular orbits — namely, of the eccentricities and 

 inclinations, and the longitudes of the nodes and perihelia, 

 their values at any epoch, their rates of change, etc. A table 

 at the Tend of the volume covering thirteen quarto pages gives 

 these data, so far as they may be required by the astronomer 

 in the prosecution of his work. 



A similar tabulation of the elements of the earth's orbit of 

 .;8 



