ON THE ORBIT OF URANUS. 



INTRODUCTION. 



The connection of the planet Uranus with the most brilliant astronomical 

 achievement of the century lends a peculiar interest to its theory. The researches 

 of Adams and Le Verrier showed that the observed motions of that planet were 

 represented, at least approximately, by the action of a theoretical planet having 

 the longitude of Neptune, Peirce showed that the action of Neptune itself 

 accounted for these motions within the limits of possible error of the observations 

 used by Le Verrier. It remains to be seen whether the agreement between theory 

 and observation still subsists when the comparatively few observations used by those 

 investigators are reduced with tlic more refined data now at our disposal, and when 

 the great mass of additional observations made both before and since the date of 

 Le Verrier's researches are included. 



The circumstances connected with the discovery of Neptune have been so 

 exhaustively recounted by a number of authors that it would be difficult to add 

 anything not already familiar to astronomers witliout transcending our present 

 limits. I shall therefore confine myself to such an account of previous researches 

 on the theory of Uranus as may give an idea of their nature and extent, and facili- 

 tate their comparison with the methods and results of the present investigation. 



The perturbations used by Bouvard in his tables are those of the Mecanique 

 Celeste. Although not affected with any striking error, the numerical methods 

 adopted in their computation are necessarily too rough to allow of much interest 

 attaching to their comparison with the results of the more recent researches. 



It is essential to a clear understanding of subsequent researches that we classify 

 the methods which have been or may be adopted in the computation of the 

 general perturbations of the planets. This computation comprises two distinct 

 operations: (1) the development of the disturbing forces, or some quantities of 

 which these forces are functions ; (2) the integration of the equations of motion 

 under the influence of these forces. In each of these operations three methods 

 have been employed. 



In developing the perturbative function, we have first the purely analytic method 

 used by the great geometers of the last century. In this method tliis function is 

 developed in powers of the eccentricities and mutual inclination of the orbits of 

 the two planets, and the numerical coefficients are found by substituting the values 

 of the elements in these expressions. It is only applicable when the eccentricities 



1 March, 1873. ( 1 ) 



