THE ORBIT OF URANUS. 



167 



The following aro the approximate normals to which these equations give rise. 

 "Inac ( -unicics being detected in several of the equations of condition after these 

 normals were formed, they do not accurately correspond to those equations as written. 



'Hie values of the unknown quantities deduced from these normals worn substi- 

 tuted in the equations of condition, and a farther approximation was made by 

 solving the equations given by the residuals. The following arc the first approxi- 

 mations given by the normals, and the finally concluded corrections 



Preliminary. Final. 



lp\ 15.00 13.44 

 0.52 0.36 

 4.25 - 4.04 



= K- 4.73 4.33 



3 f ', _ 3.78 3.44 



Mass of Neptune ^^ TsltTf 



The final values of the corrections being substituted in the equations leave the 

 following system of residuals, or outstanding excesses of the observed longitudes 

 over theory. Column /M gives the residual of the equation itself; the probable 

 error of which has always been judged to be 0."5, while in column }1 this residual 

 is divided by / to obtain the residual correction of the longitude itself. The values 

 of the factors /are. found with the original equations on pages 159 and 160- 



