THE ORBIT OF NEPTUNE. 71 



Substituting these values of the corrections in equations (7), we have the fol- 

 lowing residuals, which are grouped, as before, according to the time of year of the 

 normals on which the equations were founded. Thus, the first residual of each 

 series of modern observations corresponds to positions of Neptune observed when 

 the planet culminated after 13'' 30"' during the years to which the series belongs. 



The second, to observations between 10 30 and 13 30 

 The third, to observations between 7 30 and 10 30 

 • The fourth, before 7 30 



We first give the residuals from the equations (7), each of which is supposed, 

 to be of equal precision ; then the numl^ers by which the errors of observation 

 are multiplied to reduce them to the assumed standard of precision derived from 

 (6), column M. ; and, finally, the apparent errors of the theory derived from ob- 

 servations themselves, formed by dividing the residuals of the equations by the 

 measures of precision. 





Residuals of equations. 





Actual mean residuals or ap- 

 parent errors of theory. 



1st series, 

 1795, 



1 + o"58 



— l.Sfi 



i 



+ 2.3 



-7'. 2ft 





/ — 0.7 



— O.Qfi 



2 



— 0.35 



— 0.30ft 



2d series, 



— 0.2 



— 0.8^ 



5 



— 0.04 



— 0.16ft 



1846-1847, 



+ 2.4 



+ 1.5^. 



5 



+ 0.48 



+ 0.30ft 





(, — 1.1 



+ 1.4^ 



4 



— 0.28 



+ 0.35ft 





/ — 2.3 



— 0.7^ 



2 



— 1.15 



— 0.35ft 



3d series, 



+ 0.5 



— 1.5^ 



7 



+ 0.07 



— 0.21(1 



1848-1850, 



+ 1.0 



+ 0.6^ 



8 



+ 0.12 



+ 0.08ft 





\ — 0.7 



+ 0.8^ 



5 



— 0.14 



+ 0.16ft 





/+ 0.8 



— 0.4^ 



1 



+ 0.80 



— 0.40ft 



4 th series. 



— 0.6 



— 0.5ff 



6 



— 0.10 



— 0.08ft 



1851-1853, 



— 1.6 





5 



— 0.32 







\ — 2.5 



+ 0.6^ 



4 



— 0.62 



+ 0.15ft 





/ — 1.0 



— 2.7fi 



8 



— 0.12 



— 0.34ft 



5th series. 



— 0.2 



— 1.2^ 



7 



— 0.03 



— 0.17ft 



1854-1856, 



— 1.4 



+ 0.4^ 



7 



— 0.20 



+ 0.06ft 





1+0.4 



+ 1.0^ 



7 



+ 0.06 



+ 0.14ft 





/+2.8 



-1.7ft 



8 



+ 0.35 



— 0.21ft 



6 th series, 



+ 3.7 



— 0.4ft 



9 



+ 0.41 



— 0.05ft 



1857-1859, 



+ 1.8 



+ 0.9f. 



8 



+ 0.22 



+ 0.11ft 





\ — 3.4 



+ 2.2f^ 



9 



— 0.38 



+ 0.24ft 





/+ 2.8 



— 1.4ft 



7 



+ 0.40 



— 0.20ft 



7th series. 



— 0.6 



— 0.5ft 



9 



— 0.07 



— 0.06ft 



1860-1862, 



— 2.5 



+ 1.6ft 



9 



— 0.28 



+ 0.18ft 





1+2.2 



+ 1.7// 



7 



+ 0.31 



+ 0.24ft 



