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. 



h. m. h. m. 



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 numbers by which the errors of observation 

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

 (G), 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. Acfual cfln resi<1 ^ or ap- 



parent errors ot theory. 



