74 



THE ORBIT OF NEPTUNE. 



position of the plane of the orbit, '\ve shall divide the residuals of latitude into 

 five groups, the last one including three years, and each of the others four years. 

 To find the heliocentric angular distance of the planet above the plane of its 

 assumed orbit, we shall take an indiscriminate mean of the errors of geocentric 

 latitude of each group, multiply it by 0.98 to reduce it to heliocentric error, and 

 correct it for the mean error in longitude. 



The mean errors of geocentric latitude, with the equations to which they give 

 rise, are as follows. The probable errors of each modern mean is estimated at 

 0".15 : so that the Lalaude position is entitled to a precision of ^ 



The solution of which by least squares gives 



lip = 0".73 ; Sq = 0".41. 



The residuals, multiplying the first by 10 to reduce it to actual observed error, 

 are 



1795, + o"7 



1846-49, 0.13 



1850-53, + 0.07 



1854-57, + 0.09 



1858-61, + 0.07 



1862-64, 0.10 



So that the Lalande observation is represented within 0".7, notwithstanding 

 the small weight with which it enters the equations. In fact, if p and q were 

 determined from the modern observations alone, the Lalande position would still 

 be represented within about 0".7. 



35. Concluded elements of Neptune. 



From equations (1) and (2) of this chapter, we have 



& =te + 1.77 M +0.85 87,: ; 

 0.0185A - 0.073&fe; 



the concluded corrections to the 



So that, making the mass of Uranus ^ T 

 provisional elements of 19 are 



