74 



THE ORBIT OF NEPTUNE. 



position of the plane of the orbit, we 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 Lalande position is entitled to a precision of ^^. 



Limiting Dates. 



C5/3 



Equation 



of Condition. 





1795, 



+ 1.1 



Ozz+OMUp 



— 0.058^g 



+ 0.11 



184C-49, 



— 0.97 



— 0.866 



— 0.500 



— 0.96 



1850-53, 



— 0.75 



— 0.934 



— 0.358 



— 0.75 



1854-57, 



— 0.71 



— 0.978 



— 0.208 



— 0.71 



1858-61, 



— 0.67 



— 0.999 



— 0.052 



— 0.68 



1862-64, 



— 0.79 



— 0.996 



+ 0.084 



— 0.80 



The solution of which by least squares gives 



^p = — 0".73; 8q = — 0"Al. 

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



1795, + 0.7 



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 pi 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 



^£ = & + 1.77 SJi + 0.85 ^Z.-; 

 Sn = Ssd + OMSSJi — 0.073S7i;; 



So that, making the mass of Uranus ^ x^^oq-, the concluded corrections to the 

 provisional elements of § 19 are 



