DETERMINATION OF ELLIPTIC ORBITS. 



147 



XL 



This gives the following equations for an ephemeris : 

 T=1806, June, 23*97450, Paris mean time 

 [2-8863186](* - T) = E ln 8flcond8 - [4-2216270] sin E 



Heliocentric coordinates relating to the ecliptic. 



x = + -1820700 - [0-3530366] cos E- [01827457] sin E 

 y = - -1244685 + [0*1878576] cos E- [0*3603257] sin E 

 z=- -0373970 + [9-6656346] cos E+ [9-3320292] sin E 



The differences of the values of T w , T (2) , jT (3) , from their mean T, 

 indicate the residual errors of this hypothesis. They indicate differ- 

 ences in the calculated and the observed geocentric positions which 

 are represented by the geocentric angles subtended by the path 

 described by the planet in the following fractions of a day : '00054, 

 00003, -00052. Since the heliocentric motion of the planet is about 

 one-fourth of a degree per day, and the planet is considerably farther 

 from the earth than from the sun at the times of the first and third 

 observations, the errors will be less than half a second in arc. 



If we desire all the accuracy possible with seven-figure logarithms, 

 we may form a third hypothesis based on the following corrections : 



A log Tl = 



= _ .Q000017, 



A log r 3 = M T _ T t (l] = - -0000018. 



The equations for an ephemeris will then be : 



T=1806, June 23-96378, Paris mean time 

 [2-8863140](* - T) = # in 8econd8 - [4-2216530] sin E 



Heliocentric coordinates relating to the ecliptic. 



x = + -1820765 - [0-3530261] cos E- [0-1827783] sin E 

 y=- 1244853 + [0-1878904] cos E- [0'3603153] sin E 

 z=- -0373987 + [9-6656285] cos E+ [9-3320758] sin E 



The agreement of the calculated geocentric positions with the data 

 is shown in the following table : 



