DETERMINATION OF ELLIPTIC ORBITS. 129 



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The numerical determination of this value of Sj 8 2 +s s is critical 

 only to the first degree. 



The eccentricity and the true anomalies may be determined in the 

 same way as in the correction of the formula. The position of 

 the orbit in space may be derived from the following considerations. 

 The vector @ 2 is directed from the sun toward the second position 

 of the body; the vector @ 4 from the first to the third position. 

 If we set 



5 = @4-^@ 2 . (40) 



the vector <5 5 will be in the plane of the orbit, perpendicular to 

 and on the side t 

 the length of @ 6 , 



and on the side toward which anomalies increase. If we write s 5 for 



and 



will be unit vectors. Let 3 and 3' be unit vectors determining the 

 position of the orbit, 3 being drawn from the sun toward the peri- 

 helion, and 3' at right angles to 3, in the plane of the orbit, and on 

 the side toward which anomalies increase. Then 



3= --cos^ 2 - sin^ 2 ^, (41) 



S Z S 5 



3'= -sinu, 2 +cos%?k (42) 



S 2 8 5 



The time of perihelion passage (T) may be determined from any 

 one of the observations by the equation 



k 



(t T) = EesmE, (43) 



a* 



the eccentric anomaly E being calculated from the true anomaly v. 

 The interval t T in this equation is to be measured in days. A 

 better value of T may be found by averaging the three values given 

 by the separate observations, with such weights as the circumstances 

 may suggest. But any considerable differences in the three values 

 of T would indicate the necessity of a second correction of the 

 formula, and furnish the basis for it. 



For the calculation of an ephemeris we have 



SR= -ae3 +003^03+ sin .#63' (44) 



in connection with the preceding equation. 



Sometimes it may be worth while to make the calculations for the 



correction of the formula in the slightly longer form indicated for 

 G. H. i 



