DETERMINATION OF ELLIPTIC ORBITS. 137 



Direction-cosines of semi-minor axis. 



\= 



~ , 

 p 2 H 



coav 9 

 -75 



2 S 6 



Components of the semi-axes. 



a x = al a = am a z = an 



X. 



Time of perihelion passage. 



Corrections for aberration. 



tan Jj^ = e tan J^ 6^ = - '0057613^ 



tan %E 2 = e tan Jv 2 St z = - '0057613/02 



tan \ES = e tan Jv, ^ 3 = - '0057613/03 



log '0057613 = 7-76052 



e sn 



The threefold determination of T affords a control of the exactness 

 of the solution of the problem. If the discrepancies in the values of T 

 are such as to require another correction of the formulae (a third 

 hypothesis), this may be based on the equations 



3 2 2 1 



where T (l)) T (2) , T (3) denote respectively the values obtained from the 

 first, second, and third observations, and M the modulus of common 



logarithms. 



XL 



For an ephemeris. 



a 



Heliocentric coordinates. (Components of $R.) 



x = ea x + a x cos E+b x sin E 

 y = ea y + a y cos E+ b y sin ^ 

 z= ea z +a z coaE+b z smE 



These equations are completely controlled by the agreement of the 

 computed and observed positions and the following relations between 

 the constants : 



^ A + a A + a*6, = aj+ a* + a? = a, 2 b x * + V + bf = (1 - e 2 )a 2 



