DETERMINATION OF ELLIPTIC ORBITS. 121 



Our equation may therefore be regarded as signifying that the three 

 vectors 9^, 9? 2 , ^3 ^ e ^ n one pl ane > an( l that the three triangles 

 determined each by a pair of these vectors, and usually denoted by 

 [^2^3]' [ r i r sl' [ r i r z\> are proportional to 



Since this vector equation is equivalent to three ordinary equations, 

 it is evidently sufficient to determine the three positions of the body 

 in connection with the conditions that these positions must lie upon 

 the lines of sight of three observations. To give analytical expression 

 to these conditions, we may write ( lf (5 2 , (5 3 for the vectors drawn 

 from the sun to the three positions of the earth (or, more exactly, of 

 the observatories where the observations have been made), 8 lf 3 2 > 83 

 for unit vectors drawn in the directions of the body, as observed, 

 and p lf p 2 , p 3 for the three distances of the body from the places of 

 observation. We have then 



By substitution of these values our fundamental equation becomes 



)=0, (7) 



where p lt p 2 , /o 3 , r lt r 2 , r s (the geocentric and heliocentric distances) 

 are the only unknown quantities. From equations (6) we also get, 

 by squaring both members in each, 



* s 2 



by which the values of r x , r 2 , r 3 may be derived from those of 

 Pi> P2> /3> or V ^ ce v ers d- Equations (7) and (8), which are equivalent 

 to six ordinary equations, are sufficient to determine the six quantities 



r i r z> r z> Pi> Pz> P*> or > ^ we 8U PP se the values of r x , r 2 , r B in terms 

 of p lt p 2 , p s to be substituted in equation (7), we have a single vector 

 equation, from which we may determine the three geocentric distances 



Pi> Pz> Pz- 



It remains to be shown, first, how the numerical solution of the 



equation may be performed, and secondly, how such an approximate 

 solution of the actual problem may furnish the basis of a closer 

 approximation. 



Solution of the Fundamental Equation. 



The relations with which we have to do will be rendered a little 

 more simple if instead of each geocentric distance we introduce the 



