128 



DETERMINATION OF ELLIPTIC ORBITS. 



of the heliocentric distances (say r 2 ). We need not trouble ourselves 

 farther about this change, for it will be of a magnitude which we 

 neglect in computations with seven-figure tables. That the helio- 

 centric angles thus determined may not agree as closely as they might 

 with the positions on the lines of sight determined by the first solution 

 of the fundamental equation is of no especial consequence in the 

 correction of the fundamental equation, which only requires the exact 

 fulfilment of two conditions, viz., that our values of the heliocentric 

 distances and angles shall have the relations required by the funda- 

 mental equation to the given intervals of time, and that they shall 

 have the relations required by the exact laws of elliptic motion to 

 the calculated intervals of time. The third condition, that none of 

 these values shall differ too widely from the actual values, is of a 

 looser character. 



After the determination of the heliocentric angles and the semi- 

 parameter, the eccentricity and the true anomalies of the three 

 positions may next be determined, and from these the intervals of 

 time. These processes require no especial notice. The appropriate 

 formulae will be given in the Summary of Formulae. 



Determination of the Orbit from the Three Positions and the 



Intervals of Time. 



The values of the semi-parameter and the heliocentric angles as 

 given in the preceding paragraphs depend upon the quantity s l s 2 +s 3) 

 the numerical determination of which from s lt s z , and s 3 is critical to 

 the second degree when the heliocentric angles are small. This was 

 of no consequence in the process which we have called the correction 

 of the fundamental equation. But for the actual determination of the 

 orbit from the positions given by the corrected equation or by the 

 uncorrected equation, when we judge that to be sufficient a more 

 accurate determination of this quantity will generally be necessary. 

 This may be obtained in different ways, of which the following is 

 perhaps the most simple. Let us set 



4 o 1 * \ / 



and s 4 for the length of the vector @ 4 , obtained by taking the square 

 root of the sum of the squares of the components of the vector. It is 

 evident that s z is the longer and s 4 the shorter diagonal of a parallelo- 

 gram of which the sides are s 1 and s 3 . The area of the triangle having 

 the sides s lt s 2 , s 3 is therefore equal to that of the triangle having the 

 sides s.p s 3 , s 4 , each being one-half of the parallelogram. This gives 



