126 DETERMINATION OF ELLIPTIC ORBITS. 



is combined with the correction required by the approximate nature of 

 the equation.* 



The solution of the fundamental equation gives us three points, 

 which must necessarily lie in one plane with the sun, and in the lines 

 of sight of the several observations. Through these points we may 

 pass an ellipse, and calculate the intervals of time required by the 

 exact laws of elliptic motion for the passage of the body between 

 them. If these calculated intervals should be identical with the given 

 intervals, corrected for aberration, we would evidently have the true 

 solution of the problem. But suppose, to fix our ideas, that the 

 calculated intervals are a little too long. It is evident that if we 

 repeat our calculations, using in our fundamental equation intervals 

 shortened in the same ratio as the calculated intervals have come out 

 too long, the intervals calculated from the second solution of the 

 fundamental equation must agree almost exactly with the desired 

 values. If necessary, this process may be repeated, and thus any 

 required degree of accuracy may be obtained, whenever the solution of 

 the uncorrected equation gives an approximation to the true positions. 

 For this it is necessary that the intervals should not be too great. It 

 appears, however, from the results of the example of Ceres, given 

 hereafter, in which the heliocentric motion exceeds 62 but the 

 calculated values of the intervals of time differ from the given values 

 by little more than one part in two thousand, that we have here not 

 approached the limit of the application of our formula. 



In the usual terminology of the subject, the fundamental equation 

 with intervals uncorrected for aberration represents the first hypothesis; 

 the same equation with the intervals affected by certain numerical 

 coefficients (differing little from unity) represents the second hypothesis; 

 the third hypothesis, should such be necessary, is represented by a 

 similar equation with corrected coefficients, etc. 



In the process indicated there are certain economies of labor which 

 should not be left unmentioned, and certain precautions to be observed 

 in order that the neglected figures in our computations may not 

 unduly influence the result. 



It is evident, in the first place, that for the correction of our funda- 

 mental equation we need not trouble ourselves with the position of the 

 orbit in the solar system. The intervals of time, which determine this 

 correction, depend only on the three heliocentric distances r lt r z , r s and 

 the two heliocentric angles, which will be represented by v 2 v l and 

 v 8 v z , if we write v lt v 2 , v 3 for the true anomalies. These angles 

 ( v z~~ v i an d v 8 Vg) niay be determined from r lt r 2 , r s and n v n 2 , n B , 



* When an approximate orbit is known in advance, we may correct the fundamental 

 equation at once. The formulae will be given in the Summary, xii. 



