176 DETERMINATION OF AN ORBIT FROM [BuOK II. 



the corresponding interval of time : thus two values will result for each of the 

 elements, and from their differences any two may be taken at pleasure for X and 

 Y. One advantage, not to be rejected, gives great value to this method ; it is, 

 that in the first hypotheses the remaining elements, besides the two which are 

 chosen for fixing X and Y, can be entirely neglected, and will finally be deter 

 mined in the last calculation based on the corrected values of x, y, either from 

 the first combination alone, or from the second, or, which is generally preferable, 

 from the combination of the first place with the third. The choice of those two 

 elements, which is, commonly speaking, arbitrary, furnishes a great variety of 

 solutions ; the logarithm of the semi-parameter, together with the logarithm of 

 the semi-axis major, may be adopted, for example, or the former with the eccen 

 tricity, or the latter with the same, or the longitude of the perihelion with any 

 one of these elements : any one of these four elements might also be combined 

 with the eccentric anomaly corresponding to the middle place in either calcula 

 tion, if an elliptical orbit should result, when the formulas -27-30 of article 96, 

 will supply the most expeditious computation. But in special cases this choice 

 demands some consideration ; thus, for example, in orbits resembling the parabola, 

 the semi-axis ma or or its logarithm would be less suitable, inasmuch as excessive 

 variations of these quantities could not be regarded as proportional to changes of 

 x, y: in such a case it would be more advantageous to select -. But we give less 

 time to these precautions, because the fifth method&quot;, to be explained in the follow 

 ing article, is to be preferred, in almost all cases, to the four thus far explained. 



128. 



Let us denote three radii vectores, obtained in the same manner as in articles 

 125, 126, by r, r , r&quot; ; the angular heliocentric motion in orbit from the second to 

 the third place by If, from the first to the third by 2/, from the first to the 

 second by 2/&quot;, so that we have 







next, let 



/ r&quot; sin 2f=n,r / sin 2/ = , r i&amp;gt; sin 2/&quot; = &quot; ; 



