SECT. 1.] THREE COMPLETE OBSERVATIONS. 233 



obtained from the combination of equations VII., VIII., IX., article 143, 



p _ 7? sin 5 sine&quot; _ Ssm(l l) 

 ~ R&quot; sin 8&quot; sin* lt&quot;l^ (F^l 7 ) 



In this case, therefore, the value of P is had from the data of the problem, and, 

 therefore, the positions of the points 0, C , C&quot;, will remain indeterminate. 



163. 



The method which we have fully explained from article 136 forwards, is prin 

 cipally suited to the first determination of a wholly imknown orbit : still it is em 

 ployed with equally great success, where the object is the correction of an orbit 

 already approximately known by means of three observations however distant 

 from each other. But in such a case it will be convenient to change some things. 

 When, for example, the observations embrace a very great heliocentric motion, it 



nff 



will no longer be admissible to consider and 66&quot; as approximate values of the 

 quantities P, Q : but much more exact values will be obtained from the very 

 nearly known elements. Accordingly, the heliocentric places in orbit for the 

 three times of observation will be computed roughly by means of these elements, 

 whence, denoting the true anomalies by v, v , v&quot;, the radii vectores by r, r, r&quot;, the 

 semi-parameter by p, the following approximate values will result : 



p _ r sin (v v) ,, 4 r * sin ^ (v r v) sin ^ (/ v ) 



~r&quot; sin (&quot; ) y~ p cos (v&quot; v) 



With these, therefore, the first hypothesis will be constructed, and with them, a 

 little changed at pleasure, the second and third : it would be of no advantage 

 to adopt P and Q 1 for the new values, since we are no longer at liberty to sup 

 pose that these values come out more exact. For this reason all three of the 

 hypotheses can be most conveniently despatched at the same time: the fourth will 

 then be formed according to the precepts of article 120. Finally, we shall not 

 object, if any person thinks that some one of the ten methods explained in arti 

 cles 124-129 is, if not more, at least almost equally expeditious, and prefers to 



use it. 



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