232 DETERMINATION OF AN ORBIT FROM [BOOK II. 



. w . n /sinv . ,,, ,, 



&quot; 



r sin (C - A D } = r&quot;P sin (?- 



from the combination of which with equations VIII. and IX. of article 143, the 

 quantities r, C, r&quot;, &quot; can be determined. The remaining processes bf the calcula 

 tion will agree with those previously described. 



162. 



In the second case, where B&quot; coincides with B, D&quot; will also coincide with them 

 or with the opposite point. Accordingly, we shall have AD 1 d and A&quot; I? d&quot; 

 either equal to or 180 : whence, from the equations of article 143, we obtain 



n r _ I sins ^ffsinii 



n sin E sin (z -(- A D ff) 



n r 1 sin R sin 5&quot; 



&quot; &quot; sine&quot; sin (z + A r D if) 

 R sin d sin e&quot; sin (s + yl Z&amp;gt;&quot; d ) = P7T sin d&quot; sin e sin (z + vl Z&amp;gt; d ). 



Hence it is evident that z is dcterminable by P alone, independently of Q, (un 

 less it should happen that A D&quot; A D, or = ^l Z&amp;gt; + 180, when we should have 

 the third case) : 2 being found, r will also be known, and hence, by means of 

 the values of the quantities 



n r n r 1 , n , n&quot; 



, -, also and : 



n n&quot; n n 



and, lastly, from this also 



Evidently, therefore, P and Q cannot be considered as data independent of each 

 other, but they will either supply a single datum only, or inconsistent data. The 

 positions of the points O, C&quot; will in this case remain arbitrary, if they are only 

 taken in the same great circle as O . 



In the third case, where A , B, B 1 , B&quot;, lie in the same great circle, D and D&quot; will 

 coincide with the points B&quot;, B, respectively, or with the opposite points : hence is 



